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Characterizing the AC-MOT Anholm, Melissa 2011

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Characterizing the AC-MOTbyMelissa AnholmB.Sc., University of California–Santa Barbara, 2006M.Sc., University of Wisconsin–Milwaukee, 2009A 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 2014c© Melissa Anholm 2011AbstractMagneto-Optical Traps (MOTs) have long been used to produce samplesof cold trapped neutral atoms, which can be used in the measurement ofa variety of physical quantities and theories. Until recently, one limitationof this type of trap was the necessity for the presence of a relatively largemagnetic field which would decay only slowly after the trapping mechanismwas turned off. This residual magnetic field is expected to partially destroyany atomic polarization induced, for example, by optical pumping. As aresult, the precision of any physical measurement which requires polarizationis limited. We will discuss the construction of our version of a newer typeof MOT, the AC-MOT [2], which is designed specifically so as to minimizeresidual magnetic fields. We have found that our AC-MOT has lifetimesand cloud sizes similar to those we measured in our DC-MOT. We intend touse a trap similar to this in upcoming nuclear beta decay parity-violationmeasurements. We also discuss the numerical evolution of the optical Blochequations in the presence of transverse and longitudinal magnetic fields, soas to quantify the effect of a magnetic field on atomic polarization.iiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . vii1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 The Magneto-Optical Trap . . . . . . . . . . . . . . . . . . . . 32.1 The Zeeman Shift . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Atoms in an Optical Molasses . . . . . . . . . . . . . . . . . 42.3 The Magneto-Optical Trap . . . . . . . . . . . . . . . . . . . 83 The AC-MOT–What Makes It Different? . . . . . . . . . . 134 The Offline AC-MOT . . . . . . . . . . . . . . . . . . . . . . . 154.1 Methodologies for Measuring Trap Characteristics . . . . . . 154.1.1 Measuring the Number of Trapped Atoms . . . . . . 174.1.2 Measuring the Lifetime . . . . . . . . . . . . . . . . . 194.1.3 Measuring the Trap Width and Position . . . . . . . 194.2 Laser Frequency Calibrations in the Offline AC-MOT . . . . 224.3 The Phenomenology of Lifetime Measurements in the AC-and DC-MOT . . . . . . . . . . . . . . . . . . . . . . . . . . 254.4 Measured Lifetimes and AC Frequency . . . . . . . . . . . . 254.5 Systematic Effects in the Offline AC-MOT . . . . . . . . . . 285 Turning the Trap Off . . . . . . . . . . . . . . . . . . . . . . . 305.1 Methods for Turning Off the Trap . . . . . . . . . . . . . . . 315.2 Duty Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35iiiTable of Contents5.3 Residual Magnetic Field in a Pyrex Cell . . . . . . . . . . . . 386 Calibrations of the Online Trap . . . . . . . . . . . . . . . . . 416.1 Frequency Response in the Online Setup . . . . . . . . . . . 456.2 Frequency Response in the Hall Probes . . . . . . . . . . . . 466.3 Acoustic Resonances . . . . . . . . . . . . . . . . . . . . . . . 496.4 Control of Online Power Supplies . . . . . . . . . . . . . . . 526.4.1 Determining the Number of Points for an ArbitraryWaveform . . . . . . . . . . . . . . . . . . . . . . . . 526.4.2 Adjusting Waveform Parameters . . . . . . . . . . . . 536.5 Residual Magnetic Field in the Online Chamber . . . . . . . 566.6 Magnetic Field During Optical Pumping Time . . . . . . . . 597 The Optical Bloch Equations . . . . . . . . . . . . . . . . . . 627.1 Explicit Form of the Density Matrix . . . . . . . . . . . . . . 627.2 The General Form of Rotating Coordinates . . . . . . . . . . 637.3 Derivation of a Toy-Model Set of Optical Bloch Equations . 668 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80AppendicesA Waveform Generation Code . . . . . . . . . . . . . . . . . . . 82B Very Basic Things . . . . . . . . . . . . . . . . . . . . . . . . . 86B.1 The Center of Gravity . . . . . . . . . . . . . . . . . . . . . . 86B.2 Diagonalizing the Hamiltonian . . . . . . . . . . . . . . . . . 86B.3 Rotating Coordinates . . . . . . . . . . . . . . . . . . . . . . 86B.4 Lifetimes and Half-Lifes . . . . . . . . . . . . . . . . . . . . . 87B.5 Reduced Matrix Elements . . . . . . . . . . . . . . . . . . . . 88B.6 Doppler Cooling Limit . . . . . . . . . . . . . . . . . . . . . 88ivList of Figures2.1 Zeeman Shifts in a Magnetic Field . . . . . . . . . . . . . . . 52.2 Diagram of a Magneto-Optical Trap . . . . . . . . . . . . . . 92.3 41K Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . 103.1 AC-MOT Sample Magnetic Field . . . . . . . . . . . . . . . . 143.2 AC-MOT Sample Laser Polarization . . . . . . . . . . . . . . 144.1 Photo of the Offline MOT . . . . . . . . . . . . . . . . . . . . 164.2 Geometry of the Offline MOT . . . . . . . . . . . . . . . . . . 174.3 CCD Camera Output . . . . . . . . . . . . . . . . . . . . . . 184.4 Trap Lifetime Measurement . . . . . . . . . . . . . . . . . . . 204.5 Trap Width Measurement . . . . . . . . . . . . . . . . . . . . 214.6 Offline Optical Setup . . . . . . . . . . . . . . . . . . . . . . . 234.7 Trap Frequency and Laser Power . . . . . . . . . . . . . . . . 244.8 AC and DC Lifetimes . . . . . . . . . . . . . . . . . . . . . . 264.9 Lifetimes as a Function of AC Frequency . . . . . . . . . . . . 275.1 Methods for Killing the Trap . . . . . . . . . . . . . . . . . . 325.2 Trap-killing Methods, Trap Size, and Laser Power . . . . . . 335.3 Trap-killing Methods, Lifetime, and Laser Power . . . . . . . 345.4 Lifetime and Duty Cycle . . . . . . . . . . . . . . . . . . . . . 365.5 Fluorescence and Duty Cycle . . . . . . . . . . . . . . . . . . 375.6 Optimal Waveform to Minimize Residual Magnetic Field . . . 395.7 Offline Residual Magnetic Field . . . . . . . . . . . . . . . . . 406.1 The Inside of the New Trapping Chamber . . . . . . . . . . . 426.2 Quadrupole Coil Geometry in the Online Chamber . . . . . . 436.3 Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.4 Initial Hall Probes Layout . . . . . . . . . . . . . . . . . . . . 476.5 Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 486.6 Ceramic Feedthroughs . . . . . . . . . . . . . . . . . . . . . . 506.7 Acoustic Resonances in the New Chamber . . . . . . . . . . . 51vList of Figures6.8 Improperly-Tuned Sinusoidal Start . . . . . . . . . . . . . . . 556.9 Current and Magnetic Field . . . . . . . . . . . . . . . . . . . 566.10 Current and Magnetic Field: A Closer Look . . . . . . . . . . 576.11 Current and Magnetic Field: A Much Closer Look . . . . . . 586.12 Current in Top and Bottom Coils . . . . . . . . . . . . . . . . 596.13 Quadrupole and Dipole Components of the Magnetic Field . 606.14 A Fit to the Residual Quadrupole Field . . . . . . . . . . . . 617.1 D1 Transition Strengths for I = 1/2 . . . . . . . . . . . . . . 67viAcknowledgementsSpecial thanks to my research advisor John Behr, who was always willingto spend time helping students stuck at any level of cluelessness withoutmaking them feel stupid. For some reason, he also seemed to think it wasworthwhile to pay me for doing things that were largely quite enjoyable.I would also like to thank undergraduate researchers Heather Nortonand Rhys Anderson, both of whom did a lot of the work that I wanted verymuch to avoid doing myself.viiChapter 1Introduction and Motivation1.1 MotivationSince the magneto-optical trap (MOT) was first described in 1987 by Raabet. al. [1], it has become a standard technique for confining cold samples ofneutral atoms. These cold trapped atoms may subsequently be used in themeasurement of a variety of physical quantities.The MOT necessitates the use of a magnetic field in order to produce aconfining force on the trapped atoms. However, there exist a certain class ofexperiments which require a sample of cold atoms in zero (or minimal) mag-netic field – notably any experiment in which high polarization of atomicangular momenta is needed. If a MOT is to be used in such a case, thetrapping mechanism must be intermittently shut off for a period of time.Because the atoms have been cooled in the trap, they will disperse onlyslowly after discontinuation of the MOT’s trapping forces, and it is possibleto restart the MOT before most of the atoms have moved beyond the trap-ping region. It is during this “off” time that the atoms may be polarized bya properly-tuned laser if needed, and data may be collected with minimalinterference from magnetic fields.It is useful, therefore, to find a method to eliminate the magnetic field inthe trapping region as rapidly as possible, so that a maximal amount of timecan be spent collecting data. Recently Harvey and Murray [2] have builtand described a new type of MOT designed to do just that–the AC-MOT,so named for the electrical current in the MOT’s electromagnets. Previousgeneration of MOTs had used only DC currents for that purpose. Althoughthe experimental setup for an AC-MOT is more complicated, the benefit isthat the magnetic field can be eliminated much more quickly than is thecase for a DC-MOT.This thesis will describe and characterize many aspects of two AC-MOTsbuilt for use in- or alongside nuclear beta decay experiments for the TRINATresearch group.11.2. Overview1.2 OverviewThe entirety of Chapter 2 is devoted to a description of the physical processesinvolved in a functional (AC- or DC-) MOT, and Chapter 3 describes theadditional requirements for an AC-MOT.We describe our own offline AC-MOT, including our commonly usedmeasurement techniques, and characterize some of its properties in Chap-ter 4. Chapter 5 discusses and demonstrates optimal strategies for turningoff trapping forces in the offline AC-MOT.Chapter 6 deals only with the online AC-MOT. The setup is described,and some measurements relevant specifically to the online AC-MOT arepresented.In Chapter 7, attention is given to quantifying the polarization problemscaused by residual magnetic fields. Beginning with a derivation of the well-known Optical Bloch Equations, we introduce terms into the Hamiltonian tomodel the effect of a non-zero magnetic field on the polarization of a sampleof atoms. Qualitative results of this model are discussed.Chapter 8 is a discussion which provides additional context for our re-sults, and suggests possible future work on this topic.2Chapter 2The Magneto-Optical TrapSince its invention in 1987 [1], the Magneto-optical trap (MOT) has becomea standard technique for creating samples of tightly-confined cold atoms.The key principle is to use a magnetic field gradient, in addition to laserstuned near an atomic resonance, to cause atoms to absorb photons, pushingthem toward the MOT centre and simultaneously cooling them.In order to understand the mechanism by which a MOT is able to confineatoms, we must first introduce the Zeeman effect (Section 2.1) and a descrip-tion of an optical molasses (Section 2.2). A functional MOT combines theforces resulting from these two mechanisms to trap and cool atoms.2.1 The Zeeman ShiftThe atomic Hamiltonian for an atom in the presence of a weak magneticfield (such as is present in a MOT) picks up an additional Zeeman shiftterm [3],HˆZeeman = −~µ · ~B, (2.1)where the magnetic moment ~µ is given by~µ = −µB(gS ~S + gL~L+ gI~I)(2.2)= −µB gF ~F (2.3)where µB is the Bohr magneton, and gS , gL, gI , and gF are the g-factorsassociated with electron spin, orbital angular momentum, nuclear spin, andtotal angular momentum, respectively. Typically, however, gI  gS , gL andso the nuclear spin term is neglected for the purpose of laser tuning for MOToperation. Then,~µ ≈ −µB(gS ~S + gL~L)(2.4)= −µB gJ ~J, (2.5)32.2. Atoms in an Optical Molassesand Eq. 2.1 becomesHˆZeeman = gJ µB ~J · ~B. (2.6)If we take the direction of the magnetic field as our quantization axis andlabel it as zˆ, the result is a simple perturbation to atomic energy levels [3],∆EZeeman = gJ µBMJBz, (2.7)where we note that in the case where angular momentum ~J and magneticfield ~B are parallel, the Zeeman contribution to the total energy of thatstate is positive. This result is accurate for systems in which the magneticfield can be taken as “small”, meaning that MJ and MI are good quantumnumbers.It is convenient at this point to introduce new notation to describe theZeeman contribution to a transition as a whole – including the Zeeman shiftsin the ground and excited states within a single term. Thus, we defineµ′ := (geMe − ggMg)µB, (2.8)where g× and M× are the Lande g-factor and the zˆ component of the angularmomentum, and the subscripts ‘g’ and ‘e’ refer to the ground- and excitedstates, respectively. Then the change to the energy of a transition is∆Etransition = µ′Bz. (2.9)In a MOT, the magnetic field must be quadrupolar. In other words,although the magnetic field is zero at the centre, it increases linearly withdistance in every direction. Thus, so too will the atomic energy levels varylinearly along the paths of the beams of light, as shown in Fig. 2.1.2.2 Atoms in an Optical MolassesWe will now consider a system of two-level atoms located at the intersectionof two counter-propagating laser beams, detuned slightly from the atomicresonance. Such a setup is sometimes referred to as a one-dimensional “opti-cal molasses” due to the viscous drag force induced on atomic motion, whichwill be discussed in more detail shortly.In such a system, an incident photon can excite an atom from the groundstate into the excited state, while simultaneously giving the atom a “push”proportional to the laser’s detuning from the atomic resonance. An excited42.2. Atoms in an Optical Molassesσ+-2-1+1+2 0-1+1 0-2-1+1+2 0F=1F=1F=2F=2B•zˆ→z766.7 nm3.4 MHz254.0 MHzZeeman-Shifted Energy Levels in 41KMFBz-1+1 0+2+1-1-2 0+1-1 0+2+1-1-2 0MF+1-1 0σ-Figure 2.1: A level diagram to show the perturbation to hyperfine energylevels in 41K resulting from a linear magnetic field gradient such as is usedin a MOT.atom will eventually spontaneously decay back into its ground state, emit-ting another photon in a random direction during the process. Thus, the52.2. Atoms in an Optical Molassesatom’s overall motion within an optical molasses is partly determined bythe balance of laser intensities, and part random walk.Additionally, laser light can also stimulate an excited atom to emit aphoton and decay back to its ground state. This effect is negligible providedthat the laser’s intensity is sufficiently small, but as the laser intensity in-creases, so too does this effect. In the limit of infinite intensity, stimulatedemission is just as likely to occur as stimulated excitation, and as a result,the atomic population is split evenly between the ground and excited states.To describe the transition between these two regimes, we introduce thesaturation intensity, Isat, which is the intensity at which the decay rate byspontaneous emission is equal to the decay rate from stimulated emission fora laser tuned precisely to the atomic resonance. For an atomic transition oflinewidth γ (equivalently, the linewidth γ describes the spontaneous decayrate, and is the inverse of the excited state’s lifetime, so that γ = 1/τ) andenergy ~ω0, the saturation intensity isIsat =~ω30 γ12pi c2 . (2.10)Then, we can describe the intensity I of the on-resonance light within aparticular system in terms of its relation to the saturation intensity by in-troducing s0, the on-resonance saturation parameter, so thats0 :=IIsat. (2.11)In practice, it is often useful to detune the lasers slightly from the reso-nant transition. We now introduce the detuning parameter,δ0 := ωL − ω0 (2.12)which describes the difference between the laser frequency and the atomicresonance frequency. In such a situation, we still wish to be able to describethe extent to which the transition is saturated by the laser light. We there-fore introduce the off-resonance saturation parameter, s, which is related tos0 bys := s01 + (2δ0/γ)2. (2.13)For atoms at rest in an optical molasses, the excited state populationfraction is given byρee =s2(s+ 1) =s0/21 + s0 + (2δ0/γ)2. (2.14)62.2. Atoms in an Optical MolassesThis result emerges as a steady-state solution to the two-level Optical BlochEquations. (For a more