Bi-selective pulses for large-area atom interferometry
BBi-selective pulses for large-area atom interferometry
Jack Saywell, ∗ Max Carey, Ilya Kuprov, and Tim Freegarde School of Physics & Astronomy, University of Southampton, Highfield, Southampton, SO17 1BJ, UK School of Chemistry, University of Southampton, Highfield, Southampton, SO17 1BJ, UK (Dated: April 28, 2020)We present designs for the augmentation mirror pulses of large-momentum-transfer atom in-terferometers that maintain their fidelity as the wavepacket momentum difference is increased.These bi-selective pulses, tailored using optimal control methods to the evolving bi-modal mo-mentum distribution, should allow greater interferometer areas and hence increased inertial mea-surement sensitivity, without requiring elevated Rabi frequencies or extended frequency chirps. Us-ing an experimentally validated model, we have simulated the application of our pulse designs tolarge-momentum-transfer atom interferometry using stimulated Raman transitions in a laser-cooledatomic sample of Rb at 1 µ K. After the wavepackets have separated by 42 photon recoil momenta,our pulses maintain a fringe contrast of 90% whereas, for adiabatic rapid passage and conventional π pulses, the contrast is less than 10%. Furthermore, we show how these pulses may be adapted tosuppress the detrimental off-resonant excitation that limits other broadband pulse schemes. I. INTRODUCTION
Atom interferometers [1] reverse the optical interferom-etry roles of light and matter by using pulses of laser lightto split, redirect, recombine and interfere atomic matterwaves, allowing the precise measurement of gravitationalfields [2–4], inertial motions [5–7] and more esoteric fields[8, 9] to which optical interferometers would have littleor no sensitivity.Most atom interferometers for inertial sensing useBragg [10] or Raman [11, 12] transitions driven bycounter-propagating laser pulses as the beamsplitters andmirrors that split and direct the atomic wavepackets.The two-photon recoil accompanying these stimulatedscattering processes imparts a momentum difference of2 (cid:126) k between the two interferometer paths, where k is thesingle-photon wave number. As with an optical interfer-ometer, the measurement sensitivity depends upon thespatial area enclosed. This is proportional to the mo-mentum separation, and can hence be increased by usingadditional mirror pulses - known as augmentation pulses- to impart further impulses to the atomic wavepacketsforming a large momentum transfer (LMT) interferome-ter [13].In practice, inhomogeneities such as variations in beamintensity and atomic velocity reduce the fidelity of theaugmentation pulses. The accrued effect of such imper-fections limits the fringe visibility and can reverse theLMT sensitivity gains [13–15]. Cooling the atoms or fil-tering their velocity distribution can improve the initialfidelity at the expense of a weaker signal, but a veloc-ity spread is inevitable in LMT interferometry becauseof the increasing momentum separation introduced be-tween the primary interfering arms. Techniques such asincreasing the Rabi frequency [16] and employing beamshaping optics [17] increase the experimental complexity ∗ [email protected] and power requirements of the interferometer.An alternative approach is to adopt robust atom-opticsthat retain high fringe visibility for large momentum sep-arations. Composite [18, 19] and adiabatic rapid passage(ARP) [20] Raman pulses have been investigated, achiev-ing momentum splittings of 18 (cid:126) k [14] and 30 (cid:126) k [15] re-spectively. Both these techniques can increase the veloc-ity acceptance of the augmentation pulses at the expenseof an increase in pulse duration, but in practice fringevisibility was lost after 4-7 augmentation pulses. Alter-native atom-optics to Raman transitions such as multi-photon Bragg pulses [10, 21] and Bloch oscillations withinoptical lattices [22–24] have achieved impressive momen-tum transfer, but these put even more stringent demandson the initial momentum distribution [25] and, when usedfor interferometry, similarly limit the signal-to-noise ratio(SNR) [14, 15].We have previously used optimal control techniques[26, 27] to design robust high fidelity pulses for small mo-mentum transfer interferometry [28, 29]. In this paper,we address LMT, by designing “bi-selective” pulses thatoffer high fidelity for two ranges of velocity that trackthe atoms in the two arms of the LMT interferometer,allowing high fringe contrast to be maintained for largerinterferometer areas. We compare our pulses with con-ventional composite pulses and ARP through simulationsusing an previously validated approach [15, 19, 29], andconsider an extension to reduce unwanted double diffrac-tion [30, 31]. Our method is a departure from previ-ous approaches in which the same robust pulse has beenused for every augmentation pulse in the interferometersequence, ultimately limiting the achievable momentumsplitting to the velocity acceptance of the pulse. Thetechnique has general applicability because many inter-ferometer arrangements are already set up to implementsimilar modulation sequences and, as the algorithm opti-mises tolerance of variations, the designs do not dependcritically upon experimental parameters. a r X i v : . [ phy s i c s . a t o m - ph ] A p r II. LMT ATOM INTERFEROMETERS
