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Higgs spectroscopy of superconductors : a new method to identify the superconducting gap symmetry Cheng, Nathan 2018

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Higgs Spectroscopy ofSuperconductorsA new method to identify the superconducting gapsymmetrybyNathan ChengB.Sc., The University of British Columbia, 2016A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2018c© Nathan Cheng 2018iiCommittee PageThe following individuals certify that they have read, and recommend to the Faculty of Graduateand Postdoctoral Studies for acceptance, the thesis entitled:Title: Higgs Spectroscopy of Superconductorssubmitted by Nathan Cheng in partial fulfillment of the requirements forthe degree of Master of Sciencein PhysicsExamining CommitteeMona BerciuSupervisorAndrea DamascelliAdditional ExamineriiiAbstractIn this thesis we study the response of a BCS superconductor to an external ultra-fast terahertzelectromagnetic field, which we choose so as to mimic the setup in a pump-probe experiment.We begin by considering an optical experimental setup and demonstrate that in an optical pump-probe experiment, the superconducting amplitude Higgs mode can be excited and measured withultra-fast terahertz pump pulses. Moreover, for an anistropic d-wave superconductor, there are twoHiggs mode, one at the usual 2∆ energy and one with a lower energy. The latter can be used todifferentiate the d-wave symmetry from isotropic s-wave, by varying the polarization of the pumprelative to the sample. For a linearly polarized pump with a vector potential aligned along a d-wavenode we find only a single Higgs mode, while for a direction along an antinode we find two Higgsmodes.Next, we consider an angle resolved photoemission spectroscopy (ARPES) experiment and derivea new set of equations of motion, for which we can analyze the two-time nonequilibrium Green’sfunctions. We show that the Higgs mode can also be studied in an ARPES pump-probe experiment.Moreover, we show how an ARPES pump-probe experiment can be used to differentiate betweendifferent momentum-dependent nonequilibrium Higgs modes. Our results suggest that in a d-wavesuperconductor, the second low-energy Higgs mode is of osculating, B1g character, which correspondsto a symmetry breaking along the d-wave nodal lines. Further study of the role of momentumsymmetry breaking promises to provide deeper insight into generating new nonequilibrium states.ivLay SummaryIn this thesis, we develop a theoretical framework that can be used to elucidate future experimentalfindings in terahertz time-resolved experiments on superconductors. The theoretical frameworkand experimental methods proposed in this thesis can be used to characterize different symmetriesinherent to superconductors in equilibrium and nonequilibrium. These advances in the field of Higgsspectroscopy promise to provide a deeper insight into superconducting dynamics and advance thegrowing field surrounding nonequilibrium superconductors and materials research.vPreface• A version of the work discussed in Chapter 2 is currently published as B. Fauseweh, L. Schwarz,N. Tsuji, N. Cheng, N. Bittner, H. Krull, M. Berciu, G. S. Uhrig, A. P. Schnyder, S. Kaiser,D. Manske arXiv:1712.07989. It makes use of the formalism by Papenkort, Axt and Kuhn[37].• I carried out the numerical calculations and numerical analysis in this publication, which per-tain to the interaction of the superconductor with an electromagnetic field, while the analyticwork on quenches and different nonequilibrium symmetries was primarily contributed by B.Fauseweh, L. Schwarz. and N. Tsuji. The project was primarily overseen by D. Manske andthe publication is based on collaboration with Bittner, Krull, Berciu, Uhrig, Schnyder, Kaiser.The draft of the manuscript was also written by B. Fauseweh.• I have also carried out all of the work in Chapter 3, which will be submitted for publicationshortly.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 BCS theory of superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Mean-field superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Ginzburg-Landau theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Nonequilibrium spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Nonequilibrium Superconductivity: Optical Response . . . . . . . . . . . . . . . 72.1 Introduction and analogy with Ginzburg-Landau theory . . . . . . . . . . . . . . . . 72.1.1 Superconducting symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.1 Equations of motion following the pump . . . . . . . . . . . . . . . . . . . . 122.3.2 Equations of motion following the probe . . . . . . . . . . . . . . . . . . . . 142.3.3 Initial conditions and numerical approximations . . . . . . . . . . . . . . . . 162.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4.1 Higgs oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4.2 Optical response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Table of Contents vii3 Nonequilibrium Superconductivity: ARPES . . . . . . . . . . . . . . . . . . . . . . 213.1 Introduction to Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3 Time-dependent ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3.1 Model Hamiltonian with the ansatz . . . . . . . . . . . . . . . . . . . . . . . 243.4 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.4.1 Initial conditions and numerical approximations . . . . . . . . . . . . . . . . 293.4.2 Nonequilibrium Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . 303.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.5.1 Higgs oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.5.2 Spectral Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37AppendicesA First Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41viiiList of Tables2.1 Character table for the D4 point group . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Parameters used to calculate the optical conductivity . . . . . . . . . . . . . . . . . . 173.1 Parameters used to calculate the spectral function . . . . . . . . . . . . . . . . . . . 30ixList of Figures1.1 Ginzburg-Landau free energy potential . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 Nonequilibrium superconducting oscillatory modes in the Ginzburg-Landau picture . 82.2 dx2−y2 Higgs amplitude modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Comparison of Higgs oscillations for different paring symmetries . . . . . . . . . . . 182.4 Comparison of different optical conductivity Higgs responses . . . . . . . . . . . . . 193.1 Comparison of dx2−y2 Higgs oscillations in our new formalism . . . . . . . . . . . . . 323.2 Comparison of the spectral function A(θ, ω) for a pumped dx2−y2 superconductor . . 333.3 Comparison of the spectral functions A(θ, ω, δt) for a pumped dx2−y2 superconductor 33xAcknowledgementsI am so grateful to both Dirk Manske and Mona Berciu – Dirk for our ongoing collaboration, whichhas both opened up the world of physics to me as well as an entire new world of cultures, whichI am very happy to now take part in. Your continuing advice, not just about physics, has beeninvaluable.Mona Berciu has been above and beyond and I am so thankful to have had the opportunity towork with her. To me, she has been exceptionally kind, helpful and inspiring. Her dependable, calmreassurance has been a huge buoy to my entire MSc. experience. I am extremely thankful to her forher guidance.The entirety of Mona Berciu’s group has been extremely helpful to me – especially StepanFomichev, Mirko Mo¨ller and John Sous, whom I have had countless exceptional discussions with. Iam also grateful to James Charbonneau, whose class I have had the pleasure of being a part of asa teaching assistant. Being a part of teaching has been one of my most valued parts of my Msc.Lastly, I am very thankful to Tim Jaschek, for our many scientific and unscientific discussions. Hisongoing friendship has truly made my MSc. memorable.I would like to thank all my friends who have made the past years an unforgettable experienceand finally, my parents, Mark and Elizabeth, and my sister, Kira, whose love and encouragementhas been my anchor.xiDedicationFor my sister, whose wisdom creates such richness of life for all those around her.1Chapter 1IntroductionSuperconductivity, discovered in 1911 by Kamerlingh Onnes [51], continues to be one of the mostinteresting phases of matter. Originally shown to appear in mercury below 4.2 Kelvin, it is nowknown that most elemental metals and simple metallic compounds undergo a phase transition toa superconducting state below some critical temperature in the range of up to 20 K. Remarkably,upon cooling below this transition temperature, these materials exhibit absolutely zero resistivityand, partly as a consequence, expel weak magnetic fields from the bulk – the Meissner effect. Aswould later be discovered, the source of such peculiar properties is a low energy bosonic condensatewith an energy gap corresponding to the condensate energy. However, this explanation would notbe established until 45 years later.These compounds became known as conventional superconductors when in 1986 the first so-called unconventional superconductor [7] was discovered in lanthanum doped copper oxide. Quickly,an entire class of copper oxide superconductors were discovered that continue to be an ongoing andchallenging research topic in condensed matter physics. This class of superconductors is uniquefor a number of reasons, the first being an unusually high superconducting transition temperature.Among other peculiarities, these superconductors do not have a uniform superconducting energygap. In fact, for certain points in the Brillouin zone, the superconducting gap actually closes. Since,there have been many other classes of superconductors with unconventional forms of momentumdependence beyond conventional s-wave – d-wave, s+/-, etc. – making symmetry a meaningfulmethod of distinguishing between these classes.1.1 BCS theory of superconductivityAs a testament to the difficulty of both experimentally probing and theoretically describing thisphase, a microscopic description of conventional superconductivity was not discovered until the1957 Bardeen Cooper and Schrieffer (BCS) theory of superconductivity [1–3, 11], for which theywould receive the 1972 Nobel prize in physics (for unconventional superconductivity, a conclusivemicroscopic description remains elusive). While widely known in the 50s that a weak attractivepotential can bind a pair of particles in two-dimensions, but not three, Cooper showed that certainthree-dimensional electrons, which exist in a thin shell of energy near the Fermi energy (the Fermisurface) and in the presence of an attractive electron-electron potential, will