# Enhancing strontium clock atom interferometry using quantum optimal control

Zilin Chen, Garrett Louie, Yiping Wang, Tejas Deshpande, Tim Kovachy
Department of Physics and Astronomy and Center for Fundamental Physics,
Northwestern University

###### Abstract

Strontium clock atom interferometry is a promising new technology, with
multiple experiments under development around the world to explore its
potential for dark matter and gravitational wave detection. In these
detectors, large momentum transfer (LMT) using sequences of many laser pulses
is necessary, and thus high fidelity pulses are important since small errors
become magnified. Quantum Optimal Control (QOC) is a framework for developing
control pulse waveforms that achieve high fidelity and are robust against
experimental imperfections. Resonant single-photon transitions using the
narrow clock transition of strontium involve significantly different quantum
dynamics than more established atom interferometry methods based on far-
detuned two-photon Raman or Bragg transitions, which leads to new
opportunities and challenges when applying QOC. Here, we study in simulation
QOC pulses for strontium clock interferometry and demonstrate their advantage
over basic square pulses (primitive pulses) and composite pulses in terms of
robustness against multiple noise channels. This could improve the scale of
large momentum transfer in Sr clock interferometers, paving the way to achieve
these scientific goals.

††preprint: APS/123-QED

##  I Introduction

Light-pulse atom interferometry is a technology that uses laser pulses to
split, manipulate, and recombine the motional states of atoms so that precise
measurements can be made from their interference. It has proven itself a
powerful tool for precision metrology and sensing, with applications including
tests of quantum mechanics and the equivalence principle [1, 2, 3, 4, 5, 6, 7,
8, 9, 10, 11, 12, 13, 14, 15, 16], terrestrial and spaceborne gravitational
wave detection [17, 18, 19, 20, 21, 22, 23, 24, 25, 26], precision
measurements of the fine structure constant [27, 28, 29] and gravity [30, 31],
searches for dark matter [32, 33, 34] and dark energy [35], and mobile
surveying [36, 37]. The most sensitive interferometers employ large momentum
transfer (LMT) techniques, which increase the enclosed spacetime area with
additional laser pulses. Typical LMT atom optics utilize far-detuned, multi-
photon transitions [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], where two
ground states are coupled via a short-lived excited state. The scaling of such
pulse sequences is limited by spontaneous emission, which can only be
mitigated so long as additional laser power is available. STIRAP has been used
in atom interferometry but does not currently match the performance of other
state-of-the-art atom optics, limited by other detuned excited states [49].

In contrast, alkaline earth atoms such as Sr possess long-lived excited
states, which have been leveraged to achieve state-of-the-art atomic clocks
[50, 51]. In ${}^{87}{\rm Sr}$, the ${}^{1}{\rm S}_{0}\to{}^{3}{\rm P}_{0}$
clock transition has a lifetime over 100 seconds, allowing resonant, single-
photon atom optics with greatly reduced spontaneous emission losses. This
affords an enormous increase in available pulse area, potentially scalable to
thousands of pulses before significant spontaneous emission losses [52, 26].

A clock interferometer is a specific type of atom interferometer using such
resonant, single-photon transitions. It offers improved laser phase noise
rejection in differential measurement configurations comparing multiple
interferometers over a long baseline [19] 111for atom interferometers based on
two-photon atom optics, multiple baselines can be used to achieve improved
laser noise suppression [21], which is essential for dark matter and
gravitational wave detection. Moreover, the clock transitions in alkaline
earth atoms are orders of magnitude less susceptible to magnetic fields than
in the alkalis [54]. This new generation of recently demonstrated single-
photon clock atom interferometers [55, 56, 52, 57] is poised to study hitherto
elusive phenomena such as ultralight, wavelike dark matter [32, 33, 26, 25,
24, 23, 34], tests of atom charge neutrality [58], mid-band gravitational wave
detection [26, 24, 23, 25], and tests of quantum mechanics at unprecedented
delocalization scales [26].