Figure 1 shows the space-time diagram of a typi-cal Raman LMT interferometer. While the standard π/ − π − π/ (cid:126) k eff ,where k eff = 2k is the effective wavevector of the atom-optics, a greater momentum splitting can be achieved byaugmenting the mirror and beamsplitter pulses with ad-ditional pulses with alternating wavevectors [13]. Theseaugmentation pulses are designed to transfer atomic pop-ulation between two internal states and increase the mo-mentum splitting between the wavepackets in the inter-ferometer. By extending the beamsplitter operation byN augmentation pulses, the momentum splitting may beincreased to (2N + 1) (cid:126) k eff , increasing the intrinsic phasesensitivity of the interferometer.However, this increased splitting also introduces differ-ential Doppler shifts between the interfering arms, quan-tised in multiples of the two-photon recoil shift δ recoil = (cid:126) k / m , that depend on the atomic mass m . Indeed,this is automatic, since the Doppler shift and inertial sen-sitivity have the same origin. For the n th augmentationpulse ( n = 1 , , , ..., N), the arms of the interferometerhave Raman resonance conditions separated by 4 nδ recoil [14, 32, 33] (for Raman transitions on the Rb D2 line, δ recoil ≈ π × . n th augmentation pulse may thus be visualised as twoGaussian distributions, corresponding to the Maxwell-Boltzmann temperature of the atomic source, separatedin frequency space by 4 nδ recoil . As illustrated in Figure1, the augmentation pulses are only efficient as long asthis split distribution fits within the velocity acceptanceof the pulses. For conventional π -pulses this can spanjust a few recoil momenta, fundamentally limiting themomentum transfer achievable before fringe visibility islost [13, 14].Many composite and shaped pulses have been devel-oped in the field of nuclear magnetic resonance (NMR)spectroscopy to improve the control of nuclear spins inthe presence of experimental inhomogeneities. Compos-ite pulses [18] replace single rectangular pulses with se-quences of pulses of varying durations and phases. Theoverall effect of the pulse sequence is to replicate the oper-ation of a single pulse, but with an increased tolerance ofunwanted variations in coupling strength, detuning fromresonance, or both. Dunning et al. [19] investigated thepotential fidelity improvements various established com-posite pulses might bring in atom interferometry com-pared with rectangular π pulses. Of the pulses tested,the WALTZ pulse [34], a relatively short three step ro-bust state-transfer pulse, had the highest fidelity and ve-locity acceptance. Similarly, Butts et al. [14] employedWALTZ to improve the contrast in an LMT interferom-eter, doubling the sensitivity.An alternative class of robust pulses is known as fre-quency swept adiabatic rapid passage (ARP). ARP cantransfer atomic population between the states of a two- level atom with high efficiency [15, 20, 35, 36]. Duringan ARP pulse, the driving field frequency is swept slowlythrough resonance such that the atomic state follows theevolution of the field and may be moved with high preci-sion anywhere on the surface of the Bloch sphere [37]. Inthis picture, the quantum state vector precesses about aninstantaneous rotation axis, the field vector, defined bythe amplitude, detuning, and phase of the driving field.For ARP to be efficient, the pulse must satisfy a condi-tion of adiabaticity: the motion of the field vector on theBloch sphere must be slower than the rate at which theatomic state precesses about it [20]. The simplest exam-ple of ARP is the linear frequency chirp, where the pulseamplitude, or Rabi frequency, is fixed, and the laser de-tuning is swept linearly through resonance. This pulsecan be efficient and robust, but it is necessarily long incomparison to rectangular and composite pulses, therebyincreasing the risk that coherence is lost through spon-taneous emission.By allowing the pulse amplitude to vary, variousschemes have been developed which outperform the lin-ear frequency chirp [38–42]. These pulses have high ef-ficiency and robustness, maintaining adiabaticity whilereducing pulse duration. An example is the Tanh/TanARP pulse [41], where the frequency sweep follows a tan-gential function of time and the pulse amplitude followsa hyperbolic tangential function of time. The Tanh/Tanpulse was implemented in a large-area atom interferom-eter by Kotru et al. [15] increasing the interferometercontrast over that obtained using simple π pulses andachieving a momentum splitting of 30 (cid:126) k in a 9 µ K Csatomic sample.Although ARP can obtain impressive populationtransfer efficiency, with a detuning bandwidth that in-creases with