Science, Faculty of
Physics and Astronomy, Department of
DSpace
UBCV
MacPhail-Bartley, Ian David
Doctor of Philosophy - PhD
University of British Columbia
This work outlines the control of molecular rotation in superfluid helium using nonresonant laser fields. Experiments within bulk superfluid helium-4 demonstrate control over the rotational frequency and direction of rotation of electronically excited helium dimers (excimers), which are created in nanometre-scale bubbles in the fluid. The excimers rotate for thousands of rotational periods, indicating relatively weak but nonzero coupling to the surrounding helium. Controlling the rotation of molecules therefore serves as a probe of superfluid helium, and its coupling to impurities. The weak coupling is attributed to the fact that helium dimers rotate with rotational energy well above that of the expected excitations of the surrounding helium. By studying other molecules embedded in helium nanodroplets, we are able to explore the rotation of molecules below, near, and above this energy scale. The influence of strong coupling to the helium becomes extreme when the energies are comparable, severely distorting observed rotational spectra. Results presented here demonstrate that the rotation of molecules in helium nanodroplets may be controlled in the same manner as molecules in the gas phase. Experiments using an optical centrifuge to attempt to control molecular rotation in helium nanodroplets, and analysis regarding the results, are presented, as well as the first experiments studying rotationally excited nitrogen in helium nanodroplets. Experimental results on rotationally exciting nitric oxide dimers in helium nanodroplets show that (NO)₂ may be a suitable candidate as a molecule for further study. Alignment of the molecule offers insights to its anisotropic polarizability, and upon excitation of long-lasting rotation, exhibits exhibits a stronger observable than previously-studied molecules whose rotational energy may be controlled within the desired range.
Control of Molecular Rotation in Superfluid HeliumbyIan David MacPhail-BartleyHBSc Physics, Lakehead University, 2017A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Physics)The University of British Columbia(Vancouver)June 2024© Ian David MacPhail-Bartley, 2024The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Control of Molecular Rotation in Superfluid Heliumsubmitted by Ian David MacPhail-Bartley in partial fulfillment of the require-ments for the degree of Doctor of Philosophy in Physics.Examining Committee:Valery Milner, Associate Professor, Physics, UBCSupervisorKirk Madison, Associate Professor, Physics, UBCSupervisory Committee MemberFrank Stienkemeier, Professor, Physics, University of FreiburgSupervisory Committee MemberEd Grant, Professor, Chemistry, UBCUniversity ExaminerTakamasa Momose, Professor, Chemistry, UBCUniversity ExaminerHenrik Stapelfeldt, Professor, Chemistry, Aarhus UniversityExternal ExaminerAdditional Supervisory Committee Members:Roman Krems, Professor, Chemistry, UBCSupervisory Committee MemberiiAbstractThis work outlines the control of molecular rotation in superfluid helium usingnonresonant laser fields. Experiments within bulk superfluid 4He demonstrate con-trol over the rotational frequency and direction of rotation of electronically excitedhelium dimers (excimers), which are created in nanometre-scale bubbles in thefluid. The excimers rotate for thousands of rotational periods, indicating relativelyweak but nonzero coupling to the surrounding helium. Controlling the rotation ofmolecules therefore serves as a probe of superfluid helium, and its coupling to im-purities. The weak coupling is attributed to the fact that helium dimers rotate withrotational energy well above that of the expected excitations of the surroundinghelium.By studying other molecules embedded in helium nanodroplets, we are able toexplore the rotation of molecules below, near, and above this energy scale. Theinfluence of strong coupling to the helium becomes extreme when the energies arecomparable, severely distorting observed rotational spectra.Results presented here demonstrate that the rotation of molecules in heliumnanodroplets may be controlled in the same manner as molecules in the gas phase.Experiments using an optical centrifuge to attempt to control molecular rotation inhelium nanodroplets, and analysis regarding the results, are presented, as well asthe first experiments studying rotationally excited nitrogen in helium nanodroplets.Experimental results on rotationally exciting nitric oxide dimers in helium nan-odroplets show that (NO)2 may be a suitable candidate as a molecule for furtherstudy. Alignment of the molecule offers insights to its anisotropic polarizability,and upon excitation of long-lasting rotation, exhibits exhibits a stronger observ-able than previously-studied molecules whose rotational energy may be controllediiiwithin the desired range.ivLay SummaryStrong laser fields are used to control the speed and direction of molecular rota-tion. Controlled molecular rotation is then used to investigate microscopic fluidresistance in liquid helium, which has been shown to have vanishing viscosity atlow temperatures. Fluids with no viscosity are known as “superfluids”. Our exper-iments are performed both in nanometre-scale droplets of helium, as well as in alarger volume of several hundred millilitres. Our experiments show that the rota-tion of molecules embedded in liquid helium can be controlled in a similar fashionto isolated molecules in the gas phase. By controlling the rotational energy ofthe embedded molecules, we are able to investigate how this energy is dissipatedby the liquid helium. This moves us forward towards developing a microscopicunderstanding of the nature of superfluidity.vPrefaceThe majority of this thesis is my original, unpublished, independent work.In the experiments described in Chapter 4, performed by Alexander Milner, Icontributed by assisting with the creation and characterization of the chiral pulsetrain. I also contributed through conceptualizing the experiment, technical discus-sions about the experiment, and editing the manuscript published in Physical Re-view A [125], which was mostly written by Alexander and Valery Milner. KatarinaPreocanin designed and constructed the pulse shaper with my help. Peter (Xuan-shan) Peng and Shroyon Dasgupta created software to calibrate the pulse shaper,with my guidance.Starting with an optical table in an otherwise empty room, I designed the vac-uum and optical systems used for the experimental studies described in the sub-sequent chapters. This effort involved the integration of an ultrafast laser systemwith multiple homemade optical pulse shapers, and the helium droplet apparatusoutlined in Chapter 6 and Appendix E. I designed, constructed, and characterizedthe helium droplet apparatus myself, and performed all of the related experimentsindependently. Masayoshi Kuwabara performed ion simulations and mechanicaldesign for the time of flight velocity map imaging spectrometer installed in thesystem, under supervision from myself and Jordan Fordyce.I contributed to the design and led the construction of the “slow” optical cen-trifuge, used in the experiments described in Section 2.3 and Chapter 8. I alsoperformed characterization experiments, included in our publication in Reviewof Scientific Instruments [104], working directly with Walter Wasserman. Wal-ter Wasserman and Valery Milner performed the initial calculations for the pulseshaper geometry, and Valery performed the simulations included in the publication.viAlexander Milner performed the Raman spectroscopy (shown in Figure 3.10), us-ing the apparatus I constructed. Portions of the text and figures in Chapter 2 andAppendix F are adapted from this publication. Walter and I contributed equally tothe first draft of the publication. Significant edits were then made by myself andValery, including revisions requested after submission to the journal.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xixDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Quantum Control of Molecular Rotation . . . . . . . . . . . . . . 21.3 Molecules in Fluids . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.1 Helium Droplets . . . . . . . . . . . . . . . . . . . . . . 51.3.2 Rotational Spectra in Helium Droplets . . . . . . . . . . . 72 Laser-Induced Rotational Control of Molecules . . . . . . . . . . . . 112.1 Rotational Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 11viii2.1.1 The Diatomic Rigid Rotor . . . . . . . . . . . . . . . . . 112.1.2 A Diatomic with Centrifugal Distortion . . . . . . . . . . 132.1.3 Larger Molecules and Asymmetric Tops . . . . . . . . . . 142.2 Laser-Molecule Interaction . . . . . . . . . . . . . . . . . . . . . 152.3 Optical Centrifuge . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.1 Spinnability and Limitations . . . . . . . . . . . . . . . . 182.3.2 Creating an Optical Centrifuge . . . . . . . . . . . . . . . 192.4 Kicked Molecules and Rotational Coherence . . . . . . . . . . . . 232.5 Pulse Trains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.5.1 Annihilation/Enhancement . . . . . . . . . . . . . . . . . 252.5.2 State-Selective Excitations . . . . . . . . . . . . . . . . . 263 Detecting Molecular Rotation . . . . . . . . . . . . . . . . . . . . . . 283.1 Time of Flight Velocity Map Imaging . . . . . . . . . . . . . . . 283.1.1 Time of Flight . . . . . . . . . . . . . . . . . . . . . . . 283.1.2 Velocity Map Imaging . . . . . . . . . . . . . . . . . . . 303.1.3 Coulomb Explosion Imaging . . . . . . . . . . . . . . . . 303.1.4 Time of Flight Covariance Mapping . . . . . . . . . . . . 323.2 Angular Covariance Mapping . . . . . . . . . . . . . . . . . . . . 353.3 Angle-Dependent Ionization . . . . . . . . . . . . . . . . . . . . 373.4 Rotational Coherence Spectroscopy . . . . . . . . . . . . . . . . 403.4.1 Rotational Raman Spectroscopy . . . . . . . . . . . . . . 403.5 Laser-Induced Fluorescence . . . . . . . . . . . . . . . . . . . . 443.6 Resonance-Enhanced Multi Photon Ionization . . . . . . . . . . . 454 Rotational Control of Molecules in Liquid HeliumPart 1: He2* in Bulk Superfluid 4He . . . . . . . . . . . . . . . . . . 464.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2 Experimental Technique . . . . . . . . . . . . . . . . . . . . . . 484.2.1 Generation and Probing of Helium Dimers . . . . . . . . 484.2.2 Pulse Train Generation . . . . . . . . . . . . . . . . . . . 534.2.3 Chiral Pulse Train . . . . . . . . . . . . . . . . . . . . . 574.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58ix4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 Helium Droplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.1 Supersonic Gas Jets and Cluster Formation . . . . . . . . . . . . 665.2 Helium Droplet Production . . . . . . . . . . . . . . . . . . . . . 675.3 Molecules Embedded in Helium Droplets . . . . . . . . . . . . . 695.3.1 Formation of Molecules and Molecular Complexes in He-lium Droplets . . . . . . . . . . . . . . . . . . . . . . . . 705.4 Non-Axial Recoil in Helium Droplets . . . . . . . . . . . . . . . 716 Design and Characterization of a Helium Droplet Machine . . . . . 736.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.2 Design of the Helium Droplet System . . . . . . . . . . . . . . . 756.2.1 Design of the Source Chamber . . . . . . . . . . . . . . . 766.2.2 Design of the Doping System . . . . . . . . . . . . . . . 816.3 Characterization of the Helium Droplet Apparatus . . . . . . . . . 836.3.1 Initial Alignment of Components . . . . . . . . . . . . . 836.3.2 Doping Procedure . . . . . . . . . . . . . . . . . . . . . 876.4 Considerations for Future Apparatus . . . . . . . . . . . . . . . . 887 Rotational Control of Molecules in Liquid HeliumPart 2: CS2 in Helium Droplets . . . . . . . . . . . . . . . . . . . . . 907.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 907.2 Experimental Technique . . . . . . . . . . . . . . . . . . . . . . 947.2.1 Single Kick Calibration . . . . . . . . . . . . . . . . . . 947.2.2 Double Kick . . . . . . . . . . . . . . . . . . . . . . . . 967.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 998 Centrifuged CS2 in Helium Droplets . . . . . . . . . . . . . . . . . . 1038.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1038.2 Experimental Technique . . . . . . . . . . . . . . . . . . . . . . 1078.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088.3.1 Results with the 3D Optical Centrifuge . . . . . . . . . . 1088.3.2 Increasing sensitivity by using a 2D Optical Centrifuge . . 111x8.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1159 Rotational Excitation of N2 in Helium Droplets . . . . . . . . . . . . 1209.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1209.2 Experimental Technique . . . . . . . . . . . . . . . . . . . . . . 1219.2.1 Coulomb Explosion Imaging of N2 in Helium Droplets . . 1219.2.2 Angle-Dependent Ionization of N2 . . . . . . . . . . . . . 1229.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1249.3.1 Kicked N2 . . . . . . . . . . . . . . . . . . . . . . . . . 1249.3.2 Additional Data: Dynamics of (N2)2 . . . . . . . . . . . . 1269.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12810 Alignment and Rotational Excitation of (NO)2 in Helium Droplets . 13110.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13110.1.1 (NO)2 Isomers . . . . . . . . . . . . . . . . . . . . . . . 13210.1.2 Expected Results . . . . . . . . . . . . . . . . . . . . . . 13310.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13610.2.1 Adiabatic Alignment of (NO)2 in a Linearly Polarized LaserField . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13610.2.2 Dynamics of (NO)2 after Kicking . . . . . . . . . . . . . 13910.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14111 Outlook on Controlling Molecular Rotation in Superfluids . . . . . 14411.0.1 Further Excitation of Molecules in Helium Droplets . . . 145Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147A Identifying the Nitric Oxide Dimer . . . . . . . . . . . . . . . . . . . 174A.1 Discerning that effects are not from N2O or NO2 . . . . . . . . . 174A.2 Discerning that effects are not from (NO)3 . . . . . . . . . . . . . 179B Supersonic Jet Temperature Determination with REMPI . . . . . . 183C REMPI of NO in Helium Droplets and Transport to the Surface . . 186C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186xiC.2 Experimental Technique . . . . . . . . . . . . . . . . . . . . . . 188C.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189C.3.1 REMPI . . . . . . . . . . . . . . . . . . . . . . . . . . . 189C.3.2 Transport of NO from the centre of a Helium Droplet . . . 189D Data Processing Techniques . . . . . . . . . . . . . . . . . . . . . . . 191D.1 2-body Ion Pair Filtering . . . . . . . . . . . . . . . . . . . . . . 191D.2 Error Bar Calculations . . . . . . . . . . . . . . . . . . . . . . . 192E The Vacuum System . . . . . . . . . . . . . . . . . . . . . . . . . . . 193E.1 Specific Parts Used . . . . . . . . . . . . . . . . . . . . . . . . . 194E.1.1 Droplet Source . . . . . . . . . . . . . . . . . . . . . . . 194E.1.2 Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . 196E.1.3 Droplet Doping System . . . . . . . . . . . . . . . . . . 196E.1.4 Pulsed Nozzle for Supersonic Jet . . . . . . . . . . . . . . 196E.1.5 Differential Pumping . . . . . . . . . . . . . . . . . . . . 196E.1.6 Time of Flight Velocity Map Imaging Spectrometer . . . . 197E.1.7 Vacuum Pumps . . . . . . . . . . . . . . . . . . . . . . . 198E.1.8 Vacuum Gauges . . . . . . . . . . . . . . . . . . . . . . 199E.1.9 Other . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199F Pulse Characterization . . . . . . . . . . . . . . . . . . . . . . . . . 206F.1 Peak Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206F.2 Optical Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . 207F.3 Cross-Correlation Frequency Resolved Optical Gating (XFROG) . 207xiiList of TablesTable 3.1 An angular covariance matrix with 90◦ bin widths. . . . . . . . 35Table 6.1 A droplet beam intensity measurement for various backing pres-sures and droplet sizes, from the group of Frank Stienkemeier. . 78Table 10.1 Calculated rotational constants of nitric oxide dimers. . . . . . 133Table 10.2 Calculated polarizability tensors of nitric oxide dimers. . . . . 134xiiiList of FiguresFigure 1.1 Infrared rotational spectroscopy of OCS in 4He and 3He droplets,compared to OCS in the gas phase. . . . . . . . . . . . . . . . 4Figure 1.2 Comparison between the infrared rotational spectroscopy ofOCS in 3He droplets, as different numbers of additional 4Heatoms are added. . . . . . . . . . . . . . . . . . . . . . . . . 6Figure 1.3 Calculated rotational frequencies of CS2 in helium droplets, toillustrate the effective centrifugal distortion. . . . . . . . . . . 9Figure 2.1 Depictions of symmetric top molecules. . . . . . . . . . . . . 14Figure 2.2 Depiction of the electric field vector of an optical centrifuge. . 18Figure 2.3 Schematic for creating an optical centrifuge. . . . . . . . . . . 21Figure 2.4 The two-photon excitation field of a pair of pulses. . . . . . . 27Figure 3.1 The Time of Flight Velocity Map Imaging Spectrometer. . . . 29Figure 3.2 Definition of coordinates for the VMI apparatus. . . . . . . . . 31Figure 3.3 A sample TOF mass spectrum. . . . . . . . . . . . . . . . . . 32Figure 3.4 An example of TOF covariance analysis. . . . . . . . . . . . . 34Figure 3.5 Simulated VMI and angular covariance maps for randomly ori-ented and aligned molecules. . . . . . . . . . . . . . . . . . . 36Figure 3.6 A demonstration of the angular anisotropy of the ionizationcross section of nitrogen. . . . . . . . . . . . . . . . . . . . . 38Figure 3.7 The measured laser field intensity dependence of the angularanisotropy of the ionization cross section of nitrogen. . . . . . 39Figure 3.8 Depiction of Stokes and anti-Stokes Raman scattering. . . . . 41xivFigure 3.9 Optical schematic for rotational Raman spectroscopy. . . . . . 42Figure 3.10 Rotational Raman spectroscopy of centrifuged gas molecules. 43Figure 4.1 Our cryostat for experiments in bulk liquid helium. . . . . . . 49Figure 4.2 Energy curves for helium excimer experiments in liquid helium. 50Figure 4.3 An example of the laser-induced fluorescence signal of heliumexcimers in liquid helium as a function of time, with respect tothe arrival time of the pump pulse. . . . . . . . . . . . . . . . 51Figure 4.4 Optical schematic of the pulse shaper used for creating pulsetrains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Figure 4.5 A sample pulse train. . . . . . . . . . . . . . . . . . . . . . . 56Figure 4.6 Experimental results showing that the rotation of helium ex-cimers in superfluid helium can be “turned off”. . . . . . . . . 59Figure 4.7 Experimental results showing that the direction of rotation ofhelium excimers in superfluid helium can be controlled. . . . 60Figure 4.8 The temperature dependence of the total laser-induced fluores-cence signal of helium excimers in bulk liquid helium. . . . . 62Figure 4.9 The temperature dependence of the first rotational line of he-lium excimers in bulk liquid helium. . . . . . . . . . . . . . . 63Figure 4.10 The temperature dependence of first and second sound in liq-uid helium . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Figure 5.1 The helium droplet size distribution for various nozzle condi-tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Figure 5.2 Calculated doping probability curves for helium droplets oftwo sizes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Figure 6.1 The first-generation helium droplet machine. . . . . . . . . . 74Figure 6.2 A CAD render of the helium droplet machine I designed andbuilt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Figure 6.3 Droplet beam intensity measurements from the group of GaryDouberly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Figure 6.4 Recorded pressure increases when the helium droplet nozzletemperature is decreased. . . . . . . . . . . . . . . . . . . . . 85xvFigure 6.5 A droplet beam intensity measurement in my system. . . . . . 86Figure 7.1 Calculated rotational frequencies of CS2 in helium droplets. . 91Figure 7.2 The rotational dynamics of CS2 in helium droplets, after beingkicked by pulses of various intensity. . . . . . . . . . . . . . . 93Figure 7.3 The rotational dynamics of CS2 in helium droplets, after beingkicked by a single pulse. . . . . . . . . . . . . . . . . . . . . 95Figure 7.4 The power spectrum of the rotational dynamics of CS2 in he-lium droplets, after being kicked by a single pulse. . . . . . . 96Figure 7.5 The two-photon excitation field of a pair of pulses separated by75 ps, compared to the rotational energy level spacing of CS2in helium droplets. . . . . . . . . . . . . . . . . . . . . . . . 98Figure 7.6 Numerically simulated rotational coherence of CS2 in heliumdroplets in a double kick experiment, plotted as a function ofthe separation between the two kick pulses. . . . . . . . . . . 99Figure 7.7 Experimental results demonstrating selective rotational excita-tion of CS2 in helium droplets, in a double kick experiment. . 100Figure 7.8 The power spectrum of the rotational dynamics of CS2 in he-lium droplets in a double kick experiment. . . . . . . . . . . . 101Figure 8.1 Infrared rotational spectroscopy linewidth, as a function of ro-tational energy, for various molecules in helium droplets. . . 105Figure 8.2 Comparison between the rotational dynamics of centrifugedCS2 in the gas phase versus in helium droplets. . . . . . . . . 108Figure 8.3 The rotational dynamics of CS2 in helium droplets shortly afterthe end of the centrifuge. . . . . . . . . . . . . . . . . . . . . 110Figure 8.4 Simulated VMI signal for the rotational state |J,MJ⟩= |6,6⟩. . 111Figure 8.5 A two-dimensional optical centrifuge. . . . . . . . . . . . . . 112Figure 8.6 Simulated VMI signal of 2D and 3D optical centrifuges. . . . . 113Figure 8.7 Experimental results comparing the effect of the 2D opticalcentrifuge on CS2 molecules in helium droplets and in the gasphase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114xviFigure 8.8 Experimental results showing the observed signal of CS2 in he-lium droplets with the 2D centrifuge is indistinguishable fromthat of a single linearly polarized laser field. . . . . . . . . . . 115Figure 9.1 Optical schematic for ionization rate measurements with a con-tinuously modulated kick pulse. . . . . . . . . . . . . . . . . 123Figure 9.2 Experimental results showing the rotational dynamics of N2 inhelium droplets, after a single kick. . . . . . . . . . . . . . . 124Figure 9.3 Experimental results showing the isotropic alignment signalaround the rotational revival of nitrogen, in helium droplets. . 125Figure 9.4 The rotational dynamics of N2 in the gas phase, after a singlekick. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126Figure 9.5 Experimental results showing the alignment dynamics of N2in helium droplets, after a single kick, as measured using theangular anisotropy of the ionization cross-section. . . . . . . . 127Figure 9.6 The rotational dynamics of nitrogen dimers in helium droplets,after a single kick. . . . . . . . . . . . . . . . . . . . . . . . 127Figure 10.1 The molecular structure of nitric oxide dimers. . . . . . . . . 132Figure 10.2 Depictions of the Coulomb explosion axis of nitric oxide dimers,relative to the most polarizable axis. . . . . . . . . . . . . . . 135Figure 10.3 Adiabatic alignment of nitric oxide dimers in helium droplets. 137Figure 10.4 The maximum degree of alignment of nitric oxide dimers, as afunction of the alignment pulse intensity. . . . . . . . . . . . 137Figure 10.5 Angular covariance maps of nitric oxide dimers in helium droplets,for both aligned and randomly oriented molecular ensembles. 138Figure 10.6 The rotational dynamics of nitric oxide dimers in helium droplets,after a single kick. . . . . . . . . . . . . . . . . . . . . . . . 139Figure 10.7 The power spectrum of the rotational dynamics of nitric oxidedimers in helium droplets, after a single kick. . . . . . . . . . 140Figure 10.8 Estimated rotational frequencies of nitric oxide dimers in he-lium droplets. . . . . . . . . . . . . . . . . . . . . . . . . . . 142xviiList of AbbreviationsAMU Atomic Mass UnitCEI Coulomb Explosion ImagingCFG Optical CentrifugeCRS Coherent Raman ScatteringFFT Fast Fourier TransformKER Kinetic Energy ReleaseLIF Laser-Induced FluorescenceMCP Microchannel PlatePMT Photo Multiplier TubeREMPI Resonance-Enhanced Multi Photon IonizationRGA Residual Gas AnalyzerSCFG Slow Optical CentrifugeSLM Spatial Light ModulatorTOF Time of FlightTOFVMI Time of Flight Velocity Map ImagingVMI Velocity Map ImagingXFROG Cross-Correlation Frequency-Resolved Optical GatingxviiiAcknowledgmentsI would like to thank the graduate students across the world who have laid thefoundation for this work. Without them, none of this would be possible.Thank you to Valery Milner for the opportunity to pursue cutting-edge re-search, and Alexander Milner for teaching me the laboratory skills necessary forthis work. I’ve learned a lot about solving challenging problems from each of them,and I truly enjoyed the opportunity to design and build my own laboratory space.Roman Krems, Frank Stienkemeier, and Kirk Madison have made for an ex-cellent committee. I regret not scheduling more frequent meetings, you’ve all beenable to offer great insight every time we’ve talked.Working with Walter Wasserman was an absolute joy, and throughout the design-build process he was unequivocally the most useful “assistant” I could ask for. Iwas fortunate to get to work directly with Jordan Fordyce during my first year, aninspiration. Thanks to Ryan Harvey for helping me assemble the helium dropletmachine, and keeping the dye laser running. Working with undergraduate studentsOded Aminov, Masayoshi Kuwabara, Matan Guttmann, Pedram Amani, KatarinaPreocanin, Peter (Xuanshan) Peng, and Shroyon Dasgupta has been enlightening.Thank you again to Valery for recruiting such talented students.The experience at UBC would not have been the same without my other chemphysbasement friends Audrey Scognamiglio, Erik Frieling, Perrin Waldock, Pinrui Shen,Rachel Wang, Riley Stewart, Sudhang Varshney, Avinash Deshmukh, and DenisUhland. Apologies to Pavle Djuricanin, for constantly catching you in the hall toask questions. Your thoughtful advice has helped me again and again. PHAS tech-nical services have been of great help throughout this process, particularly MattSluyter and Mark Carlson. Engineering Physics students are lucky to have Bern-xixhard Zender in the project lab, thank you for all you’ve taught me. Thanks also toJulien Beaudoin Bertrand, for help along the way.Most importantly, thanks to my family for supporting me along the way. Myparents Jan and Tony, and uncle James, have always pushed me to pursue whatinterests me. For that I am grateful. Thank you to my wonderful partner Emma,for being there for me throughout this journey, and into the next chapter. I amlooking forward to spending a lot more time with you.xx.This thesis is dedicated to my late grandparents Sam, Hilda, Donald, andEmma, as well as my wonderful dog, Banjo.xxiChapter 1Introduction1.1 OutlineChapters 2 and 3 outline the methods of controlling and detecting molecular rota-tion used in this work, followed by Chapter 4 where successful control of both thedegree and direction of molecular rotation inside bulk superfluid helium is demon-strated [125].In Chapters 5 and 6, the technical aspects of preparing a doped helium dropletto study molecular rotors in superfluid environments are described in detail, fromthe perspective of the vacuum system designer and operator. This system was usedfor the experiments discussed in the remaining chapters. The experiments in Chap-ter 7 are analogous to those of Chapter 4, where rotational state-selective excitationof molecules in helium droplets (as opposed to in bulk helium) is demonstrated forthe first time. Subsequently, Chapter 8 works towards the use of an optical cen-trifuge to spin CS2 in helium droplets, and addresses some of the obstacles thathave been encountered.Chapter 9 offers the first experimental results attempting to rotationally excitenitrogen in helium droplets. This is followed by experiments creating, aligning,and rotationally exciting nitric oxide dimers [(NO)2] in Chapter 10.11.2 Quantum Control of Molecular RotationThe modern ultrafast laser quantum control toolbox is rapidly expanding, allowingexperimentalists to control matter in many ways. Control has many meanings,but a few parameters relating to molecular rotation that can be controlled on themolecular scale are:1. The axis of molecular rotation.2. The direction of molecular rotation.3. The rotational energy state distribution.4. The phase of rotation, in terms of the alignment or orientationa of molecularaxes at specific times, or the rotational wavepacket’s phase.These parameters can be controlled using laser light, both adiabatically and nona-diabatically. Laser pulses have been used to induce field-free planar alignment[69], three-dimensional alignment [91], and the direction of molecular rotation[53, 104]. Controlling just the phase of rotation has been used to measure molec-ular properties with rotational coherence spectroscopy [30], control of rotationalenergy with phase has been utilized to probe molecular potential energy surfaces[120], and control of both the direction and energy has been used for the spec-troscopy of extremely high rotational states [84]. Controlling the directionality ofmolecular rotation has also been used to orient chiral molecules [122, 170].Laser-induced alignment has been utilized to determine molecular structures[155], and modify molecular dipole forces [142]. High degrees of controlled ro-tational excitation have demonstrated the magnetization of oxygen gas at roomtemperature [55, 121], and stability against molecular collisions, leading to gyro-scopic behaviour and an anisotropic transfer of kinetic energy to the surroundinggas [81, 117, 128, 129]. It was first shown experimentally over fifty years agothat molecular alignment/orientation influences chemical reaction rates [11], andfurther development of the quantum control toolbox offers not only the ability tocontrol molecular distributions but also intermolecular interactions.aOrientation refers to alignment where the specific direction of molecular axes are defined.2This thesis looks at several quantum control tools, and applies them to moleculesembedded in superfluids, where rotational dynamics are often vastly different thanin the free molecular systems most often studied.1.3 Molecules in FluidsA molecule in a viscous fluid will experience friction as it rotates, resulting ina transfer of angular momentum to/from the fluid, until both are in equilibrium.Comparing the timescale of this relaxation to the separation between rotational fre-quencies will determine whether rotational structure can be resolved. This posesinteresting limitations to the applicability of methods of controlling molecular ro-tation to molecules in liquids, but also highlights how controlled rotation could beused as a probe of the surroundings.A fluid with zero viscosity (and therefore no friction) is known as a “super-fluid”, which will flow with no dissipation of kinetic energy as heat. Macroscopi-cally, this allows an object to move through the fluid with no drag. Microscopically,the movement of an impurity is akin to that of a free particle. This free motionoccurs while the relevant kinetic energies are sufficiently low to not excite the su-perfluid’s elementary excitations, such as surface waves and compressive densityfluctuations, including quasiparticles such as phonons or rotons.Helium is used for many applications because it is relatively inert, light, opti-cally transparent, and has the lowest boiling point of any element at 4.2 K. Cool-ing magnets and floating balloons may be its most well-known uses, but the low-temperature properties of liquid helium have proven extraordinarily useful for study-ing molecular structures. The most stable isotopes of helium are 3He and 4He, withthe bosonic 4He being many orders of magnitude more abundant than the fermionic3He. 4He condenses to a superfluid at 2.2 K, while 3He must be cooled below 5 mKto show superfluid properties. All of the work in this thesis refers to 4He.Embedding a molecule inside liquid helium is challenging. Liquid helium haslow solubility, and molecules tend to condense on the surface of the container[166]. A notable exception is when a droplet of liquid helium exists in vacuum,where molecules may reside either on the surface or interior, which is what themajority of this thesis is about. A second exception is He2 excimers created within3Figure 1.1: The infrared rotational spectrum near the ν3 vibrational band ofOCS was measured for free molecules in a seeded argon beam (a), in0.37 K 4He droplets (b), and in 0.15 K 3He droplets (c) [148]. HigherDrel and Irel values indicate more efficient infrared photon absorption.The 4He droplets slightly broaden the rotational lines, and more notice-ably bring the lines closer together, representing slower rotation, indica-tive of an increased moment of inertia. The spectrum is still resemblingthat of a free molecule. In 3He, the lines have been broadened substan-tially, indicating the helium rapidly quenches the rotation of molecules.Figure taken from [167].4bulk liquid helium, where rotational excitations have been recently studied [124]and are discussed in Chapter 4. Helium is transparent from deep ultraviolet tofar infrared, making it suitable for studies with strong laser fields [25]. The cou-pling between molecules embedded (doped) in helium droplets and the surround-ing helium is not fully understood on a microscopic level, though experiments haveshown that the rotation behaves very differently from free molecules, or what onewould expect in a non-superfluid medium. Investigating the relationship betweenthe density of states of elementary excitations of the helium and the kinetic en-ergy of the molecule is a primary motivator for controlling molecular rotation insuperfluid helium.The infrared rotational spectra of OCS in the gas phase from a seeded molecu-lar jet, in 0.37 K 4He helium droplets, and in 0.15 K 3He helium droplets are com-pared in Figure 1.1. The broadened and totally unresolved lines in the 3He dropletswere estimated to have a rotational relaxation time of 50 ps, compared to 1.5 nsin the 4He droplets. This longer relaxation time is considered to be a signatureof superfluidity [64], as the dopant is able to undergo several full rotations beforerelaxation. The redistribution of rotational lines from the gas phase spectrum is ac-counted for by a renormalization of the moment of inertia, due to a solvation shellof several helium atoms that adiabatically follow the rotation of the dopant withinthe droplet.1.3.1 Helium DropletsHelium droplets have been used for isolation spectroscopy for decades, with mul-tiple in-depth reviews covering many topics ([35, 127, 167, 168, 191]), includinglaser-induced alignment of molecules in helium droplets [160]. Infrared rotationalspectroscopy of molecules in helium droplets has even been utilized as an under-graduate laboratory exercise [146]. The low temperature environment of heliumdroplets provides an opportunity for weak bonds to persist without the same sizelimitations of traditionally-used rigid noble gas crystal matrices. A “shell” of he-lium adapts to droplet impurities of any size, and tends to shift spectra significantlyless than in rigid crystals [167].A cluster of a few dozen 4He atoms within a 3He droplet was used to inves-5Figure 1.2: The infrared rotational spectrum near the ν3 vibrational band ofOCS molecules in 3He droplets, with an additional N¯4 4He atoms addedto the droplet. Higher relative depletion values indicate more efficientinfrared photon absorption. (A) is the same as what was depicted inFigure 1.1 (c), with (B) through (F) having an additional N¯4 4He atomsadded to the 3He droplets. The wave number change is plotted withrespect to 2061.71 cm−1. Figure taken from [64].6tigate how the size and thickness of this “shell” relates to the superfluid nature of4He at low temperatures, by observing the infrared rotational spectrum of OCS asplotted in Figure 1.2 [64]. In these experiments, it was found that with approxi-mately 60 or more 4He atoms surrounding OCS, the molecule would rotate freely.It was later discovered that much smaller clusters of 4He would still allow for freerotation of OCS [106], and that for small clusters the rotational spectrum stronglydepends on the droplet size - potentially due to the dopant’s rotational energy cou-pling to excitations on the surface of the droplet [107]. Understanding the specificsof microscopic superfluidity poses great challenges both experimentally and theo-retically.1.3.2 Rotational Spectra in Helium DropletsTwo regimes of molecules in equilibrium with superfluid helium are generally con-sidered. Either the molecule rotates slowly enough for helium atoms to form a co-rotating solvation shell, which adiabatically follows the molecule if the attractivepotential between the molecule and the shell is sufficiently strong, or the moleculerotates too quickly for any notable influence of the surrounding helium. Theseare referred to as “heavy” and “light” rotors, relating to the molecule’s moment ofinertia. This distinction is generally thought of as occurring around a rotationalconstant of 1 cm−1, which coincides with the first few excited rotational states be-ing close to the high density of states in superfluid helium around 5− 15cm−1[67].Experiments from 2018 with isotopologues b of OCS (gas phase rotational con-stant of 0.2 cm−1) have shown that the adiabatic following approximation partiallybreaks down [50]. These experiments studied the effective moment of inertia ofthe solvation shell. One would expect that in the heavy rotor limit, there shouldbe only small dependence on slight changes to the shell’s moment of inertia, fora given molecular geometry, with the potential energy surface of the molecule es-sentially dictating the density of the co-rotating helium. Their results showed anapproximately linear relationship between the molecule’s gas phase rotational con-stant and that of the solvation shell. This trend was in the opposite direction ofbMolecules that differ only in isotope composition.7what would be expected from the slight differences in the potential energy surfacesof the molecules, indicating that the trend was indeed due to the differing momentsof inertia, and that OCS is not quite in the “heavy rotor” limit.Additionally, it has been found that the effective moment of inertia of moleculesin helium droplets rapidly increases with angular momentum, occurring as themolecule’s rotational energy approaches that of the elementary excitations of itssurroundings. This results in heavy rotors having a rotational energy spectrum inhelium droplets approximating E(L) = BHDL(L+1)−DHDL2(L+1)2, withBHD ≈ 3BGas, (1.1)DHD = 0.031×B1.818HD .The use of L represents the total angular momentum of the molecule and surround-ing helium, rather than just that of the molecule, following the convention of [33].BHD and DHD are the effective rotational and centrifugal distortion constants ofthe molecule in helium droplets. Equation 1.1 describes the effective centrifugaldistortion constants, which are several orders of magnitude higher than that of thesame molecules in the gas phase. This has a dramatic effect on the rotational en-ergy level separation spectrum, as shown in Figure 1.3. Rotational energy lines aremuch closer together than they would be for a rigid rotor with D−→ 0.This interaction was probed using laser-induced molecular rotation in [33]. Gasphase molecules tend to produce periodic revivals of rotational alignment when ex-cited with an intense laser pulse of linear polarization, referred to as a “kick” anddiscussed in more detail in Section 2.4. This excitation method produces a superpo-sition of molecular quantum rotational states, by exerting a strong torque towardsthe laser polarization. Molecules embedded in helium nanodroplets produce anunusual oscillatory alignment under strong excitation, but when the degree of ex-citation is lowered, the molecules form revivals [159], with rotational coherencelasting hundreds of picoseconds [33]. The rotation and angular momentum trans-fer were studied by monitoring the time dependence of the molecular alignment. Amore precise model for the effective rotational and centrifugal distortion constantsin helium droplets was recently developed [34], which has been experimentallyverified to better match the dynamics of very light rotors (BHD ≫ 1cm−1) rotating8Figure 1.3: The gap between rotational energy levels quantized by total an-gular momentum L, for ∆L = ±2 with only even rotational states.This represents CS2 in helium droplets, with rotational energy lev-els E(L) = BHDL(L+ 1)−DHDL2(L+ 1)2, using BHD = 730MHz andDHD = 1.2MHz from [30]. The blue curve labelled “Rigid Rotor” hasDHD = 0, to illustrate the effect the centrifugal distortion has on the ro-tational energy spectrum. The vertical error bars represent the reporteduncertainty in BHD only.freely [124, 143]. This model uses the potential energy surface of the dopant toestimate the interaction strength [97], and predicts that with sufficient angular mo-mentum in the system, the solvation shell’s angular momentum will saturate androtational dynamics will once more approach that of a free molecule. A glimpse ofthis detachment was observed in [159], but was rapidly quenched within a couplerotational periods.To the best of my knowledge, no experiments have been performed whichdemonstrate the same molecule being rotationally excited below, near, and wellabove the energy range of 5− 15cm−1 - a window representing the highest den-sity of states for elementary superfluid helium excitations [67]. The iodine exper-iments in [159] are fascinating, because they demonstrate extraordinarily strongcoupling between the dopant and the droplet around this range, although the exci-tation scheme used did not allow for much higher rotational energy, which explainswhy the detachment from the solvation shell is so brief. The short rotational co-9herence lifetimes for rotational energies over this range are attributed to strongcoupling between the dopant and the helium [35]. Infrared spectroscopy is limitedto probing the initial state distribution, hence the desire to control the state popula-tions and probe the dynamics with other methods so this interaction can be studiedin detail.10Chapter 2Laser-Induced RotationalControl of Molecules2.1 Rotational SpectraThe Born-Oppenheimer approximation is based on the assumption that electronsorbiting nuclei are significantly faster, due to the fact their mass is several orders ofmagnitude lower than that of the nucleus. In this case, the electronic wavefunctionessentially follows the moving nuclei and can be treated separately from the nuclearmotion. The total energy of a molecule may, therefore, be expressed as a sum ofseparate terms:ETotal = EElectronic+EVibrational+ERotational+ENuclear Spin. (2.1)2.1.1 The Diatomic Rigid RotorIf the molecule is treated as a rigid system of masses, the energy associated with arotational energy level is found from solving the Schro¨dinger equation:HˆRotational|ψ⟩ = ERotational|ψ⟩, (2.2)HˆRotational|ψ⟩ = Jˆx22Ix +Jˆy22Iy +Jˆz22Iz . (2.3)11The eigenfunctions for a linear molecule (moments of inertia I= Ix = Iy,Iz = 0) arefound in spherical coordinates to be the spherical harmonics Y MJJ (θ ,φ), where J =0,1,2,3, ... and MJ = −J,(−J+ 1), ...,(J− 1),J are the total angular momentumquantum number and the projection of angular momentum on the quantization axis,where θ and φ are the two angles describing the spatial orientation of the molecularaxis.The eigenvalues of the rotational Hamiltonian give the rotational energy spec-trumaE(J) = BJ(J+1), (2.4)where there is no energy dependence on the projection MJ , and the rotational con-stantB =ℏ22I, (2.5)provides the energy scaling for the rotational level separation:∆E = E(J+1)−E(J) = 2B(J+1). (2.6)The allowed rotational energy levels and transitions depend on the molecule, andinteraction type [110]. As an example, consider molecules with multiple atomsof integer spin (bosons). Because the total wavefunction must remain symmetricupon atoms’ exchange, aspects of the molecular structure, such as the electronicwavefunction being symmetric (antisymmetric), will necessitate that the rotationalwavefunction consist of only symmetric (antisymmetric) spherical harmonics. CO2is an example of a molecule with only even rotational states, which holds true untilsignificant vibrational excitation breaks the symmetry and allows for odd rotationalstates [173]. O2, on the other hand, has an antisymmetric electronic wavefunction,and therefore consists of only odd rotational states. Nuclear spin isomers of thesame molecule impose similar restrictions, where the fraction of symmetric (para),and antisymmetric (ortho), spin states must be considered to determine the allowedrotational structure.The quantization of angular momentum necessitates selection rules relating toaFrom here on I will use E = ERotational12transfer of angular momentum to/from an optical field, as the total angular momen-tum must be conserved. There is a large number of possible excitation schemes[57], but our work is centred around two-photon Raman transitions. Raman tran-sitions refer to inelastic photon scattering, where the energy difference betweenthe input and output photons corresponds to the energy transferred to, or from, themolecule. This dictates the selection rules of ∆J = ±2, ∆MJ = 0,±2, dependingon light polarization. Chirality in the initial rotational state’s angular momentumdistribution or excitation process will determine the sign of ∆MJ , relative to ∆J.The optical centrifuge, described in Section 2.3, is an example of a chiral excita-tion, and centrifuged molecules undergoing unidirectional rotation are an exampleof a chiral rotational state distribution.2.1.2 A Diatomic with Centrifugal DistortionIn rotational spectroscopy, the vibrational component due to the bond stretchingis often treated as a small perturbation of the rotational spectrum, with two termsfrom the Dunham expansion [23]:E(J) = BJ(J+1)−DJ2(J+1)2, (2.7)which is describing a semiclassical system where for each (quantized) increase ofangular momentum, stretching of the bond increases the moment of inertia. Thescaling factor D (often written as DJ or De) is known as the “centrifugal distortionconstant”. What Equation 2.7 shows is that as the total angular momentum J in-creases, there will be subsequently smaller increases in E as the bond stretches. Thecentrifugal force is pulling on the bond, shifting its equilibrium position. As theseincreases in moment of inertia become significant, there may be a situation wherewhere a higher J state corresponds to a lower rotational frequency, Ω=√2E(J)I . Inthe context of experiments with an optical centrifuge, we refer to this effect as the“centrifugal wall”.13Figure 2.1: The top-left and top-right pictures show the prolate symmetric topEthane (C2H6), and oblate symmetric top Benzene (C6H6), respectively.Depictions of prolate and oblate ellipsoids are shown in the bottom-leftand bottom-right. Ethane has 120◦ (C3) rotational symmetry about thecarbon bond, and Benzene has 60◦ (C6) rotational symmetry about anaxis perpendicular to the molecular plane in the centre of the carbonring.2.1.3 Larger Molecules and Asymmetric TopsAdding a third atom introduces further rotational-vibrational coupling through bend-ing modes, affecting the effect of centrifugal distortion. However, if the moleculeis linear (CS2, N2O, OCS, CO2, etc.), then Equation 2.7 remains a good approxi-mation for free molecular rotation.If a triatomic molecule is not linear (SO2, H2O, O3, etc.), it represents an“asymmetric top”, with three unique moments of inertia and corresponding ro-tational constants, with few exceptions such as the equilateral triangular structureof the trihydrogen cation. Most polyatomic molecules are asymmetric tops. Aconcise definition of an asymmetric top is a molecule with either no rotationalsymmetry, or only instances of rotational symmetry for rotations by 180◦, such asthe bent structure of SO2 or H2O. This type of symmetry axis is referred to as a C2symmetry.14The degree of inertial asymmetry for an arbitrary molecule is quantified byRay’s asymmetry parameterκ =2By−Bx−BzBx−Bz , (2.8)where in the limit κ → −1 the molecule approaches a “prolate symmetric top”(Bx > By ≈ Bz, shown on the left of Figure 2.1). In this case, the rotational energyspectrum is approximated with:Byz =By+Bz2, E(J,K) = ByzJ(J+1)+(Bx−Byz)K2. (2.9)K =−J,−J+1...