# Spinor matterwave control with nanosecond spin-dependent kicks

Liyang Qiu<EMAIL_ADDRESS>Lingjing Ji  Jiangyong Hu  Yizun He  Yuzhuo
Wang  Saijun Wu<EMAIL_ADDRESS>Department of Physics, State Key
Laboratory of Surface Physics and Key Laboratory of Micro and Nano Photonic
Structures (Ministry of Education), Fudan University, Shanghai 200433, China.

(today)

###### Abstract

Significant aspects of advanced quantum technology today rely on rapid control
of atomic matterwaves with hyperfine Raman transitions. Unfortunately,
efficient Raman excitations are usually accompanied by uncompensated dynamic
phases and coherent spin-leakages, preventing accurate and repetitive transfer
of recoil momentum to large samples. We provide systematic study to
demonstrate that the limitations can be substantially overcame by dynamically
programming an adiabatic pulse sequence. Experimentally, counter-propagating
frequency-chirped pulses are programmed on an optical delay line to parallelly
drive five $\Delta m=0$ hyperfine Raman transitions of 85Rb atoms for spin-
dependent kick (SDK) within $\tau=40$ nanoseconds, with an $f_{\rm SDK}\approx
97.6\%$ inferred fidelity. Aided by numerical modeling, we demonstrate that by
alternating the chirps of successive pulses in a balanced fashion,
accumulation of non-adiabatic errors including the spin-leakages can be
managed, while the dynamic phases can be robustly cancelled. Operating on a
phase-stable delay line, the method supports precise, fast, and flexible
control of spinor matterwave with efficient Raman excitations.

##  I Introduction

Precise control of effective 2-level systems is instrumental to modern quantum
technology. In atomic physics, such controllable two-level systems are
naturally defined on a pair of long-lived atomic internal states and are often
referred to as atomic spins. To control the external motion, quantized photon
recoil momentum can be transferred to the atom during spin-flips driven by
Raman excitations [1, 2, 3]. By laser cooling [4], the atomic motion can be
sufficiently slowed that the momentum transfers by the rapid optical pulses
are effectively instantaneous “spin-dependent kicks” (SDK) [5, 6], with the
excitation efficiency insensitive to the atomic motion. SDKs are therefore a
class of broadband spinor matterwave control techniques with an achievable
accuracy similar to those for 2-level internal atomic spin controls [7, 8, 9].
Beyond traditional applications such as to enhance the enclosed area of light
pulse atom interferometers [3, 10, 6], SDKs emerge as an important technique
to control spin-motion entanglement and to improve the scalability of ion-
based quantum information processing [11, 12, 5, 13].

For precise spinor matterwave control, it is essential to operate SDK at high
enough speed to suppress low-frequency noises and to compose multiple
operations within a limited duration. Furthermore, except for working with
microscopically confined samples [5, 13, 8], high quality optical control
needs to be highly resilient to intensity errors associated with illumination
inhomogeneity [14, 15]. To meet the fidelity requirements by the next
generation quantum technology [16, 17, 18, 19, 20, 21], particularly for large
samples, high speed implementation of error-resilient composite techniques
[22, 23, 24] are likely required. However, the SDK speed is practically
limited by intricate requirements associated with precise Raman control,
including the following two categories.

The first type of SDK speed constraints are associated with multi-level
dynamics accompanying the Raman control of “real” atoms. To avoid spontaneous
emission, the Raman excitation bandwidth has to be much smaller than the
single-photon detuning $\Delta_{e}$ (Fig. 1(a)) to avoid populating the
excited states. But a moderate $\Delta_{e}$ is typically required for
efficient control with limited laser power, as well as for managing the
differential Stark shifts (Fig. 1(b)) [25]. In addition, as to be clarified
further, to avoid the $m$-changing transitions during Raman excitation [26],
one usually lifts the Zeeman degeneracy with a strong enough quantization
field [27]. The typically moderate field strength also limits the speed of the
Raman control.

The second type of SDK speed constraints are associated with the requirement
of controllable direction of the recoil momentum transfer. As in Fig. 1, when
counter-propagating ${\bf E}_{1,2}$ pulses are applied to drive a Raman
transition, to ensure the preference for the atom to absorb a photon from the
${\bf E}_{1}$ field followed by a stimulated emission into the ${\bf E}_{2}$
field, certain mechanism is needed to prevent the time-reversed Raman process
from occurring. For hyperfine spin control of alkaline-like atoms, the
directionality is ultimately supported by the ground-state hyperfine splitting
$\omega_{{\rm hfs},g}$ [5]. Practically, however, when the Raman
interferometry is operated in the retro-reflection geometry with established
advantages [28, 29, 30, 31, 32, 33], the directionality is typically protected
by the moderate 2-photon frequency differences introduced either by additional
frequency modulations [3, 28] or Doppler shifts of moving atoms [29], which in
turn limit the SDK speed.

Figure 1: Schematic of SDK on a hyperfine manifold, with the 85Rb D1 line as
the example in this work. (a) Cross-linearly polarized, counter-propagating
${\bf E}_{1,2}$ resonantly drive the $\Delta m=0$ and $\Delta m=\pm 2$ Raman
transitions. The optical couplings are illustrated with solid and dashed
lines, with $m=\pm 1,\pm 3$ ground states as examples. When the single-photon
detuning $\Delta_{e}$ is much larger than the excited state hyperfine
splitting $\omega_{{\rm hfs},e}$, the $\Delta m=\pm 2$ couplings through the
$F^{\prime}=2,3$ intermediate states largely cancel each other, leading to
decoupled $\\{|a_{m}\rangle,|b_{m}\rangle\\}$ sub-spin systems. (b) The
reduced level diagrams for the weakly coupled
$\\{|a_{m}\rangle,|b_{m}\rangle\\}$ systems ($m=\pm 1$). The bare levels are
relatively shifted and Raman coupled. The effective Hamiltonian parameters
$\Omega^{\pm 2}$, $\Omega_{\rm R}^{(m)}$, $\delta_{0}$ for Eqs. (1)(2)(3) are
marked. The differential Stark shift $\delta_{0}$ is largely responsible for
the SDK dynamic phase (Eq. (5)). Similarly coupled sub-spin systems composed
of $\\{|a_{m}\rangle,|b_{m}\rangle\\}$ with $m=0,\pm 2$ are not shown.  Figure
2: (a) The schematic setup. Frequency-chirped optical waveforms (illustrated
with the real part of the complex envelop function $\mathcal{E}_{1,2}$,
sharing a same color-coding as those for ${\bf E}_{1,2}$ in Fig. 1(a)) are
programmed by an optical arbitrary waveform generator (OAWG) and sent through
a single mode fiber to the delay line. An ${\bf E}_{1}$ pulse collides with a
retro-reflected ${\bf E}_{2}$ pulse at the atomic sample to drive the
$|a_{m}\rangle\leftrightarrow|b_{m}\rangle$ Raman transition while imparting
$\pm\hbar\mathbf{k}_{\rm R}$ kicks to atoms reaching $|a_{m}\rangle$ and
$|b_{m}\rangle$ respectively. Next, after a delay time $\tau_{\rm d}$, the
${\bf E}_{1}$ pulse is retro-reflected to meet ${\bf E}_{2}^{\prime}$ at the
same location to further boost the $|a_{m}\rangle$, $|b_{m}\rangle$
separation. QWP: quarter-wave plate. (b) The timing sequence for the “balanced
chirp-alternating scheme” (Eq. (11)). The two-photon detunings $\delta_{\rm
R}$ are plotted vs time. The adiabatic controls with “up-chirp” and “down-
chirp” 2-photon detuning $\delta_{\rm R}$ are referred to as $U_{\rm u}$ and
$U_{\rm d}$ respectively. Here the amplitudes for the retro-reflected pulses
are reduced by a $\kappa$ factor. Typical Bloch sphere dynamics of an
$\\{|a_{m}\rangle,|b_{m}\rangle\\}$ spinor ($m=0$) are given in (c) at various
laser intensities parametrized by the Raman pulse area $\mathcal{A}_{\rm R}$.
The weak $\Delta m=\pm 2$ couplings (Fig. 1(b), Eq. (1)) are ignored.

The purpose of this work is to improve the speed and precision of spinor
matterwave control in presence of the above mentioned technical barriers. In
particular, we study Raman adiabatic SDK [10, 6] within nanoseconds in a
retro-reflection optical setup [34, 32], for alkaline atoms with Zeeman
degeneracy prone to coherent spin leakage. As schematically summarized in Fig.
1 (level diagram) and Fig. 2 (experimental implementation), counter-
propagating frequency-chirped pulses are programmed on the optical delay line
to adiabatically drive the $F_{b}=2\leftrightarrow F_{a}=3$ Raman transitions
of 85Rb through the D1 line, while imparting $\pm\hbar\mathbf{k}_{\rm R}$
momentum “kicks” to atoms. Here $\mathbf{k}_{\rm
R}=\mathbf{k}_{2}-\mathbf{k}_{1}$ is the difference of k-vectors between the
pulse pair. The adiabatic Raman transfer technique [35, 10, 6] (Fig. 2(b)(c))
provides the intensity-error resilience for the focused laser to address a
mesoscopic sample. To enforce the directionality in the standard retro-
reflection geometry [34, 32], the regular method of resolving the 2-photon
frequency differences [3, 28, 29] are abandoned. Instead, we exploit the fact
that the nanosecond pulses are short enough to be spatially separated (Fig.
2(a)), allowing controllable collision of specific Raman pulses within the
atomic sample, at specific instances, to drive the Raman transitions.

Experimentally, equipped with a wideband optical arbitrary waveform generation
system (OAWG) [36], we drive adiabatic SDKs within tens of nanoseconds, the
fastest realization to date [10, 6], by shaping counter-propagating pulses
(Fig. 2) with merely tens of milliwatts laser power. The spin population and
momentum transfer are measured and compared with full-level numerical
simulations, with which we infer an SDK fidelity of $f_{\rm SDK}\approx
97.6(3)\%$. The $\delta f\sim 2.5\%$ infidelity, as unveiled by the full-level
numerical simulation, is primarily limited by spontaneous emission and the
$\Delta m=\pm 2$ leakage among Zeeman sublevels within the ground state
hyperfine manifold (Fig. 1(b)) at the moderate $\Delta_{e}=2\pi\times 10$ GHz
detuning.

Aided by numerical modeling, we further demonstrate that beyond the incoherent
momentum and population transfer, high-fidelity spinor matterwave “phase
gates” capable of coherently transferring many recoil momenta can be
synthesized by nanosecond adiabatic SDKs on the delay line. This is despite
the coherent spin leakage and intensity-dependent diffraction phase broadening
[25, 37, 6, 38] by individual “kicks”, at the moderate $\Delta_{e}$ [39, 8].
In particular, we demonstrate that a tailored adiabatic sequence with
alternating up and down 2-photon chirps [40] (Fig. 2(b)(c)) suppresses the
coherent accumulation of spin leakages across the nearly degenerate hyperfine
levels dressed by the cross-linearly polarized pulses (Fig. 1(b)).
Furthermore, as to be clarified further, the time-reversal symmetry of the
chirp-balanced scheme ensures robust cancellation of the dynamic, laser
intensity-dependent diffraction phases, with $\Delta_{e}>\omega_{{\rm hfs},g}$
unbounded by traditional choices [25, 41], even in presence of non-perfect
retro-reflection (Fig. 2(b)) [32].

Our work suggests that fast, large-momentum atom optics can be synthesized for
accurately manipulating multi-level atoms in the unconventional regime of
Raman control [25, 41]. Comparing with atom interferometric large-momentum
beamsplitting techniques based on high-order Bragg diffraction [42, 43], Bloch
oscillations [44, 45], as well as traditional Raman schemes [3, 10, 6], the
key advantage offered by our SDK technique is the ultra-high SU(2) operation
speed. The high speed not only helps the composite technique to avoid the
complexity of continuous motion itself, but also protects the precise
intensity-error cancellation in presence of low-frequency noises, including
those due to atomic motion in inhomogenuous laser beams. We notice that
similar advantages are shared by the recently developed high speed clock atom
interferometry [46, 47], where the atomic spins are defined directly on kHz-
level optical lines, to achieve exceptionally high momentum transfer
efficiencies. Similar to that technique, to coherently drive the hyperfine
spins within nanoseconds requires high laser intensities. The nanosecond SDK
in this work is realized by weakly focusing the milli-Watt level pulses to a
mesoscopic sample. To address larger samples for e.g. inertial sensing [34,
43] on a compact delay line, more advanced laser modulation techniques need to
be developed for generating high power wideband optical waveforms.

In the following the main part of the paper is structured into three sections.
In Sec. II, we set up a light-atom interaction framework where multi-level
couplings are treated as perturbations to the effective 2-level Raman
interaction within the ground state hyperfine manifold. Key notations and
quantities for the experimental and numerical studies are defined. In Sec.
III, we demonstrate the flexible matterwave control by programming counter-
propagating pulses on an optical delay line with OAWG [36]. Constrained by
experimental resources, the characterization of the adiabatic SDK technique is
limited to the inference of the SDK fidelity with atomic velocity and
hyperfine population measurements. However, in Sec. IV, we demonstrate
efficient suppression of coherent spin-leakage accumulation and intensity-
dependent dynamic phases by the chirp-alternating scheme, and provide
numerical evidence that the chirp-balanced SDK scheme supports coherently
control of multi-spinor matterwave with large momentum transfer, even with a
moderate laser power in the 10 mW range as in this work.

##  II Theoretical Model

###  II.1 Spin-dependent kicks on a hyperfine manifold

We consider the Fig. 1 (a) alkaline-like atom subjected to Raman excitation on
the D1 line. Similar conclusions holds for D2 excitations. The Zeeman-
degenerate $a,b$ and $e$ hyperfine states with total angular momentum
$F_{a,b}$ and $F^{\prime}_{e}$ are labeled as
$\ket{c_{m}}\equiv\ket{c,F_{c},m}$ with magnetic quantum number
$m\in\quantity[-F_{c},F_{c}]$. Here, with the nuclear spin $I>1/2$, there are
$2F_{c}+1$ Zeeman sublevels for each manifold $c\in\quantity{a,b}$, and
similarly for the $e$ manifold. The
$|a_{m}\rangle\leftrightarrow|b_{n}\rangle$ hyperfine Raman transition is
driven by pairs of counter-propagating laser pulses,
${\mathbf{E}}_{1,2}={{\mathbf{e}}}_{1,2}\mathcal{E}_{1,2}{e}^{i\quantity(\mathbf{k}_{1,2}\cdot\mathbf{r}-\omega_{1,2}t)}+c.c.$
with shaped slowly-varying amplitudes $\mathcal{E}_{1,2}(\mathbf{r},t)$ and
Raman-resonant carrier frequency difference
$\omega_{2}-\omega_{1}=\omega_{{\rm hfs},g}$.