the pulse duration, its potential utility in in-terferometry is limited by the dynamic phase imprintedon the diffracting wavepackets and the variation in ef-fective time origin [20, 32, 33, 36]. The dynamic phasedepends on the optical intensity, and rapid dephasingis therefore inevitable when the atom cloud expandsthrough variations in laser intensity. In practice this ef-fect is limited because ARP pulses applied in quick suc-cession approximately cancel the dynamic phase, but itleads to a trade-off between longer and theoretically moreefficient pulses, and dephasing caused by imperfect dy-namic phase cancellation when the beam quality is non-ideal [15]. There is therefore a need for pulses whichmatch or improve upon ARP in terms of state transferefficiency, but which are robust to temporal variations inthe Rabi frequency during the interferometer sequence.LMT interferometry requires augmentation pulses thatprovide efficient population transfer across the atomcloud and throughout the sequence. To maintain sen-sitivity and prevent loss of fringe visibility, the inter-ferometer sequence itself should impart the same phaseto all atoms, and all atoms should have the same sensi-tivity to the external influence being measured. Pulsesneed not satisfy these conditions individually, provided
AugmentationpulsesAugmentationpulsesAugmentationpulsesAugmentationpulses+k e ff -k e ff -10 -5 0 5 100 timelaserintensity longitudinal atomicposition momentum [ ħ k e ff ] π /2 ππ /2 π -pulse transfere ffi ciency2 ħ k18 ħ k FIG. 1. Atomic state trajectories (left) as a function of position within an LMT interferometer sequence showing the case wherethe beamsplitter and mirror operations are extended by a sequence of 4 augmentation pulses (18 (cid:126) k trajectory). The initialmomentum distribution (right), represented by the red shaded region, is separated into two arms. The momentum distributionseen by each pulse throughout the LMT sequence for Rb atoms with a Maxwell-Boltzmann temperature of ∼ µ K is shownby the blue shaded regions on the right. As more pulses are added to increase the interferometer area, the separation inthe resonance conditions for the arms begins to exceed the velocity acceptance of a π -pulse, shown bottom right for a Rabifrequency of 200 kHz. that subsequent cancellation achieves them for the se-quence as a whole. ARP pulses, for example, havedetuning-dependent effective time origins and dynamicphases which can be cancelled by a later pulse providedthere is no variation in optical intensity. III. OPTIMAL CONTROL TECHNIQUES
Robust pulses may be generated with optimal controltechniques [43, 44] by dividing a pulse into discrete timeslices and treating the phase and/or amplitude of eachslice as control parameters that may be adjusted to op-timise the fidelity of a desired optimisation.In a typical Raman atom interferometer, two-photontransitions are driven by lasers far from single-photonresonance so that the intermediate level can be adiabat-ically eliminated to leave an effective two-level systembetween stable states | g (cid:105) and | e (cid:105) . The atom optics then effect a rotation of the quantum state on the surface ofthe Bloch sphere for this basis at a rate ˜Ω R = (cid:112) Ω R + δ ,about an axis (the field vector) Ω = Ω R cos( φ L )ˆ x + Ω R sin( φ L )ˆ y + δ ˆ z, (1)determined by the relative laser phase φ L , the two-photon Rabi frequency on resonance Ω R , and the Ramandetuning δ = ( ω − ω ) − ω eg + δ Doppler + δ recoil (2)that includes terms for the two-photon recoil shift δ recoil = (cid:126) k / m and the Doppler shift δ Doppler = k eff · p /m , which depends on the initial momentum p of each atom in the interferometer. ω , are the laser fre-quencies and ω eg is the frequency splitting of the levels | g (cid:105) and | e (cid:105) . This rotation can be written in terms of apropagator ˆ U = (cid:18) C ∗ − iS ∗ − iS C (cid:19) (3)acting on the ( | g (cid:105) , | e (cid:105) ) basis, where C and S are the‘continuing’ and ‘scattering’ amplitudes defined as [45] C ≡ cos (cid:0) ˜Ω R τ (cid:14) (cid:1) + i (cid:0) δ (cid:14) ˜Ω R (cid:1) sin (cid:0) ˜Ω R τ (cid:14) (cid:1) S ≡ e iφ L (cid:0) Ω R (cid:14) ˜Ω R (cid:1) sin (cid:0) ˜Ω R τ (cid:14) (cid:1) . (4)The conventional beamsplitter and mirror pulses ofatom interferometry have a fixed amplitude and phase,and the durations of the interactions are set so that, onresonance, they perform π/ π rotations respectively.Composite and ARP pulses depend on varying the pulseparameters Ω R ( t ), φ L ( t ), and δ ( t ) as a functions of time t so that, for a given atom, the rotation axis and ratechange during the pulse.The action of such a pulse on the quantum state canbe evaluated efficiently by dividing the pulse into discretetimesteps of duration d t , with the propagator for an en-tire pulse given by the time-ordered product of propa-gators for each slice, calculated according to Equation(3). A pulse is then described by piece-wise constantwaveforms Ω R ( t ), φ L ( t ), and δ ( t ) that