−1,0,1...J−1,J is only a good quantum number when κ =±1.In the opposite limit of κ → +1, the molecule approaches an “oblate symmetrictop” (Bx ≈ By > Bz, shown on the right of Figure 2.1). The rotational energy of anoblate symmetric top is:Bxy =Bx+By2, E(J,K) = BxyJ(J+1)+(Bz−Bxy)K2. (2.10)2.2 Laser-Molecule InteractionThe carrier oscillation frequency of an optical field is high enough that the effects ofpermanent molecular dipoles are effectively averaged out [42]. This leaves us withthe induced dipole moment µ⃗ ind, which is typically expanded in a Taylor series inpowers of the electric field E⃗ as:µ⃗ ind =∞∑n=1αˆn · E⃗n, (2.11)where α⃗n is the n-th order polarizability tensor of the molecule in the laboratoryframe. Typically, the linear term dominates over higher order interactions andµ⃗ ind = αˆ · E⃗, (2.12)15with αˆ being the linearb polarizability tensor. The force on the system in the pres-ence of an electromagnetic field is the Lorentz force [63]:F⃗ = (⃗µind · ∇⃗)E⃗ + 1cdµ⃗inddt× B⃗ = αˆ[(E⃗ · ∇⃗)E⃗ + 1c∂ E⃗∂ t× B⃗], (2.13)which is equal to the negative gradient of the potential energy F⃗ =−∇⃗U . To writean expression for U , we use the identity(E⃗ · ∇⃗)E⃗ = ∇⃗(12E⃗ · E⃗)− E⃗× (⃗∇× E⃗), (2.14)with Maxwell’s equation∇⃗× E⃗ =−1c∂ B⃗∂ t, (2.15)and the identityE⃗× 1c∂ B⃗∂ t+1c∂ E⃗∂ t× B⃗ = 1c∂ (E⃗× B⃗)∂ t, (2.16)to rewrite Equation 2.13 with only the first termc:F⃗ = αˆ(12∇⃗E⃗2)=−∇⃗U. (2.17)As the polarizability does not explicitly depend on position, we can writeU =− αˆE⃗22. (2.18)Averaging over the carrier oscillations in E⃗(t),U =− αˆ ε⃗24, (2.19)bHigher order terms are generally referred to as the n-th order“hyperpolarizability” tensors.cThe second term has been neglected, as it describes the force along the propagation direction dueto the variation in time of the field intensity, which is not relevant for our experiments with pulsedlasers where the interaction during the rising edge of each pulse is cancelled out during the fallingedge.16where ε⃗(t) is the slowly-varying intensity profile of the field. For a symmetric topmolecule (one where two of three moments of inertia are the same) in a linearlypolarized field of amplitude E0, this interaction is succinctly written asU(θ) =−14E02∆α cos2 θ =−U0 cos2 θ , (2.20)where θ is the angle between the polarization vector of the field and the most polar-izable molecular axis, E0 is the field amplitude, and the polarizability anisotropy∆α = α∥−α⊥ is the difference of the polarizability components along (α∥) andperpendicular (α⊥) to the molecular symmetry axis. The torque τ exerted on amolecule by the potential in Equation 2.20 isτ =∂U∂θ=U0 sin(2θ), (2.21)which results in an average angular acceleration of:θ¨mol ≈ 2U0πI . (2.22)2.3 Optical CentrifugeAn optical centrifuge (CFG) is a linearly polarized laser pulse, whose polarizationvector is rotating with constant angular acceleration, as depicted in Figure 2.2. Atthe beginning of the pulse, the polarization vector is almost stationary. Owing tothe interaction potential outlined in Equation 2.20, molecules align with the field.If the field turns on much more slowly than the timescale of molecular rotation, thefield-free eigenstates are adiabatically transferred to pendular states aligned to thefield. If the polarization vector changes its direction with angular acceleration θ¨pol,the molecule will follow the field adiabatically as long as |θ¨pol| < |θ¨mol|, or fromEquation 2.22 above:|θ¨pol|< 2U0πI . (2.23)The directionality of the rotation of the field controls the directionality of molec-ular rotation, and the well-defined rotational frequency of the centrifuge at eachmoment in time makes it possible to create a narrow rotational wavepacket at a17Figure 2.2: A 3D render of an optical centrifuge. The vector of linear po-larization E⃗ follows the surface of the red corkscrew shape. The cen-trifuge shown accelerates from 0-10 THz over ∼100 ps, and is propa-gating along the wave vector k⃗ to the right. Molecules trapped by theleading edge of the centrifuge follow the accelerated rotation of E⃗, andcontinue rotating at the terminal frequency of the centrifuge after thefield has passed.target rotational state, by truncating the centrifuge at any desired frequency.Quantum mechanically, the optical centrifuge rotationally excites moleculesthrough a series of Raman transitions with ∆J,∆MJ =±2 [175]. Section 2.3.2 de-scribes the creation of the centrifuge field, comprising two laser pulses of oppositecircular polarizations with diverging optical frequency. When the optical frequencydifference between the two pulses is resonant with a molecule’s rotational energylevel separation, stimulated absorption of a photon from one field to an interme-diate virtual state, followed by stimulated emission into the other field, results inrotational excitation of the molecule.2.3.1 Spinnability and LimitationsIn [104], we expanded on Equation 2.23 in terms of a dimensionless “spinnability”factor, which depends both on the properties of the molecule and the parametersof the centrifuge. That equation is essentially a statement that “the angular accel-18eration of the molecule towards the polarization vector of the field must be greaterthan the angular acceleration of the field’s polarization vector”, resulting in the pen-dular states about the accelerating rotation of the field polarization vector. Puttingeverything together, the spinnability criterion is then:σ0 :=U0πIβ=14π× E20β× ∆αI> 1, (2.24)which is expected to be sufficient when the initial energy of rotation around therelevant molecular axis ( 12 kT , where k is the Boltzmann constant and T the temper-ature) is lower than the kinetic energy acquired by the molecule during the turn-ontime of the centrifuge ton, U0/2+ I(2β ton)2/2. All of the work in this thesis is per-formed at low temperature where Equation 2.24 applies, with the high temperaturespinnability condition outlined in [104].Centrifugal distortion will simultaneously alter ∆α and I, and in the classicalpicture, this is not an issue so long as their ratio supports a sufficient angular accel-eration for the molecule in the field. Quantum mechanically, centrifugal distortionreduces the rotational energy of subsequent angular momentum states, as describedin Section 2.1.2, which increases the rate at which Raman transitions must occurin the optical centrifuge. Eventually the centrifugal wall is reached, where thenext increase in J requires a decrease in frequency, at which point an acceleratingcentrifuge will not be able to rotationally excite the molecules any further. Whilequantum mechanical descriptions of molecular rotation in an optical centrifuge ex-ist (see [5, 87, 175, 176]), a full model accounting for centrifugal distortion has notyet been developed.The spinnability implications of perturbations to the free rigid rotor are par-ticularly relevant for experiments studying molecules in liquids, where the rate ofangular momentum transfer to the surrounding medium will increase the rate atwhich Raman transitions must occur to effectively spin a molecule in an opticalcentrifuge.2.3.2 Creating an Optical CentrifugeFollowing Karczmarek et al. [79], consider the interference of two laser pulses ofopposite circular polarizations (eˆ± = xˆ± iyˆ) and carrier frequencies shifted by ±Ω19from ω0, respectively, propagating along zˆ:E⃗+(t) =E02[xˆcos(ω0+Ω)t+ yˆsin(ω0+Ω)t] , (2.25)E⃗−(t) =E02[xˆcos(ω0−Ω)t− yˆsin(ω0−Ω)t] .The result of such interference is an optical field, whose linear polarization is ro-tating in the xy plane with a constant angular frequency Ω,E⃗⟲(t) = E⃗+(t)+ E⃗−(t) = E0 cos(ω0t) [xˆcos(Ωt)+ yˆsin(Ωt)] . (2.26)To make the polarization vector rotate with a gradually increasing angular speed(Ω = θ¨polt), the frequency difference between the two interfering fields must beincreasing with time. This can be achieved by chirping the frequencies of the twopulses in opposite directions, i.e. by adding quadratic phase factors of oppositesign,E⃗±(t) =E02[xˆcos(ω0t+β±t2)± yˆsin(ω0t+β±t2)] , (2.27)with β± ≷ 0, and the average chirp rate β ≡ (β+−β−)/2 > 0 defining the instan-taneous rotational frequency and angular acceleration as follows:Ω(t) = 2β t, θ¨pol = Ω˙(t) = 2β . (2.28)As originally proposed and implemented by Villeneuve et al. [174], the fieldof an optical centrifuge can be created by means of a pulse shaper shown in Fig-ure 2.3. The shaper consists of two standard pulse stretchers based on diffractiongratings [169], which produce the two frequency-chirped pulses E± described inEq. (2.27). The stretchers are built in a well-known ‘4 f ’ geometry [182], in whichan image of the input grating (GR0, shared by the two stretchers ) is created bymeans of a lens pair (L0/L+ and L0/L−), located four focal lengths away at pointsA and B. The 4 f geometry is utilized for two purposes. First, it enables splittingthe input beam into two spectral components, hereafter referred to as “two cen-trifuge arms”, in the Fourier plane of lens L0. Numerically modeled spectra foreach arm of an optical centrifuge are shown in inset (a) of Figure 2.3, whereasinset (b) shows an example of the corresponding temporal profiles with slightly20Figure 2.3: Centrifuge pulse shaper. Shown are the gratings, lenses and mir-rors producing two standard stretchers for the two “arms” of the cen-trifuge. The secondary gratings GR± are displaced from the focal planesof the corresponding lenses L± by distances l±. Here, subscripts ‘±’refer to the blue (up-chirped) and red (down-chirped) centrifuge fieldsE±(t), respectively. Retro-reflectors R± shift the beams to the parallelplanes above/below the plane of the figure, thus separating them spa-tially at the output port of the centrifuge. Insets on the left show ourdefinition of the grating orientation angle θ0, as well as the method ofcombining the two centrifuge arms into a single centrifuge pulse bymeans of two wave plates and a polarizing beam splitter (PBS). Ex-amples of the calculated spectral and temporal profiles of an opticalcentrifuge are shown in insets (a) and (b) on the right, respectively. Un-equal frequency chirps, and hence unequal pulse durations, as well asunequal optical path lengths, and hence mismatched rising edges, wereassumed for illustration purposes.21different durations and time delays (with respect to the input pulse) to illustratethe case of a misaligned centrifuge, often found in a real experiment. Second,the 4 f configuration makes it possible to apply frequency chirps of both positiveand negative sign. Following [169], the chirp produced by a standard double-passdouble-grating stretcher around the centre frequency ω0 can be calculated as:β± =−d2ω30 cos2(θ0)16π2l±c, (2.29)where l± is the distance between the gratings along the path of the light beam withω = ω0, 1/d is their groove density, θ0 is the orientation angle of the gratings (seeFigure 2.3), and c is the speed of light. One can see that the sign of β± is determinedby the sign of the corresponding l±. Imaging an input grating at locations A andB enables negative values for the effective distance between the gratings (here,negative l+ in the blue centrifuge arm in Figure 2.3) and hence positive values ofthe frequency chirp.According to Equation 2.29, to make the centrifuge accelerate faster (higherabsolute values of β ), one has to decrease distance l := |l±|, angle θ0, and groovedensity 1/d. From the practical perspective, this does not present any techni-cal challenges since weaker gratings are readily available and shorter distancesare easy to implement. If one wishes to use uncompressed pulses from a typi-cal Ti:Sapphire chirped pulse amplifier, as in the original work of Villeneuve etal. [174], an output frequency chirp on the order of +0.3 ps−2 must be taken intoaccount. In practice, this amounts to shifting the two gratings in both centrifugearms by the same positive distance, i.e. l± → l±+∆l, dependent on the gratinggroove density.To make the centrifuge accelerate slower (lower β ), one needs to increase thethree shaper parameters (l,θ0 and 1/d), which are all capped by hard technical con-straints. The highest available groove density producing reasonably high diffrac-tion efficiency in the spectral region of interest is 2400 mm−1. As the angle be-tween the incident beam and the normal to the grating is larger than θ0 (by thedeviation angle of a few degrees), the latter is naturally limited by ≈ 80◦. Finally,the absolute value of distance l+ must be smaller than the focal length f , or elsethe diffracted beam in the blue arm will not clear the lens (see Figure 2.3).22As depicted in Figure 2.3, the diffracted beam should be well-matched to thedimensions of the optical elements used. The spectral bandwidth of the cen-trifuge shaper ∆νsh, and hence the maximum angular frequency of the centrifugeνmax = ∆νsh/2, is determined by the groove density of the gratings and the nu-merical aperture NA of the lenses. Using the grating equation and a limit of smallNA (typical for most pulse shapers due to geometrical constraints), one arrives at∆νsh = 2dccos(θ0)NA/λ 20 . Using Equation 2.29, the terminal frequency of thecentrifuge can be expressed as a function of the chirp parameter:νmax = NA√β2lπλ0. (2.30)The spectral bandwidth of the laser source should be matched to that of the cen-trifuge shaper. If the former is too broad, valuable energy will be lost with photonswhose frequencies are outside of the shaper’s bandwidth. In the opposite limit ofinput pulses which are spectrally too narrow, there will be not enough bandwidthto accelerate molecular rotation all the way to νmax. As a practical trade-off, let usassume I(ν0±νmax) = 0.1I(ν0), i.e. that the centrifuge shaper truncates the inputlaser spectrum I(ν) at 10% of its peak value at the central frequency ν0. This as-sumption results in the following expression for the duration of the input pulses inthe Fourier transform limit (τTL, full width at half maximum):τTL =√ln(10) ln(2)π1νmax. (2.31)2.4 Kicked Molecules and Rotational CoherenceWhen a pulse is much shorter than the molecule’s rotation period, the excitationwill be nonadiabatic (and “impulsive”) rather than adiabatic, which typically leadsto a coherent superposition of a number of rotational states. Truncating a longpulse quickly also results in a similar excitation of multiple states [28, 190]. Shortpulses are often referred to as “kicks.” Gas phase laser-kicked rotors have beenstudied extensively [14, 54, 157].Classically, each molecule experiences an angle-dependent torque towards the23polarization vector. Kicking an initially isotropic distribution of molecules in theirground rotational state will result in all molecules rotating towards the field po-larization vector, with varying rotational frequencies. Because of the strong rela-tionship between their initial angular phase and post-excitation frequency, and thefrequency spectrum being discrete, the molecular wavefunction will revive period-ically in time - an effect known as “quantum revivals”.Quantum mechanically, a kick results in a series of Raman transitions with∆J = ±2, where angular momentum is directly transferred between the laser fieldand the molecule. The “rotational ladder” of the molecule can be climbed, if thelaser pulse bandwidth covers the molecule’s transition frequencies. The rotationalwavepacket broadens with each transition, as J may increase or decrease. For thesimplest case of a linear rigid rotor, the transition frequencies are:∆E = E(J+2)−E(J) = B(J+2)(J+2+1)−BJ(J+1) = B(4J+6). (2.32)Equation 2.32 describes frequencies that are harmonics of B, starting with the sixthharmonic. Each subsequent frequency increases by 4JB, meaning the first threefrequencies for ∆J = +2 from J = 0,1,2 are the sixth, tenth, and fourteenth har-monics of B. A factor of two can be removed, making these the third, fifth, andseventh harmonics of 2B. The fundamental period is then Trev = 12B , where a pe-riodic rephasing of all harmonics occurs at integer multiples of Trev. For kickedrotors, this is the quantum revival mentioned above. As the spectrum now consistsof only odd multiples of 2B, there will also be rephasing at the half revival t = Trev2 ,when the molecules are π out of phase with respect to t = 0.Centrifugal distortion breaks the harmonic structure of the set of frequencies,due to decreasing energy level separation. If the distortion is minimal, revivalfeatures will still be present at early multiples of the fundamental period, with someadditional dephasing having the effect of broadening the revival structure [22, 71].Novel pulse shaping methods have been used to minimize this dephasing [72]. Ingeneral, the quantum revival occurs at the second derivative of the rotational energywith respect to J [95, 96], resulting in a J-dependent revival time when centrifugaldistortion is considered, owing to terms with higher powers of J in Equation 2.7.Symmetric and asymmetric tops may also produce revivals [8, 85]. The addi-24tional rotational constants produce additional types of revivals, which are classifiedas J-type, A-type, or C-type [76]. The only work in this thesis with an asymmetrictop is in Chapter 10.2.5 Pulse TrainsThe series of Raman transitions, described in the previous section, can be eitherenhanced or suppressed with subsequent pulses, where the timing between pulseschanges the spectrum of the two-photon excitation field.2.5.1 Annihilation/EnhancementIf the rotational wavepacket has a quantum revival, applying a second pulse at therevival time will increase the degree of excitation, and further broaden the J-statedistribution. At the time of the revival, the molecular axes are in their original ori-entation, and so for each molecule the induced torque will be in the same directionas that of the first kick. For a given total fluence, the degree of excitation may bethe same as that of a single pulse, but by distributing the energy in time, ionizationmay be avoided [40, 89]. In fact, it has been shown to be beneficial to split theenergy between pulses, so that the alignment initiated by the first pulse can be usedto maximize the torque/energy transfer from the second pulse [93].Placing a second identical pulse at t = 12 Trev will apply an “equal and opposite”torque, which effectively cancels out (annihilates) the initial excitation [90]. Differ-ent molecules may be selectively excited, by applying the second pulse around thetime where alignment of one molecular species overlaps with the anti-alignmentof the other. The double pulse annihilation/enhancement scheme has been usedfor selective alignment of molecular spin isomers, by taking advantage of the factthat the allowance of only odd or even values of J determines whether there is netalignment or anti-alignment at t = 14 Trev and34 Trev [51]. Reference [52] used thisscheme to selectively excite 15N2 in a mixture with 14N2, where t = 63ps is ap-proximately the 7th and 7 12 th revival respectively. Pulse trains with more pulseshave also been used to selectively excite isotopologues of N2, by kicking at thefull revival period many times, and to selectively excite spin isomers (which haveidentical Trev), with the train period set slightly off 34 Trev [199]. This is noteworthy25because it combines the excitation selectivity of a double pulse scheme with thehigher degrees of rotational excitation possible using a longer pulse train.2.5.2 State-Selective ExcitationsIn the case of molecules with both odd and even J states (like N2O), the rela-tive populations of odd and even states can be manipulated in the same way [58].This points to a more general phenomenon, where the second kick can selec-tively enhance or suppress specific coherences, rather than the entire excitationas a whole. To suppress specific frequencies, one may simply adjust the pulse inthe time domain such that the Fourier transform of the two-photon Raman field[E(2)(t) = E(t)E∗(t)] has a value of zero at the target frequency. Conveniently,this is the Fourier transform of the pulse intensity profile, hence simple mecha-nisms such as adding a second pulse at a specific delay can effectively controlstate-specific transition probabilities.The Fourier transform of the intensity profile of a pair of pulses separated bytime T produces a periodic oscillating pattern (illustrated in Figure 2.4) in the fre-quency domain, under the same envelope of a single pulse. The maxima are peri-odically spaced by f = 1T , which gives some insight as to why various rotationalfrequencies of a kicked diatomic molecule are readily enhanced/annihilated [90].As was shown in [17], the excitation of specific states can be controlled this way.In addition to the possibility of controlling the excitation using beamsplittersand delay stages to create pulse trains, tailored spectral phases using a pulse shapercan be used to control transition probabilities equally well [114]. This mechanismof state-selective excitation was first demonstrated with vibrational transitions overthree decades ago [184]. It was demonstrated that this is true for any nonresonantprocess with a narrow transition bandwidth [115], including the rotational Ramantransitions for kicked molecules. There are two major implications of this. First,excitation pulse energy can be further spread out in time to reduce undesired ef-fects such as ionization without sacrificing transition probability, or creating a longsequence of nested beamsplitters. Second, spectral phase distributions can often begenerated that will enhance desired transitions, and suppress others.26Figure 2.4: Above: The intensity envelopes of one and two pulses (blue andred, respectively). Each pulse has a duration of 1 ps. The pair of pulsesis separated by 5 ps, and each has half the amplitude of the single pulse.Below: The Fourier transform of these intensity profiles, representingthe spectrum of the two-photon excitation field, E⃗ · E⃗∗. This shows thatthe pair of half-amplitude pulses has the same envelope in the frequencydomain as the single pulse, but with zero spectral intensity betweenmaxima at multiples of f = 15ps = 200GHz.27Chapter 3Detecting Molecular RotationA variety of experimental techniques have been utilized to detect molecular ro-tation. The Time of Flight Velocity Map Imaging (TOFVMI) spectrometer hasbeen my primary piece of apparatus for observing molecular rotation, includingall experiments in helium droplets and molecular jets. Rotational coherence spec-troscopy techniques were performed in the spectral domain with Coherent Ramanscattering for room temperature gasses to characterize the optical centrifuge, and inthe time domain using laser-induced fluorescence to observe the rotation of heliumexcimers in bulk liquid helium.3.1 Time of Flight Velocity Map Imaging3.1.1 Time of FlightBy accelerating ions created by a short laser pulse to a few keV in a static electricfield, and then allowing them to propagate in a field-free region as shown in Figure3.1, one can determine the mass to charge ratio simply based on the time of flight.Ions are detected by a microchannel plate (MCP), which amplifies one ion hit intoa “shower” of electrons, which can be directed to a phosphor screen and detectedby a photomultiplier, or measured directly as a small AC current on the MCP.If the potential at the position of ionization is V , the kinetic energy accrued byan ion of charge q before entering the field-free region equals qV . Solving for the28Figure 3.1: The Time of Flight Velocity Map Imaging Spectrometer. A shortlaser pulse ionizes molecules in a strong static electric field between theRepeller (R) and Extractor (E) electrodes, which accelerate ions throughthe hole in the Ground (G) electrode. Ions then travel through a vari-able length field-free flight tube, before hitting the microchannel platedetector.speed givess =√2qVm, (3.1)for a particle of mass m. If the detector is a distance d away, that gives a flight timeT =ds. (3.2)For instance, a 5 kV potential with d = 20cm will result in a time of flight of 760 nsfor an N+ fragment from the ionized N2 molecule, easily resolved from the “parent”N2+ ion at 1080 ns with relatively affordable electronics. MCP assemblies generallyhave sub-nanosecond response times, with output pulse widths on the nanosecondscale, so if the response current is measured using an oscilloscope, mass differencesof a single atomic mass unit (AMU) are resolvable. Since T ∝√m, the timingresolution ∆T ∝√m+1√m is greater for smaller masses, with an approximate 10 ns29separation at 100 AMU for this configuration.3.1.2 Velocity Map ImagingThe electrodes and applied voltages can be designed such that they form an elec-trostatic lens that directs ions towards a specific spot on the detector, depending ontheir mass and 2D velocity component parallel to the detector, relatively indepen-dent of their initial position within a small volume.The relationship between the fragment’s location r in the plane of the detector,its charge q, mass m, and velocity parallel to the detector v, is well approximatedby the empirical relation originally fit by Eppink and Parker [48]r = a · v√mq, (3.3)where the prefactor a is empirically determined for a given set of voltages in a givensetup. For many experiments, a camera with an imaging lens is pointed towards thephosphor screen, so the prefactor accounts for the magnification and pixelizationof the camera system. Once calibrated, this constitutes a “velocity map”. Calibra-tion is typically done using a well-known Coulomb explosion process, such as the2.0(5) eV kinetic energy release of the CS2 dimer [78].3.1.3 Coulomb Explosion ImagingWith the ability to separate different mass/charge ratios in time, and measure thekinetic energies of charged particles created in the “focus” of the TOFVMI lens,one can multiply ionize a molecule and image the kinetic energy release (KER) asthe charges repel each other through Coulombic forces. In the simplest case ofa diatomic molecule, the 2D projection gives a good indication of the molecule’sorientation at the time of ionization, as shown in figure 3.2. This is assumed to bethe case for most two-body Coulomb explosion processes, and is referred to as the“axial recoil approximation”.Even a strongly bonded linear triatomic molecule such as CS2 undergoes mea-surable bending, and does not recoil entirely along the bond axis. To accuratelyreport alignment factors, the effects of non-axial recoil may be deconvoluted [36].30Figure 3.2: Depiction of a diatomic molecule with some spatial orientationin three-dimensional space. In our Coulomb explosion experiments, weproject the two-dimensional velocities of each ion fragment onto theplane of the TOFVMI detector. θ2D is defined as the angle between anion fragment and the axis perpendicular to both the laser propagationand TOFVMI axes, Y and Z respectively.A more complex molecule, such as CH3CHCH2O (propylene oxide, referred toas PPO, and studied by our group), may have bending and torsional modes thatcontribute to “smearing” the possible KER of any specific fragment across manyangles. Scattering of ion fragments from helium atoms after Coulomb explosionof molecules embedded in helium droplets is another serious complication, and isdiscussed in Section 5.4.By calculating the angular distribution of fragments collected from this 2Dprojection, in combination with the assumptions and analysis relating fragmentpositions with molecular orientation, one can calculate the degree of anisotropyin the molecular distribution’s spatial orientation. The degree of alignment withrespect to a particular axis is typically quantified as ⟨cos2 θ2D⟩, with θ2D defined inFigure 3.2. In contrast, ⟨cosθ2D⟩ can be used as a measure of molecular orientation,31Figure 3.3: The Time of Flight mass spectrum when CH3CHCH2O was ion-ized by an intense femtosecond pulse. Red lines indicate mass to chargeratios of 6,12,16,26, and 31, assigned to C++, C+, O+, C2H2+, andCH3O+, respectively. Ion fragments with mass to charge ratios closeto H+ were not recorded.rather than alignment.3.1.4 Time of Flight Covariance MappingThe covariance between two variables is defined ascov(x,y) = ⟨xy⟩−⟨x⟩⟨y⟩, (3.4)where ⟨...⟩ represents averaging over the ensemble. The above expression reflectsthe number of simultaneous observations, minus the expected number of uncor-related simultaneous observations, given the individual count rates. Evidentlycov(x,y) = cov(y,x).By collecting ion mass spectra for individual laser pulses, one can study the dif-ferent fragmentation pathways. The fraction of molecules that dissociates througheach pathway can vary depending on laser wavelength and intensity. The covari-ance (unscaled correlation) between fragments provides information as to whichpathways are contributing to a given m/q. One can then construct a covariance“map”, where variables x and y correspond to different m/q ratios.32Molecular nitrogen (14 N2) can produce ion pairs with the following m/q values:(14/1 & 14/1),(14/1 & 14/2),(14/1 & 14/3),(14/2 & 14/3),etc.In a simple application, this could help identify whether a strong NO+ signal iscoming from N2O or NO2, depending on whether the NO+ signal is positively ornegatively covariant with N+ or O+. As a more complex example, in the case of alarger molecule like propylene oxide (CH3CHCH2O), it may be useful to determinewhether O+ fragments are primarily coming with other monatomic fragments, theCH3 methyl group, or the entire chain of three carbons. A TOF mass spectrumaveraged over many laser shots, as shown in Figure 3.3, informs us only that eachof these fragments are present. By analyzing the covariance between the detectionof fragments in Figure 3.4, we can obtain significantly more information. With thelines marked 12 and 16, we see that C+ and O+ primarily arrive with other smallfragments - mostly monatomic ions. Meanwhile, clusters of multiple carbon atomswith hydrogen or oxygen are not strongly correlated with any other fragments. Forpolyatomic molecules, this type of analysis is useful for simulating the Coulombexplosion process, where ion ejection vectors may vary, depending on how themolecule is fragmented.33Figure 3.4: Covariance analysis of the time of flight spectrum ofCH3CHCH2O ionized by an intense femtosecond laser pulse, recordedover 2000 individual laser pulses. This is the same data that is plottedin Figure 3.3. The colourmap is linear, and reflects the covariance ofthe intersecting mass to charge ratios. The relatively similar intensity at(16,12) and at (16,16) implies that O+ primarily arrives with C+. The(12,12) signal is much stronger, meaning that carbon ions more oftenarrive with one another, rather than with oxygen. That is not surprising,given there are three carbon atoms for every one oxygen atom in PPO,but it is useful to know that most oxygen ions come with C+, and veryfew come with larger clusters.343.2 Angular Covariance MappingAdapting Equation 3.4 to the ion count in N acquisitions in two angular bins Θaand Θb, each representing a range of θ2D values, givescov(Θa,Θb) = ⟨ΘaΘb⟩−⟨Θa⟩⟨Θb⟩= 1NΣNi=1(Θia− Θ¯a)(Θib− Θ¯b), (3.5)where Θ¯a and Θ¯b reflect the average number of ions in each of those bins over alllaser shots.Taking as an example four large 90◦ bins, the covariance map is a four-by-four matrix, shown in Table 3.1. The four angular bins are from 0◦ to 90◦, 90◦ to180◦, 180◦ to 270◦, and 270◦ to 360◦. Replacing the subscript for bin identifiersin Equation 3.5 with the two angles defining the bin for clarity, Θa −→ Θ900 wouldrepresent the angular bin between 0◦ and 90◦.cov(Θ900 ,Θ18090 ) reflects whether the observation of ions in each of these twobins simultaneously is due to a physical relationship, or merely the statistical out-come of unrelated events. In a Coulomb explosion measurement, it would be anindicator that the Coulomb explosion process is sending ions in pairs with trajec-tories separated by ∼90◦ to these specific bins in the laboratory frame.By comparing the values of cov(Θ900 ,Θ270180) and cov(Θ18090 ,Θ360270) for a diatomicmolecule, one could identify if the molecule was more likely aligned around themiddle of the angular bins at θ2D = 45◦ (and 225◦), or θ2D = 135◦ (315◦). Intypical experiments, the bin size is determined by the number of ions collected toensure good statistics, with bin widths usually around 4◦ for a few minutes of datacov(Θ900 ,Θ900 ) cov(Θ18090 ,Θ900 ) cov(Θ270180,Θ900 ) cov(Θ360270,Θ900 )cov(Θ900 ,Θ18090 ) cov(Θ18090 ,Θ18090 ) cov(Θ270180,Θ18090 ) cov(Θ360270,Θ18090 )cov(Θ900 ,Θ270180) cov(Θ18090 ,Θ270180) cov(Θ270180,Θ270180) cov(Θ360270,Θ270180)cov(Θ900 ,Θ360270) cov(Θ18090 ,Θ360270) cov(Θ270180,Θ360270) cov(Θ360270,Θ360270)Table 3.1: The four-by-four covariance matrix for 90◦ angular bins. Notethat there is symmetry about the diagonal because cov(a,b) = cov(b,a),halving the amount of computation necessary.35Figure 3.5: Left: Definition of θ2D on the VMI detector, with respect to theTOFVMI axis (Z) and laser propagation direction (Y). Middle: Sim-ulation of a randomly oriented ensemble of homonuclear diatomicmolecules, and the resulting isotropic VMI image and angular covari-ance map. The two parallel lines linking θ2D with θ2D + 180◦ indicatethat ions are found in pairs on opposite sides of the detector, for allvalues of θ2D. Right: Simulation of an ensemble of homonuclear di-atomic molecules mostly aligned to the axis along X, from which θ2Dis measured. The angular covariance map below shows narrow covari-ance between 0◦ and 180◦, in contrast to the broad covariance across allangles for the isotropic ensemble.collection.In [155], angular covariance maps were utilized to determine the structure of(C6H5Br)2 and C6H5Br-I2, two dimers created in helium droplets. For such acomplicated structure, the methodology involves simulating possible geometriesand the respective angular covariance maps for aligned molecules. In two-bodyCoulomb explosion processes, the velocity vectors must point in opposite direc-tions to conserve momentum, and so an angular covariance map can readily be used36to identify the axis of molecular alignment, as depicted in the rightmost image inFigure 3.5. I discuss the different angular covariance maps of two and three-bodyCoulomb explosion processes in Appendix A.3.3 Angle-Dependent IonizationThe anisotropy of molecular electronic configuration leads to an angle-dependentionization rate when they are exposed to a linearly polarized laser field [66, 196].The specifics depend on the electronic wavefunction. For instance, N2 is mosteasily ionized when the molecular axis is aligned to the field polarization, whereasthe ionization of O2 peaks at ∼45◦ [86]. An example of how this was measured inour laboratory is shown in Figure 3.6.As shown in Figure 3.7, the ionization saturates at higher intensities, thus ceas-ing to be an effective tool for determining the molecular alignment. For both N2and CS2, the effect is quite strong with a reasonable ion yield [13].This technique offers much lower sensitivity, but does not rely on an ultra highvacuum chamber with a sensitive detector as used in the VMI apparatus, and alsoworks with a standard TOF using just one high voltage electrode and no 2D detec-tion.37Figure 3.6: The alignment factor ⟨cos2 θ2D⟩ (above) and total ion count (be-low) as a function of probe delay around nitrogen’s half revival att = 4.2ps, after being kicked at t = 0ps. The probe is linearly polar-ized parallel to the VMI detector, i.e. along X in Figure 3.2. The bluetraces were recorded with the kick polarized parallel to the probe po-larization, while the red traces were recorded with the kick polarizationperpendicular to the probe polarization. Molecules align with the kickpolarization around 4 ps after the kick, slightly before the half revival.The ratio of the respective ion counts for two kick polarization direc-tions, along X and along Z (shown in the bottom plot), reflects the de-gree of alignment (shown in the top plot).38Figure 3.7: The “Ion Count Ratio” plotted is the ratio between the numberof N+ ions collected when N2 is aligned parallel, versus perpendicular,to the probe polarization axis. What this demonstrates is that as theprobe intensity increases, the ion count ratio saturates, and moleculesbecome equally likely to be ionized regardless of their spatial orienta-tion. A lower ion count ratio indicates greater sensitivity to molecularalignment. This was measured at the half revival after a kick (see Figure3.6). This plot includes data for nitrogen adapted from [196].393.4 Rotational Coherence SpectroscopyWhen an ensemble of molecules is rotationally “kicked” (see Section 2.4), thetransient alignment of molecules can be measured directly through various means.Measurements that observe this alignment in the time domain can be Fourier trans-formed to calculate the frequencies corresponding to excited state transitions. Twoprominent time-resolved methods of measuring molecular alignment are wavepacketimaging through CEI (Section 3.1.3), or by observing the change in polarization ofa probe pulse, due to the rotation-induced birefringence of an ensemble of rotat-ing molecules [187]. An alternative frequency-resolved detection technique relieson nonresonant coherent Raman scattering, which can measure both the frequencyand the direction of molecular rotation, by recording the frequency shift and thepolarization of the scattered light at a fixed probe delay.3.4.1 Rotational Raman SpectroscopyCoherent Raman scattering (CRS) of light from rotating molecules represents oneof the most direct detection techniques to characterize the rotational excitation byan optical centrifuge. Raman scattering refers to inelastic photon scattering. In thecase of rotational excitation, a molecule can absorb and emit two photons, eachof opposite angular momentum, resulting in a transition with ∆J = ±2, as shownin Figure 3.8. The difference in photon energy will reflect the energy differencebetween the corresponding rotational levels.As executed in most of our experiments, a circularly polarized probe can beused to identify the direction of rotation. Molecules rotating with the probe willbe “sped up” by undergoing anti-Stokes Raman scattering, and output a red-shiftedphoton, while counter-rotating molecules blue-shift the output. For a circular probepolarization, the polarization of the Raman sidebands is also circular and oppositeto the input probe polarization. Classically, these shifts (sidebands) are explainedas frequency modulation due to the rotation of a birefringent material providing atime-dependent refractive index, with the magnitude of the shift being twice therotation frequency. Generally, the bandwidth of the probe pulse determines thewidth of each sideband, forcing a trade-off between temporal and spectral reso-lution. Being a third-order nonlinear optical process, coherent Raman scattering40Figure 3.8: Stokes and anti-Stokes Raman transitions, depicted for a set ofrotational energy levels with E(J) = BJ(J+1). In our experiments, thephoton energy is much greater than the separation between rotationalstates. In Stokes (anti-Stokes) scattering, the emitted photon is lower(higher) energy with respect to the absorbed probe photon (green).requires relatively dense gas samples to generate a measurable signal [31].The Raman spectroscopy setup is shown schematically in Figure 3.9. A rota-tional excitation such as an optical centrifuge, described in Section 2.3, is focusedinside an optical cell filled with a gas at room temperature and variable pressure,although a sufficiently dense jet has also been used [14]. Probe pulses, circularlypolarized by a combination of a linear polarizer and a quarter-wave plate, followthe centrifuge after a variable time delay. Frequency doubling of probe pulsesallows us to combine and separate them from the centrifuge light using dichroicmirrors. Another set of a linear polarizer and a wave plate acts as a circular an-alyzer, separating the weak frequency-shifted Raman sidebands from the strongunshifted and oppositely polarized background. Narrowing the spectral bandwidthof probe pulses to ∼0.1 nm by means of a probe pulse shaper is useful for resolv-ing individual rotational states, as well as for distinguishing centrifuge-populatedsuperrotor states from those initially populated in a room-temperature ensemble.Examples of CRS spectra from the centrifuged gas of oxygen and nitrogenmolecules are shown in Figure 3.10 (a) and (b), respectively. Recording the spectra41Figure 3.9: Experimental configuration for characterizing the rotational ex-citation with an optical centrifuge by means of coherent Raman spec-troscopy. PBS: polarizing beam splitter, DM: dichroic mirror. An op-tical centrifuge field is produced in the centrifuge shaper shown in Fig-ure 2.3. Probe pulses are compressed pulses, split from the main laseroutput before the centrifuge shaper, and frequency doubled in a nonlin-ear crystal.as a function of the centrifuge-probe time delay provides us with a two-dimensionalRaman spectrogram, and offers a convenient way of following the rotational dy-namics of molecules inside the centrifuge, as well as the field-free dynamics atlater times. For both gases, Raman spectrograms were recorded with two differentoptical centrifuges in separate optical setups and different gas cells. The “slow cen-trifuge” (upper traces) accelerates to around 2.5 THz over 500 ps, while the “fastcentrifuge” (lower traces) used here accelerates to around 7 THz in ∼70 ps.During each centrifuge the molecules are following the accelerated rotation, asreflected by the bright tilted traces in the upper part of the slow centrifuge spectro-grams. From the tilt, one can extract the angular acceleration (5.4 GHzps−1) and42Figure 3.10: Two-dimensional Raman spectrograms from the centrifuged gasof (a) oxygen and (b) nitrogen, recorded at room temperature. Hori-zontal traces reflect the centrifuge-induced rotational coherence, whichdecays due to collisions. Tilted traces starting from the origin are dueto the linearly increasing with time rotational frequency of centrifugedmolecules. The additional lines present for N2 are explained in thebody of the text.43the corresponding value of the average frequency chirp (introduced in Equation2.27) βSCFG = 0.0170(3) ps−2, in agreement with the chirp value we extracted us-ing frequency-resolved cross-correlation in [104]. Some molecules fall behind thecentrifuge, while others are brought to the superrotor states at the end of the pulse.In the case of the fast centrifuge, the molecules spin up at around 95.5 GHzps−1,corresponding to the average frequency chirp βFCFG = 0.31(1) ps−2.The Boltzmann distribution at room temperature results in many initially pop-ulated states, resulting in some quickly-decaying Raman signal as the centrifugepasses through the transition frequencies from those states, visible over the firstfew terahertz in each plot. The multitude of lines visible at higher frequencies forN)2 in the fast centrifuge is likely due to poor alignment of the centrifuge, result-ing in many molecules “falling out” of the centrifuge before its terminal frequency.The alternating line intensity for N2 (most visible in the slow centrifuge) is dueto nuclear spin statistics. 