in-depth treatment of the Optical Bloch Equations,see Chapter 7.) It should be clear that in the limit of large s values, ρee ap-proaches 1/2 as expected. Due to the very nature of a steady state solution,the rate at which atoms are excited to the higher energy level is equal to therate at which atoms decay to the ground state. Therefore, the total scat-tering rate R for photons incident on atoms at rest in an optical molasses isgiven by [4]R = γρee =γ s0/21 + s0 + (2δ0/γ)2. (2.15)Since every photon absorbed transfers to the atom a momentum of∆~p = ~ωLc eˆL, (2.16)where eˆL is a unit vector in the direction of the laser’s propagation, we areable to write out an expression for the average force on an atom at rest withina single laser beam. Trivially, the average force must be the momentumtransferred multiplied by the rate at which this momentum transfer occurs,and so we find that in general,~F1 = R∆~p (2.17)= γ s0/21 + s0 + (2δ0/γ)2(~ωLc)eˆL. (2.18)Note that Eq. 2.18 need not include any contribution from spontaneousemission following absorption of a photon, because the (vector) average ofany such contribution must be zero, since there is no preferred direction ofspontaneous emission.Of course, we cannot assume that atoms within the optical molasses willbe at rest. For an atom moving with 3-velocity ~v, an incident laser beamwill be observed to have a Doppler shifted frequency ω′L, so thatωL → ω′L(~v) := ωL(1− ~v · eˆLc)(2.19)within the atom’s reference frame. (Note that as expected, the Dopplershifted frequency increases if the direction of atomic motion is anti-parallelto the direction of laser propagation.) This effective frequency shift propa-gates through to the equations of force, showing up everywhere the laser’sfrequency is referenced.72.3. The Magneto-Optical TrapWe now return to consideration of the average force within a single beam.With the Doppler shift taken into consideration, Eq. 2.18 becomes~F ′1(~v) =γ(s0 ~ωL/(2c))(1− ~v · eˆL/c)1 + s0 +(2γ (δ0 − (ωL/c)~v · eˆL))2 eˆL. (2.20)Therefore the net force on an atom moving with velocity ~v within twosuch counter-propagating laser beams (that is, one beam propagates in di-rection +eˆL, while the other propagates in direction −eˆL) is given by~FOM(~v) =γ(s0 ~ωL/(2c))(1− ~v · eˆL/c)1 + s0 +(2γ (δ0 − (ωL/c)~v · eˆL))2 eˆL− γ(s0 ~ωL/(2c))(1 + ~v · eˆL/c)1 + s0 +(2γ (δ0 + (ωL/c)~v · eˆL))2 eˆL. (2.21)2.3 The Magneto-Optical TrapWe now turn our focus to a description of a MOT. To create a MOT, a linearmagnetic field gradient must be applied on top of an optical molasses. Bytaking advantage of the change in absorption from the Zeeman effect’s per-turbation to atomic resonances, atoms can be preferentially pushed towardsa central region, where the Zeeman shift is zero.To create the appropriate magnetic field, one employs a set of two elec-trical coils with antiparallel currents, so that the magnetic field is zero at thetrap’s centre, but its local gradient is non-zero; in the immediately surround-ing region the magnitude of the magnetic field grows linearly with distancein any direction. The atomic energy levels are perturbed as a result, and theMOT uses a series of six laser beams (three sets of two anti-parallel ‘twin’beams), all intersecting at the trap’s centre, to take advantage of these en-ergy perturbations to produce a confining force on the atoms. The geometryof a MOT in a laboratory is shown in Fig. 2.2.The six beams of light must each consist of two frequency components,referred to here as the “trapping frequency” and the “repumper frequency”[4]. Examination of Fig. 2.3 shows why this must so – without the additionof a repumper component, it wouldn’t be long before the trapped atoms alldecayed into the “wrong” ground state, from which the original trappinglaser could not excite them. For optimal MOT function, both the trapping82.3. The Magneto-Optical TrapFigure 2.2: The necessary components of a magneto-optical trap includetwo electrical coils running anti-parallel currents, and six beams of lightintersecting at the centre of the geometry. The electrical coils produce aquadrupolar magnetic field. Each beam of light consists of two frequencycomponents – the “trapping” and “repumper” beams, and all are circularlypolarized and counter-propagating, so as to couple to specific atomic energytransitions.and repumper components should be red-detuned from their respective tran-sitions. Typically, a MOT has much more power in the trapping componentthan the repumper component.The reasons for red-detuning the lasers are twofold. The first reason92.3. The Magneto-Optical Trap-1 +10 +2-2-1 +10 +2-2-1+10-1 +10F=1P3/2P3/2F=1F=2F=2S1/2S1/2389.88MHz386.50MHz132.49MHz135.87MHzTransitions in 41KExcitation from !+ Trapping Beam Excitation from !+ Repumper BeamSpontaneous Emission from F=2 Excited StateSpontaneous Emission from F=1 Excited State Figure 2.3: Hyperfine transitions in 41K – Zeeman splitting of the hyperfinestates is not shown. This diagram shows only the absorption transitionswhich couple to σ+ light. An absorbed laser beam with σ+ polarizationmust increase the atom’s angular momentum projection along the axis ofpropagation. We see this as a change in the quantum number MF by oneunit, so as to conserve angular momentum. By a similar argument, absorp-tion of σ− light will decrease MF by one unit. For spontaneous emission,there is no requirement on the direction or polarization of the spontaneouslyemitted photon, so the process can change MF by 0, +1, or -1.is that the atoms which are moving quickly antiparallel to the directionof the laser’s propagation will see the light as being blueshifted closer toresonance; photons will then be preferentially absorbed by these atoms, andthe imparted change in momentum will cause the atom to slow. By a similarargument, atoms moving parallel to the direction of light propagation areless likely to absorb a photon. This is the mechanism at work in an opticalmolasses, and the forces involved have been quantified in Section 2.2 The102.3. The Magneto-Optical Trapsecond reason for the detuning is simply a matter of selecting which side ofthe cloud of trapped atoms is more likely to absorb a photon propagatingin a particular direction – the goal of course being that the absorbed linearmomentum should push the atoms toward the central trapping region. Thus,when the beam interacts with the atoms near the centre of the trappingregion, it has the dual effects of slowing the atoms and pushing them towardsthe centre [1].We now turn to a mathematical description of the forces within a one-dimensional MOT. Although it can be rigorously shown that a MOT is ableto confine atoms in three dimensions, as in [5], to aid clarity of description wewill consider only a one-dimensional MOT for the remainder of this section.In effect, this means that we will apply the effects of a linearly changingZeeman shift, described in Section 2.1, to the optical molasses described inSection 2.2. In these sections, no consideration was given to the polarizationof the incident laser, but this matter can no longer be ignored. As can beseen in Figs. 2.1 and 2.3, an incoming laser with σ+ polarization will interactwith a different set of atomic transitions than it would when its polarizationwas σ−, and in particular, swapping both the sign of the magnetic fieldand the polarization of the laser will lead to an indentical set of couplingsbetween the atom and the laser, with an identical set of energy perturbationsto transitions.We will proceed by adding the Zeeman shift terms into Eq. 2.20. Thepresence of a Zeeman shift will change the effective value of the atomic res-onance, ω0 (rather than ωL) in a manner which depends on the polarizationof the incident laser. Therefore, for a σ− or σ+ laser respectively, the atomicresonance to which the laser couples becomes:ω0 → ω′±0 (~r) := ω0 ± µ′Bz(~r)/~ (2.22)where µ′ is as described in Eq. 2.8. This change makes its way through thecalculations, showing up everywhere that ω0 shows up. Eq. 2.20 becomes~F±1 (~v,Bz) =γ(s0 ~ωL/(2c))(1− ~v · eˆL/c)1 + s0 +(2γ (δ0 − (ωL/c)~v · eˆL ∓ µ′Bz(~r)/~))2 eˆL. (2.23)In a MOT, we will wish to avoid the scenario in which the atoms areall optically pumped into one particular energy level, so we select one laserto have σ− polarization, and its counterpropagating twin must have σ+polarization. It is important that the correct beam be given the correct po-larization, and to do that we recall that in a MOT, we wish to preferentially112.3. The Magneto-Optical Trappush atoms toward the centre of the trap, therefore the absolute value ofthe effective detuning must be smaller in regions further from the centre. Ifwe use the magnetic field gradient shown in Fig. 2.1 and consider only thetrapping beam, then µ′ takes the same sign as the σ± polarization (We willwrite this explicitly as µ′± for clarity, remembering that µ′− = −µ′+.). Wetherefore choose the σ− light to propagate in the +zˆ direction, and the σ+light to propagate in the -zˆ direction, and Eq. 2.23 becomes~F+1 (~v,Bz) =− γ(s0 ~ωL/(2c))(1 + ~v · zˆ/c)1 + s0 +(2γ (δ0 + (ωL/c)~v · zˆ − µ′+Bz(~r)/~))2 zˆ (2.24)~F−1 (~v,Bz) =γ(s0 ~ωL/(2c))(1− ~v · zˆ/c)1 + s0 +(2γ (δ0 − (ωL/c)~v · zˆ + µ′−Bz(~r)/~))2 zˆ, (2.25)and the total average force on an atom within a MOT is simply the sum ofthese two terms. Because we have chosen to work in a system where themagnetic field along the zˆ-axis is given byBz =∂B∂z z, (2.26)we can now write the average force in terms of the atom’s position. We findthat~FMOT(~v, z) =γ s0 ~ωL2c zˆ×−1− ~v · zˆ/c1 + s0 + 4γ2(δ0 + (ωL/c)~v · zˆ − µ′+ ∂B∂z z/~)2+ 1− ~v · zˆ/c1 + s0 + 4γ2(δ0 − (ωL/c)~v · zˆ + µ′− ∂B∂z z/~)2 (2.27)gives the average force on an atom within a MOT.12Chapter 3The AC-MOT–What MakesIt Different?For our experiment, we need atoms which are not only cold and well con-fined, but spin polarized as well. Although the standard “DC” MOT haspreviously been demonstrated to trap and cool atoms in a confined region,it requires the use of a non-uniform magnetic field, which hinders our abilityto polarize the atoms. Our challenge is to create an atom trap from whichthe magnetic non-uniformities can be eliminated rapidly. To that end, wehave implemented an AC-MOT – a MOT in which the magnetic field gen-erating coils are run with an AC electrical current, and laser polarizationsare switched rapidly so as to match. This setup minimizes problems fromresidual eddy currents in nearby conductors, allowing us to eliminate themagnetic field much more rapidly [2]. With the quadrupole component ofthe magnetic field gone, the sample of cold atoms can be better polarizedbefore they disperse.To effect these goals, an “AC-MOT” was designed and constructed, ini-tially by Harvey and Murray [2]. The principle was straightforward: insteadof directing a DC current through the trap’s anti-Helmholz coils, an ACcurrent would be used. Additionally, the polarization of the trapping laserswould need to be alternated synchronously with the current. In this way,although both the polarity of the magnetic quadrupole field and the laserpolarization varied, a trapping force would remain.By using a system in which the current is expected to vary, it becomeseasier to shut the current off rapidly. In particular, it is possible to select acut-off time for the current such that the residual magnetic field from eddycurrents is minimized.13Chapter 3. The AC-MOT–What Makes It Different?Out[219]=0.0005 0.0010 0.0015 0.0020 0.0025 0.0030￿1.0￿0.50.51.0Figure 3.1: Sample waveform for use in an AC-MOT. Magnetic field (blue)lags current through quadrupole coils (yellow) slightly as a result of eddycurrents in nearby conductors.Out[239]=0.0005 0.0010 0.0015 0.0020 0.0025 0.0030￿1.0￿0.50.51.0Figure 3.2: Laser polarization (cyan) is kept in phase with the magneticfield (blue), such that a trapping force is present at all times.For systems in which the transient response time of the AC-MOT coilsis much faster than that of the surrounding materials, we can describe theeddy currents after shutting off an AC-MOT driven by a voltage-controlledamplifier, which in turn is driven by a signal of V = V0 sin(ωt+ φ), byIeddy(φ, t) = I0ωτ cos(ωt+ φ)− sin(ωt+ φ) + (sinφ− ωτ cosφ)e−t/τ1 + ω2τ2 , (3.1)where τ is a constant that depends on the properties of nearby materials.One can see from this expression that, given an equal number of half-cycles,a choice of φ = tan−1(ωτ) results in the eddy currents being zero after theAC-MOT is shut off [2].14Chapter 4The Offline AC-MOTIn order to test the usefulness of the AC-MOT for our purposes, we ini-tially implemented an AC-MOT in our “offline” geometry, which was neverintended to be used for our final experiment. This allowed us to examinesystematic effects in advance of the arrival of the “online” chamber. Addi-tionally, because our intent for the online setup was to trap 37K, which isradioactive with a half-life of 1.2s, it was not possible to work directly withthis isotope during much of the setup and optimization processes. Instead,we chose to work primarily with (stable) 41K, which was an ideal candidatebecause its hyperfine structure is similar to that of 37K.Our offline setup, shown in Fig. 4.1 included a vacuum-pumped pyrex cellwith a connected potassium dispenser, and vertically oriented anti-Helmholzcoils external to the cell. In the AC-MOT, these coils carry a sinusoidallyvarying current. The optical details (shown explicitly in Fig. 4.6) are verysimilar to those one might use in a typical DC-MOT, with the notableaddition of an electro-optic modulator (EOM), used to rapidly flip the po-larization of the laser light between two perpendicular linear polarizationstates. The geometry of the current-carrying magnetic field coils is shownin Fig. 4.2.4.1 Methodologies for Measuring TrapCharacteristicsIn order to evaluate a variety of trap properties, we used a CCD camera tocapture images of the trapping region. Typical results are shown in Fig. 4.3.Each such image requires a camera exposure that ∼ 10 - 100 of milliseconds,limiting our ability to record rapid changes to the atom cloud. Because afull AC cycle takes only ∼ 1 millisecond, this technique is not able to tell usanything about the cloud during different phases of the AC trapping cycle.Instead, we examine images taken over a period of many AC cycles to learnabout overall, time-averaged characteristics of the AC-MOT.154.1. Methodologies for Measuring Trap CharacteristicsFigure 4.1: A photo of the offline MOT’s vacuum-pumped pyrex cell andvertically-oriented current-carrying coils. The CCD camera used for fluo-rescence and lifetime measurements is visible on the right centre, and thepotassium vaporizer is located beyond the port in the lower right corner.Trim coils are used to keep the magnetic field at the centre of the pyrex cellat zero, and the green and grey trim coils are clearly visible in this picture.The vertical trim coils cannot be seen, as they are wound about the samerings that contain the vertical quadrupole coils. To complement this photo-graph, a schematic of the coils’ geometry is shown in Fig. 4.2, and the opticssetup is shown in Fig. 4.6.164.1. Methodologies for Measuring Trap Characteristics18 cm20 cm15 cm3 cm9 cm3 cm9 cm11 cm11 cm17.5 cmTop ViewSide ViewA.  Quadrupole Coils + Vertical Trim CoilsAAAA.  