Optimal control of mirror pulses for atom interferometry

Atom matterwave interferometry requires mirror and beamsplitter pulses that are robust to inhomogeneities in field intensity, magnetic environment, atom velocity and Zeeman sub-state. Pulse shapes determined using quantum control methods offer significantly improved interferometer performance by allowing broader atom distributions, larger interferometer areas and higher contrast. We have applied gradient ascent pulse engineering (GRAPE) to optimise the design of phase-modulated mirror pulses for a Mach-Zehnder light-pulse atom interferometer, with the aim of increasing fringe contrast when averaged over atoms with an experimentally relevant range of velocities, beam intensities, and Zeeman states. Pulses were found to be highly robust to variations in detuning and coupling strength, and offer a clear improvement in robustness over the best established composite pulses. The peak mirror fidelity in a cloud of $\sim 80\ \mu$K ${}^{85}$Rb atoms is predicted to be improved by a factor of 2 compared with standard rectangular $\pi$ pulses.


I. INTRODUCTION
Emerging quantum technologies require the coherent manipulation of quantum states. For example, ultraprecise cold-atom-based sensors such as gravimeters, accelerometers, magnetometers and gyroscopes [1][2][3][4] use interactions with laser pulses to form the beamsplitters and mirrors of matterwave interferometers [5], and these 'π/2' and 'π' pulses must operate with high fidelity if the best sensitivity is to be achieved by using pulse sequences to increase the interferometer area [6] or maximise the entanglement [7].
The fidelity of quantum state manipulation deteriorates when there are inhomogeneities in the interaction field, magnetic environment, atomic velocities and quantum state distributions [8]. This limits the number of control operations that can be performed before coherence is lost, so it is common to filter the atomic sample to restrict the variations experienced [6,9,10] by fewer atoms. Inhomogeneities thus limit the interferometer area and sample size, and hence the measurement contrast and sensitivity [11,12].
Various techniques have been developed in the field of NMR spectroscopy to produce control pulses that are robust to variations in the interaction strength and detuning, and such techniques should be applicable to other systems including the effective 2-level schemes of atom interferometry. Shaped pulses [13][14][15], rapid adiabatic pulses [16][17][18] and composite pulses [19][20][21][22] all use complex time-dependent interactions to reproduce the desired operation of a single pulse while compensating for the effects of inhomogeneities. For atom interferometry, McGuirk et al. [10] suggested that composite pulses could improve the augmentation pulses within Large Momentum Transfer (LMT) arrangements, and Butts et al. * j.c.saywell@soton.ac.uk [12] demonstrated that the WALTZ [23] composite inversion pulse doubled the sensitivity of a cold Cs atom interferometer.
In a feasibility study of the applicability of composite pulses to cold-atom interferometers, Dunning et al. [8] analysed the performance of various established NMR pulse sequences for inversion or mirror operations in a thermal cloud of 85 Rb in a velocity-sensitive Raman arrangement subject to intensity variations and a distribution over Zeeman sublevels. Although cold-atom arrangements differ from NMR applications in the magnitudes of different inhomogeneities and the correlations between them, most of the schemes tested improved the robustness of the inversion operation to both detuning and coupling strength variations, and there was excellent agreement between the observed performance and that predicted computationally, showing there to be no further significant perturbations or decoherence.
In this paper, we address the computational design and optimisation of mirror pulses specifically for atom interferometry. Using the well-known optimal control algorithm GRAPE [24] and the advanced spin dynamics simulation toolbox Spinach [25], we have derived pulse shapes that we predict will improve interferometric contrast following a Mach-Zehnder sequence within a 100 µK cloud of 85 Rb atoms by a factor of 1.76 compared with standard rectangular pulses of constant phase and intensity. Our pulse shapes compensate off-resonance and pulse-length errors better than established composite pulse sequences, doubling the peak mirror fidelity when simulated in the σ + − σ + Raman polarisation arrangement for a ∼ 80µK cloud. It is expected that such optimal pulses should allow for greater interferometric areas, higher contrast, warmer samples, and therefore increased interferometric sensitivities.

