# Tractor Atom Interferometry

A. Duspayev<EMAIL_ADDRESS>Department of Physics, University of Michigan,
Ann Arbor, MI 48109, USA  G. Raithel  Department of Physics, University of
Michigan, Ann Arbor, MI 48109, USA

###### Abstract

We propose a tractor atom interferometer (TAI) based on three-dimensional (3D)
confinement and transport of split atomic wavefunction components in potential
wells that follow programmed paths. The paths are programmed to split and
recombine atomic wavefunctions at well-defined space-time points, guaranteeing
closure of the interferometer. Uninterrupted 3D confinement of the interfering
wavefunction components in the tractor wells eliminates coherence loss due to
wavepacket dispersion. Using Crank-Nicolson simulation of the time-dependent
Schrödinger equation, we compute the quantum evolution of scalar and spinor
wavefunctions in several TAI sample scenarios. The interferometric phases
extracted from the wavefunctions allow us to quantify gravimeter sensitivity,
for the TAI scenarios studied. We show that spinor-TAI supports matter-wave
beam splitters that are more robust against non-adiabatic effects than their
scalar-TAI counterparts. We confirm the validity of semiclassical path-
integral phases taken along the programmed paths of the TAI. Aspects for
future experimental realizations of TAI are discussed.

##  I Introduction

Since their first demonstrations Carnal and Mlynek (1991); Keith  _et al._
(1991); Riehle  _et al._ (1991); Kasevich and Chu (1991), atom interferometers
(AI) Cronin  _et al._ (2009); Abend  _et al._ (2020) have become a powerful
tool with a broad range of applications in tests of fundamental physics
Tarallo  _et al._ (2014); Schlippert  _et al._ (2014); Kovachy  _et al._
(2015); Jaffe  _et al._ (2017); Rosi  _et al._ (2017), precision measurements
Fixler  _et al._ (2007); Hanneke  _et al._ (2008); Parker  _et al._ (2018);
Morel  _et al._ (2020); Xu  _et al._ (2019); Barrett  _et al._ (2019) and
applied sciences Ménoret  _et al._ (2018); Bongs  _et al._ (2019); Garrido
Alzar (2019). A challenge in AI design is to achieve a high degree of
sensitivity with respect to the measured quantity (e.g., an acceleration)
while minimizing geometrical footprint of the apparatus and maximizing readout
bandwidth to allow for practical applications. Previous work on AI includes
free-space Asenbaum  _et al._ (2017); Rosi  _et al._ (2014); Hamilton  _et
al._ (2015) and point-source Dickerson  _et al._ (2013); Hoth  _et al._ (2016)
AI experiments, as well as guided-wave AI experiments Wu  _et al._ (2007);
Moan  _et al._ (2020) and proposals Davis and Narducci (2008); Zimmermann  _et
al._ (2019). Free-space and point-source AIs typically employ atomic fountains
or dropped atom clouds. The point-source method supports efficient readout and
data reduction Chen  _et al._ (2020), enables high bandwidth, and affords
efficiency in the partial-fringe regime. Atomic fountains typically employed
in free-space AI maximize interferometric time and, hence, increase
sensitivity Asenbaum  _et al._ (2017); Rosi  _et al._ (2014); Hamilton  _et
al._ (2015), but require large experimental setups. Guided-wave AIs offer
compactness and are often used as Sagnac rotation sensors, but are susceptible
to noise in the guiding potentials. In both free-space and guided-wave AI,
wavepacket dynamics along unconfined degrees of freedom can cause wave-packet
dispersion and failure to close, i.e. the split wavepackets may fail to
recombine in space-time. Coherent recombination of split atomic wavefunctions
upon their preparation and time-evolution remains challenging in recent AI
studies Stickney and Zozulya (2002, 2003); Burke  _et al._ (2008); Stickney
_et al._ (2008).