Horizontal Trim CoilsBBB BBBThe Offline MOTFigure 4.2: Geometry of the Offline MOT. Note that there are three separatesets of ‘trim coils’, used to adjust the DC components of the magnetic field,such that the Zeeman shift is zero at the centre of the trap (the vertical trimcoils are separate from the quadrupole coils, despite being located within thesame plastic frame). In practice, the current in these trim coils was set so asto optimize the appearance of the cloud of trapped atoms – a method whichseemed to work reasonably well for us, given the fact that the local magneticfield in our lab could be measurably altered by the status of TRIUMF’scyclotron (below our lab), or the operation of certain equipment for otherexperiments (above our lab).4.1.1 Measuring the Number of Trapped AtomsIn order to determine the number of atoms in the trap at any given time,a smaller region of interest is selected (see Fig. 4.3), and the backgroundbrightness is subtracted off, pixel by pixel. Then, the overall trap fluores-cence is given by the sum of the brightness for each pixel within the regionof interest. Because trap fluorescence is expected to be proportional tothe number of trapped atoms, this type of measurement tells us how manyatoms we are able to trap at a particular time up to an overall scaling fac-tor, provided that parameters such as laser power and frequency are keptconstant. In the majority of cases examined within this document, we will174.1. Methodologies for Measuring Trap Characteristicsbe interested only in the way the number of trapped atoms scales – not inthe total number of atoms.Figure 4.3: Images of the trapping region as collected by a CCD camera.The image on the right shows a cloud of atoms trapped in a MOT, while theimage on the left shows the background for the same spatial region, collectedwhile the laser light was present but the magnetic field had been turned off.Each pixel from the CCD camera’s image has a ‘brightness’ value rangingfrom 0 to 255. In order to make sense of the results, it is usually necessary toconsider only some smaller region of interest immediately surrounding thecloud of atoms, and subtract the background image from the trap image,pixel by pixel, before analyzing the results. The green rectangles show atypical region of interest.184.1. Methodologies for Measuring Trap Characteristics4.1.2 Measuring the LifetimeBecause we are in a regime of low density, the number of trapped atomsN(t) in a MOT with trap lifetime τ will obey the equationdN(t)dt = N0 e+t/τ (4.1)while it is loading. We therefore chose to measure the lifetime by determiningthe rate at which atoms are loaded into the trap, as shown in Fig. 4.4.Beginning from a state with no trapped atoms, the conditions necessary fortrapping are created, and the total fluorescence in the region of interest ismonitored intermittently over a period of several seconds.In order to obtain a measurement of the trap lifetime, we record a back-ground image followed by a series of images of the trap over a period of time.To increase the signal-to-noise ratio, the cloud of trapped atoms may be de-stroyed and fluorescence measurements repeated over again several times.One notable pitfall that must be avoided when measuring the trap life-time in this manner is setting an incorrect region of interest. As one mightexpect, the cloud has been observed to move slightly when trap settingsare adjusted. While position stability is largely a non-issue in the onlineAC-MOT, the offline setup was prone to a variety of systematic effects thatwere not necessarily consistent from day to day (described further in Sec-tion 4.5). Unfortunately, if the region of interest is set to be too large,the signal is overwhelmed by the noise and cannot be used. Therefore itwas necessary to double check, for each such lifetime measurement, that thecloud was positioned where it was expected to be–within our defined regionof interest.4.1.3 Measuring the Trap Width and PositionWe are also sometimes interested in measurements of the size and position ofthe trap. In the offline AC-MOT, we were limited to using the CCD camerafor such measurements, and even in principle one CCD camera would onlyallow us to measure the width and position along two of the three spatialdimensions. In practice, however, we used only one dimension, because thepronounced vertical streaks in the camera output (see Fig. 4.3) make itdifficult to work with.Once the CCD camera has collected an image, the brightnesses of itspixels are projected along its two axes, and gaussian curves are fit to theresult, as shown in Fig. 4.5. Fig. 4.5 also demonstrates the problem inherent194.1. Methodologies for Measuring Trap Characteristics  Time (s)  Fluorescence (arb)           ! = 7.350 ± 0.162 s N0 = 30 ± 0.2 (arb)  Trap Creation as a Function of TimeFigure 4.4: A typical set of reduced fluorescence data from the CCD cam-era, used to extract AC-MOT lifetimes. The quantity plotted on the verticalaxis is the fluorescence measured only within a pre-defined region of interest,after the background brightness has been subtracted off. At t=0, when theconditions necessary for trapping begin, there is no cloud of trapped atomspresent. The fluorescence is measured at time intervals spanning a periodof several seconds, at which point the trapping mechanism is removed andthe atoms are allowed to disperse. Once the cloud has been completely de-stroyed, the measurements may be repeated again as the number of trappedatoms grows again. To obtain a plot such as this, with sufficiently cleandata, the cycle of measurements is repeated 3 to 5 times. A fit is thenperformed to extract the trap lifetime τ , which is described by Eq. 4.1.204.1. Methodologies for Measuring Trap CharacteristicsVertical FWHM = 1.73 ± 0.08 mmHorizontal FWHM = 0.64 ± 0.14 mmVertical (mm)-0.5                0.0                0.5                1.0                1.5                2.0                2.5                3.0                3.5Horizontal (mm)-4.5                  -4.0                  -3.5                  -3.0                  -2.5                  -2.0                  -1.5                  -1.0       8580757065Average Trap Projections10810610410210098969492ROI Fluorescence (arb)Figure 4.5: A typical set of trap fluorescence projections along the verticaland horizontal axes. The projections are fit to a gaussian, and the FWHMused to characterize cloud width. In these plots, the background fluorescencehas been subtracted off–however the vertical lines (as seen in Fig. 4.3) arenot steady from image to image, and so they are still prominently visiblein the horizontal projection. The result of this is that our horizontal trapwidth measurements are not particularly reliable. In this thesis, when thetrap width is mentioned without explicitly stating an axis of projection, itshould be assumed that a vertical projection is being used.in working with fluorescence projections along the horizontal axis withinour system. The ‘full width at half maximum’ (FWHM) of the verticalprojection’s gaussian is used to characterize the cloud’s width, and its centreis used to describe the cloud’s position.214.2. Laser Frequency Calibrations in the Offline AC-MOT4.2 Laser Frequency Calibrations in the OfflineAC-MOTOne aspect of optimal MOT operations is that of tuning the lasers to optimalfrequencies. In many setups, the precise parameters of the MOT’s repumperbeam are of secondary importance to the MOT’s operation. However, as canbe seen in Fig. 2.3 for 41K, the repumper component in our MOT is actuallyquite important, due to the proximity of the F = 1 and F = 2 excitedstates to one another. That is, a high fraction of the atoms excited bythe trapping laser will end up in the ‘wrong’ excited state, increasing thelikelihood that they will decay to the ‘wrong’ ground state – therefore therepumper component of the MOT’s laser beams plays a relatively large role.The optical setup in our offline AC-MOT is shown in Fig. 4.6, and thevariable subscripts in Eqs. 4.2 and 4.3 make reference to that diagram. Thetrapping frequency, ftrap and the repumper frequency, frepump in this systemcan be described byftrap = flock − fAOM A − fAOM B (4.2)frepump = ftrap + fVCO. (4.3)We chose to lock the laser to the spectral peak at approximately 277.4MHz from the centre of gravity in 39K. This peak was the result of acombination of two different D2 transitions in 39K: F = 1→ F = 1 (272.7MHz) and F = 1 → F = 2 (282.0 MHz) [6, 7]. The relative strengths ofthese two transitions in our vapor cell is unclear, therefore there exists somesystematic uncertainty in the absolute value of all measured frequencieswhich depend on the laser’s lock point.In order to produce the desired coupling to both the F = 2→ F = 1 andF = 2→ F = 2 D2 transitions (132.49 MHz and 135.87 MHz, respectively,from the centre of gravity [6, 7]), the laser frequency was adjusted from itslock point by passing it through two AOMs. To produce a laser beam tunedto +130 MHz (that is, slightly red-detuned from both resonances), AOMsA and B were adjusted to decrease the frequency of the laser light passedthrough them by a total of 147 MHz. We took this as our starting trapfrequency.Sidebands were added next – initially at fVCO = 241 MHz, and the side-band with the positive detuning was used as the repumper. This produceda repumper frequency of +371 MHz – red-detuned from the D1 transitionsat +386.5 MHz (F = 1 → F = 1) and +389.88 MHz (F = 1 → F = 2)[6, 7].224.2. Laser Frequency Calibrations in the Offline AC-MOTDiode LaserTapered Amplifierλ2λ2EOMNon-Polarizing Beam SplitterPyrex Cellmirrormirrorλ/2λ/4λ/4λ2λ/4Offline SetupVerticalBeamHorizontalBeam 1HorizontalBeam 2AOM DPhotodiodePhotodiodeLock-In AmplifierVCOPotassiumVapor CellAOM BAOM APump BeamProbe BeamsPolarizing FilterNon-Polarizing Beam Splitterλ/4Potassium VaporizerQuadrupole Coil (Top)Quadrupole Coil (Bottom)λ/4λ/4mirrorFigure 4.6: A diagram of the components of the offline AC-MOT. The geo-metric layout of the current-carrying magnetic field coils is shown in Fig. 4.2.The basic design is similar to that of a typical DC-MOT. The laser is lockedto a chosen carrier frequency by using standard saturation spectroscopytechniques, and a voltage-controlled oscillator (VCO) is used to add side-bands above and below the carrier frequency. AOM D is used only to turnthe light on and off. The EOM is used as a rapidly-changing variable waveplate. Tuned properly, it is able to adjust a polarized laser beam passingthrough it to either of two perpendicular linear polarization states, which isessential for the operation of an AC-MOT. Subsequent half-wave plates areused to adjust the polarization directions (in both states) so as to minimizeellipticity after the laser passes through beam splitters, and quarter waveplates are used immediately before the beams enter the trapping chamberso as to produce circular polarization. Thus, when the direction of linearpolarization is switched by ninety degrees at the exit from the EOM, thedirection of circular polarization of the light in the chamber is also switched.In Fig. 4.7, the trap frequency ftrap was swept while keeping the re-pumper frequency fixed by a method in which fVCO and fAOM B were swept234.2. Laser Frequency Calibrations in the Offline AC-MOTtogether, such that their combined contribution to frepump was constant.However since fVCO does not contribute to ftrap, the change in fAOM B alsoproduced a change to ftrap.120 130 140 15002000400060008000100001200014000Trap Frequency (MHz)Fluorescence (arb)F=2 → F=1 (D1)F=2 → F=2 (D2)F=2 → F=3 (D2)120 130 140 150010000200003000040000500006000070000Fluorescence (arb)F=2 → F=1 (D1)F=2 → F=1 (D2)F=2 → F=2 (D2) F=2 → F=3 (D2)0.5 1.0 1.5 2.0 2.5 3.012340.5 1.0 1.5 2.0 2.5 3.01234Power = 48 mWPower = 79 mWfrepump = 371.0 MHzLaser Power and Trap FrequencyF=2 → F=1 (D2)0.5 1.0 1.5 2.0 2.5 3.012340.5 1.0 1.5 2.0 2.5 3.012340.5 1.0 1.5 2.0 2.5 3.01234frepump = 369.5 MHzfrepump = 371.0 MHzfrepump = 372.5 MHzPower = 48 mWFigure 4.7: The relationships between trapping frequency, repumper fre-quency, laser power, and trap fluorescence are shown here. Power measure-ments were taken after the EOM (as shown in Fig. 4.6), and would includethe trapping frequency component of the laser beam as well as the twosidebands, each of which has about 10% the power of the carrier frequencycomponent. One of these sidebands functions as the repumper, and theother is simply ignored as it is not close to any atomic resonances. Fluo-rescence is measured according to the procedure described in Section 4.1.1,though we do not interpret the overall fluorescence as proportional to thenumber of trapped atoms here, since a varying laser frequency will also havea large effect on the fluorescence for any particular trapped atom.244.3. The Phenomenology of Lifetime Measurements in the AC- and DC-MOT4.3 The Phenomenology of LifetimeMeasurements in the AC- and DC-MOTIt is interesting to consider the relationship between lifetimes in AC- and DC-MOTs. Fig. 4.8 shows that lifetimes for an AC-MOT are significantly shorterthan those measured in a DC-MOT under similar operating conditions.One possible reason for the experimentally measured decreased lifetime isthat in an AC-MOT, the restorative ‘spring’ force resulting from the Zeemanshift is eliminated or decreased during some fraction of each AC cycle. Theauthor speculates that if this were the case, it might be possible to bringthe results into closer agreement if the laser frequencies were tuned closerto resonance, so as to make the trapped atoms colder. Unfortunately, thishas never been attempted.Another possible interpretation for this result is that our loss mechanismsin the offline AC-MOT are dominated by systematic effects. In particular,the gain on the two power supplies controlling each quadrupole coil indi-vidually under AC operation were seen to drift over time. If their currentoutputs became too unequal, the AC-MOT was lost entirely – that is, thecloud positions for each “side” of the AC cycle ceased to overlap with oneanother. This was a significant problem in the offline setup, but the ex-tent to which this may have contributed to our measurement of lower AClifetimes is unclear.4.4 Measured Lifetimes and AC FrequencyWe were interested to find out whether the trap lifetime in an AC-MOTchanges as a function of its (AC) frequency, and to see how lifetimes in theAC-MOT relate to the lifetimes of a comparable DC-MOT.The results are shown in Fig. 4.9. It appears that the trap lifetimes areroughly constant as a function of frequency, provided that the AC-MOT isgoing faster than some minimum frequency (around 100 Hz). The preciselocation of the cutoff drifted over the course of data collection, as we seefrom the measured 1s lifetime at 100 Hz and the 4s lifetime at 50 Hz. Thiscutoff effect has also been observed previously, in [2], where the cutoff wasobserved to be around 2 kHz.The astute reader may notice that the 500 Hz lifetimes in Fig. 4.9 appearto be inconsistent with those in Fig. 4.8 – but this does come as a surprise,as the data for the two plots was collected months apart, and any numberof trapping variables may have been changed. Parameters such as overall254.4. Measured Lifetimes and AC FrequencyÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ0 1 2 3 4 5 6 70246810Current - Quadrupole Coil HALLifetimeHsLAC and DC LifetimesV_EOM = -250VV_EOM = +302V500 Hz AC HRMS currentLDC “-”DC “+”AC 0 Hz (RMS cu t)ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ0 1 2 3 4 5 6 70246810Current - Quadrupole Coil HALLifetimeHsLAC and DC LifetimesV_EOM = -250VV_EOM = +302V500 Hz AC HRMS currentLCurrent - Quadrupole Coil (A)AC and DC if ti eLifetime (s)Figure 4.8: A comparison of the measured lifetimes in an AC trap and thetwo equivalent DC trapping states. That is, the DC “+” trap is identicalto the DC “−” trap, except for the direction of current flow in the anti-Helmholtz coils, and the sign of the laser polarization. In the AC-MOT,the laser polarization flips back and forth between these two states, and thecurrent in the anti-Helmholtz coils varies sinusoidally at 500 Hz. It can beseen from this plot that the AC-MOT lifetimes are shorter than those of acomparable DC-MOT.laser power, fraction of power in the repumper beam, frequency tuning ofeither or both of these components, laser alignment, and laser polarizationin either or both of the DC ‘states’ (both of which are needed for an AC-MOT), are all things which have been known to drift or change over longertime periods, and these can all affect the robustness of a MOT. One mustalso consider the systematic effects discussed in Section 4.5, which can altertrap characteristics on much shorter timescales.264.4. Measured Lifetimes and AC FrequencyFigure 4.9: Lifetimes as a function of AC frequency. Lifetimes are measuredas described in Section 4.1.2. For each datapoint, the amount of current ineach coil was normalized with respect to the other coil’s current through amethod in which the trap position as seen by a CCD camera was requiredto remain constant. The peak-to-peak amplitude of the magnetic field (asmeasured by a Hall probe located near one side of the trap) was also heldfixed at every data point. A