A. Two-level Representation
The Hamiltonian of a two-level atom interacting with a laser may be written in the rotating frame as [5]: Here, Ω R is the Rabi frequency, φ L is the laser phase, and δ(t) is the detuning from resonance. The detuning can be expressed as [5]: δ AC represents the AC Stark shift, ω L (t) is the laser frequency, and ω 12 is the separation of the two levels. k L is the laser wavevector,h|k L | 2 /2m is the recoil shift, and p · k L /m is the Doppler shift. The two-level system hence consists of the states |1, p and |2, p +hk L , where 1 and 2 refer to the two atomic states andhk L is the photon impulse. If the beams are kept resonant at the centre of the velocity distribution, e.g. by chirping the frequency to account for acceleration under gravity [1,14], the detuning can be written as |k L |v, where v is the relative speed of a given atom along the direction of the wavevector. Therefore, the detuning of a given atom will remain approximately fixed throughout the interferometer sequence, and the range of detunings will be due to the momentum distribution and hence temperature of the cloud. In the Bloch sphere representation, the state vector of any two-level system may be written as [26] |ψ = cos where ϑ and ϕ are the polar and azimuthal coordinates of a point on the surface of a unit sphere. Free and driven evolution is expressed in terms of rotations about axes through the centre of the Bloch sphere. The propagator, whereT is the time-ordering operator [27], describes the unitary evolution of the quantum state |ψ from time t 1 to t 2 . Taking the matrix exponential of the Hamiltonian in the rotating frame (Equation 1), it can be shown that [28]Û where σ is the vector of Pauli matrices. Therefore, the effect of a pulse of duration dt is to rotate the state on the surface of the Bloch sphere by an angle of magnitude about an axis defined by the field vector Ω(t): Recalling the following identity, the form of the propagator for a pulse with a constant field vector and duration ∆t is given by [14,28] where C and S are defined as:

B. Inhomogeneities and Composite Pulses
It is common in NMR spectroscopy to refer to offresonance and pulse-length errors. An off-resonance error arises when the detuning δ is non-zero and the field vector does not lie perfectly in the ϑ = π/2 plane of the Bloch sphere. A pulse-length error occurs when the desired total rotation angle around the field vector is incorrect, due to either an error in the pulse duration or an error in the effective coupling strengthΩ R . Detunings lead to deflection of the atomic state trajectory, and variations in the coupling strength or Rabi frequency lead to errors in the rotation angle around the field vector. These errors lead to dephasing and reductions in the fidelity of state manipulation [8].
Composite pulses are pulse sequences intended to replace a desired operation while providing robustness to off-resonance and pulse-length errors. Composite pulses compensate for such errors by using concatenated sequences with tailored phases and durations [19], which are equivalent to series of rotations on the Bloch sphere; both the combined laser phase (φ L ) and pulse duration (∆t) may be different for each pulse in the sequence. The notation θ φ is used to specify a particular rotation element, so that a composite pulse may be written as the sequence: Such pulses can be placed into two categories: 'pointto-point' (PP) rotations, and 'universal rotations' (UR). The first are only designed to perform the correct rotations for specific starting states, whereas UR pulses are intended to perform the correct rotation for all starting points. The type of pulse required depends on the role in the application. Although numerous composite pulses have been developed for NMR applications, most result from a combination of calculation and intuition and there is no way to design or tailor a sequence for a particular application automatically. Numerical optimisation of a control system has become a promising alternative for broadband pulse generation in NMR [29], and has been successfully applied to the control of Bose-Einstein condensates [30][31][32], the stabilisation of ultra-cold molecules [33], the optimisation of magneto-optical traps [34] and nitrogen-vacancy center magnetometry [35]. Optimal control theory offers methods to generate optimised pulse sequences which are tailored to specific systems and applications.