Here we propose and analyze an AI method in which there are no unconfined
degrees of freedom of the center-of-mass (COM) motion. The method relies on
confining, splitting, transporting and re-combining atomic COM quantum states
in three-dimensional (3D) quantum wells that move along user-programmed paths.
We refer to this approach as “tractor atom interferometer” (TAI). Proper
tractor path control ensures closure of the interferometer, and tight 3D
confinement at all times during the AI loop suppresses coherence loss due to
wavepacket dispersion.

FIG. 1: (Color online) Tractor control functions, $\pm z_{0}(t)$ (solid
lines), and tractor paths, $z^{I/II}(t)$ (dashed lines), for a scalar TAI. The
gray-shaded region shows the AI area. The insets (A)-(E) show how the trapping
potential $U_{1}(z,t)$ and the scalar wavefunction evolve in time. Note the
different $z$-ranges in the insets. The final state is a superposition of the
ground ($|\psi_{\nu 0}\rangle$) and first excited ($|\psi_{\nu 1}\rangle$) COM
states in the recombined well, as seen in inset (E), with the population ratio
revealing the interferometric phase $\Delta\phi_{Q}$.

##  II Quantum model

The quantum state of a single atom in the COM and spin product state space is

$|\psi\rangle=\sum_{i=1}^{i_{\rm{max}}}|\psi_{i}(t)\rangle\otimes|i\rangle\quad,$
(1)

with COM components $|\psi_{i}(t)\rangle$ in a number of spin states
$i_{\rm{max}}$. For simplicity, we assume that all elements of the spin-state
basis $\\{|i\rangle\\}$ are position- and time-independent, and that the $x$\-
and $y$-degrees of freedom of the COM are frozen out. Denoting
$\psi_{i}(z,t)=\langle z|\psi_{i}(t)\rangle$, the time-dependent Schrödinger
equation becomes

$\displaystyle{\rm{i}}\hbar\frac{\partial}{\partial t}\psi_{i}(z,t)$
$\displaystyle=$
$\displaystyle-\Big{[}\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial
z^{2}}+U_{i}(z,t)\Big{]}\psi_{i}(z,t)$
$\displaystyle+\sum_{j=1}^{i_{\rm{max}}}\frac{\hbar\Omega_{ij}(z,t)}{2}\psi_{j}(z,t)\quad,$
(2)

with $i=1,...,i_{\rm{max}}$, particle mass $m$, COM potentials $U_{i}(z,t)$
that may depend on spin, and couplings $\Omega_{ij}(z,t)$ between the spin
states.

In our examples below, we consider a scalar case, in which $i_{\rm{max}}=1$,
and a spinor case with $i_{\rm{max}}=2$. In the scalar case, the tractor-traps
of TAI are all contained in a single potential $U_{1}(z,t)$ for a scalar
wavefunction $\psi_{1}(z,t)$ (and there are no couplings $\Omega_{ij}$). In
the spinor case, the spin space can be viewed as that of a spin-1/2 particle
with spin states $\\{|\uparrow\rangle,|\downarrow\rangle\\}$. The spin states
could, for instance, represent two magnetic sublevels of the $F=1$ and $F=2$
hyperfine ground states of 87Rb. In spinor TAI, the spin states have distinct
potentials, $U_{\uparrow}(z,t)$ and $U_{\downarrow}(z,t)$, with spin-specific
potential wells, and the spinor wavefunction components are coupled via
$\Omega_{\downarrow\uparrow}=\Omega^{*}_{\uparrow\downarrow}$.

We numerically solve Eq. 2 using the Crank-Nicolson (CN) method Koonin and
Meredith (1990). We use 87Rb atoms in wells $\sim 20~{}\mu$m wide and $\sim
h\times 1$ MHz deep. We employ a time step size of $\Delta t=$ 10 ns, and a
spatial grid step size of $\Delta z=$ 10 nm. For the spinor simulations, we
have generalized our CN algorithm to cover problems with $i_{\rm{max}}>1$.