also form pairs ofbound states (Cooper pairs) between electrons of opposite momentum [11]. Together with Schriefferand Bardeen, Cooper wrote down a microscopic Hamiltonian for such an interaction, which wouldbecome known as the BCS theory of superconductivity [2, 3]:HBCS =∑k,σkc†k,σck,σ +∑k,k′Vk,k′c†k↑c†−k↓c−k′↓ck′↑ (1.1)1.1. BCS theory of superconductivity 2In the original work, the form of the potential Vk,k′ was taken to be some constant attractive termbetween electrons (holes) c†k,σ (ck,σ) of opposite spin and momentum in some band k. However,the exact potential is now known to be highly momentum dependent depending on the specificsuperconductor. For instance, in the d-wave superconductors, the potential has a d-wave dependence,which we will discuss in detail below.1.1.1 Mean-field superconductivityFortunately, for conventional superconductors in particular, electron pairs often span tens or evenhundreds of lattice sites. As such, the electron (hole) pairing density is relatively constant and amean-field approximation is well justified. Making such an approximation reduces the quartic su-perconducting Hamiltonian to a simple, quadratic Hamiltonian for a single particle and additionallydefines a complex order parameter representing the superconducting gap ∆k, which is defined interms of the mean-field parameter,∆k′ =∑k∈WVk,k′〈c−k↓ck↑〉. (1.2)Here, we also enforce the requirement that these electrons (holes) reside within a thin shellW above(below) the Fermi surface – as required for electrons (holes) to experience an attractive electron-electron (hole-hole) interaction. In this case, the mean-field Hamiltonian reduces toHMF =∑k,σkc†k,σck,σ +∑k∈W[∆kc†k↑c†−k↓ + ∆∗kc−k↓ck↑]. (1.3)It is convenient to write the Hamiltonian in terms of Bogoliubov quasiparticle operators, which aregiven by,αk = u∗kck,↑ + vkc†−k↓ (1.4a)βk = u∗kc−k↓ − vkc†k↑ (1.4b)Under these transformations and normalizing – |uk|2 + |vk|2 = 1 – the mean-field Hamiltonianwritten in terms of the Bogoliubov quasiparticles is then,HMF =∑k[Rkα†kαk −Rkβkβ†k + Ckα†kβ†k + C∗kβkαk](1.5)where Rk and Ck are given by,Rk = k(1− 2vkv∗k) + ∆∗kukvk + ∆u∗kv∗k (1.6a)Ck = −2ku∗kvk + ∆k(u∗k)2 −∆∗k(vk)2 (1.6b)For finite ∆k, the quasiparticle energy Ek =√2k + |∆k|2 is non-vanishing for all points around theFermi surface, so the quasiparticle spectrum is gapped by ∆k. As the superconducting gap ∆k closesto zero, Ck must vanish so that the Hamiltonian becomes diagonal; Rk then simply has the form ofthe band energy. We choose our normalization in such a way that uk is real and vk is complex and1.2. Ginzburg-Landau theory 3Re(∆)Im( ∆)FFigure 1.1: Ginzburg-Landau free energy potential in the superconducting phasewith respect to some global order parameter ∆.carries the phase of the gap, ∆k. To satisfy these condition and re-obtain our original mean-fieldHamiltonian uk and vk are defined in the following way:uk =√12(1 +kEk)(1.7a)vk =∆k|∆k|√12(1− kEk)(1.7b)with Ek =√2k + |∆k|2. Besides being of a much simpler form, this mean-field Hamiltonian alsoallowed the reconciliation of the BCS, microscopic theory of superconductivity with the more macro-scopic or phenomenological, Ginzburg-Landau theory of superconductivity.1.2 Ginzburg-Landau theoryOriginally formulated prior to the BCS theory, the Ginzburg-Landau theory [18, 28] attempted todescribe the continuous phase transition between the superconducting and normal states in terms ofsome global order parameter for which they gave no microscopic justification. Even so, their theorywas rather effectively able to describe certain macroscopic properties according to this complex orderparameter field. For instance, it allows one to derive two length-scales corresponding to the Londonpenetration depth of a magnetic field into the superconductor and the characteristic length scaleof superconducting density fluctuations, the superconducting coherence length. Ginzburg-Landautheory has since been derived beginning from the microscopic BCS description of superconductivity.In essence, Ginzburg-Landau theory is just an extension of Landau mean-field theory for continuous,1.3. Nonequilibrium spectroscopy 4second order phase transitions. It states that in the absence of a superconducting current, the freeenergy density in the superconducting state, Fs, is the sum of the normal state free energy densityFn and terms depending on some complex field ∆ (which we know from BCS theory corresponds tothe superconducting order parameter).Fs = Fn + α|∆|2 + β2|∆|4 (1.8)The total free energy is of course,F =∫dV Fs (1.9)Above the transition temperature Tc, this problem admits the trivial solution |∆| = 0 as a minimumsolution for α > 0. However for α < 0 and β > 0, this problem admits a second solution fortemperatures below Tc of the form,|∆|2 = −α(Tc − T )β(1.10)where the form of the temperature dependence of α = α(Tc − T ) below the transition temperatureis included. The approximate form of α and β can also be derived from BCS theory for variousthermodynamic points. This family of solutions corresponds to a nonzero minimum of F occurringexactly at |∆| and, since the phase of ∆ remains arbitrary, the free energy F has the form of theso-called ”Mexican-hat” potential (Fig. 1.1).1.3 Nonequilibrium spectroscopyResearchers have always been interested in studying the dynamics of systems. Historically, suchresearch led to the invention of the first motion picture films in the 1890s, which implementedshutter speeds on fast enough timescales to resolve and photograph the constituents of motion.To the same effect, nonequilibrium research in science has sought to capture the time-evolution ofa variety of systems ranging from the flutter of a hummingbird’s wings all the way down to themicroscopic scale motion of atoms and electrons; the latter become increasingly more difficult toimage as they require ”shutter-speeds” fast enough to capture motion at the femtosecond and evenshorter timescales. Among numerous nonequilibrium milestones, the 1967 Nobel Prize in Chemistrywas awarded to Eigen, Norrish and Porter for the visualization of rapid chemical reactions and,with the development of femtosecond spectroscopy, the 1999 Nobel Prize in Chemistry was awardedto Zewail for his work on imaging transition states in chemical reactions. Modern time-resolvedexperiments operate across a wide spectrum of energies in the femtosecond and even attosecondtimescale and are rapidly becoming more proficient in the study of quantum effects.While the study of equilibrium properties may provide the foundations needed to describe amaterial, investigating its dynamics is crucial to developing a complete understanding [10, 13, 23, 33].For instance, nonequilibrium experiments can access and probe non-thermal excited states otherwiseinaccessible in thermal equilibrium. One of the most prominent tools to excite and study a systemin nonequilibrium conditions is pump-probe spectroscopy. A pump pulse is first used to excite thesystem into a nonequilibrium state and after a short time delay, an ultra-fast probe pulse is used tomeasure the nonequilibrium state as a function of the delay time. Successive probes in turn can be1.4. Synopsis 5used to illustrate the time-resolved dynamics of the nonequilibrium state.The profile of the pump used to excite the system can be varied to induce different excitationspectra. For instance, a continuous pumping pulse can be used to coherently excite a specific state,thereby providing a degree of optical control. Other optical pumps effectively excite the sampleinto a high temperature state, from which information can be deduced by studying the decay andthe lifetime of the different induced excitations. However, both of these methods can be classifiedas adiabatic methods, as the relevant timescales are often much longer than the response timeof the system. For the rest of this thesis, we will focus on non-adiabatic pump pulses, whichoccur on timescales faster than the response time of the system. Effectively, the pump acts as aquantum quench, driving the system into a nearby state while simultaneously inducing non-adiabaticexcitations such as the Higgs amplitude mode. As a result, the effects are often highly non-linearand become increasingly difficult to study theoretically – especially in the field of strongly-correlatedmaterials, which are already difficult to model, even in equilibrium.1.4 SynopsisAn equilibrium superconducting condensate satisfies the minimum of Eq.1.8 by definition. In thefollowing chapters we will theoretically investigate the nonequilibrium effects of perturbing the con-densate and thus, the form of Eq. 1.8. We refer to this experimental technique as ”Higgs spec-troscopy”. To study and characterize the resulting phenomena – the Higgs amplitude mode – wehave developed software to solve the equations which follow, as well as a new formalism to de-scribe the superconducting dynamics in time-resolved angle resolved photoemission spectroscopy(tr-ARPES) experiments. The thesis is outlined as follows.In Chapter 2, we begin with an introduction to the nonequilibrium Ginzburg-Landau free energypicture of a superconductor. Following a brief introduction to the symmetries of the systems wewish to study – s-wave and d-wave – we derive the equations of motion for a BCS superconduc-tor interacting with an electromagnetic field according to the standard density matrix formalism.We use this form of Hamiltonian and subsequent equations to mimic the form of a pump-probeexperiment and compare the effects of exciting the Higgs amplitude mode(s) in s-wave and d-wavesuperconductors. We also calculate the relevant linear response functions for an optical experimentand predict possible techniques which can be used to experimentally measure these nonequilibriummodes. Our results reveal the possibility of exciting two out of the four Higgs amplitude modes ina d-wave superconductor. This is in contrast to an isotropic s-wave superconductor where only asingle Higgs mode can be excited; there is only one nonequilibrium Higgs mode in an isotropic su-perconductor. Therefore, differentiating the Higgs amplitude mode excitations in a superconductoris a highly effective way of directly characterizing the condensate pairing symmetry.To better understand the nature of the nonequilibrium modes, in Chapter 3 we derive a newformalism to calculate the two-time nonequilibrium Green’s functions, which can be compared withtr-ARPES spectra. We show that, as in the case of time-resolved optical experiments, tr-ARPEScan also be used to detect and distinguish s-wave and d-wave superconductors based on their Higgsamplitude mode response. Moreover, the ARPES momentum resolution can also be used to dis-tinguish the symmetry of the nonequilibrium Higgs mode that is excited. For