III. OPTIMAL CONTROL THEORY
Optimal control theory aims to obtain control parameters that allow a dynamical system to be driven so as to maximise some objective function. In quantum mechanics, this objective function often represents the accuracy with which initial states may be driven to desired final states by the available control fields, subject to constraints on the capabilities of the instruments such as maximum power and frequency.
The total Hamiltonian of a system may be written in the formĤ where the drift HamiltonianĤ (0) represents the free evolution of the system. The operatorsĤ (n) correspond to the experimental control fields whose amplitudes c (n) (t) can be set to form a given composite pulse sequence.
To determine the performance of a given pulse, we consider an example fidelity measure of the following form where |ψ D is the desired target state, |ψ 0 is the initial state, and f is a differentiable function of the projection of the resultant state onto |ψ D . The aim is to either minimise or maximise F for all members of an ensemble with varying drifts. The choice of |ψ 0 , |ψ D , and f is application dependent.
A well-known gradient-based optimisation method, first developed by Khaneja et al. [24] for the design of NMR pulse sequences, is Gradient Ascent Pulse Engineering (GRAPE). GRAPE has computationally efficient gradients using analytical derivatives [36,37], which can be used to approximate and even compute the Hessian [38] and allow for Newton-Raphson type optimisations. GRAPE begins by discretising the M control sequences c (n) (t) into N timesteps c (n) k of duration ∆t, and assuming that during each timestep the control amplitude c (n) k is a constant. The optimisation involves finding the vectors {c (n) k } such that the chosen functional F is either maximal or minimal. The form of the time-ordered propagator is then simplified to a product of k "slices" For the optimisation, derivatives of F are required with respect to the set of control coefficients {c (n) k }. GRAPE computes these derivatives in an efficient way, which can be seen by making the observation that the only element in Equation 15 that depends on c (n) k isÛ k . Therefore, computation of ∂F/∂c (n) k for all timesteps k and all control channels n requires just two simulations, namely The directional derivatives which require computation are An approximate expression for this derivative is given by Khaneja et al. [24]. First-order gradient ascent can then be used to optimise the fidelity F iteratively. Improvements were made in the use of propagator derivatives by De Fouquieres et al. [39] which use the gradient to approximate a second order optimisation method, called a quasi-Newton optimisation, improving optimisation convergence. This work uses the limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) quasi-Newton method implemented in Spinach [38,39] with analytical directional derivatives [37].
The two control coefficients c (1) (t), c (2) (t) are given by and Ω R (t) sin[φ L (t)] respectively. This form is directly analogous to the case of a spin system interacting with an applied rf field given appropriate parameter scaling of the magnitudes of each term [40]. Optimal control can be used to obtain optimal waveforms Ω R (t) and φ L (t), or the amplitude may be fixed and an optimal phase profile obtained. Initial guesses for the pulse waveform (Ω R (t) and φ L (t)) are provided and the derivatives of the relevant fidelity measure are calculated, returned and fed into the optimisation module of Spinach [38]. The control pulses are made robust to variations in detuning (off-resonance errors) by providing Spinach with an ensemble of drift Hamiltonians and maximising or minimising the average of individual fidelities. Robustness to variations in power (pulse-length errors) is achieved by providing a range of power levels, which are averaged over in the fidelity calculation [41,42]. We define the ensemble provided to the optimisation as consisting of a number of offsets in detuning and coupling strength, denoted by δ off and Ω off R respectively. Finally, penalties may be added to the objective function to restrict experimentally relevant quantities such as maximum power or enforce waveform smoothness [24,38].