new measurement at the 500 Hz baseline wastaken between datapoints at other frequencies in order to be certain thatthere was no time-dependence in the results. Although the trap lifetime isnearly constant at “high enough” frequencies, there is a sharp cutoff around50-100 Hz – though the cutoff itself appears to have changed during themeasurement process. This change could be due to any of a number ofsystematic effects described in Section 4.5. The reason for the existence ofsuch a cutoff is unclear, but the effect has previously been observed in [2] tooccur around 2kHz.274.5. Systematic Effects in the Offline AC-MOT4.5 Systematic Effects in the Offline AC-MOTThe offline AC-MOT setup was plagued by a variety of systematic problems.One noteworthy problem that had been observed in the offline MOT setupeven before it was converted into an AC-MOT was the non-uniformity ofthe windows on the pyrex cell. This had the dual effects of damaging laserpolarization, and distorting the beam profile. Thus, whenever the opticalsetup was ajusted or bumped and recalibrated, the laser interference pat-terns within the central trapping region would change as well. This couldbe observed qualitatively in images of the trapped cloud of atoms collectedwith the CCD camera–a slight recalibration of the lasers would often resultin a DC trap that had moved a few millimeters or had changed its shape.Such problems are to be expected.In the AC-MOT, however, this problem was compounded. In order toproduce a functional AC-MOT, it is necessary for the cloud’s location in oneDC-MOT state must be the same as (or at least overlapping with) the cloud’slocation in the other DC state. If this cannot be achieved, the AC-MOT willfail – and to the extent that is achieved only poorly, the characteristics of theAC-MOT will be non-optimal. The atomic cloud may be spread out, andatoms may be ejected from the trapping region, damaging the AC-MOT’sability to retain atoms. This problem of was largely corrected in the onlineAC-MOT setup, where lasers entered the chamber only through viewportsspecifically designed to be flat, and mounted on a much more solid stainlesssteel chamber.An additional source of instability specific to the offline AC-MOT wasthe power supplies controlling the current through the anti-Helmholtz coils.In the offline MOT, it was necessary to use a different set of power supplies tocontrol the AC-MOT than we had used for DC control. Although the powersupplies used for DC operation of the MOT were quite stable, they werenot able to produce an alternating current. For running the anti-Helmholtzcoils in AC mode we used a pair of Peavey CS 4080Hz audio amplifiers,which were unable to produce a DC current. It is worth noting here thatit was necessary to use two amplifiers – one for each current-carrying coil– because a single amplifier would have been unable to drive the necessarycurrent through both coils. Though I am unable to comment on the qualityof this hardware when used for its intended purpose, it turned out to bequite problematic for use in an AC-MOT.As is typical for audio amplifiers, these devices were not able to out-put a tunable DC offset to their current. In principle, they were expectedto produce symmetric positive and negative voltage outputs, automatically284.5. Systematic Effects in the Offline AC-MOTadjusting their average DC offsets to zero. In practice, oscilloscope readoutssuggested that the DC offsets output from these devices drifted over time.The gain on these amplifiers drifted over time as well – perhaps as a resultof them heating with use, though the drift on these two audio amplifierswas asynchronous. Since it was necessary to run these amplifiers at close totheir full power, it was common for the gain on one or both amplifiers todrift upward just enough for the output waveform to be clipped on one side.Therefore, it was necessary to closely monitor current output from theaudio amplifiers while trapping. Though nothing could be done about DCoffsets, their respective gains could be adjusted manually, and this sometimeshad an effect on the DC offset.Problems with the audio amplifiers dominated systematic effects in theoffline AC-MOT. Instability in the anti-Helmholtz coil currents producedinstability in the atoms’ effective (Zeeman shift) potential well. Since anAC-MOT requires that the potential minima for both ‘states’ must be inthe same spatial location, the offline AC-MOT proved to be very finicky.Lifetime measurements and trap size measurements were very different fromday to day, and somewhat different from hour to hour.Fortunately, this latter problem was mostly eliminated in the online AC-MOT. The solution was simply to buy a more expensive set of power suppliesdesigned to do what we wanted. The online setup used two Matsusada DOP25-80 power supplies, which are capable of outputting both DC and ACcurrents and voltages, and which also had extremely minimal drift. As aresult, the online AC-MOT proved to be significantly more stable than theoffline AC-MOT.29Chapter 5Turning the Trap OffThe reader will recall that the goal of this work is to obtain samples ofcold, tightly confined, spin-polarized atoms for further study. To polarize asample of atoms well, the magnetic field gradient must be zero (or as close asis achievable), and an incident optical pumping laser must be as completelypolarized as possible, and tuned precisely to the proper atomic resonance.Unfortunately, these requirements (as well as the fact that the lasers used forconfining atoms within a MOT would destroy any atomic polarization) meanthat atomic polarization cannot be obtained during the normal operation ofa MOT. The trapping lasers must be shut off and the magnetic field gradientmust be eliminated before successful optical pumping can occur.To this end, we employ a strategy of running the AC-MOT over severalAC ‘cycles’, then halting the trapping mechanism for a period of time toallow for optical pumping. The MOT forces resume again – ideally beforetoo many atoms have been lost from the trapping region – and the atomcloud is compressed once more during operation of the AC-MOT, until thetrapping forces are halted again. We will henceforth refer to this on/offcycling – and more specifically to the fraction of the total time spent withthe AC-MOT off – as the ‘duty cycle’.We wish to allow as much time as possible for optical pumping, whilesimultaneously retaining as many atoms as possible. The residual magneticfield during the optical pumping window is an additional consideration, andits optimization was the reason behind our choice to develop an AC-MOT. Itis, of course, necessary to use an integer number of AC cycles between opticalpumping times so as to allow us to shut off the current in the quadrupole coilsat a moment when the residual eddy currents in the surounding materialsare zero [2]. In this way, the magnetic field gradient is minimized so as toallow for optical pumping.The problem of optimizing laser polarization for optical pumping is ad-dressed by former co-op student Scott Smale in his end of term report [8].Here, it suffices to say that in the online trap, the optical pumping lasersetup was entirely separate from the one that was used for trapping in theMOT, and optical pumping was never implemented at all in the offline trap.305.1. Methods for Turning Off the Trap5.1 Methods for Turning Off the TrapThe trapping mechanism in a MOT can be stopped by shutting off eitherthe magnetic field or the trapping laser. It is helpful to determine whichmechanism for removing trapping forces produces the best trap lifetimesand smallest cloud sizes, as averaged over many on/off cycles during whichthe cloud of atoms is retained.The data used to generate Figs. 5.1, 5.2, and 5.3 was taken in the offlineMOT. We ran a continuous loop of 3 AC cycles at 1000 Hz, followed by 1“off” cycle – 3 ms on followed by 1 ms off. We also took a series of data inwhich the trapping mechanism was not stopped, in order to make a usefulcomparison.We are interested in understanding what happens to the cloud of trappedatoms when we turn the trap off. It is possible to halt the trapping mech-anism by turning off the magnetic field gradient, the laser, or both. Weexamine the differing effects of each of these techniques, and the results areshown in Figs. 5.1, 5.2, and 5.3.As one might expect, increasing the laser power increases trap fluores-cence, but it also heats the atoms, thereby decreasing trap lifetime. Wefind that it is possible to preserve the trap lifetime, despite removing trap-ping forces during some fraction of the time, as long as the trapping lightis removed. While the magnetic field is being turned off, it is non-zero andpoorly quantified for some length of time. Furthermore, since we attemptto eliminate the magnetic field as rapidly as possible, it is entirely reason-able to suppose that we may overcorrect the current, and thus the sign ofthe magnetic field gradient would be briefly opposite of our expectation. Ifat any point the sign of the magnetic field is reversed, the atoms that hadpreviously been trapped could be ejected upon interacting with the laser.We did not examine the presence of this mechanism beyond what is shownin this section.315.1. Methods for Turning Off the TrapÊÊÊÊ20000400006000080000100000ÊÊÊÊ20000400006000080000100000ÊÊÊÊ20000400006000080000100000ÊÊÊÊ0 2 4 6 8 10 12 14 1620000400006000080000100000ÊÊÊÊ0 5 10 15020000400006000080000100000• Always On• Kill Laser•  Kill Magnetic Field•  Kill BothÊÊÊÊ0 5 10 15020000400006000080000100l ays Onil Laser  il agnetic Field  ill BothTrap Lifetime (s)Fluorescence (arb.)Laser Power = 66 mWLaser Power = 49 mWLaser Power = 40 mWLaser Power = 21 mWTrap Killing MethodsFigure 5.1: Different methods for eliminating the trapping mechanism anddestroying the cloud of atoms were tested at several different levels of laserpower in the offline AC-MOT at fAC = 1000 Hz. The goal was to remove thetrapping mechanism for some part of the duty cycle while still maintaining along average lifetime and a high number of trapped atoms. Trap fluorescenceis treated as a stand-in for the number of atoms in the trap, and is measuredas in Section 4.1.1. Except for the points labelled as ‘always on’, the MOTis run with a duty cycle in which the trapping forces are on for 3 ms, thenoff for 1 ms.325.1. Methods for Turning Off the Trap1.6 1.8 2.0 2.2 2.4203040506070Trap Size mm⇥LaserPowermW⇥ Always On Kill B  Kill Laser  Kill Both1.4 1.6 1.8 2.0 2.2 2.4010203040506070Trap Size mm⇥LaserPowermW⇥ Always On ill B ill Laser  Kill Bothlways Onil  Bil  Laseril  BothTrap Size ( m)Laser Power (mW)Trap-killing Methods, Trap Size, and Laser PowerFigure 5.2: Different methods to eliminate the trapping mechanism, andtheir effects on trap size, are shown for several different laser power mea-surements. The AC-MOT is run at a frequency of fAC = 1000 Hz, andexcept for the points labelled as ‘always on’, the MOT is run with a dutycycle in which the trapping forces are on for 3 ms, then off for 1 ms. Laserpower is measured immediately after the EOM shown in Fig. 4.6. Trap sizeis averaged over many on/off cycles, and is measured as described in Sec-tion 4.1.3. This data seems to suggest that whether or not the magneticfield is shut off, the trap diameter is larger if the laser is shut off. Thisis consistent with what we might have guessed based on the premise thatthe optical molasses from trapping lasers would create a drag force to slowexpansion of the atom cloud once the MOT’s confining force is shut off.335.1. Methods for Turning Off the TrapTrap-killing Methods, Trap Lifetime, and Laser Power4 6 8 10 12 14203040506070Lifetime s⇥LaserPowermW⇥ Always On Kill B  Kill Laser  Kill BothLaser Power (mW)1.4 1.6 1.8 2.0 2.2 2.4010203040506070Trap Size mm⇥LaserPowermW⇥ Always On Kill B  Kill Laser  Kill Bothlways Onil  Bil  Laseril  BothLifet me ( )Figure 5.3: The AC-MOT is run at a frequency of fAC = 1000 Hz, and exceptfor the points labelled as ‘always on’, the MOT is run with a duty cycle inwhich the trapping forces are on for 3 ms, then off for 1 ms. Laser power ismeasured immediately after the EOM shown in Fig. 4.6, and trap lifetimeis measured as in Section 4.1.2, and is averaged over many on/off cycles.For this plot, trapping forces are removed from an AC-MOT operating withseveral different values of laser power (plotted on the vertical axis). Theeffect on (average) trap lifetime is shown. Whether the magnetic field isturned off or not, the lifetimes are longer when the laser is shut off. It isinteresting to note that that the trap lifetimes measured in the case wherethe trapping mechanism is always on are shorter than the trap lifetimesmeasured when the laser (and therefore the trapping mechanism) is shut offfor part of the duty cycle. The mechanism behind this effect is unclear.345.2. Duty Cycle5.2 Duty CycleIt is important to maximize the percentage of time during which the mag-netic field (which is itself needed in order to maintain the trap) is off, whilesimultaneously keeping the observed trap lifetime as long as possible (as av-eraged over many trapping/non-trapping cycles), because we are only ableto collect useful data during the time when the magnetic field is off and theatoms are polarized. The eventual goal is optimize count-limited beta-decaystatistics using (radioactive) 37K as our trapped isotope. To this end, weexamine different methods for destroying the trapping forces, and achievableduty cycles in an AC-MOT trapping 41K, which has a hyperfine structuresimilar to that of 37K.Initial results describing the effect of different duty cycles on averagetrap lifetime and fluorescence are shown in Figs. 5.4 and 5.5. Both showdata collected in the offline pyrex chamber using 41K. It is worth notingthat the most obvious way to improve on these results would be to use aset of laser beams which is more optimally balanced and more gaussian inprofile. However, in the offline vapor cell we were limited by the qualityof the chamber windows through which the laser passed. In particular, ananti-reflective coating on the windows would have likely produced a largeimprovement in beam profile and balance. This was later achieved in theonline chamber, however no comparable data has been collected in thatgeometry.355.2. Duty CycleOut[630]=∑∑∑∑∑----------*Continuous*1offê18on*2offê18on*1offê3on*2offê3on0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70123456Cycles Off ê Cycles OnLifetimeHsLTrap Lifetimes at 2 kHzCycles ff / Cycles OnLifetime (s)Trap Lifetimes at 2 kHz Varying Duty Cycles n the Offline AC-MOTut[630]=*Continuous*1offê18on*2offê18on*1offê3on*2offê3on. . . . . . . .l lLifetimeHsLFigure 5.4: Initial results describing the effect of varying the duty cycleon the (time-averaged) trap lifetime, at fAC = 2 kHz. Trapping forces areremoved by eliminating both the magnetic field and the trapping laser. Life-time measurements are taken as per the procedure described in Section 4.1.2,with the additional caveat that lifetime measurements are averaged over mul-tiple on/off cycles. The green curve is intended only to guide the eye. Onecan see that turning the trapping forces off for a sufficiently small fractionof the time does not harm trap lifetime measurements; counterintuitively,the trap lifetime may even increase slightly in some cases. This observationis consistent with the data presented in Fig. 5.3.365.2. Duty Cycle*Continuous*1offê18on*2offê18on*1offê3on*2offê3on0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7010000200003000040000500006000070000Cycles Off ê Total CyclesBrightnessHarb.LTrap Fluorescence at 2 kHz*Continuous*1offê18on*2offê18on*1offê3on*2offê3on0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.701 0002 0003 0004 0005 0006 0007 000Cycles Off ê Total CyclesBrightnessHarb.LTrap Fluor scence at 2 kHzTrap Fluorescence at 2 kHz Varying Duty Cycles in the Offline AC-MOTCycles Off / Cycles OnBrightness (arb)Figure 5.5: Trap fluorescence measured as a function of the trapping dutycycle at fAC = 2 kHz. Trapping forces are removed by eliminating both themagnetic field and the trapping laser. Brightness measurements are takenas per the procedure described in Section 4.1.1, with the additional caveatthat fluorescence measurements are averaged over multiple on/off cycles.The green curve is intended only to guide the eye. Since the laser is shut offduring the “off” cycles, one might expect that in the absence of atom loss,fluorescence should be proportional to the fraction of total time used for theAC-MOT, but this does not appear to be the case. Instead, we see thatthe brightness is roughly constant provided that the duty cycle includes asufficient percentage of time with the AC-MOT on.375.3. Residual Magnetic Field in a Pyrex Cell5.3 Residual Magnetic Field in a Pyrex CellOne of the things we attempted to optimize is the residual magnetic fieldin the magneto-optical trap. In particular, our goal was to eliminate themagnetic field in the trapping region as completely and as quickly as possibleafter turning off the trapping mechanism. As a first test, we worked withour offline AC-MOT, where we measured the residual magnetic field usinga Hall probe located outside the chamber. Typical results are shown inFigs. 