a linearly polarizedpump incident on a d-wave superconductor, our results predict that aligning the magnetic potential1.4. Synopsis 6along a d-wave antinode will excite both the Higgs breathing mode (A1g) and Higgs osculatingmode (B1g), while aligning the magnetic potential along a d-wave node will only excite the Higgsbreathing mode. Lastly, we discuss a possible explanation and extensions for exciting other Higgsamplitude modes using the symmetry breaking of the linear electromagnetic pumping term.Finally, in Chapter 4, we summarize the main results of this thesis and discuss further promisingavenues to study superconductors via Higgs spectroscopy.7Chapter 2Nonequilibrium Superconductivity:Optical Response2.1 Introduction and analogy with Ginzburg-Landau theoryIn this thesis we theoretically study the nature of a non-equilibrium superconductor that is excitedand measured using pump-probe spectroscopy. Specifically, since we want to study rather low energyfeatures, we will be interested primarily in energies in the range of a few terahertz, timescales onthe order of a few hundred femptoseconds and small enough fluence that the superconducting stateremains intact – the pump should not heat the superconductor into the normal state. Consideringthe narrow experimental window, these experiments are difficult and come close to the boundaryof allowable experimental precision given by the Heisenberg uncertainty principle. However, recentexperiments have been successful in studying non-equilibrium isotropic superconductors [29–32] andhave just begun studying anisotropic superconductors [24], thus paving the way towards directlyprobing the superconducting condensate. A myriad of theoretical proposals and experiments havebeen conducted [4–6, 8, 9, 15, 16, 25–27, 34–50, 52, 53] to study phenomena ranging from coupledsuperconductors to other exotic phenomena. In this thesis, we develop a theoretical framework toprove that experimental measurements of the Higgs amplitude mode can uncover the superconduct-ing symmetry, thus laying the foundation for studying these phenomena via the new technique of”Higgs spectroscopy”.Let us return to the Mexican hat potential, which coincidentally, also arises in high energy physicswith the Higgs boson. Since a Cooper pair is in fact a boson, a superconductor can be viewed asa condensed matter analogue to the high-energy physics Higgs-boson [22]. This equivalency ofstructure gives rise to the nomenclature of the Higgs amplitude mode. For a superconductingcondensate at the minimum of the potential, there are clearly two oscillatory modes permitted – anamplitude Higgs mode up and down the walls of the potential and a Goldstone phase mode aroundthe minimum of the Mexican hat (Fig. 2.1).To experimentally investigate the Higgs mode, pump-probe spectroscopy must be employed toinduce and measure the non-equilibrium state. Physically, a few pairs of electrons are broken bythe pumping pulse, which causes the Mexican-hat potential to shrink slightly, but does not destroythe overall superconductivity. As the potential is altered, one of two scenarios can occur. Eitherthe potential is shrunk slowly so that the condensate is only excited in an adiabatic fashion andalways remains at the bottom of the potential, or the potential is shrunk faster than the condensatecan respond. In this latter non-adiabatic case, the condensate will wind up elevated above the newpotential minimum along the potential wall. This intermediary state then relaxes towards the newequilibrium, giving rise to the Higgs oscillations.2.1. Introduction and analogy with Ginzburg-Landau theory 8Im( ∆)FRe(∆)Figure 2.1: Nonequilibrium superconducting oscillatory modes. The amplitudeHiggs mode manifests as the condensate oscillates up and down the potentialwalls (green line). The Goldstone mode manifests as the condensate phase os-cillates around the potential minimum (white line).2.1.1 Superconducting symmetriesSuperconductors are different from high-energy physics models in that they also allow further degreesof freedom – the momentum dependence of the superconducting order parameter and the symmetryof the lattice. In this sense, superconductors actually host an additional degree of freedom throughwhich one can probe the Higgs mode [4, 39, 41]. One of the key results of this work will be toinvestigate the effect of including the momentum dependence on the Higgs amplitude mode anddiscovering a second Higgs mode for certain anisotropic momentum dependencies.In this thesis, we work solely with a two-dimensional square lattice. In this case, the two super-conducting symmetries, s and dx2−y2 , belong to the A1g and B1g representations of the D4 pointgroup (Table 2.1) because they are even (odd) under C4 rotation around the z-axis out of plane, even(even) under C ′2 rotation around the x and y axes intersecting the antinodes and even (odd) underC ′′2 rotation around the xˆy axis intersecting the nodes [14]. While in an isotropic superconductor,the nonequilibrium Higgs only permits a single mode – A1g isotropic expansion and contractionof the order parameter – as momentum anisotropy is introduced, the superconductor can permitvarious nonequilibrium Higgs oscillations depending on the specific group symmetries of the orderparameter and the lattice. For instance, a dx2−y2 superconductor can admit four possible in-planenonequilibrium amplitude modes corresponding to the A1g, A2g, B1g and B2g symmetries listed inTable 2.1 and illustrated pictorially in Fig. 2.2. The focus of this thesis will be on studying differ-ences between the most common superconducting symmetries, s and dx2−y2 , though the predictionsand analysis can easily be extended to other symmetries using group theory considerations.2.1. Introduction and analogy with Ginzburg-Landau theory 9D4 Table E C2 2C4 2C′2 2C′′2A1g 1 1 1 1 1A2g 1 1 1 −1 −1B1g 1 1 −1 1 −1B2g 1 1 −1 −1 1Table 2.1: Character table for the D4 point group. For a square lattice alignedalong the x and y axes, C2 and C4 correspond to rotations about the z-axis. C′2corresponds to rotations around the x and y axes. C ′′2 corresponds to rotationsaround the xy-axes.(a) Breathing mode (b) Osculating mode (c) Rotating mode (d) Clapping modeFigure 2.2: Pictorial representation of all four possible dx2−y2 nonequilibriumHiggs amplitude modes, as allowed by group symmetry considerations. In termsof the group symmetry they are the (a) A1g breathing mode, (b) B1g osculatingmode, (c) A2g rotating mode, (d) B2g clapping mode.2.2. Model Hamiltonian 102.2 Model HamiltonianWe now consider a time-dependent Hamiltonian. In particular, we study the time-evolution of aBCS superconductor in the presence of some time-varying electromagnetic potential, which will takethe form of a realistic experimental pumping or probing laser. As a starting point, a mean-field BCSsuperconductor HMF is considered together with some electromagnetic interaction HEM . This willprove to be an auspicious starting point, as many experimental techniques probe such light-mattermaterial interactions. We are especially interested in the class of pump-probe experiments that probethe time-dependence of particular properties following some electromagnetic pump excitation, whichdisturbs the equilibrium of the material. The Hamiltonian is,H = HMF +H(1)EM +H(2)EM (2.1)where we have broken up the electromagnetic interaction term into linear and quadratic order terms.Following the discussion from the previous chapter, the BCS mean-field Hamiltonian is given by,HMF =∑k,σkc†k,σck,σ +∑k∈W[∆kc†k↑c†−k↓ + ∆∗kc−k↓ck↑]. (2.2)where c†k,σ and ck,σ are, respectively, the electron creation and annihilation operators for an electronof momentum k and spin σ. k = ~2k2/(2m)− EF is the electron dispersion for a single quadraticband with a circular Fermi surface, m is the effective mass and EF is the Fermi energy level. W isthe set of all momentum vectors k, such that |k| ≤ ~ωc for some cutoff energy ~ωc in the pairinginteraction. Finally, ∆ is the mean-field gap-parameter determined by the microscopic interactionVk,k′∆k′ =∑k∈WVk,k′〈c−k↓ck↑〉. (2.3)The electromagnetic portion of the Hamiltonian in second quantization is,H(1)EM =e~2m∑k,q,σ(2k+ q) ·Aq(t)c†k+q,σckσ (2.4a)H(2)EM =e22m∑k,q,σ∑q′Aq−q′(t) ·Aq′(t) c†k+q,σckσ (2.4b)where Aq is the electromagnetic vector potential for a momentum transfer q.We now make a change of basis to Bogoliubov quasiparticle operators, which are again given inEq. 1.4. This change of basis will be more intuitive when we consider the time-dependent equationsof motion for the superconductor. Under this change of basis, the mean-field superconductingHamiltonian (Eq. 2.2) becomes,HMF =∑k[Rkα†kαk −Rkβkβ†k + Ckα†kβ†k + C∗kβkαk](2.5)2.3. Equations of motion 11where Rk and Ck are the same as in Eq. 1.6. We now also need to make the change of basis for theelectromagnetic term HEM .H(1)EM =e~2m∑k,q(2k+ q) ·Aq(t)[(u∗k+quk + vk+qv∗k)α†k+qαk − (v∗k+qvk + u∗kuk+q)β†kβk+q+(vk+qu∗k − u∗k+qvk)α†k+qβ†k + (v∗kuk+q − v∗k+quk)βk+qαk](2.6a)H(2)EM =e22m∑k,q∑q′Aq−q′(t) ·Aq′(t)[(u∗k+quk − vk+qv∗k)α†k+qαk − (v∗k+qvk − u∗kuk+q)β†kβk+q−(vk+qu∗k + u∗k+qvk)α†k+qβ†k − (v∗kuk+q + v∗k+quk)βk+qαk](2.6b)Though the phase in a superconductor is arbitrary, we can fix a specific initial phase of ∆ for ourcalculations. In fact, we have verified that varying this initial choice of phase has no impact on thecalculation beyond the initial phase-offset. Therefore, to simplify notation, we choose our initial ∆to be real and positive and employ the following shorthand for the subsequent sections,L±k,q = uk+quk ± vk+qvkM±k,q = uk+qvk ± vk+quk(2.7)2.3 Equations of motionTo calculate the time-dependence of various quantities in the system, there are generally two differentapproaches. One can take the quasiparticle operators to be time-dependent and the states themselvesto be time-independent (the Heisenberg picture), or the quasiparticle states to be time-dependentand the operators time-independent (the Schro¨dinger picture). We will utilize the first approach,which involves solving the Heisenberg equations of motion for an operator Aˆ,ddtAˆ(t) =i~[H, Aˆ(t)] +(∂Aˆ∂t)H(2.8)Since our interest is in determining experimental observables, our choice of working in the Heisenbergpicture makes taking any expectation value with respect to our time-independent states trivial. Theparticular values we will be interested in are the four quasiparticle expectation values that appear inour Hamiltonian with various momenta: 〈α†kαk+q〉(t), 〈β†kβk+q〉(t), 〈α†kβ†k+q〉(t) and 〈αkβk+q〉(t).The α and β operators are intrinsically time-independent. Therefore, for these expectation values,the last term in Eq. 2.8 will be identically zero. The Heisenberg equations of motion we need to2.3. Equations of motion 12solve