In practice, the performance of atom interferometers is also limited by the
noise in the driving field and inhomogeneities across the atom cloud. Atom
losses and phase errors caused by these effects accumulate with repeated
pulses and thus limit the scaling of LMT systems. Simple robust control pulses
for two-level quantum systems under detuning and amplitude errors were
developed for nuclear magnetic resonance (NMR) spectroscopy [59, 60, 61] and
have more recently found popularity in quantum computation [62, 63]. More
sophisticated pulse shaping algorithms, such as gradient ascent pulse
engineering (GRAPE) [64] and chopped random basis (CRAB) optimization [65,
66], minimize a cost function that quantifies the infidelity of the operation.
As this may be an arbitrary function of hundreds of parameters, highly
modulated control pulses can now be tailor-made for driving particular quantum
dynamics with an appropriate cost metric. For instance, quantum optimal
control (QOC) is a powerful tool for realizing high-fidelity gates in quantum
computing [67, 68, 69, 70, 71] and levitated nanoparticle control [72, 73].

In the field of atom interferometry, quantum control schemes including
composite pulses [74, 75, 76], shaped pulses [77], adiabatic rapid passage
[46, 78], and numerical optimal control [79, 80, 81, 82, 83] have been applied
to Raman and Bragg transitions with alkalis, and Floquet pulse engineering has
been applied to single-photon atom optics on the ${}^{1}{\rm
S}_{0}\to{}^{3}{\rm P}_{1}$ transition of ${}^{88}{\rm Sr}$ [57]. In this
paper, we report simulation studies of numerical optimal control’s
applications to ${}^{1}{\rm S}_{0}\to{}^{3}{\rm P}_{0}$ clock interferometry
with ${}^{87}{\rm Sr}$. Using the gradient-based optimization tools in
Q-CTRL’s BOULDER OPAL package [84], we design optimized analogs of $\pi$
pulses which are robust to as many as five simultaneous noise channels.
Sequences of many of these pulses can be used to achieve enhanced LMT [18].
Under an experimentally relevant range of noise, we find that the optimized
pulses have fidelity improved beyond that of basic square (primitive) pulses
by an order of magnitude or more. Successful implementation of these pulses
may allow atom interferometers to overcome unavoidable limitations and realize
their full potential.

A challenge in clock interferometry is that imperfections during laser beam
delivery or local changes to the quantization axis from stray magnetic fields
can introduce undesired polarization components, which couple transitions to
additional magnetic sublevels (Fig. 1(a)). Larger bias fields can suppress
errors in the quantization axis, but are unfavorable due to quadratic Zeeman
shifts [26] and cannot mitigate errors already present in the beam. It is
therefore a valuable application of quantum optimal control to design pulses
that are insensitive to polarization defects, reducing the atom loss and phase
errors arising from the population of other sublevels.

##  II Optimization Protocol

In the interferometry scheme studied here, a cloud of ${}^{87}{\rm Sr}$ atoms
is initialized in the $\ket{{}^{1}{\rm S}_{0};m_{F}=9/2}$ state and driven to
the $\ket{{}^{3}{\rm P}_{0};m=9/2}$ state on the 698 nm clock transition,
which has a natural linewidth of 1 mHz [26]. For optimization, we split a
pulse of fixed duration uniformly into $N$ time segments, whose amplitude and
phase become the $2N$ independent variables—hereafter referred to as c—which
are modified during optimization. The pulse duration and the number of
segments are chosen before optimization, as is the learning rate (step size in
the gradient-based search). We vary these parameters between optimization runs
to determine what gives the best results (Fig. 1(b)). We choose maximum Rabi
frequencies of several kHz [26] and find pulse lengths of 2 ms or longer best
for optimizing against all noise channels. With the many-second interferometer
duration in tall atomic fountains [26] or spaceborne detectors [17, 22, 23],
sequences of thousands of pulses may be performed, further increased by future
upgrades to laser power [85, 86]. In the optimized pulse, the segment time is
approximately 17 $\mu s$, which is significantly longer than the response time
of an acousto-optic modulator (AOM).