5.6 and 5.7.Our methodology suffered from a variety of systematic effects. With theHall probe located outside the chamber, far from the region in which the fieldgradient could be expected to be linear, we treated the measured value of themagnetic field as being directly proportional to the size of the field gradientwithin the central trapping region. Given our knowledge of the system’sgeometry, we were able to estimate the proportionality constant, but thiswas made more difficult by the fact that the Hall probe’s mount was notstable. Both the position and orientation of the probe would change betweendatasets, and the result was a noticeable change in measured magnetic fieldstrength.Because of the amplifiers available to us, we controlled the current throughthe anti-Helmholtz coils indirectly, through their voltages. In the offlinetraps, a variety of audio amplifiers were used to drive the rapidly oscillatingAC voltages we needed. Two audio amplifiers were used at any given time –one to drive each coil – because an individual audio amplifier could not pro-duce sufficient current to drive both coils at once. This technique producedseveral systematic effects. The amplifiers’ gains drifted by approximately10% over the course of a couple hours’ worth of measurements. Addition-ally, the frequency dependence of the gain was different between any setof two amplifiers. Unfortunately, when the currents through the two coilsbecame too unequal, the trap was destroyed.The amplifiers were controlled in the voltage domain by means of anarbitrary waveform. These waveforms were generated by computer to aformat which was compatible with the Stanford Research Systems DS435function generators, to which the waveforms were uploaded. The functiongenerators were then used to control the audio amplifiers directly.385.3. Residual Magnetic Field in a Pyrex CellFigure 5.6: Optimal Waveform to Minimize Residual Magnetic Field in theOffline Trap. Channel-1 shows the readout of the Bell Labs Hall probe,while Channel-3 shows the output of an SRS-DS345 function generator asit is being used to control (one of) the anti-Helmholtz coils. On Channel-1, 1 mV is equivalent to 1µT at the probe’s location. The peak-to-peakamplitude of the magnetic field is approximately 2.2 mT.395.3. Residual Magnetic Field in a Pyrex CellFigure 5.7: This is a zoomed in view of the same output as is shown inFig. 5.6. At the time of the initial drop in magnetic field, the residualmagnetic field (Channel-1) is 3.2% of its maximum value, and decays rapidlyaway. Note that this Hall probe’s overall offset drifts by approximately 1mV.40Chapter 6Calibrations of the OnlineTrapIn addition to our offline setup, measurements and calibrations were alsodone on a new “online” MOT chamber setup. The online trapping setupdiffers from the offline setup in that the online setup includes two distinctmagneto-optical traps, separated by approximately two metres. The first isused for initial collection of a sample of atoms, either from the beamline itselfin the case of radioactive atoms, or else from a dispenser to be used for offlinetests. The trapped atoms in the first MOT are then periodically loaded intothe second MOT according to the methodology described in [9]. This is doneso as to minimize the background from decaying untrapped radioactives nearthe second trapping chamber where our detection apparatus is located. Thefirst trap is only intended to be run as a DC-MOT.The second trapping chamber was newly constructed from 316-L stain-less steel, which has very low electrical and thermal conductivity as com-pared with other metals. This material was selected because while it wasnecessary for our chamber to be sturdy enough to mount a series of deterc-tors, it was also desirable to minimize electrical eddy currents resulting fromthe AC-MOT in this chamber.Around this time, we also acquired a new model of Hall probe, andtwo new, more reliable amplifiers to drive the anti-Helmholtz coils in thesecond trap. Additionally, in the time before the chamber was closed upfor trapping, it was possible to mount the Hall probes reliably at knownpositions relative to the centre of the trapping region. This allowed for amuch more accurate estimate of the effects of eddy currents at the trapcentre than had been possible in the offline setup.The power supplies for the online trap are both Matsusada DOP 25-80,which are designed to run in the range of ±25V and ±80A. These werean upgrade from the offline MOT’s audio amplifiers, not only because oftheir greatly decreased propensity for drifting, but also because these powersupplies allowed for rapid switching between an AC signal and a DC voltageor current.41Chapter 6. Calibrations of the Online TrapFigure 6.1: A photo of the inside of the new trapping chamber, before mostof the hardware for observing β-decay daughter particles was installed. Thetwo current-carrying copper quadrupole coils are clearly visible. The cham-ber walls are made of 316-L stainless steel, a material which was selected asa compromise between strength, cost, and minimizing eddy currents in thechamber walls.This ability for the second MOT to switch between AC and DC modesof operation rapidly and without destroying the cloud of trapped atoms wasessential for two reasons, the first being the method by which the second trapacquires its atoms. The atoms are loaded directly from the first MOT, whichtransfers its atoms over to the new chamber at intervals of approximately onesecond according to the method described in [9]. The atoms are transferredby using a resonantly tuned laser to “push” the atoms over from their initialposition in the first MOT, through two optical funnels, and into the newchamber where they are re-collected in the second DC-MOT, which alreadyholds any atoms remaining from previous transfers which have not yet beenlost. The optical funnels are each comprised of two sets of current-carrying42Chapter 6. Calibrations of the Online Trap ”	 ”	 ”	 ”	IIFigure 6.2: Geometry of the quadrupole coils in the online MOT is shownhere. Water flows through the coils to cool them. These coils can be drivenin AC mode with a DC offset, so they serve a second purpose as trim coils,cancelling out one component of the ambient magnetic field within the lab(generated by the nearby cyclotron when it is in operation, and by the earthitself). There are two additional sets of trim coils (not shown) which areexternal to the chamber, and tuned with a DC offset so as to cancel themagnetic field components in the other two dimensions.43Chapter 6. Calibrations of the Online Trapcoils–effectively a two-dimensional MOT–in order to keep the atoms confinedin the transverse directions during the transfer. Once the atoms have beencollected in the second MOT, they must not be lost as we switch its operationfrom a DC-MOT to an AC-MOT.The author of this thesis speculates that it may be possible to adapt thistechnique to collect atoms in an AC-MOT directly. However, the atoms’times-of-flight are expected to be in the 10s of milliseconds, so a relativelysmall spread in transfer speed would cause arrival times to vary, spanningseveral AC cycles. It seems likely therefore that transferring atoms directlyinto an AC-MOT would result in a less robust and less efficient transfermechanism. We did not attempt this.The second reason for the importance of our ability to switch betweenAC and DC operation is that our goals for data collection require a smallcloud of polarized atoms. We polarize the atoms by optically pumping themafter the MOT has been turned off, but this is ineffective in the presenceof non-uniform and non-static magnetic fields. In order for optical pump-ing to be effective, it is necessary that any present magnetic field must beuniform and aligned along a previously chosen axis of laser propagation.Any misalignment of the magnetic field axis with the optical pumping axiswill prevent the atoms from being fully polarized. Similarly, a magneticfield which is non-uniform over the extent of the atomic cloud must producenon-uniform atomic polarization when an optical pumping laser is applied.Though we might be tempted to simply apply a large dipole field on top ofany residual quadrupole field from the MOT so as to maximally align thetwo axes, this is inadvisable. The presence of a large magnetic field willcause energy splitting between nearby hyperfine levels, which will harm op-tical pumping efficiency as the laser becomes more detuned from only someof the transitions to which it was intended to couple. A large magnetic fieldwould also cause mixing between the states we had hoped to optically pump,again damaging our ability to polarize the sample of atoms.Thus, the best solution is to eliminate the quadrupole field used in theMOT as rapidly as possible, while simultaneously applying a small constantdipole field on top of any residual non-uniformities. To rapidly eliminate thequadrupole field from the MOT, we use an AC-MOT. However, the smalldipole field is produced by generating a small DC current in the same coilsthat had been producing the quadrupole field, so the necessity of switchingbetween AC and DC operation rapidly is clear.446.1. Frequency Response in the Online Setup6.1 Frequency Response in the Online SetupThe first task was to calibrate the power supplies’ output in the relevantsetup. With this information, we can monitor the current output eitherthrough the “current output monitor” port in the power supply, or if we arerunning the power supplies in current-control mode, indirectly through theamplitude of the input signal.We first note that the power supplies’ specifications are such that theycan accept input voltages within the range [-10V, 10V], which is used tocontrol an output signal in the range [-80A, 80A], and [-25V, 25V]. Theyare also equipped with ‘voltage monitor’ and ‘current monitor’ ports, bothof which output a voltage signal in the range [-10V, 10V], to monitor thevoltage and current outputs, respectively. Therefore we write,Iout[A] =80A10V Imon[V] (6.1)Vout =25V10V Vmon= 25V10V Vin (6.2)It is useful to measure the effective inductance of the two coils when theyare both running in their trapping (anti-Helmholtz) configuration. Becauseof the limit to the amount of voltage we are able to output to the coils, itis necessary to know their effective inductance in order to determine howhigh an AC frequency we can run our MOT at, if we require some minimumof current output. We determined both the resistive and capacitive compo-nents of the impedance of the system to be negligible, and drove the powersupplies with a continuous sinusoid of amplitude Vin = 9.8V, at several dif-ferent frequencies. Recall that under harmonic excitation, a system with apurely inductive load can be described byImaxVmax= 12pifL. (6.3)The data was therefore fit toImon,max[V]Vin,max[V]=(25V80A) 12pifL, (6.4)and is shown in Fig. 6.3. The result was an effective inductance of (50.1± 0.2)µH.456.2. Frequency Response in the Hall ProbesFigure 6.3: Inductance in the online MOT setup. The coils were both drivenin voltage-control mode with a continuous sinusoidal input of amplitude9.8V. Best fit gives L = (50.1±0.2)µH, which is reasonably close to the valueof 48µH that was obtained by direct calculation in a simplified geometry.The coils were driven 180◦ out of phase with one another (as they would beduring trapping) so as to account for mutual inductance effects.This result is quite close to the value of 48µH which can be calculatedby use of the Biot-Savart law (ie, assuming that changes to the current are“slow”) in a simplified geometry – 2 sets of circular coils of 16 turns each,with a radius of 78.7 mm, separated by a distance of 137.5 mm.6.2 Frequency Response in the Hall ProbesWe are interested in the size of the magnetic field gradient produced by anAC-MOT at varying frequencies. It is helpful to know the inductance of the466.2. Frequency Response in the Hall Probessystem, but it is also possible that the Hall probes themselves may sufferfrom frequency-dependent systematic effects. We expect magnetic field tobe directly proportional to the current in the coils, so any deviation fromthat would indicate either some frequency dependence in the Hall probes’response, or (less likely) some frequency dependence in the accuracy of thepower supplies’ current monitor output.Trap CentreTurquoise Test KitGreen Test KitBell Labs ProbeBell Labs ProbeTrap CentreTurquoise Test KitGreen Test KitHall Probes:  First LayoutHall Probes:  Second LayoutFigure 6.4: Hall probe configurations used in the “online” trapping chamber.Angular tolerances on the axes of the Hall probes are all ±15◦, except whereotherwise specified. “out of page dimension” tolerances are ±6 mm. TheBell Labs probe is constrained to measure only the zˆ component of themagnetic field (that is, out of the page), while the Ametes Test Kit probesmeasure the radial component of the field–to within given tolerances–unlessotherwise specified.476.2. Frequency Response in the Hall ProbesIn order to quantify this effect, we set up all three of the Hall probes asshown in the first configuration of Fig. 6.4, and drove the coils with a seriesof pure sinusoids at different frequencies, while holding the amplitude of thecurrent output fixed. The result is shown in Fig. 6.5. Note that while somefrequency dependence clearly does exist, it is reasonably small within thefrequency range in which we are interested.0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)0.000000.000020.000040.000060.000080.000100.000120.000140.000160.00018V_probe / I_coils (V/I)Measured Probe Voltage / Coil Current vs. Frequency -- Sinusoidal WaveformBell LabsGreen Test KitTurquoise Test Kit .Figure 6.5: The ratio between the measured amplitude of the magnetic fieldand the measured amplitude of the coil current, as seen by three differentHall probes configured as on the left side of Fig. 6.4, as a function of fre-quency. Ideally, these lines would each be completely flat, indicating nofrequency dependence in the Hall probes’ outputs, however the dependenceseen is not a major concern if we only consider our region of interest for theAC-MOT (∼ 500 - 2000 Hz).486.3. Acoustic Resonances6.3 Acoustic ResonancesBecause mechanical stability is important to any precision apparatus, wehoped to choose a trap frequency for the AC-MOT that would excite fewmechanical resonances. In particular, mechanical vibrations in our current-carrying coils (located inside the vacuum chamber) have the potential todamage or shake loose the ceramic feedthroughs which maintain the vacuumseal between the inside of the trapping chamber and the outside world (seeFig. 