constitute a finitenumber of control parameters. Upon defining a suitablefidelity, such as a measure of how accurately a given ini-tial state | ψ ( t = 0) (cid:105) is driven to a target | ψ T (cid:105) by thepulse, optimal waveforms may be found using numeri-cal routines from optimal control theory. In this workwe focus on producing pulses with optimised phase andamplitude profiles φ L ( t ) and Ω R ( t ).One efficient and popular quantum control algorithm,originally developed for NMR applications, is gradientascent pulse engineering, or GRAPE [26]. GRAPE com-putes derivatives of the pulse fidelity with respect to thecontrol parameters without the need for computationallyexpensive finite-differencing methods, and can be used inconjunction with the limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) quasi-Newton method [27].By computing the gradient and estimating the Hessianof the fidelity landscape, and provided local maxima withlow-fidelity are avoided, the algorithm can find an opti-mum solution efficiently.We consider the following two possible fidelities foraugmentation pulses, F square = | (cid:104) e | ˆ U | g (cid:105) | (5) F real = Re( (cid:104) e | ˆ U | g (cid:105) ) . (6)Both of these fidelities, if maximised, will yield pulsesthat efficiently transfer population from one basis state tothe other, the essential requirement of an augmentationpulse. If the fidelities are averaged over an ensemble ofdetunings and a range of Rabi frequencies, the resultingpulse will be made robust to these specific errors. Max-imising F square leads to pulses where the phase of the overlap (cid:104) e | ˆ U | g (cid:105) is unimportant but the quantum state isrotated by 180 ◦ from pole-to-pole on the Bloch sphere.Conversely, maximising F real leads to pulses where thephase of the overlap (cid:104) e | ˆ U | g (cid:105) is well-defined.If the phase of the overlap (cid:104) e | ˆ U | g (cid:105) varies from pulse-to-pulse across an atomic sample in an LMT sequence,the resulting interference fringes can be washed out whenthe contributions from each atom in the cloud are aver-aged at the end of the interferometer. Providing the Rabifrequency does not change much between augmentationpulses, these phases approximately cancel in LMT inter-ferometers but if the Rabi frequency varies temporallythroughout the interferometer, rapid dephasing can oc-cur. This is readily observed with ARP, which requiresa high degree of cancellation in the phase factors intro-duced by each pulse in the sequence [15, 33, 36]. Op-timising F real for a range of Rabi frequencies will meanthe phase of the overlap (cid:104) e | ˆ U | g (cid:105) remains fixed even if thecoupling strength varies within that range. Therefore,we expect F real to yield augmentation pulses that are in-sensitive to dephasing caused by temporal changes in theRabi frequency due, for example, to noisy beam intensityprofiles and/or the ballistic expansion of the atom cloud. IV. BI-SELECTIVE RAMAN PULSES
For the nominal three-pulse π/ − π − π/ n th augmen-tation pulse as F A n = (cid:88) δ, Ω R ∈ L n F ( δ, Ω R ) + (cid:88) δ, Ω R ∈ U n F ( δ, Ω R ) , (7)where F = F real , square is the single-atom augmentationpulse fidelity (Equations 5 and 6) and L n and U n arethe ensembles representing the atomic frequency distri-butions of the lower and upper arms during the n th aug-mentation pulse respectively. In order to normalise thefidelity such that the maximum value is unity, we divideEquation 7 by the number of detunings and amplitudeerrors included in the entire ensemble.We compose the detuning ensembles for the lowerand upper interferometer arms for the n th augmenta-tion pulse from two uniform discrete distributions cen-tred at ± nδ recoil . The ensembles used in the optimi-sation should represent the velocity distribution of theatoms. However, in order to reduce computation timewe approximate the true distribution using two uniformdistributions each with a sample size of 20. The rangespanned by each distribution is given by 4 standard devi-ations of detuning arising from the velocity distributionalong the Raman beam axis, which we assume follows aMaxwell Boltzmann distribution.Our LMT pulses are individually optimised, which as-sumes there are no correlations in residual errors betweenpulses. However, It may be possible to develop alterna-tive measures of performance which reflect the fidelity ofthe entire interferometer and allow pulses within a se-quence to be optimised cooperatively [47], for exampleby compensating each other’s imperfections.An alternative approach to achieve bi-selectivity in-volves concatenating two inversion pulses with the ap-propriate frequency shifts to address each interferometerarm separately. While this makes the individual opti-misations simpler, the resulting pulses are longer, givedifferent velocity classes different pulse origins, and losefidelity as the classes began to overlap. A simple super-position of the two components would cure the problemsof length and origin, though not overlap, and