##  III Scalar TAI

In our scalar TAI implementation, the scalar potential $U_{1}(z,t)$ is the sum
of two identically-shaped Gaussian potential wells that are both $h\times 500$
kHz deep and have a full-width-at-half-depth of 23.5 $\mu$m. The two Gaussians
are centered at positions that are chosen to be symmetric in $z$ and are given
by programmed tractor control functions $\pm z_{0}(t)$ (red solid lines in
Fig. 1). Initially, they are co-located at $z_{0}=0~{}\mu$m, forming a single
trap that is $h\times 1$ MHz-deep. A 87Rb atom is initialized in its COM
ground state, $|\psi_{\nu 0}\rangle$, of the $h\times 1$ MHz-deep well (inset
(A) in Fig. 1). The function $z_{0}(t)$ is then gradually ramped up in order
to split the single initial well into a pair of symmetric wells, causing the
wavefunction to coherently split into two components (inset (B) in Fig. 1).
For $|z_{0}|\gtrsim 10~{}\mu$m, the split wells are about $h\times 500$ kHz
deep. The minima in $U_{1}(z,t)$ follow the paths $z^{I}(t)$ and $z^{II}(t)$
shown by dashed blue lines in Fig. 1, with $z^{I}(t)=-z^{II}(t)$. The paths
$z^{I/II}(t)$ are found by solving $(\partial/\partial z)U_{1}(z,t)=0$. After
the splitting, tunneling-induced coupling of the wavefunction components
ceases, and the components adiabatically follow the separated paths of the
potential minima. In our example, the wavefunction components stay separated
for about 1 s, with the separation held constant at a maximum of
$2z_{0}=100~{}\mu$m for a duration of 0.2 s (inset (C) in Fig. 1). The wells
and wavefunction components in them are recombined in a fashion that mirrors
the separation (insets (D) and (E)). In order to provide a sufficient degree
of adiabaticity, the ramp speed of the tractor control functions,
$|\dot{z}_{0}(t)|$, is reduced near the times when the wavefunction splits and
recombines. The total duration of the cycle is 2.2 s, which is in line with
the typical operation of modern AIs Xu  _et al._ (2019).

In scalar TAI, the tractor paths, $z^{I/II}(t)$, differ significantly from the
tractor control functions, $\pm z_{0}(t)$, during the well separation and
recombination phases, while they are essentially the same when the minima are
separated by more than the width of the wells (compare red-solid and blue-
dashed lines in Fig. 1). The AI area, visualized in Fig. 1 by a gray shading,
is the area enclosed by the paths $z^{I}(t)$ and $z^{II}(t)$.

AI closure is guaranteed by virtue of proper tractor control. This is evident
in the simulated wavefunction plots included in Fig. 1. The quantum AI phase,
$\Delta\phi_{Q}$, accumulates in the phases of the complex coefficients of the
COM ground states in the split wells at positive and negative $z$, denoted
$|\psi_{\nu 0,\pm}\rangle$ (inset (C) of Fig. 1). Upon recombination of the
pair of wells into a single well (insets (D) and (E) of Fig. 1), the atomic
state becomes mapped into a coherent superposition of the lowest and first-
excited quantum states of the combined well, $|\psi\rangle=c_{\nu 0}|\psi_{\nu
0}\rangle+c_{\nu 1}|\psi_{\nu 1}\rangle$ (inset (E) of Fig. 1). The observable
probabilities, $|c_{\nu 0}|^{2}$ and $|c_{\nu 1}|^{2}$, yield the TAI quantum
phase, $\Delta\phi_{Q}$, via the relation

$\frac{|c_{\nu 1}|^{2}}{|c_{\nu 0}|^{2}+|c_{\nu
1}|^{2}}=\sin^{2}(\frac{\Delta\phi_{Q}}{2}),\quad.$ (3)