14 N is a boson, so the molecular wavefunction must besymmetric when nuclei are exchanged. The nuclear spin is 1, resulting in a 6:3ratio of symmetric:antisymmetric spin wavefunctions, resulting in a 2:1 ratio ofeven and odd rotational wavefunctions, respectively. Each atom in O2 is a bosonwith a nuclear spin of 0, but its ground state electron configuration is antisymmetric(X3Σg – ), resulting in only odd rotational wavefunctions, and the larger line spacingin Figure 3.10.3.5 Laser-Induced FluorescenceThis detection technique is analogous to the angle-dependent ionization method,described in Section 3.3. It is based on the anisotropy of a resonant photon ab-sorption with regard to the molecular axis. Here, probe light excites moleculesto an electronic state, which decays radiatively - a phenomenon known as laser-induced fluorescence (LIF). With an anisotropic absorption cross section relativeto the molecular axis, the alignment of a molecular ensemble may be inferred froma differential signal that measures LIF for two orthogonal probe polarizations. Theexperimental scheme for our study of the rotation of excited helium dimers in bulksuperfluid helium using LIF is described in detail in Chapter 4, where LIF is utilizedfor rotational coherence spectroscopy.443.6 Resonance-Enhanced Multi Photon IonizationIn cases where radiative decay is too weak to detect via LIF, Resonance-EnhancedMulti Photon Ionization (REMPI) is often utilized, thanks to the much higher ef-ficiency of ion detection. The system is resonantly excited to an intermediate ro-vibrational state and then subsequently ionized, either by photons from the same oranother source. Both the excitation and ionization can be from one or multiple pho-tons each. This ionization scheme is typically denoted as (N +M) REMPI, whereN is the number of photons to promote the molecule to the intermediate state, andM is the number of photons to ionize post-excitation.By measuring the ion yield as a probing laser’s wavelength is scanned acrossresonances, the yield for each wavelength will scale with both the probability ofexcitation and initial populations of all resonant transitions. With a broad distribu-tion of initial states, these resonances often overlap, and REMPI spectra can becomequite congested, even without excitation, due to the numerous possible transitions.In the opposite case of a narrow distribution of states, REMPI has been used forspectroscopy of highly excited molecules using an optical centrifuge [84]. Pho-tons from a circularly polarized probe are more efficiently absorbed as the laser’selectric field vector rotation coincides with that of the molecule, hence for unidi-rectional molecular rotation there is a significant disparity in ionization rates fortwo directions of probe rotation [17, 82, 198].In the course of my work I used REMPI to measure the rotational temperature ofthe supersonic jet used for some gas phase experiments, as described in AppendixB, in addition to a brief experiment in helium droplets, documented in AppendixC.45Chapter 4Rotational Control of Moleculesin Liquid HeliumPart 1: He2* in Bulk Superfluid4HeWhile the majority of the time during my PhD was spent working with heliumnanodroplets, interesting results were also seen on the other side of the laboratorywith studies in bulk helium. This chapter provides an overview of our recent ex-periments controlling the rotation of molecules in bulk superfluid helium, which isconnected to similar experiments in helium droplets in later chapters. The tempera-ture of bulk liquid helium can be controlled through various means, such as evapo-rative cooling by pumping on the vessel, or directly heating the sample. Controlledvariation of sample temperature is essential to study the two-fluid model of super-fluidity, which describes how the superfluid fraction depends on temperature. Suchcontrolled variation is not possible in the helium droplet experiments described inlater chapters. The solubility of molecules in liquid helium is very low, making itdifficult to study the dynamics of a rotor in bulk helium, as impurities are quicklymoved to vessel surfaces [167]. One solution to this problem is to study the dynam-ics of helium dimers in bulk liquid helium. This chapter outlines those methods in46detail, and demonstrates quantum control of both the degree of rotational excitationand the direction of rotation using a carefully chosen sequence of pulses, referredto as a pulse train.4.1 IntroductionIt has been known for several decades now that ionization within bulk liquid heliumresults in a series of excitations that decay, leaving behind a significant density ofhelium dimers in their lowest metastable triplet state a3Σu+ [41, 80]. Electroni-cally excited molecules are referred to as “excimers”. The a3Σu+ lifetime in liquidhelium is found to be on the order of ten seconds [27, 109], with a rotational re-laxation time of several milliseconds [108], providing ample opportunity to studyrotational dynamics on the picosecond timescale. The lifetime is limited by bi-molecular collisions (between two dimers), the rate of which is strongly dependenton the fluid temperature [10, 80]. Roton excitations in the surrounding helium limitthe diffusion of dimers throughout the liquid, reducing the number of bimolecularcollisions. As roton energy and prevalence varies with temperature, this leads toa temperature-dependent dimer lifetime [4]. It was found that the dimers may re-side in macroscopic gas pockets, or nanometre-sized bubbles formed by the strongrepulsion between the Rydberg electron of the excimer and the surrounding closed-shell helium atom [10, 45]. Infrared absorption spectra showed that dimers in thebubble state undergo nearly free rotation [47].As the helium nuclei are identical bosons, the molecular wavefunction mustremain symmetric upon their exchange. The electronic excitation results in anantisymmetric electron orbital wavefunction. Excimers in excited singlet statescan readily decay into helium dimers in their ground state (X1Σg+), which havea binding energy on the order of 1 mK, and therefore quickly recombine with thesurrounding liquid helium. The lowest triplet excited state cannot decay withoutflipping an electron’s spin, which is suppressed due to weak spin-orbit coupling.The triplet state spin wavefunctions are symmetric, requiring odd rotational wave-functions with N = 1,3,5... to produce a symmetric molecular wavefunction. HereN represents the total angular momentum (J) excluding electronic spin (S). Thetriplet’s three possible values of S split each rotational state into three fine structure47components (J = N,N + 1,N − 1), due to spin-spin and spin-rotational coupling[100]. This splitting is on the order of a few gigahertz, while the ∆N = 2 splittingsrelevant for Raman excitations are on the terahertz scale. Both splittings result inobservable beating in time-domain studies of molecular rotation, on the nanosec-ond and picosecond timescales, respectively [124].A recent study in the Milner laboratory expanded on prior work by rotation-ally exciting the dimers with a 70 fs kick pulse, measuring a rotational coherencelifetime on the order of one nanosecond for the N = 1−→ N = 3 transition [124].Utilizing this long coherence time, we then carried out an experimental demon-stration of controlling molecular rotation inside the superfluid. A “double kick”,described in Section 2.5.2, was used to “shut off” molecular rotation after a se-lectable number of rotations, while a more complex “chiral pulse train” (explainedin Section 4.2.3) was shown to control directionality.4.2 Experimental TechniqueThe liquid helium cryostat with laser viewports is shown in Figure 4.1. Helium gasis condensed in a thermally-isolated volume of approximately 500 mL at 4.2 K.After this volume is full, the helium is evaporatively cooled by pumping away ∼ 23of the liquid helium, resulting in approximately 170 mL of liquid remaining, whencooled below ∼2 K. The rotational control experiments were performed at 1.4 K,below the superfluid transition.4.2.1 Generation and Probing of Helium DimersIt was demonstrated in [9] that a 790 nm/80 fs pulse could efficiently generatedimers in the bubble state, with a second nearly identical pulse used in [10] servingas a probe through two-photon laser-induced fluorescence (LIF). The second pulseexcites the d3Σu+ state, which then decays (with a time constant of 25 ns [9]) to theb3Πg state by radiating a photon around 640 nm, as depicted in Figure 4.2 [10].The b3Πg state then decays nonradiatively back to the a3Σu+ state, for the processto be repeated.In our experiment, a 2 ps “pump” laser pulse with a peak intensity of 1.7×1012 W/cm2 prepared an ensemble of He2* with a density around 1013 cm−3 [124].48Figure 4.1: The optical cryostat used for this work, image courtesy of LakeShore Cryotronics. The first stage of the Sumitomo RDK-408D2 coldhead chills a cryoshield that surrounds the innermost chamber, wherethe liquid helium is formed. A vacuum pump controls the pressureabove the helium, so the temperature can be lowered through evapo-ration. The cooldown process is approximately 13 hours. Afterwards,helium can be condensed for up to 14 hours, consuming approximately1 Bar from a T sized gas cylinder each hour. The operating time withsuperfluid helium is approximately equal to the condensation time.49Figure 4.2: Left: Calculated potential energy curves for He2* in liquid he-lium, as a function of the size of the bubble surrounding He2*. Whenmolecules are excited to higher Rydberg states, the bubble must grow toaccommodate it. Figure taken from [10]. Right: The laser-induced fluo-rescence scheme. The 793 nm pulses have a broad bandwidth (∼30 nm),and so are able to excite the two-photon absorption line at 800 nm,bringing the excimers to the to the d state [173], which decays to theb state by emitting a single 640 nm photon. The b state then nonradia-tively decays back to the a state.This intensity was chosen as it is sufficiently low to avoid forming macroscopic gaspockets, which were identified through the appearance of sharp rotational lines inthe LIF spectrum [124]. Macroscopic pockets disrupt spatial coherence in prop-agating beams, and generally make it more difficult to identify the interactionsbetween the surrounding fluid and solvated dimers, as the rotational structure be-comes a superposition of features from gas-phase and solvated excimers [113].The arrival time of the probe pulse with respect to the excitation pulse trainwas varied with a translation stage. Rotational coherence was measured usingthe anisotropy of the photon absorption cross section in the LIF process. The ab-sorption cross section’s dependence on the relative orientation of the (ensemble-averaged) molecular axis to the probe polarization allows one to infer the angle ofthe molecule, similar to the angle-dependent ionization technique described in Sec-50Figure 4.3: An example of the LIF signal as a function of time, with respect tothe arrival time of the pump pulse. Arrows indicate the arrival times ofthe three pulses, and the coloured traces were recorded with the beamslabeled.tion 3.3. This then allows for rotational coherence spectroscopy, where scanningthe probe delay (∆tprobe) while recording the LIF signal allows one to observe theevolution of the molecular wavepacket after rotational excitation. The polarizationof the excitation pulse is modulated with a Pockels cell at 20 Hz, thereby alternat-ing between LIF minima and maxima as a function of the fixed-polarization probe’sdelay, so that lock-in amplification can be utilized for a more sensitive “dichroism”measurement (explained later in the text).Pump, train, and probe pulses are all taken from the same femtosecond lasersource with a 1 kHz repetition rate. The pump pulse is delayed approximately 40 nsafter the train, so as to prepare dimers for the subsequent laser pulse, meaning thetrain is exciting dimers formed nearly 1 ms in advance, as shown in Figure 4.3.The train intensity is adjusted based on maximizing the efficiency of the desiredrotational excitation, while ensuring it does not produce a strong LIF signal in theabsence of the pump. The probe energy was adjusted to maximize the differencein the modulated dichroism signals, while ensuring the majority of observed fluo-rescence is dependent on the earlier pump pulse (see the difference in LIF signalstrength between the blue and red traces in Figure 4.3 during the gate).The three collinear laser beams were then focused into the cryostat down to a51beam radius of around 25 µm by an f = 25cm lens, with approximate pulse en-ergies (intensities) of 100 µJ (1.7×1012 W/cm2), 20 µJ (below 6.6×1012 W/cm2for each pulse in a double kick), and 2 µJ (1.3× 1012 W/cm2) for the 2 ps pump,variable train, and 50 fs probe respectively.The LIF signal was measured through a viewport orthogonal to the laser beampropagation, with a large condenser lens focusing the fluorescence onto a photo-multiplier tube (PMT), after a 640 nm spectral filter with 14 nm bandwidth. Theoutput from the PMT was then gated in time over a few nanoseconds (as shownto scale in Figure 4.3) after the probe with a Stanford Research Systems BoxcarAverager (SR250), and fed into a lock-in amplifier (SR830), alongside the refer-ence signal of the modulated train polarization at 20 Hz, to obtain the fluorescenceintensities for each modulated state.The linear dichroism signal for double kick experiments was defined in termsof the fluorescence intensity I as:LD =12I⇕↕ − I⇔↕(I⇕↕ + I⇔↕ ), (4.1)with the kick polarization (superscript) respectively parallel or perpendicular to thefixed linear probe polarization (subscript). Unidirectional rotation was observedusing a circular probe in a circular dichroism measurement, where the directionof rotation of the chiral pulse train was modulated. The superscript and subscriptagain refer to the train and probe directionality, respectively:CD =12I↷⟳ − I↶⟳(I↷⟳ + I↶⟳ ). (4.2)The circular probe was passed through a continuously rotating half wave plate, soas to scramble any ellipticity that may contribute to CD measurements. A Fouriertransform of LD(∆tprobe) or CD(∆tprobe) then presents the frequencies of rotation.The amplitudes of these frequencies in the power spectrum reflects the degree ofinduced rotational coherence.524.2.2 Pulse Train GenerationBoth the double kick and chiral pulse train were formed using a 4 f pulse shape witha spatial light modulator (SLM) to control the output pulse. The basic principle of4 f pulse shapers is to use dispersive elements to provide different optical paths fordifferent optical frequencies. It requires relatively well-matched pairs of diffractiongratings, and lenses or focusing mirrors, and careful alignment [181]. A foldedversion of the standard 4 f pulse shaper was utilized with a concave mirror insteadof a pair of lenses, and required just a single diffraction grating. The shaper isshown in Figure 4.4. The concave mirror is hit off-centre, much like the lenses ina typical stretcher configuration, as used for the optical centrifuge in Section 2.3.The total path length in the shaper is four times the focal length of the focusingmirror.As schematically depicted in Figure 4.4b, when the beam hits above the centreof the concave mirror (CM), the beam will be reflected down back towards the grat-ing (GR). A narrow “folding” mirror (FM) is placed at the height of the centre ofthe concave mirror, partway before beam returns to the grating, directing the beamtowards a third mirror placed at the Fourier plane (FPM). The Fourier plane mirrorreflects the beam back towards the folding mirror, where it is directed back to theconcave mirror. It will then be reflected by the concave mirror below the centre,after which it will be levelled out and travel back towards the grating below thefolding mirror. The output diffracted beam will be below the input, so it can easilybe redirected elsewhere. In practice, this requires careful consideration of the beamdiameter and component specifications, because the beam must clear above/belowthe folding mirror, while the entire vertical direction of the beam comes solely fromhitting the concave mirror off-axis. The benefits of fitting a shaper with an f = 1mfocusing element without 4 m of space on the optical table, or requiring a seconddiffraction grating, are traded against a somewhat worse output beam profile dueto the optical aberrations, stemming from off-axis focusing.A dual-layer CRI spatial light modulator with 640 linear pixels was placed atthe Fourier plane. Each layer of an SLM consists of a liquid crystal array, whosebirefringence can be controlled independently for each pixel by applying an electricpotential across the pixel. This changes the refractive index only for light polarized53(a) From above, showing the spectral distribution among components.(b) The height of the beam at each component.Figure 4.4: The folded 4 f pulse shaper used to create the pulse train. The in-put is above the output, and passes the transmissive diffraction grating(GR), then travels above the folding mirror (FM) to the spherical con-cave mirror (CM). It hits CM above the centre, which focuses the beamdown. It reflects off FM, then travels through the spatial light modulator(SLM), and is reflected back by the Fourier plane mirror (FPM) towardsFM. The beam then returns to CM below the centre, where it levels outto traverse under FM towards GR. The diffracted beam is then pickedoff by a mirror below the input beam.54along the extraordinary axis of the crystal. In a dual-layer SLM, the extraordinaryaxes of the crystals in the first and second layer are mutually orthogonal. For anSLM with a linear pixel array, the extraordinary axes are tilted ±45◦ with respectto the row of pixels. When placed at the Fourier plane of a 4 f shaper, this allowsfor the optical phase shifts along these axes, δ1 and δ2 for the first and secondlayer, respectively, to be controlled independently for each spectral component.Assuming a linearly polarized input field, E⃗in(t) = xˆcos(ωt), the resulting fieldafter passing through an SLM pixel is:E⃗out(t) = xˆcos(ωt+δ1+δ22)cos(δ1−δ22)(4.3)−yˆsin(ωt+δ1+δ22)sin(δ1−δ22).If subsequently the beam is passed through a linear polarizer, filtering out the com-ponent polarized along yˆ, the output is the same as the input, but with an additionalspectral phase ofφ =δ1+δ22, (4.4)and a modified spectral amplitude ofH = cos(δ1−δ22). (4.5)In the context of our apparatus with the SLM placed at the Fourier plane of a 4 fpulse shaper, because δ1 and δ2 are controlled separately for each optical frequencyω , φ = φ(ω) and H = H(ω). For a more detailed description of the liquid crystalmechanism, see references [180], [21], and [197]. The procedure used for cali-brating the optical phase shift along the extraordinary axes of the liquid crystals, inrelation to the applied voltage, is similar to that described in [197].A pulse train can be formed through phase modulation alone [183]. Applyingphase-only modulation ofφ(ω) = Asin[(ω−ω0)τ] (4.6)55Figure 4.5: Example of an experimental pulse train measured with standardcross-correlation techniques, with a train period of 500 fs and modula-tion amplitude of A = 2.6.results in a pulse trainE(t) =∞∑n=−∞Jn(A)ε(t+nτ)cos(ω0t), (4.7)where E(t) and ε(t) represent the electric field envelopes of the output and inputpulses, respectively. The modulation amplitude A determines the energy distribu-tion between the pulses in the train, following the Bessel functions of the first kindJn(A). The separation between pulses is determined by the train period τ , and ω0is the optical carrier frequency of the input field. An example of such a pulse train,with τ = 500fs and A = 2.6, is shown in Figure 4.5.It is not possible to create a pair of pulses using only spectral phase modulation.For that, amplitude modulation is required. Applying a periodic spectral amplitudeofH(ω) = cos((ω−ω0)τ2)(4.8)splits the input pulse into two identical pulses, separated in time by ∆t = τ .564.2.3 Chiral Pulse TrainA chiral pulse train is a periodic pulse train, in which the linear polarization ofeach pulse rotates by a constant angle with respect to the previous one [197, 198].The working principle expands on the idea that by rotationally exciting moleculeswith a periodic pulse train, the rotation of molecules will either be “sped up” or“slowed down” with each pulse, until the rotating molecular axis aligns with thepolarization of subsequent pulses in the train, at which point the induced torquebecomes zero, as was shown in Equation 2.21. By rotating the polarization of eachpulse in the train, the symmetry between clockwise and anticlockwise rotation canbe broken, and the direction of molecular rotation can be controlled.As described in detail by Sergey Zhdanovich in his PhD thesis [197], a chiralpulse train is created by applying a sinusoidal spectral phase ofφ(ω) = Asin[(ω−ω0)τ+ξ1,2] (4.9)to the two SLM layers, where ξ1,2 is some additional phase factor, applied to allpixels of the first and second layer, respectively. From Equation 4.4, it can be seenthat the output polarization changes when ξ1 ̸= ξ2. In this case, the spectral phaseof Equation 4.9 results in a pair of pulse trains polarized along the extraordinaryaxes of the two SLM layers, eˆ1 and eˆ2. These two pulse trains are similar to that ofEquation 4.7, but with an additional phase factor, which varies between pulses inthe train:E⃗out(t) =∞∑n=−∞(eˆ1[Jn(A)ε(t+nτ)cos(ω0t+ξ1)](4.10)+eˆ2[Jn(A)ε(t+nτ)cos(ω0t+ξ2)]).As before, ε(t) is the electric field envelope of the linearly polarized input pulse,and τ is the train period. Consider the nth pulse, which can be written as:57E⃗nout(t) = Jn(A)ε(t+nτ)×(xˆcos(ω0t+nξ1)+ yˆcos(ω0t+nξ1)+ yˆcos(ω0t+nξ2)− yˆcos(ω0t+nξ2)).(4.11)Inserting a quarter-wave plate into the beam, with its optical axis oriented alongthe initial field polarization (x-direction), would delay the xˆ components by π2 withrespect to the yˆ components. The resulting field is:E⃗nout(t) =−2Jn(A)ε(t+nτ)sin(ω0t+(nξ1+ξ22))(4.12)×[xˆcos(nξ1−ξ22)+ yˆsin(nξ1−ξ22)].The term in the square brackets of Equation 4.13 defines the linear polarizationof the nth pulse, which is rotated by an angle n ξ1−ξ22 with respect to the inputpolarization. Setting ξ1 = −ξ2 = ξ gives a period of polarization rotation of T =2πτξ . In our work, we used a modulation amplitude of A = 2.6 to produced a chiralpulse train consisting of four strong kicks and two weaker ones, as shown in Figure4.5.4.3 ResultsThe result of the double kick experiment on helium dimers in bulk superfluid he-lium is shown in Figure 4.6. It illustrates a high degree of rotational control withproperly timed pulse sequences. The plot on the left shows that as the separa-tion between kick pulses is varied, the amplitude of LD1,3 (the amplitude of theoscillations in LD(∆tprobe) at the frequency corresponding to the N = 1−→ N = 3transition) can be controlled. A very weak LD1,3 signal, observed for the pulse sep-aration of 220 fs (blue), demonstrates that the rotational excitation can be “turnedoff”. The plot on the right shows LD(∆tprobe) for pulse separations of 220 fs (blue)and 440 fs (red). This result reaffirms that there is minimal decoherence on the58Figure 4.6: Left: The amplitude of the linear dichroism signal at the fre-quency corresponding to the N = 1 −→ N = 3 transition (2.27 THz) ofHe2* in superfluid helium, as a function of pulse separation in the dou-ble kick experiment. Simulated data is shown with a solid line. Right: Asample LD(∆tprobe) for two pulse separations of 220 fs (blue) and 440 fs(red).picosecond timescale, indicating weak coupling between the dimers and the sur-rounding helium. This shows that the degree of excitation can be controlled in thesame manner as the rotation of molecules in the gas phase.Our results with the chiral pulse train, shown in Figure 4.7, indicate that the di-rectionality of rotation can also be controlled in a similar fashion to what we haveseen in our earlier studies in gas-phase ensembles [198]. As defined in Equation4.2, positive CD values correspond to a stronger LIF signal when the pulse trainis rotating clockwise, in comparison to when it is rotating counter-clockwise. Theinverse signs of the blue and red curves in Figure 4.7 (recorded for clockwise andcounter-clockwise circular probe polarizations, respectively) illustrates how the di-rectionality of the pulse train, with respect to the polarization of the probe, affectsthe circular dichroism measurement. The phase shift of π in (b) relative to (c)illustrates how the directionality of molecular rotation, again with respect to thepolarization of the probe, affects the measurement.59Figure 4.7: The amplitude of the circular dichroism signal at the frequencycorresponding to the N = 1 −→ N = 3 transition of He2* in superfluidhelium, as a function of pulse train period in the chiral pulse train ex-periment. Simulated data is shown with solid lines. The plots on theright show sample CD(∆tprobe) for two pulse separations of 330 fs (b)and 550 fs (c), recorded for both clockwise and counter-clockwise cir-cular probe polarization. Figure taken from [125].A pulse-to-pulse rotation of 45◦ was used here, determined to be optimal inprevious studies [198]. The optimal directionality is observed for a train period of18 and38 of the molecular rotation period. In the former case, the field polarizationfollows the molecules which are rotating in the same direction as the polarizationvector in the train. In the latter case, the molecules rotating in the opposite directionare followed by the pulses in the train. No directionality is observed for trainperiods of 14 or12 of the molecular rotational period, as shown by the overlappingpoints in Figure 4.7 at 220 fs and 440 fs.4.4 DiscussionThese experiments build off our group’s recent work, where the dynamics of laser-induced coherent molecular rotation in bulk superfluid helium were first observed[124]. In that work, the rotational decoherence rate and LIF intensity were observedto be strongly temperature-dependent. The measured temperature dependence ofthe total LIF signal is shown in Figure 4.8. The total LIF signal is dependent on the60number density of helium excimers, which is time-dependent because collidingexcimers produce neutral helium atoms [10, 80]. The time-dependent number den-sity N(t) depends on the initial number density N0, and the temperature-dependentreaction rate K(T ):N(t) =N01+K(T )N0t. (4.13)Excimers in bubbles scatter off of thermal rotons, and so a higher roton densityresults in a slower He2* diffusion rate. This results in longer-lived excimers, dueto the reduced bimolecular collision rate as the diffusion rate decreases. The rotondensity depends on both the liquid temperature and roton energy (∆(T )), which de-creases with increased temperature. The roton density grows with√T exp[−∆(T )/T ],resulting in longer excimer lifetimes at higher temperatures. A fit to Equation 4.13is plotted with a solid line in Figure 4.8, with N0 as the only fitting parameter.The temperature dependence of the total LIF signal is starkly different fromthe temperature dependence of LD1,3, which is plotted in red circles in Figure 4.9for a fixed probe delay. A decrease in LD1,3 while the total LIF signal increasesis explained by a rotational decoherence rate which varies with temperature, as aseparate effect. This cannot be attributed to bimolecular collisions, as the aver-age displacement of excimers on the time scale of this experiment is an order ofmagnitude lower than the average intermolecular separation [124]. If decoherencewere limited by helium atoms moving at the temperature-dependent velocity offirst sound, v1(T ), thenLD1,3(t,T ) = LD(0)1,3 exp[−γ1,3(T )t], (4.14)where γ1,3(T ) = Neqn (T )σ1,3(T )v1(T ) is the decoherence rate, Neqn (T ) is the equi-librium atom number density of the normal fluid, and σ1,3(T ) is the scattering crosssection. σ1,3(T ) is unknown, so was treated as a single fitting parameter that didnot vary with temperature. Fits for LD1,3(t,T ) (thick dashed blue) and γ1,3(T ) (thindashed blue) using Equation 4.14 are shown in Figure 4.9. This model predicts asignificantly slower decoherence rate at 1.95 K than the ∼1 GHz measured, anddoes not predict the degree of flattening of the curve observed at lower tempera-tures.61Figure 4.8: The temperature dependence of the total LIF intensity of heliumexcimers in the bubble phase in bulk liquid helium. The LIF is mea-sured approximately 1 ms after the 2 ps pump pulse used to produce theexcimers. Owing to the increased roton density at higher temperatures,the bimolecular collision rate decreases, resulting in a higher excimernumber density. The solid line is a fit to Equation 4.13. Figure takenfrom [124].To reconcile this, the model was adapted for a nonequilibrium normal fluidnumber density. This is justified because the impulsive Raman excitation tran-siently promotes the excimer to a higher electronic state, with a significantly higherinteraction potential with the surrounding helium - about 100 K higher than that ofthe a3Σu+ state [18]. This injection of energy microscopically heats the surround-ing liquid, increasing the normal fraction. Considering the flow of the normalfraction away from the excimer as a coherent pulse of second sound with width w,the nonequilibrium density of the normal fraction around the molecule isNneqn (T, t) = N exp[−[v2(T )t]2w2], (4.15)where N is the total density of the liquid - entirely normal at time zero. The fit using62Figure 4.9: Red circles show the measured temperature dependence of LD1,3for kicked helium excimers, recorded at a fixed probe delay of 850 ps.The equilibrium fit for a constant normal fraction using Equation 4.14is shown with a thick dashed blue line on the left axis, and correspondcalculated decoherence rate with a thin dashed blue line on the rightaxis. The nonequilibrium fit (with a time-dependent normal fraction)is shown with a thick solid red line, again with the thin solid red lineshowing the calculated decoherence rate on the right axis. The nonequi-librium model better fits the flattening of the curve observed around1.6 K. Figure taken from [124].63Figure 4.10: Experimentally observed temperature dependence of first andsecond sound in liquid helium, where Tc ≈ 2.17K. First sound cor-responds to fluctuations in the total density of the fluid, while sec-ond sound relates to fluctuations in entropy/temperature. Fluctuationsin temperature vary the normal/superfluid fractions, oscillating out ofphase with each other. Figure taken from [165].Equation 4.14 with Neqn (T ) replaced by Nneqn (T, t) is plotted with solid red linesin Figure 4.9, with the thick and thin lines representing LD1,3(t,T ) and γ1,3(T ),respectively. The speed of second sound has a complex temperature dependence,with a local minimum around 0.5Tc≈ 1K, as shown in Figure 4.10. A faster secondsound speed results in the high normal fraction being more quickly replaced by thefrictionless superfluid component, resulting in slower decoherence. This explainsthe observed maximum in LD1,3(t,T ) around 1.8 K, following the local maximumin v2(T ). Our current apparatus cannot cool below 1.4 K, hence the lack of dataaround the inflection points in v2(T ) at lower temperatures.The double kick experiment could offer an exciting opportunity to generate andprobe second sound. With two kicks separated by 220 fs, the rotational populationremains in the N = 1 state. Pulses of second sound could then be probed on thepicosecond scale at arbitrary times after the double kick, using an independent third64kick.Generally, effects such as ionization limit the maximum intensity that can beused for rotational excitation, in which case spreading an excitation across mul-tiple pulses is beneficial. In this experiment, the LIF intensity is roughly linearwith the pulse energy, owing to the two-photon nature of the excitation. This posesa significant challenge if one wants to excite molecular superrotors using a longpulse train, as increasing the train energy by increasing the number of weaker non-ionizing pulses will still lead to increasing, and eventually, saturating, the LIF sig-nal. Perhaps with an alternative detection scheme, such as the polarization dragdue to the mechanical Faraday effect [123], a long high intensity pulse train couldbe used [15]. With a different laser source, the excitation could be performed us-ing lower energy photons, which would no longer efficiently populate the d3Σu+state. Superrotors in liquid helium offer an alternative scheme to probe the bub-ble dynamics, and molecules with a high degree of oriented angular momentum(unidirectional rotation) may be able to control the formation of quantized vortices[60, 147, 164, 179], hence the desire for a higher degree of excitation.65Chapter 5Helium DropletsThis chapter provides an overview of the production and utilization of heliumdroplets, for the experiments I performed in our laboratory.5.1 Supersonic Gas Jets and Cluster FormationGasses undergoing supersonic expansion from a region of high pressure to vacuumthrough a narrow orifice will adiabatically cool during the expansion process. Thedegree of cooling depends on the choice of gas, the stagnation temperature andpressure, the expansion vacuum chamber’s pressure, and the specific nozzle ge-ometry and size. The molecular jet can then be sampled by a “skimmer” to selectthe central part of the jet, and allow propagation into lower pressure experimen-tal chambers through the sections of differential pumping. Subsequent skimmersresult in “geoemetric cooling”, where particles are simply unable to pass througha second skimmer downstream if they have significant velocities perpendicular tothe propagation of the beam.To allow for higher expansion pressures, pulsed nozzles are often used, espe-cially when paired with pulsed laser experiments. Parker Hannifan / General Valveswith orifices exceeding 100 µm can easily produce pulsed jets of a few Kelvin at10 Hz with a modest turbomolecular pump. More exotic plunger-type valves suchas the Even-Lavie valve can produce much shorter pulses, and operate at variablefrequencies as high as 1 kHz with higher pressures and larger pumps. Piezo valves,66such as the Amsterdam Piezo Valve, have gone up to and beyond 5 kHz whilemaintaining jet temperatures around 1 K.In our laboratory, a continuous beam of oxygen, expanding from a pressure ofa few atmospheres into a vacuum chamber with a pumping speed of 1300 Ls−1through a 5 µm orifice, resulted in a gas temperature of 10 K [56]. Densities varyimmensely depending on the distance from the nozzle, the nozzle geometry, ap-plied pressure, and other factors. The range typically extends from 1010 atoms/cm3to 1016 atoms/cm3 [49, 130, 144].As the atoms/molecules undergo collisions throughout the expansion, they mayform clusters when the kinetic energies are low enough relative to the bindingenergy. Weakly bound Van der Waals complexes are readily formed in supersonicjets [29]. At low temperatures, even helium will condense and form clusters, whichare referred to as “helium droplets”.5.2 Helium Droplet ProductionHelium droplets are generally produced in a log-normal distribution of sizes, withthe mean number of helium atoms in a droplet (⟨NHe⟩) controllable with the nozzletemperature and backing pressure (T0, P0), as depicted in Figure 5.1 [38]. The widthof the size distribution is broad, comparable to its mean [98]. Most researchers usea well-known size dependence on temperature and pressure, shown in Figure 5.1,as a look-up table to estimate the droplet size in their apparatus, though thermo-dynamic scaling relations have been developed for helium droplets below ∼10 nmdiameter [25, 145].As the temperature is decreased, the flux of helium leaving the nozzle increases,with an increase in helium density in the droplet beam approaching an order ofmagnitude when T0 (and P0) produce liquid helium before exiting the nozzle. Tworegimes of helium droplet formation are considered: supercritical, when the initialstate is liquid, and subcritical, when it is still gaseous [167]. The former contributesto the “Rayleigh Break-up of Liquid Jet” and “Fragmentation of Liquid” depictedin Figure 5.1, while subcritical refers to the “Condensation of Gas” regime. Heliumdroplets are often referred to as “nanodroplets”, as they are typically produced onthe nanometre scale. High backing pressures with powerful crycoolers can pro-67Figure 5.1: The average helium droplet size (⟨NHe⟩), for various nozzle tem-peratures and backing pressures using a standard 5 µm nozzle. Figureadapted from [167].duce droplets with an average size ⟨NHe⟩ exceeding 1011 atoms/droplet, with sizesreaching a few micrometres [172]. In the supercritical regime the size distributionbecomes bimodal, but is similarly broad [24].685.3 Molecules Embedded in Helium DropletsMolecules can be embedded in helium droplets either by being mixed with the he-lium before expansion through the nozzle, or by being “picked up” upon collidingwith a droplet. Most molecules condense at temperatures well above the nozzletemperatures used for droplet formation, in which case they will clog and poten-tially destroy the nozzle if mixed with the helium prior to expansion. At highertemperatures, this technique has been used to produce clusters of a few dozen he-lium atoms doped with a carbonyl sulfide (OCS) at P0 ≤ 100atm, T0 > 130K [106].On the other hand, the “pickup” procedure is quite simple: if an atom ormolecule collides with a droplet, it may be solvated by the droplet. The droplet willthen rapidly cool the dopant down to equilibrium at 0.37 K, by “boiling” heliumatoms off the droplet surface. Evaporating each helium atom takes approximately5 cm−1 of energy [20].The pickup process follows a Poisson distribution P(k), where the probabilityof capturing k dopants in a beam with average droplet size ⟨NHe⟩ is controlled withthe dopant gas number density (ρ = NV =PkbT, where T is the temperature and kb isthe Boltzmann constant) and length L of the doping region [166]:λ = (15.5A˚2PdkbTL)⟨NHe⟩ 23 ,P(k)≈ λkk!e−λ . (5.1)Figure 5.2 shows P(k) for two droplet sizes. The overlapping curves for differentvalues of k make it impossible to select a single target number of dopants perdroplet and have precisely that number for all droplets. When only a single dopantis desired, experimentalists may set a relatively low doping cell pressure to reducethe probability of picking up more dopants, somewhere around P(1) ≈ 0.2 on theblue curves. For other experiments with multiple dopants, this technique of settingthe doping cell pressure below that which would maximize the pickup probabilityfor the target number of dopants is still useful, with an example of optimizing thedimer signal while minimizing the trimer signal outlined in Appendix A.Whether a molecule will be embedded inside or reside on the surface of a69Figure 5.2: The pickup probability given by Equation 5.1 for two differentdroplet sizes, with a doping region of length L = 7.6cm. Each tracerepresents the pickup probability of k dopants. Note the different expo-nents on the horizontal pressure scales.droplet depends on the balance of attractive and repulsive forces. This is discussedin detail in [3]. In general, this is a question of whether the attraction of the dopantnuclei to He exceeds the electron-He repulsion. The repulsion tends to dominatefor electronically excited states [3, 127, 140], and for species with unpaired elec-trons, such as alkali metal atoms or dimers in triplet ground states [68] .5.3.1 Formation of Molecules and Molecular Complexes in HeliumDropletsIf a droplet captures multiple dopants, the low temperature of the droplet and rapidcooling of dopants allows for very weak attractive forces to form molecular bonds.The specifics of the interaction will be dependent on the species in question. Thesurrounding helium has almost no effect on the interaction forces themselves, asthey are primarily due to permanent and induced electric multipole interactions.Isomers that cannot readily be observed in supersonic jets may be formed andstudied in helium droplets. In a study of (OCS)2 isomers [116], it was explainedthat