requiresamplitude modulation in addition [48].The concept of bi-selective pulses for LMT interferom-etry bears resemblance to the technique of band-selectivepulses in NMR spectroscopy [48–51], where pulses havebeen designed to excite or invert nuclear spins withinsingle or multiple frequency bands, but suppress the re-sponse of spins outside the desired frequency range. Typ-ically, these pulses require smooth waveforms becausethe response at large resonance offsets is determined bythe Fourier transform of the pulse shape [52]. As a re-sult, composite pulses, which are composed of concate-nated sequences of constant amplitude pulses with dis-crete phase shifts, lead to non-negligible excitation faroff-resonance.The suppression of the action of a pulse outside aspecific frequency range may be useful in interferometergeometries where, for example, the counter-propagatingRaman beams are obtained by retroreflection of a singlebeam with both Raman frequencies and there are neces-sarily four frequency components that may interact withthe atom cloud. When there is no acceleration of theatoms along the beam axis then double-diffraction inter-ferometry schemes can be be employed that make use ofall of the frequency components simultaneously [30, 31] .However, in vertically-orientated, ground-based atom in- terferometers, the Doppler shift caused by gravitationalacceleration is commonly used to isolate a single fre-quency pair by shifting the other pair off resonance [14].However, broadband pulses such as composite or ARPpulses often have non-zero transfer efficiency at large de-tunings [32, 52], meaning one must wait longer to ensurenegligible excitations from the off-resonant pair, costingtime which may otherwise be used to enhance the sensi-tivity.By directly suppressing the transfer efficiency outsidea specific range of frequencies, such off-resonant excita-tion with broadband pulses may be avoided. We achievethis suppression of unwanted excitation by modifying ourbi-selective pulse fidelity (Equation 7), adding the follow-ing penalty term which represents the suppression bandof detunings where the transfer efficiency should be min-imised, F suppress = δ max (cid:88) Ω R ,δ = δ min | (cid:104) g | ˆ U | g (cid:105) | + − δ max (cid:88) Ω R ,δ = − δ min | (cid:104) g | ˆ U | g (cid:105) | . (8)Here, δ min , max represent the initial and final Raman de-tuning values for the two suppression bands. V. RESULTS
We have optimised phase-modulated, constant ampli-tude, bi-selective augmentation pulses with GRAPE, us-ing both the phase insensitive fidelity F square and thephase sensitive fidelity F real . We have investigated theirperformance through simulation of the resulting interfer-ometer contrast in a laser-cooled sample of Rb at 1 µ K.In all optimisations, the timestep of the pulses was 25 n s,and the effective Rabi frequency was 200 kHz, meaningthe duration of a rectangular π pulse ( t π ) should be 2.5 µ s in the absence of any detuning or amplitude error.The pulses were optimised for an atomic temperature of1 µ K, and a range of amplitude errors of ±
10 % the ef-fective Rabi frequency. The length of the augmentationpulses was fixed to be 4 t π . This duration allowed for asufficiently good pulse to be obtained when using bothfidelities, although optimising F real typically requires alonger pulse to reach an equivalent fidelity to the phase-insensitive case. The initial guess for the phase profile ofthe pulses was in each case a sequence of random phases.We keep the pulse amplitude constant when optimisingthe bi-selective fidelity F A n (Equation 7) because allowingthe amplitude to vary does not lead to a higher terminalfidelity in this case.The NMR spin simulation software suite for MATLAB, Spinach [53], and its optimal control module, were mod-ified to optimise bi-selective pulses using the L-BFGSGRAPE method [27]. Each pulse optimisation was setto terminate following 300 iterations or when either thenorm of the fidelity gradient became smaller than 10 − orthe norm of the step size dropped below 10 − . The result- π π π P h a s e ( r a d . ) n = 1 − −
200 0 200 4000 . . . T r a n s f e r e ffi c i e n c y π π π P h a s e ( r a d . ) n = 4 − −
200 0 200 4000 . . . T r a n s f e r e ffi c i e n c y π π π P h a s e ( r a d . ) n = 7 − −
200 0 200 4000 . . . T r a n s f e r e ffi c i e n c y Time (t/ t π ) -1 π π π P h a s e ( r a d . ) n = 10 − −
200 0 200 400