Figure 1: (a) Substructure of the ${}^{1}{\rm S}_{0}$ and ${}^{3}{\rm P}_{0}$
manifolds, with transition strengths indicated. Parasitic $\sigma_{+}$ and
$\sigma_{-}$ polarizations couple to states outside the desired $m_{F}=9/2\to
m_{F}^{\prime}=9/2$ two-level system. (b) Optimization procedure. Randomly
sampled noise values for multiple noise channels define a batch of noise
trajectories. The unitary evolution of the system is calculated under each,
yielding a batch of infidelities which are averaged to give the final cost
function. The scope of robustness is tuned by modifying the sampling
distributions. (c) The optimized pulse starts and ends with a smooth switching
(shown in the insets). The phase and amplitude modulation are smoothed with a
Gaussian-weighted moving average filter. The frequency components higher than
$4\times 10^{4}$ Hz are filtered out. The frequency spectrum is within the
modulation bandwidth of acousto-optic modulators (AOMs), and the switching on
and off process is within the rise time achievable by AOMs. The smoothed pulse
is used for all simulations. Figure 2: Comparison of the optimized pulse to
primitive pulse and various composite pulses [79, 59]. Single pulse infidelity
with noise parameters’ variation are compared among the optimized pulse,
primitive pulse, and composite pulses. $\Delta B$ is the deviation of the bias
magnetic field from its nominal value $B_{0}=1$ G, so that the total bias
field is $B=B_{0}+\Delta B$. $\beta_{v}$ is the detuning due to the Doppler
shift in units of 100 Hz. The units of the horizontal axes for the
polarization and amplitude plots correspond to fractional errors. In contrast
to all other pulses, the optimized pulse maintains a high degree of robustness
against all noise channels and outperforms all other pulses in robustness
against polarization noise.

To achieve robust control, we use a cost function based on random samplings of
noise trajectories (one trajectory is the Hamiltonian with a particular set of
noise parameters) [87]. We include five channels of static noise: $\sigma_{+}$
and $\sigma_{-}$ polarization components, which couple sublevels outside the
desired two-level system; amplitude noise on the drive; detuning errors from
Doppler shifts in the atom cloud; and variation in the bias field, which
changes the Zeeman splitting of the sublevels. Each noise channel is
associated with a Hamiltonian term (see Appendix A). For each of these
operators, we randomly sample $N_{B}$ sets of noise amplitudes—denoted as
$\bm{\beta}^{(i)}=(\epsilon_{+}^{(i)},\epsilon_{-}^{(i)},\beta_{A}^{(i)},\beta_{v}^{(i)},\beta_{B}^{(i)})$
for the $i$th noise trajectory in the batch—from Gaussian distributions whose
predefined width determines the scope of desired robustness.

For each of these Hamiltonians $\hat{H}^{(i)}(\textbf{c},t)$, we calculate the
unitary evolution, from which we determine an infidelity
$\mathcal{I}^{(i)}(\textbf{c})$. The infidelity is defined as

$\mathcal{I}^{(i)}(\textbf{c})=1-\left|\frac{\operatorname{Tr}\left(U_{\text{target
}}^{\dagger}U^{(i)}(\textbf{c})\right)}{\operatorname{Tr}\left(U_{\text{target
}}^{\dagger}U_{\text{target }}\right)}\right|^{2},$ (1)

where $U_{\text{target }}$ is the target state evolution, and $U^{(i)}$ is the
evolution due to the Hamiltonian under the $i$th noise trajectory [84]. This
metric is sensitive to both the population transfer and the phase imprinted by
the pulse, which is important for atom interferometry. All 20 ground and
excited sublevels are included in our simulations. To construct the total cost
function, we average the infidelities from all trajectories:

$\mathcal{C}(\textbf{c})=\frac{1}{N_{B}}\sum\limits_{i=1}^{N_{B}}\mathcal{I}^{(i)}(\textbf{c}),$
(2)

where we typically use a batch size $N_{B}$ of 200. We found that increasing
to a larger batch size did not provide significant improvement. The gradient-
based optimizer in BOULDER OPAL determines the control variables c which
minimize this cost function. Thus, the pulse waveforms (Fig. 1(c)) are
determined which retain low infidelity even in the presence of noise.

Alternative methods of optimizing composite pulses for multilevel systems have
also been previously investigated by other groups. Caneva et al. [66]
investigated the efficiency of the chopped random basis (CRAB) technique in
optimizing different quantum processes. Genov and Vitanov [88] also used
composite pulses to achieve efficient population transfer in multi-state
quantum systems. Compared to this study, our QOC method, instead of analyzing
the system’s mathematical process in a Taylor expansion approximation to
determine the optimal pulses, numerically searches in a large parameter space
to find the most robust pulse shape. In the future work, it will be
interesting to compare the results of different optimization methods applied
to clock atom interferometers.

