Spinor matterwave control with nanosecond spin-dependent kicks

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 $^{85}$Rb 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 [3,6,10], SDKs emerge as an important technique to control spin-motion entanglement and to improve the scalability of ion-based quantum information processing [5,[11][12][13].
For precise spinor matterwave control, it is essential to operate SDK at high enough speed to suppress lowfrequency noises and to compose multiple operations within a limited duration. Furthermore, except for working with microscopically confined samples [5,8,13], 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 errorresilient 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 ∆ e ( Fig. 1(a)) to avoid populating the excited states. But a moderate ∆ 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 counterpropagating E 1,2 pulses are applied to drive a Raman transition, to ensure the preference for the atom to absorb a photon from the E 1 field followed by a stimulated emission into the 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 groundstate hyperfine splitting ω hfs,g [5]. Practically, however, when the Raman interferometry is operated in the retroreflection 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.
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 [6,10] within nanosec- onds in a retro-reflection optical setup [32,34], 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  Fig. 1(a)) are programmed by an optical arbitrary waveform generator (OAWG) and sent through a single mode fiber to the delay line. An E1 pulse collides with a retro-reflected E2 pulse at the atomic sample to drive the |am⟩ ↔ |bm⟩ Raman transition while imparting ±ℏkR kicks to atoms reaching |am⟩ and |bm⟩ respectively. Next, after a delay time τ d , the E1 pulse is retro-reflected to meet E ′ 2 at the same location to further boost the |am⟩, |bm⟩ separation. QWP: quarter-wave plate. (b) The timing sequence for the "balanced chirp-alternating scheme" (Eq. (11)). The twophoton detunings δR are plotted vs time. The adiabatic controls with "up-chirp" and "down-chirp" 2-photon detuning δR are referred to as Uu and U d respectively. Here the amplitudes for the retro-reflected pulses are reduced by a κ factor. Typical Bloch sphere dynamics of an {|am⟩, |bm⟩} spinor (m = 0) are given in (c) at various laser intensities parametrized by the Raman pulse area AR. The weak ∆m = ±2 couplings ( Fig. 1(b), Eq. (1)) are ignored.
programmed on the optical delay line to adiabatically drive the F b = 2 ↔ F a = 3 Raman transitions of 85 Rb through the D1 line, while imparting ±ℏk R momentum "kicks" to atoms. Here k R = k 2 − k 1 is the difference of k-vectors between the pulse pair. The adiabatic Raman transfer technique [6,10,35] (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 [32,34], 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 [6,10], by shaping counterpropagating 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 SDK ≈ 97.6(3)%. The δf ∼ 2.5% infidelity, as unveiled by the full-level numerical simulation, is primarily limited by spontaneous emission and the ∆m = ±2 leakage among Zeeman sublevels within the ground state hyperfine manifold ( Fig. 1(b)) at the moderate ∆ e = 2π × 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 [6,25,37,38] by individual "kicks", at the moderate ∆ e [8,39]. 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 ∆ e > ω 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 [3,6,10], 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 lowfrequency 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 lightatom 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
A. 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 ′ e are labeled as |c m ⟩ ≡ |c, F c , m⟩ with magnetic quantum number m ∈ [−F c , F c ]. Here, with the nuclear spin I > 1/2, there are 2F c + 1 Zeeman sublevels for each manifold c ∈ {a, b}, and similarly for the e manifold. The |a m ⟩ ↔ |b n ⟩ hyperfine Raman transition is driven by pairs of counter-propagating laser pulses, E 1,2 = e 1,2 E 1,2 e i(k1,2·r−ω1,2t) + c.c. with shaped slowlyvarying amplitudes E 1,2 (r, t) and Raman-resonant carrier frequency difference ω 2 − ω 1 = ω hfs,g .