Due to uninterrupted 3D confinement, TAI eliminates free-particle wave-packet
dispersion. There is, however, a possibility of non-adiabatic transitions into
excited COM states during the wavefunction splitting and recombination, which
would reduce the interferometer’s contrast and introduce spurious signals. To
show that under conditions such as in Fig. 1 there is no significant coherence
loss due to non-perfect closure or non-adiabatic effects, we have performed
wavefunction simulations for 17 values of a small acceleration, $a$, along the
$z$-direction. The acceleration adds a potential $U_{g}=m\,a\,z$ to the TAI
potential $U_{1}(z,t)$. We vary $a$ over a range from 0 to 10 mGal ($10^{-4}$
m/s2). From the wavefunction simulations, we determine
$\sin^{2}(\Delta\phi_{Q}/2)$ according to Eq. 3 as a function of $a$, and plot
the results in Fig. 2 (a) (symbols). The visibility of the expected sinusoidal
dependence reaches near-unity, providing evidence for near-perfect closure and
absence of coherence loss due to non-adiabatic COM excitations. The
acceleration sensitivity is discussed in Sec. VI.

FIG. 2: (Color online) Simulated measurement of
$\sin^{2}(\frac{\Delta\phi_{Q}}{2})$ vs a small background acceleration, $a$,
using TAI with scalar (a) and spinor (b) wavefunctions. Squares and dots in
(a) and (b) are from wavefunction simulation data, while solid lines are
semiclassical path-integral data. No fitting is performed.

##  IV Spinor TAI

The scalar scheme in Sec. III serves well to describe the TAI concept. At the
splitting, the initial COM state is supposed to evolve into the even-parity
superposition of the ground states in the split wells, $(|\psi_{\nu
0,+}\rangle+|\psi_{\nu 0,-}\rangle)/\sqrt{2}$, without populating the odd-
parity superposition and other excited COM states. However, under conditions
that are less ideal than in Sec. III, scalar TAI is prone to non-adiabatic
excitation of unwanted COM states at the times when the wells split and
recombine. The splitting and, similarly, the recombination are fragile because
the potential is very soft at the splitting and recombination times, and non-
adiabatic mixing can easily occur (see Sec. VI).

FIG. 3: (Color online) Example for spinor TAI. The tractor control functions,
$z_{0,\downarrow}(t)$ and $z_{0,\uparrow}(t)$, are identical with the
respective paths, $z^{II}(t)$ and $z^{I}(t)$. The interferometric area is
shaded in gray. The insets (A), (C), (E) show spin-dependent tractor
potentials and spinor wavefunctions at different instants of the evolution
time. Note the different $z-$ranges in these insets. The Bloch spheres in
insets (B) and (D) visualize the $\pm\pi/2$-splitting and recombination pulses
of spinor TAI. The final-state populations in the spin states reveal the
interferometric phase $\Delta\phi_{Q}$.

The fragility of scalar TAI is avoided in our second, improved method that
operates on a two-component spinor system ($i_{\rm{max}}=2$ in Eqs. 1 and 2)
with a pair of spin-dependent potentials. The atomic wavefunction is initially
prepared in the COM ground state $|\psi_{\nu 0,\downarrow}\rangle$ of a spin-
down tractor potential (inset (A) in Fig. 3). With initially overlapping and
identical spin-down and -up tractor potentials, $U_{\downarrow}(z,t=0)$ and
$U_{\uparrow}(z,t=0)$, a short $\pi/2$ coupling pulse with Rabi frequency
$\Omega_{\downarrow\uparrow}$ (see Eq. 2) prepares a coherent superposition
$(|\psi_{\nu 0,\downarrow}\rangle+|\psi_{\nu 0,\uparrow}\rangle)/\sqrt{2}$ of
the COM ground states of the potentials $U_{\downarrow}(z,t=0)$ and
$U_{\uparrow}(z,t=0)$. The $\pi/2$-pulse can be realized by a microwave or a
momentum-transfer-free optical Raman transition. In the simulated case, the
Rabi frequency $\Omega_{\downarrow\uparrow}$ is position-independent, has a
fixed magnitude $\Omega_{\downarrow\uparrow}(t)=2\pi\times 178$ kHz, and is on
for a coupling-pulse duration of $\Delta t_{R}=1.4~{}\mu$s, which is short on
the time scale of the interferometer. The resultant TAI splitting is depicted
on the Bloch sphere in the inset (B) of Fig. 3. After the $\pi/2$-pulse, the
spin-dependent potentials and the spinor wavefunction components in them are
translated following symmetric tractor control functions
$z_{0,\uparrow}(t)=z_{0}(t)=-z_{0,\downarrow}(t)$. After a holding time of 0.2
s at a maximal separation of 100 $\mu$m (inset (C) in Fig. 3), the splitting
is reversed and the wavefunction components become overlapped again. Closure
occurs via an exit $-\pi/2$ pulse (inset (D) in Fig. 3). In the case of spinor
TAI, the tractor paths and control functions are identical,
$z^{I}(t)=z_{0,\downarrow}(t)$ and $z^{II}(t)=z_{0,\uparrow}(t)$. The AI area
is visualized by the gray shaded region in Fig. 3.