##  III Results and Comparison

###  III.1 Single optimized pulse

To visualize the performance of various pulse schemes, we scan the infidelity
across values of one (Fig. 2) or two (Fig. 6) noise channels. In Fig. 2, the
optimized pulse performs up to an order of magnitude (or more) better than the
primitive and composite pulses for the range of polarization errors considered
here while maintaining strong robustness against amplitude and detuning
errors. The range of noise values in optimizations and these plots follow what
is expected in relevant experiments. In dark matter and gravitational wave
experiments, it is favorable for the atom cloud’s spatial extent to remain
within the central region of the laser beam to help mitigate systematic errors
such as those due to laser wavefront perturbations and residual AC Stark
shifts from far-detuned transitions [26]. Therefore the laser amplitude
variation across the atom cloud is assumed to be within several percent. We
also assume detunings to be on the order of 100 Hz, which is consistent with
atom clouds lensed to sub-nK effective temperatures [89, 90, 26] that are
desirable for systematic error mitigation [26]. A 0.1 polarization amplitude
error gives a reasonable $1\%$ of optical power in unwanted components. In
general, the amplitudes of the $\sigma_{+}$ and $\sigma_{-}$ polarization
components can be complex. While Fig. 2 plots values of these amplitudes over
a real domain to ease visualization, we note that using complex values for
these coefficients does not significantly affect the scale of the
infidelities. We also evaluate the impact of variations in the bias magnetic
field on the infidelity compared with other noise channels.

Figure 3: Phase deviation induced by single optimized pulse. In the optimized
pulse, time-dependent noises are included (the root mean square (RMS) of the
noises is $0.5\%$ in the amplitude and $1\%$ in the phase). A set of 10
simulations are performed with random time-dependent noises for different
noise parameters. (a), (c), (e) are mean values, and (b), (d), (f) are
corresponding standard deviations of the simulation set.

In Fig. 3, we show the phase deviation induced by a single optimized pulse as
a function of different noise channels (i.e., the difference between the
actual and ideal final state phase value). An atom interferometer with
$\sim$1000 optimized pulses would introduce a total $\sim$ rad scale phase
deviation, due to the $\sim$ mrad scale phase deviation of each pulse for the
various noise channels. Atomic dark matter and gravitational wave detectors
typically look for time varying signals in a particular frequency band, such
as 0.3 - 3 Hz [26]. If, for example, the fractional fluctuation of the
polarization errors are at the level of $0.1\%/\sqrt{\text{Hz}}$ in this
frequency band, the corresponding interferometer phase noise would be at the
level of $\sim\text{mrad}/\sqrt{\text{Hz}}$. Such phase fluctuations may be
further suppressed if both arms of the interferometer, or both interferometers
in a differential gradiometer configuration [26], experience close to the same
errors.

Figure 4: Time-dependent noise impact on infidelities. For the same optimized
pulse, adding noise in the control pulse will negatively impact the robustness
which is determined by measuring the effective area (red) where the infidelity
is a certain value (0.0007 for the polarization map (b), 0.0004 for the
amplitude/detuning ($\beta_{v}$) map (d) and 0.0002 for the $\sigma_{-}$
polarization/field deviation $\Delta B$ map (f). The robustness on the
polarizations [(a)], the amplitude/detuning [(c)], and polarization/magnetic
field [(e)] are worse if RMS of the noise increases. A value $x$ of the RMS of
noise on the horizontal axis of the plot corresponds to RMS fractional
amplitude noise of $x$ and RMS phase noise of $x$ rad. We note that the
response to noise is not perfectly symmetric. We do not necessarily expect the
optimization to provide perfect symmetry here as we might for a two-level
system since the dynamics are significantly more complex, with many different
states that have different coupling strengths and different detunings. Figure
5: (a) A schematic diagram of momentum change in both arms of the
interferometer. Primitive pulses are used in the highlighted parts to avoid
unwanted off-resonant interaction with the untransferred arm. The maximum
momentum in our simulation setup is 500 $\hbar k$. (b) A schematic diagram of
a differential measurement between two interferometers spatially separated
over a baseline.