Although motion of the multi-level atom in the pulsed E 1,2 fields is quite complicated (Appendix A), the internal state dynamics can be substantially simplified when the laser polarization e 1,2 are chosen to be cross-linear ( Fig. 2(a)) [30,48] and the single-photon detuning ∆ e is much larger than the excited state hyperfine splitting ω hfs,e . In this case, the destructive interference of Raman couplings through the intermediate |e l ⟩ levels prevent efficient ∆m = ±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 with [49]. Therefore, in the large ∆ 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 {|a m ⟩ , |b m ⟩}, labeled by magnetic quantum number m (See Fig. 1(b) for the m = ±1 examples.). For each mspin, the Rabi frequency driven by the Raman pulse can be written as The close-to-unity factor χ (m) is determined by the associated Clebsh-Gordan coefficients, normalized at χ (0) = 1 and decreases slowly with |m| [50]. The Hamiltonian for the m-spin is given by Here the detuning δ 0 is the (m-independent) differential Stark shift between the two hyperfine ground states that can be nullified at suitable ∆ e /ω hfs,g and Ω a,b combinations [25]. The Pauli matrices {σ [51]. For notation convenience in the following, we further define projection operators 1 (m) ≡ σ amam + σ bmbm and1 (m) = 1 − 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 to describe the Raman-dressed ground state dynamics. Apart from the ∆m = ±2 couplings specified by Eq. (1), the V ′ term includes non-Hermitian and non-adiabatic |e l ⟩ couplings, the m−sensitive light shifts [26], and the "counter-rotating" couplings involving detuned Raman excitations (Eq. (A3)).
With the atomic position 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 kinetiĉ P 2 /2M operator, see Appendix B for implementation (P = iℏ∇ 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 The φ + is a diffraction phase offset [52] which is generally Zeeman state m-and laser intensity |E| 2 -dependent. In the following our goal is to construct perfect U K (k R ) operation as in Eq. (5) from the full Hamiltonian in Eq. (4), and furthermore to design multiple SDKs with suppressed m and |E| 2 dependence. For the purpose, we first define a fidelity f SDK to qualify the implementation.

B. Non-ideal SDK
To qualify the Fig. 2 SDK implementation modelled by Eq. (4) Hamiltonian, we refer the imperfect realization of SDK asŨ (k R ; η) with η 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 ⟩},{|a m ⟩} states as Here theŨ (k R ; η)|c m ⟩ is compared to U K (k R )|c m ⟩ across the atomic sample with an ⟨...⟩ z average. An ensemble average of the mode-squared fidelity is then performed with ⟨...⟩ η 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 φ + 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 ⟨...⟩ cm 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 ⟩, |b m ⟩} by V ′ (r, t), we define an average spin leakage probability by the non-ideal SDK as By comparing Eq. (6) with Eq. (7) it is easy to verify f SDK ≤ 1 − ε leak , that is, any spin leakage leads to inefficient control. The leakage probability ε leak by Eq. (7) includes contributions from ∆m = ±2 transition as well as those due to spontaneous emission, as ε leak = ε ∆m + ε sp . Here the spontaneous emission probability ε sp during the SDK control is obtained by evaluatingŨ (k R ; η) with the stochastic wavefunction method [54,55] (Appendix B). For the ∆m = ±2 leakage, we find ε ∆m ∝ ω 2 hfs,e /∆ 2 e , as by Eq. (1), and is thus suppressible at large ∆ e similar to the suppression of spontaneous emission. However, one should note that unlike spontaneous emission which simply leads to decoherence, the ∆m = ±2 leakage is a coherent process. A sequence of such coherent leakages may quickly add up in amplitudes to become significant, even if ε ∆m for a single SDK is negligibly small.