IV. INTERFEROMETER FIDELITY
Although mirror pulses may be optimised individually, we obtain measures of mirror pulse fidelity by considering the role of the mirror within a 3-pulse Mach-Zehnder sequence. In addition to the ability of a mirror pulse to transfer as many atoms from ground to excited state as possible, variations in the atomic phase of the final state following a mirror pulse must be minimised to preserve interferometric contrast, as shown by Luo et al. [14]. The excited state population after a sequential application of pulse propagators corresponding to the π/2 − π − π/2 sequence acting on an atom initially in the ground state can be obtained by following the analytical treatment given by Stoner et al. [28], where C and S are defined in Equation 10 and refer to elements of the pulse propagators. We have assumed the initial and final beamsplitters in the Mach-Zehnder sequence to be identical and the dwell times between pulses to be the same. The subscripts ' π 2 ' and 'π' refer to the beamsplitters and mirrors respectively. φ i is an interferometric phase term which gives information about the inertial forces acting during the sequence, but which is modified by pulse-dependent phase shifts. This phase must be preserved when averaging over an ensemble of atoms with a distribution of velocities and coupling strengths. The interferometer output may be written as [14] where φ p (δ) is a phase shift introduced by the pulses and A(δ) and B(δ) represent the interferogram offset and contrast respectively. In the ideal case A(δ), B(δ), and φ p (δ) should be constant for all detunings present in an atomic sample. For maximum contrast following thermal averaging, A and B must be unity for all atoms. These requirements can be used to obtain a measure of fidelity for the mirror pulse, which we can use to numerically optimise pulse shapes with these properties across an ensemble. The phase shift due to the pulses may be expanded as and the contrast may be written as Assuming rectangular and perfectly resonant beamsplitter pulses across the atomic ensemble (|C π 2 | 2 |S π 2 | 2 → 1/4) implies that the following constraints on the mirror pulse for an ensemble of atoms will maximise contrast of the Mach-Zehnder output after thermal averaging: We therefore consider the following two mirror pulse fidelities: Maximising fidelity F π real or F π imag constrains the phase of the overlap 2|Û π |1 . All the conditions of the optimal inversion pulse (Equations 23) are satisfied. Further, it can be shown that these performance functions (Equations 24 and 25) are equivalent to obtaining the universal 180 • rotations explored by Kobzar et al. [42].
Previous work by Dunning et al. [8] defined the following fidelity for composite mirror pulses: where c 2 is the final excited state amplitude of the twolevel system and |ψ F is the final state. The fidelity in Equation 26 does not constrain the phase of the final state and yields pulses which are point-to-point (PP) rotations between the ground and excited states. Since the mirror pulse in the Mach-Zehnder sequence is to be applied to atoms which ideally lie at a range of points on the equator of the Bloch sphere, we expect that pulses which maximise fidelity F π square will lead to poor interferometric contrast when averaging over an ensemble. Our approach is to use GRAPE to generate pulses which maximise a given mirror fidelity, F π real , F π imag , or F π square for ensembles with experimentally relevant ranges of detunings and coupling strengths. We then determine through simulation the effect of the cloud temperature alone upon the fringe contrast that these pulses could yield in an interferometric application.

V. SYSTEM AND MODEL PARAMETERS
Although our approach may be applied to any spin system, we evaluate it for a cold-atom light-pulse interferometer such as that in [8], wherein a thermal cloud of several million atoms of 85 Rb at a temperature of order 10 − 100 µK is addressed by counter-propagating 780 nm laser beams tuned to a Raman transition between the hyperfine levels |5 2 S 1/2 , F = 2 and |5 2 S 1/2 , F = 3 (levels 1 and 2 respectively). The Raman beams are detuned far from single photon resonance with the intermediate |5 2 P 3/2 state, which may theoretically be adiabatically eliminated so that each atom can be described as an effective two-level system [5]. A description of our experimental arrangement is given elsewhere [8]. Our effective two-state system evolves under the Hamiltonian in Equation 1, with Ω R replaced by an effective two-photon Rabi frequency Ω 1 Ω 2 /(2∆), where Ω 1 and Ω 2 represent the coupling of each laser to levels 1 and 2 respectively, and ∆ represents the single photon detuning of both lasers from the intermediate level. The laser frequency becomes ω L = ω L1 − ω L2 , where ω L1,L2 are the frequencies of the counter-propagating Raman beams formed by lasers 1 and 2, φ L becomes an effective combined laser phase φ L1 + φ L2 , and k L is the effective wavevector k L1 − k L2 .