either helium droplets are “the only medium that provides a gentle enough70environment for the self-assembly of this elusive species,” or that tetramers wereobfuscating the signal in the gas phase. Other experiments found that long chainsof HCN seemed to bond head-to-tail in helium droplets, due to the dipole-dipoleinteraction being sufficiently strong to reorient the molecules before bonding [131],a result not seen in other environments. This mechanism of “freezing” into a localminimum was experimentally demonstrated by doping helium droplets with twoargon atoms and one tetracene molecule [141], where the resulting structure varieddepending on which species was picked up by the droplet first.When produced through supersonic jet expansion, clusters may form beforethe final temperature has been reached. This strongly favours the creation of themost stable isomer, even when the final jet temperature is well below the bindingenergy of less stable configurations. This demonstrates the usefulness of heliumdroplets for novel chemistry. Dimers of various species including C6H5Br-I2 havebeen imaged in helium droplets through Coulomb explosion imaging, with angu-lar covariance maps (discussed in Section 3.2) permitting structural determination[137, 138, 155].5.4 Non-Axial Recoil in Helium DropletsCoulomb explosion experiments remain a useful tool for studying molecules inhelium droplets, with a complication that after the explosion process, an ion frag-ment must escape the helium droplet. It will scatter off helium atoms as it does so[7, 186], and some of the original Coulomb explosion emission angle informationwill be lost. Heavier fragments with more kinetic energy will be reoriented to alesser degree as they scatter while travelling through the droplet. In addition to theloss of angular information, it has been shown that the probability of an ion beingejected diminishes if the droplet size is too large [6].Deconvolution techniques using angular covariance maps may help reconstruct-ing the distribution of emission angles [36], allowing for an estimate of the molecu-lar ⟨cos2 θ2D⟩, introduced in Figure 3.2, but this is not a solution for the case wherethe ion scattering is so strong that it results in a completely isotropic VMI signal. Inthis case, smaller droplets may be used, though a conventional helium droplet sys-tem generally produces a low density of doped smaller droplets resulting in a weak71signal. This is largely due to the evaporation of helium atoms necessary to cool thedopant becoming an increasingly large fraction of the droplet for smaller dropletsizes [163]. I am not aware of any quantitative analysis of the number density ofhelium droplets after initial skimming as a function of the droplet size. Table 6.1and Figure 6.3 in the next chapter both show a higher beam intensity for smallerdroplets, as compared to larger droplets at the same backing pressures, indicatinga higher number density of smaller droplets.One possible solution to the dopant-helium scattering problem is to study moleculesthat are on the surface of the droplet. Recent studies in the Stapelfeldt grouphave successfully produced and observed the Coulomb explosion of sodium dimers(Na2) on the surface of helium droplets [88], and later rotationally excited with ashort pulse [85]. Alternatively, the rotational dynamics of deuterium in heliumdroplets were recently studied using angle-dependent ionization [143]. Relative togas phase D2, a substantial decrease in kinetic energy in the D+ fragments was at-tributed to efficient inelastic scattering from the helium atoms, and was connectedto a significant decrease in the anisotropy of ions collected.72Chapter 6Design and Characterization of aHelium Droplet MachineThis chapter outlines the technical considerations that went into the design, con-struction, and characterization of the helium droplet system used in later chapters.6.1 IntroductionA typical helium droplet apparatus consists of at least three chambers: the sourcechamber, the doping chamber, and the detection chamber. Frequently, an addi-tional chamber is placed between doping and detection to add an extra stage ofdifferential pumping, and occasionally, more detectors are placed in additionalchambers. In principle, reducing the nozzle-detector distance is desirable, becausethe helium droplet beam is diverging as the gas expands. Distances on the orderof 1 m offer number densities estimated to be around∼1×107 doped droplets/cm3[39, 111, 162]. While a more dense beam is desirable, a larger distance adds signif-icantly more working room in the chambers for accessing components, and allowsfor a more versatile experimental apparatus. More compact systems may also havehigher vacuum chamber pressures, due to poor conductance, if the chamber geom-etry is not carefully considered. This can lead to beam destruction and reduceddensities.The first-generation helium droplet system in the Milner laboratory, shown in73Figure 6.1: The previous system used in [56] doped the droplets by floodingthe only differential pumping chamber with gas, and had the nozzle∼15 cm from the VMI detector. The nozzle was deep in a narrow cavity,far from the vacuum pump. This led to weak droplet signals, and a highamount of effusive gas leaking from the doping chamber towards thedetector.Figure 6.1, was a modification of the system designed by Aleksey Korobenkofor pulsed supersonic gas experiments, used in [83]. The modified system wascharacterized by Jordan Fordyce in [56]. That system was originally designed tominimize the nozzle-detector distance. It proved to create relatively weak heliumdroplet beam, and a significant amount of doping gas found its way to the detector,resulting in a very low signal-to-background ratios.Providing a better environment for working with smaller droplets (to studymolecules like nitrogen while avoid the effects of scattering of ion fragments fromhelium atoms after Coulomb explosion, described in Section 5.4), and reducingthe effusive background of dopant molecules in the detection chamber, were themain objectives for the new chamber design. The former revolves primarily aroundpumping capabilities, as the helium droplet beam density increases with backingpressure, while the latter is a combination of isolating the detection chamber from74the doping gas inlet, as well as reducing the ultimate pressure in the detectionchamber.Helium droplets produced in the first-generation apparatus would give a sourcechamber ion gauge readinga of 1.2× 10−4 Torr for helium droplets with ⟨NHe⟩ ≈3000atoms, produced at 24 Bar backing pressure and 14.5 K nozzle temperature[56]. The exact pressure near the nozzle was unknown, due to chamber geometry,but was assumed to be significantly greater than 1.2× 10−4 Torr, given the com-pactness of the system and poor beam quality. The mean free path (average dis-tance travelled between collisions with other gas molecules) scales inversely withchamber pressure, and for the room temperature helium in the chamber the meanfree path is around 10 cm at a pressure of 1×10−3 Torr. Helium cluster formationtakes place up to around 1000 nozzle diameters from the source [167], so with typ-ical 5 µm nozzles this chamber pressure would be on the upper limit of acceptableoperating regimes, as the beam can be transferred to the subsequent chamber afteraround 1 cm, one order of magnitude shorter than the mean free path.6.2 Design of the Helium Droplet SystemI designed the new system with four vacuum chambers, hereafter referred to as:Source, Doping, Differential, Detection (VMI).Source: The helium droplet nozzle and cryocooler assembly mounted in alarge chamber with high pumping speeds, and designed in such a way as to keepthe nozzle and the skimmer far from the chamber walls. A gate valve after theskimmer allows for measurements without droplets, although a small beam shutterwould suffice for those experiments.Doping: Two doping cells connected to precision leak valves, with a liquidnitrogen cryotrap in between the cells. The metal mounting assembly should beable to accommodate a high-temperature oven, if solid dopants were to be used.Differential: To reduce the flux of dopant gas towards the detector, a simpledifferential pumping stage is necessary. A gate valve was placed after the dopingchamber, so that the presence of the background dopant molecules in the detectionchamber could be distinguished between effusive molecules travelling through theaUncorrected for helium, whose pressure is around five times greater than the gauge reading.75beam skimmers between chambers, and molecules that migrate backwards throughthe turbomolecular pumps. A cylindrical liquid nitrogen cryotrap can be attachedto the top flange, with the droplet beam passing through, to help prevent effusivegas molecules from migrating downstream.Detection (VMI): This chamber was designed with eight ports in the horizon-tal plane facing the middle of the chamber, so that the helium droplet beam could beintersected with either two orthogonal laser beams (each hitting the droplet beamat ±45◦) or one beam at 90◦. Each arrangement has the option of lenses or in-vacuum mirrors. A vacuum port opposite the helium droplet beam can be used toeither transfer the droplets to a possible subsequent chamber, or enter an installedResidual Gas Analyzer (RGA) directly. The VMI detector hangs from the top, withall feedthroughs attached to the mounting flange. The bottom of the chamber hasa large flange, with ample open space below, so that a closed-cycle helium cry-opump could be added, if better pumping of molecules such as N2 was needed toreduce the background signals. Additional ports were added wherever possible forpumping, gauging, and observation.6.2.1 Design of the Source ChamberThe source chamber pressure for a given droplet size scales roughly linearly withthe backing pressure. This would be true for the droplet beam density at the detec-tor, if beam quality conditions were independent of backing pressure. However, ahigher source chamber pressure will lead to a greater degree of beam destructiondue to collisions, particularly around the skimmer, resulting in a less-than-linearincrease in signal strength at the detector. The group of Frank Stienkemeier pro-vided the intensity measurements, shown in Table 6.1, demonstrating that the levelof beam destruction may be dependent on the droplet size, and that quadruplingthe backing pressure will not necessarily quadruple the signal strength. Comparingeach pair of values in the rightmost column, quadrupling the backing pressure in-creases the intensity by a factor of 2.4×, 3.5×, and 4.8× for ⟨NHe⟩= 1000, 10000,and 50000, respectively.Figure 6.3 shows data adapted from Figure 3.6 of the 2021 thesis of Travis M.Jones, from the group of Gary Douberly. Extrapolating the data for ⟨NHe⟩= 100076Figure 6.2: Solidworks CAD model cross-sectional view of the heliumdroplet chamber constructed for this work, as described in the text.From left to right are the Source, Doping, Differential, and VMI cham-bers. This render does not depict the pulsed valve or liquid nitrogencryotraps. The flange in between the Doping and Differential chambershas a 5 mm diameter flat aperture, rather than a conical skimmer.up to 80 Bar would provide a similar beam intensity increase of 2.8× greater thanat 20 Bar, comparable to the factor of 2.4× found from Table 6.1. However, herefor the largest droplets of ⟨NHe⟩ = 200000, the signal approximately triples as thepressure is increased fourfold, while in Table 6.1, the signal more than quadrupledfor the largest size of ⟨NHe⟩= 50000, indicating more intricate dependence on othersystem parameters.In all these measurements, the source chamber pressure was kept low enoughfor the mean free path to exceed the nozzle-skimmer distance by an order of magni-tude. Localized effects, such as skimmer clogging, are assumed to be the cause ofthe sub-linear increase in beam intensity as a function of backing pressure. Theseeffects are unlikely to be greatly remedied with better vacuum pumps, and are pri-marily a question of geometry. One very influential factor, which was not variedin these experiments, is the nozzle-skimmer distance. The maximum transmis-sion depends on both the skimmer orifice size, and incoming beam density [195].This means that one would expect the optimal skimmer distance to be different for77Nozzle ⟨NHe⟩ Source Pressure Detection Pressure “Intensity”Conditions (atoms / droplet) (mBar) Rise (mBar) (mBar / ⟨NHe⟩)20 Bar / 17 K 1000 6×10−5 2×10−9 2×10−1280 Bar / 28.5 K 1000 1.8×10−4 4.8×10−9 4.8×10−1220 Bar / 12 K 10000 7.5×10−5 4×10−9 0.4×10−1280 Bar / 17.7 K 10000 2.8×10−4 1.4×10−8 1.4×10−1220 Bar / 10.4 K 50000 8.5×10−5 5×10−9 1×10−1380 Bar / 15 K 50000 3.1×10−4 2.4×10−8 4.8×10−13Table 6.1: Measuring the detection chamber pressure increase due to thedroplet beam, and dividing by the average droplet size, provides a roughmeasure of the beam intensity. Here it is shown that quadrupling thebacking pressure for a given droplet size approximately quadruples thebeam intensity for larger droplets, but is closer to doubling the intensityfor smaller droplets, indicating greater beam destruction. Data pointsare taken from a figure provided by Frank Stienkemeier, from a heliumdroplet system with 2500 Ls−1 source chamber pump, with a standard5 µm helium droplet nozzle.20 Bar and 80 Bar backing pressures.To optimize signal strength, a manipulator was constructed for our systemthat allowed the nozzle-skimmer distance to be varied while observing the heliumdroplet signal. The source skimmer was mounted on a cone, to centrally locate itin the source chamber, as well as to maximize the line-of-sight pumping behindthe skimmer in the subsequent chamber. There is a significant flux of bare heliumatoms passing through the skimmer, that must be pumped away to reduce beamdestruction.The gas throughput calculation is a simple linear relation between the targetpressure (P) and pumping speed (S), which reads Q = P× S [126]. The samethroughput needs to be matched by the forevacuum pump, accounting for the con-ductance of whatever tubing connects it to the high vacuum pump, using the tubeconductance equations, outlined below. It is evident that to maximize the possi-ble helium droplet beam density, the source chamber pumping speed should bemaximized. The effective pumping speed of a modern turbomolecular pump isessentially a question of the inlet cross-sectional area. For these reasons, thesource chamber was designed to accommodate two turbomolecular pumps, directly78Figure 6.3: The left plot uses the raw RGA pressures for each droplet size,while on the right they have been rescaled based on the lowest back-ing pressure provided, with the baseline subtracted. This system has a7500 Ls−1 vacuum pump on the source chamber, and a standard 5 µmhelium droplet nozzle, with source chamber ion gauge readings from2.5×10−5 Torr to an extrapolated estimate of ∼3×10−4 Torr through-out. Adapted from data in figure 3.6 of [77].aboveb and below the nozzle and skimmer. Port sizes suitable for pumping speedsaround 2000 Ls−1 were chosen, because they offered the maximum speed-per-cost.Only a single pump was purchased, with a 1 inch thick acrylic viewport placed onthe top of the chamber, where a second large pump could be located in the future.The chamber was designed with large side access ports, which could accommo-date two additional large pumps, as well as keeping future options open in termsof mounting a beam shutter, or additional viewports.A target source chamber pressurec of around 2×10−4 Torr was chosen, basedbOwing to the ballistic (line-of-sight) molecular flow regime, placing the two pumps on the op-posite sides of the chamber offers better pumping speed than attaching them to the same side using atee flange.cA target real pressure rather than uncalibrated gauge reading.79on the operating conditions used in previous laser-induced alignment experiments:7.5×10−5 Torr at 25 Bar / 12 K [37], and 2.3×10−4 Torr at 25 Bar / 16 K [136], intwo different systems. With a pumping speed of 2000 Ls−1, that gives a throughputof 2× 10−4 Torr× 2000Ls−1 = 0.4TorrL/s. The target forevacuum pressure isdetermined by the specific high vacuum pumps chosen, whose pumping speedscan be greatly diminished if the forevacuum pressure is too high.A throughput of 0.4 TorrL/s, with a forevacuum target of around 0.05 Torr, re-quires an effective forevacuum pumping speed of 0.4TorrL/s0.05Torr = 8Ls−1. The effectivepumping speed (Se) is related to the chamber conductance (C) and the forevacuumpump’s pumping speed (Sfv) by1Se=1C+1Sfv, (6.1)where the conductance for a series of parts is1C=1C1+1C2+1C2..., (6.2)while connections in parallel increase conductance by direct summation [126].Corners, narrow orifices such as valves, and other components will further re-duce conductance. Forevacuum systems are typically operating in the viscous flowregime, where the conductance (in litres per second) of a cylindrical pipe of lengthL and diameter D (both in cm), with an average pressure of Pavg (in Torr) of roomtemperature helium, is expressed in the equation below [126]:C = (105Ls−1cm−3 Torr−1)D4LPavg. (6.3)A crossover into the molecular flow regime changes this to:C = (35Ls−1cm−4)D3L, (6.4)suggesting the ways of increasing conductivity are to use either a very short pipe,or a large diameter pipe [126]. Using equations 6.3 and 6.4, a standard KF40vacuum hose with 2 inch inner diameter running 1 m to a pump would result in a80conductance of 35− 45Ls−1. Plugging this into Equation 6.1, with Se = 8Ls−1,gives Sfv = 10Ls−1 as the necessary speed.A 4 inch diameter, 10 m long pipe provided slightly higher conductance, andallowed the forevacuum pump to be placed in a separate room, to keep the mainlaboratory quieter. Two 4 inch diameter lines were installed, to minimize the riskof throttling. The specifics of their construction are discussed in more detail inAppendix E.The helium droplet nozzle itself is prone to damage from debris, oil, and thefreezing of other molecular/atomic species. Maintaining a flux of helium above5 Bar helps prevent clogging, as does limiting the use of this nozzle to high pu-rity helium. When gas cylinders are exchanged, it is important to check with theRGA that the beam has not been contaminated, before turning the cryocooler on.This is done by monitoring helium, nitrogen, and oxygen pressures on the RGAwhile raising the backing pressure, with the droplet nozzle at room temperature. Ifthere is any increase in nitrogen or oxygen, the helium line should be checked forleaks. The section that is exposed to atmosphere upon switching cylinders shouldbe pumped and purged a few times.To reduce the likelihood of atmospheric gasses clogging the nozzle in the eventof a power failure, solenoid valves were set up at the back of the turbomolecularpumps for sealing the chamber from the atmosphere. With the return of electri-cal power, a scroll pump would start pumping the chamber immediately. This isdepicted in Figure E.1 of Appendix E.6.2.2 Design of the Doping SystemThe Poisson doping distribution (Equation 5.1) identifies that for a given averagedopants-per-droplet target (⟨k⟩), there is an inverse relationship between the re-quired dopant gas pressure and the length of the doping region. A larger dopingvolume will consume more dopant gas, and a lower dopant pressure will requiremore precise pressure stabilization to maintain stable doping conditions. On theother hand, a higher dopant pressure will produce a stronger effusive gas beamthrough the cell orifices, which should be minimized in diameter, to reduce the ef-fusive beam contribution, and dopant consumption. The cell orifices must also be81large enough to not clip the droplet beam. Here, I will show the calculations forthe effect the doping cell geometry has on the doping chamber pressure, and thenecessary pumping speed for that chamber.The conductance for an aperture of area A (cm2) in the molecular flow regimeis dependent on the temperature (T , Kelvin) and molecular weight (M, AMU)CA = (3.7Ls−1K−12 AMU12 )×A(TM) 12, (6.5)and induces a gas load, dependent on the doping cell pressure QA =CA×Pd, as-suming the surrounding vacuum chamber is kept at a significantly lower pressure[126]. This is assumed because, per Equation 5.1, the average number of dopantspicked up ⟨k⟩ is proportional to the pressure, and so the design is generally suchthat this occurs only within the doping cell. A 5 mm orifice diameter and 3 inch(0.076 m) doping cell length were chosen, based on various other helium dropletsystems. For nitrogen, this gives a total conductance between the doping cell andthe doping chamber of CA = (3.7Ls−1K−12 AMU12 )× (30028 )12 = 12Ls−1.For helium droplets with N = ⟨NHe⟩= 1000, with an average number of dopantsper droplet of ⟨k⟩ = 1, solving Equation 5.1 with these parameters for the dopingcell pressure givesPd = ⟨k⟩ KbT15.5A˚2LN23= 1× 1.38×10−23 JK−1300K15.5A˚20.076m(100023 )= 7.6×10−4 Pa = 2.6×10−5 Torr,and would result in a gas load (through both orifices) of QA = 2×12Ls−1×2.6×10−5 Torr = 6.2× 10−4 TorrL/s. Applying the same aperture throughput calcula-tions for the 2×10−4 Torr ambient pressure in the source chamber, with a typical1 mm diameter skimmer aperture, provides a gas load of ∼5× 10−5 TorrL/s, asmall increase relative to the doping flux. Maintaining a pressure an order of mag-nitude below Pd would therefore require a pumping speed of S =6.2×10−4 TorrL/s2.6×10−6 Torr ≈240Ls−1.Lengthening the doping cell could reduce the necessary pumping speed, butas we had a spare 1000 Ls−1 turbomolecular pumpd, the 3 inch cells with 5 mmdThe pump used has a very poor compression ratio for helium, so was backed by the sourcechamber. See Appendix E for more details.82orifices were sufficient. The doping cells are 1 inch in diameter, chosen to pro-vide ample room for mounting, based on the available space in an existing vacuumchamber. The doping cell has a welded Swagelock ferrule connector on the side,which is then connected to a leak valve. For more details on the doping cell con-struction, see Appendix E. It is important that connections on the atmospheric sidebe leak-tight, so the dopants are not contaminated. A helium leak detector was usedto check these connections. Liquids (such as CS2) were placed in a ∼12 inch longquarter-inch stainless steel tube as a sample holder, connected to the leak valve.The liquid sample holder was heated, to maintain a pressure at the leak valve inletof around 10 PSi absolute pressure (below atmospheric pressure), as lower inletpressures were more difficult to adjust carefully with the leak valve. Gas cylinderswere regulated to 20− 30PSi at the inlet. Suggested improvements are discussedin Section 6.4.As described in Section 5.2, the pick-up process involves the evaporation ofseveral helium atoms from the droplet. This means that a second doping cell couldbe filled with helium, to reduce ⟨NHe⟩ after the initial doping, where the kineticenergy of the room temperature helium atoms is dissipated by droplets through theremoval of several dozen helium atoms per single picked up helium atom. Thiswould allow one to study small droplets, while utilizing a lower dopant cell pres-sure due to the larger pick-up cross-section of the initially large droplets. I am notaware of any characterization or utilization of this technique.6.3 Characterization of the Helium Droplet ApparatusThis section covers technical steps involved in the characterization of the heliumdroplet apparatus. First, the steps to mechanically align the droplet beam throughthe system are discussed, followed by a comparison to the first-generation heliumdroplet system in this laboratory, and then details of the doping procedure.6.3.1 Initial Alignment of ComponentsInitially, a HeNe laser was directed towards the nozzle from the centre of the VMIchamber, with no skimmers present, ensuring that it passed through the centre ofthe front and back vacuum ports, as well as being centred between the TOFVMI re-83peller and extractor plates. Then, skimmers were placed in the Differential chamberand Source chambers, ensuring they were centred around the HeNe beam. Next,the nozzle was positioned ∼2 cm from the first skimmer, and centred on the HeNebeam. Finally, the final viewport was replaced with the Residual Gas Analyzer(RGA, a quadrupole mass spectrometer), to enable the measurement of the dropletbeam intensity.Helium dimers in their ground state have a very weak binding energy, on theorder of 1 mK [65], with that of trimers and tetramers being on the 200−400mKscale [177]. This results in very low RGA readings for helium clusters Hen froma room temperature ensemble, such as the background gas in the helium dropletsystem. The RGA has an electron impact ionizer that fragments incident heliumdroplets into many Hen+ cations. The He2+ cation is stable, and therefore providesa strong signal indicative of helium droplets [135]. Maximizing the 8 AMU signalon the RGA is how many research groups optimize their helium droplet system. Fora given backing pressure, there will be a large increase in droplet beam flux whenthe temperature is lowered to the “supercritical” regime (see Figure 5.1), where liq-uid helium forms within the nozzle itself [162]. This results in an increased dropletbeam density downstream, which serves as an indicator for both gas beam align-ment through the vacuum system, as well as the successful formation of heliumdroplets.The alignment with the HeNe laser beam resulted in the helium droplet beammissing the final skimmer, due to imperfect alignment of the system. This wasdetermined by observing no helium dimer (8 AMU) increase on the RGA, despite asignificant pressure rise of 3.8×10−8 Torr in the penultimate Differential pumpingchamber, when a 10 Bar / 12 K helium droplet beam was introduced. After thisskimmer was removed, there was a strong helium dimer signal on the RGA. Thechamber had to be opened to atmosphere and pumped down repetitively, so that theskimmer could be positioned such that it did not greatly diminish the dimer signal.After that was done, the same procedure was performed to install the skimmer inthe doping chamber. We assumed that the cause of the misalignment using theHeNe laser procedure was that the nozzle is slightly tilted, which would preventthe droplet beam from travelling through the exact centre of each vacuum chamber.At a backing pressure of 35 Bar, the helium dimer signal increases by more84Figure 6.4: The recorded increase in Source and VMI chamber pressures,as well as the RGA helium and helium dimer signals while the heliumdroplet nozzle temperature is decreased. Recorded with a backing pres-sure of 35 Bar. The sharp rise in the mass 8 signal around 11 K indicatesthe transition from subcritical to supercritical droplet formation, wherethe helium in the nozzle is transforming from gas to liquid within thenozzle itself, resulting in a much higher beam density.than an order of magnitude as the supercritical regime is entered, as shown withthe red trace in the rightmost plot of Figure 6.4, while the source chamber pressureincreases by just a factor of two, as shown in the leftmost plot. The source chamberpressure increase is gradual across the temperature range scanned, while the heliumdimer signal rises sharply at the supercritical transition.Adjusting the nozzle-skimmer distance by a few millimetres increased the RGAreadings another ∼ 20 %, with the optimal distance being around 2 cm for 40 Bar.With this complete, a comparison at different backing pressures and temperatures85Figure 6.5: The helium dimer mass 8 signal on the RGA was measured asthe backing pressure was increased, with the nozzle temperature variedfrom 14 K up to 20 K to maintain ⟨NHe⟩ ≈ 5000. The pressure valueshad the baseline of the signal with no droplet beam subtracted, so theincrease can be plotted directly. The nozzle-skimmer distance was ad-justed for 40 Bar, and not varied for this measurement.adjusted for a constant droplet size (as in Table 6.1) was performed. This is plottedin Figure 6.5, and shows that for ⟨NHe⟩ ≈ 5000 there is an approximately linearrelationship between the backing pressure and the beam intensity.What is particularly exciting about the results in Figure 6.5 is that the pres-sure increase is far above what we were able to achieve in our previous system.The 20 Bar / 14 K pressure increases in each system are comparable, but the newsystem’s RGA reading of 1.8×10−8 Torr at 60 Bar is about five times greater thanthe maximum 3.8× 10−9 Torr reading at 20 Bar in the old system for this dropletsize. This confirms that, as designed, the new system is capable of creating a muchhigher droplet beam density. The signal increasing 5× for a backing pressure in-crease of 3× points to a high degree of beam destruction in the previous system,even at the lower pressure.Figure 6.5 shows an approximately one-to-one relationship between dropletbeam intensity and backing pressure, while the results in both Table 6.1 and Figure6.3 show a more gentle increase in intensity. An optimistic interpretation would be86that this new system has less beam destruction for this droplet size. Both of thosesystems utilize 300 µm skimmers in the source chamber, while I have used a 1 mmskimmer, following the group of Henrik Stapelfeldt. Smaller droplets are moresensitive to skimmer interference [61], hence the combination of a larger skimmerwith the ability to tune the nozzle-skimmer distance, are important factors to con-sider here. Later tests in our system found a consistent 20−50% increase in signalwhen the nozzle was moved a few millimetres closer to (away from) the skimmerfor lower (higher) backing pressures. The manipulator design unintentionally tiltsthe nozzle assembly as it is adjusted, making a measurement of this effect ardu-ous. The fact that the nozzle position was fixed at the 40 Bar position for the datapresented in Figure 6.5 means the 20 Bar signal increase would be even higherthan recorded, if this distance were optimized. This would make the results moreconsistent with the measurements in Table 6.1 and Figure 6.3.6.3.2 Doping ProcedureThe procedure used for doping droplets here has been to evacuate the doping cell,in order to allow a pure undoped helium droplet beam to pass through the vacuumsystem and enter the Residual Gas Analyzer (RGA). The mass 8 helium dimersignal, as well as the dopant and cluster masses, are monitored.As an example, for the nitrogen experiments described in Chapter 9, with he-lium droplets produced at 60 Bar / 21 K, the RGA reads 8.3× 10−10 Torr of mass8, and about 3× 10−11 Torr of mass 30 (15N2), with no doping. Blocking thegate valve in front of the helium droplet source chamber brings these down to1×10−12 Torr and 3×10−12 Torr, respectively, indicating that some portion of themass 30 signal is from the undoped droplet beam.A low flux of 15N2 to the doping cell is then adjusted, until there is an increasein the dimer (15N2)2 signal at mass 60. Per the Poisson doping process depictedin Figure 5.2, there exists a doping cell pressure of 15N2 that will maximize themass 60 signal, which can be used to locate current conditions. The ideal monomerdoping conditions occur when the dopant flux is lowered to reduce the dimer signalto slightly below 50% of the maximum. The impact of dimers when attempting tomeasure monomers varies by experiment. In some cases, the dopant flux must be87further decreased to increase the ratio of monomers to dimers, at the expense ofabsolute monomer signal strength.If the molecule is too heavy to be detected by the RGA, 100 AMU in our case,an approximate rule of thumb is that monomer doping conditions correspond to a10% reduction in the RGA He2+ signal, for helium droplets with ⟨NHe⟩ ≈ 103−104.This is because the likelihood of a charge from the electron impact of the RGAionizer transferring to the dopant is approximately 50% over this range of dropletsizes [44, 99]. This reduces the probability of the charge residing on He2. Asshown in Figure 5.2, monomer doping (with a low fraction of dimers) occurs withapproximately one dopant in every five droplets, so the depletion is approximately50%5 = 10% of He2+. If the goal is to create molecular dimers in the droplets (e.g.(NO)2 as discussed in Chapter 10), one expects a dimer in approximately one infive droplets, and a monomer in every third droplet. This means that around halfthe droplets are doped, therefore reducing the He2+ signal by ∼ 25%.Another practical way to gauge doping conditions is to look at the monomermass with velocity map imaging. Monomers should have zero kinetic energy re-leased upon ionization, and will be focused to a single point on the VMI detector.This point will be blurred and shifted slightly by the velocity distribution of thehelium droplet beam. On the other hand, dimers and trimers undergoing Coulombexplosion produce distinct signals at the monomer mass with non-zero velocities.This is discussed in detail with experimental results using the NO dimer in Ap-pendix A.6.4 Considerations for Future ApparatusSingle-use overpressure “burst disks” that fracture to connect the chamber to atmo-sphere if the vacuum chamber exceeds atmospheric pressure would help preventdamage in circumstances such as a gas line break, the nozzle orifice rupturing, orif valves to isolate chambers from their forevacuum pumps are erroneously closedfor an extended period of time with a high flux of helium.A notable improvement would be to have a pressure reading directly on thedoping cells, rather than only on the chamber which holds the cells. This wouldallow one to “dial in” a doping pressure using Equation 5.1, to dope for monomer88or dimer droplet conditions. With a pressure reading there may still be some uncer-tainty on the doping, due to potential variations in the helium droplet size based onexact nozzle orifice size and beam destruction around the skimmer, but this tech-nique would be quite useful for replicating conditions on a day-to-day basis, andupon switching gasses.89Chapter 7Rotational Control of Moleculesin Liquid HeliumPart 2: CS2 in Helium DropletsThis chapter describes experiments where CS2 in helium droplets is excited usinga double kick scheme, as described in Section 2.5.2. They are presented as the firstdemonstration of a rotational control experiment in helium droplets, with prelimi-nary results demonstrating that some degree of state-selective excitation is possiblein helium droplets.7.1 IntroductionRotational coherence spectroscopy of kicked molecules has been used to measurethe effective rotational and centrifugal distortion constants of OCS, CS2, I2, andD2 in helium droplets [30, 143]. Oscillations in molecular alignment signals last-ing several periods were Fourier transformed to obtain the transition frequencies,which are fitted with Equation 2.7 to find effective rotational constants. J is re-placed by L to represent the total angular momentum, including that of the sur-rounding helium, following the convention of reference [33]:∆EL,L+2 = E(L+2)−E(L) = 4BL+6B−8DL3−36DL2−60DL−36D. (7.1)90Figure 7.1: The first six frequencies for the rotational energy transition struc-ture of CS2 in helium droplets, calculated from Equation 7.1 usingB = 0.024cm−1 and D = 3× 10−5 cm−1 from [30]. Due to the largeamount of centrifugal distortion, a “centrifugal wall” is formed around20 GHz, where rotational states with higher angular momentum mayhave lower rotational frequencies. The error bars account for the re-ported uncertainty in B, but not D.The oscillations in droplets were significantly slower than in the gas phase forOCS, CS2, and I2, resulting in an approximately threefold decrease in the fit for therotational constant B, and therefore a threefold increase in the effective momentsof inertia calculated from B (Equation 2.5), while the rotational frequencies of D2in helium droplets were approximately the same as isolated D2 in the gas phase.The frequencies found for CS2 in helium droplets are plotted in Figure 7.1.The expectation is that there will be minimal decoherence if the molecule’srotational energy is either below the energy of a roton excitation in the heliumbath (∼6 cm−1, 180 GHz), as in [30], or if the molecule is in a sufficiently highrotational state. The latter would imply that dissipating rotational energy wouldrequire a large number of excitations of the helium, as is the case for D2 [143].The intuitive explanation for this effect, known as the energy gap law, is that theprobability of energy transfer between the molecule and the helium bath is sig-nificantly reduced when multiple bath excitations must occur at the same locationsimultaneously in order for the molecule to transfer energy.91Rotational decoherence in kick experiments manifests through an exponentialdecay in the oscillating alignment traces. Figure 7.2, taken from [33], shows themeasured alignment of CS2 in helium droplets for various kick strengths. It can beseen that higher frequency oscillations, corresponding to higher levels of rotationalenergy resulting from stronger kicks, are decaying more rapidly. The interpreta-tion is that because the excitations accessible with a kick for CS2 in helium cannotgreatly exceed the roton energy, the coupling and corresponding angular momen-tum/energy transfer to the droplet increases as the degree of excitation is increasedwithin this range. The maximum excitation from a kick can be estimated classi-cally by comparing the maximum torque on a molecule exposed to the field relativeto the moment of inertia of the molecule. Restricting the pulse duration to be muchshorter than the pre-excitation rotational period of the molecule (to stay in the im-pulsive limit [157]), this results in a rotational frequency offmax =12π∆αFkick2Iεc, (7.2)where ∆α is molecule’s anisotropic polarizability, Fkick is the total fluence of thekick pulse, I is the molecule’s moment of inertia, ε is the electric permittivity, andc is the speed of light [94, 159]. The maximum fluence that can be used dependson the molecule, and is limited by ionization, as well as the restriction on the pulseduration. For CS2, this limit is approximately a 2 ps kicka with an average intensityof 1×1013 W/cm2. This results in a fluence of ∼20 J/cm2, and the correspondingmaximum rotational frequency isfmax =12π∆αFkick2Iεc≈ 0.8THz = 26cm−1, (7.3)using the gas phaseb rotational constant B = 0.109cm−1, and ∆α ≈ 9A˚3 for CS2.This is not significantly greater than the roton energy, and would be expected todecay relatively quickly, consistent with the results shown in Figure 7.2, as well asthe results in [32] which extend up to 14.2 J/cm2.a2 ps is ∼ 1% of the revival period.bThe gas phase constant was used to illustrate that even if the molecule completely detached fromthe helium, rotational transition energies are still comparable to the roton energy.92Figure 7.2: Left: the alignment factor ⟨cos2 θ2D⟩ of kicked CS2 in 25 Bar /14 K helium droplets, measured through Coulomb explosion imagingof the S+ fragments. The probe delay was scanned with respect to thekick at t = 0ps. The different subplots correspond to varying the fluence(Falign) of the 2.3 ps kick. Right: the power spectra of the correspond-ing alignment traces. The peaks are the rotational frequencies of CS2in helium droplets. The broad line around 20 GHz corresponds to the“centrifugal wall” shown in Figure 7.1, where several rotational energylevels are clustered together. Figure taken from [33].93What these experiments demonstrate is that when the excitation is kept wellbelow the roton energy, the dynamics are not overwhelmingly different from in gasphase, aside from the renormalized rotational constants. This implies that molecu-lar rotation can be controlled using similar excitation schemes, allowing for selec-tive excitation of different rotational states, selective rotational control in molecularmixtures including chiral enantiomers [170, 189], as well as controlling the direc-tionality of molecular rotation [17, 53]. Given that molecules, some of which arenot readily available in standard gas phase ensembles, may be synthesized in coldhelium droplets (e.g. NO dimer, described in Chapter 10), making use of thesetechniques may prove necessary for structural analysis. State-selective enhance-ment/suppression also offers one possible method to experimentally measure howthe decoherence rates vary for different rotational states in helium droplets.The experiment presented here is to verify that specific rotational coherencesproduced by a single kick, as shown in Figure 7.2, can be “turned off” with asecond kick.7.2 Experimental Technique7.2.1 Single Kick CalibrationHelium droplets were produced at 25 Bar / 13.5 K, and doped with CS2 using theprocedure outlined in Section 6.3.2. A simple test using only the probe beam veri-fied that the droplet size was appropriate for ⟨cos2 θ2D⟩measurements, by checkingthat S+ ions can escape the droplets with minimal impact from the scattering (seeSection 5.4). To check this, we measure ⟨cos2 θ2D⟩ with the probe polarized par-allel and perpendicular to the VMI detector. The molecule ejects S+ ions primarilyalong the axis of the probe polarization, due to the preferential ionization when themolecule is aligned with the laser polarization, as described in Section 3.3 [1, 13].This results in an isotropic signal with ⟨cos2 θ2D⟩ = 0.5 when the probe is polar-ized perpendicular to the VMI detector, and an anisotropic signal when the probepolarization is parallel to the detector. The increase in ⟨cos2 θ2D⟩ for the parallelprobe, relative to ⟨cos2 θ2D⟩ for the perpendicular probe, gives a rough estimate asto how the S+ ions scattering off of helium atoms after Coulomb explosion affects94Figure 7.3: ⟨cos2 θ2D⟩ calculated from the S+ ion signal from CS2 kicked witha 4 J/cm2 2 ps pulse in 25 Bar / 14 K helium droplets, ionized with a200 ps probe of average intensity of 1015 W/cm2. The shaded red arearepresents the temporal profile of the kick pulse relative to the observeddynamics. As will be the case with most ⟨cos2 θ2D⟩ measurements pre-sented in this thesis, the baseline is not the expected isotropic value of⟨cos2 θ2D⟩= 0.5. This is due to a combination of defects on the VMI de-tector, and effusive background molecules whose velocity distributionis centred at a different location on the VMI detector (due to their zeroaverage velocity), which results in a skewed VMI distribution and artifi-cially anisotropic ⟨cos2 θ2D⟩.the ⟨cos2 θ2D⟩ observable. Under these conditions, the effect of modulating theprobe polarization resulted in a ⟨cos2 θ2D⟩ increase of approximately 0.18, slightlyless than the increase of 0.2 reported in [36], but otherwise indicating good func-tionality of the VMI-based analysis of molecular alignment.Figure 7.3 shows the alignment factor ⟨cos2 θ2D⟩ as a function of probe delayfor a single kick, as measured in this apparatus. The fluence of 4 J/cm2 was cho-sen to maximize the strength of the second oscillation around 70 ps, which in ananalogous experiment in [33] had occurred with a kick fluence closer to 2 J/cm2(Figure 7.2). This discrepancy may be attributed to poor beam quality in the focus,due to a strong spherical aberration in the kick beam used. The second oscilla-tion in Figure 7.3 has an amplitude of about 0.01, much less than the amplitude95Figure 7.4: The Fourier transform of the alignment factor ⟨cos2 θ2D⟩ after onekick (the signal plotted in Figure 7.3), taken between 10 ps and 303 psafter the kick, with a Hamming window applied.of nearly 0.1 in Figure 7.2 at ∼2 J/cm2. This discrepancy can not be due to S+fragments scattering off of helium atoms as they exit the droplets, as evidenced bythe ⟨cos2 θ2D⟩ increase of ∼ 0.18 in the probe-only experiment discussed abovec.It likely stems from ionizing a large number of molecules in helium droplets out-side the kick’s focus, pulling ⟨cos2 θ2D⟩ down towards the isotropic value of 0.5.In hindsight, rather than increasing the intensity of the probe to increase the countrate, the beam waists should have been increased.The dominant oscillations in Figure 7.3 with a period around 50 ps