In the absence of non-adiabatic transitions into excited COM states in the
spin-dependent potentials, the final state is of the form
$|\psi\rangle=c_{\downarrow}|\psi_{\nu
0,\downarrow}\rangle+c_{\uparrow}|\psi_{\nu 0,\uparrow}\rangle$. The AI phase
$\Delta\phi_{Q}$ is encoded in the final populations in the two spin states
(inset (E) in Fig. 3) and follows

$\frac{|c_{\uparrow}|^{2}}{|c_{\downarrow}|^{2}+|c_{\uparrow}|^{2}}=\sin^{2}(\frac{\Delta\phi_{Q}}{2}),\quad.$
(4)

In experimental implementations, the $|c_{\downarrow}|^{2}$ and
$|c_{\uparrow}|^{2}$ can be measured, for instance, via state-dependent
fluorescence to yield $\sin^{2}(\frac{\Delta\phi_{Q}}{2})$.

Similar to the scalar case in Sec. III, we have performed wavefunction
simulations for a set of accelerations $a$ along the $z$-direction, which add
identical gravitational potentials $U_{g}=m\,a\,z$ to both spin-dependent
potentials. From the simulated spinor wavefunctions we extract $c_{\uparrow}$
and $c_{\downarrow}$, compute $\sin^{2}(\Delta\phi_{Q}/2)$ according to Eq. 4,
and plot the results in Fig. 2 (b) (symbols). The results again provide
evidence for near-perfect closure and absence of coherence loss due to non-
adiabatic COM transitions.

##  V Comparison of quantum and semiclassical phases

Using the path integral formalism, the semiclassical phase of an AI loop,
$\Delta\phi_{S}$, in 1D is Cronin  _et al._ (2009):

$\Delta\phi_{S}=\frac{\int_{t_{a}}^{t_{b}}[\mathcal{L}^{II}(z,\dot{z},t)-\mathcal{L}^{I}(z,\dot{z},t)]~{}dt}{\hbar},$
(5)

where $\Delta\phi$ is in rads, $\mathcal{L}^{II/I}$ are the Lagrange functions
on the paths $z^{II/I}(t)$ of the centroids of the split atomic wavefunction
components, and $t_{a}$ and $t_{b}$ are the splitting and recombination times.

A key feature that distinguishes TAI from other AIs is that the paths
$z^{II/I}(t)$ are pre-determined by the system controls (and therefore do not
have to be computed prior to using Eq. 5). Simultaneous arrival of the split
wavefunction components at the recombination point is achieved by proper
programming of the tractor paths.

The guaranteed closure of TAI in space-time is related to the fact that the
number of generalized Lagrangian coordinates in TAI is zero. Other AIs
typically have at least one generalized coordinate, along which the classical
motion is unconstrained and along which quantum wavepackets may disperse. The
AI can then, in principle, fail to achieve closure due to a difference in
classical propagation times along the AI paths between splitting and
recombination. A propagation time difference can be caused by uncontrollable
conditions, such as an erratic background acceleration. In TAI, closure is
guaranteed by virtue of uninterrupted 3D control of the interferometric paths
and suitable tractor programming.