6.6). The chamber must be maintained at ultra-high vacuum duringany science run, so the prospect of loosening the ceramic feedthroughs withpersistent acoustic vibrations was a very real concern.We can see from Fig. 4.9 that below some cutoff, the trap frequency doesnot have a particularly large effect on trap lifetime, so we were largely freeto select a trap frequency based on this secondary criterion – minimizingmechanical resonances.In the new chamber, we chose to run the AC-MOT at a frequency offAC = 1/(9.824× 10−4s) ≈ 1017.9 Hz. This is near a minimum of acousticnoise, as determined by Fig. 6.7. One possible systematic problem withthis set of measurements is that they were all collected while the secondhalf large flange on the chamber was absent. The simple presence of thisflange would change the geometry whose resonance we are interested in, andeliminating the atmosphere within the chamber (as during operation of theMOT) would remove one source of the resonances observed.496.3. Acoustic ResonancesWater FlowCurrent FlowFigure 6.6: Ceramic feedthroughs to the top coil. The current-carryingquadrupole coils must be kept electrically isolated from the stainless steelchamber as they are fed voltages from the outside. These coils are keptcool inside the vacuum chamber by pumping water through them as theyrun. Each of these connections between the inside and outside of the vacuumchamber uses such a feedthrough. There are eight of these in total, and theyare relatively brittle; they could be broken by repeated stresses between thequadrupole coils and the ceramic, such as may result from the varying forcesof an AC-MOT.506.3. Acoustic ResonancesAcoustic ResonancesFigure 6.7: Acoustic resonances in the online chamber. This data wastaken by driving the AC-MOT magnetic field coils with a continuous si-nusoidal waveform at a series of frequencies, as shown on the horizontalaxis, and using a nearby microphone connected to a computer with signalprocessing software to measure the overall amplitude of the resulting acous-tic noise. Note that this data is likely sampled at a rate higher than theprocessing software’s built-in Nyquist limit. Datapoints are taken at 2 Hzintervals, while the Nyquist limit was “probably” 5 Hz. (The software’sdocumentation claimed a limit 5 Hz, however the program also included abug in which the measured frequencies were reported as being twice theirtrue value. Although the software’s frequency spectrum analysis capabili-ties were not used explicitly, these limitations may still be relevant in de-termining the Nyquist limit to the resolution of any frequency-spectrumplots that resulted from any use of the software.) This oversampling givesthe illusion of presenting more information than is actually available, andwould have the effect of visually broadening the appearance of fine reso-nances on the plot. We eventually chose to run the trap at a frequencyof fAC = 1/(9.824× 10−4s) ≈ 1017.9 Hz., near a local minimum of acousticnoise.516.4. Control of Online Power Supplies6.4 Control of Online Power SuppliesThe newly purchased Matsusada DOP 25-80 power supplies’ specificationsallow for them to be controlled by an input signal in either “voltage-control”or “current-control” mode. That is, the power supplies accept an inputvoltage signal in the range ±10V, and translate this linearly, either directlyinto an output voltage (in the range ±25V), or into an output current (inthe range ±80A).Although we had initially intended to run the power supplies in current-control mode so as to have more direct control over the shape of the magneticfield, it became immediately obvious upon setup that this would not producethe desired results. We found that when controlled by a typical waveformin current-control mode, the output current contained large artifacts, theshape of which was strongly dependent not only on the frequency of the(AC) waveform and the lengths of time spent on (with a large AC signal) andoff (with either zero current or a small DC current) within each waveform,but also on the overall amplitude of the current output as well. This wouldhave been extremely challenging to work with had we continued with ouroriginal plan – however the power supplies functioned much as could havebeen expected when used in voltage-control mode, with no such artifactspresent in the output. Therefore, the power supplies were always run involtage-control mode, and the generated input waveforms were all createdspecifically to be used in this mode of operation.The waveforms used for controlling the Matusada DOP 25-80 powersupplies are themselves produced by two SRS DS345 function generators(one for each coil). The waveform shapes are created by running a Pythonscript (see Appendix A) and output in a format to match the SRS DS345specifications, then uploaded to the function generators.6.4.1 Determining the Number of Points for an ArbitraryWaveformThe first thing to determine about any arbitrary waveform is the number ofpoints comprising it (N_points) – a matter which turns out to be less trivialthan one might have imagined. An arbitrary waveform may contain up to16,300 points, and the function generator can sample those points at ratesofsamplerate = 4.0× 107/ N_sample (points/second). (6.5)where N_sample is an integer chosen by us. The function generator extrapo-lates its output voltage linearly between any set of two adjacent points in526.4. Control of Online Power Suppliesan arbitrary waveform.It is clearly advantageous for the purpose of precise control to use awaveform with as many points as possible, so as to be able to sample pointsas rapidly as possible. We also would like for each individual sinusoidalperiod to utilize the same number of points (points_per_cycle), so that theexact results of shutting off the AC-MOT are reproducible independently ofthe number of AC cycles in the arbitrary waveform. We therefore requirethe waveform to obey the expressionN_points = N_cycles * points_per_cycle (6.6)N_points ≤ 16300 (6.7)This also constrains the values of AC frequency (f_AC) we may use. Inparticular,f_AC = sampleratepoints_per_cycle(6.8)=(4.0× 107 Hz) N_cyclesN_sample * N_points(6.9)where we hope to keep f_AC near some desired value while simultaneouslymaximizing N_points and minimizing N_sample. N_cycles is determined by thedesired duty cycle of the AC-MOT, such thatN_cycles = cycles_on + cycles_off. (6.10)Note that while cycles_on is required to have an integer value in order forthe AC-MOT to function properly, there is no similar a priori requirementon cycles_off. In fact, for any given f_AC, we are able to adjust the valueof cycles_off while simultaneously changing N_points, provided only that(N_cycles / N_points) remain fixed at a value that works. N_sample shouldnot be adjusted without undergoing a major overhaul, as the sample rateis relevant to the shape of the “fast” features near the start and end ofthe waveform – features which use a small number of points, but which arecritical to properly controlling the magnetic field.6.4.2 Adjusting Waveform ParametersIn addition to the AC-frequency of a waveform, there were four parametersof the waveform which were varied in order to produce the most optimalmagnetic field for use in the AC-MOT. phi_start and phi_end controlled thephase angle at which the sinusoidal waveform begins and ends, respectively.536.4. Control of Online Power SuppliesThe need for these parameters is consistent with the description of an AC-MOT given in [2]. Additionally, t_startdelta and t_enddelta control thelength of time allocated to a maximum-amplitude voltage spike at the be-ginning and end of the sinusoidal component of the waveform, respectively.These were needed to overcome limitations specific to our power supplies.Since t_startdelta produced a voltage spike in the same direction as thebeginning of the sinusoid itself, its effect on the magnetic field was similarto starting the sinusoid up with a decreased starting phase phi_start Theprimary effects of both of these two parameters were on the shape of thesinusoidal part of the current output, rather than on the residual magneticfield after the driving sinusoidal voltage was removed (see Fig. 6.8).546.4. Control of Online Power SuppliesFigure 6.8: Ch1 shows the current through the quadrupole coils, while Ch3shows the driving voltage. Although this data was collected in the offlinetrap, it shows clearly the results of using poorly-tuned starting parametersfor the sinusoidal part of the waveform. In particular, note the non-uniformamplitude of the sinusoid, as well as its varying DC offset – both of whichhave been shown to harm the AC-MOT’s ability to collect and retain atoms.One can also see the large residual current still present while the AC-MOTis “off”.556.5. Residual Magnetic Field in the Online Chamber6.5 Residual Magnetic Field in the OnlineChamberAs is noted elsewhere, our goal is to produce a uniform magnetic field alignedwith the axis of optical pumping as rapidly as possible after switching fromthe (non-uniform) quadrupole field that is needed to run the MOT.mTBell Labs BzGreen BxTurquoise BxCurrent (Amp)Monday 15 April, 2013mTBell Labs BzGreen BxTurquoise BxCurrent (Amp)Monday 15 April, 2013mTBell Labs BzGreen BxTurquoise BxCurrent (Amp)onday 15 April, 2013mTBell Labs BzGreen BxTurquoise BxCurrent (Amp)Monday 15 April, 2013mTBell Labs BzGreen BxTurquoise BxCurrent (Amp)Monday 15 April, 2013mTBell Labs BzGreen BxTurquoise BxCurrent (Amp)Thursday 2 May, 2013Figur 6.9: Current and magnetic field are shown as read out on an oscillo-scope. The probes were positioned as in the second layout shown in Fig. 6.4.The waveform has been optimized to minimize the residual magnetic field ina ‘3 cycles on/2 cycles off’ duty cycle at fAC ≈ 1017.9 Hz, and entire scale ofthe readouts is shown. Yellow shows the current readout directly from theoscilloscope, while blue shows the magnetic field readout taken directly fromthe oscilloscope and converted into mT. The cyan curves which lie nearly ontop of the readouts are fits. In all cases, the background has been subtractedoff.In Fig. 6.10, The magenta vertical lines show the time at which theAC-MOT part of the waveform goes on or off, as taken from the larger-566.5. Residual Magnetic Field in the Online ChambermTBell Labs BzGreen BxTurquoise BxCurrent (Amp)Monday 15 April, 2013mTBell Labs BzGreen BxTurquoise BxCurrent (Amp)  il, mTBell Labs BzGreen BxTurquoise BxCurrent (Amp)Monday 15 April, 2013mTBell Labs BzGreen BxTurquoise BxCurrent (Amp)Monday 15 April, 2013mTBell Labs BzGreen BxTurquoise BxCurrent (Amp)Monday 15 April, 2013mTBell Labs BzGreen BxTurquoise BxCurrent (Amp)Monday 15 April, 2013mTBell Labs BzGreen BxTurquoise BxCurrent (Amp)Monday 15 April, 2013Thursday 2 May, 2013Figure 6.10: This plot shows data equivalent to that shown in Fig. 6.9 –however in this figure, the oscilloscope has been set to a more sensitive scalesuch that we are able to examine the magnetic field more closely during thetime in which we intend it to be off. Note that this results in a systematiczero offset to the output. The backgrounds, as read out on this scale, havebeen subtracted off. The cyan curves show the fits as found from the dataat a less precise scale in Fig. 6.9. On this intermediate scale, we see clearlythat at least two of the three Hall probes pick up a spike when the sinusoidalpart of the waveform begins or ends. These spikes are unlikely to accuratelyportray the magnetic field, as nothing to produce such readings is shown inthe current readout from the coils. The coils are driven in voltage controlmode, and a more plausible explanation is that the Hall probes or their leadsare able to pick up on the voltage spikes sent to the coils when the AC-MOTpart of the waveform begins or ends.scale fit to the current. The grey vertical lines show the time at which thesinusoidal waveform resumes, as taken from the fits to the waveforms onthe Hall probes. (The green and turquoise test kit Hall probes are very576.5. Residual Magnetic Field in the Online Chambersimilar, however there is not an obvious voltage spike on the green test kit’sreadout. We use the measurements from the turqoise test kit to producethe vertical grey lines on both the green and turquoise test kits’ plots.) Thetime delay between the magenta and grey lines gives us an estimate for howlong the Hall probes are affected by a voltage spike. Therefore, we considerthe waveform to be an inaccurate representation of the magnetic field duringthis same time delay at both the end and the beginning of the “magneticfield off” time. We fit the residual magnetic field only to parts of the readoutwhich have not been excluded by this method.Figure 6.11: This plot shows the same data that can be seen in Fig. 6.10,however on this scale we are better able to see the features of the residualmagnetic field. Fits to this data (for times after the grey lines) are shownin red.586.6. Magnetic Field During Optical Pumping Time6.6 Magnetic Field During Optical PumpingTimeThe goal is to make a uniform dipole field while the AC-MOT is off, with assmall a transverse field as possible, so as to allow optical pumping to betterpolarize the atoms. The results are shown below. All the figures in thissection were created while using a waveform with three cycles on and twocycles off, at fAC ≈ 1017.9 Hz.Figure 6.12: Current in the top and bottom power supplies during the opticalpumping phase, as read out on the oscilloscope. The blue outline shows thesystematic uncertainty due to limitations in the precision with which thepower supplies were calibrated.Unfortunately, it was not possible to measure the optical pumping fielddirectly with a Hall probe, so estimates of the magnetic field must be madeonly from current data output from the power supplies. This fails to takeinto account any effects from eddy currents in the surrounding chamber,596.6. Magnetic Field During Optical Pumping TimeFigure 6.13: A model of the magnetic field resulting from the current in thecoils. The blue outline shows the systematic uncertainty due to limitationsin the precision with which the power supplies were calibrated. This wascreated by combining the current measurements in the coils (as shown inFig. 6.12) with a COMSOL model of the coil geometry showing that eachcoil produces a magnetic field of magnitude 1.1 G/Amp at the centre, anda magnetic field gradient (along the vertical axis) of 0.2 G/cm/Amp at thecentre. The current in the two coils may be run in parallel or antiparallelconfigurations to cancel out the quadrupole or dipole components of thefield.which is an important effect, so the quality of this method as an estimatorfor the magnetic field is not entirely clear.606.6. Magnetic Field During Optical Pumping TimeOptical pumping begins at 200 µs.Fit:  @B@z = Aet/⌧ + CA = (6.8± 1.5)⇥ 101B = (3.22± 0.04)⇥ 102⌧ = (1.85± 0.03)⇥ 104CMonday 15 April, 2013Figure 6.14: A closer look at the residual magnetic quadrupole field shownin Fig. 6.13. The field decays with a time constant of τ = 185µs. Theoptical pumping laser is turned on 200mus after the AC-MOT is shut off,so as to avoid dealing with the worst of the residual quadrupole field duringoptical pumping.61Chapter 7The Optical Bloch EquationsThe optical Bloch equations are useful to describe the evolution of the quan-tum states in a sample of atoms in an optical molasses. In particular, we areinterested in quantizing the extent to which the polarization of our sampleof atoms is destroyed by stray magnetic fields.The Optical Bloch Equations are specifically concerned with an atomicsystem including one valence electron in an electromagnetic field, usuallyproduced by a laser. Though we speak of the density of states for a singleatom, the description developed here may just as easily be applied to astatistical ensemble of many such atoms, provided that they do not interactwith one another.7.1 Explicit Form of the Density MatrixRecall the standard definition of the density matrix,ρˆ(t) := |ψ(t)〉 〈ψ(t) | . (7.1)We wish to describe ρˆ(t) and its time evolution explicitly in a basis chosen tosimplify calculations. We select a basis such that the basis kets, | i〉, containno time dependence. Thus, we write:|ψ(t)〉 =∑ici(t) | i〉 (7.2)〈ψ(t) | =∑ic∗i (t) 〈 i | (7.3)andρˆ(t) =∑i, jci(t) c∗j (t) | i〉〈j | . (7.4)627.2. The General Form of Rotating CoordinatesEquivalently, we show the matrix representation for a system of n basisstates:|ψ 〉 ←→c0c1...cn(7.5)〈ψ | ←→(c∗0 c∗1 · · · c∗n)(7.6)ρˆ ←→c0c∗0 c0c∗1 · · · c0c∗nc1c∗0 c1c∗1 · · · c1c∗n... ... . . . ...cnc∗0 cnc∗1 · · · cnc∗n. (7.7)7.2 The General Form of Rotating CoordinatesWe begin with a statement of the Schro¨dinger equation and its Hermitianconjugate.ddt |ψ 〉 =1i~Hˆ |ψ 〉 (7.8)ddt 〈ψ | = −1i~ 〈ψ | Hˆ (7.9)We wish to describe the time-evolution of the density operator, ρˆ, which wedefine asρˆ := |ψ 〉〈ψ | . (7.10)It must then be the case thatdρˆdt =ddt(|ψ 〉)〈ψ | + |ψ 〉 ddt(〈ψ |)(7.11)= 1i~Hˆ |ψ 〉〈ψ | −1i~ |ψ 〉〈ψ | Hˆ (7.12)= 1i~(Hˆρˆ− ρˆHˆ)(7.13)= 1i~[Hˆ, ρˆ]. (7.14)We now consider the Hamiltonian, Hˆ for our system. Although in Sec-tion 7.1 we have constrained the basis kets of our system to have no time-dependence, we cannot guarantee that this will result in a time-independentHamiltonian – in general, this will not be the case.637.2. The General Form of Rotating CoordinatesAlthough Eq. 7.14 is true for any Hamiltonian Hˆ, we will henceforth takeHˆ0 = Hˆ to be the atomic Hamiltonian for a hydrogen-like atom with no ex-ternal field and we consider a generalized change of coordinates described bya unitary transformation, Uˆ . In general, Uˆ = Uˆ(t) may be time-dependent,but it is useful to remember that any unitary transformation is only a gen-eralized rotation matrix, and so the condition for unitarity isUˆ Uˆ † = Uˆ †Uˆ = I˜ . (7.15)We define a new operator, H˜0, byH˜0 := Uˆ †Hˆ0Uˆ . (7.16)It follows trivially thatHˆ0 = UˆH˜0Uˆ †, (7.17)and so we return to considering the evolution of the density matrix ρˆ. FromEq. 