correspondto a broad ∼20 GHz peak in the frequency domain, as shown in Figure 7.4. Thisis consistent with the results in [33], albeit with significantly fewer data points re-sulting in a low quality spectrum. The intention with the double kick experiment isto suppress this ∼20 GHz peak, hence rather than improving the frequency resolu-tion, data collection time was allocated to improving the signal-to-noise ratio overthe first few periods.7.2.2 Double KickA double kick was produced using a Michelson interferometer to vary the sepa-ration between kicks with a translation stage. The 2 ps kick pulses each have acLater alignment experiments in Chapter 8 with the same doped helium droplet conditions alsoshow an increase in ⟨cos2 θ2D⟩ well above 0.01.96nominal fluence of 4 J/cm2, when focused on the CS2-doped helium droplet beam.A second kick pulse results in quantum interference that can enhance or sup-press different frequency components in the power spectrum, depending on theevolution of the initial wavepacket at the time of the second kick. In a model wheredifferent rotational energy levels have different lifetimes, monitoring the amplitudeof transition frequencies in the power spectrum as a function of temporal separationbetween kicks would provide a measurement of the lifetime of each state. The ro-tational spectrum of CS2 in helium droplets begins with a few roughly equidistantlines around 5, 10, and 15 GHz, as shown in the top two panels of the right columnin Figure 7.2, followed by a series of lines tightly spaced in the 18−23GHz rangeshown in lower panels, which appear as the broad peak around 20 GHz in our data(Figure 7.4), due to the low frequency resolution.Owing to the non-equidistant rotational spectrum of a molecule in the droplet(due to the much stronger centrifugal distortion, caused by the helium solvationshell), there is no fundamental period for a full quantum revival to occur across allpossible frequencies, unlike the rigid rotor case where Trev = 12B . As the quantumrevival occurs at the lowest common multiple of all periods, a state-dependent fre-quency spacing results in a revival period that depends on the specific rotationalstates populated, and gets pushed further out in time as more and more differentfrequency spacings are added [14, 16, 22].To illustrate how significant this effect can be, consider two frequencies presentin Figure 7.1: 18 GHz and 23 GHz. Their periods are 55.56 ps and 43.48 ps, re-spectively. The lowest common multiple of these two periods is 60394 ps. Thisis several orders of magnitude longer than the delay scans shown in Figure 7.2,and does not even account for matching the periods of all the other frequenciespresent. Such strong “centrifugal distortion” results in no observable rotational re-vivals. Regardless, even for a rigid rotor with an energy level spacingd of 5 GHz,the revival at 600 ps is far too late for the apparent decoherence timescalee.Classically, to enhance rotation, a second kick needs to be applied when themolecule’s rotational phase is such that it is rotating towards, rather than awaydRemember ∆E = B(4J+6) from Equation 2.32, so Trev = 62∆E .eThe traces in Figure 7.2 were recorded out to 1200 ps, but the later signal is isotropic at highfluences due to decoherence, so the authors plotted truncated scans [33].97Figure 7.5: The Fourier transform of the intensity envelope of two 1.4 pspulses separated by 75 ps. This represents the two-photon excitationfield (see Section 2.5.2). The first six frequencies from Figure 7.1 areshown with red vertical lines.from, the polarization axis of the kick. Thus, enhancement (suppression) of a par-ticular frequency can be initiated by the second kickf arriving around 12(34)of thecorresponding rotational period. Due to the symmetry of CS2, an alignment signalsuch as ⟨cos2 θ2D⟩ will be modulated at half the classical rotational period. Thismeans the observed oscillations around 20 GHz can be thought of as correspondingto a classical rotational period of Trot =∼100ps. A double kick separation around∼75 ps was chosen for this experiment, because it was expected to suppress therotations around 20 GHz (75ps = 34 Trot).Quantum mechanically, a 75 ps pulse separation offers an excitation in the fre-quency domain with significantly lower intensity of the two-photon field, respon-sible for the rotational excitation around 20 GHz, as shown in Figure 7.5. This re-sults in a suppressed excitation probability at 20 GHz, and will generate rotationalwavepackets that reflect this, as described in Section 2.5.2.The results of numerical simulations of the rotational excitation of CS2 molecules,embedded in helium droplets, are shown in Figure 7.6. In the simulation, we solvethe time-dependent Schro¨dinger equation with the interaction Hamiltonian, out-lined in Equation 2.20. The rotational wavefunction, expanded in the basis offAssuming the second kick has the same polarization axis as the first kick.98Figure 7.6: Numerically simulated rotational coherences of the first five fre-quencies after the second kick, plotted as a function of the separationbetween the two kick pulses. With a kick separation of around 78 ps, allfrequencies are effectively suppressed (black vertical line). State ener-gies were calculated using the rotational constants measured in [30].spherical harmonics, isψ(t) = ∑L,McL,Me−iELt/ℏψL,M, (7.4)where the numerical simulation provides us with the complex amplitudes cL,M im-mediately after the interaction with the field. The amplitudes of rotational coher-ences, oscillating at the frequency ∆EL,L+2 after the interaction, are then calculatedas the real parts of the off-diagonal products cL,Mc∗L+2,M, summed over all Ms.From this, the double kick pulse separation of 78 ps was chosen for experiments,as it showed the greatest degree of suppression at all rotational frequencies.7.3 ResultsThe results of the double kick experiment to suppress the ∼20 GHz frequenciesare shown in Figure 7.7. Unfortunately, these measurements proved to be very99Figure 7.7: CS2 kicked with a pair of 4 J/cm2 2 ps pulses, in 25 Bar / 13.5 Khelium droplets. The two curves (average of 20 scans each) were pro-duced over the same three hours, by alternating the single and doublepulse excitations every 30 s.time consuming, with poor signal-to-noise in the ⟨cos2 θ2D⟩ measurements, evenafter performing the scan for multiple hours. Nevertheless, in Figure 7.7 it canbe seen that there is a distinct difference between the results with one or bothkicks. The oscillations with a period of ∼50 ps are no longer present in the signal,corresponding to the kick separation of 78 ps.The power spectrum of Figure 7.7 is plotted in Figure 7.8, and even with thisrelatively coarse measurement it is clear that the dominant ∼20 GHz contributionis indeed suppressed - although the low resolution scan cannot resolve the separatefrequencies such as the ∼10 GHz oscillation present in the blue curve in Figure7.7. This makes it unclear whether the 78 ps separation between pulses is indeedoptimal for the maximum suppression of CS2 rotation, as suggested by our simu-lations, or whether other factors (such as centrifugal distortion, not included in thesimulations) dictate different kick sequences.While this experiment successfully demonstrated that some degree of rotationalcontrol in helium droplets using a double kick is possible, significantly longer datacollection time would have to be spent to obtain usable estimates of state lifetimes.It is expected that increasing the probe focal volume to better match that of the kick100Figure 7.8: The Fourier transform of the alignment factor ⟨cos2 θ2D⟩ after oneor two kicks (as plotted in Figure 7.7), starting with a probe delay of10 ps. A Hamming window was applied. The disparity between thecurves around 20 GHz shows that molecular rotation at this frequencywas effectively suppressed.would greatly improve both the signal-to-noise ratio and count rate, in which casescanning a few more rotational periods may provide better frequency resolution forfurther analysis.The preliminary results, presented above, show that a double kick scheme mayserve as one possible method to measure the different decoherence rates of rota-tional energy levels, where the impact of an earlier kick decreases as the pulseseparation increases, and angular momentum is transferred from the dopant to thehelium droplet before the second kick arrives. These results also imply that the ro-tational control techniques used in gas phase molecular ensembles could be appliedto molecules in helium droplets [122, 170, 189]. The potential to selectively excitemolecules in a mixture is particularly relevant when novel species are synthesizedin helium droplets. When dimers are formed in helium droplets from two dif-ferent species, such as forming C6H5Br-I2 by doping the droplets separately withC6H5Br and I2 in [155], it is possible to minimize the contribution of (C6H5Br)2and (I2)2. This is done by reducing the pressure in the independent doping cells tominimize the production of those dimers directly. If one wanted to study a trimerformed from three separate species it would not be possible to do so without form-ing a significant number of dimers from each of the three possible pairs of species,101necessitating the observation of a mixture of molecules. Rotational control tech-niques could provide an opportunity to study the dynamics of the trimer of interest,while suppressing signal contributions from the dimers.102Chapter 8Centrifuged CS2 in HeliumDropletsThis chapter outlines experimental attempts to spin carbon disulphide in heliumdroplets with an optical centrifuge.8.1 IntroductionPrevious experiments did not directly examine the dynamics of each rotational stateindividually, to observe how the angular momentum transfer, and correspondingloss of coherence between states, differ for different rotational states. As describedin Section 2.4, the previously-studied impulsive/nonadiabatic excitations populatean increasingly broad distribution of states as the average level of excitation isincreased. Infrared rotational spectroscopy has been limited to studying the ini-tially populated states, which makes it impossible to study high rotational states ofmolecules in helium droplets. Direct control over the degree of rotational excita-tion is a primary motivator for using an optical centrifuge on molecules in heliumdroplets, so that rotational dynamics can be investigated state-by-state.Additionally, the results from [159] demonstrated that I2 (as well as OCS andCS2, analyzed more thoroughly in [78]) may briefly (for less than 5 ps) break freefrom the solvation shell, when kicked strongly. The model presented was thatthe molecule begins to rotate too fast for the helium solvation shell to follow, but103quickly transfers energy back to its surroundings. In those experiments, a sharpincrease of ⟨cos2 θ2D⟩ occurred immediately after the kick, on a similar timescaleboth in and outside of helium droplets, as if the molecule has detached from thehelium. In this case, it is expected that the full rotational spectrum would approachthat of the gas phase, if the decoupling were long-lived [33]. Gas phase molecularsuperrotors in extreme rotational states exhibit more than an order of magnitude de-crease in collisional decoherence rates than rotors with lower angular momentum[117, 118, 128]. In simple terms, it is not as “easy” for a superrotor to lose rota-tional energy when the surrounding environment has a low density of states withmatching energies. The expectation is that if a molecule were to be driven wellabove the energy of elementary excitations in superfluid helium (such as rotons),the decoherence would be suppressed. This effect has recently been demonstratedin bulk helium (by our group [124]), and also in helium droplets [143]. This begetsthe question of what happens if a molecule in helium droplets is driven above thespeed necessary to decouple from the superfluid, and up to much higher degrees ofexcitation?CS2 is a popular molecule for alignment experiments, and the S+ ion fragmentis readily observed from helium droplets of a few thousand atoms. The rotationalenergy spectrum of CS2 in helium droplets leads to rotational energy state sepa-rations (corresponding to rotational frequencies) shown previously in Figure 7.1,where after just a few states an increase in angular momentum results in dimin-ishing increases in rotational energy [30]. This occurs due to the coupling withthe surrounding helium, as isolated CS2 has steadily increasing rotational frequen-cies up to and beyond 1 THz. The Angulon model of molecule-droplet interaction[34, 151] describes how excitations of the surrounding helium bath may lead to thefirst few states of a dopant’s rotational spectrum behave as that of a simple rotor(that of Equation 2.7), albeit with significantly modified rotational and centrifugaldistortion constants. The bath excitations are quasiparticles referred to as “angu-lons”. The “CS2 + angulon” spectrum as depicted in Figure 7.1 may hit a “cen-trifugal wall”, where subsequent rotational transitions are no longer occuring athigher frequencies, at which point the molecule would no longer be able to followthe rotation of an optical centrifuge with a constantly increasing angular frequency(this effect was studied by our group in the gas phase [120]). The distortion can104Figure 8.1: The rotational energy linewidths for various molecules in he-lium droplets, plotted as a function of rotational energy transitions ∆E.Broadened lines correspond to faster decoherence, interpreted as fasterenergy transfer from the molecule to the droplet. Overlaid is the den-sity of states for excitations in bulk superfluid helium. The three peakscorrespond to rotons, maxons, and two-roton excitations. From the PhDthesis of Gary Douberly [59], appearing also in [35].be interpreted as variations in the helium solvation shell surrounding the molecule[34], which takes on some of the angular momentum transferred to the molecule,reducing the energy separation between levels, similar to the effect of centrifugaldistortion increasing a molecule’s moment of inertia.Predicting how transitions will (or will not) occur in an optical centrifuge as therotational structure changes is a challenge [5], with the spinnability arguments inSection 2.3.1 being insufficient for a nonrigid rotor, not to mention a rotor coupledto a many-body environment. Nevertheless, the first few states of CS2 in helium105droplets are approximately quadratically spaced in rotational energy, and thus havelinearly increasing transition frequencies, which ought to be suitable for excitationwith an optical centrifuge.Based on experimental data, a molecule detaches from its solvation shell whenthe kinetic energy of a helium atom, attached to the molecule at the end of themolecular bond, exceeds the binding energy [159]. For He – CS2, this is estimatedto be 20.8 cm−1, with the helium atom located 3.4 A˚ away from the molecularcentre of mass [193]. This would not occur until a rotational frequency exceedingf =√2EbondmHe2πrHe=356ms−12π3.41A˚= 166GHz, (8.1)well beyond the uppermost frequencies in Figure 7.1, where subsequent states fol-lowing this model begin to have transitions at continually lower frequencies.The mechanism described in [159] and [78] has the kick impulsively transferenergy to the molecule and its solvation shell, followed by the centrifugal forcebreaking the linkage between the molecule and the helium solvation shell, simi-lar to the centrifugal dissociation of molecules when excited to extreme rotationalstates [46, 174].With increased rotational energy, the interaction between the solvation shelland surrounding helium increases as the energies overlap with the region with ahigh density of states near the roton energy [92], which in turn increases the effec-tive moment of inertia of the molecule with its shell, thereby reducing the maxi-mum angular acceleration of the two. The interaction with the laser in Equation2.20 has a well depth of around U0 = 200cm−1 for CS2 with the average centrifugeintensity of ∼2×1012 W/cm2 - much stronger than a single He – CS2 bond, but anorder of magnitude weaker than the well depth of the more intense 330 fs pulseneeded to show detachment with a kick in [78].In practice, this means the question of “spinnability” comes down to not justthe centrifuge field strength and rotational acceleration in relation to the molecule’smoment of inertia and anisotropic polarizability, but also a question of whether thatinteraction “pushes through” the centrifugal wall, due to the interaction with thehelium solvation shell that adds inertia. The goal here is to better understand the106rotational dynamics when CS2 is rotationally excited in helium droplets, and howthe interaction with the helium affects the ability of an optical centrifuge to spinmolecules.8.2 Experimental TechniqueThe construction of the “slow” optical centrifuge (SCFG), discussed in Section 2.3and [104], was motivated largely from preliminary experiments using a “fast” op-tical centrifuge to spin CS2 in helium droplets with Jordan Fordyce [56], and theresults of Anders Vestergaard Jørgensena, which had been discussed in his PhD[78]. In those experiments, dynamics were described as weak adiabatic alignmentto the centrifuge field, indistinguishable from that of a single circularly polarizedfieldb. A circularly polarized pulse aligns molecular axes to the polarization plane,rather than the polarization vector, due to the circular symmetry of the field onthe timescale of molecular rotation. It was not determined whether this was dueto a “centrifugal wall” being reached, where increasing frequencies cannot excitethe molecule, or if the increased effective moment of inertia in droplets was sim-ply outside the bounds to satisfy the “spinnability” criteria for adiabatic excitation,outlined in Section 2.3.1. The much slower acceleration of SCFG reduces the ratiobetween the necessary torque applied to the molecule relative to its moment of in-ertia to follow the rotation of the centrifuge by a factor of∼ 20, far greater than thehelium-induced increase in the moment of inertia by a factor of ∼ 3. Classically,that ratio determines the angular acceleration of the molecule aligning towards thefield, hence the desire to ensure the angular acceleration of the molecule towardsthe field is greater than the angular acceleration of the field itself.The rotational dynamics are observed through Coulomb explosion imaging,with the molecule’s orientation inferred by calculating ⟨cos2 θ2D⟩ from the S+ frag-ments. Helium droplets of various sizes were doped with CS2 using the heatedsample holder and leak valve as described in Chapter 6.aUnpublished results from Anders show the same effect for OCS in helium droplets.bThe optical centrifuge is comprised of two circularly polarized fields.1078.3 ResultsExperiments with the slow optical centrifuge demonstrate that CS2 in helium dropletsdoes not follow the centrifuge into extreme rotational states.8.3.1 Results with the 3D Optical CentrifugeFigure 8.2 shows the alignment factor ⟨cos2 θ2D⟩ of S+ ions from Coulomb-explodedCS2 as a function of probe delay during the centrifuge, for molecules both embed-ded in helium droplets and in the gas phase from a seeded jet. These results showthat after the ∼500 ps pulse envelope of SCFG, the alignment factor returns to theinitial value. This is sufficient to make the statement that molecules are not un-dergoing the same uni-axial rotation after the centrifuge as is shown for gas phaseFigure 8.2: The alignment factor ⟨cos2 θ2D⟩ for S+ ions from CS2 as a func-tion of probe delay for a 500 ps SCFG with an average intensity of∼2× 1012 W/cm2, shown both for free CS2 molecules in a supersonicjet (a) and in helium droplets produced at 40 Bar / 17 K (b). The 140 fsprobe has a peak intensity of 3× 1015 W/cm2. The shaded area repre-sents the intensity envelope of the centrifuge, measured with the stan-dard cross-correlation technique.108CS2 in Figure 8.2 (a). The terminal frequency of SCFG of ∼2.3 THz correspondsto putting gas phase CS2 in J,MJ ≈ 140, which would result in a very high degreeof alignment. The angular momentum state |J,MJ⟩= |140,140⟩ has an expectationvalue of ⟨cos2 θ2D⟩ above 0.9, although as shown in Figure 8.2 this degree of align-ment was not recorded for isolated CS2 under these conditions, due to the low butfinite initial temperature, resulting in multiple initial rotational states populated.The energy separation between the rotational states for a molecule spun upto the end of the centrifuge is above 70 cm−1, well above the roton energy. Thisexperiment has been repeated in various conditions, and consistently the maximumdegree of alignment in droplets is below the maximum retrievable due to scatteringof S+ fragments from helium atoms as they escape the droplet (described in Section5.4, an increase in ⟨cos2 θ2D⟩ of nearly 0.2 in these conditions), and below thatobtained under ideal conditions with a kick.SCFG accelerates to 2.3 THz over around 500 ps, which means that for CS2(with the rotational constants measured in helium droplets, providing the rotationalfrequencies plotted in Figure 7.1) the centrifugal wall around 20 GHz is reachedwithin 5 ps. Previous experiments in the gas phase demonstrated that keeping thecentrifuge field on after the molecules stop following it does not affect the inducedcoherent rotation [2]. After a molecule lags behind, the continuation of the cen-trifuge field appears to a molecule as a circularly polarized field, rotating too fastto support adiabatic following. Consider the decomposition of a circularly polar-ized field into two orthogonal linearly polarized fields in the same plane. As alinear molecule, CS2 is attracted to both vectors, and is therefore aligned withinthe plane with no preferential orientation. When the polarization is rotating tooquickly to be relevant, in regards to the rotational energy spectrum of the molecule,it makes sense to consider the centrifuge as a circularly polarized field whose effecton molecular alignment is to adiabatically hold CS2 in the plane of rotation. Thisis why it is not of concern that the centrifuge field extends beyond the centrifu-gal wall, so long as the necessary set of rotational excitations to get there occur.We experimentally checked that the observed behaviour is indeed insensitive to thetruncation of the cfg field. Figure 8.3 was recorded with SCFG truncated to∼70 ps,and as in Figure 8.2, the alignment increases during the pulse, and decays shortlyafter.109Figure 8.3: The alignment factor ⟨cos2 θ2D⟩ for S+ ions from CS2 as a func-tion of probe delay, for a truncated ∼2× 1012 W/cm2 SCFG (FWHMof 2 nm after ∼70ps), shown both in helium droplets and in a super-sonic jet. The 140 fs probe has a peak intensity of 1× 1016 W/cm2.The ⟨cos2 θ2D⟩ baseline is reduced for the measurement (relative to theisotropic value of 0.5 and that of Figure 8.2) in droplets, because theprobe is ionizing a significant number of room temperature backgroundmolecules. The kinetic energy release upon Coulomb explosion fromthe droplet beam is shifted on the VMI detector relative to the back-ground by the velocity of the helium droplet beam, and ⟨cos2 θ2D⟩ iscalculated relative to the shifted position. The seeded jet is more dense,so the background contribution is less significant.This experiment has the centrifuge accelerating more than an order of mag-nitude above the 20− 25GHz frequencies excited by a kick, while also having alesser degree of alignment. It appears the molecules are not following the cen-trifuge even to that extent, as the wavefunction for |J,MJ⟩ = |6,6⟩ would be ex-pected to provide higher degrees of alignment. Estimates from a toy model iondistribution generated with the spherical harmonic Y 66 (θ ,φ), shown in Figure 8.4,give ⟨cos2 θ2D⟩ ≈ 0.7.110Figure 8.4: Left: the spherical harmonic |J,MJ⟩ = |6,6⟩, with the centrifugeshown propagating along the Y axis (magenta). Y 66 (θ ,φ) is shown withthe red disk, where θ is the angle between the molecular axis and thepropagation axis of the centrifuge (the Y axis), and φ is from the Zaxis. The image below shows the numerically simulated ion projectionon the VMI detector along the time of flight axis (Z). Right: the samenumerical simulation of the 2D VMI ion image for |J,MJ⟩ = |6,6⟩, forwhich ⟨cos2 θ2D⟩= 0.7 is calculated with θ2D as the angle with respectto the plane of the centrifuge rotation (red line). The 2D ion image is a2D histogram of ion counts at each pixel coordinate, with the colourmapreflecting the relative number of ions in each bin.8.3.2 Increasing sensitivity by using a 2D Optical CentrifugeBy utilizing the ”two-dimensional” optical centrifuge (2D CFG) described in [119],the effect of the centrifuge can be modulated without turning the high-intensity cen-trifuge beam on and off. This is important as the latter introduces thermal effectsin various optical components, increasing the noise level in our detection. Modu-lation allows for a more sensitive comparison to the baseline ⟨cos2 θ2D⟩ value. A2D CFG is formed by passing a CFG through a linear polarizer, with the resultingfield being the 2D projection of the rotating polarization vector. This polariza-tion can then be modulated by passing the beam through a half wave plate, whoseorientation is controlled with a computer. The projection is an accelerating pulsetrain, where both the duration of each pulse and separation between pulses is de-111Figure 8.5: A simulation of the electric field of the first 50 ps of SCFG (a)and its 2D projection (b), referred to as the 2D centrifuge, which can bethought of as an accelerating pulse train. (c) shows the intensity profileof the 2D CFG.creasing, as depicted in Figure 8.5. Each pulse “kicks” the molecules to rotate ata slightly higher frequency, hence the need to decrease the time between pulses.High degrees of adiabatic rotational excitation can still be reached [119], thoughthe rotation is no longer unidirectional. Restraining the polarization of the cen-trifuge to linear changes the selection rules, from ∆J = 2, ∆MJ = 2 for the full 3Dcentrifuge, to ∆J = 2, ∆MJ = 0 for the 2D centrifuge.By modulating whether the 2D CFG is polarized parallel or perpendicular tothe VMI detector, the 2D projection of the aligned molecules will be modulated be-tween being circularly anisotropic or isotropic, respectively. In the perpendicularcase, there should be no ⟨cos2 θ2D⟩ increase as the molecules align to the polariza-tion vector, while with the field parallel to the detector, ⟨cos2 θ2D⟩ would increasewith molecular alignment. This provides a useful comparison, to ensure that anysmall change in ⟨cos2 θ2D⟩ is in fact due to molecular alignment, rather than someother experimental factor. The polarization was modulated with a motorized halfwave plate placed after the linear polarizer. An example of the different ion distri-112Figure 8.6: By modulating the 2D CFG polarization, ⟨cos2 θ2D⟩will be modu-lated between isotropic and anisotropic distribution. The magneta arrowshows the centrifuge propagation direction, and the shaded blue curvesdepict the electric fields of (a) a 3D centrifuge, as well as two orthogo-nal 2D centrifuges in (b) (c). The resulting molecular axis distributions(spherical harmonics) are shown “in front” of the centrifuge for illustra-tive purposes, with simulated 2D VMI ion distributions projected ontothe plane below.113Figure 8.7: The alignment factor ⟨cos2 θ2D⟩ for S+ ions from CS2 as a func-tion of probe delay for a truncated ∼2×1012 W/cm2 SCFG (FWHM of∼0.7 nm after ∼25ps) shown both in helium droplets and in a super-sonic jet. Parallel refers to the 2D CFG being polarized parallel to theVMI detector, with perpendicular being along the TOFVMI axis. As wasshown in Figure 8.6, perpendicular (parallel) 2D centrifuge polarizationresults in an isotropic (anisotropic) VMI signal.butions anticipated for different centrifuges is shown in Figure 8.6.Figure 8.7 shows ⟨cos2 θ2D⟩ as a function of probe delay for the 2D CFG, in thegas phase and in helium droplets. The maximum degree of alignment in heliumdroplets is much closer to that in gas phase for this centrifuge truncation. However,one can see that the increase in ⟨cos2 θ2D⟩ during the pulse is significantly higherthan the sustained ⟨cos2 θ2D⟩ both in the jet and helium droplets - a feature that isnot typical of the expected rotational excitation, gradually increasing in time. The25 ps 2D SCFG used consists of 3−4 pulses, whose widths, timings, and intensitiesvary randomly from pulse to pulse due to the lack of interferometric stability, whichresults in random relative phase between the two centrifuge arms. The last twopulses are approximately 5 ps apart, thus resonant with a frequency of 200 GHz.This is again significantly faster than the highest frequency recorded for CS2 inhelium droplets of ∼23 GHz [33].In principle, this excitation is similar to the double kick used in Chapter 7, but114Figure 8.8: Comparison between the effect of the full 2D CFG and a singlelinearly polarized centrifuge arm, with the same pulse energy and tem-poral profile. The excitation field envelope (measured with the standardcross-correlation technique) is shown as a shaded curve.with the shot-to-shot fluctuations in the 2D CFG “pulse train”, the rotational coher-ence as a function of probe delay after the 2D CFG will be “blurred” on averageacross all measurements. A comparison scan shown in Figure 8.8 using a singlehorizontally polarized arm of the CFG, with the same overall pulse energy and trun-cation, was recorded in helium droplets. The features are indistinguishable withinerror bars, and therefore it seems the pulsing of the 2D CFG field is not leading tothe expected rotational acceleration.The ∼25 ps duration is half of the ∼20 GHz oscillation period seen in CS2 inhelium droplets, making it unreasonable to consider this as a single “short” kick.Higher resolution delay scans after the field were not performed for this reason.8.4 DiscussionWhile these experiments have not answered questions regarding the decoherencemechanism and timescale, they have demonstrated that despite the rotational con-stant of CS2 decreasing in helium droplets by a factor of 3, a compensatory 20×decrease in optical centrifuge acceleration is not sufficient to force the molecule115follow the centrifuge into an extreme superrotor state.Consider the centrifuge intensity, as its rotational frequency passes the resonantrotational frequencies of CS2 in helium droplets. The rising edge of the slow cen-trifuge pulse was measured to be 8−10ps, at which point the centrifuge is alreadyspinning at nearly 50 GHz. As the centrifuge has constant angular acceleration,this means the intensity as the centrifuge passes 10 GHz (the second frequency inFigure 7.1) will be close to 14 of its peak intensity. This provides a well depth ofU0 ≈ 50− 100cm−1 as the centrifuge passes 10 GHz for the experiments with anaverage intensity over the whole pulse of 2×1012 W/cm2, still a few times higherthan the 20.8 cm−1 binding energy of He – CS2. Same is valid also for the firstrotational frequency of ∼5 GHz.At these lower frequencies, the model is that a “nonsuperfluid shell” of heliumatoms is adiabatically following the rotation of the molecule without any significantdeformation. This means that for these first few frequencies, the question shouldbe about whether the centrifuge can apply sufficient torque to rotate the moleculewith the helium shell, and its corresponding increase in moment of inertia, notnecessarily a question of detaching from the shell [34]. Substituting a conserva-tive estimate for the intensity of the centrifuge at the earlier times, when it passesthese lower frequencies, of 0.2×2×1012 W/cm2 = 4×1011 W/cm2 into Equation2.22 (the average torque), for a moment of inertia three times that of CS2 with theanisotropic polarizabiltiy of ∆α ≈ 9A˚, provides an average angular acceleration ofθ¨mol ≈ 2U0πI = 32radGHz/ps, (8.2)which is similar to the angular acceleration of the field ofθ¨pol = 2β = 2×18radGHz/ps = 36radGHz/ps, (8.3)measured in [104]. Given the above estimate, the spinnability criterion is of orderunity, and might have dropped below this threshold value due to unaccounted-forexperimental imperfections.In previous experiments by Anders Vestergaard Jørgensen, the rise time of thecentrifuge was approximately 5 ps [78]. With the ∼ 20× greater angular accelera-116tion, the centrifuge he used has exceeded a rotational frequency of 10 GHz withina tenth of a picosecond - 2% of the rise time - resulting in a minuscule intensity asthe centrifuge passes the first two rotational frequencies of CS2 in helium droplets.The spinnability at 10 GHz for his centrifuge with the same peak intensity wouldbe one hundredthc of the value used here, which again motivated us to design andbuild the much slower optical centrifuge.In references [30] and [33], the coherence of the excited low rotational statessurvives for hundreds of picoseconds after a short kick - and higher energy oscil-lations around 15− 20GHz are also still present, even after 600 ps. One wouldexpect centrifuged molecules at these frequencies to maintain their alignment for asimilar time, if they had been able to follow the centrifuge this far.With these considerations, it would be expected that an optical centrifuge de-signed to have appropriated intensity and angular acceleration at 5 GHz and 10 GHzwould be able to generate a long-lasting rotational wavepacket of CS2 with |J,MJ⟩=|4,4⟩, if not |6,6⟩, before the effective centrifugal distortion prevents further excita-tion by forming the centrifugal wall. Equations 8.2 and 8.3 suggest that a reductionin the angular acceleration of SCFG by a factor of less than 2 would be sufficient,but as was laid out in [104], it is challenging (though not impossible) to reducethe chirp rate below that of the “slow” optical centrifuge used in this work, andionization poses an upper limit to how strong a centrifuge pulse can be. Technicalaspects of the slow centrifuge shaper used in this work limited the terminal fre-quency of the centrifuge well above 10 GHz, making it impossible to increase theintensity at the time of the first rotational resonances of CS2 without imposing amuch stronger ionizing field for ∼10 ps afterwards. Rather than trying to increasethe angular acceleration of the molecule towards the field, the acceleration of thefield must be lowered to match that of the molecule.The relevant adiabaticity requirements of the optical centrifuge are that the ro-tating linear polarization vector of the centrifuge must sweep out an angle of atleast 180◦ during the rise time of the field intensity, as well as that the angularacceleration of the molecule towards the polarization vector of the laser field mustcTaking 10 GHz to be passed at 2% rather than 10% of the rising edge, and the angular accelera-tion being 20× faster.dFrom the spinnability criterion in Equation 2.24.117be greater than the angular acceleration of the polarization vector itself [79]. How-ever, the latter has always been described in terms of the induced torque relativeto the molecule’s moment of inertia, with the moment of inertia considered con-stant. Classically, even if the centrifuge has passed the first excitation frequency,the slowly-varying strong field eventually “catches” the molecule later during therise time as it sweeps past. If the spinnability criterion would not be met at the fre-quency of the centrifuge as it polarization sweeps past the molecule, the moleculewill not be able to follow. That the spinnability may unintuitively be decreasingduring the rising edge of the pulse, due to the increasing centrifugal distortion asthe rotational frequency increases, has not been previously taken into account inthe experiments with an optical centrifuge, as the gas phase molecules previouslystudied exhibit much weaker centrifugal distortion.In conclusion, our interpretation of why the experiments at both UBC andAarhus University did not see long-lasting rotation of the first few rotational stateswith the centrifuge, is that the spinnability criterion was not met due to the ini-tial capture of molecules happening during the rising edge of the centrifuge field,where the intensity is still low. Not meeting the spinnability criterion is distinctfrom hitting the “centrifugal wall”. After the centrifugal wall, subsequent rota-tional transitions occur with decreasing, rather than increasing, transition frequen-cies [120]. Spinning past the centrifugal wall would require careful considerationof the transition frequencies, while meeting the spinnability criterion for rotationalstates below the wall would not require anything other than a slower optical cen-trifuge.In “A simple model for high rotational excitations of molecules in a superfluid”the authors state“One might need to redefine the adiabaticity criterion of molecule-laser interactions and to use non-linear ramp pulses in order to accountfor the threshold of the states” [34],referring to the threshold where the molecule’s rotational energy starts to couplestrongly to the bath, and exchanges energy with the droplet.So what about breaking through the centrifugal wall by detaching the moleculefrom its solvation shell to reduce the effective centrifugal distortion? Centrifugal118distortion, and other mechanisms that decrease the rotational energy level spacing,increase the effective moment of inertia, thereby reducing the angular accelerationof the molecule in the field. In the case of a molecule coupled to another systemwhich can absorb energy, such as a molecule embedded in a liquid, the timescale ofenergy and angular momentum transfer poses an additional restriction for the rateof excitation, and therefore impact the question of spinnability. The estimated max-imum rotational frequency of∼250 GHz from CS2 rotating in a helium droplet, af-ter having detached from the solvation shell after a strong kick, decohered within afew picoseconds [78, 159]. This is to be expected, given its proximity to the rotonenergy (∼200 GHz). With the spinnability requirement enforcing slow accelera-tions, optical centrifuge durations are typically on the order of 100 ps. This meansthe energy transfer to the surroundings on the picosecond scale poses an additionalconstraint that must be considered for the effective spinnability in a fluid, whereit would be beneficial to climb the rotational ladder faster, to minimize the lossof rotational energy to the surroundings. Transfer of energy to the surroundings isrelevant both for the case of the molecule detached from the solvation shell, andthe solvation shell adiabatically following the molecule.119Chapter 9Rotational Excitation of N2 inHelium DropletsThis chapter describes experiments attempting to observe rotational excitation ofmolecular nitrogen in helium droplets. Rotational excitation of lighter moleculeswith rotational energy level separations comparable to the energy of elementaryexcitation in superfluid helium offers the possibility to probe the molecule-dropletcoupling directly. Previous rotational excitation experiments in helium dropletsstudied molecules with rotational constants either an order of magnitude above orbelow that of N2. The energy of the former can not be brought down to that ofa roton, and no rotational excitation of the latter well above the roton energy hasbeen documented - attempts to do so were discussed in Chapter 8.9.1 IntroductionNitrogen is not infrared active, so infrared rotational spectroscopy techniques arenot able to measure its rotational constant in helium droplets. To my knowledgethere are no published REMPI rotational spectroscopy experiments of N2 in heliumdroplets, either. In its ground rotational state, nitrogen is expected to weakly coupleto the helium environment, with a rotational constant close to that of gas phase N2,around 2 cm−1, much like CO and NO [97]. However, with a small excitation atthe bottom of the rotational ladder, the energy separation ∆E = B(2J+2) becomes120comparable to the roton energy of 6 cm−1, which would result in strong coupling tothe superfluid and correspondingly rapid rotational decoherence. At a larger degreeof excitation, the separation between rotational energy levels becomes much largerthan the roton energy, suppressing decoherence due to the increasing number ofbath excitations necessary for the molecule to transfer energy to the droplet [33].Coulomb explosion imaging of ion fragments such as N+ in helium dropletsproves challenging, due to scattering off of helium atoms as the ion exits thedroplet. Generally, with a diatomic molecule, the ion emission angle after Coulombexplosion provides an excellent indication of the orientation of the molecular axis.With sufficient scattering, this angle is scrambled, hence all orientation informationis lost. The experiment outlined in this chapter tests whether rotational coherencesof kicked N2 can be observed in helium droplets.9.2 Experimental TechniqueTwo different detection methods were used to observe the rotational dynamics afternitrogen molecules in helium droplets were kicked by a 200 fs pulse. The same kickpulse readily excites N2 in the supersonic jet with strong rotational revivals, so itis known to be suitable for exciting a molecule with the anisotropic polarizabilityand rotational constant of nitrogen in the gas phase.9.2.1 Coulomb Explosion Imaging of N2 in Helium DropletsAs N2 molecules are more likely to be ionized when oriented along the polarizationaxis of the probe than perpendicular to it, referred to as “geometric” alignment, asimple experiment using angle-dependent ionization can check whether the scatter-ing of N+ fragments as they exit the droplet is sufficiently small for the orientationof the molecular axis to be inferred. This is done by comparing ⟨cos2 θ2D⟩ whenthe probe polarization is parallel or perpendicular to the VMI detector. With heliumdroplets produced under high nozzle pressure and temperature (60 Bar, 21 K), itwas possible to observe an increase of ⟨cos2 θ2D⟩ of about 0.03 in velocity mapimaging experiments for N+ fragments from N2 in helium droplets. This is signifi-cantly lower than the increase of 0.2 seen in Chapter 7 for the heavier S+ fragmentsof CS2, and results in much lower sensitivity.121Although these conditions should be producing droplets with ⟨NHe⟩ ≈ 5000atoms [167], the average droplet size may be significantly smaller due to high pres-sures of helium in the source and doping chambers. With the pressure of heliumin the doping chamber exceeding 1× 10−5 Torr under these conditions, the likeli-hood of a droplet absorbing one (or a few) “hot” helium atoms is significant (seeEquation 5.1). Upon absorption, the droplets boil off several helium atoms, similarto the doping process outlined in Section 5.3, reducing ⟨NHe⟩. Smaller droplets areused to minimize the effect of ion fragments scattering off of the helium atoms asthey exit the droplets. The doped-droplet signal in this regime is so weak that thebackground gas, whose pressure in the detection chamber is 5× 10−9 Torr, con-tributes significantly to the signal. For this reason, the much less abundant isotopo-logue 15N2 was used, with a much lower background stemming from atmosphericgasses. In practice, these conditions give an approximate 1:1 signal-to-backgroundion count ratio for Coulomb explosion measurements, an order of magnitude im-provement compared to when 14N2 was used.9.2.2 Angle-Dependent Ionization of N2By taking advantage of the increased likelihood of ionization when a nitrogenmolecule is aligned with the axis of laser polarization, as described in Section 3.3,molecular alignment can be observed, even in the case where the scattering of ionfragments render an anisotropic Coulomb explosion process spherically symmetricupon detection. This allows for the use of larger helium droplets, leading to fasterdata acquisition and a higher signal-to-background ratio.The downside of this technique is that one must necessarily use a lower probeintensity, so as to not saturate the ionization signal. Moreover, variations in ioniza-tion rate are typically much less sensitive than ⟨cos2 θ2D⟩ measurements, as theyare limited by the degree of angular anisotropy in the ionization cross-section. Theionization rate when the pump and probe beams overlap in time may also be influ-enced by the nonlinearity of the ionization process.Differential measurements provide greater certainty that the observed effectsare due to the rotational excitation. Minor deviations in ⟨cos2 θ2D⟩, and the ion-ization rate, may arise due to small misalignment of the translation stage, used122Figure 9.1: Schematic of the polarization modulation configuration. The kickis linearly polarized with a polarizing beam splitter (PBS), and thencontinuously rotated with a half wave plate (λ2 ), before being combinedwith the linearly polarized probe using a non-polarizing beamsplitter(BS). The two beams are then focused on the droplet beam in the Timeof Flight mass spectrometer (TOF) with a lens (L), where the ionizationsignal is collected using a photomultiplier tube (PMT). The reflected(from the combining BS) part of the kick pulse is separated from theprobe with a PBS, and its polarization modulation reference signal isrecorded from scattered light reflecting off of a beam dump (BD) with aphotodiode (PD). The signals from the PMT and PD are then gated witha boxcar averager for each laser shot before being digitized.for scanning probe delay, or voltage drift/calibration errors of the TOFVMI. Slightchanges in the vapour pressure of the doping gas due to thermal fluctuations in thelaboratory also affect count rates. To implement a differential detection scheme, wemodulated the kick polarization with a half wave plate, rotating continuously witha frequency of about 2 Hz. The experimental configuration is shown in Figure 9.1.The ionization rate is recorded with a photomultiplier tube (PMT), measuring theintensity of the TOFVMI total signal for each laser shot, with the PMT current aftereach pulse averaged using a boxcar averager. A reference signal was determinedfrom a deflected laser beam passing through a linear polarizer, recorded with a pho-todiode through another boxcar averager. The series of measurements at the laser123repetition rate of 1 kHz were then Fourier-transformed in post-processing, with thephase and exact frequency of modulation extracted from the reference signal.9.3 ResultsThe results of alignment measurements of kicked 15N2 in helium droplets as afunction of probe delay using Coulomb explosion imaging are plotted in Figure9.2. The observed values of ⟨cos2 θ2D⟩ show a single alignment peak shortly afterthe kick, before reverting back to the isotropic pre-excitation value.9.3.1 Kicked N2One notable feature is the delayed increase of ⟨cos2 θ2D⟩ with respect to the in-crease in ionization around t = 0. As the strong fields of the kick and probe overlap,Figure 9.2: ⟨cos2 θ2D⟩ (top) and ion count (bottom) as a function of probedelay for the mass 15 ion of 15N2 monomer-doped 60 Bar / 21 K dropletsafter excitation by a ∼110 J/cm2 pulse of ∼200 fs. Only a single scanwithout droplets was collected, as the background contribution appearssmall. This represents about four hours of data collection. The kickpulse is shown with a shaded red area.124Figure 9.3: ⟨cos2 θ2D⟩ of kicked 15N2 in helium droplets, around the gasphase rotational revival of 15N2 at∼9 ps. This is plotted separately fromFigure 9.2 as it was recorded later, with a different isotropic baseline dueto minor defects on the TOFVMI detector.the increase in the total ionization rate, and the anisotropic ionization cross-section,will result in an increase of ⟨cos2 θ2D⟩, even if the molecular distribution remainsisotropic. The fact that the ⟨cos2 θ2D⟩ peak is slightly delayed from the kick pulseindicates that this is because the molecules take some time to align to its polariza-tion axis, rather than merely due to the preferential ionization along its axis. Themaximum increase in ⟨cos2 θ2D⟩ here is about 0.02, slightly less than the differenceof 0.03 observed with just the probe.With no sustained increase of ⟨cos2 θ2D⟩ after the initial peak, our attempts todetect any revival signals around 9 ps proved unsuccessful, as shown in Figure 9.3.For reference, rotational revivals in gas phase 14N2 are shown in Figure 9.4.Differential ionization signal for two orthogonal probe polarizations for kicked15N2 in helium droplets as a function of probe delay are plotted in Figure 9.5. Anotable feature is the short increase in the Fast Fourier Transform (FFT) componentat the frequency of the probe polarization modulation occurring approximately 11 ps after the ∼200 fs kick and probe pulses overlap, which promptly returns tothe pre-excitation value. The data in this figure represents 21 scans with 39 pointscollecting data for one minute each - nearly 14 hours of data collection. Similar125Figure 9.4: The first two rotational revivals of 14N2 in the gas phase around8.3 ps and 16.6 ps, as well as the half revivals around 4.2 ps and 12.5 ps.A much later revival around 763 ps is shown to the right, to illustratethat the alignment dynamics continue unperturbed.measurements from other days are consistent in finding the ∼1 ps peak, and nodynamics were observed at the expected half revival around 4.5 ps.9.3.2 Additional Data: Dynamics of (N2)2If the droplets are overdoped, the RGA will measure significantly stronger mass 60signal, and the 15 + ion signal will be contaminated by the signal stemming fromN+ fragments originating from N2 dimers (and trimers), which will have differentrotational dynamics. Figure 9.6 shows ⟨cos2 θ2D⟩ after a kick in droplets with asignificant number of nitrogen dimers. The increase in ⟨cos2 θ2D⟩ is very low.