In our examples we have considered tractor paths in which the kinetic energy
terms in $\mathcal{L}^{II/I}$ are equal, i.e.
$\dot{z}^{II}(t)=-\dot{z}^{I}(t)$, and we have added a gravitational potential
$U_{g}=maz$. In that case, Eq. 5 simplifies to

$\Delta\phi_{S}=\frac{\int_{t_{a}}^{t_{b}}[U_{g}(z^{I}(t))-U_{g}(z^{II}(t))]~{}dt}{\hbar}=\frac{m\,a\,C}{\hbar}.$
(6)

with a parameter

$C=\int_{t_{a}}^{t_{b}}[z^{I}(t)-z^{II}(t)]~{}dt\quad,$ (7)

that only depends on the programmed tractor paths $z^{I}(t)$ and $z^{II}(t)$.
Note there is no atom dynamics to be solved for. The $z^{I}(t)$ and
$z^{II}(t)$ are either identical with the tractor control functions
$z_{0,*}(t)$ themselves (spinor case), or they are found by solving an
equation of the type $(\partial/\partial z)U_{1}(z,t)=0$ (scalar case).

We compare the semiclassical phases $\Delta\phi_{S}$ with the quantum phases
$\Delta\phi_{Q}$ over a range of accelerations, $a$. The $\Delta\phi_{S}(a)$
that follow from Eqs. 5-7 after utilization of the appropriate tractor paths
$z^{I}(t)$ and $z^{II}(t)$ are shown in Figs. 2 (a) and (b) as solid lines. We
find in both cases that $\Delta\phi_{S}=\Delta\phi_{Q}$, with minor
discrepancies that are not visible in the figure. Quantum and semiclassical
phases are both offset-free, i.e. in our case of symmetric controls and for
$a=0$ it is $\Delta\phi_{S}=\Delta\phi_{Q}=0$. This indicates that the exact
quantum dynamics does not add an offset splitting and recombination phase.

##  VI Sensitivity, computation accuracy and non-adiabatic effects

The scalar and spinor implementations simulated in Secs. III and Sec. IV
exhibit similar sensitivities to the acceleration $a$. The sensitivities are
not the same because the cases happen to have slightly different AI areas
(shaded regions in Figs. 1 and 3). Assuming a phase resolution of $2\pi/100$,
the acceleration sensitivity of the sequences in Figs. 1 and 3 is on the order
of several hundreds of microGals ($\sim 10^{-7}g$), which is about a factor of
100 short of the level of modern gravimeters Ménoret  _et al._ (2018). TAI
could reach that level by a ten-fold increase of the interferometer time,
$T_{I}=t_{b}-t_{a}$, ([1 s, 1.2 s] in Fig. 1 and [0.5 s, 0.7 s] in Fig. 3) and
a ten-fold increase of the spatial separation between the tractor potential
wells.

We have checked that a reduction of the grid spacing $\Delta z$ in the
simulation does not noticeably affect the accuracy of the results, whereas a
reduction of the time step size $\Delta t$ does improve the agreement of
$\Delta\phi_{S}$ with $\Delta\phi_{Q}$. Therefore we attribute the minor
differences between quantum and semiclassical phases (too small to be seen in
Figs. 2 (a) and (b)) mostly to the finite step size, $\Delta t=10$ ns, in the
CN simulation. The step-size parameters chosen in our work reflect a trade-off
between accuracy of $\Delta\phi_{Q}$ and simulation time needed.