Raman Detuning (kHz) . . . T r a n s f e r e ffi c i e n c y FIG. 2. Bi-selective phase-modulated augmentation pulses optimised with the phase-sensitive fidelity F real for a temperatureof 1 µ K. The resulting phase profiles for the 1 st , 4 th , 7 th , and 10 th augmentation pulses are shown in the left panel. The rightpanel shows the simulated transfer efficiency of each bi-selective pulse (blue solid curve), the ARP Tanh/Tan pulse of equivalentlength (orange dashed curve), and the WALTZ composite pulse (purple dotted curve) as a function of the Raman detuning.The corresponding detuning distributions seen by each augmentation pulse for the primary interferometer arms are shown bythe shaded regions. The range of detunings included in each optimisation is shown by the hatched regions. ing waveforms, optimising the fidelity F real , are shownin Figure 2, for pulses tailored to the velocity distribu-tions expected for the 1 st , 4 th , 7 th , and 10 th augmenta-tion pulses of the LMT sequence. Efficient populationtransfer is achieved in each case. Interestingly, smoothand symmetrical waveforms are found despite there be-ing no constraint on symmetry or waveform smoothnessincluded in the optimisation.Figure 3 shows the phase profile and simulated trans-fer efficiency of the 10 th augmentation pulse whenthe fidelity F square is instead optimised. Results arealso shown for the WALTZ composite pulse and theTanh/Tan ARP pulse with durations of 4 t π and 8 t π . Thecorresponding detuning distributions for the upper andlower interferometer arms are also shown as shaded re-gions. Whereas the efficiencies of the WALTZ and ARPaugmentation pulses are limited by the finite velocity ac-ceptances, the bi-selective pulse is tailored for the splitmomentum distribution of the two primary interferome-ter arms, allowing it to achieve a higher efficiency than aTanh/Tan ARP pulse of twice its duration. Many of theprofiles found optimising F square have a strong parabolic component corresponding to a frequency chirp, althoughwe are far from the adiabatic regime; and some includea π phase step near the temporal midpoint: we find thatthis combination alone gives the main features but doesnot produce the flat regions of high efficiency achievedusing optimal control. All the pulses found optimising F real are time-symmetric. A. LMT contrast
We have simulated LMT interferometers of differentLMT orders, and compared the performance using ourbi-selective augmentation pulses with that using rectan-gular π and Tanh/Tan ARP augmentation pulses. Thecontrast in each case is averaged over a thermal distribu-tion of atoms and a uniform distribution of Rabi frequen-cies that is either kept temporally constant – to representa non-uniform laser intensity distribution – or else variedfrom pulse-to-pulse – to represent the motion of atomsacross such a distribution. Following the approach takenby Kotru [32], only the primary interfering paths are in- . . . . . . . . . Time (t/ t π ) -3 π -2 π -1 π π P h a s e p r o fi l e ( r a d . ) A bi-sel. 4 t π − −
200 0 200 4000 . . . . . T r a n s f e r e ffi c i e n c y B bi-sel. 4 t π − −
200 0 200 400
Raman detuning (kHz) . . . . . T r a n s f e r e ffi c i e n c y C ARP 4 t π ARP 4 t π WALTZ 3 t π FIG. 3. Phase profile (A) and simulated transfer efficiency(B) for the 10 th bi-selective pulse found optimising the fidelity F square (blue solid line). The transfer efficiency for the ARPTanh/Tan pulse (orange dashed line and green dot-dashedline) and the WALTZ pulse (purple solid line) are shown inpanel C. The detuning distributions of the two primary in-terferometer arms, corresponding to a temperature of 1 µ Kduring the 10 th augmentation pulse are shown by shaded re-gions. The hatched regions represent the bands of detuningwhere the transfer efficiency is maximised using GRAPE. cluded in the calculation to reduce computation time.The simulation results for an atomic temperature of1 µ K are shown in Figure 4. To meaningfully comparethe performance of the GRAPE bi-selective pulses withARP, the total duration of the sequences is kept the same.GRAPE bi-selective pulses far outperform basic rectan-gular π pulses, and Tanh/Tan ARP pulses of equivalentduration. When no thermal expansion is modelled, thedynamic phase cancellation of the ARP pulses is perfect,meaning the reduction in contrast with increasing LMTorder is purely due to the limited velocity acceptance ofthe pulses as the arms separate in detuning. For theARP contrast to match that achievable with bi-selectivepulses, the ARP pulse duration must be increased, thusincreasing the susceptibility to spontaneous emission (notmodelled currently) and dynamic phase dephasing [15].This highlights one key advantage of our adaptive ap-proach.If the Rabi frequency varies from pulse-to-pulse for dif- ferent atoms, the dynamic phase imprinted on the atomsby ARP is not cancelled in the interferometer, meaningdifferent atoms obtain different phases and the interfer-ence is washed out. Although ARP pulses have a ve-locity acceptance which increases with pulse duration,longer ARP pulses become more susceptible to errors indynamic phase cancellation [15].When the Rabi rate is instead varied randomly be-tween pulses in the simulation to emulate noise in theRaman beam intensity profile (panel B in Figure 4), thebi-selective pulses found optimising the phase-sensitive fi-delity F real are able to maintain significantly higher con-trast than other pulse sequences. This is because thephase variation of the wavepackets is minimised with re-spect to variations in the Rabi frequency. This is not thecase with pulses found