(N2)2 may produce N+ ions with significantly higher kinetic energy than N2, dueto the increased likelihood of higher levels of ionization, as well as the occasionalN2++ fragment which will have significantly less scattering as it exits the droplet.This means the lower ⟨cos2 θ2D⟩ is likely indicative of a lesser degree of molecularalignment, rather than scattering off of helium atoms. This could be explainedby the dimers having a lower ratio between the anisotropic polarizability and themoment of inertia. The longer/slower dynamics relative to the monomer-dopedsignal in Figure 9.2 points to a higher moment of inertia, as would be expected for(N2)2.126Figure 9.5: The FFT component of the ionization signal at the frequency ofkick polarization modulation, with a kick strength of 11 J/cm2. 15N2was doped in helium droplets produced at 60 Bar / 14 K. The insetshows the scale of the oscillation relative to the peak at t = 0ps, andthe shaded area represents the kick pulse.Figure 9.6: ⟨cos2 θ2D⟩ of the 15+ ion fragment of helium droplets overdopedwith 15N2 and kicked by a ∼200 fs pulse with a fluence of ∼50 J/cm2.The signal-to-background ratio was about ten to one under these condi-tions.1279.4 DiscussionThe broader angular distribution of lower angular momentum states produce aweaker alignment observable, hence lower frequencies will tend to have lower⟨cos2 θ2D⟩ oscillation amplitudes, similarly to the case of CS2 in Figure 7.2 inChapter 7. For this reason, attempts were made to verify the ∼1 ps oscillationthrough ⟨cos2 θ2D⟩ with a strong kick, rather than to vary the kick strength andsearch for weaker oscillations using the angle-dependent ionization technique. Thelack of consistency between the results obtained with these two methods (c.f. ab-sence of the peak around 1 ps in Figure 9.2), is concerning.The amplitude of the PMT signal around t = 0 in Figure 9.5 is difficult to inter-pret, because it is the combination of alignment and nonlinear ionization effects.This can be seen in the ion count of the Coulomb explosion measurement in Figure9.2. There is a large increase in ion count as the beams overlap in time, beginningearlier than the increase in ⟨cos2 θ2D⟩. This makes it not trivial to normalize theseseparate measurements to compare the amplitude of the ∼1 ps oscillation in thePMT measurement to the isotropic signal’s uncertainty in the ⟨cos2 θ2D⟩ measure-ment.An oscillation with a period of just under 1 ps, as was recorded in figure 9.5,would indicate a transition frequency of ∼1 THz. This is more than five timesgreater than the roton energy. Nuclear spin statistics results in a three-to-one ratioof ortho-to-para 15N2, whose spectra consist of only odd and even rotational states,respectively. This means the signal will primarily consist of only the odd rotationalstates. Kicked nitrogen would therefore produce rotational frequencies starting at600 GHz, and increasing in steps of 480 GHz, for the gas phase rotational constantof ∼2 cm−1. A single oscillation cannot be directly assigned to a specific coher-ence when multiple frequencies are present, but ∼1 THz is close to what wouldbe expected for the second frequency of ortho 15N2 of 1.08 THz. A slight renor-malization of the rotational constant, due to the interaction with the surroundinghelium, would be consistent with other light molecules such as NO and CO [97].If this is the second frequency between odd states, with a rotational constantaround 2 cm−1, then the upper state’s energy is E ≈ 2cm−1×5(5+1) = 60cm−1.As stated previously, this is well above the roton energy of 6 cm−1. Energy transfer128to the surrounding helium with a ∆J = 2 transition would require an energy transferof 36 cm−1 (six times the roton energy). In the deuterium experiments in [143],a transition frequency with an energy of 179 cm−1 (30 times the roton energy)provided rotation lasting at least 100 ps with minimal decoherence. Together, thiswould indicate that to suppress decoherence, the transition energy must exceed theroton energy by more than a factor of six, but not necessarily more than thirty. In[132], the first excited state of HF in helium droplets, with a transition energy of40 cm−1, had an estimated lifetime of 12 ps, inferred from its infrared linewidth.A 36 cm−1 transition would have to decay over a slightly shorter timescale to beconsistent with Figure 8.1.Recent experiments in bulk superfluid helium provide an additional intermedi-ate measurement. When helium dimers rotate in small cavities (“bubbles”) of bulkliquid helium [124], a rotational coherence frequency of 2.3 THz was observedto have a lifetime on the order of 1 ns. 2.3THz = 77cm−1 is slightly more thantwelve times the roton energy. It seems improbable that such a small increase inthe number of roton excitations, necessary for a rotational state to decay, wouldincrease the coherence lifetime by three orders of magnitude, from ∼1 ps all theway to 1 ns. For this reason we conclude that the 1 ps oscillation peak observedhere is most probably an experimental artifact.An alternative interpretation of the observed peak at ∼1 ps could be that thisis the initial alignment after the kick, resulting from the interference of multiplerotational frequencies. This would only be possible if nitrogen’s rotational constantwere significantly renormalized in helium droplets. One can crudely estimate thatthe highest rotational excitation stems from the molecules experiencing the largestkick, and thus initially oriented at 45◦ from the kick polarization. A rotation by45◦ in 1 ps would require a rotational frequency of 125 GHz, much less than thegas phase value for the first rotational frequency of ortho 15N2 of 600 GHz.Further reducing the background contribution with the use of a cryopump, andenlarging the interaction region from the ∼16 µm probe beam waist, would im-prove the data acquisition process. Smaller helium droplets could be used to furtherminimize the effects of N+ fragments scattering off of helium atoms in ⟨cos2 θ2D⟩experiments. Benchmark measurements using an adiabatic alignment pulse to en-sure that one detects 15N2 from helium droplets should be performed, in order to129improve the described technique based on the anisotropy of the ionization cross-section.Finally, rather than kicking with a short intense pulse, the use of a pulse witha longer rise edge that adiabatically aligns the molecules prior to being rapidlytruncated, could allow for stronger kicks that may be able to excite the moleculesto sufficiently high states to suppress decoherence [28]. More importantly, sincethe efficiency of adiabatic alignment is much higher at lower temperatures, thedynamics of hot background molecules would not be contributing significantly to⟨cos2 θ2D⟩ measurements in this experimental approach [156].130Chapter 10Alignment and RotationalExcitation of (NO)2 in HeliumDropletsThis chapter outlines experiments on the rotational excitation of nitric oxide dimersin helium droplets. First, possible dimer structures and previous work are intro-duced, followed by the experimental techniques used to observe the dimers, and ananalysis of our experimental observations.10.1 IntroductionIn the search of a molecule whose rotational energy in helium droplets can becontrolled and observed on a state-specific level below, near, and above the rotonenergy, the ability to synthesize molecules/complexes in helium droplets extendsthe list of practicable molecules to study. When using molecular alignment asan observable by calculating ⟨cos2 θ2D⟩ through Coulomb explosion imaging, itseems that smaller but heavier ion fragments are more readily observed withoutundergoing significant scattering off of helium atoms as they travel through thedroplet after the explosion process. Sulfur (mass 32) fragments are known to besufficiently heavy for CEI in helium droplets of a few thousand atoms, hence dimersformed from light diatomics such as NO, O2, and N2 with similar monomer masses131Figure 10.1: Depictions of the cis and trans isomers of ONNO. Oxygenatoms are shown in red, with nitrogen atoms in grey. The table be-low shows the average of the calculated and experimental geometricparameters, presented in [192].Isomer r(N−N) r(N−O) ∠(ONN)acyclic cis-ONNO (C2v) 2.25 A˚ 1.17 A˚ 93.4◦acyclic trans-ONNO ( C2h) 2.19 A˚ 1.19 A˚ 109.4◦may be suitable.(NO)2 is a good candidate because of the lower atmospheric prevalence of NOrelative to N2 and O2, though the dynamics of rotationally excited (NO)2 have notpreviously been studied, nor have its rotational constants in helium droplets beenmeasured.10.1.1 (NO)2 IsomersIt is known from calculations that (NO)2 has a number of stable isomers [134],with the acyclic cis-ONNO (C2v geometry, shown on the left in Figure 10.1) beingthe most stable. The barrier that must be overcome for the acyclic cis-ONNO toisomerize to another stable isomer, acyclic trans-ONNO (C2h geometry, on the rightin Figure 10.1), has a calculated value of 30.8 kJmol−1 (0.32 eV). This is aboveboth calculated and experimental dissociation energies, generally reported to bebelow 14 kJmol−1 [62, 74]. The barrier to reverse this isomerization process wasinitially reported as 8.01 kJmol−1 in reference [26], but more recent calculationshave reported it as low as 1.21 kJmol−1 (0.013 eV) [150]. I will hereby refer tothese as the cis and trans isomers.132Isomer Bx By Bz κCis 0.848 0.229 0.188 −0.88Trans 1.843 0.149 0.137 −0.99Table 10.1: Rotational constants (in cm−1) of (NO)2 isomers, as wellas Ray’s asymmetry parameter κ , from NIST CCCBDB with theB2PLYP=FULLultrafine method using the cc-pVDZ basis set.Neither isomer is prevalent at room temperature [12], with the mole fractionof dimers in liquid NO decreasing rapidly with increased temperature above 100 K[105]. Only the cis dimer has been observed in seeded gas supersonic jets [149,171, 178]. When the dimer is formed at 0.37 K in helium droplets (see Section5.3.1), individual NO molecules may have low enough kinetic energy for the trans−→ cis isomerization to occur. Previous studies of other molecules have shown thatcluster formation in helium droplets may give surprisingly different structures thanexpected [131]. Infrared spectroscopy experiments of (NO)2 in helium dropletshave tentatively assigned one observed line to the trans dimer depicted in Figure10.1 [70].Both isomers are near-prolate asymmetric tops. The calculated gas phase rota-tional constants of each are shown in Table 10.1. For both isomers, the rotationalconstants are comparable to those of the often studied CS2 (0.109 cm−1) and OCS(0.2 cm−1).In [70], infrared spectroscopy with ∼0.25 cm−1 resolution did not resolve ro-tational lines of (NO)2 in helium droplets, indicating the rotational constants inhelium droplets are smaller than in the gas phase. The rotational constant renor-malization in helium droplets varies significantly between molecules, so a preciseprediction is unreliable [97]. In helium droplets, this specific cis isomer is expectedto be more prevalent than other isomers, based on the infrared spectroscopy in [70].10.1.2 Expected ResultsPrevious alignment work of Van der Waals complexes in helium droplets has ef-fectively modelled the polarizability of the complex as the vector sum of the po-larizability of the base molecules, assuming negligible interaction between them[136, 154]. This tends to underestimate components parallel to the weak inter-133cis αˆi j j = x j = y j = zi = x 5.627 0 0i = y 0 3.011 0i = z 0 0 1.415trans αˆi j j = x j = y j = zi = x 2.343 0.394 0i = y 0.394 6.954 0i = z 0 0 1.436Table 10.2: The polarizability tensors αˆi j (in A˚3) of (NO)2 isomers, takenfrom NIST CCCBDB with the B2PLYP=FULLultrafine method usingthe cc-pVDZ basis set.molecular bond [75]. For the cis and trans NO dimers, this estimate results inan anisotropic polarizability of about 1.7 A˚3, as the two NO molecules are near-parallel in each configuration. The cis isomer is covalently bonded, so the contri-bution along the bond to the polarizability tensor is likely significant, though thebinding energy is far weaker than typical covalent bonds [43, 194]. Calculatedpolarizability tensors are shown in Table 10.2, where one can see that the highestpolarizability axis is along the NO-NO intermolecular bond.Assuming the calculated polarizability, the cis NO dimer would align with theN-N axis along the field polarization vector, as shown in Figure 10.2 (a). UponCoulomb explosion, the NO+ fragments would be ejected along the polarizationaxis of the alignment field, as depicted in Figure 10.2 (a). Therefore, the ion dis-tribution would be similar to a VMI image of an aligned diatomic molecule (as wasshown in Figure 3.5).For the aligned trans dimer, the NO+ fragments will also be emitted alongthe polarization axis of the alignment field. The intermolecular axis of the transdimer is not parallel to the most polarizable axis, so as the dimer rotates aboutthe alignment axis the ions will be emitted in a “cone” about the polarization axis,as depicted in Figure 10.2 (b). This means that for the same degree of molecularalignment, the angular spread of the ion distribution will be slightly more broadfor the trans dimer, relative to the cis dimer. In both cases (cis and trans), theexact angular spread will depend on the structure and magnitude of the dimer’spolarizability tensor, as well as the alignment field intensity.Molecular bonds are not rigid. Experiments with CS2 dimers demonstratedthat the ∼90◦ angle between the two CS2 molecules would bend around 5◦ whenaligned to a strong adiabatic field [138]. Robust theoretical calculations of molec-134Figure 10.2: (a) cis and (b) trans (NO)2 isomers aligned with their most po-larizable axis along the field polarization vector (black arrow). Nitro-gen atoms are shown in blue, with oxygen atoms in red. The magentaarrow shows the direction between the centre of mass of the complexand that of each NO fragment, thus representing the axis of Coulombexplosion. Free rotation about the alignment axis will maintain a ∼0◦angle between the Coulomb explosion and alignment axis in the 2Dprojection for the cis isomer (left), but will result in a more broad dis-tribution of angles for the trans isomer (right), as illustrated with thecoloured cone. The misalignment between the most polarizable axisand the Coulomb explosion axis of the trans isomer has been exagger-ated for illustrative purposes.ular structure are outside the scope of this work, but comparing the cis-to-trans iso-merization energy barrier to the well depth of the angular potential of the moleculein the laser field provides a rough upper-bound estimate of the torsion. Taking thewell depth from Equation 2.20, a field strength of 5×1011 W/cm2 provides a welldepth of ∼0.001 eV for an anisotropic polarizability of 1.7 A˚3, two orders of mag-nitude below the cis-to-trans isomerization barrier. Therefore, it is reasonable toassume that the cis and trans isomers will maintain their approximate structures inthe presence of an intense field.13510.2 ResultsOur preliminary experimental results indicate successful alignment, with rich rota-tional dynamics, of NO dimers in helium droplets.10.2.1 Adiabatic Alignment of (NO)2 in a Linearly Polarized LaserFieldHelium droplets were produced at 30 Bar / 16 K, with ⟨NHe⟩ ≈ 3000 atoms perdroplet, then doped with NO to produce dimers. Appendix A details the productionand identification of dimers.The degree of alignment of (NO)2 during a linearly polarized adiabatic align-ment pulse is plotted in Figure 10.3. ⟨cos2 θ2D⟩ slowly increases during the pulse,and then decreases shortly after. The ion distribution indicates that molecules wereconfined to the axis of laser polarization, which we interpret as both isomers align-ing along the N-N axis, consistent with the calculated polarizability tensors shownin Table 10.2. Figure 10.4 shows the maximum degree of alignment for two lowerpulse energies, with ⟨cos2 θ2D⟩ rapidly approaching the baseline value of 0.5 asthe pulse is weakened. A strong dependence on the intensity is expected for one-dimensional laser field alignment [161]. The alignment pulse is not ionizing inthis experiment, and so it is expected a much higher degree of alignment could beachieved with tighter focusing.The angular covariance map shown in Figure 10.5 (b), recorded with the align-ment pulse, demonstrates that mass 30 fragments are indeed coming in pairs whichare aligned to the field. This means that the signal in Figure 10.3 cannot be due toNO2 or N2O contamination. The parallel lines shown in Figure 10.5 (a), recordedwithout the alignment pulse, is the expected result for randomly oriented moleculessubject to two-body Coulomb explosion, as described in Section 3.2. Appendix Cshows simulated angular covariance maps for NO dimers and trimers of variousgeometry, for comparison.136Figure 10.3: ⟨cos2 θ2D⟩ calculated from NO+ fragments from Coulomb ex-plosion of the NO dimer during a linearly polarized adiabatic pulsewith a peak intensity of ∼1× 1012 W/cm2. The shaded area repre-sents the approximate 170 ps pulse profile. The ∼200 fs probe pulsewith a peak intensity ∼1.5× 1015 W/cm2 was scanned in time, withrespect to the alignment pulse. The beam diameters were∼4.2 mm and∼19 mm respectively prior to being focused with an f = 30cm lens.The baseline value is above 0.5 because of an anisotropic backgroundsignal, shifted relative to the ⟨cos2 θ2D⟩ calculations by the velocity ofthe helium droplet beam, and some defects on the VMI detector.Figure 10.4: The maximum degree of alignment (⟨cos2 θ2D⟩) for three differ-ent alignment pulse intensities. The baselines were subtracted with anisotropic value of ⟨cos2 θ2D⟩= 0.5 added, so scans from multiple dayscould more easily be compared.137Figure 10.5: (a) An angular covariance map recorded with randomly orientedmolecules in helium droplets, measured from NO+ ion fragments. Thestrong covariance between θ0+ (0◦, 180◦) indicates that the ions areemitted in pairs. (b) an angular covariance map recorded at the maxi-mum of a∼170 ps pulse with a peak intensity of 1×1012 W/cm2. Thealignment pulse is linearly polarized along θ = 0◦, suggesting that ionfragments are emitted along the most polarizable axis of the molecule.13810.2.2 Dynamics of (NO)2 after KickingThe experiments in Section 10.2.1 demonstrate that the helium droplet dopant canbe aligned and observed through Coulomb explosion imaging. With the same he-lium droplet dimer doping conditions, kick experiments were performed to studyrotational dynamics. A ∼400 fs kick pulse with a fluence of ∼10 J/cm2 irradiatedthe doped droplet beam. The arrival time of the 200 fs probe pulse with a peakintensity of ∼1.5× 1015 W/cm2 was scanned using a translation stage. Figure10.6 shows the observed rotational dynamics, which last for at least 600 ps. Evi-dently, the alignment decays substantially over the first hundred picoseconds. Thisis likely due to a combination of rotational decoherence from energy (and angularmomentum) transfer to the surrounding helium, and substantial centrifugal distor-tion, which results in strong interference from many similar rotational frequencies.Separating these effects requires further investigation.Figure 10.6: The degree of alignment of (NO)2 in 30 Bar / 18 K heliumdroplets after a ∼400 fs kick with a fluence of ∼10 J/cm2. The probehad a peak intensity of ∼1.5× 1015 W/cm2. This is the average overten scans recorded across multiple days, with the scans renormalizedto have the same mean ⟨cos2 θ2D⟩ and amplitude of the first peakaround 12 ps. A moving average filter with 6 ps width was also used tosmoothen each individual scan.139Figure 10.7: The power spectrum of ⟨cos2 θ2D⟩ of (NO)2 after a kick in he-lium droplets. The magenta trace is the Fourier Transform of all delaysstarting 2 ps after the kick, with the blue and red traces starting at 32 psand 62 ps, respectively. All three are plotted with the same parameters,so the power spectra can be compared directly. These are calculatedfrom the data plotted in Figure 10.6.Figure 10.7 shows the Fourier transform of this measurement, shown both in-cluding points immediately after the kick, and starting 32 ps and 62 ps later. Thetime dependence of the Fourier spectrum may be indicative of different decoher-ence rates for different rotational states, but will require further investigation. Themain peak around 26 GHz decays rapidly. In all previous experiments, decoherenceoccurs more quickly as the molecule’s rotational energy approaches the energy ofliquid helium’s Roton excitations (∼185 GHz). This has resulted in a maximumrotational frequency around 20−30GHz for (NO)2, CS2, I2, and OCS [33].To obtain a measurement of the rotational constants, it would be necessary torecord higher quality scans with longer acquisition times out to later probe delays,to enhance the power spectra resolution. In previous work, measurements withdifferent kick fluences were used to map out possible transition frequencies [33].With the weakest fluence, only the lowest frequency is present. As the fluenceincreases and more states are populated, overlapping spectral lines are not easilyidentified. A fitting technique for a series of measurements with different fluenceswas provided in the supplemental material of [33].14010.3 DiscussionOur experiments show the alignment of NO dimers along the polarization vector,confirming the calculated geometry of the polarizability tensor in Table 10.2. Tomy knowledge, this is the first experimental verification that there is significantlygreater polarizability along the NO-NO intermolecular bond than along each indi-vidual N-O axis.The gas phase cis isomer is similar to a prolate symmetric top, with Ray’sasymmetry parameter κ = −0.88. For a prolate symmetric top (Equation 2.9),rotation about the secondary axis produces an energy level splitting of 2J+1. Evenat 0.37 K, this quickly results in a large number of populated states as the molecularmoments of inertia increase, providing a convoluted rotational spectrum.The lowest possible rotational frequency corresponds to the (J = 0,K = 0)−→(J = 2,K = 0) transition, and has a frequency of 6Byz. The observation of a fre-quency around 10−14GHz in the power spectra recorded at later delays in Figure10.7 would then give an upper bound on the possible rotational constants. Thislimits Byz < 2.3GHz (0.08 cm−1). This is close to one third of those rotationalconstants in the gas phase (Table 10.1), which agrees with a similar shift to thatof other “heavy” rotors of a factor of 2-6. This is comparable to the rotationalconstant of OCS in helium droplets (0.073 cm−1), and greater than that of CS2(0.024 cm−1).With a rotational constant of B = 0.08cm−1 in helium droplets, the empiricalscaling relation from Equation 1.1 gives an effective centrifugal distortion constant(due to the to the effective “spreading out” of the nonsuperfluid helium solvationshell) of D = 3× 10−4 cm−1. This would lead to a rotational spectrum where thefrequencies begin to decrease with increasing J after just a few dozen gigahertz, asplotted in Figure 10.8. With a rotational spectrum of this character, rotational co-herence spectroscopy results in several rotational frequencies piling up into a singlebroad line [33], similar to the ∼26 GHz line in Figure 10.7. Interestingly, for thethree other molecules studied, I2, CS2, and OCS, this peak occurs at the similarfrequencies of 19 GHz, 22 GHz, and 32 GHz, respectively [32]. (NO)2 fits in withthis trend, where larger rotational constants have this peak occurring at higher fre-quencies. Rotational energies above ∼20 GHz having shorter coherence lifetimes141Figure 10.8: The rotational energy transition structure using Equation 2.7with B = 0.08cm−1 and D = 3×10−4 cm−1. Both even and odd statesare included as the symmetry of the molecule in question is not known.is consistent with results from infrared rotational spectroscopy of molecules in he-lium droplets [35], due to energy/angular momentum transfer to the surroundinghelium [33].A larger rotational constant reduces the number of rotational states below theroton energy, and would reduce the complexity of selective excitation using a dou-ble kick, or pulse train, similar to the experiments in Chapter 7. With the renormal-ization of the rotational constants, at 0.37 K the NO dimer will have a significantportion of the population initially in the (J = 1,K = ±1) and (J = 2,K = ±1)states, resulting in many additional frequencies after excitation, even with a rela-tively large rotational constant. This means controlling the rotation of (NO)2 couldbe challenging, but at the same time selective rotational excitation could potentiallybe used to help isolate specific frequencies, to measure the rotational constants. Itwould be interesting to investigate how the degree of centrifugal distortion differsabout the different axes, as the potential surface is not circularly symmetric aboutthe X axis in Figure 10.1.Assuming the coupling to the helium for (NO)2 is similar to that of othermolecules studied, and that these new results from kicked dimers are indeed show-ing a substantial renormalization of the rotational constants, these results would142indicate that attempts to spin (NO)2 with an optical centrifuge in helium dropletswould likely encounter the same “centrifugal wall” as described for CS2 in Chapter8. As the frequencies begin to decrease around 40 GHz in Figure 10.8, an opticalcentrifuge with constantly increasing frequency would not be able to excite thehigher angular momentum states that occur with lower frequency.(NO)2 is an appealing molecule for further studies, for various reasons. It pro-duces fragments that can be emitted from the droplets with high ⟨cos2 θ2D⟩ whenaligned, while requiring a relatively small amount of additional angular momen-tum in order to exceed the roton energy. As NO dimers are not present in theeffusive background signal, the NO dimer also provides a background-free signal,whose effects can be isolated for using a coincidence filter if necessary (see Ap-pendix D.1). And for each molecule, two NO fragments are produced, improvingion detection efficiency relative to OCS or ICH3.143Chapter 11Outlook on ControllingMolecular Rotation inSuperfluidsThe results of Chapters 4 and 7 demonstrate that molecules in superfluid heliummay be controlled using similar methods conventionally applied to gas-phase en-sembles. There are some technical limitations to extending the exact detectionscheme used here to create superrotors in bulk liquid helium, such as a lower break-down threshold and correspondingly lower available excitation strengths, whichlimited the excitation pulse energy. It’s possible that by using pulse shaping tech-niques, one may be able to further excite helium excimers. One possible methodwould be a spectrally shaped optical centrifuge which consisted of only the rota-tional transition frequencies of He2*, with all other spectral components which donot contribute to rotational excitation missing. The long-lasting free rotation andrelatively high molecular density offer an exciting opportunity to study molecularsuperrotors in a liquid, and the superfluid nature of liquid helium offers the possi-bility of controlling collective angular momentum states such as quantized vortices[179].Our preliminary results on the selective suppression of individual rotationalstates of CS2 in helium droplets offers the possibility to measure the decoherencerates in a state-specific way, and reaffirms that the created rotational wavepackets144exhibit the same quantum interference seen in the gas phase. Applying the samemechanism to the nitric oxide dimer might prove interesting, as preliminary resultsindicate a higher rotational constant, thereby reducing the number of rotationalfrequencies below the roton energy.The experiments presented in Chapter 9 with nitrogen are a loose end, and de-serve more consideration. The results of references [6] and [143] suggest that sig-nificantly smaller helium droplets should be utilized, even for angle-dependent ion-ization measurements, and that higher doped-droplet number densities in the inter-action region, produced at larger droplet sizes with higher backing pressures, maynot outweigh the greater detection efficiencies that occur with smaller droplets. Ifthe result that N2 is not undergoing sustained rotation after a single kick is con-firmed, a pulse train tailored to push past the first few rotational states may be ableto drive the molecule into a long-lived state with higher rotational energy, such aswhat was shown for D2 and He2*.The possibility of using REMPI to observe rotational excitation in helium dropletsis appealing. However, the droplet environment shifts absorption spectra [152], andas is briefly discussed in appendix C, the possibility of excited states strongly inter-acting with the droplet and ejecting the molecule must be considered. N2 and O2may be suitable, though their small absorption cross sections may result in weaksignals. We were not successful in detecting a REMPI signal from O2 in heliumdroplets in our laboratory, within the detection capabilities of our first-generationhelium droplet system [56].11.0.1 Further Excitation of Molecules in Helium DropletsAs was discussed in Chapter 8, rotational excitation of molecules embedded inhelium droplets is challenging. For heavy molecules, it may be possible to use anoptical centrifuge to control rotation well below the roton energy. This could beachieved either by using significantly slower chirp rates, or by increasing the fieldintensity. The approach might enable the study of long-lived field-free alignmentin the droplets, but only with rotational frequencies well below the roton energy of∼185 GHz.To break through the “centrifugal wall”, where subsequent angular momentum145states occur with decreasing rotational frequency, a more specific rotational controlscheme is necessary. The goal would be to break through the wall, and then excitethe molecule to rotate well beyond the rotational frequency at which the centrifugalforce prevents helium from attaching to the molecule. A shaped optical centrifugewhose spectro-temporal profile followed the sequence of transitions for ∆J = +2,while avoiding any potential ∆J =−2 transitions during the slow-down, could po-tentially be used to break through the centrifugal wall. As the angular momentumtransfer to the surrounding helium happens on the picosecond scale, rotational ex-citation would likely have to occur on a similar timescale. This would require amuch “faster” optical centrifuge. To satisfy the spinnability criterion, this wouldrequire correspondingly higher field intensity. To increase the field intensity whileavoiding ionization, it may be possible to attenuate spectral components which donot contribute to rotational excitation, as suggested for the rotational excitation ofHe2* in bulk liquid helium. Our group has previously shown that shaped opticalcentrifuge pulses can efficiently excite molecules, even with portions of the opticalspectrum attenuated [2].The spectro-temporal profile of the shaped optical centrifuge would have to bedesigned for rotational excitation of one specific molecule, but in principle could beutilized with many molecules. Nitrogen and oxygen would require the least amountof angular momentum transfer from the field, due to their high rotational constants,but the stronger observables from CS2, OCS, and (NO)2 are much easier to workwith. A smaller rotational constant also results in a more convoluted frequencyspectrum, which is a relevant consideration for creating a spectrally shaped opticalcentrifuge better matched to excite a non-rigid rotor.146Bibliography[1] A. S. Alnaser, S. Voss, X. M. Tong, C. M. Maharjan, P. Ranitovic,B. Ulrich, T. Osipov, B. Shan, Z. Chang, and C. L. Cocke. Effects ofmolecular structure on ion disintegration patterns in ionization of O2 andN2 by short laser pulses. Phys. Rev. Lett., 93:113003, Sep 2004.doi:10.1103/PhysRevLett.93.113003. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.93.113003. → page 94[2] P. Amani, A. A. Milner, and V. Milner. Selective rotational control inmixtures of molecular super-rotors. The Journal of Chemical Physics, 155(12):124201, 09 2021. ISSN 0021-9606. doi:10.1063/5.0062051. URLhttps://doi.org/10.1063/5.0062051. → pages 109, 146[3] F. Ancilotto, P. B. Lerner, and M. W. Cole. Physics of solvation. Journal ofLow Temperature Physics, 101(5):1123–1146, 1995.doi:10.1007/BF00754527. URL https://doi.org/10.1007/BF00754527. →page 70[4] K. H. Andersen, J. Bossy, J. C. Cook, O. G. Randl, and J.-L. Ragazzoni.High-resolution measurements of rotons in 4He. Phys. Rev. Lett., 77:4043–4045, Nov 1996. doi:10.1103/PhysRevLett.77.4043. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.77.4043. → page 47[5] T. Armon and L. Friedland. Quantum versus classical dynamics in theoptical centrifuge. Phys. Rev. A, 96:033411, Sep 2017.doi:10.1103/PhysRevA.96.033411. URLhttps://link.aps.org/doi/10.1103/PhysRevA.96.033411. → pages 19, 105[6] J. D. Asmussen, L. Ben Ltaief, K. Sishodia, A. R. Abid, B. Bastian,S. Krishnan, H. B. Pedersen, and M. Mudrich. Dopant ionization andefficiency of ion and electron ejection from helium nanodroplets. TheJournal of Chemical Physics, 159(3):034301, 1/7/2024 2023.147doi:10.1063/5.0160171. URL https://doi.org/10.1063/5.0160171. → pages71, 145[7] J. D. Asmussen, K. Sishodia, B. Bastian, A. R. Abid, L. Ben Ltaief, H. B.Pedersen, S. De, C. Medina, N. Pal, R. Richter, T. Fennel, S. Krishnan, andM. Mudrich. Electron energy loss and angular asymmetry induced byelastic scattering in superfluid helium nanodroplets. Nanoscale, 15:14025–14031, 2023. doi:10.1039/D3NR03295G. URLhttp://dx.doi.org/10.1039/D3NR03295G. → page 71[8] D. Baek, H. Hasegawa, and Y. Ohshima. Unveiling the nonadiabaticrotational excitation process in a symmetric-top molecule induced by twointense laser pulses. The Journal of Chemical Physics, 134(22):224302,9/29/2023 2011. doi:10.1063/1.3598962. URLhttps://doi.org/10.1063/1.3598962. → page 24[9] A. V. Benderskii, R. Zadoyan, N. Schwentner, and V. A. Apkarian.Photodynamics in superfluid helium: Femtosecond laser-inducedionization, charge recombination, and preparation of molecular Rydbergstates. The Journal of Chemical Physics, 110(3):1542–1557, 12/15/20231999. doi:10.1063/1.477796. URL https://doi.org/10.1063/1.477796. →page 48[10] A. V. Benderskii, J. Eloranta, R. Zadoyan, and V. A. Apkarian. A directinterrogation of superfluidity on molecular scales. The Journal of ChemicalPhysics, 117(3):1201–1213, 12/22/2023 2002. doi:10.1063/1.1485955.URL https://doi.org/10.1063/1.1485955. → pages 47, 48, 50, 61[11] R. J. J. Beuhler, R. B. Bernstein, and K. H. Kramer. Observation of thereactive asymmetry of methyl iodide. crossed beam study of the reaction ofrubidium with oriented methyl iodide molecules. Journal of the AmericanChemical Society, 88(22):5331–5332, 11 1966. doi:10.1021/ja00974a059.URL https://doi.org/10.1021/ja00974a059. → page 2[12] J. Billingsley and A. B. Callear. Investigation of the 2050 A˚ system of thenitric oxide dimer. Trans. Faraday Soc., 67:589–597, 1971.doi:10.1039/TF9716700589. URLhttp://dx.doi.org/10.1039/TF9716700589. → page 133[13] C. Z. Bisgaard, O. J. Clarkin, G. Wu, A. M. D. Lee, O. Geßner, C. C.Hayden, and A. Stolow. Time-resolved molecular frame dynamics offixed-in-space CS2 molecules. Science, 323(5920):1464–1468, 2009.148doi:10.1126/science.1169183. URLhttps://www.science.org/doi/abs/10.1126/science.1169183. → pages37, 94[14] M. Bitter. Quantum coherent control of laser-kicked molecular rotors. PhDthesis, The University of British Columbia, Department of Physics andAstronomy, October 2016. → pages 23, 41, 97, 193[15] M. Bitter and V. Milner. Generating long sequences of high-intensityfemtosecond pulses. Appl. Opt., 55(4):830–833, Feb 2016.doi:10.1364/AO.55.000830. URLhttps://opg.optica.org/ao/abstract.cfm?URI=ao-55-4-830. → page 65[16] M. Bitter and V. Milner. Rotational excitation of molecules with longsequences of intense femtosecond pulses. Phys. Rev. A, 93:013420, Jan2016. doi:10.1103/PhysRevA.93.013420. URLhttps://link.aps.org/doi/10.1103/PhysRevA.93.013420. → page 97[17] C. Bloomquist, S. Zhdanovich, A. A. Milner, and V. Milner. Directionalspinning of molecules with sequences of femtosecond pulses. Phys. Rev. A,86:063413, Dec 2012. doi:10.1103/PhysRevA.86.063413. URLhttps://link.aps.org/doi/10.1103/PhysRevA.86.063413. → pages 26, 45, 94[18] N. Bonifaci, Z. Li, J. Eloranta, and S. L. Fiedler. Interaction of heliumrydberg state molecules with dense helium. The Journal of PhysicalChemistry A, 120(45):9019–9027, 11 2016. doi:10.1021/acs.jpca.6b08412.URL https://doi.org/10.1021/acs.jpca.6b08412. → page 62[19] N. B. Brauer, S. Smolarek, E. Loginov, D. Mateo, A. Hernando, M. Pi,M. Barranco, W. J. Buma, and M. Drabbels. Critical landau velocity inhelium nanodroplets. Phys. Rev. Lett., 111:153002, Oct 2013.doi:10.1103/PhysRevLett.111.153002. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.111.153002. → pages186, 188, 189[20] D. M. Brink and S. Stringari. Density of states and evaporation rate ofhelium clusters. Zeitschrift fu¨r Physik D Atoms, Molecules and Clusters, 15(3):257–263, 1990. doi:10.1007/BF01437187. URLhttps://doi.org/10.1007/BF01437187. → page 69[21] T. Brixner and G. Gerber. Femtosecond polarization pulse shaping. Opt.Lett., 26(8):557–559, Apr 2001. doi:10.1364/OL.26.000557. URLhttps://opg.optica.org/ol/abstract.cfm?URI=ol-26-8-557. → page 55149[22] D. W. Broege, R. N. Coffee, and P. H. Bucksbaum. Strong-field impulsivealignment in the presence of high temperatures and large centrifugaldistortion. Phys. Rev. A, 78:035401, Sep 2008.doi:10.1103/PhysRevA.78.035401. URLhttps://link.aps.org/doi/10.1103/PhysRevA.78.035401. → pages 24, 97[23] J. M. Brown and A. Carrington. Rotational Spectroscopy of DiatomicMolecules. Cambridge Molecular Science. Cambridge University Press,2003. doi:10.1017/CBO9780511814808. → page 13[24] H. Buchenau, E. L. Knuth, J. Northby, J. P. Toennies, and C. Winkler. Massspectra and time-of-flight distributions of helium cluster beams. TheJournal of Chemical Physics, 92(11):6875–6889, 12/20/2023 1990.doi:10.1063/1.458275. URL https://doi.org/10.1063/1.458275. → page 68[25] C. Callegari, K. K. Lehmann, R. Schmied, and G. Scoles. Heliumnanodroplet isolation rovibrational spectroscopy: Methods and recentresults. The Journal of Chemical Physics, 115(22):10090–10110,9/19/2023 2001. doi:10.1063/1.1418746. URLhttps://doi.org/10.1063/1.1418746. → pages 5, 67[26] J. F. Canty, E. G. Stone, S. B. H. Bach, and D. W. Ball. Matrix isolationand theoretical studies of onno: Assignment of a new combination bandand density functional calculations. Chemical Physics, 216(1):81–89,1997. doi:https://doi.org/10.1016/S0301-0104(96)00371-0. URLhttps://www.sciencedirect.com/science/article/pii/S0301010496003710.