7.13, we find that:dρˆdt =1i~(Hˆ0ρˆ− ρˆHˆ0)(7.18)= 1i~(UˆH˜0Uˆ †ρˆ− ρˆ Uˆ H˜0Uˆ †)(7.19)Uˆ † dρˆdt Uˆ =1i~(Uˆ †UˆH˜0Uˆ †ρˆ Uˆ − Uˆ †ρˆ Uˆ H˜0Uˆ †Uˆ)(7.20)= 1i~(H˜0 Uˆ †ρˆ Uˆ − Uˆ †ρˆ Uˆ H˜0). (7.21)At this point, we begin to suspect that it may be advantageous to define a“transformed” density matrix, ρ˜, asρ˜ := Uˆ †ρˆ Uˆ . (7.22)Then,Uˆ † dρˆdt Uˆ =1i~(H˜0ρ˜− ρ˜H˜0), (7.23)but we will wish to eliminate all references to ρˆ, instead writing this expres-647.2. The General Form of Rotating Coordinatession in terms of only ρ˜. Thus, remembering that Uˆ may be time-dependent,dρ˜dt =ddt(Uˆ †ρˆ Uˆ)(7.24)= dUˆ†dt ρˆ Uˆ + Uˆ† dρˆdt Uˆ + Uˆ†ρˆ dUˆdt (7.25)= 1i~(H˜0ρ˜− ρ˜H˜0)+ dUˆ†dt ρˆ Uˆ + Uˆ†ρˆ dUˆdt (7.26)= 1i~(H˜0ρ˜− ρ˜H˜0)+ dUˆ†dt UˆUˆ†ρˆ Uˆ + Uˆ †ρˆ Uˆ Uˆ †dUˆdt (7.27)= 1i~(H˜0ρ˜− ρ˜H˜0)+ dUˆ†dt Uˆ ρ˜+ ρ˜ Uˆ†dUˆdt . (7.28)We must now recall the unitarity condition, Eq. 7.15, which allows us tofurther simplify our expression for dρ˜dt , usingddt(Uˆ Uˆ †)= ddt(Uˆ †Uˆ)= ddt(I˜)= 0 (7.29)dUˆ †dt Uˆ = −Uˆ†dUˆdt (7.30)Uˆ dUˆ†dt = −dUˆdt Uˆ†. (7.31)Then Eq. 7.28 becomesdρ˜dt =1i~(H˜0ρ˜− ρ˜H˜0)+ dUˆ†dt Uˆ ρ˜− ρ˜dUˆ †dt Uˆ (7.32)= 1i~[H˜0, ρ˜]+[dUˆ †dt Uˆ , ρ˜]. (7.33)Thus, we will find it useful to define an “effective Hamiltonian”, H˜ ′0, byH˜ ′0 := H˜0 + i~dUˆ †dt Uˆ (7.34)= Uˆ †Hˆ0Uˆ + i~dUˆ †dt Uˆ . (7.35)This notation considerably simplifies our expression for the evolution of ρ˜.Indeed, we find thatdρ˜dt =1i~[H˜ ′0, ρ˜], (7.36)657.3. Derivation of a Toy-Model Set of Optical Bloch Equationsjust as we might have guessed from Eq. 7.14.It would be nice, at this stage, to select a transformation Uˆ will beuseful. We’ll assume from this point forward that Hˆ is a basis in which theHamiltonian is already diagonalized. We hope to find a Uˆ such that dρ˜dt = 0,which will simplify evaluating the time evolution of our system. Therefore,from Eq. 7.33,[H˜0, ρ˜]= − i~[dUˆ †dt Uˆ , ρ˜], (7.37)which can be solved by settingdUˆ †dt = −1i~ H˜0 Uˆ† = − 1i~ Uˆ†Hˆ0. (7.38)Then Uˆ † has the form of an exponential. In particular,Uˆ † = A e+iHˆ0t/~Uˆ = A∗e−iHˆ0t/~,and enforcing unitarity, we note that A∗A = 1, so without loss of generalitywe choose A = 1 and find that our equation is solved byUˆ † = e+iHˆ0t/~ (7.39)Uˆ = e−iHˆ0t/~. (7.40)7.3 Derivation of a Toy-Model Set of OpticalBloch EquationsWe’ll derive the model used in Ref. [10], and also extend it. We’ll use anI = 1/2 atom (eg, Hydrogen), and consider only the F = 0 excited stateand F = 1 ground states. See Fig. 7.1, which was taken from Ref. [4].We’ll want to find a Hamiltonian, H, which describes all the relevantphysical processes in our system. Therefore, we write,H = H0 +HL +HB, (7.41)where H0 is the unperturbed atomic Hamiltonian, HL represents the interac-tion between the laser and the atoms, and HB is the part of the Hamiltonianresulting from magnetic effects.667.3. Derivation of a Toy-Model Set of Optical Bloch EquationsFigure 7.1: Transition strengths for a Hydrogen-like atom with nuclear spinI = 1/2, in a σ+ radiation field [4]. To find the σ− transition strengths,multiply all the m values by −1.Initially, we will use the following as our basis states:|Fg = 1,mz = +1〉 =1000 ; |Fg = 1,mz = 0〉 =0100|Fg = 1,mz = −1〉 =0010 ; |Fe = 0,mz = 0〉 =0001 . (7.42)Then, our unperturbed atomic Hamiltonian is given byH0 =0 0 0 00 0 0 00 0 0 00 0 0 ~ω0 . (7.43)Note that we have selected a system in which the spin-orbit coupling is iden-tically zero for the ground states, and may be considered already included677.3. Derivation of a Toy-Model Set of Optical Bloch Equationsin the excited state energy, ~ω0.We now consider magnetic perturbations to our Hamiltonian. In general,HB = −~µ · ~B (7.44)= gµ0 ~F · ~B. (7.45)It is standard practice to take the magnetic field as being directed along theaxis of quantization, and to label this direction as zˆ. Instead, we shall referto the orientation of the magnetic field as zˆ′, as it is not, in general, exactlyaligned with the axis of optical pumping, which will later be referred to aszˆ. Furthermore, primes will be used to designate terms which are describedunder the basis in which the zˆ′ axis is used as the axis of quantization.Under this convention, the perturbation to atomic energy levels is simply∆EB′ = gµ0B Fz′ . (7.46)Letting γ = µ0g, we also define the Larmor frequency ΩL according toΩL = γB = µ0g B, (7.47)so that our Hamiltonian in this basis becomesHB′ =~ΩL 0 0 00 0 0 00 0 −~ΩL 00 0 0 0 . (7.48)This result must still be transformed into the laboratory frame coordi-nate basis so that it can be combined with the other terms in the completeHamiltonian. We seek to apply a rotation operator of the formHB = R†HB′R (7.49)= eiφJnˆ HB′ e−iφJnˆ (7.50)for a rotation of angle φ about the nˆ-axis, where Jnˆ is the generator of thisrotation, and is normalized to obey the commutation relation,[Jiˆ, Jjˆ]= i ijk Jkˆ. (7.51)Our Hamiltonian combines an F = 1 and F = 0 representation, thereforewe define the following generators of rotation:Jxˆ =1√20 1 0 01 0 1 00 1 0 00 0 0 0 , (7.52)687.3. Derivation of a Toy-Model Set of Optical Bloch EquationsJyˆ =1√20 −i 0 0+i 0 −i 00 +i 0 00 0 0 0 , (7.53)Jzˆ =+1 0 0 00 0 0 00 0 −1 00 0 0 0 . (7.54)Without loss of generality, we take the transverse component of ~B to bedirected along the yˆ axis. That is,~B = B zˆ′ (7.55)= B cosφ zˆ +B sinφ yˆ, (7.56)so that the zˆ′-axis is related to the zˆ-axis by a rotation of angle φ about thexˆ = xˆ′ axis. Explicitly,HB =12(1 + cosφ) i√2 sinφ12(cosφ− 1) 0i√2sinφ cosφ i√2sinφ 012(cosφ− 1) i√2 sinφ12(1 + cosφ) 00 0 0 1~ΩL 0 0 00 0 0 00 0 −~ΩL 00 0 0 0×12(1 + cosφ) −i√2 sinφ12(cosφ− 1) 0−i√2sinφ cosφ −i√2sinφ 012(cosφ− 1) −i√2 sinφ12(1 + cosφ) 00 0 0 1, (7.57)and we find thatHB =~ΩLcosφ −i√2 ~ΩLsinφ 0 0i√2~ΩLsinφ 0 −i√2 ~ΩLsinφ 00 i√2~ΩLsinφ −~ΩLcosφ 00 0 0 0. (7.58)We now turn our attention to the system’s interaction with the laserbeam. Quite generally, this part of the Hamiltonian may be written asHL = −~d · ~E (7.59)697.3. Derivation of a Toy-Model Set of Optical Bloch Equationswhere ~E is the electric field, and ~d is the induced dipole moment betweensets of atomic states, and is taken to always be parallel to ~E. It is nowincumbent upon us to find a proper mathematical description for both ~Eand ~d.We begin by considering the dipole operator, ~d. Ref. [10] is kind enoughto suggest a decomposition of the xˆ-component of the dipole operator intoits raising and lowering operator components, as:dx ≡ ~d · xˆ =1√2(d−1 − d+1) , (7.60)and we define the related quantity,dy ≡ ~d · yˆ =i√2(d−1 + d+1) , (7.61)which is chosen so as to satisfy properties of linear independence and Her-miticity.We will now consider the electric field, which we treat as a semiclassicalelectromagnetic wave oscillating in the xy-plane. For the case of circularlypolarized light in which we are interested, we write~E = 1√2E0 [cos(ωLt)xˆ+ cos(ωLt± pi/2)yˆ] (7.62)= 1√2E0 [cos(ωLt)xˆ∓ sin(ωLt)yˆ] (7.63)= 1√2E0[12(e+iωLt + e−iωLt)xˆ∓ 12i(e+iωLt − e−iωLt)yˆ](7.64)for σ− and σ+ polarizations respectively, and the atom-laser Hamiltonianfrom 7.59 becomesHL = −12E0 [(d−1 − d+1) cos(ωLt)∓ i (d−1 + d+1) sin(ωLt)] (7.65)= −12E0[12(e+iωLt + e−iωLt)(d−1 − d+1)∓12(e+iωLt − e−iωLt)(d−1 + d+1)]. (7.66)If we consider only one mode of circular polarization, this result may besimplified. We will now substitute HL = H− or HL = H+ (where thesubscript denotes the handedness of polarization), and we find thatH− = E0(e+iωLt d+1 − e−iωLt d−1)(7.67)H+ = E0(e−iωLt d+1 − e+iωLt d−1). (7.68)707.3. Derivation of a Toy-Model Set of Optical Bloch EquationsWe’ll need to find our raising and lowering operators, d+1 and d−1 inthis basis. We define the Rabi frequency (the factor of√3 is specific to thissystem) asΩR ≡E0~√3〈F1 ||d||F2 〉 . (7.69)The Wigner-Eckart Theorem states that〈F1m1 ||d±||F2m2 〉 =~√3E0ΩR(−1)F1−m1(F1 1 F2−m1 ±1 m2), (7.70)and therefore,d+1 =~ΩRE00 0 0 00 0 0 00 0 0 +1−1 0 0 0 (7.71)d−1 =~ΩRE00 0 0 +10 0 0 00 0 0 00 0 −1 0 . (7.72)Our atom-laser interaction Hamiltonians becomeH− = ~ΩR0 0 0 +e+iωLt0 0 0 00 0 0 −e−iωLt+e−iωLt 0 −e+iωLt 0 (7.73)H+ = ~ΩR0 0 0 +e−iωLt0 0 0 00 0 0 −e+iωLt+e+iωLt 0 −e−iωLt 0 . (7.74)Finally, we are in a position to add together all the pieces of the Hamil-tonian. We find, for polarizations σ− and σ+:H = ~ΩLcosφ −i√2 ΩLsinφ 0 +ΩR e±iωLti√2ΩLsinφ 0 −i√2 ΩLsinφ 00 i√2ΩLsinφ −ΩLcosφ −ΩR e∓iωLt+ΩR e∓iωLt 0 −ΩR e±iωLt ω0. (7.75)We are now ready to make the customary change to rotating coordinates.We choose as the basis for the coordinate change,U = e−iAt (7.76)717.3. Derivation of a Toy-Model Set of Optical Bloch EquationsandA =0 0 0 00 0 0 00 0 0 00 0 0 ωL , (7.77)and define the following quantities:ρ˜ ≡ U †ρ U (7.78)H˜ ≡ U †HU − ~A. (7.79)In terms of these new variables, the time evolution equation isdρ˜dt =1i~[H˜, ρ˜]. (7.80)We write out the full rotating-coordinate Hamiltonians explicitly:H˜tot− = ~ΩLcosφ −i√2 ΩLsinφ 0 ΩRi√2ΩLsinφ 0 −i√2 ΩLsinφ 00 i√2ΩLsinφ −ΩLcosφ −ΩR e−2iωLtΩR 0 −ΩR e+2iωLt (ω0 − ωL)(7.81)H˜tot+ = ~ΩLcosφ −i√2 ΩLsinφ 0 +ΩR e−2iωLti√2ΩLsinφ 0 −i√2 ΩLsinφ 00 i√2ΩLsinφ −ΩLcosφ −ΩRΩR e+2iωLt 0 −ΩR (ω0 − ωL). (7.82)The “fast-rotating” terms which will be discarded to make the rotating waveapproximation immediately suggest themselves, and we find thatH˜tot− ≈ ~ΩLcosφ −i√2 ΩLsinφ 0 ΩRi√2ΩLsinφ 0 −i√2 ΩLsinφ 00 i√2ΩLsinφ −ΩLcosφ 0ΩR 0 0 (ω0 − ωL)(7.83)H˜tot+ ≈ ~ΩLcosφ −i√2 ΩLsinφ 0 0i√2ΩLsinφ 0 −i√2 ΩLsinφ 00 i√2ΩLsinφ −ΩLcosφ −ΩR0 0 −ΩR (ω0 − ωL). (7.84)727.3. Derivation of a Toy-Model Set of Optical Bloch EquationsWe will henceforth take the incident light to be entirely σ+ polarized, andtake H˜ = H˜tot+ as the Hamiltonian of the system, dropping the subscripts.The master equation (7.80) is still incomplete without additional termsto describe spontaneous decay from the excited state to the ground states.These must be added in by hand. In general, we are able to describe thisdephasing behavior by∂ρ∂t∣∣∣∣spont= ∂ρ∂t∣∣∣∣relax+ ∂ρ∂t∣∣∣∣repop, (7.85)where∂ρ∂t∣∣∣∣relax= −12(Γˆρ+ ρ Γˆ)(7.86)∂ρ∂t∣∣∣∣repop= Λˆ, (7.87)and where Γˆ and Λˆ are both diagonal matrices [10]. In the system weare considering, there is only a single excited state with equal coupling toeach of the three ground states. Therefore, if atoms in the excited statespontaneously decay at a rate Γρex = Γρ33 = Γρ˜33,Γˆ =0 0 0 00 0 0 00 0 0 00 0 0 Γ . (7.88)Because each of the three ground states are repopulated at an equal rateas a result of spontaneous emission from the population ρ33 = ρ˜33, we findthatΛˆ = 13Γρ331 0 0 00 1 0 00 0 1 00 0 0 0 . (7.89)It should be noted that the form of Eqs. (7.86), (7.87), (7.88), and (7.89)remains unaltered under a coordinate transformation between laboratoryand rotating coordinates. The master equation becomesdρ˜dt =1i~[H˜, ρ˜]+ ∂ρ˜∂t∣∣∣∣spont(7.90)= 1i~[H˜, ρ˜]− 12(Γˆρ˜+ ρ˜ Γˆ)+ Λˆ. (7.91)737.3. Derivation of a Toy-Model Set of Optical Bloch EquationsWe write the matrices of Eq. 7.91 out in explicit detail:ddtρ˜00 ρ˜01 ρ˜02 ρ˜03ρ˜10 ρ˜11 ρ˜12 ρ˜13ρ˜20 ρ˜21 ρ˜22 ρ˜23ρ˜30 ρ˜31 ρ˜32 ρ˜33 =−iΩLcosφ −i√2 ΩLsinφ 0 0i√2ΩLsinφ 0 −i√2 ΩLsinφ 00 i√2ΩLsinφ −ΩLcosφ −ΩR0 0 −ΩR (ω0 − ωL)ρ˜00 ρ˜01 ρ˜02 ρ˜03ρ˜10 ρ˜11 ρ˜12 ρ˜13ρ˜20 ρ˜21 ρ˜22 ρ˜23ρ˜30 ρ˜31 ρ˜32 ρ˜33+iρ˜00 ρ˜01 ρ˜02 ρ˜03ρ˜10 ρ˜11 ρ˜12 ρ˜13ρ˜20 ρ˜21 ρ˜22 ρ˜23ρ˜30 ρ˜31 ρ˜32 ρ˜33ΩLcosφ −i√2 ΩLsinφ 0 0i√2ΩLsinφ 0 −i√2 ΩLsinφ 00 i√2ΩLsinφ −ΩLcosφ −ΩR0 0 −ΩR (ω0 − ωL)−120 0 0 00 0 0 00 0 0 00 0 0 Γρ˜00 ρ˜01 ρ˜02 ρ˜03ρ˜10 ρ˜11 ρ˜12 ρ˜13ρ˜20 ρ˜21 ρ˜22 ρ˜23ρ˜30 ρ˜31 ρ˜32 ρ˜33−12ρ˜00 ρ˜01 ρ˜02 ρ˜03ρ˜10 ρ˜11 ρ˜12 ρ˜13ρ˜20 ρ˜21 ρ˜22 ρ˜23ρ˜30 ρ˜31 ρ˜32 ρ˜330 0 0 00 0 0 00 0 0 00 0 0 Γ+13Γρ331 0 0 00 1 0 00 0 1 00 0 0 0 , (7.92)and the result is given in Eq. 7.96 on the following page (displayed in land-scape orientation due to its size).Equation 7.96 is quite large and any solution would need to be obtainednumerically. However, we can still gain a bit of insight from considering itsform. We are particularly interested in effects resulting from the transversecomponents of the magnetic field, which enter the equation as factors ofΩL sinφ.From Eq. 7.96, we extract the component ρ˜00 and find that˙˜ρ00 =−1√2ΩLsinφ (ρ˜01 + ρ˜10) +13Γρ˜33, (7.93)and furthermore,ρ˜01 + ρ˜10 =∫ t0( ˙˜ρ01 + ˙˜ρ10)dt′ (7.94)= 1√2ΩLsinφ∫ t0(2(ρ˜00 − ρ˜11)− (ρ˜02 + ρ˜20)) dt′. (7.95)747.3. Derivation of a Toy-Model Set of Optical Bloch Equations   ˙˜ ρ 00˙˜ ρ 01˙˜ ρ 02˙˜ ρ 03˙˜ ρ 10˙˜ ρ 11˙˜ ρ 12˙˜ ρ 13˙˜ ρ 20˙˜ ρ 21˙˜ ρ 22˙˜ ρ 23˙˜ ρ 30˙˜ ρ 31˙˜ ρ 32˙˜ ρ 33   =                     (−1√2Ω Lsinφ(ρ˜01+ρ˜ 10)+1 3Γρ˜33)(1 √2Ω Lsinφ(ρ˜00−ρ˜ 11−ρ˜ 02)−iΩLcosφρ˜ 01)(1 √2Ω Lsinφ(ρ˜ 01−ρ˜ 12)−iΩRρ˜ 03−2iΩ Lcosφρ˜ 02) −1√2Ω Lsinφρ˜ 13−1 2Γρ˜03−i(ωL−ω 0+Ω Lcosφ)ρ˜03−iΩRρ˜ 02 (1 √2Ω Lsinφ(ρ˜00−ρ˜ 11−ρ˜ 20)+iΩLcosφρ˜ 10)(1 √2Ω Lsinφ(ρ˜01+ρ˜ 10−ρ˜ 12−ρ˜ 21)+1 3Γρ˜33)(1 √2Ω Lsinφ(ρ˜ 11−ρ˜ 22+ρ˜ 02)−iΩRρ˜ 13−iΩLcosφρ˜ 12) 1 √2Ω Lsinφ(ρ˜ 03−ρ˜ 23)−1 2Γρ˜13−iΩRρ˜ 12−i(ωL−ω 0)ρ˜13 (1 √2Ω Lsinφ(ρ˜10−ρ˜ 21)+iΩRρ˜ 30+2iΩ Lcosφρ˜ 20)(1 √2Ω Lsinφ(ρ˜ 11−ρ˜ 22+ρ˜ 20)+iΩRρ˜ 31+iΩLcosφρ˜ 21) 1 √2Ω Lsinφ(ρ˜ 12+ρ˜ 21)+1 3Γρ˜33−iΩR(ρ˜23−ρ˜ 32)  1 √2Ω Lsinφρ˜13−1 2Γρ˜23−iΩR(ρ˜22−ρ˜ 33)−i(ωL−ω 0−Ω Lcosφ)ρ˜23  −1√2Ω Lsinφρ˜ 31−1 2Γρ˜30+i(ωL−ω 0+Ω Lcosφ)ρ˜30+iΩRρ˜ 20  1 √2Ω Lsinφ(ρ˜ 30−ρ˜ 32)−1 2Γρ˜31+iΩRρ˜ 21+i(ωL−ω 0)ρ˜31  1 √2Ω Lsinφρ˜ 31−1 2Γρ˜32+iΩR(ρ˜22−ρ˜ 33)+i(ωL−ω 0−Ω Lcosφ)ρ˜32 (iΩR(ρ˜23−ρ˜ 32)−Γρ˜33)                     (7.96)757.3. Derivation of a Toy-Model Set of Optical Bloch EquationsSubstitution using Eqs. 7.93, 7.94, and 7.95 produces˙˜ρ00 = −12Ω2Lsin2 φ∫ t0(2(ρ˜00 − ρ˜11)− (ρ˜02 + ρ˜20)) dt′ +13Γρ˜33. (7.97)While the behavior of (ρ˜02 + ρ˜20) over any period of time is not immediatelyobvious, it must be real, though there is no requirement that it be positive.The magnitude of this term is also limited by maintaining normalization. Bycontrast, if we expect the system to approach a steady state of some sort–and we do–the term 2(ρ˜00 − ρ˜11) must itself approach some constant value.Over many cycles, any oscillations in either term will average themselvesout, and we find thatρ˜00 ∼ Ω2L sin2 φ. (7.98)Note that we do not rule out further dependence on ΩLsinφ buried in theother terms, but it is clear that the dominant effect is proportional to thesquare of the transverse component of the magnetic field. Similar argumentsmay be also made regarding the populations ρ˜11 and ρ˜22.76Chapter 8Conclusions8.1 ResultsWe have developed an AC-MOT in hope that it would be useful to help usobtain samples of cold, confined, spin-polarized hydrogen-like atoms, partic-ularly 37K, which is the subject of our experiments on nuclear beta decay.As this isotope is radioactive and fairly short-lived (t1/2 ≈ 1.2s), it is notnormally available to us for testing purposes. As a result, we chose to use(stable) 41K for testing and calibrations of the AC-MOT, due to its similarhyperfine structure.The majority of the measurements in this thesis were performed in the“offline” AC-MOT, which was never intended to be used with radioactiveisotopes. This was necessary because the “online” apparatus was not yetconstructed at the time. Unfortunately, the offline AC-MOT was plaguedby systematic problems, making it very difficult to produce reproduciblemeasurements. This data is therefore of questionable value to the scientificcommunity as a whole, however