are:i~ddt〈α†kβ†k+q〉 = −〈[H,α†kβ†k+q]〉= −〈[Hsc +H(1)em +H(2)em, α†kβ†k+q]〉, (2.9a)i~ddt〈αkβk+q〉 = −〈[H,αkβk+q]〉 = −〈[Hsc +H(1)em +H(2)em, αkβk+q]〉, (2.9b)i~ddt〈α†kαk+q〉 = −〈[H,α†kαk+q]〉= −〈[Hsc +H(1)em +H(2)em, α†kαk+q]〉, (2.9c)i~ddt〈β†kβk+q〉 = −〈[H,β†kβk+q]〉= −〈[Hsc +H(1)em +H(2)em, β†kβk+q]〉(2.9d)If we return to our Hamiltonian, this means that the order parameter describing the system willincur an explicit time-dependence. Rewriting ∆(t) in terms of Bogoliubov quasiparticle expectationvalues, we find∆k′(t) =∑k∈WVk,k′[ukvk(〈α†kαk〉(t) + 〈β†kβk〉(t)− 1)− u2k〈αkβk〉(t)− v2k〈α†kβ†k〉(t)](2.10)where the explicit time-dependence has been included for all of the constituents.2.3.1 Equations of motion following the pumpBefore we proceed with the explicit calculation, it is convenient to choose a specific electromagneticfield profile, which can help to simplify the number of equations we need to solve. The particularchoice we make for the electromagnetic field, which will constitute the pumping laser in a pump-probe experiment, is a classical monochromatic laser source – an electromagnetic field with a singlewell-defined frequency and momentum. The exact time-dependent profile need not be fixed, howeverwe choose a Gaussian shape, which should be representative of most experimental setups. We definethe vector potential for the pump, Aq(t),Aq(t) = Ap exp−(2√ln2tτp)2 (δq,q0e−iωpt + δq,−q0eiωpt) (2.11)where the amplitude of the pump is Ap and the full width at half maximum is τp. The pumpingfrequency and momentum are given by ωp and qp. We also choose a linear polarization for Ap suchthat the momentum vector qp and the vector potential Aq(t) are orthogonal vectors as required byelectromagnetic theory.Taking the commutator with the Hamiltonian, the time-dependent equations of motion for ourfour quasiparticle expectation values are given by:2.3. Equations of motion 13i~ddt〈α†kβ†k′〉 = −(Rk +Rk′)〈α†kβ†k′〉+ C∗k′〈α†kαk′〉+ C∗k(〈β†k′βk〉 − δk′,k)+e~2m∑q′=±q02k ·Aq′(t)[−L+k,q′〈α†k+q′β†k′〉+ L+k′,−q′〈α†kβ†k′−q′〉−M−k′−q′〈α†kαk′−q′〉+M−k,q′(〈β†k′βk+q′〉 − δk′−k,q′)]+e22m∑q′=0,±2q0∑qi=±q0Aq′−qi(t) ·Aqi(t)[−L−k,q′〈α†k+q′β†k′〉 − L−k′,−q′〈α†kβ†k′−q′〉−M+k′−q′〈α†kαk′−q′〉+M+k,q′(−〈β†k′βk+q′〉+ δk′−k,q′)](2.12)i~ddt〈αkβk′〉 = +(Rk +Rk′)〈αkβk′〉+ Ck′〈α†k′αk〉+ Ck(〈β†kβk′〉 − δk′,k)+e~2m∑q′=±q02k ·Aq′(t)[+L+∗k,q′〈αk+q′βk′〉 − L+∗k′,−q′〈αkβk′−q′〉−M−∗k′−q′〈α†k′−q′αk〉+M−∗k,q′(〈β†k+q′βk〉 − δk′−k,q′)]+e22m∑q′=0,±2q0∑qi=±q0Aq′−qi(t) ·Aqi(t)[+L−∗k,q′〈αk+q′βk′〉+ L+∗k′,−q′〈αkβk′−q′〉−M+∗k′−q′〈α†k′−q′αk〉+M+∗k,q′(−〈β†k+q′βk〉+ δk′−k,q′)](2.13)i~ddt〈α†kαk′〉 = +(Rk′ −Rk)〈α†kαk′〉+ Ck′〈α†kβ†k′〉+ C∗k〈αk′βk〉+e~2m∑q′=±q02k ·Aq′(t)[−L+k,q′〈α†k+q′αk′〉+ L+∗k′,−q′〈α†kαk′−q′〉+M−k,q′〈αk′βk+q′〉+M−∗k′,−q′〈α†kβ†k′+q′〉]+e22m∑q′=0,±2q0∑qi=±q0Aq′−qi(t) ·Aqi(t)[−L−k,q′〈α†k+q′αk′〉+ L−∗k′,−q′〈α†kαk′−q′〉−M+k,q′〈αk′βk+q′〉 −M+∗k′,−q′〈α†kβ†k′+q′〉](2.14)2.3. Equations of motion 14i~ddt〈β†kβk′〉 = +(Rk′ −Rk)〈β†kβk′〉+ Ck′〈α†kβ†k′〉+ C∗k〈αk′βk〉+e~2m∑q′=±q02k ·Aq′(t)[+L+k,−q′〈β†k−q′βk′〉 − L+∗k′,q′〈β†kβk′+q′〉−M−k,q′〈αk′βk+q′〉 −M−∗k′,−q′〈α†kβ†k′+q′〉]+e22m∑q′=0,±2q0∑qi=±q0Aq′−qi(t) ·Aqi(t)[−L−k,−q′〈β†k−q′βk′〉+ L−∗k′,q′〈β†kβk′+q′〉−M+∗k′,q′〈α†k′+q′β†k〉 −M+k,−q′〈αk−q′βk′〉](2.15)After integrating these differential equations, one can determine the pump induced changes in thesuperconducting order parameter and by extension, the amplitude mode Higgs oscillations, via Eq. Equations of motion following the probeNext, we want to consider the equations of motion following a second probing pulse, which willmeasure experimental observables. In particular, we are interested in the optical response, whichcan be related to the current density induced by the probe, jqpr . Again we split up the responseinto two parts,j−qpr = j(1)−qpr + j(2)−qpr (2.16a)j(1)−qpr =−e~2mV∑k,σ(2k+ qpr)c†k,σck+qpr,σ (2.16b)j(2)−qpr = −e2mV∑k,q,σAqpr−qc†k,σck+qpr,σ (2.16c)where V is the normalization volume. The second term j(2)qpr , is the diamagnetic current density,which has previously been shown [37, 38] to only lead to an offset in the imaginary part of thespectrum and therefore may be neglected. Performing the same transformation to the Bogoliubovquasiparticle basis, the current density j(1)qpr is given by,j(1)−qpr =−e~2mV∑k(2k+ qpr)[(u∗kuk+qpr + vkv∗k+qpr)α†kαk+qpr − (v∗kvk+qpr + u∗k+qpruk)β†k+qprβk+(vku∗k+qpr− u∗kvk+qpr)α†kβ†k+qpr + (v∗kuk+qpr − v∗k+qpruk)αk+qprβk](2.17)With the current density, we can then calculate other quantities such as the optical conductivity σ,σ(ω) =〈jqpr〉(ω)iωAqpr(ω)(2.18)by taking the Fourier transform of j(t) and Aqpr(t), where the probing vector potential takes the2.3. Equations of motion 15same functional form as the pumping vector potential in Eq. 2.11. The exact parameters of theprobe will differ from those used in the pump.We now turn to the equations of motion following the probing pulse. As is the case with mostoptical experimental setups, we work with a probing pulse polarized perpendicular to the pumpingpulse so that the off-diagonal terms are decoupled between the two pulses. To simplify this portionof the calculation, we assume that the probe pulse is sufficiently weak that it need only be calculatedto linear order. Second, we assume that the prominent excitations are induced by the pump, so thatwe can approximate 〈α†k+qprβk+qpr〉 ' 〈α†k+qpβk+qp〉, where we have used the subscripts pr and pto represent the probe and pump respectively.The differential equations for the expectation values of the quasiparticles needed to calculate theoptical response of the probe are as follows:i~ddt〈α†kβ†k+qpr〉 = −(Rk +Rk+qpr)〈α†kβ†k+qpr〉+ C∗k+qpr〈α†kαk+qpr〉+ C∗k〈β†k+qprβk〉+e~2m2k ·Aqpr(t)[−L+k,qpr〈α†k+qpβ†k+qp〉+ L+k,qpr〈α†kβ†k〉+M−k,qpr〈α†kαk〉+M−k,qpr(〈β†k+qpβk+qp〉 − 1)](2.19)i~ddt〈αk+qprβk〉 = +(Rk +Rk+qpr)〈αk+qprβk〉+ Ck〈α†kαk+qpr〉+ Ck+qpr〈β†k+qprβk〉+e~2m2k ·Aqpr(t)[+L+k,qpr〈αkβk〉 − L+k,qpr〈αk+qpβk+qp〉−M−k,qpr〈α†k+qprαk+qpr〉 −M−k,qpr(〈β†kβk〉 − 1)](2.20)i~ddt〈α†kαk+qpr〉 = −(Rk −Rk+qpr)〈α†kαk+qpr〉+ Ck+qpr〈α†kβ†k+qpr〉+ C∗k〈αk+qprβk〉+e~2m2k ·Aqpr(t)[−L+k,qpr〈α†k+qpαk+qp〉+ L+k,qpr〈α†kαk〉+M−k,qpr〈αk+qpβk+qp〉 −M−k,qpr〈α†kβ†k〉](2.21)i~ddt〈β†k+qprβk〉 = +(Rk −Rk+qpr)〈β†k+qprβk〉+ Ck〈α†kβ†k+qpr〉+ C∗k+qpr〈αk+qprβk〉+e~2m2k ·Aqpr(t)[L+k,qpr〈β†kβk〉 − L+k,qpr〈β†k+qpβk+qp〉+M−k,qpr〈αkβk〉 −M−k,qpr〈α†k+qpβ†k+qp〉](2.22)2.3. Equations of motion 162.3.3 Initial conditions and numerical approximationsThe Bogoliubov quasiparticles are fermions and obey Fermi statistics. Therefore, we can calculatethe occupation values for a given temperature prior to the pump or probe pulse as follows,〈α†kαk′〉 =1eEkkBT + 1δk,k′ (2.23a)〈β†kβk′〉 =1eEkkBT + 1δk,k′ (2.23b)〈α†kβ†k′〉 = 0 (2.23c)〈αkβk′〉 = 0 (2.23d)where Ek =√k + ∆k is the quasiparticle energy. For our calculations, we limit ourselves to theT = 0 case. Qualitatively, the results are similar for sufficiently low temperatures.Next, we choose the superconducting symmetries to consider. We primarily study the two wellknown superconductor symmetries: s-wave and d-wave. In this case, our interaction term in Eq.2.10, Vk,k′ , is of the formVk,k′ = V (2.24)orVk,k′ = V (cos kx − cos ky)(cos k′x − cos k′y) (2.25)for s-wave and d-wave (dx2−y2) respectively. However, it would be trivial to extend our formalismto other interaction symmetries in one and two-dimensions, such as p-wave.We can now proceed to fix V , or ∆ and the other superconducting properties. When choosingwhich type of pump and probe, the pumping timescale should be on the same scale as the responsetime of the superconductor, which turns out to be in the order of a few hundred femptoseconds forsuperconducting gaps in the order of a few meVs. As discussed previously, this is required to inducea non-adiabatic excitation such as the Higgs amplitude mode. The superconducting parametersused throughout this thesis were historically used to study a lead superconductor, however we haveinvestigated the effects of varying superconducting parameters and for a reasonable choice of pumpand probe pulses, the effects are purely quantitative and have do not change the qualitative results.Therefore, we proceed with the same parameters for both s-wave and d-wave for comparison. Theexact values used in our calculation are presented in Table 2.2.We also need to make certain choices to simplify the numerical calculations, so long as they donot affect the qualitative results of our calculations. We take a 2-D momentum grid with spacingalong the pumping momentum direction equal to the pumping momentum transfer. This may seemunjustified at first, but our pumping momentum is extremely small (see Table 2.2) and we havealso numerically verified that choosing a smaller discretization does not affect the calculation. Inthe second momentum direction, we choose to rather discretize the angle for simplicity. This turnsout to being equivalent to choosing Chebyshev points along the second direction. Furthermore, thisgives us the most equal distribution of points between nodes for the case of a d-wave superconductor.For the d-wave case, we approximate the pairing as ∆k = ∆0 cos 2θ. We also limit the number ofoff-diagonal terms to four. In Eq. 2.3.1, this amounts to restricting |k − k′| < 5q0. Lastly, the2.4. Results 17ParametersSuperconductor Pump Probea 10−10 m ~ωp 3.0 meV ~ωpr 2.5 meVEF 9470 meV ωp 4.56×10−3 fs-1 ωpr 3.80×10−3 fs-1m 1.9me Ap 7.0× 10−8 J sC m Apr 0.7× 10−8 J sC mwc 8.3 meV τp 400 fs-1 τpr 250.0 fs-1∆0 1.35 meV qp a 1.52× 10−6 qpr a 1.26× 10−6Table 2.2: Parameters for our calculations. a is the lattice spacing, EF is theFermi energy, m is the re-normalized mass and wc is the cutoff frequency. ∆0 isthe initial size of the gap before the pump is turned on. ωp/pr , Ap/pr, τp/pr andqp/pr are the pump/probe frequency, amplitude, full width at half maximum andmomentum.standard fourth-order Runge-Kutta method is used to time-evolve the density matrices accordingto the equations of motion.2.4 Results2.4.1 Higgs oscillationsFollowing the application of a pumping laser, the magnitude of the order parameter decreases andoscillates due to changes in the free energy potential of the superconductor. The degree to which thepotential is altered depends both on the amplitude and length of the pump. For longer duration andlarger amplitudes, more Cooper pairs are depleted, which decreases the central value about whichthe order parameter oscillates. The strongest signal occurs for a pumping laser on the order of afew hundred femptoseconds, or faster than the intrinsic superconductor response time. Additionally,the amplitude or fluence of the pump should not be so large that the number of Cooper pairs is sosignificantly depleted that oscillations cannot occur.The results for our specific parameters are presented in Fig. 2.3. At t = 0, the pump is turned onand the magnitude of the order parameter rapidly decreases as Cooper pairs are broken; the orderparameter then begins to oscillate