using F square and the contrast issignificantly reduced in such cases.We have repeated the bi-selective optimisation andcontrast simulation for a hotter atomic temperature of5 µ K and for a longer pulse duration of 5 t π . The re-sults are summarised in Table I. We also show the resultsfor the WALTZ composite pulse and a phase-modulatedpulse obtained using GRAPE maximising F square for alarge range of detunings centred on resonance. This non-selective pulse was optimised following the procedure out-lined in [28]. Longer pulses can achieve higher terminalfidelities, and using the phase-sensitive fidelity F real re-quires a longer duration than F square to reach an equiv-alent terminal fidelity. This is shown most clearly whenoptimising for the hotter cloud. There is no guaranteethat we have found the global maximum for each choiceof duration and temperature. Even so, pulses were foundwhich outperform the composite and ARP alternativestested. TABLE I. Simulated contrast values at temperatures of 1and 5 µ K for different LMT orders with no Rabi frequencyvariation between pulses. The π , WALTZ, and F square non-selective pulses are previous results with either defined ( π ,WALTZ) or previously chosen (non-selective) durations. Theresults for the ARP and bi-selective pulses are presented fortwo different durations. Bold values indicate the best per-forming pulse at each temperature. Length 1 µ K Contrast 5 µ K ContrastPulse ( T/t π ) 10 (cid:126) k 26 (cid:126) k 42 (cid:126) k 10 (cid:126) k 26 (cid:126) k 42 (cid:126) k π F square non-selective 4 0.92 0.72 0.38 0.85 0.67 0.344 0.95 0.62 0.06 0.88 0.52 0.04ARP 5 0.96 0.79 0.19 F real bi-selective 5 0.95 0.87 0.82 0.85 0.60 0.504 F square bi-selective 5 0.96 LMT order (¯ h k) . . . . . . C o n t r a s t A LMT order (¯ h k) . . . . . . B F square F real ARP π FIG. 4. Simulated interferometer contrast for a cloud of Rb at 1 µ K as a function of LMT order for different pulse sequences.The contrast is numerically averaged over Raman detunings due to a velocity distribution corresponding to a temperature of1 µ K and a uniform distribution of static Rabi rate errors of ±
10% (A) or a random temporal variation of Rabi rate errorsof ±
10% from pulse-to-pulse (B). Contrast values for bi-selective GRAPE pulses found with fidelities F square and F real (bluedotted line with diamonds and red solid line with diamonds respectively) are shown. The results for π augmentation pulses(black dotted line with rectangles) and the Tanh/Tan ARP pulse (dashed orange line with circles) are also shown. B. Suppression of off-resonant excitation
When the Doppler shift from gravitational accelera-tion is used to discriminate between retroreflected fre-quency pairs in vertically-orientated interferometers, off-resonant excitations from broadband atom-optics canlead to unwanted double-diffraction. For example, fol-lowing a 10 ms drop time [14], the resonance conditionsof the two frequency pairs will shift by 2 g k eff ×
10 ms ≈
500 kHz for Rb, where g is the local gravitational ac-celeration. Conventional robust pulses can result in anon-zero transfer at comparable detunings (Figure 5),potentially leading to double-diffraction if the unwantedfrequency pair has not been shifted far enough from res-onance [14, 32].We have optimised bi-selective pulses which suppressthe transfer efficiency outside the detuning bands of in-terest using a suitable modification of our bi-selective fi-delity (Equation 8). Figure 5 shows the preliminary re-sults. In order to achieve a good level of suppression atlarge detunings we have found it necessary to allow boththe phase and amplitude of the pulse to vary in the opti-misation procedure. We have also added penalties in theoptimisation to limit the maximum intensity during thepulse and enforce smooth waveforms [54]. VI. CONCLUSIONS
We have presented a new Raman pulse scheme for theaugmentation pulses in large-area atom interferometry,whereby individually tailored Raman pulses found usingoptimal control techniques maintain resonance with the diverging wavepackets as the Raman detuning increasesbetween the interferometer arms. We optimise our bi-selective pulses to provide maximum state-transfer whileminimising variation in the interferometer phase acrossthe atomic ensemble. The pulses can be made robustto large spatial and temporal variations in the Rabi rate,potentially allowing LMT interferometry in non-ideal ex-perimental environments such as those with warmer atomclouds and inhomogeneous laser beam fronts. Our simu-lations show that our pulses can maintain contrast at sig-nificantly higher momentum splittings than interferom-eters that have employed the best augmentation pulsesdemonstrated to-date, including the WALTZ compositepulse and the Tanh/Tan adiabatic rapid passage pulse,whose finite velocity acceptances limit the LMT momen-tum range. For large LMT orders, our pulses can beconsiderably shorter than ARP or composite equivalentsof the same efficiency, reducing the susceptibility to spon-taneous emission.