Since time-dependent noise in the laser amplitude and phase will also occur in
the physical system, we studied its impact on the infidelity of the optimized
pulse. The added white noise is constructed separately for amplitude and phase
by

$f(t)=\sum_{i}^{1000}A_{i}\cos(\omega_{i}t)+B_{i}\sin(\omega_{i}t),$ (3)

where $A_{i}$ and $B_{i}$ are randomly generated amplitudes and $\omega_{i}$
are random frequencies uniformly distributed from 50 Hz to $10^{5}$ Hz. We
normalize each $f(t)$ to the desired root mean square (RMS) value. The
spectrum of the noise is chosen in this range since lower or higher
frequencies are tested to have less impact on the robustness. A thousand
frequency components are used to guarantee sufficient sampling density in the
targeted noise spectrum. Higher sampling numbers do not change the result
significantly. Due to the randomly generated noise for each simulation, the
impact on the infidelity undergoes fluctuations between different simulation
runs. We measure this impact by observing the trend of the area in the contour
plot whose infidelity is under a certain value (effective area). By repeating
the simulation, the impact fluctuations are indicated by the standard
deviation as error bars, and the trend is shown in Fig. 4(a), (c), (e).
Overall, with bigger noise both in amplitude and phase, the effective area is
smaller in the infidelity map for different noise channels (Fig. 4(b), (d),
(f)). For the noise RMS of approximately 1 percent (defined in Fig. 4 caption)
which is an achievable level in experiments 222For instance, using
commercially available laser systems from Menlo Systems, the effective area
reduction is relatively modest (a factor of approximately 2 or less),
indicating that the optimized pulses can remain effective in the presence of
noises. This is further verified by the full interferometer simulation
described below, which shows that good total transfer efficiency (including
the net effect of all pulses) and well-controlled phase deviations can be
maintained for experimentally relevant combinations of static and time-
dependent noises, even for interferometers with thousands of total pulses.

Figure 6: Full interferometer simulation. The results from the optimized pulse
are presented. The time-dependent noise level is the same as in Fig. 3. (a),
(d), (g): the transfer efficiencies (including the net effect of all pulses)
with static noise on polarizations, Doppler detuning, magnetic field
deviation, and amplitude. (b), (e), (h): the mean phase difference (rad)
between two arms in the interferometer. (c), (f), (i): the standard deviation
of the phase difference (rad) with random noise (half percent RMS in the
amplitude and one percent RMS in the phase). Figure 7: The impact of different
noise channels on interference contrast in a full interferometer. The time-
dependent noise level is the same as in Figs. 3 and 6. In each panel, the
vertical axis is the interference contrast and the horizontal axis is the
width $w$ of the zero-centered Gaussian probability distribution from which
the corresponding noise parameter $x_{i}$ is sampled. An example of a
simulated interference pattern is shown (at the upper-right corner, below the
legend) to illustrate the contrast calculation. In contrast to other pulses,
the optimized pulse maintains a high degree of robustness against all noise
channels, and the optimized pulse outperforms all other pulses in robustness
against polarization noise.

###  III.2 Full interferometer

The optimized pulse has been tested in a full atom interferometer simulation
(Fig. 5(a)) with total 2000 pulses (each arm experiences 1000 resonant pulses
and 1000 off-resonant pulses) as a final feasibility test. In the simulation,
we calculate the finite time step unitary evolution of the states using

$\displaystyle u^{(j)}=e^{-i\hat{H}(t_{j})\Delta t_{j}/\hbar},$ (4)
$\displaystyle\Psi_{f}=u^{(M)}...u^{(2)}u^{(1)}\Psi_{i},$ (5)

where $\Psi_{i}$ and $\Psi_{f}$ are initial and final states. In the finite
time approximation, the Hamiltonian $\hat{H}(t_{j})$ keeps the same from
$t_{j}$ to $t_{j}+\Delta t_{j}$, so that the evolution operator for the finite
time $\Delta t_{j}$ is $u^{(j)}$. We calculate the full evolution by applying
a chain of finite time operators (Eq. (5)). The optimized pulse is divided
into $M=2500$ segments (step time is 0.8 $\mu$s, further time step decrease
does not significantly influence the results). Since the switching on and off
are faster than pulse variations, 50 segments are specially assigned on each
switching process to assure their smoothness (see Fig. 1 (c)). In our LMT
scheme for clock interferometry [18] (Fig. 5(a)), we utilize 20 primitive
pulses for initial momentum splitting, 480 optimized pulses for continuing the
atoms’ acceleration, 480 optimized pulses for deceleration, and 20 primitive
pulses for symmetrical deceleration. Then, the laser is tuned to be resonant
with the other arm, and the same process is repeated on the other arm. During
the LMT process, pulses are applied from alternating directions [18]. The arms
are recombined at the end, and we measure the total transfer efficiency and
the phase difference between the two arms. The reason for including the
primitive pulses is that, in creating the full interferometer, the optimized
pulses have unwanted off-resonant interaction with the untransferred arm when
the relative Doppler shift between the two arms is small. This effect can be
reduced by using primitive pulses with appropriate Rabi frequency at small
momentum separation. As the relative velocity between the arms increases, the
increasing relative Doppler shift causes the pulses to become further detuned
from the other arm, and the deleterious effect on the untransferred arm
decreases. At this point, we switch to the optimized pulse to boost the
momentum splitting to large values.