after the pump is turned off. In Fig. 2.3(a) , the Higgs oscillationsof an isotropic, s-wave superconductor interacting with a laser are presented. As previously foundfor the case of an isotropic superconductor, ([37, 38], etc.) the oscillations decay with a characteristic1/√t dephasing of the Bogoliubov quasiparticles and the oscillations themselves have a characteristic2∆ frequency (Fig. 2.3(c)), where ∆ is the new order parameter in the t = ∞ limit. Increasingthe total amount of incident energy, will further decrease this value of the order parameter, as wellas the frequency of the oscillations. These results are contrasted with the Higgs oscillations of adx2−y2 superconductor (Fig. 2.3(b)). Most notably, the dx2−y2 superconductor oscillations dependstrongly on the angle between the superconductor and polarization of the laser. φ = 0 correspondsto a magnetic vector potential aligned along one of the dx2−y2 antinodes (and subsequently, the2.4. Results 180 5 10 15 20time (ps)∆(t) (meV)(a) s-wave0 5 10 15 20time (ps)∆(t) (meV)φ = 0φ = pi/16φ = pi/8φ = 3 pi/16φ = pi/4(b) dx2−y20 1 2 3 4Frequency (meV)0123456Fourier transform ∆(t)×10 -4sdx2-y 2 φ = 0dx2-y 2 φ = pi/16dx2-y 2 φ = pi/8dx2-y 2 φ = 3 pi/16dx2-y 2 φ = pi/42∆(c)Figure 2.3: Comparison of s-wave (a) and d-wave (b) Higgs oscillations followingthe pumping pulse for the parameters given in Table 2.2. (c) Fourier transformof (a) and (b). φ is the relative angle between the vector potential and thedx2−y2 antinode.momentum vector also points perpendicularly along an adjacent node), while φ = pi/4 correspondsto having the vector potential aligned along one of the nodes. Moreover, the oscillations themselvesare much smaller in amplitude and have a much more rapid decay, which has partly contributed tothe difficulty thus far in detecting Higgs oscillations in anisotropic superconductors. The differencesare perhaps best illustrated in the Fourier transform of the oscillations (Fig. 2.3(c)). While theisotropic, s-wave superconductor has a single sharp oscillation frequency at 2∆, depending on thepolarization angle φ, the dx2−y2 superconductor has as many as 2 oscillation frequencies. As thepolarization angle φ is rotated from the node to the antinode, the amplitude of the 2∆ mode decreasesand a second mode develops below 2∆. The frequency of this second mode also depends stronglyon the amplitude and particularly the duration of the pump. The exact symmetry of this secondmode will become more apparent in the following chapter.2.4.2 Optical responseAfter the pump, a probing pulse is applied to measure the oscillations of the gap parameter ∆(t). Inreality, many probing pulses are applied to measure the oscillation at various time-delays δt between2.4. Results 192345δ t (ps)(a)5101520251 2 3 4 5Energy (meV)2345δ t (ps)(d) 2 3 4 5Energy (meV)(e) 2 3 4 5Energy (meV)(f)0.511.52∆ 2∆2∆ 2∆2∆2∆Figure 2.4: Comparison of different optical conductivity Higgs responses be-tween (a) an s-wave superconductor and (b-f) a dx2−y2 superconductor for theparameters in Table 2.2. The relative angle φ between the vector potential andthe dx2−y2 antinode is (b) φ = 0 (c) φ = pi/16 (d) φ = pi/8 (e) φ = 3pi/16 (f)φ = pi/4 .the pump and the probe, each of which define a different set of differential equations to be solved.δt is a measure of the time-difference between the centers of the pump and probe pulses. For pumpand probe time-scales on the order of a few hundred femptoseconds, this means that the smallestallowable time-delay, without significant pump and probe overlap, is around one picosecond.The results for our parameters are presented in Fig. 2.4. For an isotropic s-wave superconductor(Fig. 2.4 (a)), there is clearly only a single, sharp mode in the conductivity spectra at 2∆. Theoscillation frequency in delay-time δt is also equal to 2∆ as expected from our analysis of the orderparameter after the pump. In contrast, the conductivity response of a dx2−y2 superconductor is verydifferent. For a linear pump polarization such that the vector potential is aligned along the dx2−y2antinode (φ = 0), there is a broad mode below 2∆. The exact energy of the second mode depends onthe specifics of the pump, but not on the geometry. As the vector potential of the pump is rotatedtowards the node (away from the anti-node), the intensity of the low energy mode decreases, whilethe intensity of the 2∆ mode increases until around an angle of pi/8 between the vector potentialand the d-wave anti-node, at which point the second mode is no longer visible in the conductivityspectra. The intensity of the 2∆ mode increases to a maximum when the vector potential andd-wave node become perfectly aligned (φ = pi/4). As for the oscillations in time-delay, they haveboth the frequencies of the 2∆ mode and the mode below 2∆. Lastly, the signal has an eight-foldreflection symmetry as the pump polarization is rotated around the remaining 7pi/4 degrees; thesignal is identical to the reflection across the pi/4, C ′′2 axis and the C′4 axis. In other words, thereis no differentiation between positive and negative antinodes, nor is there a difference for points ofequal amplitude across a single antinode.2.5. Discussion 202.5 DiscussionAs we have shown, the Higgs amplitude mode of a nonequilibrium superconductor can be excitedand measured in an optical pump-probe experiment with terahertz pumping energies and a pumpingduration on the order of a few hundred femptoseconds. Moreover, the Higgs amplitude mode canbe used to differentiate between isotropic and anisotropic superconductors, simply by rotating thesample relative to the polarization of the pump. For a dx2−y2 superconductor, this corresponds to theexcitation of an additional mode below the isotropic 2∆ mode as the vector potential is rotated to liealong an antinode. In an optical experiment, this can be detected either by noting the appearance ofa low energy mode below the 2∆ energy or by observing changes to the frequency of the oscillationsas a function of time-delay between the pump and probe. Our results also can easily be extendedto different symmetries, for instance s± or p symmetries, which we expect will also be simple todifferentiate from the two cases presented here, but will follow similar trends. Therefore, amongother possibilities, exciting the Higgs amplitude mode via Higgs spectroscopy offers an unambiguousmethod for studying and differentiating between the different superconducting symmetries in amaterial.What remains is to identify the specific symmetries of the nonequilibrium modes. While weexpect the 2∆ mode in the dx2−y2 superconductor to correspond to the isotropic breathing modesince this is the only possible mode for the isotropic s-wave superconductor, the mode below 2∆ in thedx2−y2 superconductor is not quite as easy to identify using momentum integrated spectroscopy. Wedo not expect the nonequilibrium mode to acquire any angular momentum, therefore the osculatingmode is the most likely candidate. However, this does not rule out the potential of generating theseadditional nonequilibrium modes by introducing additional pumping pulses of various geometries,which could produce the angular momentum necessary to generate other nonequilibrium modes. Todetermine the exact symmetry which is broken in this setup, it will be necessary to study the Higgsmodes with momentum resolved spectroscopy, presented in the next chapter.21Chapter 3Nonequilibrium Superconductivity:ARPES3.1 Introduction to Green’s functionsWe turn now to the study of nonequilibrium superconductivity beyond the context of optical ex-periments. Namely, we want to study how the Higgs oscillation(s) will manifest in various otherexperimental contexts. In condensed matter physics, one of the most powerful tools for describinga system is the set of Green’s functions, which are closely related to various experimental observ-ables such as those measured in angle resolved photoemission spectroscopy (ARPES) [12, 13, 17]or scanning tunneling microscopy (STM). Since we will be working at zero temperature, only thezero-temperature Green’s functions are presented. The two-time electron Green’s function for atranslationally invariant system is given by,Gσ,σ′(k′, t, t′) = −i〈T ck,σ(t), c†k,σ′(t′)〉 (3.1)where T is the time ordering operator and c†k,σ (ck,σ) creates an electron (hole) with momentumk and spin σ. For the sake of simplicity and relevance to our problem, we only consider Green’sfunctions for states of equal spin, since our Hamiltonian is spin symmetric. The Green’s function isalso closely related to four other quantities, the nonequilibrium greater and lesser Green’s functions,G≶(k, t, t′) and the retarded and advanced Green’s functions, GR(k, t, t′) and GA(k, t, t′). They aredefined as follows:G>(k′, t, t′) = −i〈ck,σ(t), c†k,σ(t′)〉 (3.2a)G<(k′, t, t′) = +i〈c†k,σ(t′), ck,σ(t)〉 (3.2b)GR(k′, t, t′) = −iΘ(t− t′)〈{ck,σ(t), c†k,σ(t′)}〉 (3.2c)GA(k′, t, t′) = +iΘ(t′ − t)〈{c†k,σ(t), ck,σ(t′)}〉. (3.2d)In non-equilibrium photoemission spectroscopy, the ARPES intensity IARPES , is related to thespectral function, A(k, ω) of the non-equilibrium Green’s function. Often, this is presented simplyin terms of the lesser Green’s function, G<(k, ω), which gives us information about the electronicdensity (as opposed to information regarding the hole density, which is given by G>(k, ω)).IARPES ∝ Im G<(k, ω, δt) (3.3)where G<(k, ω, δt) is the Fourier transform of G<(k, t′ − t, δt) and δt is the time-difference between3.2. Model Hamiltonian 22the pump and the probe. The non-equilibrium spectral function also relates the retarded andadvanced Green’s functions to the lesser and greater Green’s functions according toA(k, t, t′) = G>(k, t, t′)−G<(k, t, t′) = GR(k, t, t′)−GA(k, t, t′). (3.4)Therefore, in order to study properties of the nonequilibrium ARPES signal, we must develop aformalism for evaluating the expectation value for two times rather than a single time as we havedone previously.3.2 Model HamiltonianSince we want to study the same phenomena and how it manifests in different experiments, we beginwith the same Hamiltonian as in the previous chapter – a mean-field BCS superconductor placed inan arbitrary electromagnetic field.H = HMF +H(1)EM +H(2)EM (3.5)For clarity, the relevant equations for the Hamiltonian are presented again. In terms of the Bogoli-ubov quasiparticles,αk = u∗kck,↑ + vkc†−k↓ (3.6a)βk = u∗kc−k↓ − vkc†k↑ (3.6b)the mean-field Hamiltonian is again of the form,HMF =∑k[Rkα†kαk −Rkβkβ†k + Ckα†kβ†k + C∗kβkαk](3.7)where Rk and Ck are the same as in Eq. 1.6. The electromagnetic Hamiltonian terms are,H(1)EM =e~2m∑k,q(2k+ q) ·Aq(t)[(u∗k+quk + vk+qv∗k)α†k+qαk − (v∗k+qvk + u∗kuk+q)β†kβk+q+(vk+qu∗k − u∗k+qvk)α†k+qβ†k + (v∗kuk+q − v∗k+quk)βk+qαk](3.8a)H(2)EM =e22m∑k,q∑q′Aq−q′(t) ·Aq′(t)[(u∗k+quk − vk+qv∗k)α†k+qαk − (v∗k+qvk − u∗kuk+q)β†kβk+q−(vk+qu∗k + u∗k+qvk)α†k+qβ†k − (v∗kuk+q + v∗k+quk)βk+qαk](3.8b)The same shorthand is employed as before,L±k,q = uk+quk ± vk+qvkM±k,q = uk+qvk ± vk+quk(3.9)3.3. Time-dependent ansatz 233.3 Time-dependent ansatzWe need to develop a time-dependent operator ansatz, which will allow us to calculate the valuesof α†k(t), β†k(t) – and thus also c†k,σ(t) and ck,σ(t) and all relevant two-time Green’s functions. Ourtime-dependent ansatz is of the form [19–21],α†k(t) =∑δ[ak+δ(t)α†k+δ + bk+δ(t)βk+δ + ... (3.10)where we maintain our original time-independent operator basis, but take into account time-dependentprefactors and mixing with the other operators in our basis. This ansatz arises from the Heisen-berg picture for the time-dependence of the Hamiltonian. In the Heisenberg picture, due to thetime-dependence of the Hamiltonian, H(t), a time-dependent operator, U(t, t0) is introduced givenby,U(t, t0) = exp(−i∫ tt0H(t′)dt′)(3.11)which yields a simple expression for the time-dependence of U(t, t0),i~∂∂tU(t, t0) = H(t)U(t, t0) (3.12)The time-dependence of a given operator can then be calculated from some initial value t0, whichcan be set to zero for simplicity,A(t) = U†(t)A(0)U(t) (3.13)This results in the very simple equation for the time evolution of A(t),i~ddtA(t) = i~ddtU†(t)A(0)U(t) + i~U†(t)A(0)ddtU(t) (3.14a)= −U†(t)H(t)A(0)U(t) + U†(t)A(0)H(t)U(t) (3.14b)= [U†(t)A(0)U(t), U†(t)H(t)U(t)] (3.14c)= [A(t), H˜(t)] (3.14d)with H˜(t) given by, H˜(t) = U†(t)H(t)U(t). Under this transformation, it is possible to calculatethe new Hamiltonian in terms of the time-dependent operators.Our ansatz for the time-dependence of the operators arises from this equation in the following way.If we begin with the simple assumption that for the time-dependence of our operator A(t) = a(t)A,then by calculating the commutation relations, depending on the details of the Hamiltonian, weeither can arrive at a closed form for the time-dependence of our prefactor a(t), or introduce furthertime-dependent prefactors for another operator in the system. If the latter is the case, our firstassumption is adjusted by adding the new operator and prefactor, and now take into account thatthe Hamiltonian, H˜(t) has changed as well. This process can then be repeated to improve theaccuracy of the time-dependence by adding in additional terms for more operators in the system.Alternatively, one can make a truncation by commuting the current ansatz for the time-dependencewith the newly derived Hamiltonian.3.3. Time-dependent ansatz 243.3.1 Model Hamiltonian with the ansatzFortunately, our Hamiltonian is bilinear. Therefore, when we apply this method to our Hamiltonian,the time-dependent ansatz for our operators depends only on two of our time-independent operators,α†k(t) =∑δlk,δ(t)α†k+δ +mk,δ(t)βk+δ (3.15a)β†k(t) =∑δnk,δ(t)β†k+δ + ok,δ(t)αk+δ (3.15b)with the initial conditions that lk(0) = 1, nk(0) = 1, lk,δ = nk,δ = 0 ∀ δ s.t. δ 6= 0 and mk,δ(0) =ok,δ(0) = 0 ∀ δ. δ can, in general, correspond to some arbitrary momentum transfer from thecommutator with the electromagnetic part of the Hamiltonian. However, in the next section, welimit the pump to a single momentum, which will restrict δ to integer multiples of the momentumq corresponding to to the chosen value for the pumping pulse Aq.Though in the previous chapter it was beneficial to break up the electromagnetic Hamiltonianinto linear and quadratic parts to differentiate between the quadratic excitations generated by thepump and the linear response of the probe, we will now focus solely on the former and so work witha single electromagnetic term. Our Hamiltonian in terms of our ansatz is now,H˜ = H˜MF + H˜EM (3.16)with H˜MF given now in terms of the new ansatz for our operators asH˜MF =∑k,δ,η[α†k+δαk+η((lk,δl∗k,η − o∗k,δok,η)Rk + (lk,δok,η)Ck + (l∗k,ηo∗k,δ)C∗k)+β†k+ηβk+δ((−m∗k,ηmk,δ + nk,ηn∗k,δ)Rk + (−mk,δnk,η)Ck − (m∗k,ηn∗k,δ)C∗k)+α†k+δβ†k+η((lk,δm∗k,η − nk,ηo∗k,δ)Rk + (lk,δnk,η)Ck + (o∗k,δm∗k,η)C∗k)+βk+δαk+η((mk,δl∗k,η − n∗k,δok,η)Rk + (mk,δok,η)Ck + (n∗k,δl∗k,η)C∗k)+δδ,η((−o∗k,δok,η +m∗k,ηmk,δ)Rk + (mk,δnk,η)Ck + (m∗k,ηn∗k,δ)C∗k)].(3.17)The explicit time-dependence of l,m, n and o has been excluded for compactness. We can simplifythe Hamiltonian slightly by dropping the last term in the Hamiltonian, as it only amounts to aconstant energy offset and does not effect the dynamics of the system. H˜EM given in terms of thenew ansatz for our operators is,3.3. Time-dependent ansatz 25H˜EM =[e~2m∑k,q,δ,η(2k+ q)Aq]×[α†k+q+δαk+η((lk+q,δl∗k,η + o∗k+q,δok,η)L+k,q + (−lk+q,δok,η + o∗k+q,δl∗k,η)M−k,q)+β†k+ηβk+q+δ((−m∗k,ηmk+q,δ − nk,ηn∗k+q,δ)L+k,q + (+mk+q,δnk,η − n∗k+q,δm∗k,η)M−k,q)+α†k+q+δβ†k+η((lk+q,δm∗k,η + nk,ηo∗k+q,δ)L+k,q + (−lk+q,δnk,η + o∗k+q,δm∗k,η)M−k,q)+βk+q+δαk+η((mk+q,δl∗k,η + n∗k+q,δok,η)L+k,q + (−mk+q,δok,η + n∗k+q,δl∗k,η)M−k)+δδ+q,η((−o∗k+q,δok,η +m∗k,ηmk+q,δ)L+k,q + (mk+q,δnk,η + n∗k+q,δm∗k,η)M−k,q)]+e22m∑k,q,δ,η(∑q′Aq−q′ ·Aq′)[α†k+q+δαk+η((lk+q,δl∗k,η − o∗k+q,δok,η)L−k,q − (lk+q,δok,η + o∗k+q,δl∗k,η)M+k,q)+β†k+ηβk+q+δ((−m∗k,ηmk+q,δ + nk,ηn∗k+q,δ)L−k,q − (−mk+q,δnk,η − n∗k+q,δm∗k,η)M+k,q)+α†k+q+δβ†k+η((lk+q,δm∗k,η − nk,ηo∗k+q,δ)L−k,q − (lk+q,δnk,η + o∗k+q,δm∗k,η)M+k,q)+βk+q+δαk+η((mk+q,δl∗k,η − n∗k+q,δok,η)L−k,q − (mk+q,δok,η + n∗k+q,δl∗k,η)M+k)+δδ+q,η((−o∗k+q,δok,η +m∗k,ηmk+q,δ)L+k,q + (mk+q,δnk,η + n∗k+q,δm∗k,η)M−k,q)](3.18)Again, we can drop the last term as it only amounts to a constant offset in our Hamiltonian.We can also write an expression for the order parameter ∆k(t) in terms of the ansatz for ournew operators. Beginning with a modified Eq. 2.10 for the now explicit time-dependence of ouroperators,∆k′(t) =1N∑k∈WVk,k′(ukvk(1− 〈α†k(t)αk(t)〉 − 〈β†k(t)βk(t)〉) + u2k〈βk(t)αk(t)〉 − v2k〈α†k(t)β†k(t)).(3.19)We must calculate the expectation values for our new operators in terms of our ansatz. The firstexpectation value is given by,〈α†k(t)αk(t)〉 = 〈(∑ρlk,ρ(t)α†k+ρ +mk,ρ(t)βk+ρ)(∑δl∗k,ρ(t)αk+ρ +m∗k,ρ(t)β†k+ρ)〉 (3.20)and due to the fact that, for general operators A and B, 〈A†B〉 = 0 and 〈AB†〉 = δA,B we get a3.4. Equations of Motion 26simple expression for 〈α†k(t)αk(t)〉 and the three subsequent expectation values,〈α†kαk〉 =∑ρ|mk,ρ(t)|2 (3.21a)〈β†kβk〉 =∑ρ|ok,ρ(t)|2 (3.21b)〈βkαk〉 =∑ρn∗k,ρ(t)m∗k,ρ(t) (3.21c)〈α†kβ†k〉 =∑ρnk,ρ(t)mk,ρ(t) (3.21d)Clearly the final two expectation values are the complex conjugate of each other, since l,m, n and oare simple complex numbers. Therefore, we can rewrite our equation for the gap parameter ∆k′(t)as,∆k′(t) =1N∑k∈W,ρVk,k′(ukvk(1− |mk,ρ(t)|2 − |ok,ρ(t)|2) + u2kn∗k,ρ(t)m∗k,ρ(t)− v2knk,ρ(t)mk,ρ(t))(3.22)It is a simple matter to also calculate the other four expectation values for the remaining two operatorcombinations, which will be useful for calculating the Green’s functions later:〈αk(t)α†k(t)〉 =∑ρ|lk,ρ(t)|2 (3.23a)〈βk(t)β†k(t)〉 =∑ρ|nk,ρ(t)|2 (3.23b)〈αk(t)βk(t)〉 =∑ρl∗k,ρ(t)o∗k,ρ(t) (3.23c)〈β†k(t)α†k(t)〉 =∑ρlk,ρ(t)ok,ρ(t) (3.23d)3.4 Equations of MotionWe now derive the new equations of motion in terms of the time-dependent operators. In fact, thetime-dependent equations of motion are now equations of motion for the prefactors in our ansatz.Following the same methodology as in the previous chapter, we work with the Heisenberg equationsof motion. Our operators have an explicit time-dependent term now, however we assume that thechange in our operators is small compared to the commutator term. This should be justified giventhat we imposed the time-dependence as part of our ansatz; had we not, the operators would have noexplicit time-dependence. This should be equivalent to the time-independent basis of the previouschapter, since in both cases no time-dependence will be incurred beyond terms coming from the3.4. Equations of Motion 27commutator with the Hamiltonian. The equations of motion are:i~ddtα†k(t) =[α†k(t), H˜](3.24a)i~∑ρ( ddtlk,ρ(t)α†k+ρ +ddtmk,ρ(t)βk+ρ)=[α†k(t), H˜sc + H˜em], (3.24b)i~ddtβ†k(t) =[β†k(t), H˜](3.25a)i~∑ρ( ddtnk,ρ(t)β†k+ρ +ddtok,ρ(t)αk+ρ)=[β†k(t), H˜sc + H˜em]. (3.25b)ρ will be some integer multiple of the transferred momentum q from the electromagnetic portionof the Hamiltonian. For a relatively small total incident energy, the additional terms in ρ areproportionally small to the power of the integer, so sum will converge on short time-scales beyondsome sufficiently large cutoff integer.As before, we define the vector potential for the pump Aq(t),Aq(t) = Ap exp−(2√ln2tτ)2 (δq,q0e−iωpt + δq,−q0eiωpt) (3.26)where the amplitude of the pump is Ap and the full width at half maximum is τp. The pumpingfrequency and momentum are given by ωp and qp. We choose a linear polarization for Ap suchthat the momentum vector qp and the vector potential Aq(t) are perpendicular vectors as requiredby electromagnetic theory. Since we have once again chosen the form of a monochromatic pump(and vector potential), the equations of motion will only couple terms of integer multiples of thetransferred momentum q.While we now only have two explicit equations to solve, our operators (and our Hamiltonian) aresignificantly more complicated. The α†k commutator with the electromagnetic part of the Hamilto-nian is as follows,3.4. Equations of Motion 28[α†k′(t), H˜em]=[e~2m∑k,q,δ,η,γ(2k+ q)Aq]×[α†k+q+δ(−δk′+γ,k+η)(lk′,γ((lk+q,δl∗k,η + o∗k+q,δok,η)L+k,q + (−lk+q,δok,η + o∗k+q,δl∗k,η)M−k,q)+mk′,γ((lk+q,δm∗k,η + nk,ηo∗k+q,δ)L+k,q + (−lk+q,δnk,η + o∗k+q,δm∗k,η)M−k,q))+βk+q+δ(δk′+γ,k+η)(mk′,γ((−m∗k,ηmk+q,δ − nk,ηn∗k+q,δ)L+k,q + (+mk+q,δnk,η − n∗k+q,δm∗k,η)M−k,q)− lk′,γ((mk+q,δl∗k,η + n∗k+q,δok,η)L+k,q + (−mk+q,δok,η + n∗k+q,δl∗k,η)M−k,q))]+[e22m∑k,q,δ,η,γ(∑q′Aq−q′ ·Aq′)]×[α†k+q+δ(−δk′+γ,k+η)(lk′,γ((lk+q,δl∗k,η − o∗k+q,δok,η)L−k,q − (lk+q,δok,η + o∗k+q,δl∗k,η)M+k,q)+mk′,γ((lk+q,δm∗k,η − nk,ηo∗k+q,δ)L−k,q − (lk+q,δnk,η + o∗k+q,δm∗k,η)M+k,q))+βk+q+δ(δk′+γ,k+η)(mk′,γ((−m∗k,ηmk+q,δ + nk,ηn∗k+q,δ)L−k,q − (−mk+q,δnk,η − n∗k+q,δm∗k,η)M+k,q)− lk′,γ((mk+q,δl∗k,η − n∗k+q,δok,η)L−k,q − (mk+q,δok,η + n∗k+q,δl∗k,η)M+k,q))](3.27)and for β†k we obtain a similar expression. Collecting like terms, we then get the following differentialequations for the pre-factors of α†k′(t) by choosing k′ + ρ = k + δ and k′ + ρ = k+ q + δ in thesubsequent 2 sums:3.4. Equations of Motion 29i~ddtlk′,ρ(t)α†k′+ρ =−∑δ,γα†k′+ρ(lk′,γ((lk′+ρ−δ,δl∗k′+ρ−δ,γ+δ−ρ − o∗k′+ρ−δ,δok′+ρ−δ,γ+δ−ρ)Rk′+ρ−δ+ (lk′+ρ−δ,δok′+ρ−δ,γ+δ−ρ)Ck′+ρ−δ + (l∗k′+ρ−δ,γ+δ−ρo∗k′+ρ−δ,δ)C∗k′+ρ−δ)+mk′,γ((lk′+ρ−δ,δm∗k′+ρ−δ,γ+δ−ρ − nk′+ρ−δ,γ+δ−ρo∗k′+ρ−δ,δ)Rk′+ρ−δ+ (lk′+ρ−δ,δnk′+ρ−δ,γ+δ−ρ)Ck′+ρ−δ + (o∗k′+ρ−δ,δm∗k′+ρ−δ,γ+δ−ρ)C∗k′+ρ−δ))−[e~2m∑q,δ,γ(2(k′ − q+ ρ− δ) + q)Aq]×α†k′+ρ(lk′,γ((lk′+ρ−δ,δl∗k′+ρ−δ−q,γ+δ+q−ρ + o∗k′+ρ−δ,δok′+ρ−δ−q,γ+δ+q−ρ)L+k′−q+ρ−δ,q+ (−lk′+ρ−δ,δok′+ρ−δ−q,γ+δ+q−ρ + o∗k′+ρ−δ,δl∗k′+ρ−δ−q,γ+δ+q−ρ)M−k′−q+ρ−δ,q)+mk′,γ((lk′+ρ−δ,δm∗k′+ρ−δ−q,γ+δ+q−ρ + nk′+ρ−δ−q,γ+δ+q−ρo∗k′+ρ−δ,δ)L+k′−q+ρ−δ,q+ (−lk′+ρ−δ,δnk′+ρ−δ−q,γ+δ+q−ρ + o∗k′+ρ−δ,δm∗k′+ρ−δ−q,γ+δ+q−ρ)M−k′−q+ρ−δ,q))−[e22m∑q,δ,γ(∑q′Aq−q′ ·Aq′)]×α†k′+ρ(lk′,γ((lk′+ρ−δ,δl∗k′+ρ−δ−q,γ+δ+q−ρ − o∗k′+ρ−δ,δok′+ρ−δ−q,γ+δ+q−ρ)L−k′−q+ρ−δ,q− (lk′+ρ−δ,δok′+ρ−δ−q,γ+δ+q−ρ + o∗k′+ρ−δ,δl∗k′+ρ−δ−q,γ+δ+q−ρ)M+k′−q+ρ−δ,q)+mk′,γ((lk′+ρ−δ,δm∗k′+ρ−δ−q,γ+δ+q−ρ − nk′+ρ−δ−q,γ+δ+q−ρo∗k′+ρ−δ,δ)L−k′−q+ρ−δ,q− (lk′+ρ−δ,δnk′+ρ−δ−q,γ+δ+q−ρ + o∗k′+ρ−δ,δm∗k′+ρ−δ−q,γ+δ+q−ρ)M+k′−q+ρ−δ,q))(3.28)and again similar expressions are found for the βk, αk and β†k operators.3.4.1 Initial conditions and numerical approximationsIn order to limit the number of equations in our system, one must choose an upper bound N suchthat, |γ|, |δ|, |ρ|, η ≤ Nq ∀ ηq , γq , δq , ρq , N ∈ Z. We will set the coefficients outside of this boundto zero. In practice, N is chosen so that the equations are sufficiently convergent within the timedomain of interest. Convergence will, in general, not be expected for all forms of an electromagneticterm, however, we will enforce a pump with a sufficiently small amplitude that convergence can bereached for a reasonable number of terms. This is because nonzero integer terms will scale as powersof Az0. With this choice of N , our sum now becomes limited in the following way:1) Since we have chosen k′ + γ = k+ η and k′ + ρ = k+ δ in the first sum of Eq. 3.28 and A.4, wehave the following additional restriction that, given ρ, δ, γ ∈ [−Nq, Nq] s.t. |ρ− δ − γ| ≤ Nq.2) In the subsequent sums of Eq. 3.28 and A.4, we choose k′ + γ = k+ η and k′ + ρ = k+ q+ δ so3.4. Equations of Motion 30ParametersSuperconductor Pumpa 10−10 m ~ωp 6.0 meVEF 9470 meV ωp 9.12×10−3 fs-1m 1.9me Ap 1.0× 10−8 J sC mwc 8.3 meV τp 100 fs-1∆0 1.35 meV qp a 3.04× 10−6Table 3.1: Parameters for our calculations. a is the lattice spacing, EF is theFermi energy, m is the re-normalized mass and wc is the cutoff frequency. ∆0 isthe initial size of the gap before the pump is turned on. ωp/pr , Ap/pr, τp/pr andqp/pr are the pump/probe frequency, amplitude, full width at half maximum andmomentum.that we get the additional restriction that, given ρ, δ, γ ∈ [−Nq, Nq] s.t. |ρ− δ − q− γ| ≤ Nq3) Since we have chosen k′ + γ = k+ δ and k′ + ρ = k+ η in the first sum of Eq. A.5 and A.6, wehave the following additional restriction that, given ρ, η, γ ∈ [−Nq, Nq] s.t. |ρ− η − γ| ≤ Nq.4) In the subsequent sums of Eq. A.5 and A.6, we choose k′ + ρ = k+ η and k′ + γ = k+ q+ δ sothat we get the additional restriction that, given ρ, η, γ ∈ [−Nq, Nq] s.t. |ρ− η − q− γ| ≤ Nq. Itturns out that choosing Nq in the order of ∼ 10 provides reasonable convergence for the parameterspresented in Table 3.1. However, this number rapidly increases with increasing pumping incidentenergy, as compared to the number of off-diagonal terms we needed in the previous formalism.To limit the number of equations to a reasonable number, it is necessary to use a smaller intensityof pump than in the previous chapter. We also choose a smaller time-constant in order to decreasethe length of the calculations as well. Nevertheless, we expect the same qualitative physics as we arestill in the same non-adiabatic, small fluence regime as in the previous chapter. Moreover, similarmodifications of the pump parameters in the previous formalism resulted in equivalent qualitativebehaviour to the parameters used in Chapter 1 (Table 2.2). The parameters used here are presentedin Table Nonequilibrium Green’s functionsThe solutions of the equations of motion can then be used to calculate all Green’s function G(k, t, t′),which we want to relate to the ARPES spectra. Rewriting the operators in terms of the Bogoliubovquasiparticle operators,〈ck,↑(t), c†k,↑(t′)〉 = 〈(ukαk(t)− vkβ†k(t))(ukα†k(t′)− vkβk(t′))〉 (3.29a)〈ck,↓(t), c†k,↓(t′)〉 = 〈(v−kα†−k(t) + v−kβ−k(t))(v−kα−k(t′) + u−kβ†−k(t′))〉. (3.29b)However, as our Hamiltonian is spin symmetric, it will suffice to consider only the spin-up regime.3.5. Results 31Thus, for the non-equilibrium Green’s functions,G>(k, t, t′) = −i〈(ukαk(t)− vkβ†k(t))(ukα†k(t′)− vkβk(t′))〉 (3.30a)G<(k, t, t′) = +i〈(ukα†k(t′)− vkβk(t′))(ukαk(t)− vkβ†k(t))〉. (3.30b)Finally, using Eqs. 3.21 and 3.23, in terms of the ansatz, the non-equillibrium Green’s functions are,G>(k, t, t′) =− i∑ρ[u2kl∗k,ρ(t)lk,ρ(t′)− vkukok,ρ(t)lk,ρ(t′)− vkukl∗k,ρ(t)o∗k,ρ(t′) + v2kok,ρ(t)o∗k,ρ(t′)](3.31a)G<(k, t, t′) =+ i∑ρ[u2kmk,ρ(t′)m∗k,ρ(t)− vkukmk,ρ(t′)nk,ρ(t)− vkukn∗k,ρ(t′)m∗k,ρ(t) + v2kn∗k,ρ(t′)nk,ρ(t)](3.31b)and for the retarded and advanced Green’s functions,GR(k, t, t′) =Θ(t− t′)∑ρ[u2kl∗k,ρ(t)lk,ρ(t′)− vkukok,ρ(t)lk,ρ(t′)− vkukl∗k,ρ(t)o∗k,ρ(t′) + v2kok,ρ(t)o∗k,ρ(t′)− u2kmk,ρ(t′)m∗k,ρ(t) + vkukmk,ρ(t′)nk,ρ(t) + vkukn∗k,ρ(t′)m∗k,ρ(t)− v2kn∗k,ρ(t′)nk,ρ](3.32a)GA(k, t, t′) =Θ(t′ − t)∑ρ[− u2kl∗k,ρ(t)lk,ρ(t′) + vkukok,ρ(t)lk,ρ(t′) + vkukl∗k,ρ(t)o∗k,ρ(t′)− v2kok,ρ(t)o∗k,ρ(t′)+ u2kmk,ρ(t′)m∗k,ρ(t)− vkukmk,ρ(t′)nk,ρ(t)− vkukn∗k,ρ(t′)m∗k,ρ(t) + v2kn∗k,ρ(t′)nk,ρ](3.32b)3.5 Results3.5.1 Higgs oscillationsSimilarly to the previous chapter, following the application of a pumping laser, the magnitude ofthe order parameter decreases and oscillates due to changes in the free energy potential of thesuperconductor. The degree to which the potential is altered again depends both on the amplitudeand length of the pump. Since we are interested particularly in understanding the second Higgsmode in a dx2−y2 superconductor, we will focus immediately on those results. The results for ourspecific parameters are presented in Fig. 3.1 and agree well with the results from the previousformalism (Fig. 2.3). There is once again a well-defined oscillation frequency at 2∆ and a secondlow-energy frequency below this threshold. This shows that our ansatz reproduces the same physicsfrom the previous chapter.3.6. Discussion 320 5 10 15 20time (ps)1.34951.35∆(t) (meV)dx2-y 2  φ = 0dx2-y 2  φ = pi/4(a)0 1 2 3 4Frequency (meV) transform ∆(t)×10 -5dx2-y 2  φ = 0dx2-y 2  φ = pi/4(b)Figure 3.1: (a) Absolute value of a dx2−y2 order parameter ∆(t) as a functionof time following the pump in the new formalism. (b) Fourier transform of theorder parameter in (a). φ is the relative angle between the antinode of the dx2−y2order parameter and the direction of the vector potential of the pump.3.5.2 Spectral FunctionTo determine the symmetry of the oscillations and thus, the order parameter, it is most useful tovisualize the spectral function within the energy scale of the order parameter ∆. These resultsare presented in Fig. 3.2 after a time-delay of 0.55 picoseconds and for a pump polarized so thatthe vector potential lies along θ = pi/2 in the polar plots. The dx2−y2 character is easily evident.However, for a vector potential aligned along the antinode (Fig. 3.2(a)), the xˆy intensity symmetryis broken between the dx2−y2 nodes, along the C ′′2 axis, which is indicative of the osculating, B1gnonequilibrium mode (Fig. 2.2 d). In contrast, when the vector potential is aligned along the node(Fig. 3.2(b)), the symmetry between antinodes is not broken. Rather, the C ′2 symmetry is brokenwithin a single antinode. Together, these facts point towards the second Higgs oscillation modecoming from an osculating origin.This symmetry breaking is further exemplified when looking at the time-delay data for variouspoints around the Fermi surface and near to the antinode. In Fig. 3.3, we show the oscillations inthe spectral function as a function of time-delay. Notably, the oscillations of the spectral functionare fixed in energy and only the amplitude oscillates in our calculations. These oscillations againhave characteristic frequencies corresponding to the two modes in Fig. 3.1. For the vector potentialaligned along the antinode (Fig. 3.3(a)), there is a single sharp peak at ∆(θ) for angles away fromthe θ = pi/2 (θ = 3pi/2) antinode. However, near to this antinode, the peak becomes broad and anadditional peak is clearly visible in the spectral function at the antinode. In contrast, for a vectorpotential aligned along the node (Fig. 3.3(b)), while there is broadening along the θ = pi/2 direction,there is no additional peak in the spectral function along a specific antinodes or antinodes.3.6 DiscussionThe ansatz proposed in this section is clearly a powerful tool for determining additional informationbeyond what can be achieved by the density matrix formalism, while still maintaining the same3.6. Discussion 3300.10pipi/2EFEF -∆EF+ ∆3pi/2(a)00.10EF+ ∆EFEF -∆pi/2pi3pi/2(b)Figure 3.2: Comparison of the spectral functions A(θ, ω) for a pumped dx2−y2superconductor after 0.55 picoseconds, with the vector potential aligned (a) ver-tically along the antinode (b) vertically along the node.(a) (b)Figure 3.3: Comparison of the spectral functions A(θ, ω, δt) near the antinodefor a pumped dx2−y2 superconductor with the vector potential lying along the(a) antinode and (b) node, as a function of delay-time δt. The plots correspondto various polar angle cuts near to the pi/2 antinode in Fig. 3.2(a) and the 3pi/4antinode in Fig. 3.2(b).3.6. Discussion 34underlying physics. In our model, we are able to use this ansatz to reproduce the spectral functionfor a superconductor with arbitrary symmetry interacting with an electromagnetic field and predictthe ARPES response based on the nonequilibrium spectral function. Having this information isparticularly useful for evaluating the symmetry-breaking terms since ARPES (and the spectralfunction) contain relevant experimental momentum information, which is integrated over in opticalexperiments simulated in the previous chapter on optical response.Considering the group symmetry of our lattice, we know the oscillations must fall into thecategory of either the A1g, A2g, B2g or B1g symmetries. We can now identify the mode at 2∆ asthe A1g, or breathing mode from our analysis in the previous chapter and comparisons with isotropicsuperconductors. In the spectral function, as the vector potential of the pump is aligned along thedx2−y2 antinode there is a broken xˆy symmetry. That is, the C4 and the C ′′2 symmetry are broken,while the C ′2 is not. This is more clear upon examination of the electromagnetic coupling term.While the second order term does not couple to the momentum, the first order term does couple tothe yˆ momentum along the direction of polarization, which breaks the C ′2 momentum symmetry ofthe lattice. Accordingly, this shows that our second mode corresponds to the osculating mode inFig. 