Appendix: Numerical model
We model Rb atoms undergoing Raman transitionsas two level systems, described by the basis states | g , p (cid:105) and | e , p + (cid:126) k eff (cid:105) with corresponding time-dependentamplitudes c g , e ( t ). Pulses are described by propagatorsof the form given by Equation 3, and are given by time-ordered products of the propagators for individual slicesin the case of shaped and composite pulses, where theamplitude, detuning (frequency), and phase may be var-ied for each step.The basis states for each pulse in an LMT sequence π -2 π -1 π π P h a s e ( r a d . ) A − − − −
500 0 500 1000 1500 20000 . . . T r a n s f e r e ffi c i e n c y B F square suppress F square non-suppress0 1 2 3 4 Time (t/ t π ) . . A m p li t ud e C − − − −
500 0 500 1000 1500 2000
Raman detuning (kHz) . . . T r a n s f e r e ffi c i e n c y D ARPWALTZ
FIG. 5. Panels A and C show the phase and amplitude profiles for two bi-selective pulses maximising F square . The dashedblue curve shows the case where no suppression of off-resonant excitation is included in the optimisation and the solid redcurve shows the case where off-resonant excitation is suppressed. The transfer efficiency of these two pulses as a function ofdetuning is shown in panel B, and panel D shows the transfer efficiency of the WALTZ and Tanh/Tan pulses. The diagonallyhatched regions coinciding with the shaded detuning distributions show the optimisation bands of detuning during the 8 th augmentation pulse. The outer hatched regions show the range of the detunings for which the state transfer is suppressed( δ min , δ max = 500 , vary. For example, during the initial π/ | g , p (cid:105) , | e , p + (cid:126) k eff (cid:105) , but for the first augmentation pulse(where the wavevector is reversed) the upper arm has abasis described by the states | g , p + 2 (cid:126) k eff (cid:105) , | e , p + (cid:126) k eff (cid:105) ,and the lower arm has a basis described by the states | g , p (cid:105) , | e , p − (cid:126) k eff (cid:105) . This means the resonance conditionsare separated for the two interferometer arms by 4 δ recoil .Furthermore, to reduce computation time, only the am-plitudes following the two primary interferometer armsare included in the calculation, as shown in Figure 6.This is a good assumption when the pulses providing thelarge-momentum transfer are efficient, or when there isno interference from atoms that have not followed theprimary interferometer paths [32].We simulate how the interferometer contrast is affectedby the atomic temperature and variations in the Rabirate across the atom cloud. In order to do this, we draw asample of atomic velocities (and hence Raman detunings)from a Maxwell-Boltzmann distribution for Rb at agiven temperature and a uniform distribution of Rabifrequencies in the range ±
10% of the intended Rabi rate.Alternatively, the Rabi rate may be randomly varied frompulse-to-pulse within ±
10% of the intended Rabi rate inorder to explore the robustness of sequences to temporalvariation throughout the interferometer.The contrast following an interferometer pulse se-quence may be calculated by evolving the state ampli-tudes for an atom initially in the ground internal state, | g (cid:105) , by applying the relevant pulse propagators in thecorrect order with the correct Raman detunings for eachpulse and interferometer arm. Fringes were calculatedfor each atom in the ensemble by repeating the evolu- FIG. 6. Diagram of the first half of an N=1 LMT pulse se-quence, shown up to the central mirror pulse, indicating howthe momentum and atomic state vary in each arm of the in-terferometer during the sequence. The resonant Raman fre-quency depends upon the pair of states coupled at each stage.‘A’ represents an augmentation pulse. Single and double ar-rows represent the internal states | g (cid:105) and | e (cid:105) respectively. tion of the final rectangular beamsplitter and varyingthe phase, φ bs , between 0 and 2 π . The ensemble aver-age interferograms were fitted to the sinusoidal function0 . A + B cos( φ bs + C )), where A is an offset, B is thecontrast, and C is a possible phase shift. The effect ofspontaneous emission is not modelled at present but willultimately limit the achievable contrast with extendedpulse sequences in LMT interferometers.0The sweep parameters used to define the Tanh/TanARP pulse are the same as those used by Kotru et al. [15]. The Tanh/Tan ARP pulse waveforms were dividedinto 2500 time-steps in the simulations to ensure thepiece-wise constant approximation was sufficiently accu-rate to model the frequency and amplitude sweep. ACKNOWLEDGMENTS
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