Although motion of the multi-level atom in the pulsed ${\bf E}_{1,2}$ fields
is quite complicated (Appendix A), the internal state dynamics can be
substantially simplified when the laser polarization ${\bf e}_{1,2}$ are
chosen to be cross-linear (Fig. 2(a)) [30, 48] and the single-photon detuning
$\Delta_{e}$ is much larger than the excited state hyperfine splitting
$\omega_{{\rm hfs},e}$. In this case, the destructive interference of Raman
couplings through the intermediate $|e_{l}\rangle$ levels prevent efficient
$\Delta m=\pm 2$ transitions (The rule in Fig. 1(a) is that efficient Raman
couplings are only composed by pairs of dipole couplings drawn with a same
type of solid or dashed lines.) [26]. The residual “leaking” Raman Rabi
frequency (Fig. 1(b)) scales as

$\Omega^{\pm 2}=\mathcal{O}\quantity(\frac{\omega_{\rm
hfs,e}}{\Delta_{e}})\Omega_{\rm R},$ (1)

with $\Omega_{\rm R}=\Omega^{1*}_{a}\Omega_{b}^{2}/2\Delta_{e}$,
$\Omega_{a(b)}^{1(2)}=\frac{\mathcal{E}_{1(2)}}{\hbar}\frac{1}{\sqrt{3}}\absolutevalue{\bra{J_{g}}\\!\absolutevalue{\mathbf{d}}\\!\ket{J_{e}}}$
[49].

Therefore, in the large $\Delta_{e}$ limit, the Raman dynamics in the ground-
state manifold is decomposed into those within $2F_{b}+1$ copies of pseudo
spin-$1/2$ sub-spaces $\quantity{\ket{a_{m}},\ket{b_{m}}}$, labeled by
magnetic quantum number $m$ (See Fig. 1(b) for the $m=\pm 1$ examples.). For
each $m$-spin, the Rabi frequency driven by the Raman pulse can be written as

$\Omega_{\rm R}^{(m)}=\chi^{(m)}\Omega_{\rm R}.$ (2)

The close-to-unity factor $\chi^{(m)}$ is determined by the associated Clebsh-
Gordan coefficients, normalized at $\chi^{(0)}=1$ and decreases slowly with
$|m|$ [50]. The Hamiltonian for the $m$-spin is given by

$H_{0}^{(m)}(\mathbf{r},t)=\hbar\quantity(\frac{\delta_{0}}{2}\sigma_{z}^{(m)}+\frac{\Omega_{\rm
R}^{(m)}}{2}e^{i\mathbf{k}_{\rm R}\cdot{\bf
r}}\sigma_{+}^{(m)}+\mathrm{h.c.}).$ (3)

Here the detuning $\delta_{0}$ is the ($m$-independent) differential Stark
shift between the two hyperfine ground states that can be nullified at
suitable $\Delta_{e}/\omega_{{\rm hfs},g}$ and $\Omega_{a,b}$ combinations
[25]. The Pauli matrices
$\\{\sigma_{x}^{(m)},\sigma_{y}^{(m)},\sigma_{z}^{(m)}\\}$ are defined by
$\sigma_{+}^{(m)}\equiv\sigma^{a_{m}b_{m}}$ [51]. For notation convenience in
the following, we further define projection operators
${\bm{1}}^{(m)}\equiv\sigma^{a_{m}a_{m}}+\sigma^{b_{m}b_{m}}$ and
$\bar{\bm{1}}^{(m)}=1-{\bm{1}}^{(m)}$ respectively.

With the major part of the resonant Raman interaction identified, the D1
multi-level dynamics (Fig. 1(a)) governed by the effective light-atom
interaction Hamiltonian, as detailed in Appendix A, can be written as

$H_{\rm
eff}(\mathbf{r},t)=\sum_{m=-F_{b}}^{F_{b}}H^{(m)}_{0}(\mathbf{r},t)+V^{\prime}(\mathbf{r},t)$
(4)

to describe the Raman-dressed ground state dynamics. Apart from the $\Delta
m=\pm 2$ couplings specified by Eq. (1), the $V^{\prime}$ term includes non-
Hermitian and non-adiabatic $|e_{l}\rangle$ couplings, the $m-$sensitive light
shifts [26], and the “counter-rotating” couplings involving detuned Raman
excitations (Eq. (17)).

With the atomic position ${\bf r}$ parameter in Eq. (4) as a quantum
mechanical operator acting on the external atomic wavefunction, the Eq. (4)
Hamiltonian can propagate the spinor matterwave together with the kinetic
$\hat{P}^{2}/2M$ operator, see Appendix B for implementation
($\hat{P}=i\hbar\nabla$ and $M$ the atomic mass.). During nanosecond
intervals, however, the matterwave dispersion for the laser-cooled atoms is
negligibly small. It is therefore possible to design broadband matterwave
controls for iterative applications. In particular, a spin-dependent kick is a
transfer of photon momentum to atoms accompanied by a spin-flip [5, 6]. We
refer an ideal SDK as

$U_{\rm K}(\mathbf{k}_{\rm
R})=\prod_{m=-F_{b}}^{F_{b}}\left({e}^{i\varphi^{+}}{e}^{i\mathbf{k}_{\rm
R}\cdot\mathbf{r}}\sigma_{+}^{(m)}-{\rm h.c.}+\bar{\bm{1}}^{(m)}\right).$ (5)

The $\varphi^{+}$ is a diffraction phase offset [52] which is generally Zeeman
state $m$\- and laser intensity $|\mathcal{E}|^{2}$-dependent.

In the following our goal is to construct perfect $U_{\rm K}(\mathbf{k}_{\rm
R})$ operation as in Eq. (5) from the full Hamiltonian in Eq. (4), and
furthermore to design multiple SDKs with suppressed $m$ and
$|\mathcal{E}|^{2}$ dependence. For the purpose, we first define a fidelity
$f_{\rm SDK}$ to qualify the implementation.

###  II.2 Non-ideal SDK

To qualify the Fig. 2 SDK implementation modelled by Eq. (4) Hamiltonian, we
refer the imperfect realization of SDK as $\tilde{U}(\mathbf{k}_{\rm R};\eta)$
with $\eta$ to generally represent relevant Hamiltonian parameters. We define
an average fidelity [53] over the whole atomic ensemble for a single SDK
acting on the $\\{|b_{m}\rangle\\}$,$\\{|a_{m}\rangle\\}$ states as

$f_{\rm SDK}=\left\langle\left|\left\langle\langle c_{m}|U_{\rm
K}^{\dagger}(\mathbf{k}_{\rm R})\tilde{U}(\mathbf{k}_{\rm
R};\eta)|c_{m}\rangle\right\rangle_{z}\right|^{2}\right\rangle_{\eta,c_{m}}.$
(6)

Here the $\tilde{U}({\bm{k}}_{\rm R};\eta)|c_{m}\rangle$ is compared to
$U_{\rm K}({\bm{k}}_{\rm R})|c_{m}\rangle$ across the atomic sample with an
$\langle...\rangle_{z}$ average. An ensemble average of the mode-squared
fidelity is then performed with $\langle...\rangle_{\eta}$ over the
Hamiltonian parameters of interest. Assuming light intensity hardly varies
along $z$ over the wavelength-scale distance of interest (Fig. 2(a)), the SDK
fidelity defined in this way becomes insensitive to the diffraction phase
$\varphi^{+}$ and therefore provides a convenient measure for the quality of
controlling incoherent observables, such as for the recoil momentum and
hyperfine population transfer to be experimentally measured next. The
$\langle...\rangle_{c_{m}}$ instead performs average over the $2(2F_{b}+1)$
initial states of interest with $m=-F_{b},...,F_{b}$ and $c=a,b$.

To quantify the leakage of an atomic state out of each spin sub-space
$\\{|a_{m}\rangle,|b_{m}\rangle\\}$ by $V^{\prime}({\bf r},t)$, we define an
average spin leakage probability by the non-ideal SDK as

$\varepsilon_{\rm leak}=1-\left\langle\langle
c_{m}|\tilde{U}^{\dagger}(\mathbf{k}_{\rm R},\eta){\bf
1}^{(m)}\tilde{U}(\mathbf{k}_{\rm
R};\eta)|c_{m}\rangle\right\rangle_{\eta,c_{m}}.$ (7)

By comparing Eq. (6) with Eq. (7) it is easy to verify $f_{\rm SDK}\leq
1-\varepsilon_{\rm leak}$, that is, any spin leakage leads to inefficient
control. The leakage probability $\varepsilon_{\rm leak}$ by Eq. (7) includes
contributions from $\Delta m=\pm 2$ transition as well as those due to
spontaneous emission, as $\varepsilon_{\rm leak}=\varepsilon_{\Delta
m}+\varepsilon_{\rm sp}$. Here the spontaneous emission probability
$\varepsilon_{\rm sp}$ during the SDK control is obtained by evaluating
$\tilde{U}(\mathbf{k}_{\rm R};\eta)$ with the stochastic wavefunction method
[54, 55] (Appendix B). For the $\Delta m=\pm 2$ leakage, we find
$\varepsilon_{\Delta m}\propto\omega_{\rm hfs,e}^{2}/\Delta_{e}^{2}$, as by
Eq. (1), and is thus suppressible at large $\Delta_{e}$ similar to the
suppression of spontaneous emission. However, one should note that unlike
spontaneous emission which simply leads to decoherence, the $\Delta m=\pm 2$
leakage is a coherent process. A sequence of such coherent leakages may
quickly add up in amplitudes to become significant, even if
$\varepsilon_{\Delta m}$ for a single SDK is negligibly small.

###  II.3 Adiabatic SDK

To achieve a uniformly high SDK fidelity across an intensity-varying sample
volume, particularly for $m-$spin on a hyperfine manifold with different
$\Omega_{\rm R}^{(m)}$ (Fig. 1, Eq. (2)), a standard technique is to exploit
the geometric robustness of 2-level system by inducing an adiabatic rapid
passage (Fig. 2(a)) [56, 57, 35, 10, 6]. For a particular
$\\{|a_{m}\rangle,|b_{m}\rangle\\}$ sub-spin, if we ignore $V^{\prime}$ in Eq.
(4), then the Raman adiabatic passage can easily be generated by the
$H_{0}^{(m)}$ Hamiltonian. In particular, we parametrize the Rabi frequency of
the two pulses as $\Omega_{a(b)}^{1(2)}=C_{a(b)}e^{i\phi_{1(2)}}$, and
specifically consider the time-dependent phase difference and the amplitude
profile as $\Delta\phi(t)=\phi_{0}\sin(\pi t/\tau_{\rm c})$,
$C_{a(b)}(t)=C_{a(b)}^{(0)}\sin(\pi t/\tau_{\rm c})$ respectively for
$t\in\quantity[0,\tau_{\rm c}]$ (Fig. 3(b) inset). With a large enough
2-photon sweep frequency $\delta_{\rm swp}=\pi\phi_{0}/\tau_{\rm c}$,
$|\delta_{\rm swp}|\gg\delta_{0}$ to let the 2-photon detuning $\delta_{\rm
R}=\Delta\dot{\phi}$ cover the differential Stark shift $\delta_{0}$ (Fig.
2(b)), a strong enough Raman Rabi amplitude $C_{\rm
R}^{(0)}=C_{a}^{(0)}C_{b}^{(0)}/2\Delta_{e}=2\chi^{(m)}\mathcal{A}_{\rm
R}/\tau_{\rm c}$ with a peak Raman pulse area $\mathcal{A}_{\rm R}\gg 1$, and
matched magnitudes between $C_{\rm R}^{(0)}$ and $\delta_{\rm swp}$, SDK can
be generated in a quasi-adiabatic manner [58, 59], i.e., with atomic state
$|\psi_{m}(t)\rangle\approx
c_{a}(0)e^{i\varphi_{a}(t)}|\tilde{a}_{m}(t)\rangle+c_{b}(0)e^{i\varphi_{b}(t)}|\tilde{b}_{m}(t)\rangle$
following the adiabatic basis
$\\{|\tilde{a}_{m}(t)\rangle,|\tilde{b}_{m}(t)\rangle\\}$ which are simply the
eigenstates of the instantaneous Hamiltonian. Population inversion are thus
driven quasi-adiabatically during $0<t<\tau_{\rm c}$ with the efficiency
insensitive to the laser intensity, detuning, and their slow deviations from
the specific time-dependent forms. Putting back the ${\bf r}$-dependence, the
diffraction phase accompanying the population inversion,
$\tilde{\varphi}^{+}({\bf r})\equiv\tilde{\varphi}^{+}+\mathbf{k}_{\rm
R}\cdot{\bf r}$, is evaluated as $\tilde{\varphi}^{+}({\bf
r})=\varphi_{a}(\tau_{\rm c})-\varphi_{b}(\tau_{\rm c})$ which includes not
only a geometric phase $\varphi_{\rm G}=\pi/2+\mathbf{k}_{\rm R}\cdot{\bf r}$
[60, 61], but also a dynamic phase $\varphi_{{\rm
D},m}\propto\int_{0}^{\tau_{\rm
c}}(\langle\tilde{a}_{m}|H_{0}^{(m)}|\tilde{a}_{m}\rangle-\langle\tilde{b}_{m}|H_{0}^{(m)}|\tilde{b}_{m}\rangle)d\tau$
sensitive to the control parameters $\Omega_{a(b)}^{1(2)}$ characterized by
the laser intensity profiles.

###  II.4 Dynamic phase cancellation

The $U_{\rm K}({\bf k}_{\rm R})$ operation (Eq. (5)) prints matterwave with
state $m$\- and laser intensity $|\mathcal{E}|^{2}$-dependent diffraction
phases $\varphi^{+}$ in general. For interferometrically useful coherent
matterwave controls, these dynamic parts of $\varphi^{+}$ need to be
suppressed. It is worth noting that the dynamic phase for the adiabatic SDK
survives at vanishing $\delta_{0}$ [25, 61]. Therefore, coherent matterwave
control with adiabatic SDK requires certain dynamic phase cancellation in
general.

The dynamic phases for perfect SDKs can be cancelled in pairs. To avoid
trivial operations during the pairing, the 2-photon wavevector
$\mathbf{k}_{\rm R}$ can be inverted [3, 10, 6], leading to

$\begin{split}U^{(2N)}_{\rm K}(\mathbf{k}_{\rm R})&=\underbrace{U_{\rm
K}(-\mathbf{k}_{\rm R})U_{\rm K}(\mathbf{k}_{\rm R})...U_{\rm
K}(-\mathbf{k}_{\rm R})U_{\rm K}(\mathbf{k}_{\rm R})}_{2N}\\\
&=\prod_{m=-F_{b}}^{F_{b}}\left((-1)^{N}e^{i2N\mathbf{k}_{\rm R}\cdot{\bf
r}\sigma_{z}^{(m)}}+\bar{\bf 1}^{(m)}\right).\end{split}$ (8)

Within each $m$-spin space span by $\\{|a_{m}\rangle,|b_{m}\rangle\\}$, the
$U^{(2N)}_{\rm K}(\mathbf{k}_{\rm R})$ acts as a position-dependent phase gate
to pattern the two components of arbitrary spinor matterwave with
$\pm(2N\mathbf{k}_{\rm R}\cdot{\bf r})$ phases, i.e., to coherently transfer
opposite photon recoil momenta to the spin components.