The time-dependent noise composed from Eq. (3) varies in each pulse in one
simulation run. We keep it the same for different simulation runs within a
single static noise channel scan, in which we vary two static noise parameters
while keeping others the same (Fig. 6). We carry out repeated static noise
channel scans, each with randomly generated time-dependent noise, to evaluate
the phase difference between the interferometer arms. Typical applications of
clock interferometry involve differential measurements between two atom
interferometers in a gradiometer configuration using a common laser (see Fig.
5(b)) [18, 32]. Therefore, in each scan, a differential phase is calculated by
subtracting out the phase of an interferometer with zero static noise (i.e.
set the center of the map as zero). The mean value and the standard deviation
of the differential phase are computed for each static noise point. If two
interferometers in a gradiometer experience the same static noise, the
influence of the time-dependent noise on the phase will completely cancel as a
common mode [18]. By comparing different static noise points, one can
determine the residual differential phase noise if the two interferometers in
the differential measurement are not matched perfectly in terms of
polarization, amplitude, detuning, and magnetic field.

To demonstrate the advantage of the optimized pulse, following a similar
approach to other papers studying the application of quantum optimal control
to atom interferometry in different contexts [79, 83], the contrast of the
interference pattern after the full interferometer (Fig. 7) is compared with
primitive and other composite pulses. The interference pattern is a
probability summation of the single-atom interference patterns from an
ensemble of atoms. Each atom in the ensemble experienced different static
noise according to a particular distribution. For a certain noise channel, we
use a Monte Carlo method to construct the interference pattern $F(\Phi)$ by

$\displaystyle
f_{i}(\Phi)=\frac{1}{2}+\frac{1}{2}T(x_{i})\cos\left(\Phi+\Delta\phi(x_{i})\right),$
(6) $\displaystyle F(\Phi)=\frac{1}{N}\sum^{N}_{i=1}f_{i}(\Phi),$ (7)

where $\Phi$ is in a range bigger than $2\pi$ to show the interference pattern
contrast. In a single-atom interference pattern $f_{i}(\Phi)$, we assign a
Gaussian probability distribution whose width is $w$ and pick noise parameter
$x_{i}$ randomly according to the distribution. The transfer efficiency
$T(x_{i})$ is the probability of atoms being in the correct state to
contribute to the contrast (for example, Fig. 6 (a), (d), (g)). The phase
deviation $\Delta\phi(x_{i})$ is a value sampled from a Gaussian distribution
whose mean value and standard deviation are determined by the noise parameter
$x_{i}$ (for example, Fig. 6 (b), (e), (h) for mean value, and (c), (f), (i)
for standard deviation). The sampling number $N=50000$ is big enough to give
small statistical fluctuation in determining the contrast. The normalized
interference pattern $F(\Phi)$ (an example is shown in Fig. 7) is the
probability summation of a large number of $f_{i}(\Phi)$. The contrast we
measure is

$C=F_{max}-F_{min}.$ (8)

where $F_{max}$ and $F_{min}$ are the maximum and the minimum values of
$F(\Phi)$. The contrast value $C$ and its change with the width $w$ for every
static noise channel are presented in Fig. 7.

##  IV Conclusion

In summary, we composed an optimized pulse for ${}^{87}\rm{Sr}$ clock
interferometry to improve the robustness against multiple noise channels using
the QOC method. This work demonstrates the promise of quantum optimal control
for extending the scientific reach of strontium clock atom interferometers,
potentially paving the way for these interferometers to detect gravitational
waves at currently unexplored frequencies and wavelike dark matter. We will
study the experimental implementation of the optimized pulses in the future.
To achieve the best performance, it may prove valuable to tailor the noise
model used in the optimization algorithm to specific, experimentally measured
noise properties and spectra. The application of closed-loop quantum optimal
control [92], in which experimental measurements of pulse fidelities guide the
optimization process, to atom interferometry has the potential to offer
further improvements.