In our atom cloud, there is Zeeman degeneracy over sub-states distinguished by the quantum number m F , which give rise to multiple coupling strengths, and both the laser intensity and residual magnetic field vary across the atom cloud. The counter-propagating Raman beams give the interaction a Doppler sensitivity, which we use elsewhere for velocimetry and inertial measurement [43]. Different atoms thus see different coupling strengths and Doppler, Zeeman and AC Stark shifts according to their internal and external states. Both pulse-length and off-resonance errors are present in our system. For an experimentally-measured Rabi frequency of Ω eff ≈ 2π × 360 kHz we find coupling strength variations of approximately 0.3 Ω eff and off-resonance errors due to a Gaussian velocity distribution with a full width at half maximum of approximately 1.5 Ω eff in a ∼ 80 µK cloud [44].

VI. RESULTS
All pulse optimisations were carried out using GRAPE and Spinach, constraining the effective Rabi frequency to correspond to a limited laser power and fixed singlephoton detuning, and with a discretisation timestep of 100 ns. Optimisations were carried out for ensembles with various ranges of detunings and coupling strengths (see Table I), using fidelities F π real , F π imag , and F π square . Longer pulses can excite atoms from the ground to excited states for larger detunings than shorter pulses. Mirror pulses are compared based on their ability to transfer atoms from state to state and the magnitude of the atomic phase variations over the simulated ensemble of atoms. The mirror pulse is characterised by a timevarying phase, initially set to be flat, and optimisation was continued until convergence. Different initial pulse shapes were often found to lead to the same optimal pulse, and pulses often exhibited symmetry about the mid-point -a phenomenon also seen in NMR [41,42,45]. We begin by presenting the results obtained by considering a general ensemble of pulse-length and off-resonance errors in our optimisation, before applying our chosen optimal control method to an ensemble representing the experimental inhomogeneities present in a particular Raman polarisation arrangement.
A. Mirror pulse optimisation Figure 1 shows the phase profile of a constantamplitude mirror pulse obtained by maximising F π real subject to a constrained duration of 20 µs, for a detuning range of ±1.5 Ω eff and a power range of ±0.1 in Ω eff . If the duration of the pulse and the ensemble detuning range are increased, further variations in phase appear symmetrically around the centre of the pulse shape. In practice, a balance must be struck between pulse duration and performance, dependent on the required application and capabilities of the experimental apparatus to produce complex waveforms. Table I summarises the performance of composite mirror pulses and GRAPE optimisations carried out for different ensembles and fidelities.