Practically, however, swapping the k-vectors of ${\bf E}_{1,2}$ can affect
other pulse parameters [62, 29, 28, 32]. Here, the k-vector swapping is
achieved on the delay line with retro-reflection (Fig. 2) by programming the
carrier frequencies $\omega_{1}\leftrightarrow\omega_{2}$ for the delayed
pulses. The imperfect reflection with $\kappa<1$ generally leads to unbalanced
diffraction phases [34] $\tilde{\varphi}^{+}$ associated with
$\tilde{U}(\pm\mathbf{k}_{\rm R})$. For the spinor matterwave control, we
quantify the impact of dynamic phase by comparing the phase of imperfectly
controlled $\tilde{U}^{(2N)}|c_{m}\rangle$ with the ideal $U_{\rm
K}^{(2N)}|c_{m}\rangle$ ($c=a,b$),

$\begin{array}[]{l}\varphi^{(2N)}_{{\rm D},m}={\rm arg}[\langle
a_{m}|U^{(2N)\dagger}_{\rm K}(\mathbf{k}_{\rm
R})\tilde{U}^{(2N)}(\mathbf{k}_{\rm R};\eta)|a_{m}\rangle]-\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm arg}[\langle
b_{m}|U^{(2N)\dagger}_{\rm K}(\mathbf{k}_{\rm
R})\tilde{U}^{(2N)}(\mathbf{k}_{\rm R};\eta)|b_{m}\rangle].\end{array}$ (9)

###  II.5 Chirp-alternating adiabatic SDK schemes

A central goal of this work is to synthesize perfect $U^{(2N)}_{\rm
K}(\mathbf{k}_{\rm R})$ spinor matterwave phase gates with $2N$ imperfect
$\tilde{U}(\pm\mathbf{k}_{\rm R})$, even in presence of the unbalanced dynamic
phase $\varphi_{{\rm D},m}^{(2N)}$ and the $\Delta m=\pm 2$ leakage. For the
purpose, we refer the experimentally realized non-ideal adiabatic SDKs with
equal amount of positive ($\delta_{\rm swp}>0$) and negative ($\delta_{\rm
swp}<0$) 2-photon sweeps as $\tilde{U}_{\rm u}(\pm\mathbf{k}_{\rm R})$ and
$\tilde{U}_{\rm d}(\pm\mathbf{k}_{\rm R})$ respectively (Fig. 2(b)). The
associated control Hamiltonians are $H_{\rm u}({\bf r},t)$ and $H_{\rm d}({\bf
r},t)$. In Sec. IV.3 we show that a chirp-alternating sequence

$\tilde{U}_{\rm ud}^{(2N)}(\mathbf{k}_{\rm R})=\underbrace{\tilde{U}_{\rm
u}(-\mathbf{k}_{\rm R})\tilde{U}_{\rm d}(\mathbf{k}_{\rm
R})\cdots\tilde{U}_{\rm u}(-\mathbf{k}_{\rm R})\tilde{U}_{\rm
d}(\mathbf{k}_{\rm R})}_{2N},$ (10)

fairly efficiently suppress the $\Delta m=\pm 2$ leakage. Next, as detailed in
Sec. IV.3, Sec. IV.4, by combining the $\tilde{U}_{\rm ud}^{(2N)}$ and
$\tilde{U}_{\rm du}^{(2N)}$ to form a balanced chirp-alternating sequence,

$\tilde{U}_{\rm uddu}^{(4N)}(\mathbf{k}_{\rm R})=\tilde{U}_{\rm
ud}^{(2N)}(\mathbf{k}_{\rm R})\tilde{U}_{\rm du}^{(2N)}(\mathbf{k}_{\rm R}),$
(11)

the dynamic phases by $\tilde{U}_{\rm du}^{(2N)}(\mathbf{k}_{\rm R})$ and
$\tilde{U}_{\rm ud}^{(2N)}(\mathbf{k}_{\rm R})$ robustly cancel each other.
The Eq. (11) scheme can therefore faithfully realize a $U^{(4N)}_{\rm K}({\bf
k}_{\rm R})$ phase gate.

For comparison, here the “traditional” chirp-repeating sequences [63, 10, 6]
are defined as

$\tilde{U}_{\rm uu}^{(2N)}(\mathbf{k}_{\rm R})=\underbrace{\tilde{U}_{\rm
u}(-\mathbf{k}_{\rm R})\tilde{U}_{\rm u}(\mathbf{k}_{\rm
R})\cdots\tilde{U}_{\rm u}(-\mathbf{k}_{\rm R})\tilde{U}_{\rm
u}(\mathbf{k}_{\rm R})}_{2N}.$ (12)

Similar $\tilde{U}_{\rm dd}^{(2N)}(\mathbf{k}_{\rm R})$ controls are defined
with $\mathrm{u}\rightarrow\mathrm{d}$ replacements.

To conclude this section, we note that while the D1 line is chosen in this
work (Fig. 1), the conclusions on the ground state hyperfine control dynamics
can be applied in a straightforward manner to the D2 line. Comparing with the
D1 operation, Raman SDK at a same single-photon detuning $\Delta_{e}$ on the
D2 line suffers more spontaneous emission loss due to the excitations of the
cycling transitions [64]. On the other hand, a reduced $m-$changing rate (Eq.
(1)) is expected to help the D2 performance due to the typically smaller
$\omega_{{\rm hfs},e}$ for the intermediate $P_{3/2}$ states.

##  III Experimental Implementation

###  III.1 Nanosecond SDK on a delay line

The adiabatic SDK is implemented on the 85Rb $5S_{1/2}-5P_{1/2}$ D1 line as
depicted in Fig. 1, with counter-propagating chirp pulses programmed by a
wideband optical waveform generator [36] on an optical delay line [15]. The
cross-linear polarization is realized by double-passing the light beam with a
quarter waveplate before the end mirror (Fig. 2(a)) which converts the
incident ${\bf e}_{x}$ polarization to ${\bf e}_{y}$. With the OAWG output
peak power limited to $P_{\rm max}\approx 20~{}$mW, the incident control pulse
$\mathcal{E}_{1,2}$ is weakly focused to a waist radius of
$w\approx$13\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$ to reach a peak
Rabi frequency of $\Omega_{a(b)}\approx 2\pi\times 2$ GHz. The imperfect
retro-reflection with $\kappa\approx 0.7$ ($r=|\kappa|^{2}\approx 50\%$)
reflectivity, primarily limited by increased focal beam size due to wavefront
distortion, leads to decreased $\Omega_{b(a)}=\kappa\Omega_{b(a)}$ for the
reflected pulses seen by the atomic sample. We set the single-photon detuning
to be $\Delta_{e}=2\pi\times 10$ GHz to achieve a peak Raman Rabi frequency of
$C_{\rm R}^{(0)}\approx
2\pi\times\kappa$300\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$ estimated at the
center of the Gaussian $\mathcal{E}_{1,2}$ beams. An $\tau_{\rm d}=140.37$ ns
optical delay is introduced by the $L\approx 20$ m folded delay line, which is
long enough to spatially resolve the counter-propagating nanosecond pulses. To
form the counter-propagating $\mathcal{E}_{1,2}$ pulse pair, we pre-program
$\mathcal{E}_{1,2}(t)$ and $\mathcal{E}_{2,1}(t-\tau_{\rm d})$, with a
relative delay matching the optical delay line, to ensure the pulse pair with
proper carrier frequency $\omega_{1,2}$ meeting head-on-head in the atomic
cloud. To continue multiple SDKs, additional, individually shaped pulses with
alternating $\omega_{1,2}$ can be applied, as in Fig. 2(b), with a $T_{\rm
rep}=\tau_{\rm d}$ periodicity.

Importantly, to periodically generate multiple Raman SDKs with the delay line,
every shaped pulse contribute twice to the SDK sequence. To properly shape the
Raman coupling $\Omega_{\rm
R}(t)\propto\mathcal{E}^{*}_{1}(t)\mathcal{E}_{2}(t)$ (Eq. (2)), therefore,
one needs to program the incident $\mathcal{E}_{1(2)}^{\prime}$ pulse
according to the retro-reflected $\mathcal{E}_{2(1)}$ pulse. To clear out such
pulse-history dependence in a long sequence, the retro-reflection cycles can
be interrupted a few times by increasing the inter-pulse delays beyond
$\tau_{\rm d}$. Here, if our goal is to alternate Raman chirps $\delta_{\rm
R}$, as in Fig. 2(b) and Eq. (11), then the frequency-sweeping amplitudes for
every other incoming optical pulse needs to be increased by an additional
$\delta_{\rm swp}$ (Sec. II.3). As a result, in the balanced chirp-alternating
scheme (Eq. (11)) the first $2N$ pulses becomes more and more chirped, before
a reversal of the process to rewind back the rate. High optical chirping rates
affect the $V^{\prime}$-couplings (Eq. (4)). In our experiments, the single-
photon detuning
$\Delta_{e}=2\pi\times$10\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ is much
larger than $\delta_{\rm swp}$, and we have numerically confirmed that the
increasingly chirped waveforms do not significantly affect the Raman dynamics
in the $\tilde{U}^{(4N)}_{\rm uddu}$ scheme up to $N=6$. Nevertheless, to
avoid systematic errors associated with the digital OAWG pulse shaping [36],
the chirp-rate accumulation is interrupted in this work by separating
$\tilde{U}^{(4N)}_{\rm uddu}$ into $N$ sets of 4-pulse sequences, as described
above.

We note that in the delay-line based SDK scheme, there are extra “pre-pulses”
and “post-pulses” that interact with the atomic sample alone without any
counter-propagating pulse to help driving the Raman transition (e.g. the first
and last pulses in Fig. 2b). Although these extra pulses impact negligibly the
atomic hyperfine population and momentum transfer in this work, they cause
extra dynamic phases which need to be precisely compensated for during future
coherent matterwave controls. Similar to the frequency domain Stark shift
compensation [43], for the nanosecond SDKs here one can fire additional pulses
with suitable single-photon detunings to trim the overall dynamic phase (Eq.
(9)). Within nanoseconds, cold atoms hardly move to change the local laser
intensity. We therefore expect the dynamic phase compensation to function well
in the time domain.

###  III.2 Optimizing Adiabatic SDK

Figure 3: (a) SDK momentum transfer measurement with a double imaging method:
For atomic sample subjecting $n$ SDKs ($n=25$ here) starting with
$\tilde{U}_{\rm u}(\mathbf{k}_{\rm R})\tilde{U}_{\rm u}(-\mathbf{k}_{\rm R})$,
a sequence of two absorption images are taken at $\bar{t}_{1}=10~{}\mu$s and
$\bar{t}_{2}=\bar{t}_{1}+t_{\rm tof}$, separated by $\tau_{\rm
tof}=$160\text{\,}\mu\mathrm{s}$$ time-of-flight and with a hyperfine
repumping pulse along $x$ applied in between (see main text). The center-of-
mass position shift $\delta z$ is retrieved with a typical $\pm 2~{}\mu$m
accuracy by fitting the optical depth (OD). The recoil velocity $v_{n}=\delta
z/\tau_{\rm tof}$ is then estimated during typical adiabatic SDK parameter
scan as in (b). Here, with $n=25$, $\tau_{\rm c}=60~{}$ns SDK pulses applied,
$v_{n}$ is optimized as a function of sweep frequency $\delta_{\rm swp}$, and
peak laser intensity parametrized by an estimated peak Raman pulse area
$\mathcal{A}_{\rm R}=C_{\rm R}^{(0)}\tau_{\rm c}/2$. Blue, green and orange
lines correspond to estimated peak $\mathcal{A}_{\rm R}$ of approximately
$6\pi,9\pi,12\pi$ respectively. The corresponding peak Rabi frequencies
$C_{\rm R}^{(0)}$ are marked with arrows on the top along the $\delta_{\rm
swp}$ axis. The error bars give statistical uncertainties of $v_{n}$ in
repeated measurements.

We prepare $N_{A}\sim 10^{5}$ 85Rb atoms in a compressed optical dipole trap
at a temperature of $T\sim$200\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{K}$$
[36]. The atomic sample is optically pumped into the $F=2$, $|b_{m}\rangle$
hyperfine states, elongated along $z$, with a characteristic radius of
$\sigma\approx$7\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$ in the x-y
plane (Fig. 3(a)). Immediately after the atoms released from the trap,
multiple SDKs programmed on the optical delay line with alternating
$\pm\mathbf{k}_{\rm R}$, $\mathbf{k}_{\rm R}=2k_{0}{\bf e}_{z}$ are applied to
transfer photon momentum by repetitively inverting the atomic population
between $F=2$, $|b_{m}\rangle$ and the $F=3$, $|a_{m}\rangle$ hyperfine
states. Here $k_{0}=2\pi/\lambda$ is the wavenumber of the D1 line SDK pulses
at $\lambda=795$ nm.

We use a double-imaging technique to characterize the performance of the
adiabatic SDKs, by simultaneously measuring the spin-dependent momentum
transfer and population inversion (Appendix D). Specifically, immediately
after the last of $n$ SDK pulses, a probe pulse resonant to the D2 line
$F=3-F^{\prime}=4$ hyperfine transition is applied for $\tau_{\rm
p}=$20\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$$ to record the spatial
distribution of atoms in state $|a\rangle$, in the $x-z$ plane, with
calibrated absorption imaging [15]. For atoms in $|b\rangle$, this probe is
far-detuned and weak enough not to perturb the motion. Next, after a
$\tau_{\rm tof}=$160\text{\,}\mu$$s free-flight time, the 2nd $\tau_{\rm
p}=20~{}\mu$s probe pulse is applied to image all the atoms. For the purpose,
during the time of flight an additional
$50\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ pulse along ${\bf e}_{x}$,
resonant to $F=2-F^{\prime}=3$ transition, repumps the $|b\rangle$ atoms to
$|a\rangle$ for the 2nd imaging. By comparing atom number $N_{a}$ in state
$|a\rangle$ and the total atom number $N_{A}=N_{a}+N_{b}$, inferred from the
first and second images respectively, the probability of atoms ending up in
$|a\rangle$ can be measured as a function of the number of SDKs $n$ as
$\rho_{aa,n}=N_{a}/(N_{a}+N_{b})$. In addition, by fitting both images to
locate the center-of-mass vertical positions $z_{1,2}$, the atomic velocity
$v_{n}=\delta z/t_{\rm tof}$ can be retrieved to estimate the photon momentum
transfer $p_{n}=Mv_{n}$ in unit of $\hbar k_{\rm eff}$.