###### Acknowledgements.

We thank Jens Koch, Yunwei Lu, and Q-CTRL for valuable discussions. This
material is based upon work supported by the U.S. Department of Energy, Office
of Science, National Quantum Information Science Research Centers,
Superconducting Quantum Materials and Systems Center (SQMS) under contract
number DE-AC02-07CH11359.

##  Appendix A System Hamiltonian

We break the Hamiltonian in a rotating frame into two constant terms $H_{B}$
(Zeeman shifts) and $H_{D}$ (Doppler shifts) corresponding to free evolution,
and a time-dependent control term $H_{c}$:

$\displaystyle
H(\bm{\beta})=H_{B}(\beta_{B})+H_{D}(\beta_{v})+H_{c}(\epsilon_{+},\epsilon_{-},\beta_{A}),$
(9)

where $\beta_{v}$ changes the Doppler shift, $\beta_{B}$ is the fractional
change of the magnetic field, $\beta_{A}$ is a complex coefficient that
characterizes the deviation of the amplitude and the phase of the actual
control drive from those of the ideal control drive, and $\epsilon_{\pm}$ are
fractional amplitudes of $\sigma_{\pm}$ polarization terms.

The Hamiltonian term denoted as $H_{B}$ can be mathematically divided into two
distinct components. The first component is responsible for determining the
energy splitting of the ${}^{3}P_{0}$ and ${}^{1}S_{0}$ hyperfine levels,
assuming that the laser is resonant with the transition from
$|{}^{1}S_{0};9/2\rangle$ to $|{}^{3}P_{0};9/2\rangle$. This resonance
condition ensures that the energies of these two levels are equal in the
rotating frame and are therefore set to zero in the Hamiltonian. The energy
ladder of the sublevels can be expressed as follows:

$\left\langle m_{F};{}^{1}{\rm S}_{0}\left|H_{B1}\right|{}^{1}{\rm
S}_{0};m_{F}\right\rangle-B\hbar\mu_{0}g_{S}\left(\frac{9}{2}-m_{F}\right),$
(10)

$\left\langle m_{F};{}^{3}{\rm P}_{0}\left|H_{B1}\right|{}^{3}{\rm
P}_{0};m_{F}\right\rangle=-B\hbar\mu_{0}g_{P}\left(\frac{9}{2}-m_{F}\right),$
(11)

where $B=B_{0}+\Delta B=(1+\beta_{B})B_{0}$ is the total magnetic field,
$m_{F}$ is the magnetic quantum number, $\hbar$ is the reduced Planck
constant, $\mu_{0}$ is the vacuum permeability, and $g_{S}$, $g_{P}$ are the
Landé g-factors for the ${}^{1}S_{0}$ and ${}^{3}P_{0}$ states, respectively.
To fully evaluate robustness against magnetic field changes $\Delta B$, we
consider the impact of such a magnetic field change for a fixed laser
frequency. As a result of the associated Zeeman shift, the magnetic field
change causes the laser to become slightly off-resonant by inducing additional
energy shifts of $\Delta Bg_{S}\mu_{0}9/2$ and $\Delta Bg_{P}\mu_{0}9/2$ for
the $|{}^{1}S_{0};9/2\rangle$ and $|{}^{3}P_{0};9/2\rangle$ energy levels,
respectively. Consequently, it becomes necessary to introduce a second
component in the Hamiltonian to account for these energy shifts. To maintain
clarity, the energy shift is symmetrically divided between the ground and
excited states, as only the total energy difference $\Delta E=\Delta
Bg_{P}\mu_{0}9/2-\Delta Bg_{S}\mu_{0}9/2\equiv\beta_{B}B_{0}\hbar\omega_{B}$
between the two levels affects the evolution results. Therefore, the complete
Hamiltonian term $H_{B}$ can be written as

$\begin{split}\left\langle m_{F};{}^{1}{\rm S}_{0}\left|H_{B}\right|{}^{1}{\rm
S}_{0};m_{F}\right\rangle=-\beta_{B}B_{0}\frac{\hbar\omega_{B}}{2}\\\
-\left(1+\beta_{B}\right)B_{0}\hbar\mu_{0}g_{S}\left(\frac{9}{2}-m_{F}\right),\end{split}$
(12)