The response purely as a function of the detuning from resonance is shown in Figure 2. The detuning here would be experimentally due to Doppler and Zeeman shifts. Both the excited state probability after application of the pulse to atoms in the ground state, and the phase response of the S element of the propagator are shown. GRAPE maintains a transition probability >0.976 over the optimised range of offsets (±1.5 Ω eff ). For comparison, we show the corresponding results for the rectangular π pulse and for WALTZ and KNILL pulses [23,46] of the same intensity. The WALTZ pulse is highly robust to detunings, but is designed as a point-to-point operation for a particular starting state and hence shows large variations in the atomic phase as the detuning is varied; the GRAPE pulse, in contrast, shows smaller phase vari- ations over the range of detunings. The conventional π pulse shows an ideal, flat phase response but very narrow range of detunings over which it is efficient. GRAPE can be used to optimise a mirror pulse for an experimental range of off-resonance errors and the pulselength errors which arise from the degenerate Zeeman sublevels in the σ + − σ + Raman polarisation arrangement. We simulate the pulse profile of Figure 1 using a model which accounts for the AC Stark shift, m F levels, measured atomic momentum distribution, and experimental parameters in the σ + − σ + polarisation arrangement for a cloud with temperature ∼ 80 µK (see Figure 3). Such an arrangement has large off-resonance errors and is therefore a good choice to compare pulse performance. The numerical model used in this simulation was shown to have good agreement with experiment [8,44]. In this simulation, the final excited state population is calculated numerically and averaged over a range of atomic momenta from the measured distribution, and the five Zeeman sublevels. The GRAPE simulations show an improvement in peak fidelity by a factor of 1.2 compared with WALTZ, and a factor of 2 compared with the rectangular π pulse. As the laser detuning is scanned, the peak excited state population depends on how well a mirror pulse can excite atoms from the ground to excited state for the distribution of coupling strengths and velocity classes in the ensemble. Figure 3 demonstrates GRAPE's ability to compensate for the off-resonance and pulse-length errors present in this system. Contour plots (Figure 4) demonstrate the superior robustness to both error classes offered by our chosen optimal control method.  Figure 1 in a σ + − σ + Raman arrangement, as the laser detuning δL (defined as ω12 − ωL) is scanned. This simulation uses an effective Rabi frequency of 2π × 360 kHz, and accounts for both the coupling strengths of 5 mF sublevels and the Stark shift. Both the simulation parameters and model are described elsewhere [8,44]. The offset in peak is due to the Stark shift, and the magnitude of the peak gives an indication of a pulse's ability to excite atoms across the momentum distribution for all mF sublevels, which we assume are equally populated.

B. Simulated interferometric contrast
The temperature-dependent contrast resulting from the Mach-Zehnder interferometer sequence was determined following the procedure outlined by Luo et al. [14] and integrating over Maxwell-Boltzmann velocity distributions with temperatures in the range (10 −1 − 10 3 µK). A uniform pulse intensity and single coupling strength, with no m F -dependent inhomogeneity, were assumed throughout. The beamsplitter and recombiner pulses were taken to be rectangular π/2 pulses with a fixed effective Rabi frequency of 2π × 200 kHz, and hence limit the achievable fringe contrast as shown by their combination with a perfect π mirror pulse that is taken to perform an ideal rotation for all atoms. With a realistic mirror formed by a rectangular π pulse with the same Rabi frequency as the beamsplitter and recombiner, the interferometer contrast is limited by the imperfections in the rectangular π pulse to around 0.8 at a temperature of 20µK.
The GRAPE pulse of Figure 1 offers an improvement in fringe contrast by a factor of 1.76 over that with the rectangular π pulse at 100µK, and approaches the fidelity predicted with a perfect π pulse. This limit should be approached more closely as the GRAPE pulse duration is increased, as longer pulses can compensate for a greater range of off-resonance errors. As noted by Luo et al. [14,47], the WALTZ pulse performs badly due to its non-uniform phase response, as it was designed as a point-topoint pulse whereas the mirror operation must have high fidelity irrespective of the starting point on the equator of the Bloch sphere. The point-to-point F π square GRAPE pulse also performs poorly, as expected. We find the KNILL pulse, designed as a universal 180 • rotation, offers a slight improvement over the sequence of rectangular pulses for the simulated range of temperatures, but is outperformed by GRAPE at all temperatures. The beamsplitters in each case are rectangular π/2 pulses. There is a large increase in contrast due to the GRAPE UR mirror pulse in Figure 1 which improves the rectangular π pulse by a factor of 1.76 at 100 µK, approaching the theoretical limit of a sequence with a 'perfect' π pulse resonant for all atoms in the sample. Comparisons with different sequences are made at same Rabi frequency of 2π × 200 kHz.