Typical $v_{n}$ measurement results are given in Fig. 3(b). Here, for atoms
prepared in $|b\rangle$ states subjected to an $n=25$ SDK sequence starting
with $\tilde{U}_{\rm u}(\mathbf{k}_{\rm R})\tilde{U}_{\rm u}(-\mathbf{k}_{\rm
R})$ (a $2N=24$ double-SDK followed by an additional kick to drive the final
Raman transition), the atomic population is largely in $|a\rangle$ while
$v_{n}$ is unidirectional along ${\bf e}_{z}$. We optimize $v_{n}$ by varying
the peak Raman coupling amplitude $C_{\rm R}^{(0)}$ and sweep frequency
$\delta_{\rm swp}$ of the adiabatic SDK pulses at fixed $\tau_{\rm c}$. As in
Fig. 3(b), for a fixed peak Raman pulse area $\mathcal{A}_{\rm R}$, we
generally find $\delta_{\rm swp}$ to be optimized for efficient photon
momentum transfer when it matches $C_{\rm R}^{(0)}$ (See the arrow markers in
Fig. 3(b)). However, unlike 2-level transfer [58] where an increased pulse
area always leads to improved adiabaticity and population inversion
robustness, here we find the peak $\mathcal{A}_{\rm R}\approx 9\pi$ reaches
optimal to ensure the resilience of adiabatic SDK against the up to $50\%$
intensity variation in the setup. Larger $\mathcal{A}_{\rm R}$ is accompanied
by slow decrease of $v_{n}$, due to increased probability of spontaneous
emission. Here, for $\tau_{\rm c}=60~{}$ns, we need to attenuate $C^{(0)}_{\rm
R}\approx 2\pi\times 150$ MHz to keep the peak $\mathcal{A}_{\rm R}\approx
9\pi$. By using the full $C^{(0)}_{\rm R}\approx 2\pi\times 200$ MHz available
in this work, we are able to reduce $\tau_{\rm c}$ to 40 nanoseconds while
maintaining nearly identical momentum transfer efficiency at $\delta_{\rm
swp}=2\pi\times$150\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$.

###  III.3 Inference of $f_{\rm SDK}$

Figure 4: Measurements of recoil momentum and hyperfine population transfer by
multiple SDKs optimized at $\tau_{\rm c}=40$ ns. Normalized momentum transfer
$p_{n}$ and hyperfine population $\rho_{aa,n}$ vs kick number $n$ are plotted
in (a,b)(c,d) respectively. The atoms are initialized in $F=2$,
$|b_{m}\rangle$ states. Results for the $\tilde{U}_{\rm uu}^{(n)}(\pm{\bf
k}_{\rm R})$ SDK sequence, starting with $\tilde{U}_{\rm u}(\mathbf{k}_{\rm
R})\tilde{U}_{\rm u}(-\mathbf{k}_{\rm R})$ and $\tilde{U}_{\rm
u}(-\mathbf{k}_{\rm R})\tilde{U}_{\rm u}(\mathbf{k}_{\rm R})$, are plotted in
top (a,b) and bottom (c,d) panels with disk symbols respectively. Results for
the balanced chirp-alternating $\tilde{U}^{(4N)}_{\rm uddu}(\pm{\bf k}_{\rm
R})$ (Eq. (11)) with $N=2-6$ are plotted via open squares. For comparison,
spontaneous-emission-limited $p_{n}$ and $\rho_{aa,n}$ according to numerical
simulation are plotted with black dashed lines. A phenomenological fit (solid
red and blue curves) suggests hyperfine Raman transfer efficiency $f_{\rm
R}=98.8(3)\%$ for the experimental data. See discussions in the main text for
inference of $f_{\rm SDK}=97.6(3)\%$ from the data. (e) Top: State-selective
absorption images of atomic sample in $F=3$, $\\{|a_{m}\rangle\\}$ states,
probed immediately after $n=1-25$ SDK $\tilde{U}_{\rm uu}^{(n)}({\bf k}_{\rm
R})$ pulses. Bottom: Corresponding absorption images of optically repumped
atomic samples with $\tau_{\rm
tof}=$160\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$$. Notice the $n$-odd
measurements with higher ${\rm OD}_{1}$ suffer stronger probe-induced heating
and thus wider ${\rm OD}_{2}$ distributions probed later, as expected.

With the optimal $\delta_{\rm
swp}=2\pi\times$150\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$ and peak
$\mathcal{A}_{\rm R}\approx 9\pi$ at $\tau_{\rm
c}=$40\text{\,}\mathrm{n}\mathrm{s}$$, we now apply $n=1-25$ SDKs to
characterize the momentum transfer $p_{n}=Mv_{n}$ and normalized population
$\rho_{aa,n}$ as a function of kicking number $n$. Typical results are given
in Fig. 4. Here the momentum change $p_{n}$ along ${\bf e}_{z}$ is again
unidirectional along $z$ as in Figs. 4(a). The direction is conveniently
reversed by programming $\tilde{U}_{\rm u}(-\mathbf{k}_{\rm R})$ first in the
$\pm\mathbf{k}_{\rm R}$ alternating sequence, resulting in acceleration of
atoms along $-{\bf e}_{z}$ instead as in Figs. 4(c). In contrast, the
hyperfine population $\rho_{aa,n}$ is suppressed and revived after an even and
odd number of SDKs respectively, as demonstrated in Fig. 4(c,d). In addition,
we program a balanced chirp-alternating sequence $n=4N$ that combines
$\tilde{U}_{\rm u}(\pm\mathbf{k}_{\rm R})$ with $\tilde{U}_{\rm
d}(\mp\mathbf{k}_{\rm R})$ as $\tilde{U}_{\rm uddu}^{(4N)}(\pm\mathbf{k}_{\rm
R})$ according to Eq. (11), a sequence which will be detailed in Sec. IV for
interferometric applications, with $p_{n}$ and $\rho_{aa,n}$ measurement
results also given in Fig. 4 to demonstrate similar momentum and population
transfer efficiency.

We estimate $f_{\rm SDK}$ by comparing the $p_{n}$ and $\rho_{aa,n}$
measurements as in Fig. 4 with precise numerical modeling detailed in Appendix
B, taking into account the finite laser beam sizes and imperfect reflection
with reflectivity $r=|\kappa|^{2}$. The comparison is facilitated by fitting
both the measurement and simulating data according to a phenomenological model
(Appendix C), which assumes that errors between successive SDKs are
uncorrelated and are solely parametrized by $f_{\rm R}$, a hyperfine Raman
population transfer efficiency. The model predicts exponentially reduced
increments $|p_{n+1}-p_{n}|/\hbar k_{\rm
eff}=|\rho_{aa,n+1}-\rho_{aa,n}|=f_{\rm R}(2f_{\rm R}-1)^{n}$ for $f_{\rm
R}\approx 1$ by each SDK. From measurement data in Fig. 4(a-d), $f_{\rm
R}\approx 98.8\%$ can be estimated in both $\tilde{U}_{\rm u}(\mathbf{k}_{\rm
R})$ and $\tilde{U}_{\rm u}(-\mathbf{k}_{\rm R})$ kicks (Fig. 4(a,c)),
slightly less than spontaneous-emission-limited $f_{\rm R}\approx 99.2\%$
predicted by numerical simulation of the experiments, assuming perfect
reflection with $\kappa=1$.

We simply attribute the slightly reduced $f_{\rm R}$ from the theoretical
value to imperfect retro-reflection (Appendix E). In particular, we
numerically find $f_{\rm R}$ reduces with $f_{\rm SDK}$ when
$\mathcal{E}_{1,2}$ are unbalanced in amplitudes, so that both spin leakage
and spontaneous emission are increasingly likely to occur. Taking into account
the independently measured $r\approx 0.5$ in this work and with numerically
matched $f_{\rm R}$, $\varepsilon_{\rm leak}\approx 2.5\%$ with
$\varepsilon_{\Delta m}\approx 0.5\%$ and $\varepsilon_{\rm sp}\approx 2\%$
can be estimated respectively. We therefore infer from the combined analysis
an $f_{\rm SDK}=97.6(3)\%$ for the adiabatic SDK in this experiment, slightly
less than the spontaneous-emission-limited $f_{\rm SDK}\approx 98\%$ for the
nearly perfect adiabatic SDK. The numbers can be improved further by
increasing $\Delta_{e}$ to suppress spontaneous emission as well as the
$\Delta m$-leakage. It is important to note that while the imperfect
reflection only moderately reduce $f_{\rm SDK}$, the resulting imperfection of
$\mathbf{k}_{\rm R}\leftrightarrow-\mathbf{k}_{\rm R}$ swapping in successive
$\tilde{U}_{\rm u}(\pm\mathbf{k}_{\rm R})$ control can greatly compromise the
cancellation of dynamic phase $\varphi_{\rm D}$ in double SDK (Eq. (8)) [10,
6], a topic to be detailed in Sec. IV.4.

The adiabatic SDK demonstrated in this experimental section is the fastest
realization to neutral atoms to date [10, 6]. By equipping a more powerful
laser, the SDK time can be reduced to a few nanoseconds or less [5]. A one-
meter level compact delay line would then be able to resolve the counter-
propagating pulses. Here, it is worth pointing out that the motion of the
retro-reflecting mirror (Fig. 2(a)) shifts the SDK diffraction phases (Eq.
(5)) [34]. For interferometric measurements with long interrogation time (Sec.
IV.5), the end mirror for the compact delay line can be vibration-isolated as
those in traditional atom interferometry [34, 28, 29, 30, 31, 32, 33].

##  IV Coherent control of spinor matterwave

In the previous section, we experimentally characterize nanosecond adiabatic
SDK by measuring the transfer of photon momentum and hyperfine population by a
SDK sequence. A natural question to ask is whether it is possible to exploit
the technique for coherent spinor matterwave control. As discussed in Sec.
II.4, for ideal SDKs the double SDK sequence (Eq. (8)) can be constructed to
perform position-dependent phase gates. However, as to be illustrated in the
following, the operation degrades rapidly with kick number $N$ in presence of
coherent spin leakage among the $2F_{b}+1$ $\\{|a_{m}\rangle,|b_{m}\rangle\\}$
sub-spins (Sec. IV.2), or when the non-perfect $\mathbf{k}_{\rm
R}\leftrightarrow-\mathbf{k}_{\rm R}$ swapping introduces additional dynamic
phases (Sec. IV.4).

Nevertheless, in this section we demonstrate that the $\Delta m$-leakage and
dynamic phases can be efficiently suppressed by the balanced chirp-alternating
SDK sequence $\tilde{U}^{(4N)}_{\rm uddu}(\mathbf{k}_{\rm R})$ introduced in
Sec. II.5 (Eq. (11)) for faithful implementation of the $U^{(4N)}_{\rm
K}(\mathbf{k}_{\rm R})$ phase gate by Eq. (8) to finely enable high-efficiency
large momentum transfer [3]. We further numerically demonstrate the utility of
the tailored adiabatic SDK sequence by simulating an area-enhanced atom
interferometry sequence, using the experimental parameters both within and
beyond this experimental work.

We notice control of spin leakage in quasi-two-level systems is an important
topic in quantum control theory [65, 66, 67]. Previous studies on the topic
typically involve a Morris–Shore transformation of the interaction matrix to
decompose the multi-level dynamics [68]. However, as being schematically
summarized in Fig. 1(b), the spin leakages are coherently driven through
multiple paths with multiple Raman couplings to preclude a straightforward
transformation, nor a direct application of the associated leakage-suppression
techniques [65, 66, 67].

###  IV.1 Average gate fidelity

To evaluate an imperfect double-SDK sequence $\tilde{U}^{(2N)}(\mathbf{k}_{\rm
R};\eta)$ as a quantum gate, we define an average fidelity for its performance
on arbitrary spinor matterwave states of interest, e.g., spatially within the
focal laser beam (Fig. 2(a)) and internally span the
$\\{|a_{m}\rangle,|b_{m}\rangle\\}$ sub-spin space. The average fidelity is
written as [53]

$\mathcal{F}^{(2N)}_{m}=\left|\langle\psi_{m,j}|U^{(2N)\dagger}_{\rm
K}(\mathbf{k}_{\rm R})\tilde{U}^{(2N)}(\mathbf{k}_{\rm
R};\eta)|\psi_{m,j}\rangle\right|^{2}_{\eta,j}.$ (13)

Similar to Eq. (6), here $\eta$ represents the Hamiltonian parameter in Eq.
(4) including the atomic position ${\bf r}$ of interest. For internal states
the evaluation samples $|\psi_{m,j}\rangle$ as the six eigenstates of
$\sigma_{x,y,z}^{(m)}$, listing as
$\\{|\psi_{m,j}\rangle\\}=\\{|b_{m}\rangle,|a_{m}\rangle,(|b_{m}\rangle\pm|a_{m}\rangle)/\sqrt{2},(|b_{m}\rangle\pm
i|a_{m}\rangle)/\sqrt{2}\\}$. With external motion ignored,
$\mathcal{F}^{(2N)}_{m}\approx 1$ ensures that the effectively instantaneous
$\tilde{U}^{(2N)}({\bf k}_{\rm R};\eta)$ faithfully apply the phase gate to
arbitrary spinor matterwave states near the beam focus, with certain laser
intensity of interest.

For the convenience of related discussions, we define an average $\Delta
m$-leakage probability, similar to Eq. (7), as

$\varepsilon^{(n)}_{m,{\Delta
m}}=1-\left\langle\langle\tilde{\psi}_{m,j}^{(n)}|{\bf
1}^{(m)}|\tilde{\psi}_{m,j}^{(n)}\rangle\right\rangle_{\eta,j}$ (14)

where $|\tilde{\psi}_{m,j}^{(n)}\rangle=\tilde{U}^{(n)}(\mathbf{k}_{\rm
R};\eta)|\psi_{m,j}\rangle$ is defined as the final atomic state after the
imperfect control. To exclude spontaneous emission, we simply renormalize to
have $\langle\tilde{\psi}_{m,j}^{(n)}|\tilde{\psi}_{m,j}^{(n)}\rangle=1$
before the evaluation of the $\Delta m$-leakage. Similar to Eq. (14), here the
$\Delta m$-leakage probability is also evaluated for a specific $m$-sub-spin
of interest. Similar to Eqs. (6)(7), here we expect $\mathcal{F}^{(n)}_{m}\leq
1-\varepsilon^{(n)}_{m,{\Delta m}}$, i.e., any spin leakage results in gate
infidelity.

###  IV.2 Simple double-SDK: spin leakage

Figure 5: Numerical results of spin leakage
$\varepsilon=\varepsilon^{(n)}_{m,\Delta m}$ for the traditional
$\tilde{U}^{(n)}_{\rm uu}$ (left) and chirp-alternating $\tilde{U}^{(n)}_{\rm
ud}$ (right) controls as a function of pulse number $n$, The peak Raman pulse
area is $\mathcal{A}_{\rm R}=9\pi$. The corresponding 2-photon detuning
profiles are illustrated on top of (a,b). Additional parameters for the
simulations are according to the experimental Sec. III.3. Figure 6:  Numerical
results of spin leakage $\varepsilon=\varepsilon^{(4N)}_{m,\Delta m}$ for the
traditional $\tilde{U}^{(4N)}_{\rm uu}$ (left) and chirp-alternating
$\tilde{U}^{(4N)}_{\rm ud}$ (right) as a function of kicking number $4N$ and
$\mathcal{A}_{\rm R}$, with $m=0$ (a,d), $m=\pm$ (b,e), $m=\pm 2$ (c,f). The
2-photon detuning profiles are illustrated on top of (a,d). Additional
parameters for the simulations are according to the experimental Sec. III.3.

We first consider the chirp-repeating double-SDK sequences
$\tilde{U}^{(2N)}_{\rm uu}(\mathbf{k}_{\rm R})$ and $\tilde{U}^{(2N)}_{\rm
dd}(\mathbf{k}_{\rm R})$ with Eq. (12). The simple sequences are widely
applied in previous works to drive multiple recoil momentum transfer through
both optical [58, 69] and Raman excitations [10, 6]. In absence of the spin
leakage, we expect both $\tilde{U}^{(2N)}_{\rm uu}(\mathbf{k}_{\rm R})$ and
$\tilde{U}^{(2N)}_{\rm dd}(\mathbf{k}_{\rm R})$ to amplify the momentum
transfer with cancelled dynamic phase [10, 6], as suggested by Eq. (8).