It remains to be seen in the future whether quantum and Lagrangian phases may
exhibit offsets of a physical nature. Such offsets may occur, for instance, if
the tractor control functions $z_{0,*}(t)$ were asymmetric, if the tractor-
trap paths $z^{I/II}(t)$ had accelerations large enough to cause wavepacket
drag in the 3D tractor wells, or if trajectory bundles of the semi-classical
AI traversed through caustics or focal spots that lead to quantum phase
shifts.

We have noted that scalar TAI generally is more susceptible to non-adiabatic
COM excitations in the splitters and recombiners than spinor TAI,
necessitating longer splitter- and recombiner durations with reduced slopes
$|\dot{z_{0}}|$ near the critical time points when the single well splits into
two and vice versa (see Fig. 1). This entails a longer overall AI sequence, a
reduced reading bandwidth, and additional susceptibilities to noise (such as
vibrations) during the splitting and recombination. These shortcomings are
naturally avoided in spinor TAI, where the tractor potentials do not soften at
the splitting and recombination times.

To quantify the non-adiabaticity in both TAI cases, we have run a series of
simulations of splitting sequences with smooth tractor control functions
$z_{0}(t)=50~{}\mu$m$\times\sin^{2}{[\pi t/(2T)]}$ (as in Fig. 3) for a range
of splitter durations $T$ and acceleration $a=0$. The non-adiabaticy is given
by $1-p_{0}(T)$, where $p_{0}$ is the COM ground-state probability after the
splitting. In Fig. 4 we plot $1-p_{0}(T)$ vs $T$ and the peak value,
$a_{S}=25\mu$m$\,(\pi/T)^{2}$, of the splitter acceleration $|\ddot{z}_{0}|$
for both TAI cases. The wavefunction densities in the inset visualize the
contrast between adiabatic (inset (A); no COM excitation) and non-adiabatic
splitting (inset (B); substantial COM excitation). The results underscore that
for scalar TAI it is crucial to reduce the slope $|\dot{z}_{0}|$ at the times
when the wells split and recombine. For the control-function type used in Fig.
4, spinor TAI allows for rapid splitting, $T\sim 10$ ms, while scalar TAI
requires splitter times $T\sim 1$ s.

FIG. 4: (Color online) Non-adiabaticity vs splitting duration $T$ (top axis)
and peak tractor acceleration $a_{S}$ (bottom axis) for a tractor control
function given in the text, for scalar (green squares) and spinor TAI (blue
circles). The dashed lines connecting the respective data points are to guide
the eye. It is seen that spinor TAI (blue circles) allows about a factor of
100 faster splitting than scalar TAI. The insets show representative
wavefunction densities after the splitting for a tractor acceleration in which
spinor-TAI performs very well while scalar TAI fails.

For Sagnac rotation interferometry Wu  _et al._ (2007); Moan  _et al._ (2020);
Dickerson  _et al._ (2013); Chen  _et al._ (2020), the tractor paths can be
programmed to circumscribe a non-zero geometric area $A$, and the paths can be
traversed $N$ times between splitting and recombination. For a sensitivity
estimate for the angular rotation rate $\Omega$, we assume TAI loop parameters
of $A=1$ cm2 and $N=300$, which seems feasible. For rubidium it then is
$\Delta\phi/\Omega\sim mA/\hbar\approx 4\times 10^{7}$ rad/(rad/s). Assuming a
phase resolution $\Delta\phi=2\pi/100$, the rotation sensitivity would be
$\sim$1 nrad/s.

##  VII Discussion

In this Section, we consider aspects regarding experimental implementations of
TAI. Useful methods will likely include elements of pioneering demonstrations
of AI with Bose-Einstein condensates Shin  _et al._ (2004), of proposals for
AI in magnetic microtraps Hänsel  _et al._ (2001a); Hinds  _et al._ (2001);
Hänsel  _et al._ (2001b), and of emerging techniques of moving (arrays of)
ultracold atoms on arbitrary trajectories Endres  _et al._ (2016); Barredo
_et al._ (2016); AU Kim  _et al._ (2016); Ohl de Mello  _et al._ (2019);
Schymik  _et al._ (2020). To reduce quantum projection noise, the utilized
method must allow for parallel operation of many identical loops in a small
overall volume Kovachy  _et al._ (2010). Compact, 3D-confining and
parallelizable platforms that allow dynamic tractor control include optical
lattices Mandel  _et al._ (2003); Kuhr  _et al._ (2001); Schmid  _et al._
(2006); Kumar  _et al._ (2018) and optical tweezers Endres  _et al._ (2016);
Barredo  _et al._ (2016); AU Kim  _et al._ (2016); Schymik  _et al._ (2020)
that may use an array of micro-lenses Ohl de Mello  _et al._ (2019).