2.2. In terms of the D4 character table, the coupling to our electromagnetic field, which hasbeen polarized such that the vector potential lies along the yˆ axis, perturbs the equilibrium BCSHamiltonian with a momentum anisotropy that is of A1g and B1g character.For a pump polarization where the vector potential lies along the nodal direction, the C ′′2 (xˆy)symmetry is not broken. Instead, the C ′2 symmetry is broken due to the pi/4 sample rotation.However, there is no clear indication of a B2g mode as would be the case for this symmetry breaking.This should be somewhat expected because the pump should not induce any modes associated withan angular momentum symmetry breaking, as is the case of the A2g and B2g modes. Moreover, themomentum that incurs the maximum symmetry-breaking is along the nodal lines, which presentsa further explanation regarding the absence of this mode in the superconducting amplitude Higgsmode oscillations.35Chapter 4Conclusions and OutlookThe subject of this thesis is the nonequilibrium response of superconductors with different pair-ing symmetries. The experimentally motivated models studied in this thesis have revolved aroundthe interaction of a BCS superconductor with a short-time electromagnetic field, modeled so as toemulate real experimental pump-probe spectroscopy. Our main findings show how this novel tech-nique, Higgs spectroscopy, can be used both to excite and study the nonequilibrium superconductingmode(s), as well as distinguish and study the inherent superconducting symmetries of the system invarious experimental setups.In chapter 2, we used two short-time electromagnetic fields to emulate the pump and proberespectively. We found that the first interaction with the electromagnetic field (the pump) firstreduced the size of the order parameter and then induced Higgs oscillations of the order parameter.As discussed in section 2.1, the change in value and the oscillations can be attributed to nonadiabaticchanges to the free energy potential of the superconductor. An isotropic nonequilibrium oscillatingmode at 2∆ was present regardless of the superconducting symmetry while a second lower energymode appeared for a dx2−y2 , anisotropic superconductor depending on the relative orientation of thesample and polarization of the pump. In addition, a second short-time electromagnetic interactionwas included to simulate an experimental probing pulse after some time-delay. With this secondpulse, we can calculate the optical response (conductivity) of the system. For a dx2−y2 supercon-ductor, we can then isolate the two different responses by either orienting the vector potential alonga dx2−y2 antinode to induce only the low energy mode, or along the dx2−y2 node to induce onlythe 2∆ mode. As a function of delay-time, the conductivity also exhibited oscillations in amplitudewith both the low energy and 2∆ frequencies. Overall, we showed how an optical experiment canbe used to measure various symmetry-dependent nonequilibrium Higgs modes and actually resolvethe momentum pairing symmetry of a superconductor.In chapter 3, we first developed a formalism to study the spectral function and determine thetwo-time Green’s functions for a superconductor interacting with an electromagnetic field. Thisframework allowed us to additionally make predictions regarding the time-resolved ARPES (tr-ARPES) response. The results from our approximation compare well with the results from themore established density matrix formalism in chapter 2 and we were able to reproduce the sameangle-dependent Higgs modes for a dx2−y2 superconductor. Moreover, this new formalism allowedus to study the momentum dependence of the time-resolved response. Particularly, we were ableto identify that a vector potential aligned along one of the antinodes of a dx2−y2 superconductorinduced an xˆ−yˆ symmetry breaking, which is characteristic of the osculating nonequilibrium mode ofa dx2−y2 superconductor. In contrast, a vector potential aligned alone one of the nodes of a dx2−y2superconductor did not induce the same symmetry breaking. From this, we were able to positthat the low energy Higgs mode, generated by pumping with a vector potential aligned along thedx2−y2 antinode, corresponds to the osculating nonequilibrium Higgs mode and that the 2∆ mode,Chapter 4. Conclusions and Outlook 36which can be seen regardless of the polarization or superconductor symmetry, corresponds to theisotropic breathing mode. Thus, beyond already being able to identify aspects of the superconductingsymmetry in equilibrium, utilizing momentum and time-resolved experimental techniques, such astr-ARPES, is crucial to experimentally verifying the exact momentum-dependent nature of thenonequilibrium Higgs oscillations in a superconductor.Higgs spectroscopy presents a novel experimental technique for directly probing and studyingproperties of the superconducting condensate regardless of the pairing symmetry. One of the mostpromising extensions of this work would be to exploit the coupling between the pump and the su-perconducting momentum by applying series of ultra-fast pumps with different geometry to exciteangular momentum-dependent modes or study the effects of depopulating electron-pairs along par-ticular momenta and breaking the superconducting symmetry along different axes. The methodsin this thesis can easily be extended to study other superconducting symmetries than consideredhere, such as p-wave and anisotropic s-wave. In addition, the methods can also be used to studythe phase of the superconducting condensate and phase-dependent nonequilibrium response, whichcan be a powerful tool for determining the phase differences present in s± superconductors. As anexperimental technique, with refinement and experience, Higgs spectroscopy will likely uncover morephysics beyond the scope of this thesis.In conclusion, the novel field of Higgs spectroscopy is just emerging. 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Lett., 96:179905, May 2006.41Appendix AFirst AppendixThe full derivation of the equations of motion for the operators in Chapter 3 are presented here.Beginning with α†k′ and the commutator with the superconducting portion of the Hamiltonian (thetime-dependence of the pre-factors l,m, n and o is taken to be implicit for simplified notation),[α†k′(t), H˜sc]=∑k,δ,η,γ[α†k+δ(−δk′+γ,k+η)(lk′,γ((lk,δl∗k,η − o∗k,δok,η)Rk + (lk,δok,η)Ck + (l∗k,ηo∗k,δ)C∗k)+mk′,γ((lk−δm∗k,η + nk,ηo∗k,δ)Rk + (lk,δnk,η)Ck + (o∗k,δm∗k,η)C∗k))+βk+δ(δk′+γ,k+η)(mk′,γ((−m∗k,ηmk,δ + nk,ηn∗k,δ)Rk + (−mk,δnk,η)Ck − (m∗k,ηn∗k,δ)C∗k)− lk′,γ((mk,δl∗k,η − n∗k,δok,η)Rk + (mk,δok,η)Ck + (n∗k,δl∗k,η)C∗k))](A.1)and for β†k and the commutator with the superconducting portion of the Hamiltonian,[β†k′(t), H˜sc]=∑k,δ,η,γ[αk+η(δk′+γ,k+δ)(ok′,γ((lk,δl∗k,η − o∗k,δok,η)Rk + (lk,δok,η)Ck + (l∗k,ηo∗k,δ)C∗k)+ nk′,γ((mk,δl∗k,η − n∗k,δok,η)Rk + (mk,δok,η)Ck + (n∗k,δl∗k,η)C∗k))+β†k+η(δk′+γ,k+δ)(−nk′,γ((−m∗k,ηmk,δ + nk,ηn∗k,δ)Rk + (−mk,δnk,η)Ck − (m∗k,ηn∗k,δ)C∗k)+ ok′,γ((lk,δm∗k,η − nk,ηo∗k,δ)Rk + (lk,δnk,η)Ck + (o∗k,δm∗k,η)C∗k))](A.2)and for β†k and the commutator with the EM portion of the Hamiltonian,Appendix A. First Appendix 42[β†k′(t), H˜em]=[e~2m∑k,q,δ,η,γ(2k+ q)Aq]×[αk+η(δk′+γ,k+δ+q)(ok′,γ((lk+q,δl∗k,η + o∗k+q,δok,η)L+k,q + (−lk+q,δok,η + o∗k+q,δl∗k,η)M−k,q)+ nk′,γ((mk+q,δl∗k,η + n∗k+q,δok,η)L+k.q + (−mk+q,δok,η + n∗k+q,δl∗k,η)M−k,q))+β†k+η(δk′+γ,k+δ+q)(−nk′,γ((−m∗k,ηmk+q,δ − nk,ηn∗k+q,δ)L+k,q + (+mk+q,δnk,η − n∗k+q,δm∗k,η)M−k,q)+ ok′,γ((lk+q,δm∗k,η + nk,ηo∗k+q,δ)L+k,q + (−lk+q,δnk,η + o∗k+q,δm∗k,η)M−k,q))]+[e22m∑k,q,δ,η,γ(∑q′Aq−q′ ·Aq′)]×[αk+η(δk′+γ,k+δ+q)(ok′,γ((lk+q,δl∗k,η − o∗k+q,δok,η)L−k,q − (lk+q,δok,η + o∗k+q,δl∗k,η)M+k,q)+ nk′,γ((mk+q,δl∗k,η − n∗k+q,δok,η)L−k.q − (mk+q,δok,η + n∗k+q,δl∗k,η)M+k,q))+β†k+η(δk′+γ,k+δ+q)(−nk′,γ((−m∗k,ηmk+q,δ + nk,ηn∗k+q,δ)L−k,q − (−mk+q,δnk,η − n∗k+q,δm∗k,η)M+k,q)+ ok′,γ((lk+q,δm∗k,η − nk,ηo∗k+q,δ)L−k,q − (lk+q,δnk,η + o∗k+q,δm∗k,η)M+k,q))](A.3)and for βk′ with k′ + ρ = k+ δ in the first sum and k′ = k+ q+ δ in the subsequent 2 sums:Appendix A. First Appendix 43i~ddtmk′,ρ(t)βk′+ρ =−∑δ,γβk′+ρ(mk′,γ((m∗k′+ρ−δ,γ+δ−ρmk′+ρ−δ,δ − nk′+ρ−δ,γ+δ−ρn∗k′+ρ−δ,δ)Rk′+ρ−δ+ (mk′+ρ−δ,δnk′+ρ−δ,γ+δ−ρ)Ck′+ρ−δ + (m∗k′+ρ−δ,γ+δ−ρn∗k′+ρ−δ,δ)C∗k′+ρ−δ)+ lk′,γ((mk′+ρ−δ,δl∗k′+ρ−δ,γ+δ−ρ − n∗k′+ρ−δ,δok′+ρ−δ,γ+δ−ρ)Rk′+ρ−δ+ (mk′+ρ−δ,δok′+ρ−δ,γ+δ−ρ)Ck′+ρ−δ + (n∗k′+ρ−δ,δl∗k′+ρ−δ,γ+δ−ρ)C∗k′+ρ−δ))−[e~2m∑q,δ,γ(2(k′ − q+ ρ− δ) + q)Aq]×βk′+ρ(mk′,γ((m∗k′+ρ−δ−q,γ+δ+q−ρmk′+ρ−δ,δ + nk′+ρ−δ−q,γ+δ+q−ρn∗k′+ρ−δ,δ)L+k′−q+ρ−δ,q+ (−mk′+ρ−δ,δnk′+ρ−δ−q,γ+δ+q−ρ + n∗k′+ρ−δ,δm∗k′+ρ−δ−q,γ+δ+q−ρ)M−k′−q+ρ−δ,q)+ lk′,γ((mk′+ρ−δ,δl∗k′+ρ−δ−q,γ+δ+q−ρ + n∗k′+ρ−δ,δok′+ρ−δ−q,γ+δ+q−ρ)L+k′−q+ρ−δ,q+ (−mk′+ρ−δ,δok′+ρ−δ−q,γ+δ+q−ρ + n∗k′+ρ−δ,δl∗k′+ρ−δ−q,γ+δ+q−ρ)M−k′−q+ρ−δ,q))−[e22m∑q,δ,γ(∑q′Aq−q′ ·Aq′)]×βk′+ρ(mk′,γ((m∗k′+ρ−δ−q,γ+δ+q−ρmk′+ρ−δ,δ − nk′+ρ−δ−q,γ+δ+q−ρn∗k′+ρ−δ,δ)L−k′−q+ρ−δ,q− (mk′+ρ−δ,δnk′+ρ−δ−q,γ+δ+q−ρ + n∗k′+ρ−δ,δm∗k′+ρ−δ−q,γ+δ+q−ρ)M+k′−q+ρ−δ,q)+ lk′,γ((mk′+ρ−δ,δl∗k′+ρ−δ−q,γ+δ+q−ρ − n∗k′+ρ−δ,δok′+ρ−δ−q,γ+δ+q−ρ)L−k′−q+ρ−δ,q− (mk′+ρ−δ,δok′+ρ−δ−q,γ+δ+q−ρ + n∗k′+ρ−δ,δl∗k′+ρ−δ−q,γ+δ+q−ρ)M+k′−q+ρ−δ,q))(A.4)Appendix A. First Appendix 44For αk′ we collect like terms by choosing k′ + ρ = k+ η:i~ddtok′,ρ(t)αk′+ρ =∑η,γαk′+ρ(ok′,γ((lk′+ρ−η,γ+η−ρl∗k′+ρ−η,η − o∗k′+ρ−η,γ+η−ρok′+ρ−η,η)Rk′+ρ−η+ (lk′+ρ−η,γ+η−ρok′+ρ−η,η)Ck′+ρ−η + (l∗k′+ρ−η,ηo∗k′+ρ−η,γ+η−ρ)C∗k′+ρ−η)+ nk′,γ((mk′+ρ−η,γ+η−ρl∗k′+ρ−η,η − n∗k′+ρ−η,γ+η−ρok′+ρ−η,η)Rk′+ρ−η+ (mk′+ρ−η,γ+η−ρok′+ρ−η,η)Ck′+ρ−η + (l∗k′+ρ−η,ηn∗k′+ρ−η,γ+η−ρ)C∗k′+ρ−η))+[e~2m∑q,η,γ(2(k′ + ρ− η) + q)Aq]×αk′+ρ(ok′,γ((lk′+ρ−η+q,γ+η−q−ρl∗k′+ρ−η,η + o∗k′+ρ−η+q,γ+η−q−ρok′+ρ−η,η)L+k′+ρ−η,q+ (−lk′+ρ−η+q,γ+η−q−ρok′+ρ−η,η + l∗k′+ρ−η,ηo∗k′+ρ−η+q,γ+η−q−ρ)M−k′+ρ−η,q)+ nk′,γ((mk′+ρ−η+q,γ+η−q−ρl∗k′+ρ−η,η + n∗k′+ρ−η+q,γ+η−q−ρok′+ρ−η,η)L+k′+ρ−η,q+ (−mk′+ρ−η+q,γ+η−q−ρok′+ρ−η,η + n∗k′+ρ−η+q,γ+η−q−ρl∗k′+ρ−η,η)M−k′+ρ−η,q))+[e22m∑q,η,γ(∑q′Aq−q′ ·Aq′)]×αk′+ρ(ok′,γ((lk′+ρ−η+q,γ+η−q−ρl∗k′+ρ−η,η − o∗k′+ρ−η+q,γ+η−q−ρok′+ρ−η,η)L−k′+ρ−η,q− (lk′+ρ−η+q,γ+η−q−ρok′+ρ−η,η + l∗k′+ρ−η,ηo∗k′+ρ−η+q,γ+η−q−ρ)M+k′+ρ−η,q)+ nk′,γ((mk′+ρ−η+q,γ+η−q−ρl∗k′+ρ−η,η − n∗k′+ρ−η+q,γ+η−q−ρok′+ρ−η,η)L−k′+ρ−η,q− (mk′+ρ−η+q,γ+η−q−ρok′+ρ−η,η + n∗k′+ρ−η+q,γ+η−q−ρl∗k′+ρ−η,η)M+k′+ρ−η,q))(A.5)Appendix A. First Appendix 45and for β†k′ , k′ + ρ = k+ η:i~ddtnk′,ρ(t)β†k′+ρ =∑η,γβ†k′+ρ(nk′,γ((m∗k′+ρ−η,ηmk′+ρ−η,γ+η−ρ − nk′+ρ−η,ηn∗k′+ρ−η,γ+η−ρ)Rk′+ρ−η+ (mk′+ρ−η,γ+η−ρnk′+ρ−η,η)Ck′+ρ−η + (m∗k′+ρ−η,ηn∗k′+ρ−η,γ+η−ρ)C∗k′+ρ−η)+ ok′,γ((lk′+ρ−η,γ+η−ρm∗k′+ρ−η,η − nk′+ρ−η,ηo∗k′+ρ−η,γ+η−ρ)Rk′+ρ−η+ (lk′+ρ−η,γ+η−ρnk′+ρ−η,η)Ck′+ρ−η + (o∗k′+ρ−η,γ+η−ρm∗k′+ρ−η,η)C∗k′+ρ−η))+[e~2m∑q,η,γ(2(k′ + ρ− η) + q)Aq]×β†k′+ρ(nk′,γ((m∗k′+ρ−η,ηmk′+ρ−η+q,γ+η−q−ρnk′+ρ−η,η + n∗k′+ρ−η+q,γ+η−q−ρ)L+k′+ρ−η,q+ (−mk′+ρ−η+q,γ+η−q−ρnk′+ρ−η,η + n∗k′+ρ−η+q,γ+η−q−ρm∗k′+ρ−η,η)M−k′+ρ−η,q)+ ok′,γ((lk′+ρ−η+q,γ+η−q−ρm∗k′+ρ−η,η + nk′+ρ−η,ηo∗k′+ρ−η+q,γ+η−q−ρ)L+k′+ρ−η,q+ (−lk′+ρ−η+q,γ+η−q−ρnk′+ρ−η,η + o∗k′+ρ−η+q,γ+η−q−ρm∗k′+ρ−η,η)M−k′+ρ−η,q))+[e22m∑q,η,γ(∑q′Aq−q′ ·Aq′)]×β†k′+ρ(nk′,γ((m∗k′+ρ−η,ηmk′+ρ−η+q,γ+η−q−ρ − nk′+ρ−η,ηn∗k′+ρ−η+q,γ+η−q−ρ)L−k′+ρ−η,q− (mk′+ρ−η+q,γ+η−q−ρnk′+ρ−η,η + n∗k′+ρ−η+q,γ+η−q−ρm∗k′+ρ−η,η)M+k′+ρ−η,q)+ ok′,γ((lk′+ρ−η+q,γ+η−q−ρm∗k′+ρ−η,η − nk′+ρ−η,ηo∗k′+ρ−η+q,γ+η−q−ρ)L−k′+ρ−η,q− (lk′+ρ−η+q,γ+η−q−ρnk′+ρ−η,η + o∗k′+ρ−η+q,γ+η−q−ρm∗k′+ρ−η,η)M+k′+ρ−η,q))(A.6)


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