Here, we numerically investigate the $\Delta m$-leakage probability
${\varepsilon}^{(n)}_{m,{\Delta m}}$ and the gate fidelity
$\mathcal{F}^{(n)}_{m}$, for an atom initialized in an
$\\{|a_{m}\rangle,|b_{m}\rangle\\}$ sub-spin being subjected to
$\tilde{U}^{(n)}_{\rm uu}(\mathbf{k}_{\rm R})$. With $\Delta_{e}\gg\delta_{\rm
swp}$, the results for $\tilde{U}^{(n)}_{\rm dd}(\mathbf{k}_{\rm R})$ are
effectively identical. The Hamiltonian parameters in the simulation follow the
experimental setup described in Sec. III for the 85Rb D1 line scheme (in
particular $\delta_{\rm swp}=2\pi\times 150$ MHz), except here the reflective
coefficient is set as $\kappa=1$ for simplicity, and $\Gamma_{e}=0$ to focus
on the coherent control dynamics. To elucidate the role of laser intensity for
the coherent control, we repeat the simulation while scanning the ${\bf
E}_{1,2}$ laser intensities in proportion, parametrized by the Raman pulse
area $\mathcal{A}_{\rm R}$ in the following. Typical results for 85Rb are
shown in the left panels of Fig. 5, Fig. 6 and Fig. 7.

Figure 7:  Numerical results of gate infidelity $1-\mathcal{F}$ for the
traditional $\tilde{U}^{(4N)}_{\rm uu}$ (left panels) and balanced chirp-
alternating $\tilde{U}^{(4N)}_{\rm uddu}$ (right panels) controls. With
$\mathcal{F}=\mathcal{F}_{m}^{(4N)}$ and $m=0$ (a,d), $m=\pm 1$ (b,e), $m=\pm
2$ (c,f). The 2-photon detuning profiles are illustrated on top of (a,d).
Additional parameters for the simulations are according to Sec. III.3.

We first discuss the $\Delta m$-leakage probability
$\varepsilon^{(n)}_{m,\Delta m}$ as a function of kick number $n$ for typical
$\mathcal{A}_{\rm R}=9\pi$ presented in Fig. 5(a). Here, for $n=1$, the tiny
$\varepsilon^{(n)}_{m,\Delta m}$ for all the $m-$ sub-spins are close to the
single kick leakage of $\sim 0.5\%$ as being inferred experimentally. However,
with increased $n$, $\varepsilon^{(n)}_{m,\Delta m}$ increases rapidly to
approach unity for merely $n\sim 20$, i.e., atoms starting from any of the
sub-spins $\\{|a_{m}\rangle,|b_{m}\rangle\\}$ have a substantial probability
of ending up in a different $m-$subspace.

To investigate the laser intensity dependence, the $\Delta m$-leakage
${\varepsilon}^{(2N)}_{m,\Delta m}$ is further plotted in Figs. 6(a-c) (left
panels) as a function of both the kicking number $n=4N$ and the pulse area
$\mathcal{A}_{\rm R}$, for $m=0,\pm 1,\pm 2$ sub-spins respectively. The
choice of $n=4N$ is for comparison with the balanced chirp-alternating
sequence to be discussed shortly. We see that over a broad range of pulse area
$\mathcal{A}_{\rm R}$, the spin-leakage probability
$\varepsilon^{(4N)}_{m,{\Delta m}}$ increases rapidly with $4N$ in oscillatory
fashions. There is hardly any continuous region of laser intensity with
${\varepsilon}^{(4N)}_{m,\Delta m}<0.1$.

It is important to note that although the $\Delta m$-leakages hardly affect
the recoil momenta and hyperfine population transfers (Sec. III), they do
limit the gate fidelity for faithful spinor matterwave control. The impact of
spin leakage to average gate fidelity, $\mathcal{F}^{(4N)}_{m}$, is
demonstrated by comparing Figs. 6(a-c) with Figs. 7(a-c). Here we see the gate
infidelity $1-\mathcal{F}^{(4N)}_{m}$ closely follows
$\varepsilon^{(4N)}_{m,{\Delta m}}$ to hardly reach $0.1$ over most laser
intensities, except when the laser intensity is too low to adiabatically drive
the Raman transition at all (with $\mathcal{A}_{\rm R}<3\pi$ here) where we
instead find $\mathcal{F}^{(4N)}_{m}\approx 0.5$, as expected.

Clearly, the coherent accumulation of $\Delta m$-leakage error as in Figs.
6(a-c) needs to be suppressed before the adiabatic SDK sequence can be
exploited for coherent control of spinor matterwave. Traditional methods for
such suppression include applying a moderate quantization field to lift the
Zeeman degeneracy [27, 10, 6, 70]. This Zeeman shift approach rules out the
possibility of parallel multi-Zeeman spin wave coherent control prescribed by
Eq. (8). Furthermore, since the Zeeman shifts need to be substantially larger
than the SDK bandwidth, a uniform field at a kilo-Gauss level is likely
required for the nanosecond operations, which is not favored in precision
metrology with alkaline atoms.

###  IV.3 Balanced chirp-alternating SDK scheme

We now demonstrate that the direction of chirps in an adiabatic sequence can
be programmed to suppress the coherent accumulation of $\Delta m$-leakage.
Furthermore, by balancing $\tilde{U}_{\rm ud}^{(2N)}$ with $\tilde{U}_{\rm
du}^{(2N)}$ as those in Eq. (11), the dynamic phase can be cancelled in
$\tilde{U}_{\rm uddu}^{(2N)}$ to support faithful spinor matterwave phase
gate. As to be clarified in the following, both the non-adiabatic spin-leakage
suppression and dynamic phase cancellation are supported by time-reversals of
the driven spin dynamics, much like those in the traditional spin-echo
schemes, but is achieved here in the adiabatic SDK sequence to also ensure the
laser intensity-error resilience. We note the benefit of alternating the chirp
directions was also discovered in the context of 2-level atom slowing with
adiabatic pulses [40], but otherwise rarely explored before.

To understand why the accumulation of $\Delta m$-leakage error can be partly
suppressed by the chirp-alternating $\tilde{U}_{\rm ud}^{(2N)}(\mathbf{k}_{\rm
R})$ or $\tilde{U}_{\rm du}^{(2N)}(\mathbf{k}_{\rm R})$ sequence prescribed by
Eq. (10), we come back to Eq. (4) to better understand the $\Delta m=\pm 2$
leakage itself. In particular, for atom starting in $|a_{m}\rangle$ or
$|b_{m}\rangle$ and subjected to a close-to-ideal adiabatic SDK control, with
an $H_{\rm u}({\bf r},t)$ Hamiltonian during $0<t<\tau_{\rm c}$ as prescribed
in Sec. II.5, the $\Delta m=\pm 2$ transitions driven by $V^{\prime}$ are
often a result of non-adiabatic couplings among sub-spins with nearly equal
Stark shifts (for example, the $m=\pm 1$ sub-spins in Fig. 1b). By reversing
the time-dependence of the Hamiltonian, here with $H_{\rm d}({\bf r},t)=H_{\rm
u}({\bf r},\tau_{\rm d}-t)$ within $\tau_{\rm d}<t<\tau_{\rm d}+\tau_{\rm c}$,
the sign of the non-adiabatic couplings are reversed. Such a sign reversal
would lead to complete cancellation of the non-adiabatic transitions if the
adiabatic states involved in the couplings are truly degenerate. Here, for
atoms being addressed by the cross-linear polarized light, the fact that the
$m-$sub-spins are nearly degenerate makes the sign reversal efficient for the
suppression of the unwanted leakages.

The simple picture of coherent leakage suppression is verified with numerical
simulation in Fig. 5(b) for the chirp-alternating $\tilde{U}_{\rm
ud}^{(2N)}(\mathbf{k}_{\rm R})$ sequence. Here, in contrast to the
$\tilde{U}_{\rm uu}^{(2N)}(\mathbf{k}_{\rm R})$ case in Fig. 5(a), the spin
leakage $\varepsilon_{m,\Delta m}^{(n)}$ for all the $m=0,1,2$ oscillates and
is overall efficiently suppressed. Comparing with $m=\pm 1$ where the leakage
is suppressed for even $n$, the leakage suppression from the $m=0,\pm 2$ sub-
spin follows a more complicated pattern with an approximate periodicity of 4
to 5, suggesting more complex multi-level dynamics. We further investigate the
leakage dynamics by plotting $\varepsilon_{m,\Delta m}^{(n)}$ in 2D vs $n$ and
pulse area $\mathcal{A}_{\rm R}$. The results for $n=4N$ are presented in Fig.
6(d-f) to be compared with those in Fig. 6(a-c). For the $m=\pm 1$ sub-spins,
the $\Delta m$-leakage is suppressed to $\sim 1\%$, limited by their residual
couplings to the $|a_{m}\rangle$ states with $m=\pm 3$ (Fig. 1(a)). In
Appendix F we demonstrate that the spin-leakage suppression becomes perfect in
species with nuclear spin $I=1.5$ (87Rb-like) for the pair of $m=\pm 1$ sub-
spins. On the other hand, for $m=0,\pm 2$ sub-spins here, there are stripes of
$\mathcal{A}_{\rm R}$-region (around $\mathcal{A}_{\rm R}=12\pi$ and
$\mathcal{A}_{\rm R}=22\pi$ here for example) where the leakage still
accumulate with increased $n=4N$, even though the suppression still works
fairly well ($\varepsilon_{m,\Delta m}^{(4N)}<10\%$) for most of other
$\mathcal{A}_{\rm R}$. The intricate $\mathcal{A}_{\rm R}$-dependent
$\varepsilon^{(2N)}_{m,\Delta m}$ as in Figs. 6(d)(f) suggests that the
difference of dynamic phases among $m=0,\pm 2$ sub-spins by each SDK is large
enough to affect the coherent leakage cancellation [70].

We leave a detailed investigation of the intricate coupling dynamics for
future work. Here, to construct faithful spinor matterwave control, we simply
combine the $\tilde{U}_{\rm ud}^{(2N)}$ and $\tilde{U}_{\rm du}^{(2N)}$
sequence in Eq. (11) to form the balanced chirp-alternating SDK sequence. The
idea is to exploit the time-reversal dynamics again to let the dynamic phases
by the leakage-suppressing $\tilde{U}_{\rm du}^{(2N)}(\mathbf{k}_{\rm R})$ and
$\tilde{U}_{\rm ud}^{(2N)}(\mathbf{k}_{\rm R})$ cancel each other.

We evaluate the average gate fidelity for realizing the quantum gate for
$U_{\rm K}^{(4N)}(\mathbf{k}_{\rm R})$ (Eq. (8)) with $\tilde{U}_{\rm
uddu}^{(4N)}(\mathbf{k}_{\rm R})$. The results of $\mathcal{F}_{m}^{(4N)}$,
with otherwise identical Hamiltonian parameters as those in Fig. 7(a-c), are
shown in Fig. 7(d-f). Similar to Figs. 7(a-c), here the infidelity
$1-\mathcal{F}_{m}^{(4N)}$ for the $\tilde{U}_{\rm
uddu}^{(4N)}(\mathbf{k}_{\rm R})$ sequence largely follows
$\varepsilon_{m,\Delta m}^{(4N)}$ for $\tilde{U}_{\rm ud}^{(4N)}$. Therefore,
the improvement is most significant for the $m=\pm 1$ sub-spins to approach
$\mathcal{F}_{m}^{(4N)}\approx 98\%$ at $4N=80$. For $m=0,\pm 2$ sub-spins, we
also see improved gate fidelity $\mathcal{F}^{(4N)}_{m}>90\%$ span a
substantial range of intensity for $4N$ up to 80, with lower $\mathcal{F}$
values for $\mathcal{A}_{\rm R}>6\pi$ confined around specific
$\mathcal{A}_{\rm R}$ only (deep-blue area). By moderately increasing
$\Delta_{e}$, in Appendix F we demonstrate $\mathcal{F}>99\%$ with more
confined low-$\mathcal{F}$ region after many kicks. Experimentally, the narrow
low-$\mathcal{F}$ intensity region should be avoided for faithful multi-Zeeman
spinor matterwave control of 85Rb with the balanced chirp-alternating scheme.

Finally, it is interesting to note that for the nearly perfect SDK, the gate
infidelity $1-\mathcal{F}^{(4N)}_{m}$ is dominantly due to the spin leakage,
as demonstrated by the remarkably similar $1-\mathcal{F}^{(4N)}_{m}$ and
$\varepsilon^{(4N)}_{m,\Delta m}$ data in Fig. 6 and Fig. 7. This is a result
of dynamic phase cancellation in the adiabatic limit for both the
$\tilde{U}_{\rm uu}^{(4N)}$ and $\tilde{U}_{\rm uddu}^{(4N)}$ sequences, when
the $\pm\mathbf{k}_{\rm R}$ swapping as detailed next is perfect.

###  IV.4 Robust cancellation of dynamic phase

Figure 8:  Unbalanced dynamic phase $\varphi_{\rm D}=\varphi^{(4)}_{{\rm
D},m}$ according to Eq. (9) vs pulse area $\mathcal{A}_{\rm R}$ for $m=0$
(a,d), $m=\pm 1$ (b,e) and $m=\pm 2$ (c,f) subjected to $\tilde{U}_{\rm
uu}^{(4)}(\mathbf{k}_{\rm R})$ (left) and $\tilde{U}_{\rm
uddu}^{(4)}(\mathbf{k}_{\rm R})$ (right) controls. The corresponding 2-photon
detuning profiles are illustrated on top of (a,d). Parameters for the
simulations are according to the experimental Sec. III.3, except here the
mirror reflectivity $r$ varies. The red dashed lines give $f_{\rm SDK}$ for
single kicks (different $r$-curves closely overlap).

The numerical results in Fig. 7 demonstrate that precise dynamic phase
cancellation can be achieved by pairing $\tilde{U}^{(2N)}_{\rm
ud}(\mathbf{k}_{\rm R})$ with $\tilde{U}^{(2N)}_{\rm du}(\mathbf{k}_{\rm R})$
into the balanced $\tilde{U}^{(4N)}_{\rm uddu}(\mathbf{k}_{\rm R})$. In fact,
we find that the dynamic phase cancellation in the balanced chirp-alternating
scheme is substantially more robust than the traditional double-SDK by Eq.
(8), as following.

As in Sec. II.4, the traditional method of dynamic phase cancellation [6]
requires perfect $\mathbf{k}_{\rm R}\leftrightarrow-\mathbf{k}_{\rm R}$
swapping for the successive $U_{\rm K}(\mathbf{k}_{\rm R})$ $U_{\rm
K}(-\mathbf{k}_{\rm R})$ controls. Practically the k-vector swapping is
typically accompanied by a modification of ${\bf E}_{1,2}$ intensity ratio.
For example, in the retro-reflection setup (Fig. 2(a)), the amplitude of the
reflected beam is reduced by a $\kappa<1$ factor due to the imperfect
reflection, leading to unbalanced dynamic phases $\varphi_{\rm D}$ associated
with $\tilde{U}(\pm\mathbf{k}_{\rm R})$ to compromise their cancellation in
the traditional double-SDK (Eq. (8)) [10, 6]. In fact, this systematic exists
quite generally in retro-reflection setups since the 2-photon shift
$\delta_{0}$ is sensitive to the laser intensities ratios [25, 62].