TAI can fail if the acceleration to be measured is too large, causing splitter
asymmetry and exaggerated non-adiabatic effects. In Kovachy  _et al._ (2015),
$a$ needed to be $\lesssim 10^{3}$ Gals $\approx$1 $g$. With the quantum
confinement in 3D traps, afforded by TAI, the tractor traps can be designed
steep enough to avoid such effects, which can be an advantage in high-$g$
scenarios. As seen in Fig. 4, where traps of a depth of $h\times$500 kHz and
well sizes of 20 $\mu$m are used, spinor-TAI is especially promising in that
regard. Deeper and tighter traps in optical lattices with well sizes
$<1~{}\mu$m are expected to afford closure and robustness under high-$g$
conditions.

As in guided-wave AI, in TAI it is important to avoid differential
fluctuations in depth and position of the tractor potentials. In experimental
implementations, this will present a serious challenge. For instance, the
sensitivity to differential noise in the potential depths, $\delta\Delta
V=\delta(V^{I}-V^{II})$, with $V^{I}$ and $V^{II}$ denoting the depths of the
individual tractor wells, scales with the interferometer time
$T_{I}=t_{b}-t_{a}$, which is $\sim 1$ s in our examples. The requirement
$\delta\Delta VT_{I}<h$ then leads to a requirement of $\delta\Delta V<h\times
1$ Hz. For wells that are $h\times 100~{}$kHz deep, the allowed variation in
differential tractor depth therefore is in the range of $10^{-5}$ of the well
depth. This estimation becomes considerably more favorable for tractor wells
that are approximations of deep square wells with an absolute trapping
potential near zero inside the wells. One may envision, for instance, traps
formed by blue-detuned optical lattices or box potentials, in which the atoms
are trapped near locations of minimal light intensity.

Along similar lines, in sensing applications where common-mode fluctuations do
not affect $\Delta\phi$ Chiow  _et al._ (2009), TAI could be implemented with
tractor controls that act symmetrically in both TAI paths. The symmetry
requirement extends to several types of noise, including noise in the
differential trap depths of the tractor wells, $\delta\Delta V$, in the
differential potential energy of the tractor paths, $\delta\Delta
U=m{\bf{a}}\cdot\delta({\bf{r}}^{I}-{\bf{r}}^{II})$, in constant-acceleration
background potentials, and in the differential kinetic energy along the
tractor paths. The latter aspect translates into effects of differential
mirror vibrations, phases of optical-lattice laser beams, etc. Future
experimental work as well as detailed simulations that include various types
of noise will help in exploring the opportunities afforded by TAI as well as
in establishing its practical limitations.

In summary, our theoretical study and the discussion regarding experimental
feasibility show that TAI is a promising concept for future experimental
demonstration. TAI offers a set of values that we believe is hard to match
with free-space and partially-confining atom interferometers, including
guaranteed closure in space-time, even under rough conditions with large and
variable background accelerations and rotations, suppression of wavepacket
dispersion due to uninterrupted 3D confinement, and use of user-programmable
loops with arbitrary hold times and flexible geometries for a variety of
sensing applications.

## ACKNOWLEDGMENTS

The work was supported by NSF under Grant No. PHYS-1806809 and NASA under
Grant No. NNH13ZTT002N NRA. We thank Lu Ma and Vladimir Malinovsky for useful
discussions.

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