our qualitative findings are sufficient toprovide useful guidance for us within the context of the larger nuclear betadecay experiment.In particular, we have found that the AC-MOT has trap lifetimes thatare long enough to be useful to us, even in spite of the many systematiceffects that could not be fixed in the offline setup. We don’t require aparticularly long trap lifetime – the necessary consideration is that the traplifetime must be significantly longer than the radioactive lifetime of 37K (τ =t1/2/ ln(2) ≈ 1.8s ) so that the dominant loss mechanism is radioactive decay,rather than something we might be able to improve upon by adjustments tothe apparatus.Furthermore, the AC-MOT is robust enough to allow for it to be shutoff intermittently for short periods of time – sufficient to allow us to op-tically pump the atoms and observe them before they disperse – withoutdoing too much damage to the “average” trap lifetime. We’ve found thatit is possible to have the AC-MOT off for a full 40% of the time withoutincreasing the atom loss rate beyond what is tolerable, which in turn allows778.2. Future Workus to spend (nearly) 40% of our running time collecting beta decay dataon spin-polarized atoms. This result is comparable to the 45% ‘MOT offtime’ that has been used previously in the TRINAT research group with aDC-MOT, as in Dan Melconian’s thesis work [11]. It may still be possibleto push the duty cycle further within the AC-MOT.Additionally, in the AC-MOT, the residual magnetic quadrupole fieldnot only decays rapidly, it starts out from close to zero as well, which is agreat improvement on what could be accomplished with a DC-MOT. Thisin turn allows for better polarization of atoms within the cloud.The included work on the Optical Bloch Equations is intended to be usedas a qualitative theoretical ‘sanity check’ for numerical evolution codes [12]developed separately by our collaborator Ben Fenker in his Master’s the-sis [13], and based on the description given in [14] .8.2 Future WorkThe TRINAT online setup, which is mainly designed for use in nuclear betadecay experiments, presents us with some novel possibilities for character-izing and working with the AC-MOT.The first item of note is that the trapped atom cloud may be placed in anelectrical potential, such that when an atom is ionized (or when it decays),the negatively charged electrons will be accelerated towards one side of thechamber, while the positively charged ions are accelerated towards the other.Negatively charged ions are unstable in an electric field of that magnitude(≥ 350 V/cm). This is useful in combination with additional hardware: ateither end of the chamber, we have placed a stack of microchannel plates (tomeasure the time at which a particle hits), and a delay line (to determinethe position of the hit).Additionally, the online setup is equipped with a pulsed laser whichis used to selectively photoionize trapped atoms from their excited state.This process is largely non-destructive, since the intensity of the laser is lowenough that only a tiny fraction of the excited state atoms are photoionized.In combination, this additional equipment will allow us to create a two-dimensional image of the atom cloud, with time-domain accuracy of ∼5 ns– a great improvement from the 10s of milliseconds needed for a single CCDimage. With AC frequencies of ∼1 kHz such as we have been able to achieve,we will be able to image the trap at a wide range of phases during the AC-MOT cycle.It is also possible to use the online AC-MOT as a crude electron spec-788.2. Future Worktrometer – a fact that was discovered accidentally. When an atom betadecays within the trap, one or more orbital electrons also become separatedfrom the daughter atom, and these electrons and ions are accelerated by theelectric field to microchannel plates on opposite ends of the chamber. In thepast, the sudden approximation has been used to approximate the energies ofsuch “shake-off” electrons. After TRINAT’s December 2012 run, we noticedthat the number of electrons collected in the microchannel plate varied withthe magnitude of the magnetic field. This effect can be explained by takinginto account the characteristic helical motion of a charged particle within amagnetic field, and so a larger magnetic field had the effect of decreasing thenumber of shake-off electrons impacting the microchannel plate. It may bepossible to use this effect to measure the energy spectrum of these shake-offelectrons directly, and thereby test the validity of the sudden approximationin this specific circumstance.One problem we discovered in the online setup that had not been presentin the offline AC-MOT (or even in shorter test runs with the online AC-MOT) was inductive heating of other materials within the chamber. As theonline AC-MOT had much more metal than the offline MOT, this perhapsshould not have come as such a surprise. Unfortunately, this inductiveheating caused problems in the strip detectors (positioned directly aboveand below the quadrupole coils, outside of the vacuum chamber) whichwere intended to measure the energy spectrum of betas from decays withinthe trap. It may be possible to alleviate this problem by cooling the stripdetectors. A second problem resulting from inductive heating is that itappears to cause materials within the vacuum system to outgas hydrogen,which is damaging to the MOT. However, it should be possible to mitigatethe problem by running the AC-MOT at a lower frequency. Both of thesefixes must be implemented before further use of the AC-MOT in TRINAT’sbeta decay experiments.79Bibliography[1] E. L. Raab, M. Prentiss, A. Cable, S. Chu, and D. E. Pritchard, Phys.Rev. Lett. 59, 2631 (1987).[2] M. Harvey and A. J. Murray, Phys. Rev. Lett. 101, 173201 (2008).[3] A. Corney, Atomic and Laser Spectroscopy (Oxford University Press,New York, 1977).[4] Harold J. Metcalf, Peter van der Straten, Laser Cooling and Trapping(Springer-Verlag New York, Inc., New York, 1999).[5] V. Balykin, V. Minogin, and V. Letokhov, Rep. Prog. Phys. 63, 1429(2000).[6] F. Touchard, P. Buimbal, S. Bu¨ttgenbach, R. Klapisch, M. De SaintSimon, J. M. Serre, C. Thibault, H. T. Duong, P. Juncar, S. Liberman,J. Pinard, J. L. Vialle, Physics Letters 108B 3 (1982).[7] S. Falke, E. Tiemann, and C. Lisdat. Phys. Rev. A 74, 032503 (2006).[8] S. Smale, Summer 2012 Coop Report (unpublished), 13 December 2012.[9] T. B. Swanson, D. Asgeirsson, J. A. Behr, A. Gorelov, D. Melconian, J.Opt. Soc. Am. B Vol. 15, No. 11 2641 (1998).[10] Marcis Ausinsh, Dmitry Budker, Simon M. Rochester, Optically Polar-ized Atoms: Understanding Light-Atom Interactions (Oxford UniversityPress, New York, 2010).[11] Dan G. Melconian (2000). Measurement of the Neutrino Asymmetry inthe Beta Decay of Laser-Cooled, Polarized 37K. (Ph.D. Thesis).[12] Benjamin Fenker (2013). Measurement of Asymmetry Parameters in37K – Optical Pumping of Alkali Atoms. (Master’s thesis). Retrieved fromhttp://people.physics.tamu.edu/fenkerbb/Masters.pdf .80[13] B. Fenker, Optical Bloch Equations Code, retrieved fromhttp://code.google.com/p/optical-bloch-equations .[14] P. Tremblay and C. Jacques, Phys. Rev. A 41, 4989 (1990).[15] R. Shankar, Principles of Quantum Mechanics, 2nd ed. (Plenum Press,New York, 1994).81Appendix AWaveform Generation Code1 # Python 2.7.32 import os3 import numpy # numpy version 1.6.24 import matplotlib # matplotlib version 1.2.05 from matplotlib import pyplot6 import math78 pi = numpy.pi9 deg = pi /180.01011 plotting = False # Show the waveform on screen too?12 maxpoints = 16300.0 # Maximum number of points. 16300 ismax. 7500 is editable.13 minpoints = 500.0 # Minimum points to allow in a waveform. No idea what coarseness is acceptable here.14 maxsample = 40000000.0 # Max sample rate (Hz). Allowablevalues are this number / N_sample. N_sample has some max ,too.1516 t_AC = 9.824e-4 # Seconds.17 f_AC = 1.0/ t_AC # Hz. Must be a float.18 cycles_on = 3.019 cycles_off = 2.020 A = 1.0 # Amplitude of the output waveform.2122 sinfile = "s_15q.dat"23 phi_start = 0.0* deg + 1.0*pi/2.024 t_startdelta = 1.2e-5 # Time for the delta -spike at thebeginning of the sinusoid.25 phi_end = -6.0*deg + 1.0*pi/2.026 t_enddelta = 6.0e-6 # Time for the delta -spike at the endof the sinusoid.2728 def fileclear(filename):29 fileclear1 = open(filename , ’w’)30 fileclear1.close()31 print filename , "cleared."32 return 03382Appendix A. Waveform Generation Code34 def main():35 # I’m not sure why Python requires some global variables butnot others36 # to be explicitly declared within a function , but whatever.37 global phi_start38 global phi_end3940 print "f_AC =", f_AC41 if cycles_off == 0.0:42 # This would be a continuous sinusoid. No point in messingit up43 # with extra parameters that I forgot to adjust.44 phi_start = pi/2.045 phi_end = pi/2.046 print "phi_start - pi/2 =", (phi_start - 1.0*pi /2.0)/deg , "degrees."47 print "phi_end - pi/2 =", (phi_end - 1.0*pi /2.0)/deg , "degrees."48 N_cycles = cycles_on + cycles_off4950 # Figure out how many data points to use.51 N_points = 16000.0 # 16000.0 (float !!) will work for lotsof setups.52 N_sample = (maxsample*N_cycles) / (f_AC*N_points) # All ofthese things must be floats s.t. N_sample will be a float.53 points_per_cycle = N_points / N_cycles54 # If the guess fails , try to find some other number of pointsto use.55 if maxsample/f_AC != float(int(maxsample/f_AC)):56 # maxsample/f_AC should be a whole number , otherwise thetrue trap frequency will be wrong.57 print "maxsample/f_AC =", maxsample/f_AC58 print "*** Yeah , this isn’t going to work. Adjust trapfrequency."59 # It’s better to just adjust the trap frequency you’regoing for manually (and pick something that will work),so that you actually know what you’re going to get.60 return61 elif ( (float(int(N_sample)) != N_sample) or (float(int(points_per_cycle)) != points_per_cycle) ):62 N_points = maxpoints63 points_per_cycle = N_points / N_cycles6465 # N_sample = (maxsample*N_cycles) / (f_AC*N_points)66 # This is N_sample as in "samplerate =40 ,000 ,000.0/ N_sample ".67 N_sample = maxsample / ( points_per_cycle*f_AC)68 while (( float(int(N_sample)) != N_sample) or (float(int(points_per_cycle)) != points_per_cycle)) and (N_points83Appendix A. Waveform Generation Code>= minpoints):69 N_points = N_points - 1.0 # We are iterating over afloat. Crazy , right?70 points_per_cycle = N_points / N_cycles71 N_sample = maxsample / ( points_per_cycle*f_AC)72 if (N_points == minpoints - 1.0):73 print "A waveform cannot be generated with theseparameters."74 print "Adjust the trap frequency , the number of trapcycles in the waveform , and/or the minimum acceptablenumber of points."75 return # No point continuing when we’ve already failed!76 N_points = int(N_points)7778 # Now find the sample rate.79 # awc.exe will *round* samplerate to the nearest value of40 ,000 ,000/N, and reads to 4 places past the decimal.80 # For N_sample < ~100 ,000 , there is no ambiguity. This ismuch higher N_sample than we require.81 samplerate = maxsample / N_sample # float.8283 delta_t = delta_t = 1.0/ samplerate84 print "* The waveform is", N_points*delta_t , "seconds long."85 n_on = int( points_per_cycle*cycles_on + points_per_cycle *(phi_end - phi_start)/(2.0* pi) )8687 if cycles_off == 0.0:88 n_enddelta = 089 n_startdelta = 090 else:91 n_enddelta = int(t_enddelta / delta_t)92 n_startdelta = int(t_startdelta / delta_t)9394 n_off = N_points - n_on - n_enddelta - n_startdelta95 print "n_off =", n_off , "; n_on =", n_on96 print "n_enddelta =", n_enddelta97 print "n_startdelta =", n_startdelta98 print "N_points =", N_points99100 v_arr = numpy.zeros(N_points)101 t_arr = numpy.zeros(N_points)102 for i in range(N_points):103 t_arr[i] = float(i)*delta_t104105 # Make sure voltage is well -zeroed.106 # For some reason , this seems to actually matter.107 for i in range(N_points):108 # Make every point this way. We’ll over -write some later.109 if (i%2 == 0):84Appendix A. Waveform Generation Code110 v_arr[i] = 0.00000000001000111 else:112 v_arr[i] = -0.00000000001000113114 # Make the starting spike.115 for i in range(n_off , n_off+n_startdelta):116 v_arr[i] = A117 # Make the ending delta -spike.118 if n_enddelta != 0:119 for i in range(N_points -n_enddelta -1, N_points):120 v_arr[i] = -1.0*A121 # Make datapoints for the sinusoid itself.122 for i in range(n_off+n_startdelta , N_points -n_enddelta):123 v_arr[i] = A*numpy.sin(phi_start + 2.0*pi*f_AC*( t_arr[i] -t_arr[n_off+n_startdelta ]) )124125 line1 = "%i" % N_points126 line2 = "%8.4f" % samplerate127 line3 = "%i" % 0 # ’0’ works.128 line4 = "%8.4f" % 1000.0000 # ’1000.0’ works.129130 # Write out the waveform , header first.131 fileclear(sinfile) # Clear any files with the same namesthat may have already been there.132 sinhandle = open(sinfile , ’a’)133 sinhandle.write(line1+’\n’)134 sinhandle.write(line2+’\n’)135 sinhandle.write(line3+’\n’)136 sinhandle.write(line4+’\n’)137 numpy.savetxt(sinhandle , v_arr , fmt=’%1.14f’)138 sinhandle.close()139 print sinfile , "saved.\n"140141 if plotting == True:142 print "Beginning to plot."143 fig1 = pyplot.figure ()144 t_arr = numpy.linspace (0.0, N_points*delta_t , num=N_points ,endpoint=False)145 p1 = fig1.add_subplot (1, 1, 1)146 p1.grid(True)147 p1.plot(t_arr , v_arr)148 p1.set_ylabel("Voltage")149 pyplot.show()150 return 0151152 if __name__ == ’__main__ ’:153 main()wavegen new3.py85Appendix BThings that Are VeryObviousB.1 The Center of GravityIt’s what happens when you set A = 0 and B = 0. That is all.B.2 Diagonalizing the HamiltonianGiven a Hermitian matrix Ωˆ, there exists a unitary matrix Uˆ such thatUˆ †ΩˆUˆ is diagonalized. Solving for this matrix Uˆ is, in this case, equivalent tosolving the eigenvalue problem for Ωˆ [15]. As it turns out, Uˆ is the matrix ofeigenvectors of Ωˆ, by which I mean that the eigenvectors are column vectors,and they’re all squished together to make Uˆ . It doesn’t matter what orderyou put them in, but they probably have to be normalized.Then, ifΩˆ′ := Uˆ †ΩˆUˆ , (B.1)we find that Ωˆ′ is the diagonal matrix with its elements being the eigenvalues.They’re in the same order as the eigenvectors we squished together to makeUˆ previously.Also, a unitary operator, Uˆ is one which satisfies:Uˆ Uˆ † = Uˆ †Uˆ = I˜ . (B.2)B.3 Rotating CoordinatesIf by Ωˆ we really mean the Hamiltonian Hˆ, and by Uˆ we really mean a coor-dinate change that takes us to rotating coordinates such that we can easilymake a rotating wave approximation, we must take more things into ac-count. In particular, we find that our new rotating-coordinate Hamiltonian,H˜ is given by:H˜ = Uˆ †HˆUˆ − ~Aˆ (B.3)86B.4. Lifetimes and Half-LifeswhereUˆ := e−iAˆt. (B.4)This additional term arises from our statement of the Schrodinger equation,Hˆ |ψ 〉 = i~ ∂∂t |ψ 〉 . (B.5)In particular, note that B.5 includes only a partial derivative of the wave-function. I derive this result explicitly in Chapter 7.2. See also Ref. [10],pg. 195.B.4 Lifetimes and Half-LifesSince different people use different notation to describe exponential decay ofa physical quantity, it is useful to be able to relate two of the most commonmethods for describing the decay. We begin with the rate equation,dNdt = −γ N, (B.6)where it is clear that the “rate” of decay must be γ N . If we initially haveN0 of the quantity in question, then Eq. B.6 has as its solutionN(t) = N0 e−γ t. (B.7)Note that the physical interpretation of γ is the “linewidth”.We’ll wish to convert γ into other quantities of interest. In particular,we can re-write the solution B.7 asN(t) = N0 e−t/τ , (B.8)where τ = 1/γ is referred to as the “lifetime”. Then, we find the half-life t1/2by enforcing the fact that it is the time at which the number of remainingatoms is equal to half of what was originally present. Therefore,N(t1/2) = N0e−t1/2/τ =12N0 (B.9)e−t1/2/τ = 1/2 (B.10)t1/2/τ = ln(2). (B.11)Thus, we see thatt1/2 = ln(2) τ, (B.12)where τ is the “lifetime” of the state, and t1/2 is its “half-life”.87B.5. Reduced Matrix ElementsB.5 Reduced Matrix ElementsThe Wigner-Eckart Theorem says, for vector operator V q,〈α′j′m′|V q|αjm〉 = 〈j′m′|j1mq〉〈α′j′‖V ‖αj〉. (B.13)The point being that 〈α′j′‖V ‖αj〉 is the same for all m and q.B.6 Doppler Cooling LimitHere it is!kTD =12~Γ (B.14)88

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