In contrast, here we notice that in the $\tilde{U}^{(4N)}_{\rm
uddu}(\mathbf{k}_{\rm R})$ sequence (Eq. (11)) the dynamic phase by any
$\tilde{U}^{(2)}_{\rm du}(\mathbf{k}_{\rm R})$ pair is expected to be
cancelled by a $\tilde{U}^{(2)}_{\rm ud}(\mathbf{k}_{\rm R})$ pair later. In
the adiabatic limit the cancellation is guaranteed, since for free atom
starting from any specific 2-level spin state, the sign of the chirp frequency
$\delta_{\rm swp}$ dictates the adiabatic quantum number [61] and thus the
sign of the dynamic phase in the adiabatic limit.

To demonstrate the robust dynamic phase cancellation, in Fig. 8, we compare
$\varphi_{\rm D}$ according to Eq. (9) for the $\tilde{U}^{(4)}_{\rm
uu}(\mathbf{k}_{\rm R})$ and $\tilde{U}^{(4)}_{\rm uddu}(\mathbf{k}_{\rm R})$
controls. The $f_{\rm SDK}$ values are given in the same plots, with which we
see that the high $f_{\rm SDK}$ in the quasi-adiabatic regime to be hardly
affected even by a poor reflectivity $r=|\kappa|^{2}=0.5$. On the other hand,
in contrast to Fig. 8(a-c) (left) where $r\approx 1$ is required for precise
suppression of $\varphi_{\rm D}$ (blue line), in Fig. 8(d-f) (right) the
$\varphi_{\rm D}$ is largely suppressed so long as $f_{\rm SDK}\approx 1$. It
is worth noting that the residual variation of $\varphi_{\rm D,m}^{(4)}$ for
$\tilde{U}^{(4)}_{\rm uddu}(\mathbf{k}_{\rm R})$ in Fig. 8(d-f) in the quasi-
adiabatic regime are coupled to the coherent spin leakage dynamics (Fig.
6(d-f)). Nevertheless, even for $r=0.5$ similar to the experimental situation
in this work, the residual phase variations are still limited to
$|\varphi_{\rm D,m}^{(4)}|<$50 mrad for the $m=0,\pm 2$ sub-spins, and
$|\varphi_{{\rm D},m}^{(4)}|<$3 mrad for the $m=\pm 1$ sub-spins,
respectively.

###  IV.5 Area-enhancing atom interferometry

So far in this section, we have shown that matterwave phase gate prescribed by
the ideal double-SDK (Eq. (8)) can be faithfully implemented by the balanced
chirp-alternating sequence $\tilde{U}^{(4N)}_{\rm uddu}(\mathbf{k}_{\rm R})$,
to coherently shift any $|a_{m}\rangle$, $|b_{m}\rangle$ components of
hyperfine spinor matterwave with opposite $\pm 4N\hbar\mathbf{k}_{\rm R}$
momentum within nanoseconds. For control parameters in this experimental
demonstration, our numerical results already suggest fairly high gate fidelity
with efficient suppression of the coherent spin leakage and inhomogenuous
dynamic phase. In future work, by increasing $\Delta_{e}/\Gamma$ and
$\Delta_{e}/\Delta_{\rm hfs,e}$ ratios and the laser intensities $I_{1,2}$ in
proportion (Appendix F), the residual imperfections can be further suppressed
to meet the exquisite requirements in the applications of quantum information
processing [11, 12, 16, 17] and quantum enhanced atom interferometry [18, 19,
20, 21]

Figure 9: Enhancing the enclosed area of an atom interferometer with four
$\tilde{U}_{\rm uddu}(\pm\mathbf{k}_{\rm R})$ SDK sequences. (a): Schematic of
the interferometry scheme. The dashed lines at $t=0,T,2T$ represent regular
Raman interferometry controls for $R_{1}=R_{\varphi}(\pi/2)$ splitter,
$R_{2}=R_{\varphi}(\pi)$ mirror and $R_{3}=R_{\varphi}(\pi/2)$ combiner of the
spinor matterwave respectively with $\varphi=\mathbf{k}_{\rm R}\cdot{\bf r}$.
The atomic wavefunctions in $|b\rangle$ and $|a\rangle$ states are represented
by the blue and red lines [3]. The four thick vertical lines at
$t=\tau_{1},T-\tau_{1},T+\tau_{1},2T-\tau_{1}$ with red curved arrows
represent $\tilde{U}_{1}=\tilde{U}_{\rm uddu}^{(4N)}(\mathbf{k}_{\rm R})$,
$\tilde{U}_{2}=\tilde{U}_{\rm uddu}^{(4N)}(-\mathbf{k}_{\rm R})$,
$\tilde{U}_{3}=\tilde{U}_{\rm uddu}^{(4N)}(-\mathbf{k}_{\rm R})$ and
$\tilde{U}_{4}=\tilde{U}_{\rm uddu}^{(4N)}(\mathbf{k}_{\rm R})$ sequences
respectively. We consider $\tau_{1}\ll T$. By properly choosing $\tau_{1}/T$
ratio, spurious interference by multiple imperfect controls can be suppressed,
and are not included in the simulations. (b-d): The interferometry phase
offset $\delta\Phi$ and contrast $\mathcal{C}$ as a function of SDK number
$n=4N$ and pulse area $\mathcal{A}_{\rm R}$. Here the single-photon detuning
is chosen as $\Delta_{e}=-3.3~{}\omega_{{\rm hfs},g}$. The simulations average
over $m=-2,-1,0,1,2$ states, and include $\Gamma_{e}=0.017~{}\omega_{{\rm
hfs},e}$ as for the case of 85Rb. The phase offsets and interferometry
contrasts for the simulated signals locally averaged over a $50\%$ intensity
distribution are given in (c,d). The data in Fig. (e-g) are similar to Fig.
(b-d), but with an increased single-photon detuning of
$\Delta_{e}=-6.6~{}\omega_{{\rm hfs},g}$.

Here, to demonstrate the utility of the adiabatic SDK sequence for precision
measurements, we numerically investigate a simple atom interferometry scheme
[3, 10, 6] where an “enclosed area” $A$ is enhanced by the
$\tilde{U}^{(4N)}_{\rm uddu}(\pm\mathbf{k}_{\rm R})$ sequences. As in the
Mach-Zehnder configuration in Fig. 9(a), we consider the two spinor matterwave
components forming a loop to interfere at $t=2T$ (the dashed lines). When
sensing a constant force field, the difference of matterwave phase shifts
along the two paths, $\Delta\Phi\propto A$ associated with the path-dependent
diffraction phase [43], is read out interferometrically [44, 71]. Here $A$ is
the spatial-temporal “area” enclosed by the loop. When the duration of the
pulsed rotations are negligibly short relative to the “interrogation time”
$T$, then $A=v_{\rm R}T^{2}$ is easily evaluated. Here $v_{\rm R}=\hbar k_{\rm
eff}/M$ is the photon recoil velocity. Clearly, when the atomic interferometer
acts as a force sensor, its sensitivity increases with $A$. In fact, to
achieve as large “area” $A$ as possible within a measurement time $T$ is of
general importance to precision measurements with light pulse atom
interferometry [34, 3, 44].

More specifically, the enclosed area is defined as $A=\int_{0}^{2T}\Delta
z(t)dt$ during a 3-pulse Raman interferometry sequence by integrating the
relative displacement $\Delta z(t)$ between the two matterwave diffraction
paths under the three operations as splitter ($t=0$), mirror ($t=T$) and
combiner ($t=2T$). We generally refer the idealized local spin rotations as
$R_{\varphi}(\theta)=\cos(\theta/2){\bf
1}+i\sin(\theta/2)(e^{i\varphi}\sigma_{+}+e^{-i\varphi}\sigma_{-})$ for the
Raman interferometer, for any spin state within
$\\{|a_{m}\rangle,|b_{m}\rangle\\}$. Here $\varphi=\mathbf{k}_{\rm R}\cdot{\bf
r}$ is the local Raman optical phase. Notice the spatial-dependent
$R_{\varphi}(\theta)$ rotation can in principle be generated by the Eq. (4)
Hamiltonian [72] as phase-coherent “half” and “full” kicks. The splitter and
mirror operations in the standard light Raman interferometer can then be
expressed as $R_{1}=R_{\varphi}(\pi/2)$, $R_{2}=R_{\varphi}(\pi)$ and
$R_{3}=R_{\varphi}(\pi/2)$ respectively to manipulate the spin states while
imparting the $\pm\hbar\mathbf{k}_{\rm R}$ photon recoil momentum.

We now consider enhancing the enclosed area $A$ of the standard 3-pulse Raman
interferometer with the chirp-alternating SDK sequence. In particular, we
consider the Fig. 9(a) scheme with the spinor matterwave diffraction paths
marked with solid lines: a $\tilde{U}_{1}=\tilde{U}^{(4N)}_{\rm
uddu}(\mathbf{k}_{\rm R})$ is first applied at $t=\tau_{1}$ to increase the
momentum displacement between the two interfering paths from $\Delta{\bf
p}=\hbar\mathbf{k}_{\rm R}$ by $R_{\varphi}(\pi/2)$ to $\Delta{\bf p}=(2\times
4N+1)\hbar\mathbf{k}_{\rm R}$ with the spin-dependent kicks. This $\Delta{\bf
p}$-enhancement is followed by an opposite
$\tilde{U}_{2}=\tilde{U}^{(4N)}_{\rm uddu}(-\mathbf{k}_{\rm R})$ at
$t=T-\tau_{1}$ before the $R_{2}$-operation to recover the initial $\Delta{\bf
p}$. To ensure that the interfering paths spatially overlap at $t=2T$, an
additional pair of opposite momentum boosts, $\tilde{U}_{3,4}$ are applied at
$t=T+\tau_{1}$ after the $R_{2}$ and $t=2T-\tau_{1}$ before the $R_{3}$
operation respectively. By properly choosing the $\tau_{1}/T$ ratio, spurious
interference by imperfect $R_{1,2,3}$ and $\tilde{U}_{1,2,3,4}$ controls can
be suppressed [73, 74]. With $T\gg\tau_{1}$, the enclosed area of the
resulting interfering loop is enhanced to $A^{\prime}=(2\times 4N+1)A$.

For the numerical simulation, we consider at $t=0$ the atomic state to be
initialized at certain $|b_{m}\rangle$ and subjected to the
$\tilde{U}^{(4N)}_{\rm uddu}(\mathbf{k}_{\rm R})$-enhanced 3-pulse
interferometry sequence. The output atomic state, right before the final
matterwave combiner $R_{3}$, can then be written as $|\psi_{m}({\bf
r},2T^{-})\rangle=U_{\rm AI}|c_{m}\rangle$, with $U_{\rm
AI}=U_{f}(\tau_{1})\tilde{U}_{4}U_{f}(T-2\tau_{1})\tilde{U}_{3}R^{\prime}_{2}\tilde{U}_{2}U_{f}(T-2\tau_{1})\tilde{U}_{1}R^{\prime}_{1}$
to be ${\bf r}$-dependent. Here $U_{f}(t)$ designates free propagation of
matterwave for time $t$. The $R^{\prime}_{1}=U_{f}(\tau_{1})R_{1}$,
$R^{\prime}_{2}=U_{f}(\tau_{1})R_{2}U_{f}(\tau_{1})$ takes into account the
free propagation of matterwave between the standard $R_{j}$ and area-enhancing
$\tilde{U}_{j}$ sequences. We numerically evaluate
$|\psi_{m}(z,2T^{-})\rangle$ for 1D spinor matterwave between $0<z<\lambda/2$
($\lambda$ is the optical wavelength), as described in Appendix B. To focus on
the performance of SDK, we set $R_{1,3}$ as perfect $\pi/2$ pulses and $R_{2}$
as perfect $\pi$ mirror pulse respectively. A further simplification sets
$\mathbf{k}_{\rm R}=0$ for the idealized $R_{1,2,3}$ controls, with which we
numerically evaluate
$\expectationvalue{\Sigma_{j}}=\langle\psi_{m}(z,2T^{-})|\Sigma_{j}|\psi_{m}(z,2T^{-})\rangle_{z,m}$
for an initially unpolarized atomic sample. Here
$\Sigma_{j}=\sum_{m^{\prime}=-F_{b}}^{F_{b}}\sigma_{j}^{(m^{\prime})}$ are
summed over all $m$ sub-spins for Pauli matrices with $j=x,y,z$. $\Sigma_{j}$
corresponds to observables of experimental measurements in which Zeeman
sublevels are not resolved, as in most atom interferometry experiments with
hyperfine state-dependent fluorescence readouts [34].

With $U_{f}$ chosen as free 1D propagation, the values of $\tau_{1}$ and $T$
only affect contributions of spurious interfering paths into the final
readouts [73, 74] in the simulation. With $R_{1,2,3}$ set as ideal, the
spurious interfering paths are from imperfect $\tilde{U}_{1,2,3,4}$
diffractions only. Notably, since successive adiabatic SDKs here within each
$\tilde{U}_{j}$ last merely tens of nanoseconds, the spatial displacements
among the spurious interfering paths are negligibly small comparing with the
typical coherence length of cold atom samples, and therefore do not alter the
matterwave dynamics [75, 76]. We have randomly sampled the atomic initial
position and velocity to numerically verify that the residual spurious
interference are indeed suppressible. Practically, to generate the results in
Fig. 9(b-d) with all spurious interference removed in an efficient manner, we
simply apply a digital filter to remove unwanted diffraction orders after each
$\tilde{U}_{j}$ sequence.