$\begin{split}\left\langle m_{F};{}^{3}{\rm P}_{0}\left|H_{B}\right|{}^{3}{\rm
P}_{0};m_{F}\right\rangle=\beta_{B}B_{0}\frac{\hbar\omega_{B}}{2}\\\
-\left(1+\beta_{B}\right)B_{0}\hbar\mu_{0}g_{P}\left(\frac{9}{2}-m_{F}\right),\end{split}$
(13)

where $B_{0}=1$ G is the nominal bias field and $\omega_{B}=2\pi\times 491$
Hz/G. In the second term $\mu_{0}g_{S}=2\pi\times 182$ Hz/G, and
$\mu_{0}g_{P}=2\pi\times 291$ Hz/G [93]. This term gives the energy ladder of
all sublevels with a fractional field change of $\beta_{B}$.

Similarly, the $H_{D}$ matrix elements are given by

$\left\langle m_{F};{}^{1}{\rm S}_{0}\left|\hat{H}_{D}\right|{}^{1}{\rm
S}_{0};m_{F}\right\rangle=-\beta_{v}\hbar\omega_{D}/2,$ (14)

$\left\langle m_{F};{}^{3}{\rm P}_{0}\left|\hat{H}_{D}\right|{}^{3}{\rm
P}_{0};m_{F}\right\rangle=\beta_{v}\hbar\omega_{D}/2,$ (15)

where $\omega_{D}=2\pi\times 100$ $\rm{Hz}$ is chosen as a factorization so
that $\beta_{v}$ is in units of 100 Hz in all the figures. This term accounts
for the energy shift due to the Doppler effect. The sign of $H_{D}$ term will
flip in the rotating frame when the laser pulse direction switches.

For the laser control Hamiltonian term $H_{C}$, the matrix elements are

$\left\langle m_{F};{}^{3}{\rm P}_{0}\left|\hat{H}_{C}\right|{}^{1}{\rm
S}_{0};m_{F}\right\rangle=\left(1+\beta_{A}\right)\frac{C^{\pi}_{m_{F}}}{C^{\pi}_{9/2}}\frac{\hbar\Omega(t)\sqrt{1-{\epsilon}^{2}}}{2},$
(16)

$\left\langle m_{F}+1;{}^{3}{\rm P}_{0}\left|\hat{H}_{C}\right|{}^{1}{\rm
S}_{0};m_{F}\right\rangle=\left(1+\beta_{A}\right)\frac{C^{+}_{m_{F}}}{C^{\pi}_{9/2}}\frac{\hbar\Omega(t)\epsilon_{+}}{2},$
(17)

$\left\langle m_{F}-1;{}^{3}{\rm P}_{0}\left|\hat{H}_{C}\right|{}^{1}{\rm
S}_{0};m_{F}\right\rangle=\left(1+\beta_{A}\right)\frac{C^{-}_{m_{F}}}{C^{\pi}_{9/2}}\frac{\hbar\Omega(t)\epsilon_{-}}{2},$
(18)

where ratios $C^{\pi}_{m_{F}}$, $C^{+}_{m_{F}}$, $C^{-}_{m_{F}}$ are Clebsch-
Gordon coefficients accounting for the different transition strengths which
are derived from the electric dipole matrix element between the ${}^{1}{\rm
S}_{0}$ and ${}^{3}{\rm P}_{0}$ states. These elements arise from an admixture
between ${}^{1}{\rm P}_{1}$ and ${}^{3}{\rm P}_{0}$ states due to spin-orbit
coupling and hyperfine interactions (the dipole matrix elements
$\left\langle{}^{3}{\rm P}_{0}\left|\hat{H}_{C}\right|{}^{1}{\rm
S}_{0}\right\rangle$ are proportional to $\left\langle{}^{1}{\rm
P}_{1}\left|\hat{H}_{C}\right|{}^{1}{\rm S}_{0}\right\rangle$) [93]. The
calculation of such electric dipole moments is described, for example, in
Metcalf and Van der Straten’s book [94]. $\beta_{A}(t)$ is composed of a
constant deviation $\beta_{A0}$ and white noises on both amplitude and phase
(see Eq. (3)). $\Omega(t)$ is the Rabi frequency which varies with the control
pulse amplitude, and its peak value is $2\pi\times 3\times 10^{3}$ Hz.
$\epsilon^{2}=|\epsilon_{-}|^{2}+|\epsilon_{+}|^{2}$ conserves the total
amplitude. All other matrix elements not explicitly listed are zero.

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