→ page 132[27] C. F. Chabalowski, J. O. Jensen, D. R. Yarkony, and I. Lengsfield, Byron H.Theoretical study of the radiative lifetime for the spin-forbidden transitiona3Σ+u −→ X1Σ+g in He2. The Journal of Chemical Physics, 90(4):2504–2512, 1/28/2024 1989. doi:10.1063/1.455993. URLhttps://doi.org/10.1063/1.455993. → page 47[28] A. S. Chatterley, E. T. Karamatskos, C. Schouder, L. Christiansen, A. V.Jørgensen, T. Mullins, J. Ku¨pper, and H. Stapelfeldt. Communication:Switched wave packets with spectrally truncated chirped pulses. TheJournal of Chemical Physics, 148(22):221105, 06 2018. ISSN 0021-9606.doi:10.1063/1.5028359. URL https://doi.org/10.1063/1.5028359. → pages23, 130[29] A. S. Chatterley, M. O. Baatrup, C. A. Schouder, and H. Stapelfeldt.Laser-induced alignment dynamics of gas phase CS2 dimers. Phys. Chem.150Chem. Phys., 22:3245–3253, 2020. doi:10.1039/C9CP06260B. URLhttp://dx.doi.org/10.1039/C9CP06260B. → pages 67, 191[30] A. S. Chatterley, L. Christiansen, C. A. Schouder, A. V. Jørgensen,B. Shepperson, I. N. Cherepanov, G. Bighin, R. E. Zillich, M. Lemeshko,and H. Stapelfeldt. Rotational coherence spectroscopy of molecules inhelium nanodroplets: Reconciling the time and the frequency domains.Phys. Rev. Lett., 125:013001, Jul 2020.doi:10.1103/PhysRevLett.125.013001. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.125.013001. → pages2, 9, 90, 91, 99, 104, 117[31] J.-X. Cheng and X. S. Xie. Coherent Raman Scattering Microscopy. CRCPress, Boca Raton, 1st edition, 2012. ISBN 9780429193675.doi:https://doi.org/10.1201/b12907. URL https://doi.org/10.1201/b12907.→ page 41[32] I. N. Cherepanov, G. Bighin, L. Christiansen, A. V. Jørgensen, R. Schmidt,H. Stapelfeldt, and M. Lemeshko. Far-from-equilibrium dynamics ofangular momentum in a quantum many-particle system, 2019. → pages92, 141[33] I. N. Cherepanov, G. Bighin, C. A. Schouder, A. S. Chatterley, S. H.Albrechtsen, A. V. n. Mun˜oz, L. Christiansen, H. Stapelfeldt, andM. Lemeshko. Excited rotational states of molecules in a superfluid. Phys.Rev. A, 104:L061303, Dec 2021. doi:10.1103/PhysRevA.104.L061303.URL https://link.aps.org/doi/10.1103/PhysRevA.104.L061303. → pages8, 90, 92, 93, 95, 96, 97, 104, 114, 117, 121, 140, 141, 142[34] I. N. Cherepanov, G. Bighin, C. A. Schouder, A. S. Chatterley,H. Stapelfeldt, and M. Lemeshko. A simple model for high rotationalexcitations of molecules in a superfluid. New Journal of Physics, 24(7):075004, aug 2022. doi:10.1088/1367-2630/ac8113. URLhttps://dx.doi.org/10.1088/1367-2630/ac8113. → pages8, 104, 105, 116, 118[35] M. Y. Choi, G. E. Douberly, T. M. Falconer, W. K. Lewis, C. M. Lindsay,J. M. Merritt, P. L. Stiles, and R. E. Miller. Infrared spectroscopy of heliumnanodroplets: novel methods for physics and chemistry. InternationalReviews in Physical Chemistry, 25(1-2):15–75, 01 2006.doi:10.1080/01442350600625092. URLhttps://doi.org/10.1080/01442350600625092. → pages 5, 10, 105, 142151[36] L. Christensen, L. Christiansen, B. Shepperson, and H. Stapelfeldt.Deconvoluting nonaxial recoil in Coulomb explosion measurements ofmolecular axis alignment. Phys. Rev. A, 94:023410, Aug 2016.doi:10.1103/PhysRevA.94.023410. URLhttps://link.aps.org/doi/10.1103/PhysRevA.94.023410. → pages 30, 71, 95[37] L. Christiansen. Laser Induced Alignment of Molecules Embedded inHelium Droplets: Exploration and Potential Applications. PhD thesis,Aarhus University, Department of Physics and Astronomy, July 2015. →page 80[38] R. M. Cleaver and C. M. Lindsay. Detailed design and transport propertiesof a helium droplet nozzle from 5 to 50 K. Cryogenics, 52(7):389–397,2012. doi:https://doi.org/10.1016/j.cryogenics.2012.03.004. URLhttps://www.sciencedirect.com/science/article/pii/S0011227512000562.→ page 67[39] A. Conjusteau. Spectroscopy in Helium Nanodroplets: Studying RelaxationMechanisms in Nature’s Most Fascinating Solvent. PhD thesis, PrincetonUniversity, Department of Chemistry, June 2002. → page 73[40] J. P. Cryan, P. H. Bucksbaum, and R. N. Coffee. Field-free alignment inrepetitively kicked nitrogen gas. Phys. Rev. A, 80:063412, Dec 2009.doi:10.1103/PhysRevA.80.063412. URLhttps://link.aps.org/doi/10.1103/PhysRevA.80.063412. → page 25[41] W. S. Dennis, E. Durbin, W. A. Fitzsimmons, O. Heybey, and G. K.Walters. Spectroscopic identification of excited atomic and molecularstates in electron-bombarded liquid helium. Phys. Rev. Lett., 23:1083–1086, Nov 1969. doi:10.1103/PhysRevLett.23.1083. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.23.1083. → page 47[42] C. M. Dion, A. Keller, O. Atabek, and A. D. Bandrauk. Laser-inducedalignment dynamics of HCN: Roles of the permanent dipole moment andthe polarizability. Phys. Rev. A, 59:1382–1391, Feb 1999.doi:10.1103/PhysRevA.59.1382. URLhttps://link.aps.org/doi/10.1103/PhysRevA.59.1382. → page 15[43] A. L. L. East. The 16 valence electronic states of nitric oxide dimer (NO)2.The Journal of Chemical Physics, 109(6):2185–2193, 12/13/2023 1998.doi:10.1063/1.476786. URL https://doi.org/10.1063/1.476786. → page134152[44] A. M. Ellis and S. Yang. Model for the charge-transfer probability inhelium nanodroplets following electron-impact ionization. Phys. Rev. A,76:032714, Sep 2007. doi:10.1103/PhysRevA.76.032714. URLhttps://link.aps.org/doi/10.1103/PhysRevA.76.032714. → page 88[45] J. Eloranta, N. Schwentner, and V. A. Apkarian. Structure and energetics ofHe2* bubble-states in superfluid 4he. The Journal of Chemical Physics,116(10):4039–4053, 9/17/2023 2002. doi:10.1063/1.1447884. URLhttps://doi.org/10.1063/1.1447884. → page 47[46] T. Elsaesser, S. Mukamel, M. M. Murnane, and N. F. Scherer, editors.Centrifugal Dissociation of a Molecule Using the Optical Centrifuge,Berlin, Heidelberg, 2001. Springer Berlin Heidelberg. ISBN978-3-642-56546-5. → page 106[47] V. B. Eltsov, S. N. Dzhosyuk, A. Y. Parshin, and I. A. Todoshchenko.Spectroscopic studies of the triplet-state 4He2* molecules in superfluidhelium up to the solidification pressure. Journal of Low TemperaturePhysics, 110(1):219–224, 1998. doi:10.1023/A:1022503926297. URLhttps://doi.org/10.1023/A:1022503926297. → page 47[48] A. J. B. Eppink and D. H. Parker. Velocity map imaging of ions andelectrons using electrostatic lenses: Application in photoelectron andphotofragment ion imaging of molecular oxygen. Review of ScientificInstruments, 68(9):3477–3484, 12/26/2023 1997. doi:10.1063/1.1148310.URL https://doi.org/10.1063/1.1148310. → page 30[49] U. Even. The Even-Lavie valve as a source for high intensity supersonicbeam. EPJ Techniques and Instrumentation, 2(1):17, 2015.doi:10.1140/epjti/s40485-015-0027-5. URLhttps://doi.org/10.1140/epjti/s40485-015-0027-5. → page 67[50] T. Faulkner, I. Miller, and P. L. Raston. Quantum cascade laserspectroscopy of OCS isotopologues in 4He nanodroplets: A test ofadiabatic following for a heavy rotor. The Journal of Chemical Physics,148(4):044308, 9/17/2023 2018. doi:10.1063/1.5009908. URLhttps://doi.org/10.1063/1.5009908. → page 7[51] S. Fleischer, I. S. Averbukh, and Y. Prior. Selective alignment of molecularspin isomers. Phys. Rev. Lett., 99:093002, Aug 2007.doi:10.1103/PhysRevLett.99.093002. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.99.093002. → page 25153[52] S. Fleischer, I. S. Averbukh, and Y. Prior. Selective control of molecularrotation. Journal of Physics B: Atomic, Molecular and Optical Physics, 41(7):074018, mar 2008. doi:10.1088/0953-4075/41/7/074018. URLhttps://dx.doi.org/10.1088/0953-4075/41/7/074018. → page 25[53] S. Fleischer, Y. Khodorkovsky, Y. Prior, and I. S. Averbukh. Controllingthe sense of molecular rotation. New Journal of Physics, 11(10):105039,2009. doi:10.1088/1367-2630/11/10/105039. URLhttps://dx.doi.org/10.1088/1367-2630/11/10/105039. → pages 2, 94[54] S. Fleischer, Y. Khodorkovsky, E. Gershnabel, Y. Prior, and I. S. Averbukh.Molecular alignment induced by ultrashort laser pulses and its impact onmolecular motion. Israel Journal of Chemistry, 52(5):414–437, 2012.doi:https://doi.org/10.1002/ijch.201100161. URLhttps://onlinelibrary.wiley.com/doi/abs/10.1002/ijch.201100161. → page 23[55] J. Floß, C. Boulet, J.-M. Hartmann, A. A. Milner, and V. Milner. Electronicspin polarization with an optical centrifuge. Phys. Rev. A, 98:043401, Oct2018. doi:10.1103/PhysRevA.98.043401. URLhttps://link.aps.org/doi/10.1103/PhysRevA.98.043401. → page 2[56] J. A. M. Fordyce. Towards rotational control of molecules in heliumnanodroplets. MSc thesis, The University of British Columbia, Departmentof Physics and Astronomy, June 2020. → pages 67, 74, 75, 107, 145, 198[57] D. M. Friedrich. Two-photon molecular spectroscopy. Journal of ChemicalEducation, 59(6):472, 06 1982. doi:10.1021/ed059p472. URLhttps://doi.org/10.1021/ed059p472. → page 13[58] Y. Gao, C. Wu, N. Xu, G. Zeng, H. Jiang, H. Yang, and Q. Gong.Manipulating molecular rotational wave packets with strong femtosecondlaser pulses. Phys. Rev. A, 77:043404, Apr 2008.doi:10.1103/PhysRevA.77.043404. URLhttps://link.aps.org/doi/10.1103/PhysRevA.77.043404. → page 26[59] J. Gary E. Douberly. Infrared Laser Spectroscopy of Dopants In and OnHelium Nanodroplets: Rotational and Vibrational Dynamics. PhD thesis,University of North Carolina at Chapel Hill, Department of Chemistry,2006. → page 105[60] O. Gessner and A. F. Vilesov. Imaging quantum vortices in superfluidhelium droplets. Annual Review of Physical Chemistry, 70(1):173–198, 071542019. doi:10.1146/annurev-physchem-042018-052744. URLhttps://doi.org/10.1146/annurev-physchem-042018-052744. → page 65[61] L. F. Gomez, E. Loginov, R. Sliter, and A. F. Vilesov. Sizes of large Hedroplets. The Journal of Chemical Physics, 135(15):154201, 12/20/20232011. doi:10.1063/1.3650235. URL https://doi.org/10.1063/1.3650235. →page 87[62] R. Gonza´lez-Luque, M. Mercha´n, and B. O. Roos. A theoreticaldetermination of the dissociation energy of the nitric oxide dimer.Theoretica chimica acta, 88(6):425–435, 1994. doi:10.1007/BF01113292.URL https://doi.org/10.1007/BF01113292. → page 132[63] J. P. Gordon. Radiation forces and momenta in dielectric media. Phys. Rev.A, 8:14–21, Jul 1973. doi:10.1103/PhysRevA.8.14. URLhttps://link.aps.org/doi/10.1103/PhysRevA.8.14. → page 16[64] S. Grebenev, J. P. Toennies, and A. F. Vilesov. Superfluidity within a smallhelium-4 cluster: The microscopic andronikashvili experiment. Science,279(5359):2083–2086, 1998. doi:10.1126/science.279.5359.2083. URLhttps://www.science.org/doi/abs/10.1126/science.279.5359.2083. →pages 5, 6, 7[65] R. E. Grisenti, W. Scho¨llkopf, J. P. Toennies, G. C. Hegerfeldt, T. Ko¨hler,and M. Stoll. Determination of the bond length and binding energy of thehelium dimer by diffraction from a transmission grating. Phys. Rev. Lett.,85:2284–2287, Sep 2000. doi:10.1103/PhysRevLett.85.2284. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.85.2284. → page 84[66] J. L. Hansen, L. Holmegaard, J. H. Nielsen, H. Stapelfeldt, D. Dimitrovski,and L. B. Madsen. Orientation-dependent ionization yields fromstrong-field ionization of fixed-in-space linear and asymmetric topmolecules. Journal of Physics B: Atomic, Molecular and Optical Physics,45(1):015101, dec 2011. doi:10.1088/0953-4075/45/1/015101. URLhttps://dx.doi.org/10.1088/0953-4075/45/1/015101. → page 37[67] M. Hartmann, F. Mielke, J. P. Toennies, A. F. Vilesov, and G. Benedek.Direct spectroscopic observation of elementary excitations in superfluid hedroplets. Phys. Rev. Lett., 76:4560–4563, Jun 1996.doi:10.1103/PhysRevLett.76.4560. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.76.4560. → pages 7, 9155[68] J. Higgins, C. Callegari, J. Reho, F. Stienkemeier, W. E. Ernst,M. Gutowski, and G. Scoles. Helium cluster isolation spectroscopy ofalkali dimers in the triplet manifold. The Journal of Physical Chemistry A,102(26):4952–4965, 06 1998. doi:10.1021/jp981115g. URLhttps://doi.org/10.1021/jp981115g. → page 70[69] M. Z. Hoque, M. Lapert, E. Hertz, F. Billard, D. Sugny, B. Lavorel, andO. Faucher. Observation of laser-induced field-free permanent planaralignment of molecules. Phys. Rev. A, 84:013409, Jul 2011.doi:10.1103/PhysRevA.84.013409. URLhttps://link.aps.org/doi/10.1103/PhysRevA.84.013409. → page 2[70] H. Hoshina, M. Slipchenko, K. Prozument, D. Verma, M. W. Schmidt,J. Ivanic, and A. F. Vilesov. Infrared spectroscopy and structure of (NO)nclusters. The Journal of Physical Chemistry A, 120(4):527–534, 2016.doi:10.1021/acs.jpca.5b10228. URLhttps://doi.org/10.1021/acs.jpca.5b10228. PMID: 26756475. → pages133, 179, 181, 186[71] Y. Huang, G. Liu, and S. Xu. Rotational dephasing of the molecularalignment by centrifugal distortion [invited]. Chinese Optics Letters, 20(11):100005, 10 2022. URLhttps://www.researching.cn/articles/OJ94b292b6bc90e424. → page 24[72] Y. Huang, H. Chen, G. Liu, and S. Xu. Eliminating molecular-alignmentdephasing with a phase-modulated femtosecond laser pulse. Phys. Rev. A,107:053108, May 2023. doi:10.1103/PhysRevA.107.053108. URLhttps://link.aps.org/doi/10.1103/PhysRevA.107.053108. → page 24[73] W. M. Huo, K. P. Gross, and R. L. McKenzie. Optical Stark effect in thetwo-photon spectrum of NO. Phys. Rev. Lett., 54:1012–1015, Mar 1985.doi:10.1103/PhysRevLett.54.1012. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.54.1012. → pages 184, 189[74] J. Ivanic, M. W. Schmidt, and B. Luke. High-level theoretical study of theNO dimer and tetramer: Has the tetramer been observed? The Journal ofChemical Physics, 137(21):214316, 9/29/2023 2012.doi:10.1063/1.4769226. URL https://doi.org/10.1063/1.4769226. → page132[75] L. Jensen, P.-O. A˚strand, A. Osted, J. Kongsted, and K. V. Mikkelsen.Polarizability of molecular clusters as calculated by a dipole interaction156model. The Journal of Chemical Physics, 116(10):4001–4010, 11/12/20232002. doi:10.1063/1.1433747. URL https://doi.org/10.1063/1.1433747. →page 134[76] P. W. Joireman, L. L. Connell, S. M. Ohline, and P. M. Felker.Characterization of asymmetry transients in rotational coherencespectroscopy. The Journal of Chemical Physics, 96(6):4118–4130,9/29/2023 1992. doi:10.1063/1.461867. URLhttps://doi.org/10.1063/1.461867. → page 25[77] T. M. Jones. A helium nanodroplet instrument for infrared spectroscopy ofmolecular ions. MSc thesis, The University of Georgia, Department ofChemistry, 2021. → page 79[78] A. V. Jørgensen. Building an Optical Centrifuge: Rotation of IsolatedMolecules and Molecules in Helium Nanodroplets. PhD thesis, AarhusUniversity, Department of Physics and Astronomy, August 2018. → pages30, 103, 106, 107, 116, 119[79] J. Karczmarek, J. Wright, P. Corkum, and M. Ivanov. Optical centrifuge formolecules. Phys. Rev. Lett., 82:3420–3423, Apr 1999.doi:10.1103/PhysRevLett.82.3420. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.82.3420. → pages 19, 118[80] J. W. Keto, F. J. Soley, M. Stockton, and W. A. Fitzsimmons. Dynamicproperties of neutral excitations produced in electron-bombarded superfluidhelium. I. the He(23S) and He2(a3Σ) atomic and molecular metastablestates. Phys. Rev. A, 10:872–886, Sep 1974.doi:10.1103/PhysRevA.10.872. URLhttps://link.aps.org/doi/10.1103/PhysRevA.10.872. → pages 47, 61[81] Y. Khodorkovsky, U. Steinitz, J.-M. Hartmann, and I. S. Averbukh.Collisional dynamics in a gas of molecular super-rotors. NatureCommunications, 6(1):7791, 2015. doi:10.1038/ncomms8791. URLhttps://doi.org/10.1038/ncomms8791. → page 2[82] K. Kitano, H. Hasegawa, and Y. Ohshima. Ultrafast angular momentumorientation by linearly polarized laser fields. Phys. Rev. Lett., 103:223002,Nov 2009. doi:10.1103/PhysRevLett.103.223002. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.103.223002. → page 45157[83] A. Korobenko. Control of molecular rotation with an optical centrifuge.PhD thesis, The University of British Columbia, Department of Physicsand Astronomy, July 2016. → pages 74, 198[84] A. Korobenko, A. A. Milner, J. W. Hepburn, and V. Milner. Rotationalspectroscopy with an optical centrifuge. Phys. Chem. Chem. Phys., 16:4071–4076, 2014. doi:10.1039/C3CP54598A. URLhttp://dx.doi.org/10.1039/C3CP54598A. → pages 2, 45[85] L. Kranabetter, H. H. Kristensen, A. Ghazaryan, C. A. Schouder, A. S.Chatterley, P. Janssen, F. Jensen, R. E. Zillich, M. Lemeshko, andH. Stapelfeldt. Nonadiabatic laser-induced alignment dynamics ofmolecules on a surface. Phys. Rev. Lett., 131:053201, Aug 2023.doi:10.1103/PhysRevLett.131.053201. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.131.053201. → pages 24, 72[86] P. Krause and H. B. Schlegel. Angle-dependent ionization of smallmolecules by time-dependent configuration interaction and an absorbingpotential. The Journal of Physical Chemistry Letters, 6(11):2140–2146, 062015. doi:10.1021/acs.jpclett.5b00929. URLhttps://doi.org/10.1021/acs.jpclett.5b00929. → page 37[87] R. V. Krems. Molecular Rotations Under Control, chapter 5, pages136–140. John Wiley & Sons, Ltd, 2018. ISBN 9781119382638.doi:https://onlinelibrary.wiley.com/doi/abs/10.1002/9781119382638.ch5.URL https://onlinelibrary.wiley.com/doi/abs/10.1002/9781119382638.ch5.→ page 19[88] H. H. Kristensen, L. Kranabetter, C. A. Schouder, J. Arlt, F. Jensen, andH. Stapelfeldt. Laser-induced Coulomb explosion imaging of alkali-metaldimers on helium nanodroplets. Phys. Rev. A, 107:023104, Feb 2023.doi:10.1103/PhysRevA.107.023104. URLhttps://link.aps.org/doi/10.1103/PhysRevA.107.023104. → page 72[89] K. F. Lee, I. V. Litvinyuka, P. W. Dooley, M. Spanner, D. M. Villeneuve,and P. B. Corkum. Two-pulse alignment of molecules. Journal of PhysicsB: Atomic, Molecular and Optical Physics, 37(3):L43, 2004.doi:10.1088/0953-4075/37/3/L02. URLhttps://dx.doi.org/10.1088/0953-4075/37/3/L02. → page 25[90] K. F. Lee, E. A. Shapiro, D. M. Villeneuve, and P. B. Corkum. Coherentcreation and annihilation of rotational wave packets in incoherent158ensembles. Phys. Rev. A, 73:033403, Mar 2006.doi:10.1103/PhysRevA.73.033403. URLhttps://link.aps.org/doi/10.1103/PhysRevA.73.033403. → pages 25, 26[91] K. F. Lee, D. M. Villeneuve, P. B. Corkum, A. Stolow, and J. G.Underwood. Field-free three-dimensional alignment of polyatomicmolecules. Phys. Rev. Lett., 97:173001, Oct 2006.doi:10.1103/PhysRevLett.97.173001. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.97.173001. → page 2[92] K. K. Lehmann. Rotation in liquid He4 lessons from a highly simplifiedmodel. The Journal of Chemical Physics, 114(10):4643–4648, 1/5/20242001. doi:10.1063/1.1334620. URL https://doi.org/10.1063/1.1334620. →page 106[93] M. Leibscher, I. S. Averbukh, and H. Rabitz. Molecular alignment by trainsof short laser pulses. Phys. Rev. Lett., 90:213001, May 2003.doi:10.1103/PhysRevLett.90.213001. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.90.213001. → page 25[94] M. Leibscher, I. S. Averbukh, and H. Rabitz. Enhanced molecularalignment by short laser pulses. Phys. Rev. A, 69:013402, Jan 2004.doi:10.1103/PhysRevA.69.013402. URLhttps://link.aps.org/doi/10.1103/PhysRevA.69.013402. → page 92[95] C. Leichtle, I. S. Averbukh, and W. P. Schleich. Generic structure ofmultilevel quantum beats. Phys. Rev. Lett., 77:3999–4002, Nov 1996.doi:10.1103/PhysRevLett.77.3999. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.77.3999. → page 24[96] C. Leichtle, I. S. Averbukh, and W. P. Schleich. Multilevel quantum beats:An analytical approach. Phys. Rev. A, 54:5299–5312, Dec 1996.doi:10.1103/PhysRevA.54.5299. URLhttps://link.aps.org/doi/10.1103/PhysRevA.54.5299. → page 24[97] M. Lemeshko. Quasiparticle approach to molecules interacting withquantum solvents. Phys. Rev. Lett., 118:095301, Feb 2017.doi:10.1103/PhysRevLett.118.095301. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.118.095301. → pages9, 120, 128, 133[98] M. Lewerenz, B. Schilling, and J. Toennies. A new scattering deflectionmethod for determining and selecting the sizes of large liquid clusters of1594He. Chemical Physics Letters, 206(1):381–387, 1993. ISSN 0009-2614.doi:https://doi.org/10.1016/0009-2614(93)85569-A. URLhttps://www.sciencedirect.com/science/article/pii/000926149385569A. →page 67[99] W. K. Lewis, C. M. Lindsay, R. J. Bemish, and R. E. Miller. Probingcharge-transfer processes in helium nanodroplets by optically selectedmass spectrometry (osms): Charge steering by long-range interactions.Journal of the American Chemical Society, 127(19):7235–7242, 05 2005.doi:10.1021/ja042489s. URL https://doi.org/10.1021/ja042489s. → page88[100] W. Lichten, M. V. McCusker, and T. L. Vierima. Fine structure of themetastable a3Σ+u state of the helium molecule. The Journal of ChemicalPhysics, 61(6):2200–2212, 5/22/2024 1974. doi:10.1063/1.1682292. URLhttps://doi.org/10.1063/1.1682292. → page 48[101] S. Linden, H. Giessen, and J. Kuhl. XFROG — A new method foramplitude and phase characterization of weak ultrashort pulses. physicastatus solidi (b), 206(1):119–124, 01 1998.doi:https://doi.org/10.1002/(SICI)1521-3951(199803)206:1⟨119::AID-PSSB119⟩3.0.CO;2-X. URLhttps://doi.org/10.1002/(SICI)1521-3951(199803)206:1⟨119::AID-PSSB119⟩3.0.CO;2-X. → page 207[102] S. Linden, J. Kuhl, and H. Giessen. Amplitude and phase characterizationof weak blue ultrashort pulses by downconversion. Opt. Lett., 24(8):569–571, Apr 1999. doi:10.1364/OL.24.000569. URLhttps://opg.optica.org/ol/abstract.cfm?URI=ol-24-8-569. → page 207[103] R. Ma, C. Wu, N. Xu, J. Huang, H. Yang, and Q. Gong. Geometricalignment of CH3I in an intense femtosecond laser field. Chemical PhysicsLetters, 415(1):58–63, 2005.doi:https://doi.org/10.1016/j.cplett.2005.08.070. URLhttps://www.sciencedirect.com/science/article/pii/S0009261405011905.→ page 188[104] I. MacPhail-Bartley, W. W. Wasserman, A. A. Milner, and V. Milner. Lasercontrol of molecular rotation: Expanding the utility of an optical centrifuge.Review of Scientific Instruments, 91(4):045122, 04 2020. ISSN 0034-6748.doi:10.1063/1.5140358. URL https://doi.org/10.1063/1.5140358. → pagesvi, 2, 18, 19, 44, 107, 116, 117, 208160[105] S. Marinakis, C. Cockrell, K. Trachenko, T. F. Headen, and A. K. Soper.Microscopic structure of liquid nitric oxide. The Journal of PhysicalChemistry B, 126(47):9860–9870, 12 2022. doi:10.1021/acs.jpcb.2c05384.URL https://doi.org/10.1021/acs.jpcb.2c05384. → page 133[106] A. R. W. McKellar, Y. Xu, and W. Jaeger. Spectroscopic exploration ofatomic scale superfluidity in doped helium nanoclusters. Phys. Rev. Lett.,97:183401, Nov 2006. doi:10.1103/PhysRevLett.97.183401. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.97.183401. → pages 7, 69[107] A. R. W. McKellar, Y. Xu, and W. Jaeger. Spectroscopic studies ofOCS-doped 4He clusters with 9-72 helium atoms: Observation of broadoscillations in the rotational moment of inertia. The Journal of PhysicalChemistry A, 111(31):7329–7337, 08 2007. doi:10.1021/jp070618w. URLhttps://doi.org/10.1021/jp070618w. → page 7[108] D. N. McKinsey, C. R. Brome, J. S. Butterworth, S. N. Dzhosyuk, P. R.Huffman, C. E. H. Mattoni, J. M. Doyle, R. Golub, and K. Habicht.Radiative decay of the metastable He2(a3Σ+u ) molecule in liquid helium.Phys. Rev. A, 59:200–204, Jan 1999. doi:10.1103/PhysRevA.59.200. URLhttps://link.aps.org/doi/10.1103/PhysRevA.59.200. → page 47[109] D. N. McKinsey, C. R. Brome, S. N. Dzhosyuk, R. Golub, K. Habicht, P. R.Huffman, E. Korobkina, S. K. Lamoreaux, C. E. H. Mattoni, A. K.Thompson, L. Yang, and J. M. Doyle. Time dependence of liquid-heliumfluorescence. Phys. Rev. A, 67:062716, Jun 2003.doi:10.1103/PhysRevA.67.062716. URLhttps://link.aps.org/doi/10.1103/PhysRevA.67.062716. → page 47[110] I. R. McNab. Rotational Spectroscopy, Theory, pages 978–987. AcademicPress, Oxford, 2017. ISBN 978-0-12-803224-4.doi:https://doi.org/10.1016/B978-0-12-803224-4.00273-9. URL https://www.sciencedirect.com/science/article/pii/B9780128032244002739. →page 12[111] C. Medina, A. Ø. Lægdsmand, L. B. Ltaief, Z. Hoque, A. H. Roos,L. Jurkovicˇova´, O. Hort, O. Finke, M. Albrecht, J. Nejdl, F. Stienkemeier,J. Andreasson, E. Klimesˇova´, M. Krikunova, A. Heidenreich, andM. Mudrich. Long-lasting xuv activation of helium nanodroplets foravalanche ionization. New Journal of Physics, 25(5):053030, may 2023.doi:10.1088/1367-2630/acd5f6. URLhttps://dx.doi.org/10.1088/1367-2630/acd5f6. → page 73161[112] T. P. Melia. Decomposition of nitric oxide at elevated pressures. Journal ofInorganic and Nuclear Chemistry, 27(1):95–98, 1965.doi:https://doi.org/10.1016/0022-1902(65)80196-8. URLhttps://www.sciencedirect.com/science/article/pii/0022190265801968. →page 174[113] L. G. Mendoza-Luna, N. M. K. Shiltagh, M. J. Watkins, N. Bonifaci,F. Aitken, and K. von Haeften. Excimers in the lowest rotational quantumstate in liquid helium. The Journal of Physical Chemistry Letters, 7(22):4666–4670, 11 2016. doi:10.1021/acs.jpclett.6b02081. URLhttps://doi.org/10.1021/acs.jpclett.6b02081. → page 50[114] D. Meshulach and Y. Silberberg. Coherent quantum control of two-photontransitions by a femtosecond laser pulse. Nature, 396(6708):239–242,1998. doi:10.1038/24329. URL https://doi.org/10.1038/24329. → page 26[115] D. Meshulach and Y. Silberberg. Coherent quantum control of multiphotontransitions by shaped ultrashort optical pulses. Phys. Rev. A, 60:1287–1292, Aug 1999. doi:10.1103/PhysRevA.60.1287. URLhttps://link.aps.org/doi/10.1103/PhysRevA.60.1287. → page 26[116] I. Miller, T. Faulkner, J. Saunier, and P. L. Raston. Observation of theelusive “oxygen-in” OCS dimer. The Journal of Chemical Physics, 152(22):221102, 10/1/2023 2020. doi:10.1063/5.0010716. URLhttps://doi.org/10.1063/5.0010716. → page 70[117] A. A. Milner, A. Korobenko, J. W. Hepburn, and V. Milner. Effects ofultrafast molecular rotation on collisional decoherence. Phys. Rev. Lett.,113:043005, Jul 2014. doi:10.1103/PhysRevLett.113.043005. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.113.043005. → pages 2, 104[118] A. A. Milner, A. Korobenko, K. Rezaiezadeh, and V. Milner. Fromgyroscopic to thermal motion: A crossover in the dynamics of molecularsuperrotors. Phys. Rev. X, 5:031041, Sep 2015.doi:10.1103/PhysRevX.5.031041. URLhttps://link.aps.org/doi/10.1103/PhysRevX.5.031041. → page 104[119] A. A. Milner, A. Korobenko, and V. Milner. Field-free long-livedalignment of molecules with a two-dimensional optical centrifuge. Phys.Rev. A, 93:053408, May 2016. doi:10.1103/PhysRevA.93.053408. URLhttps://link.aps.org/doi/10.1103/PhysRevA.93.053408. → pages 111, 112162[120] A. A. Milner, A. Korobenko, J. W. Hepburn, and V. Milner. Probingmolecular potentials with an optical centrifuge. The Journal of ChemicalPhysics, 147(12):124202, 1/4/2024 2017. doi:10.1063/1.5004788. URLhttps://doi.org/10.1063/1.5004788. → pages 2, 104, 118[121] A. A. Milner, A. Korobenko, and V. Milner. Ultrafast magnetization of adense molecular gas with an optical centrifuge. Phys. Rev. Lett., 118:243201, Jun 2017. doi:10.1103/PhysRevLett.118.243201. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.118.243201. → page 2[122] A. A. Milner, J. A. M. Fordyce, I. MacPhail-Bartley, W. Wasserman,V. Milner, I. Tutunnikov, and I. S. Averbukh. Controlled enantioselectiveorientation of chiral molecules with an optical centrifuge. Phys. Rev. Lett.,122:223201, Jun 2019. doi:10.1103/PhysRevLett.122.223201. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.122.223201. → pages 2, 101[123] A. A. Milner, U. Steinitz, I. S. Averbukh, and V. Milner. Observation ofmechanical Faraday effect in gaseous media. Phys. Rev. Lett., 127:073901,Aug 2021. doi:10.1103/PhysRevLett.127.073901. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.127.073901. → page 65[124] A. A. Milner, V. A. Apkarian, and V. Milner. Dynamics of molecular rotorsin bulk superfluid helium. Science Advances, 9(26):eadi2455, 2023.doi:10.1126/sciadv.adi2455. URLhttps://www.science.org/doi/abs/10.1126/sciadv.adi2455. → pages5, 9, 48, 50, 60, 61, 62, 63, 104, 129[125] A. A. Milner, I. MacPhail-Bartley, K. Preocanin, S. Dasgupta, X. Peng, andV. Milner. Coherent control of molecular rotation in superfluid helium.Phys. Rev. A, 109:013110, Jan 2024. doi:10.1103/PhysRevA.109.013110.URL https://link.aps.org/doi/10.1103/PhysRevA.109.013110. → pagesvi, 1, 60[126] J. H. Moore, C. C. Davis, and M. A. Coplan. Building Scientific Apparatus.Cambridge University Press, Cambridge, 4th edition, 2009. ISBN9780521878586. doi:DOI:10.1017/CBO9780511609794. URLhttps://www.cambridge.org/core/books/building-scientific-apparatus/52BB9BC3EDF3A8F604EF95D83901AA00. → pages 78, 80, 82[127] M. Mudrich and F. Stienkemeier. Photoionisaton of pure and doped heliumnanodroplets. International Reviews in Physical Chemistry, 33(3):301–339, 07 2014. doi:10.1080/0144235X.2014.937188. URLhttps://doi.org/10.1080/0144235X.2014.937188. → pages 5, 70163[128] M. J. Murray, H. M. Ogden, C. Toro, Q. Liu, and A. S. Mullin. Impulsivecollision dynamics of co super rotors from an optical centrifuge.ChemPhysChem, 17(22):3692–3700, 2016.doi:https://doi.org/10.1002/cphc.201600871. URLhttps://chemistry-europe.onlinelibrary.wiley.com/doi/abs/10.1002/cphc.201600871. → pages 2, 104[129] M. J. Murray, H. M. Ogden, and A. S. Mullin. Anisotropic kinetic energyrelease and gyroscopic behavior of CO2 super rotors from an opticalcentrifuge. The Journal of Chemical Physics, 147(15):154309, 1/4/20242017. doi:10.1063/1.4997701. URL https://doi.org/10.1063/1.4997701. →page 2[130] E. Narevicius and M. G. Raizen. Toward cold chemistry with magneticallydecelerated supersonic beams. Chemical Reviews, 112(9):4879–4889, 092012. doi:10.1021/cr2004597. URL https://doi.org/10.1021/cr2004597. →page 67[131] K. Nauta and R. E. Miller. Nonequilibrium self-assembly of long chains ofpolar molecules in superfluid helium. Science, 283(5409):1895–1897,1999. doi:10.1126/science.283.5409.1895. URLhttps://www.science.org/doi/abs/10.1126/science.283.5409.1895. →pages 71, 133[132] K. Nauta and R. E. Miller. Metastable vibrationally excited HF (v=1) inhelium nanodroplets. The Journal of Chemical Physics, 113(21):9466–9469, 1/29/2024 2000. doi:10.1063/1.1319965. URLhttps://doi.org/10.1063/1.1319965. → page 129[133] K. Nauta and R. E. Miller. Rotational and vibrational dynamics of CO2 andN2O in helium nanodroplets. The Journal of Chemical Physics, 115(22):10254–10260, 11/8/2023 2001. doi:10.1063/1.1416492. URLhttps://doi.org/10.1063/1.1416492. → page 174[134] J. K. Park and H. Sun. Relative stabilities of (NO)2. Chemical Physics, 263(1):61–68, 2001. doi:https://doi.org/10.1016/S0301-0104(00)00350-5.URLhttps://www.sciencedirect.com/science/article/pii/S0301010400003505.→ page 132[135] L. Pauling. The normal state of the helium molecule-ions He2+ andHe2++. The Journal of Chemical Physics, 1(1):56–59, 1/01/1933 1933.164doi:10.1063/1.1749219. URL https://doi.org/10.1063/1.1749219. → page84[136] J. D. Pickering. Alignment and Imaging of Molecular ComplexesEmbedded in Helium Nanodroplets. PhD thesis, Aarhus University,Department of Chemistry, September 2018. → pages 80, 133[137] J. D. Pickering, B. Shepperson, L. Christiansen, and H. Stapelfeldt.Femtosecond laser induced Coulomb explosion imaging of aligned OCSoligomers inside helium nanodroplets. The Journal of Chemical Physics,149(15):154306, 11/7/2023 2018. doi:10.1063/1.5049555. URLhttps://doi.org/10.1063/1.5049555. → page 71[138] J. D. Pickering, B. Shepperson, B. A. K. Hu¨bschmann, F. Thorning, andH. Stapelfeldt. Alignment and imaging of the CS2 dimer inside heliumnanodroplets. Phys. Rev. Lett., 120:113202, Mar 2018.doi:10.1103/PhysRevLett.120.113202. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.120.113202. → pages71, 134[139] E. Polyakova. Multiphoton Ionization of Molecules Embedded inSuperfluid Liquid Helium Droplets. PhD thesis, University of SouthernCalifornia, Department of Chemistry, August 2005. → pages 186, 188[140] E. Polyakova, D. Stolyarov, and C. Wittig. Multiple photon excitation andionization of NO in and on helium droplets. J Chem Phys, 124(21):214308,Jun 2006. ISSN 0021-9606 (Print); 0021-9606 (Linking).doi:10.1063/1.2198844. → pages 70, 176, 186, 187, 188, 189, 190[141] N. Po¨rtner, A. F. Vilesov, and M. Havenith. The formation ofheterogeneous Van der Waals complexes in helium droplets. ChemicalPhysics Letters, 343(3):281–288, 2001. ISSN 0009-2614.doi:https://doi.org/10.1016/S0009-2614(01)00648-0. URLhttps://www.sciencedirect.com/science/article/pii/S0009261401006480.→ page 71[142] S. M. Purcell and P. F. Barker. Tailoring the optical dipole force formolecules by field-induced alignment. Phys. Rev. Lett., 103:153001, Oct2009. doi:10.1103/PhysRevLett.103.153001. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.103.153001. → page 2[143] J. Qiang, L. Zhou, P. Lu, K. Lin, Y. Ma, S. Pan, C. Lu, W. Jiang, F. Sun,W. Zhang, H. Li, X. Gong, I. S. Averbukh, Y. Prior, C. A. Schouder,165H. Stapelfeldt, I. N. Cherepanov, M. Lemeshko, W. Jaeger, and J. Wu.Femtosecond rotational dynamics of D2 molecules in superfluid heliumnanodroplets. Phys. Rev. Lett., 128:243201, Jun 2022.doi:10.1103/PhysRevLett.128.243201. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.128.243201. → pages9, 72, 90, 91, 104, 129, 145[144] R. Rajeev, S. V. Raja, T. Madhu Trivikram, K. P. M. Rishad, andM. Krishnamurthy. Measurement of the spatio-temporal gas density profileof a supersonic jet. Journal of Applied Physics, 114(8):083112, 10/2/20232013. doi:10.1063/1.4819448. URL https://doi.org/10.1063/1.4819448. →page 67[145] P. L. Raston. Hends: A program for calculating average heliumnanodroplet sizes. SoftwareX, 14:100703, 2021. ISSN 2352-7110.doi:https://doi.org/10.1016/j.softx.2021.100703. URLhttps://www.sciencedirect.com/science/article/pii/S2352711021000480.→ page 67[146] P. L. Raston. Helium nanodroplet isolation spectroscopy in anundergraduate teaching laboratory. Journal of Molecular Spectroscopy,388:111676, 2022. ISSN 0022-2852.doi:https://doi.org/10.1016/j.jms.2022.111676. URLhttps://www.sciencedirect.com/science/article/pii/S0022285222000947.→ page 5[147] G. W. Rayfield and F. Reif. Quantized vortex rings in superfluid helium.Phys. Rev., 136:A1194–A1208, Nov 1964.doi:10.1103/PhysRev.136.A1194. URLhttps://link.aps.org/doi/10.1103/PhysRev.136.A1194. → page 65[148] B. G. Sartakov, J. P. Toennies, and A. F. Vilesov. Infrared spectroscopy ofcarbonyl sulfide inside a pure 3He droplet. The Journal of ChemicalPhysics, 136(13):134316, 9/29/2023 2012. doi:10.1063/1.3697475. URLhttps://doi.org/10.1063/1.3697475. → page 4[149] K. Sato, Y. Achiba, and K. Kimura. Photoelectron spectroscopic evidencefor photodissociation of the nitrogen oxide dimer in supersonic jet.Chemical Physics Letters, 126(3):306–310, 1986.doi:https://doi.org/10.1016/S0009-2614(86)80088-4. URLhttps://www.sciencedirect.com/science/article/pii/S0009261486800884.→ page 133166[150] R. Sayo´s, R. Valero, J. M. Anglada, and M. Gonza´lez. Theoreticalinvestigation of the eight low-lying electronic states of the cis- andtrans-nitric oxide dimers and its isomerization using multiconfigurationalsecond-order perturbation theory (caspt2). The Journal of ChemicalPhysics, 112(15):6608–6624, 9/29/2023 2000. doi:10.1063/1.481234.URL https://doi.org/10.1063/1.481234. → page 132[151] R. Schmidt and M. Lemeshko. Rotation of quantum impurities in thepresence of a many-body environment. Phys. Rev. Lett., 114:203001, May2015. doi:10.1103/PhysRevLett.114.203001. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.114.203001. → page 104[152] R. Schmied, P. C¸arc¸abal, A. M. Dokter, V. P. A. Lonij, K. K. Lehmann, andG. Scoles. UV spectra of benzene isotopomers and dimers in heliumnanodroplets. The Journal of Chemical Physics, 121(6):2701–2710,1/8/2024 2004. doi:10.1063/1.1767515. URLhttps://doi.org/10.1063/1.1767515. → page 145[153] C. Schouder. Laser-induced Alignment and Structure Determination ofMolecular Dimers embedded in Helium Nanodroplets. PhD thesis, AarhusUniversity, Department of Physics and Astronomy, September 2019. →page 191[154] C. Schouder, A. S. Chatterley, F. Calvo, L. Christiansen, and H. Stapelfeldt.Structure determination of the tetracene dimer in helium nanodropletsusing femtosecond strong-field ionization. Structural Dynamics, 6(4):044301, 1/6/2024 2019. doi:10.1063/1.5118005. URLhttps://doi.org/10.1063/1.5118005. → page 133[155] C. Schouder, A. S. Chatterley, M. Johny, F. Hu¨bschmann, A. F. Al-Refaie,F. Calvo, J. Ku¨pper, and H. Stapelfeldt. Laser-induced Coulomb explosionimaging of (C6H5Br)2 and C6H5Br–I2 dimers in helium nanodroplets usinga TPX3Cam. Journal of Physics B: Atomic, Molecular and OpticalPhysics, 54(18):184001, 2021. doi:10.1088/1361-6455/ac04c4. URLhttps://dx.doi.org/10.1088/1361-6455/ac04c4. → pages 2, 36, 71, 101[156] T. Seideman. On the dynamics of rotationally broad, spatially aligned wavepackets. The Journal of Chemical Physics, 115(13):5965–5973, 10 2001.ISSN 0021-9606. doi:10.1063/1.1400131. URLhttps://doi.org/10.1063/1.1400131. → pages 130, 189[157] T. Seideman and E. Hamilton. Nonadiabatic Alignment by Intense Pulses.Concepts, Theory, and Directions, volume 52, pages 289–329. Academic167Press, 2005. ISBN 1049-250X.doi:https://doi.org/10.1016/S1049-250X(05)52006-8. URLhttps://www.sciencedirect.com/science/article/pii/S1049250X05520068.→ pages 23, 92[158] B. Shepperson, J. Liu, A. M. Ellis, and S. Yang. Ionization of doped heliumnanodroplets: Residual helium attached to diatomic cations and theirclusters. The Journal of Physical Chemistry A, 115(25):7010–7016, 062011. doi:10.1021/jp112204e. URL https://doi.org/10.1021/jp112204e. →page 176[159] B. Shepperson, A. A. Søndergaard, L. Christiansen, J. Kaczmarczyk, R. E.Zillich, M. Lemeshko, and H. Stapelfeldt. Laser-induced rotation of iodinemolecules in helium nanodroplets: Revivals and breaking free. Phys. Rev.Lett., 118:203203, May 2017. doi:10.1103/PhysRevLett.118.203203. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.118.203203. → pages8, 9, 92, 103, 106, 119[160] A. Slenczka and J. P. Toennies, editors. Molecules in Superfluid HeliumNanodroplets: Spectroscopy, Structure, and Dynamics. SpringerInternational Publishing, 2022. ISBN 978-3-030-94896-2.doi:https://link.springer.com/book/10.1007/978-3-030-94896-2. URLhttps://link.springer.com/book/10.1007/978-3-030-94896-2. → page 5[161] H. Stapelfeldt and T. Seideman. Colloquium: Aligning molecules withstrong laser pulses. Rev. Mod. Phys., 75:543–557, Apr 2003.doi:10.1103/RevModPhys.75.543. URLhttps://link.aps.org/doi/10.1103/RevModPhys.75.543. → page 136[162] F. Stienkemeier and K. K. Lehmann. Spectroscopy and dynamics in heliumnanodroplets. Journal of Physics B: Atomic, Molecular and OpticalPhysics, 39(8):R127, apr 2006. doi:10.1088/0953-4075/39/8/R01. URLhttps://dx.doi.org/10.1088/0953-4075/39/8/R01. → pages 73, 84[163] F. Stienkemeier, O. Bu¨nermann, R. Mayol, F. Ancilotto, M. Barranco, andM. Pi. Surface location of sodium atoms attached to 3He nanodroplets.Phys. Rev. B, 70:214509, Dec 2004. doi:10.1103/PhysRevB.70.214509.URL https://link.aps.org/doi/10.1103/PhysRevB.70.214509. → page 72[164] M. R. Sultanova, I. A. Remizov, L. P. Mezhov-Deglin, and A. A.Levchenko. Generation of quantum vortices by waves on the surface ofsuperfluid helium. JETP Letters, 118(8):585–590, 2023.168doi:10.1134/S0021364023602841. URLhttps://doi.org/10.1134/S0021364023602841. → page 65[165] E. Taylor, H. Hu, X.-J. Liu, L. P. Pitaevskii, A. Griffin, and S. Stringari.First and second sound in a strongly interacting fermi gas. Phys. Rev. A, 80:053601, Nov 2009. doi:10.1103/PhysRevA.80.053601. URLhttps://link.aps.org/doi/10.1103/PhysRevA.80.053601. → page 64[166] J. P. Toennies and A. F. Vilesov. Spectroscopy of atoms and molecules inliquid helium. Annual Review of Physical Chemistry, 49(1):1–41, 1998.doi:10.1146/annurev.physchem.49.1.1. URLhttps://doi.org/10.1146/annurev.physchem.49.1.1. PMID: 15012423. →pages 3, 69[167] J. P. Toennies and A. F. Vilesov. Superfluid helium droplets: A uniquelycold nanomatrix for molecules and molecular complexes. AngewandteChemie International Edition, 43(20):2622–2648, 2004.doi:https://doi.org/10.1002/anie.200300611. URLhttps://onlinelibrary.wiley.com/doi/abs/10.1002/anie.200300611. → pages4, 5, 46, 67, 68, 75, 122[168] J. P. Toennies, A. F. Vilesov, and K. B. Whaley. Superfluid HeliumDroplets: An Ultracold Nanolaboratory. Physics Today, 54(2):31–37, 022001. ISSN 0031-9228. doi:10.1063/1.1359707. URLhttps://doi.org/10.1063/1.1359707. → page 5[169] E. Treacy. Optical pulse compression with diffraction gratings. IEEEJournal of Quantum Electronics, 5(9):454–458, 1969.doi:10.1109/JQE.1969.1076303. → pages 20, 22[170] I. Tutunnikov, J. Floß, E. Gershnabel, P. Brumer, I. S. Averbukh, A. A.Milner, and V. Milner. Observation of persistent orientation of chiralmolecules by a laser field with twisted polarization. Phys. Rev. A, 101:021403, Feb 2020. doi:10.1103/PhysRevA.101.021403. URLhttps://link.aps.org/doi/10.1103/PhysRevA.101.021403. → pages2, 94, 101[171] A. van Deursen and J. Reuss. Molecular beam intensities and collisioncross-sections of small Ne, H2, N2, NO and O2 clusters. InternationalJournal of Mass Spectrometry and Ion Physics, 23(2):109–122, 1977.doi:https://doi.org/10.1016/0020-7381(77)80093-4. URLhttps://www.sciencedirect.com/science/article/pii/0020738177800934. →page 133169[172] D. Verma and A. F. Vilesov. Pulsed helium droplet beams. ChemicalPhysics Letters, 694:129–134, 2018.doi:https://doi.org/10.1016/j.cplett.2018.01.035. URLhttps://www.sciencedirect.com/science/article/pii/S0009261418300447.→ page 68[173] F. Vestin, K. Nilsson, and P.-E. Bengtsson. Validation of a rotationalcoherent anti-Stokes Raman spectroscopy model for carbon dioxide usinghigh-resolution detection in the temperature range 294-1143 K. Appl. Opt.,47(11):1893–1901, Apr 2008. doi:10.1364/AO.47.001893. URLhttps://opg.optica.org/ao/abstract.cfm?URI=ao-47-11-1893. → pages12, 50[174] D. M. Villeneuve, S. A. Aseyev, P. Dietrich, M. Spanner, M. Y. Ivanov, andP. B. Corkum. Forced molecular rotation in an optical centrifuge. Phys.Rev. Lett., 85:542–545, Jul 2000. doi:10.1103/PhysRevLett.85.542. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.85.542. → pages 20, 22, 106[175] N. V. Vitanov and B. Girard. Adiabatic excitation of rotational ladder bychirped laser pulses. Phys. Rev. A, 69:033409, Mar 2004.doi:10.1103/PhysRevA.69.033409. URLhttps://link.aps.org/doi/10.1103/PhysRevA.69.033409. → pages 18, 19[176] N. V. Vitanov, T. Halfmann, B. W. Shore, and K. Bergmann. Laser-inducedpopulation transfer by adiabatic passage techniques. Annual Review ofPhysical Chemistry, 52(1):763–809, 2001.doi:10.1146/annurev.physchem.52.1.763. URLhttps://doi.org/10.1146/annurev.physchem.52.1.763. PMID: 11326080. →page 19[177] L. Vranjesˇ and S. Kilic´. Helium trimers and tetramers in two dimensionsand quasi-two-dimensions. Phys. Rev. A, 65:042506, Apr 2002.doi:10.1103/PhysRevA.65.042506. URLhttps://link.aps.org/doi/10.1103/PhysRevA.65.042506. → page 84[178] E. A. Wade, J. I. Cline, K. T. Lorenz, C. Hayden, and D. W. Chandler.Direct measurement of the binding energy of the NO dimer. The Journal ofChemical Physics, 116(12):4755–4757, 2/17/2024 2002.doi:10.1063/1.1459702. URL https://doi.org/10.1063/1.1459702. → page133170[179] R. H. Walmsley and C. T. Lane. Angular momentum of liquid helium.Phys. Rev., 112:1041–1047, Nov 1958. doi:10.1103/PhysRev.112.1041.URL https://link.aps.org/doi/10.1103/PhysRev.112.1041. → pages 65, 144[180] A. Weiner, D. Leaird, J. Patel, and J. Wullert. Programmable shaping offemtosecond optical pulses by use of 128-element liquid crystal phasemodulator. IEEE Journal of Quantum Electronics, 28(4):908–920, 1992.doi:10.1109/3.135209. → page 55[181] A. M. Weiner. Femtosecond pulse shaping using spatial light modulators.Review of Scientific Instruments, 71(5):1929–1960, 2/29/2024 2000.doi:10.1063/1.1150614. URL https://doi.org/10.1063/1.1150614. → page53[182] A. M. Weiner. Ultrafast optical pulse shaping: A tutorial review. OpticsCommunications, 284(15):3669–3692, 2011.doi:https://doi.org/10.1016/j.optcom.2011.03.084. URLhttps://www.sciencedirect.com/science/article/pii/S0030401811003750.