We are particularly interested in the interferometry contrast $\mathcal{C}$
and diffraction phase offset $\delta\Phi$. The contrast $\mathcal{C}$ sets the
quality of the final matterwave interference fringes. The phase offset
$\delta\Phi$, stemming from the unbalanced dynamic phase by the four
$\tilde{U}_{\rm uddu}$ sequences, enters the interferometry readout as
systematic bias against any precision measurements or controls. We evaluate
the interferometry contrast as
$\mathcal{C}=\sqrt{\expectationvalue{\Sigma_{x}}^{2}+\expectationvalue{\Sigma_{y}}^{2}}$,
keeping in mind the ideal $R_{3}$ rotates ${\bf\Sigma}=\\{\Sigma_{j}\\}$ by
$90^{\circ}$ for the final population readouts. The diffraction phase offset
is instead evaluated as
$\delta\Phi=\arg\quantity[\expectationvalue{\Sigma_{x}}+i\expectationvalue{\Sigma_{y}}]-\Phi_{0}$
with $\Phi_{0}$ to be the relative phase between $\ket{a_{m}}$ and
$\ket{b_{m}}$ right after the ideal $R_{1}$ splitter. Typical numerical
results are presented in Figs. 9(b-d). The simulation is again performed on
the 85Rb D1 line, here with spontaneous emission included. For Figs. 9(b-d) on
the left panel, the control laser parameters are chosen close to this
experimental work, with $\Delta_{e}=-3.3~{}\omega_{{\rm hfs},g}$ so both
$\varepsilon_{\rm sp}$ and $\varepsilon_{\Delta m}$ are quite substantial.
Nevertheless, we find four $n=4N=12$ chirp-alternating SDKs can be applied for
a 25-fold enhancement of interferometry enclosed area, at a moderate expense
of reduced contrast to $\mathcal{C}\approx 0.5$ (which is still highly useful
[34]), with $\delta\Phi<0.01$ to be experimentally calibrated when necessary.
Here, to avoid excessive spontaneous emission and coherent leakage (Fig.
6(d-f), the peak $\mathcal{A}_{\rm R}\approx 6-8\pi$ needs to be chosen (Fig.
9(b-d)). On the other hand, by doubling the single photon detuning to
$\Delta_{e}=-6.6~{}\omega_{{\rm hfs},g}$ (with laser intensity increased in
proportion to maintain the Raman Rabi frequency), $\varepsilon_{\rm sp}$ are
halved, while the impact of $\varepsilon_{\Delta m}$ leakage are dramatically
suppressed in the $\tilde{U}_{\rm uddu}$ sequence (Figs. 9(e-g), also see
Appendix F) [77]. The further detuned $\tilde{U}_{\rm uddu}$ sequence should
thus support up to 50-fold enhancement of interferometry enclosed area, with
spontaneous-emission-limited $\mathcal{C}>0.5$ contrast.

##  V Discussions

Significant aspects of advanced quantum technology today are based on
controlling alkaline atoms through their center-of-mass motion and ground-
state hyperfine interaction. The two long-lived degrees of freedom can be
entangled optically by transferring photon recoil momentum with Raman
excitations. The tiny atomic recoil effect can be amplified by the repetitive
application of such excitations. The spin-dependent large momentum transfer is
expected to improve the scalability for precision measurements in atom
interferometry [3, 10, 6] and for quantum information processing with trapped
ions [11, 12, 5, 13, 17]. Practically, unlike interrogating microscopically
confined single ions where Raman excitation with multiple-THz single-photon
detuning is feasible [39, 8], addressing larger samples prefers efficient
excitations at moderate single-photon detunings. The seemingly unavoidable
imperfections associated with spin-leakages and dynamic phases need to be
managed in non-traditional ways, for achieving the high speed, high fidelity
control with intensity-error resilience required by the next generation
quantum technology [16, 17, 18, 19, 20, 21].

In this work, we have demonstrated a novel configuration of adiabatic SDK
implemented on an optical delay line, which is able to reach the speed limit
of Raman SDK control [5, 39], featuring robust intensity-error resilience,
while maintaining various advantages of optical retro-reflection established
for precision atom interferometry. The experimental characterization of the
technique is limited to the hyperfine population and spin-dependent momentum
transfers, but we clarify in Sec. IV that high precision phase gates enabling
spin-dependent large momentum transfer can be efficiently realized at the
moderate single-photon detuning, by properly programming the chirp direction
to suppress the accumulation of coherent errors. We have provided numerical
evidence that the chirp-balanced SDK scheme support faithful, parallel $\Delta
m=0$ control of multi-Zeeman spinor matterwave, with giant spin-dependent
forces applied within nanoseconds to rapidly shift the phase-space spin
separation, even with the $\sim$10 mW laser power as in this work. Since
within nanoseconds various low-frequency noises including those due to
matterwave dispersion are negligible for cold atoms, we expect accurate
implementation of the full scheme to be benchmarked in future interferometric
measurements.

Our work suggests that Raman SDK can be dynamically perfected against multi-
level couplings and dynamic phase broadening, for error-resilient parallel
control of multiple hyperfine spinors within nanoseconds, with exquisite
precision. Operating in the unconventional regime of Raman control [25, 41],
the technique requires relatively moderate laser intensity when comparing with
similar techniques for controlling microscopically confined ions [5, 16, 17].
Nevertheless, we note the high speed SDK demonstrated in this work is realized
by weakly focusing the milli-Watt level output from OAWG [36] to a mesoscopic
sample to reach the required intensity. Our technique is immediately useful
for coherently controlling mesoscopic ultra-cold samples for quantum
simulation [78, 79] and atom interferometry [80, 81, 82, 83]. By improving the
peak power of the nanosecond pulses [84, 85, 86, 14, 87], our method may
drastically enhance the practical benefits of large-momentum beamsplitting in
Raman atom interferometry [3, 10, 6], and to support ultra-precise matterwave
control for quantum enhanced technologies [18, 19, 21, 20].

## Acknowledgements

We thank Yidi Ma and Xing Huang for experimental assistance. We are grateful
to Prof. Yiqiu Ma for insightful comments to the manuscript, and to Prof.
Xiaopeng Li and Prof. Haidong Yuan for helpful discussions. We acknowledge
support from National Key Research Program of China under Grant No.
2017YFA0304204 and No. 2016YFA0302000, from NSFC under Grant No. 12074083.

##  Appendix A Full Hamiltonian

The theoretical analysis as well as numerical simulation in this work is based
on the full light-atom interaction Hamiltonian on the D1 line. Following the
notation in the main text, the effective, non-Hermitian Hamiltonian is written
as

$\begin{split}H_{\rm
eff}(\mathbf{r},t)=&\hbar\sum_{e}\quantity(\omega_{e}-\omega_{e0}-i\Gamma_{e}/2)\sigma^{e_{l}e_{l}}+\\\
&\hbar\sum_{c=a,b}\quantity(\omega_{c}-\omega_{g0})\sigma^{c_{m}c_{m}}+\\\
&\frac{\hbar}{2}\sum_{c={a,b}}\sum_{e}\Omega^{j}_{c_{m}e_{l}}(\mathbf{r},t)\sigma^{c_{m}e_{l}}+{\rm
h.c.}\end{split}$ (15)

Here $\omega_{e0},\omega_{g0}$ are decided by the energy of reference level in
the excited and ground state manifolds respectively, chosen as the top
hyperfine levels in this work. The laser Rabi frequency,

$\begin{split}\Omega^{j}_{e_{l}a_{m}\quantity(b_{n})}(\mathbf{r},t)&\equiv-\frac{\matrixelement{e_{l}}{\mathbf{d}\cdot{\mathbf{e}}_{j}\mathcal{E}_{j}(\mathbf{r},t)}{a_{m}\quantity(b_{n})}}{\hbar},\end{split}$
(16)

is accordingly written in the $\omega_{e0,g0}$ frame under the rotating wave
approximation. $\mathbf{d}$ to be the atomic electric dipole operator. The
$\sigma^{a_{m}e_{l}}=\outerproduct{a_{m}}{e_{l}}$,
$\sigma^{e_{l}a_{m}}=\outerproduct{e_{l}}{a_{m}}$ are the raising and lowering
operators between states $|a_{m}\rangle$ and $|e_{l}\rangle$. Similar $\sigma$
operators are defined for all the other $|a_{m}\rangle$, $|b_{n}\rangle$ and
$|e_{l}\rangle$ state combinations.

The full $H_{\rm eff}$ in Eq. (15) is rewritten as Eq. (4) in the main text.
For weak off-resonant pulses and to facilitate understanding of ground-state
Raman interaction, we can also adiabatically eliminate the excited states to
approximately have

$\begin{split}H_{\rm eff}(\mathbf{r},t)&\approx\\\
&\hbar\sum_{e}\sum_{j=1,2}\frac{\Omega^{j}_{a_{m}e_{l}}\Omega^{j*}_{a_{n}e_{l}}}{4\quantity(\nu_{j}-\omega_{ea})}\sigma^{a_{m}a_{n}}+\frac{\Omega^{j}_{b_{m}e_{l}}\Omega^{j*}_{b_{n}e_{l}}}{{4\quantity(\nu_{j}-\omega_{eb})}}\sigma^{b_{m}b_{n}}\\\
&+\hbar\sum_{e}\frac{\Omega^{1}_{a_{m}e_{l}}\Omega^{2*}_{b_{n}e_{l}}}{4\Delta_{e}}{e}^{i\mathbf{k}_{\rm
R}\cdot\mathbf{r}}\sigma^{a_{m}b_{n}}+{\rm h.c.}\\\
&+\hbar\sum_{e}\frac{\Omega^{2}_{a_{m}e_{l}}\Omega^{1*}_{b_{n}e_{l}}}{4\Delta_{e}}{e}^{i\mathbf{k}_{\rm
R}\cdot\mathbf{r}}{e}^{i2\omega_{ab}t}\sigma^{b_{n}a_{m}}+{\rm
h.c.},\end{split}$ (17)

with the convention of summing over repeated $m,n,l$ indices [51]. The single
photon detuning is defined as
$\Delta_{e}=\omega_{1}-\omega_{ea}=\omega_{2}-\omega_{eb}$. The vector
$\mathbf{k}_{\rm R}=\mathbf{k}_{2}-\mathbf{k}_{1}$ is the k-vector associated
with the Raman transition driven by the counter-propagating pulses.

The Raman coupling associated with $\sigma^{b_{n}a_{m}}$ in line 2 of Eq. (17)
is accompanied by a $\pm\hbar\mathbf{k}_{\rm R}$ momentum transfer to the
spinor matterwave. Similarly, the “counter-rotating” term in line 3 of Eq.
(17) leads to an opposite, $\mp\hbar\mathbf{k}_{\rm R}$ momentum transfer. For
smooth laser pulses with bandwidth $\delta\omega\ll\omega_{ab}$ to be
discussed, this “counter-rotating” Raman process is energetically suppressed.
The regime of resonant Raman interaction with directional momentum transfer is
the focus of this work.

##  Appendix B Numerical model

We numerically simulate the evolution of 1D spinor matterwaves with full D1
light-atom interactions [64, 88] driven by the counter-propagating Raman
pulses as in Fig. 1 and 2. To account for radiation damping, we follow a
stochastic wavefunction method [89] to evaluate the wavefunction $|\psi({\bf
r},t)\rangle$ for atom at location ${\bf r}$ under the non-Hermitian
Hamiltonian $H_{\rm eff}$ (see Eq. (15) with non-Hermitian part
$\hbar\sum_{e}i\Gamma_{e}\sigma^{e_{l}e_{l}}/2$ in the first line). Here
$\Gamma_{e}$ is the natural linewidth of the D1 line. The simulations treat
both the internal and external motion of the spinor matterwave quantum
mechanically. For the purpose, we sample $|\psi({\bf r},t)\rangle$ densely
over a uniform grid within $0<z<\lambda/2$ and sparsely in the $x-y$ plane,
and follow a split-operator method to evaluate internal/external atomic motion
numerically with interleaved steps. Taking advantage of the short $\tau_{\rm
c}$ for single SDK, the internal state dynamics is evaluated within a single
step with frozen external motion under a local $|\psi({\bf r})\rangle$ basis,
with atomic position ${\bf r}$ treated as a parameter of $H_{\rm eff}$. The
evaluation of observables later is normalized by $\mathcal{N}=\sum_{\bf
r}\langle\psi({\bf r},t=0)|\psi({\bf r},t=0)\rangle$, with corrections from
stochastic contributions to be discussed shortly. Between SDKs, a Fourier
transform along ${\bf e}_{z}$ can be performed to evolve the free-flying
spinor matterwave along $z$ if necessary. To save computation resources, the
relatively simple atomic dynamics in the x-y plane is ignored. To evaluate
momentum distribution of spinor matterwave, we simply perform a Fourier
transform to the space dependent $\langle c_{m}|\psi({\bf r},t)\rangle$ for
any specific spin state $|c_{m}\rangle$.

Beyond the coherent evolution, the simulation complexity is substantially
reduced by skipping the evaluation of stochastic trajectories heralded by a
single “quantum jump” [54, 55]. Specifically, after each pulsed interaction,
the trajectories suffering a quantum jump are simply assumed to repopulate
$\\{|a\rangle,|b\rangle\\}$ in a uniform manner with properly shifted photon
recoil momentum. Without further evolution, these trajectories contribute to
the evaluation of incoherent, single-time observables such as hyperfine
population and photon momentum transfer. The overall probability of
spontaneous emission is determined by the norm of the final wavefunction,
$\varepsilon_{\rm sp}=1-\frac{1}{\mathcal{N}}\sum_{\bf r}\langle\psi({\bf
r},\tau_{\rm tot})|\psi({\bf r},\tau_{\rm tot})\rangle$, after a total
evolution time $\tau_{\rm tot}$. The simplification is generally justified for
evaluating coherent observables of interest, since the expectation values
shifted stochastically lead to zero coherent contributions. For the incoherent
observables such as average photon momentum and hyperfine population, the
simplification is supported by the simple D1 structure under consideration
here [64], where a single spontaneous emission effectively randomizes the
following Raman interaction dynamics.

##  Appendix C A Markovian model for $f_{\rm SDK}$ estimation

For atoms subjected to multiple SDKs, the dynamics of spinor matterwave that
deviates from the ideal control can be depicted as diffusing in a “momentum-
lattice” [15]. Our numerical simulation suggests that with fair efficiency of
single adiabatic pulse to achieve hyperfine transfer efficiency of $f_{\rm
R}>95\%$, the resulting average momentum $p_{n}$ and population
$\rho_{aa/bb,n}$ roughly follow a simple Markovian model. The model assumes
that both the momentum and population transfer by the next kick are decided by
the present population difference $\rho_{aa}-\rho_{bb}$ only. The details of
the Markovian model are described as following.

Suppose that after $n$ kicks, the atomic ensemble is with momentum $p_{n}$ (in
unit of $\hbar\mathbf{k}_{\rm R}$) and population contrast
$\mathcal{C}_{n}\equiv|\rho_{aa,n}-\rho_{bb,n}|$, then the next kick will
impart momentum as

$\Delta p_{n+1}=p_{n+1}-p_{n}=f_{0}(1-\varepsilon_{\rm sp}/2)\mathcal{C}_{n},$
(18)

where $f_{0}$ is the hyperfine population transfer efficiency in absence of
the spontaneous emission. Here we have assumed that during the single pulse
process, the spontaneous emission occurs with a uniform distribution of
probability, thus the associated population recycled to the ground states
acquires half of $\hbar\mathbf{k}_{\rm R}$ momentum on average.