VII. DISCUSSION
Tolerance of pulse-length and off-resonance errors is essential for the pulse operations in atom interferometers, where a range of velocities, beam intensities and Zeeman substates may be encountered. We have used the optimal control technique of gradient ascent pulse engineering (GRAPE) to obtain robust mirror pulses, tailored to accommodate the inhomogeneities found in cold-atom matterwave interferometers, and find such pulses to outperform all the other composite pulses that we have tested. By using a numerical model which has been shown to agree well with experiment [8], we have simulated the performance of an atom interferometer for the σ + − σ + polarisation Raman arrangement within a ∼ 80 µK cloud of atomic rubidium subject to realistic intensity inhomogeneities and Zeeman substate distributions, and find that our GRAPE pulses show a peak fidelity twice that obtained with a basic rectangular π pulse, and 1.2 times that achieved using the WALTZ sequence, with significantly greater tolerance of variations in the atom velocity.
The improved fidelity should allow improvements in the sensitivity of interferometric measurements by permitting greater use of augmentation pulses for large momentum transfer interferometers [12], while the tolerance to atom velocity variations will lower measurement noise by allowing the use of warmer atom clouds, and hence higher atom numbers, without incurring the losses of further cooling or filtering. Our GRAPE pulses should provide, for example, transfer efficiencies above 0.9 for a detuning range 1.4 times that tolerated by the WALTZ sequence, which was otherwise the best pulse tested for this system [8].
Replacing the basic rectangular π pulse with a GRAPE mirror in a Mach-Zehnder arrangement at 100µK is sufficient to improve the simulated interferometer contrast by a factor of 1.76, or to achieve the same contrast as a basic π pulse for a 15µK atom cloud. At higher atom temperatures, we see that the interferometer fidelity is limited principally by the fidelity of the beamsplitter operations.
Our optimal control approach depends upon an appropriate choice of the measure of performance. We find that those used for broadband UR 180 • pulses, such as F π real (Equation 24) and equivalents considered by Kobzar et al. [42], are able to preserve the interferometric phase and increase contrast, and when our optimisation is carried out for small detuning ranges we produce similar pulse shapes. Measures of performance more suitable for point-to-point operations, such as F π square (Equation 26), conversely yield lower interferometer contrast, as does the WALTZ point-to-point composite pulse.
While GRAPE should also be applicable to the design of beamsplitter pulses, we expect that a more fruitful approach will be to consider symmetries in the Mach-Zehnder sequence and compensate in the second beamsplitter operation for errors introduced in the first, so as to optimise the interferometer as a whole. Such cooperative pulse optimisation was investigated by Braun et al. [48,49] for Ramsey-style experiments in NMR, and allows greater freedom in the optimisation as individual beamsplitter pulses are permitted imperfections provided that they are cancelled elsewhere in the interferometer sequence. This should allow shorter pulse sequences, desirable for interferometric sensors operating in dynamic environments [28] and is attractive for optimisation of the π/2 − π − π/2 sequence used for inertial sensing applications. Nonetheless, we expect the mirror optimisation described here to suffice for a large contrast improvement in many current configurations.
Our future work will involve experimental demonstration of GRAPE mirror pulses in our experiment, and analysis of how features present in pulse profiles from optimal control methods allow for error compensation. This will involve characterising the dynamics and evolution of atomic trajectories on the Bloch sphere. I. Performance of GRAPE pulses compared with composite mirror pulses. Detuning tolerance is measured in units of the effective Rabi frequency Ω eff and represents the range of detuning for which the final excited state probability is >0.5 and >0.9 after application to atoms in the ground state. The maximum phase response variation is taken over a range of ± Ω eff in δ. Optimisation parameters are provided for GRAPE pulses including the offsets optimised for pulse-length and off-resonance errors, and the number of offsets used. In each optimisation, ensembles were weighted by 8 additional detuning offsets near resonance. The best values for the standard composite sequences tested and GRAPE sequences are in bold.