→ page 20[183] A. M. Weiner and D. E. Leaird. Generation of terahertz-rate trains offemtosecond pulses by phase-only filtering. Opt. Lett., 15(1):51–53, Jan1990. doi:10.1364/OL.15.000051. URLhttps://opg.optica.org/ol/abstract.cfm?URI=ol-15-1-51. → page 55[184] A. M. Weiner, D. E. Leaird, G. P. Wiederrecht, and K. A. Nelson.Femtosecond pulse sequences used for optical manipulation of molecularmotion. Science, 247(4948):1317–1319, 1990.doi:10.1126/science.247.4948.1317. URLhttps://www.science.org/doi/abs/10.1126/science.247.4948.1317. → page26[185] C. M. Western. Pgopher: A program for simulating rotational, vibrationaland electronic spectra. Journal of Quantitative Spectroscopy and RadiativeTransfer, 186:221–242, 2017. ISSN 0022-4073.doi:https://doi.org/10.1016/j.jqsrt.2016.04.010. URLhttps://www.sciencedirect.com/science/article/pii/S0022407316300437.Satellite Remote Sensing and Spectroscopy: Joint ACE-Odin Meeting,October 2015. → page 184[186] F. Wiegandt, F. Trinter, K. Henrichs, D. Metz, M. Pitzer, M. Waitz,E. Jabbour al Maalouf, C. Janke, J. Rist, N. Wechselberger, T. Miteva,171S. Kazandjian, M. Scho¨ffler, N. Sisourat, T. Jahnke, and R. Do¨rner. Directobservation of interatomic coulombic decay and subsequent ion-atomscattering in helium nanodroplets. Phys. Rev. A, 100:022707, Aug 2019.doi:10.1103/PhysRevA.100.022707. URLhttps://link.aps.org/doi/10.1103/PhysRevA.100.022707. → page 71[187] N. Xu, C. Wu, Y. Gao, H. Jiang, H. Yang, and Q. Gong. Measurement ofthe field-free alignment of diatomic molecules. The Journal of PhysicalChemistry A, 112(4):612–617, 01 2008. doi:10.1021/jp075728h. URLhttps://doi.org/10.1021/jp075728h. → page 40[188] A. Yabushita, T. Fuji, and T. Kobayashi. SHG FROG and XFROG methodsfor phase/intensity characterization of pulses propagated through anabsorptive optical medium. Optics Communications, 198(1):227–232,2001. ISSN 0030-4018.doi:https://doi.org/10.1016/S0030-4018(01)01510-3. URLhttps://www.sciencedirect.com/science/article/pii/S0030401801015103.→ page 207[189] A. Yachmenev and S. N. Yurchenko. Detecting chirality in molecules bylinearly polarized laser fields. Phys. Rev. Lett., 117:033001, Jul 2016.doi:10.1103/PhysRevLett.117.033001. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.117.033001. → pages94, 101[190] Z.-C. Yan and T. Seideman. Photomanipulation of external molecularmodes: A time-dependent self-consistent-field approach. The Journal ofChemical Physics, 111(9):4113–4120, 09 1999. ISSN 0021-9606.doi:10.1063/1.479184. URL https://doi.org/10.1063/1.479184. → page 23[191] S. Yang and A. M. Ellis. Helium droplets: a chemistry perspective. Chem.Soc. Rev., 42:472–484, 2013. doi:10.1039/C2CS35277J. URLhttp://dx.doi.org/10.1039/C2CS35277J. → page 5[192] I. I. Zakharov, V. F. Anufrienko, O. I. Zakharova, S. A. Yashnik, and Z. R.Ismaguilov. Ab initio calculation of nitrogen oxide dimer structure and itsanion-radical. Journal of Structural Chemistry, 46(2):213–219, 2005.doi:10.1007/s10947-006-0033-1. URLhttps://doi.org/10.1007/s10947-006-0033-1. → page 132[193] L. Zang, W. Dai, L. Zheng, C. Duan, Y. Lu, and M. Yang. Theoreticalprediction of the linear isomers for rare gas-carbon disulfide complexes:172He-CS2, Ne-CS2, and Ar-CS2. The Journal of Chemical Physics, 140(11):114310, 03 2014. ISSN 0021-9606. doi:10.1063/1.4868325. URLhttps://doi.org/10.1063/1.4868325. → page 106[194] H. Zhang, G.-L. Zheng, G. Lv, Y.-Z. Geng, and Q. Ji. Covalentintermolecular interaction of the nitric oxide dimer (NO)2. Chinese PhysicsB, 24(9):093101, 2015. doi:10.1088/1674-1056/24/9/093101. URLhttps://dx.doi.org/10.1088/1674-1056/24/9/093101. → page 134[195] Z. Zhang and U. Even. Pulsed supersonic beams from high pressuresource: Simulation results and experimental measurements. Advances inChemistry, 2014:636042, 2014. doi:10.1155/2014/636042. URLhttps://doi.org/10.1155/2014/636042. → page 77[196] Z. X. Zhao, X. M. Tong, and C. D. Lin. Alignment-dependent ionizationprobability of molecules in a double-pulse laser field. Phys. Rev. A, 67:043404, Apr 2003. doi:10.1103/PhysRevA.67.043404. URLhttps://link.aps.org/doi/10.1103/PhysRevA.67.043404. → pages 37, 39[197] S. Zhdanovich. Control of atoms and molecules with shaped broadbandpulses. PhD thesis, The University of British Columbia, Department ofPhysics and Astronomy, July 2012. → pages 55, 57, 198[198] S. Zhdanovich, A. A. Milner, C. Bloomquist, J. Floß, I. S. Averbukh, J. W.Hepburn, and V. Milner. Control of molecular rotation with a chiral train ofultrashort pulses. Phys. Rev. Lett., 107:243004, Dec 2011.doi:10.1103/PhysRevLett.107.243004. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.107.243004. → pages45, 57, 59, 60, 188[199] S. Zhdanovich, C. Bloomquist, J. Floß, I. S. Averbukh, J. W. Hepburn, andV. Milner. Quantum resonances in selective rotational excitation ofmolecules with a sequence of ultrashort laser pulses. Phys. Rev. Lett., 109:043003, Jul 2012. doi:10.1103/PhysRevLett.109.043003. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.109.043003. → page 25173Appendix AIdentifying the Nitric OxideDimerThis appendix outlines how the NO+ signals in Chapter 10 were determined tobe from nitric oxide dimers. The most likely contaminants were assumed to beN2O, NO2, or (NO)3. This was important because the rotational constant of N2Oin helium droplets, 0.07 cm−1 [133], is comparable to one third of the expectedsecond and third gas phase rotational constants of (NO)2.A.1 Discerning that effects are not from N2O or NO2NO stored under pressure may form NO2 [112]. The same gas cylinder was con-nected to the pulsed jet system to verify with the RGA that NO was far more preva-lent than N2O or NO2, by two to three orders of magnitude, as shown by comparingthe intensities of masses 30, 44, and 46 in Figure A.1. There are a few additionalindicators that these measurements are indeed measuring NO+ fragments from NOclusters. In doping the helium droplets with N2O, it was found that with sim-ilar RGA mass 44 readings there was a significantly weaker NO+ ion signal inCoulomb explosion measurements. NO2 doping was not attempted, as it was farless prevalent as shown below in Table A.1, and does not appear to be a significantcontaminant, as shown in Figure A.1. The time of flight mass spectrum plottedin Figure A.2 can also be analyzed. There are significant peaks at both the N2O174Figure A.1: The same NO gas cylinder was connected to the Parker pulsedvalve to create a jet to measure the gas composition. Shown is thedifference between with and without the pulsed jet. No significant NO2was detected, and the increase of N2O of 1×10−9 Torr is significantlyless than the 8×10−8 Torr increase of NO.He2 NO N2O NO2 (NO)2 (N2O)2 (NO)3Mass 8 30 44 46 60 88 90With Droplets 5.5×10−10 1.8×10−10 3.9×10−11 7.5×10−12 1.6×10−11 7.4×10−12 1.4×10−12Without Droplets 1.1×10−12 1.1×10−11 2×10−11 2×10−12 1.6×10−12 1.3×10−12 1.1×10−12Table A.1: RGA pressure readings (in Torr) for NO dimer doping experimentsof droplets produced by 30 Bar / 18 K.Figure A.2: Time of flight mass spectrum recorded with NO-doped heliumdroplets. Red lines indicate masses 14 and 16 for N+ and O+ fragments,as well as 30 to 48 in increments of 4.and NO2 masses, 44 and 46 AMU respectively. The intensity of these peaks beingcomparable to the dimer signal at 60 may be concerning!175More information can be gained by creating a time of flight covariance map(Figure A.3) as was demonstrated for CH3CHCH2O in Section 3.1.4. The redline at mass to charge 30 shows that NO+ fragments very rarely arrive with N+or O+ at 14 or 16. Figure A.3 also shows that NO+ more frequently arrives withmasses 34,38, and 42 than with 14 or 16. These higher masses are attributed toion fragments that have helium attached, NOHe2+ etc. These fragments have beenobserved previously in [140] and [158].It is perhaps more convenient to analyze an individual row of Figure A.3 tomake this conclusion. Figure A.4 takes two slices of Figure A.3, with the magnifiedinset showing the covariance of 30 + with 14 + and 16 +. These results indicate thatif one were to measure the signal at 34 +, they would almost certainly be measuringfragments that came with 30 +. Measuring ion fragments with He attached is acommon method to isolate the signal that comes from molecules in helium droplets.In combination with the other measurements, it seems most probable that 30 + isindeed almost entirely due to (NO)2, and so the stronger 30 + signal was measuredfor all reported Coulomb explosion imaging measurements instead, to increase theion count rate.176Figure A.3: A time of flight covariance map made from the individual lasershots used to create Figure A.2. The colourmap intensity represents thecovariance at the intersecting mass to charge ratios. Red lines indicatethe masses of nitrogen, oxygen, and clusters of NO attached to He.177Figure A.4: The covariance between 30 + and 34 + with respect to othermasses. The vertical scale is consistent between the measurements andis a linear measurement of the covariance, but its absolute value is arbi-trary depending on the signal amplification of the system. In the top plotthe covariance between 30 + and 34 + is 0.2, nearly four times greaterthan the covariance between 30 + and 14 + or 16 +. In the bottom plot itcan be seen that the covariance between 34 + and 30 + is more than anorder of magnitude greater than the covariance between 34 + and 14 +or 16 +.178A.2 Discerning that effects are not from (NO)3An understanding of the angular covariance mapping technique provides an oppor-tunity to verify both that the doping and ion detection region are appropriate forthe study of molecular dimers. This is particularly relevant for NO, as the trimer ispredicted to be an isosceles triangle [70] with the NO molecules oriented perpen-dicular to the trimer plane, rather than a linear chain. The following discussion isspecifically regarding Coulomb explosion of equally heavy fragments with equalcharge.Unlike monomers, which have zero kinetic energy release (KER), multiply ion-ized dimers and trimers may Coulomb explode into pairs and trios of NO+. Two-body Coulomb explosion can only produce singly-ionized pairs 180◦ apart, whilethe structure of a trimer affects the angle between fragments. An angular covari-ance map of randomly oriented molecules will show dimers as a pair of parallelFigure A.5: Simulated angular covariance maps for a randomly oriented en-semble of dimers and trimers. 60◦, 30◦ and 15◦ refers to the vertex(unique) angle of the simulated isosceles triangles. The trimer struc-ture calculated in [70] is an isosceles triangle with a vertex angle ofapproximately 48◦, so one would expect to observe four distinct paral-lel lines.179Figure A.6: Simulated angular covariance map for a randomly oriented en-semble of trimers with an equilateral triangle structure.lines for the 180◦ pairs, while a trimer will produce two or four parallel lines, de-pending on the geometry. An equilateral triangle structure will produce ion trios atθ0+ (0◦, 120◦, 240◦) at the maximum kinetic energy, which occurs when the planeof the trimer’s triangular structure is parallel to the VMI detector. Simulated angu-lar covariance maps of randomly oriented dimers and trimers are shown in FigureA.5, with the equilateral triangle structure in the top-right corner.When a trimer is aligned such that one fragment is ejected along the TOFVMIaxis, the other two fragments will again be at θ0+ (0◦, 180◦), for molecules freelyrotating around the TOFVMI axis. In this case the 2D projection is essentially thesame as that of a linear chain. For a randomly oriented ensemble, these moleculeswill contribute an additional set of lines to the trimer’s angular covariance map.However, the positive covariance from trimers aligned this way contributes signif-icantly less than the negative covariance at θ0+ (0◦, 180◦), from the much largernumber of molecules not aligned to this axis, and the resulting covariance alongthese angles is negative. Similarly, it is unlikely (impossible) for two fragmentsto be ejected at the same angle, except in the limiting case where the vertex anglegoes to zero, so there is strong negative covariance around θ0, as shown in FigureA.6.To find appropriate radii (kinetic energies on the VMI detector) for ⟨cos2 θ2D⟩measurements, angular covariance measurements of the NO+ ion fragment wererecorded for trimer doping. In Figure A.8 it is shown that for higher kinetic ener-180Figure A.7: Trimer doping withNO+ measured at lowerkinetic energy, interpreted asthe dimer radii.Figure A.8: Trimer doping athigher kinetic energy, inter-preted as the trimer radii.Compare to the top-right ofFigure A.5.gies (labelled as trimer radii) there are four parallel lines, while in Figure A.7 atlower kinetic energies (labelled as dimer radii) there are two broad lines. Adjust-ing the threshold between high/low KER, to improve the clarity of the four parallellines, provides an upper bound for the kinetic energy range to use for ⟨cos2 θ2D⟩calculations of dimers. The broad pair of lines in Figure A.7 are the combinationof ions originating from dimers, as well as those from trimers that are not orientedparallel to the detector plane. This is consistent with the triangular trimer structurefrom [70].This provides a good starting point for finding the dimer doping regime, bylowering the doping pressure until there are few ions in the trimer radii. FigureA.10 shows no strong covariance at the trimer radii while the pair of lines fromdimer radii are now much more narrow as is expected for two-body Coulomb re-pulsion.181Figure A.9: Dimer doping calcu-lated at dimer radii. Compareto the top-left of Figure A.5.Figure A.10: Calculated over thetrimer radii.Mass 8 Mass 30 Mass 44 Mass 60 Mass 90Dimer Doping 5.5×10−10 3.8×10−10 4×10−11 7.5×10−11 2.6×10−11Trimer Doping 4.2×10−10 2.3×10−10 2.9×10−11 9.2×10−11 4.1×10−11Table A.2: RGA pressure readings (in Torr) for NO dimer and trimer dopingangular covariance map experiments of droplets produced by 30 Bar /16 K. For measurements in Section 10.2 the doping was slightly changedto use 30 Bar / 18 K droplets to reduce the effects of scattering.182Appendix BSupersonic Jet TemperatureDetermination with REMPIResonant-enhanced multi photon ionization (see Section 3.6) was used to measurethe rotational temperature of the supersonic jet, used for gas phase studies. Nitricoxide (NO) was utilized because the dye laser was already set up for it in a sepa-rate experiment by Ryan Harvey. The X2Π ground vibrational state is resonantlyexcited to the A2Σ+ state by a single photon around 226 nm, and readily ionizedafterwards with another photon.Figure B.1: The energy level diagram for the X2Π−→ A2Σ+ −→ NO+ + e–(1+1) REMPI scheme.183Higher initial temperature will broaden the initial distribution of rotational lev-els that are populated, increasing the number of resonant transitions. In practicethis means that as the laser frequency is scanned around 226 nm (44190 cm−1),the rotational temperature will determine the relative degree of ionization at eachfrequency. An ion yield spectrum can then be compared to simulation at varioustemperatures, typically using PGOPHER [185], and the rotational temperature ofthe sample can be estimated with the best fit.In experiment, NO was diluted approximately ten-to-one with helium and ap-plied to the Parker pulsed valve with approximately 2 Bar of backing pressure. The226 nm beam was produced with a dye laser using Coumarin 47 (Sigma Aldrich)dye diluted in methanol with a concentration of 0.25 gL−1 in the oscillator, andone eighth that in the amplifier. This produced a ∼452 nm beam, which was thenfrequency-doubled using the Sirah SHG-215 BBO! stage. Measurements with arecently serviced valve produced a rotational temperature close to 6 K, while ex-periments two years later with a heavily used valve were closer to 15 K. Eachmeasurement is only a few seconds, and no compensation for shot-to-shot laserfluctuation leads to high noise, but the strong nonlinearity in the ionization ratemakes peak identification relatively straightforward. (2+ 2) REMPI was also at-tempted, using the fundamental 452 nm beam without the doubling stage, but itwas found that with the necessary intensities for a strong signal there was substan-tial line broadening due to the AC Stark effect [73].184(a) Recorded in January 2021.(b) Recorded in May 2023.Figure B.2: (1+ 1) REMPI spectra of NO in seeded helium jets, with sim-ulated PGOPHER spectra for approximate bounds on the temperatureestimation.185Appendix CREMPI of NO in Helium Dropletsand Transport to the SurfaceThis appendix outlines brief experiments to study nitric oxide (NO) using resonantionization, following [140], and the possibility of transporting the molecule to thesurface of the droplet as a means of studying its rotational dynamics. Recent workhas shown that many (perhaps all) of these molecules are actually ejected from thedroplet [19]. My attempts fell short of reproducing the results of [140], [139], and[19], and the intended extension of those experiments was not realized.C.1 IntroductionResonance-Enhanced Multi Photon Ionization (REMPI, described further in Sec-tion 3.6) is one possible method to measure the rotational excitation of molecules.In helium droplets, quantifying molecular alignment through Coulomb explosionimaging is sometimes impossible, due to the scattering of ion fragments off of he-lium atoms as they exit the droplet. This is especially relevant when looking atlighter molecules such as NO, whose ion fragments are more prone to scatteringthan heavier fragments such as S+ from CS2. The rotational constant of NO inhelium droplets has been measured using infrared spectroscopy to be 1.25 cm−1[70], and NO in helium droplets has been previously studied using REMPI [140].It was found the excitation to the lowest Rydberg state was blueshifted by several186Figure C.1: Figure taken from [140]. By scanning a broad range in 30 cm−1steps Polyakova et. al. were able to observe several resonances of NOin helium droplets, with a substantial shift from the gas phase attributedto strong electron repulsion of the helium. Peaks highlighted by arrowsare attributed to intermediate surface-bound high Rydberg states.thousand wavenumbers in helium droplets, and the spectral lines were significantlybroadened .While a linewidth on the order of ∼10 cm−1 would eclipse the first few ro-tational lines with B = 1.25cm−1, one can still expect a significant increase inionization rate when a molecule undergoing uni-directional rotation is exposed toa probe of circular polarization with the same direction of rotation, relative to a187counter-rotating probe [198].The broad REMPI line in the inset of Figure C.1 was attributed to excitation ofa low Rydberg state, which caused a small bubble to form around the molecule inthe helium droplet, which then migrated to the droplet’s surface [139, 140]. Thisexcitation was later used in reference [19] to demonstrate that the movement ofthe bubble to the surface was limited to the Landau critical velocity of ∼58 ms−1.Crossing the radius of a 12 nm droplet at 60 ms−1 would take just 100 ps, nearlytwo orders of magnitude shorter than the laser pulse duration used. This made itreasonable to conclude the subsequent ionization took place after the molecule hasreached the surface of the droplet. This “transport” effect was considered as anopportunity to avoid the effect of N+ and O+ ions scattering as they are expelledfrom the droplet in VMI experiments.C.2 Experimental TechniqueHelium droplets were produced at 25 Bar / 12 K before being doped with NO.A dye was needed to scan between 24200-24500 cm−1 (408.2-413.2 nm), and soExalite 411 was chosena. The ionization signal was collected using the TOFVMIcamera.For the transport experiments, the dye laser set to a fixed wavelength to movethe dopant away from the centre of the droplet, followed by an 800 nm ∼200 fsprobe for Coulomb explosion imaging of the molecular axis. The probe will bothboth preferentially ionize molecules depending on the angle between the molec-ular axis and the probe’s polarization axis, and slightly torque molecules to ro-tate towards that axis. These two phenomena are referred to as “geometric” and“dynamic” alignment, respectively [103], the former being the cause of the angle-dependent ionization described in Section 3.3, and the latter being negligible inour experiments. This allows for a simple experiment where ⟨cos2 θ2D⟩ of the frag-ments is expected to vary depending on the probe polarization, only if the fragmentdoes not undergo significant scattering, as should be the case for NO molecules ei-ther on the surface, or expelled from, the helium droplet.aThe dye did not dissolve well in ethanol, and so 1,4-Dioxane was used, as recommended bySirah.188C.3 ResultsC.3.1 REMPIUnfortunately, these experiments proved extraordinarily difficult, due to stabilityissues with the dye laser. No measurements that fully encompassed the target rangewhile modulating the droplet beam as in [140] were able to be performed withsuitable signal levels.The slight increase in ionization around 24500 cm−1 was reproducible, but dueto the dye curve the scan could not be extended to higher wavenumbers. No linearound 24320 cm−1 was observed, though it should be noted that this excitationwas successfully excited by another research group in reference [19]. Dye lasercalibration is within 50 cm−1, so this is not the issue. Broadening due to the ACStark effect of gas phase NO is significant at this intensity [73], but this would notexplain why in Figure C.2b the signal resembling a peak around 24500 cm−1 is∼200 cm−1 blueshifted from Figure C.1, rather than broadening of the 24320 cm−1peak.C.3.2 Transport of NO from the centre of a Helium DropletThe background signals from 14N2 and 16O2 fragments proved overwhelminglyhigh, and no dependence on the dye laser could be observed in ⟨cos2 θ2D⟩ of eitherthe N+ or O+ fragment. These experiments were performed with the dye laser setat both 24320 cm−1 as well as ∼24500 cm−1 due to the results of Figure C.2.The decreased ionization potential of an excited Rydberg state may stronglyinfluence the angle-dependent ionization rates, and so if this experiment were tobe repeated it would be worthwhile to include an adiabatic alignment pulse, ratherthan to rely on geometric alignment from the probe. As the degree of alignment fora long linearly polarized pulse has a strong inverse dependence on the temperature,this would also mitigate the influence of the thermal background [156].189(a) REMPI scan across the inset from Figure C.1. Approximately 5 mJ pulse energy acrosseach scan.(b) Across the largest peak from above. ∼2.6 mJ across each scan.Figure C.2: Attempts to recreate the inset in Figure C.1 from reference [140]proved unsuccessful, with the “peak” being consistently centred closerto 24480 cm−1 than at 24320 cm−1. The ∼10 ns dye laser beam was∼11 mm in diameter and focused with an f = 30cm lens.190Appendix DData Processing TechniquesD.1 2-body Ion Pair FilteringWhen a signal is not background-free, it can be beneficial to implement lossy filtersthat preferentially reduce the background. For diatomic molecules, one techniqueto ensure that changes in the alignment factor ⟨cos2 θ2D⟩ are caused by the moleculeof interest is to only accept ion fragments that come in pairs with equal and oppositevelocities. For example, this could easily separate the dynamics of nitrogen oroxygen in a mixture with nitric oxide.This is also useful for studying clusters, where rotational dynamics of the car-bon disulphide dimer (CS2)2 have been studied with this type of coincidence filter[29], to eliminate the contributions of the trimer (CS2)3 .In helium droplets, the dopant pick-up statistics outlined in section 5.3 illustratethat there will be some trimers when studying dimers. While their prevalence canbe assessed using the angular covariance map as outlined in Appendix A, a 2-bodyfilter is also useful in instances such as ensuring the NO+ signal being observed isindeed from the nitric oxide dimer (NO)2, rather than nitrous oxide (N2 O).The filter’s sensitivity can be optimized from a covariance map of isotropicmolecules, described in detail in Constant Schouder’s PhD thesis [153]. Even witha filter that should keep 95 % of pairs based on the molecule’s KER distribution andemission angle distribution, this can be quite lossy due to non-unitary detectionefficiencies.191D.2 Error Bar CalculationsThe error bars on each plot are the standard error of the mean of several repeatedmeasurements. In most measurements, this is a few seconds or minutes for eachprobe delay, with the probe returning to the earliest delay for the subsequent mea-surement after the latest delay. This is done so that in cases of something like a3 h measurement consisting of many 10 min scans, any effects due to laser overlapor power drift, or changes to the helium droplet doping, are approximately equalat earlier and later delays. In some experiments, the measurement is paused in be-tween scans so the overlap can be verified, by redirecting the laser beams after thefocusing lens onto a camera sensor. This is particularly important for kick exper-iments where the fluence is of interest, because if the kick and probe are slightlymisaligned, the overall effect will be the observation of molecules kicked by aslightly different fluence, which was shown to dramatically alter the dynamics inhelium droplets. In some experiments repeated scans across multiple days werecombined.192Appendix EThe Vacuum SystemThis appendix outlines some specific components of the vacuum system. The de-sign of this apparatus is described in chapter 6.Originally, the 90◦ horizon port in the detection chamber was to have a smallsource chamber for an Even Lavie pulsed valve. This chamber would be on the op-tical table, designed so that the valve could be used for both Raman spectroscopyimmediately after the nozzle (as done in reference [14]), and gas phase VMI exper-iments. This idea was abandoned before a turbomolecular pump for that chamberwas ordered, solely because a quick experiment of hanging a Parker pulsed valvein the Differential chamber provided a sufficiently dense and cold molecular jetfor my planned experiments. Experiments with CS2-doped helium droplets did notseem to be limited by the signal-to-background ratio, thanks to the combination ofan additional differential pumping stage, and the use of doping cells, so the lossof the Differential chamber’s planned liquid nitrogen cryotrap did not seem detri-mental. The cryotrap built for the Doping chamber had a small vacuum leak in thewelds, but was not repaired for the same reason, and did not get installed.193Figure E.1: The Doping, Differential, and VMI chamber turbopumps eachhad their outlet connected to the Source chamber’s inlet, using metalKF and conflat bellows. The outlet of the Source chamber’s turbop-ump was connected to a tee, with one side connected to a “No VoltRelease” (NVR) switch that would isolate the chamber from the Ed-wards iGX 600L forevacuum pump in the event of a power failure.The other connection had a vacuum-actuated vale that would shut withthe loss of power and open when it sensed power and vacuum on thescroll pump side. This configuration will isolate the chamber from bothpumps when the power shuts off, and then begin pumping on it with thescroll pump when the power comes on.E.1 Specific Parts UsedA more technical run-through of the specific components used for the vacuum sys-tem used in this work.E.1.1 Droplet SourceA Sumitomo RDK-408D2 cryocooler with F-70L compressor was chosen becauseit is very common, both in neighbouring laboratories and in other helium dropletmachines. Comparable models from ARS, Leybold, and others would have beensuitable too. The cooling power is excessive, but allows for the use of a nozzlemade in Freiburg attached on a long piece of thin copper L-stock, where thermalconduction between the cryocooler and the nozzle is quite poor. In some laborato-194Figure E.2: The nozzle’s heater (no polarity) is on the central two pins, withthe two sets of four DT-670A1-CU pins connected with rotational sym-metry, meaning that i+ from one diode is adjacent to i− of the other.ries, a piece of stainless steel is placed in between the cryocooler and the nozzle tointentionally reduce the thermal conductivity, allowing for higher temperatures tobe more easily controlled with a less powerful heater.A standard 5 um platinum Scanning Electron Microscope aperture from PlanoGmbH was used as a nozzle, so that the known size distribution of droplets could beused. In principle a different nozzle size could have been used to more effectivelyproduce smaller droplets.The helium (6.0 purity) passes through a Swagelok SS-4F-05 particulate fil-ter (0.5 µm stainless steel) before entering the vacuum system, and again througha similar SS-2F-05 filter on the cryocooler immediately before entering the noz-zle. The backing pressure is controlled with a Swagelok KCP1MRB2A2P60000regulator (with two PGI-50M-BG100-LAVX gauges for inlet and outlet pressure)shortly before entering the vacuum chamber.The nozzle temperature is controlled with a Lakeshore 325 temperature con-troller, which reads the temperature from a DT-670A1-CU diode on the nozzle, aswell as monitoring a second sensor on the copper cryoshield. It heats the nozzle byapplying current to a 50Ω resistor (Digikey A131965-ND) on the copper nozzlebody. The resistor is not at all intended for use in Vacuum.195E.1.2 ManipulatorThe manipulator functions quite well, although the nozzle tilts significantly whenthe Z axis is adjusted. Figure E.3 (a) shows the KF40 vacuum flanges on theinstrumentation “skirt” that the cold head is attached to. The nozzle’s tempera-ture sensors and heater are connected with an electric feedthrough on one of theseflanges, as is the helium gas feedthrough.E.1.3 Droplet Doping SystemThe doping system was fabricated in the department’s machine shop. It consistsof two stainless steel cells (1 inch diameter, 3 inch length, 5 mm orifice) with 2Dmanipulation at each end allowing them to be positioned around the droplet beam.An all-metal leak valve (Kurt J. Lesker VZLVM263R) is connected to the cellswith 18 inch stainless steel tubing with Swagelok ferrule connectors on each end.Gasses that are even mildly corrosive to stainless steel should use a valve with adifferent sealing surface, such as sapphire.E.1.4 Pulsed Nozzle for Supersonic JetInitially, an Even Lavie pulsed valve (EL-5-C-S.S.-2020) capable of producing< 50µs pulses of gas, was to be used in a separate chamber, pumped via bel-lows connected to the “Differential” pumping chamber. Instead, a Parker Series9 valve (General Valve) with a 200 µm orifice was used, with an IOTA ONE con-troller. These valves produce much longer pulses (a few hundred microseconds,adjustable), which introduce much higher gas loads, so require either larger pumps,lower repetition rates, or lower backing pressures.Vacuum-rated Kapton-coated copper wires were used, and despite a small amountof standard electrical tape this chamber still reaches pressures below 1×10−8 Torrwhen there is no gas load applied.E.1.5 Differential PumpingUpwards of 99% of the helium must be skimmed away after the nozzle, and soa Beam Dynamics Model 50.8 with a 2 mm was used. Originally, a 3 mm ori-fice model 2 skimmer was used in the Doping and Differential pumping chambers,196although the Doping chamber skimmer was replaced with 5 mm diameter flat aper-ture (manufactured with a water jet), due to alignment difficulties. The 3 mm ori-fice in the Differential pumping chamber skimmer was formed by filling a damaged300 µm skimmer with epoxy, and using a mill in very gentle passes to cut away thetip, before heating the skimmer with a heat gun to cause it to expand, and releasefrom the epoxy. This skimmer could not be replaced with an aperture, as it wasnecessary for forming a jet with the pulsed nozzle.With so many degrees of freedom, alignment was quite tricky and requiredmany steps of iteration, which required opening and closing chambers frequently.E.1.6 Time of Flight Velocity Map Imaging SpectrometerThe TOFVMI consists of a series of stainless steel electrodes clamped together, withinsulating Macor spacers held in a Mu-metal shield, a series of high voltage powersupplies, a microchannel plate (MCP) with a phosphor screen, a scientific camerawith lens for imaging the phosphor screen, and a photomultiplier tube for time offlight measurements. Colour filters were placed on both the PMT and Camera tofilter out laser (800 nm and ambient light. The PMT was typically recorded directlywith an oscilloscope but for the lock-in detection described in section 9.2.2 thePMT was connected to a boxcar averager, and recorded with a Natural InstrumentsData Acquisition card.• Camera: FLIR BlackFly BFS-U3-04S2M-CS with a 720x540 1/2.9” sensor.It can run at 1 kHz when cropped to 200x200.• Lens: Edmund Optics 33-302 8 mm UC Series f/1.8• Photo Multiplier Tube: Hamamatsu H10721-01• MCP with Phosphor: MCP-050-D-L-P46F-F-V emitting at 520 nm and adecay time of 65 ns to 10 % of peak.• Colour Filters: Thorlabs BG39• Oscilloscope: Tektronix TDS2012C• Boxcar Averager: Stanford Research Systems SR250197Pump Model Pumping Speed (Ls−1) Ultimate Pressure (Torr) Compression Ratio (He)Forevacuum Edwards IGX 600L 167 5×10−4 xFV Backup Varian TriScroll 300 4 1×10−2 xSource Leybold 2200 iPL 2050 < 1×10−8 5×104Doping Leybold 1000C 1000 < 1×10−8 5×104Differential Leybold 250i 250 < 5×10−10 1×108VMI Leybold 250i 250 < 5×10−10 1×108Table E.1: Vacuum pumps used. The vintage 1000C was “Made in West Ger-many” and also previously used in ref [197] and [83], and the Differentialchamber’s pump was replaced with a similar Pfeiffer TMU 260 from thesystem used in [56] after an issue with the Leybold power supply.• Data Acquisition card: NI DAQ PCI-6221 with BNC-2110• High Voltage for MCP: 2 kV iSEG DPp 20 405 24 5 M for the microchannelplate and 6 kV DPp 60 105 24 5 M for the phosphor.• High Voltage for MCP: 30 kV iSEG CPp 300 304 24 5 for both the repellerand extractor electrodes.The high voltage power supplies are modular units, installed into rack mountboxes, and controlled with an Arduino set up by the department’s electronics shop.According to their output feedback, they are outputting ∼5 µA of current, in stan-dard operating conditions.E.1.7 Vacuum PumpsLeybold turbopumps offered the best pumping speeds for helium at a given price.The Edwards iGX600L was chosen because it works well in Henrik Stapelfeldt’sgroup in Aarhus, and would be able to pump multiple vacuum systems simultane-ously, unlike the more-appropriate iGX100L.The iGX 600L was placed in a separate room, and connected with 4 inch(4.5 inch) inner (outer) diameter Schedule 40 PVC pipes, using custom KF adaptersto connect to standard vacuum components. Vanstone plumbing flanges with 4 inchviton 150 lb VP820 low torque gaskets proved to seal effectively, and significantlyeased assembly. P70 purple primer and 711 gray PVC cement were used. Twolines were run to the droplet machine, about 15 m each. It was experimentally198Gauge Model Type Pressure Range (Torr)Forevacuum Hastings DV-6M Thermocouple 1×10−3 to 2Source MKS 972B DualMag Cold Cathode Vacuum Transducer 1×10−8 to 760Doping Top MKS 972B DualMag Cold Cathode Vacuum Transducer 1×10−8 to 760Doping Side MKS 500 Cold Cathode 1×10−10 to 1×10−2Differential MKS 500 Cold Cathode 1×10−10 to 1×10−2VMI MKS 500 Cold Cathode 1×10−10 to 1×10−2Table E.2: Vacuum gauges used.determined that due to the poor compression ratio for helium, the doping cham-ber Leybold 1000C pump should be backed by a lower pressure, and the outlet wasconnected to the source chamber by a 2.75 Inch Conflat bellows. This increased thehelium droplet beam signal by an order of magnitude. Eventually, the Differentialand VMI chambers were also connected in this mannerE.1.8 Vacuum GaugesGauges were chosen based on price and range, with little regard to accuracy. TheMKS 500 series have some issues where they refuse to turn on occasionally, buthave otherwise been reliable. The MKS 972B gauges are nice in that they can turnon at atmosphere, but some read as high as 900 Torr at atmosphere (760 Torr) andthey must be read with a differential ADC, so one can not simply connect a cheapermodel of Arduino with a screen.E.1.9 OtherCustom pneumatic gate valves to isolate chambers were ordered from VAT, witha viewport so that a HeNe laser could be used for aligning the skimmers, dopingcells, and nozzle to the VMI. As the nozzle was seemingly slightly tilted, thisproved difficult and in hindsight, was not necessary. The gate valve between theSource and Doping chamber is not meant for Ultra High Vacuum, and uses a simpleelastomer seal on the top of the body, although these chambers have sufficientlyhigh pumping speeds that this does not seem to limit their ultimate pressure. Thevalve between the Doping and Differential chambers is UHV-rated.As the Doping chamber contains multiple elements to align, it was very helpfulto be able to keep the other chambers under high vacuum while iterating adjust-199ments.The VMI chamber has a very old Stanford Research Systems Residual GasAnalyzer 100 facing the helium droplet beam.Gas lines that were only used above atmospheric pressure were constructedusing Swagelok SS-1RS4 needle valves, and SS-4H bellows valves were used forlines below atmospheric pressure.200(a) An isometric view of the nozzle X −Y −Z manipulator. Bolts and set screws are notshown. Flat-tipped set screws in the holes marked X and Y push against steel blocks, to movethe whole assembly sideways and vertically, and then the long ACME-threaded rod turns toslide the cold head assembly into/out of the chamber.(b) Cross-section view of the manipulator with the positions of the single X −Y and pairof Z o-rings shown in red.Figure E.3: The nozzle and cold head manipulator.201(a) The complete doping assembly with space for a cryotrap in between cells.(b) A CAD render of the side profile demonstrating how the parts fit together.Figure E.4: The doping cell mount was designed to simplify the manufac-turing steps, rather than a more elegant construction. Most of the partsrequire only 2D machining. It is designed for assembly through a single8 inch Conflat port.202Figure E.5: The valve is simply hanging from 14 inch stainless steel tubing,attached to a feedthrough on a linear manipulator to raise and lower itinto the droplet beam path. The valve is not precisely centred on theskimmer, and the exact skimmer-nozzle distance is unknown.203Figure E.6: Cross section view of the 2D skimmer and aperture manipula-tors built for this system. The thin brim of the skimmers was rigidlyclamped in the yellow piece by the threaded ring shown in green. Itclosely fit in the circular stainless steel piece with radial clearance sothat the radial screws shown in white could manipulate the skimmerposition around a larger bored hole.This assembly is bolted to a double-sided conflat flange which was placed between chambers.204Figure E.7: The VMI electrode assembly.The repeller on the bottom has asmall hole so that a molecular beam could pass from the bottom.205Appendix FPulse CharacterizationF.1 Peak IntensityA laser beam with wavelength λ of 1/e2 diameter D and M2 = 1, focused by anelement of focal length f , produces a spot size of radiusr =2λ fπD. (F.1)The spot size could also be measured with a beam profiler or by scanning a pinholeacross the focus, when focusing with a lens. A pulse which is Gaussian in time andspace has the intensity profileI(x,y, t) = Ipeak exp[−2x2r2x]exp[−2y2r2y]exp[−4ln2t2τ2], (F.2)where rx and ry (both 1/e2 radii) account for beams which are not circularly sym-metric, and τ is the full width half max duration of the pulse. Integrating over timeand space,Etotal = Ipeak[rx√π2][ry√π2][τ2√πln2], (F.3)where Etotal is the total energy of the pulse, as measured on a power meter. Solvingfor the peak intensity:206Ipeak =Etotalτrxry4√ln2π 32. (F.4)F.2 Optical AutocorrelationA Spectra-Physics PulseScout autocorrelator was used to estimate the duration ofthe compressed and slightly-stretched pulses used as probes and kicks. It splits asingle pulse in two, and varies the delay of one with respect to the other, whileoverlapping the two pulses on a nonlinear crystal. The sum frequency generationsignal is then measured with a photodiode, and plotted as a function of delay time.This provides an estimate of the pulse duration, although the multitude of waysan autocorrelation measurement can be misleading have been well documented[188]. Ultimately, none of the experiments presented in this thesis are expectedto be sensitive to the pulse duration, or temporal structure, within this expecteduncertainty.F.3 Cross-Correlation Frequency Resolved OpticalGating (XFROG)Cross-Correlation Frequency Resolved Optical Gating (XFROG) is a well knownmethod of characterizing laser pulses [101, 102], schematically shown in Fig-ure F.1(a). The method is based on sum frequency generation. Here, however, thefield of the pulse in question is mixed with a shorter (generally Fourier-transformlimited) reference pulse. The spectrum of the XFROG field,EXFROG,±(t,∆t) ∝ E±,x(t+∆t/2)ETL,x(t−∆t/2), (F.5)is recorded as a function of the time delay ∆t. Reference field ETL(t) corre-sponds to a compressed reference pulse. When ETL(t) is much shorter than thepulse of interest (∼ 1 ps vs > 100 ps, respectively), the recorded XFROG spec-trograms |EXFROG,±(ω,∆t)|2 are direct time-frequency maps of the pulse beingstudied. They reflect the time dependence of the instantaneous frequency of E±(t),shifted to 2ω0 due to the SFG mixing, i.e. 2ω0 + 2β±∆t. Hence, the slope of an207Figure F.1: (a) Experimental layout for the cross-correlation frequency re-solved optical gating (XFROG) characterization of the centrifuge field.Centrifuge pulses are linearly polarized with a polarizing beam splitter(PBS) and focused on nonlinear barium borate (BBO) crystal, togetherwith the linearly polarized short reference pulses. Sum frequency gener-ation spectrum is recorded with a spectrometer as a function of the timedelay ∆t. (b) top: XFROG scans of the red (tilted up) and blue (tilteddown) centrifuge arms of a slow centrifuge; bottom: temporal profilesobtained from the traces above.XFROG spectrogram corresponds to twice the frequency chirp of the target pulse,β±.An example of the experimentally measured XFROG trace for our slow cen-trifuge (from [104]) is shown in the upper part of Figure F.1(b). Integrating overwavelength provides one with a temporal profile of each arm, shown in the lowerpart of Figure F.1(b).From the two initial slopes of−2.9(1) pm/ps and 3.1(1) pm/ps, one can extractthe frequency chirps of β+ = 0.0171(5) ps−2 and β− = −0.0182(5) ps−2, for theblue and red arms, respectively. The average value of β = 0.0176(7) ps−2.When used to characterize a long chirped pulse, like that of an optical cen-trifuge, it is useful to spectrally narrow the probe pulse to improve the spectralresolution of the XFROG trace, and in Figure F.1 the 7.6 nm probe was narrowedbelow 0.5 nm, at the expense of some temporal resolution due to the Fourier trans-form limit. When only the duration of the pulse being characterized is in question,it often makes more sense to use the shortest reference pulse possible, and XFROG208is sometimes more convenient than the autocorrelator in section F.2. A collinearcross-correlation configuration is also a very convenient way to overlap two pulsesin time, as the nonlinear increase in blue light when two 800 nm pulses overlap inspace and time on a barium borate crystal is easily noticed by eye, even when thetime delay between pulses is being rapidly scanned such that they only overlap fora small fraction of a second. When the approximate pulse durations are known, itis also easy to simply record the signal with a photodiode throughout an automatedscan to find the overlap.209
Thesis/Dissertation
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Physics
Vancouver : University of British Columbia Library
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Graduate
Milner, Valery
Control of molecular rotation in superfluid helium
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