Similarly, the population distribution can be written as

$\begin{split}\rho_{aa,n+1}=&(1-\varepsilon_{\rm
sp})\quantity[(1-f_{0})\frac{1+\mathcal{C}_{n}}{2}+f_{0}\frac{1-\mathcal{C}_{n}}{2}]\\\
&+\varepsilon_{\rm sp}/2\\\ \rho_{bb,n+1}=&(1-\varepsilon_{\rm
sp})\quantity[f_{0}\frac{1+\mathcal{C}_{n}}{2}+(1-f_{0})\frac{1-\mathcal{C}_{n}}{2}]\\\
&+\varepsilon_{\rm sp}/2,\end{split}$ (19)

so that

$\mathcal{C}_{n+1}=(1-\varepsilon_{\rm sp})(2f_{0}-1)\mathcal{C}_{n}.$ (20)

We define Raman transfer efficiency as $f_{\rm R}=f_{0}(1-\varepsilon_{\rm
sp}/2)$. When both $f_{0}$ and $1-\varepsilon_{\rm sp}$ are close to unity,
Eqs. (18)(20) can be approximated as

$\begin{split}&p_{n+1}-p_{n}=f_{\rm R}\mathcal{C}_{n},\\\
&\mathcal{C}_{n+1}=(2f_{\rm R}-1)\mathcal{C}_{n}.\end{split}$ (21)

With the recursion relations by Eq. (21), we arrive at

$\begin{split}&p_{n}=f_{\rm R}\frac{1-(2f_{\rm R}-1)^{n}}{1-(2f_{\rm
R}-1)},\\\ &\mathcal{C}_{n}=(2f_{\rm R}-1)^{n}.\end{split}$ (22)

Finally, we remark that for the Raman SDK, there is a slight difference of
spontaneous emission loss for single kicks between the $a\rightarrow b$ and
$b\rightarrow a$ process. As we consider repetitive SDK with $n$ up to a quite
large number (e.g. $n_{\rm max}=25$ in our experiment), we effectively set the
same $\varepsilon_{\rm sp}$ parameter for the opposite population transfer
processes.

##  Appendix D Absorption imaging analysis

In Sec. III we have introduced the double imaging technique. This section
provides details on deriving recoil momentum $p_{n}$ and population transfer
$\rho_{aa}$ from the imaging data.

Our atomic sample is prepared in a cross-dipole trap with slight asymmetry.
When deriving the momentum transfer from the double images as in Fig. 3, we
found that neither before nor after the time-of-flight, the absorption profile
can fit perfectly to a 2D Gaussian. In addition, due to the relatively short
exposure time of 20$~{}\mu$s to the weak probe ($s=1$), there is substantial
photon shot noise in single-shot images. To faithfully retrieve central
position and atom number from each pair of double-image, we take the following
procedure. First, we repeat the same type of measurements for $N=80$ shots,
and do a principal components analysis to all pairs of double-images after
background subtractions. The first three components are kept for the following
analysis. We then do a 2D Gaussian fit, expanding the fitted Gaussian profiles
to $1.5$ times the waist into a wide enough step-wise mask. The population
ratios $\rho_{aa/bb}$ are evaluated by the ratio of total counts between the
two images within the mask, where the center-of-mass (COM) positions $z_{1}$
and $z_{2}$ are also directly evaluated.

We note that the first image senses atoms at $F=3$ “$|a_{m}\rangle$” states
only. In other words, the atoms kicked to the “visible” (or “invisible”,
depending on whether the kicking number $n$ is an even or odd number)
hyperfine levels are post-selected. Since there is a small interval
$\tau_{\rm{p},1}$ between SDK and the first image, there is a bias to the COM
position $z_{1}$ of the whole cloud due to this post-selection. To correct for
the bias, we rewrite the position difference as
$z_{2}-z_{1}=\tilde{v}_{n}\tau_{\rm tof}+\rho_{bb,n}\tilde{v}_{b,n}\tau_{{\rm
p},1}$. Here $\tilde{v}_{n}$ and $\tilde{v}_{b,n}$ are the mean velocity for
the whole atomic ensemble and for the atoms in $F=2$ after $n$ SDKs,
respectively. The correction to the post-selection induced velocity bias is
then given by

$\tilde{v}_{n}=v_{n}\quantity(1-\rho_{bb,n}v_{b,n}\frac{\tau_{1}}{\tau_{\rm
tof}})\equiv
v_{n}\quantity(1-\xi\quantity(n,v_{n})\frac{\tau_{\rm{p},1}}{\tau_{\rm
tof}}).$ (23)

Here, $\xi\quantity(n,v_{n})$ depends on the relation between $v_{n}$ and
$\rho_{bb,n},v_{b,n}$. The relation can be approximated with the model in
Appendix C. With
$\tau_{\rm{p},1}\sim$15\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$$ in our
experiment, this correction is typically $1-\tilde{v}_{n}/v_{n}\sim\pm 5\%$
(the $\pm$ signs depend on $n$), which impacts $f_{\rm R}$ at $3\%$ level. The
correction is model-dependent. We correct for the bias in our final estimation
of $f_{\rm SDK}$ and leave $3\%$ as the dominant uncertainty in our $f_{\rm
SDK}$ estimation.

##  Appendix E Impact of mirror optical loss to $f_{\rm SDK}$

Figure A1: (a) Numerical simulations for $f_{\rm R}$ and $f_{\rm SDK}$ vs
Raman pulse area $\mathcal{A}_{\rm R}$ for different reflectivity
$r=\absolutevalue{\kappa}^{2}$ of the retro-reflection mirrors. The red bars
show estimated distribution of atoms subject to different pulse areas under
the Gaussian beam illumination in this experiment. (b) $n(\mathcal{A}_{\rm
R})$-weighted average $f_{\rm R}$ and $f_{\rm SDK}$ vs reflectivity $r$. The
experimentally measured $f_{\rm R}$ and inferred $f_{\rm SDK}$ are marked with
error bars.

Numerical simulations suggest that the reflectivity
$r=\absolutevalue{\kappa}^{2}$ of the retro-reflection mirror (Fig. 2) affects
$f_{\rm SDK}$ at the $1\%$-level. To investigate the effect, we sample $0<r<1$
during the simulation, and calculate Raman transfer efficiency $f_{\rm R}$ and
SDK fidelity $f_{\rm SDK}$ with peak Raman pulse area $\mathcal{A}_{\rm R}$ as
in Fig. A1(a). In light of the mixed state nature of the experimental
measurements, here the results are again averaged over the initial states
$|b_{m}\rangle$. For all the simulation, we evaluate peak Rabi frequency
$\Omega_{a,b}$ as in previous work of electric dipole transition control [15]
where the laser beam waist and the atomic ensemble size are also carefully
characterized. Based on the geometry parameters, we plot the fractions of
atoms as a function of peak Raman pulse area in Fig. A1(a) in histograms. The
fractions are applied to weight the average over all pulse areas for the
evaluation of $\expectationvalue{f_{\rm R}}$ and $\expectationvalue{f_{\rm
SDK}}$ at various reflectivity $r$ in Fig. A1(b). On the plot, the measured
Raman transfer efficiency $f_{\rm R}$ suggest $r\sim 55(5)\%$. This
reflectivity is consistent with experimental measurements on the ratio of
Stark shifts by the incident and reflected beams.

##  Appendix F Different nuclear spins

In the main text we have discussed the nanosecond SDK for 85Rb atoms featuring
$m=0,\pm 1,\pm 2$ weakly-coupled sub-spins (Fig. 1). In particular, in Sec. IV
we have shown that within the experimentally explored parameter regime, a
balanced chirp-alternating sequence $\tilde{U}_{\rm uddu}^{(4N)}({\bf k}_{\rm
R})$ (Eq. (11)) quite efficiently suppress the leakage among the five sub-
spins to achieve phase gate fidelity $\mathcal{F}_{m}^{(4N)}\approx 90\sim
98\%$ at large $N$, for most of light intensities (Fig. 7). As clarified in
Sec. IV.3, the leakage suppression exploits the approximate time-reversal
symmetry to cancel the non-adiabatic tunnelings among the sub-spins.
Practically, the leakage dynamics supported by the non-degenerate sub-spins is
complex enough to merit future study by itself. Instead of attempting a full
understanding of the dynamics, in this section we provide numerical examples
of coherent matterwave control for other alkaline species. The simulations are
according to those outlined in Appendix B and in parallel to Fig. 7, in
particular with spontaneous emission ignored by setting $\Gamma_{e}=0$ on the
D1 line. The examples with nuclear spin $I=1.5,2.5,3.5$ are given in Fig. A2,
Fig. A3, Fig. A4 respectively. To compare with the Fig. 7 results, we set the
hyperfine splittings $\omega_{{\rm hfs},g}$, $\omega_{{\rm hfs},e}$ to be
identical to 85Rb for all the simulations. In addition, we choose
$\Delta_{e}=-6.6\omega_{{\rm hfs},g}$, twice as those for Fig. 7, to
demonstrate the substantially enhanced spin-leakage suppression.

Figure A2: Numerical results of gate infidelity $1-\mathcal{F}$ for the
traditional $\tilde{U}^{(4N)}_{\rm uu}$ (left panels) and balanced chirp-
alternating $\tilde{U}^{(4N)}_{\rm uddu}$ (right panels) controls as those in
Fig. 7, $\mathcal{F}=\mathcal{F}^{(4N)}_{m}$, except for
$\Delta=-6.6\omega_{{\rm hfs,g}}$ here and for a fictitious 85Rb atom with
nuclear spin $I=1.5$. The 2-photon detuning profiles are illustrated on top of
(a,d). The exceptionally high fidelity demonstrated in Fig. (e) is a result of
$m=\pm 1$ sub-spin degeneracy in the Raman control dynamics. For comparison,
$1-f^{\prime}_{\rm SDK}$ for a single kick is plotted in both (c)(f) by
setting $\Delta_{{\rm hfs},e}=0$ to suppress the spin-leakage according to Eq.
(1).

Figure A2 compares the SDK phase gate fidelity by the traditional
$\tilde{U}^{(4N)}_{\rm uu}$ (left) and the chirp-alternating
$\tilde{U}^{(4N)}_{\rm uddu}$ (right) sequences, for the 87Rb-like atom. With
$F_{b}=1$ for $I=1.5$ (Fig. 1(a)), there are three sub-spins with $m=0,\pm 1$.
The $m=0$ sub-spin is only coupled to single $|a_{m}\rangle$ states with
$m=\pm 2$ (similar to the $m=\pm 3$ end-couplings in Fig. 1(a)). The gate
infidelity for $\tilde{U}^{(4N)}_{\rm uu}$ is slightly compromised by the
resulting leakage ($\mathcal{F}_{m}^{(4N)}$ between $98.5\%$ and $99.9\%$ at
$4N=80$ ), as in Fig. A2(a). For comparison, in absence sub-spin leakage to
fix, the gate fidelity for the chirp-alternating $\tilde{U}^{(4N)}_{\rm uddu}$
in Fig. A2(d) is even slightly worse ($\mathcal{F}_{m}^{(4N)}$between $98.5\%$
and $99.5\%$ at $4N=80$). Nevertheless, we note the $\tilde{U}^{(4N)}_{\rm
uddu}$ is still preferred practically, not only because it supports the $m=\pm
1$ sub-spin performance to be discussed next, but also due to the robust
dynamic phase cancellation (Sec. IV.4).

In contrast, the resonant leakages between the degenerate $m=\pm 1$ sub-spins
is substantial which strongly affect the gate fidelity for $\tilde{U}_{{\rm
uu}}^{(4N)}$. This is shown in Fig. A2(b) similar to the Fig. 7(b) results.
Interestingly, in absence of the $m=\pm 3$ end-couplings for 85Rb (Fig. 1(a)),
here the leakage-suppression by chirp-alternating the $\tilde{U}_{\rm
uddu}^{(4N)}$ sequence is essentially perfect for the 87Rb-like atom. As in
Fig. A2(e), the $\mathcal{F}_{m}^{(4N)}$ reaches $99.99\%$ level at large $N$,
which appears to be only limited by the SU(2) non-adiabaticity of single
adiabatic SDK [59], as suggested by $f^{\prime}_{\rm SDK}$ in Fig. A2(c,f)
evaluated according to Eq. (6), after setting $\omega_{{\rm hfs},e}=0$ to
remove the $m$-changing couplings (Eq. (1)).

Figure A3: Numerical results of gate infidelity $1-\mathcal{F}$ for the
traditional $\tilde{U}^{(4N)}_{\rm uu}$ (left panels) and balanced chirp-
alternating $\tilde{U}^{(4N)}_{\rm uddu}$ (right panels) controls as those in
Fig. 7, $\mathcal{F}=\mathcal{F}^{(4N)}_{m}$, except for
$\Delta=-6.6\omega_{{\rm hfs,g}}$ here. The corresponding 2-photon detuning
profiles are illustrated on top of (a,d).

Next, Figure A3 investigates SDK phase gate fidelity for 85Rb featuring five
nearly degenerate sub-spins with $m=0,\pm 1,\pm 2$, as in the main text. Here
the single-photon detuning $\Delta_{e}=-6.6\omega_{{\rm hfs},g}$ is doubled to
be compared with the Fig. 7 results. We see that for both $\tilde{U}_{\rm
uu}^{(4N)}$ and $\tilde{U}_{\rm uddu}^{(4N)}$ sequences, at the larger
$\Delta_{e}$ the intensity-dependent $\mathcal{F}^{(4N)}_{m}$ oscillatory
features due to the spin-leakages are slowed (with respect to the increasing
$n=4N$) and sharpened. With the $\Omega^{\pm 2}$ couplings halved (Eq. (1))
relative to the Fig. 7 case, the infidelity for the $\tilde{U}_{\rm
uddu}^{(4N)}$ sequence is now below $1\%$ at the large $N$, for most of large
laser intensities. The low-$\mathcal{F}$ region are now narrowed within
$11<\mathcal{A}_{R}/\pi<13$.

Figure A4: Numerical results of gate infidelity $1-\mathcal{F}$ for the
traditional $\tilde{U}^{(4N)}_{\rm uu}$ (left panels) and balanced chirp-
alternating $\tilde{U}^{(4N)}_{\rm uddu}$ (right panels) controls as those in
Fig. 7, $\mathcal{F}=\mathcal{F}^{(4N)}_{m}$, except for
$\Delta=-6.6\omega_{{\rm hfs,g}}$ here and for a fictitious 85Rb atom with
nuclear spin $I=3.5$. The corresponding 2-photon detuning profiles are
illustrated on top of (a,e).

Finally, Figure A4 investigates SDK phase gate fidelity for 133Cs-like atom
featuring seven nearly degenerate sub-spins with $m=0,\pm 1,\pm 2,\pm 3$. With
the large number of sub-spins involved, the spin-leakage dynamics are more
complex. Nevertheless, comparing with the gate fidelity for $\tilde{U}_{\rm
uu}^{(4N)}$ in Figs. A4(a-d), in Figs. A4(e-h) the drastic fidelity
enhancement due to the spin-leakage suppression by $\tilde{U}_{\rm
uddu}^{(4N)}$ is evident. For the seven sub-spin system, the strips of
low$-\mathcal{F}$ intensity region become quite widely distributed.
Consequently, comparing to the 87Rb-like atom with Fig. A2(d)(e) and the 85Rb-
like atom with Fig. A3(d-f), it becomes more difficult to find a suitable
range of laser intensities for the high-fidelity multi-$m$ SDK phase gates to
operate. More generally, as the nuclear spin $I$ increases, it becomes more
difficult to achieve $m-$independent phase gate prescribed by Eq. (8) in the
main text with composite SDK. To improve the aspect, variations of additional
pulse parameters beyond the simple chirp-alternation as in this work may be
explored. In addition, in measurements where coherent control of a single
$m-$spinor matterwave, such as the $m=0$ clock states, is all that is needed,
then circularly polarized nanosecond pulses can be applied to naturally
suppress the $m$-changing transitions during the spinor matterwave control
[70]. To this end, we expect the delay-line based nanosecond SDK to
fascilitate the accurate implementation of $m-$preserving Raman control with
circularly polarized pulses [62].

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