C. 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 Ω (m) R ( 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)) [6,10,35,56,57]. For a particular {|a m ⟩, |b m ⟩} sub-spin, if we ignore V ′ in Eq. (4), then the Raman adiabatic passage can easily be generated by the H Hamiltonian. In particular, we parametrize the Rabi frequency of the two pulses as Ω (2) , and specifically consider the timedependent phase difference and the amplitude profile as Fig. 3(b) inset). With a large enough 2-photon sweep frequency δ swp = πϕ 0 /τ c , |δ swp | ≫ δ 0 to let the 2-photon detuning δ R = ∆φ cover the differential Stark shift δ 0 ( Fig. 2(b)), a strong enough Raman Rabi amplitude C peak Raman pulse area A R ≫ 1, and matched magnitudes between C (0) R and δ swp , SDK can be generated in a quasi-adiabatic manner [58,59], i.e., with atomic state |ψ m (t)⟩ ≈ c a (0)e iφa(t) |ã m (t)⟩ + c b (0)e iφ b (t) |b m (t)⟩ following the adiabatic basis {|ã m (t)⟩, |b m (t)⟩} which are simply the eigenstates of the instantaneous Hamiltonian. Population inversion are thus driven quasi-adiabatically during 0 < t < τ c with the efficiency insensitive to the laser intensity, detuning, and their slow deviations from the specific time-dependent forms. Putting back the rdependence, the diffraction phase accompanying the population inversion,φ + (r) ≡φ + + k R · r, is evaluated as φ + (r) = φ a (τ c ) − φ b (τ c ) which includes not only a geometric phase φ G = π/2+k R ·r [60,61], but also a dynamic phase φ D,m ∝ The U K (k R ) operation (Eq. (5)) prints matterwave with state m-and laser intensity |E| 2 -dependent diffraction phases φ + in general. For interferometrically useful coherent matterwave controls, these dynamic parts of φ + need to be suppressed. It is worth noting that the dynamic phase for the adiabatic SDK survives at vanishing δ 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 k R can be inverted [3,6,10], leading to Within each m-spin space span by {|a m ⟩, |b m ⟩}, the U (2N ) K (k R ) acts as a position-dependent phase gate to pattern the two components of arbitrary spinor matterwave with ±(2N k R ·r) phases, i.e., to coherently transfer opposite photon recoil momenta to the spin components.
Practically, however, swapping the k-vectors of E 1,2 can affect other pulse parameters [28,29,32,62]. Here, the k-vector swapping is achieved on the delay line with retro-reflection ( Fig. 2) by programming the carrier frequencies ω 1 ↔ ω 2 for the delayed pulses. The imperfect reflection with κ < 1 generally leads to unbalanced diffraction phases [34]φ + associated withŨ (±k R ). For the spinor matterwave control, we quantify the impact of dynamic phase by comparing the phase of imperfectly E. Chirp-alternating adiabatic SDK schemes A central goal of this work is to synthesize perfect U (2N ) K (k R ) spinor matterwave phase gates with 2N im-perfectŨ (±k R ), even in presence of the unbalanced dynamic phase φ (2N ) D,m and the ∆m = ±2 leakage. For the purpose, we refer the experimentally realized non-ideal adiabatic SDKs with equal amount of positive (δ swp > 0) and negative (δ swp < 0) 2-photon sweeps asŨ u (±k R ) and U d (±k R ) respectively ( Fig. 2(b)). The associated control Hamiltonians are H u (r, t) and H d (r, t). In Sec. IV C we show that a chirp-alternating sequencẽ fairly efficiently suppress the ∆m = ±2 leakage. Next, as detailed in Sec. IV C, Sec. IV D, by combining theŨ to form a balanced chirp-alternating sequence, the dynamic phases byŨ ud (k R ) robustly cancel each other. The Eq. (11) scheme can therefore faithfully realize a U For comparison, here the "traditional" chirp-repeating sequences [6,10,63] are defined as 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 ∆ 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 ω hfs,e for the intermediate P 3/2 states.

A. Nanosecond SDK on a delay line
The adiabatic SDK is implemented on the 85 Rb 5S 1/2 − 5P 1/2 D1 line as depicted in Fig. 1, with counterpropagating 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 e x polarization to e y . With the OAWG output peak power limited to P max ≈ 20 mW, the incident control pulse E 1,2 is weakly focused to a waist radius of w ≈ 13 µm to reach a peak Rabi frequency of Ω a(b) ≈ 2π × 2 GHz. The imperfect retro-reflection with κ ≈ 0.7 (r = |κ| 2 ≈ 50%) reflectivity, primarily limited by increased focal beam size due to wavefront distortion, leads to decreased Ω b(a) = κΩ b(a) for the reflected pulses seen by the atomic sample. We set the single-photon detuning to be ∆ e = 2π × 10 GHz to achieve a peak Raman Rabi frequency of C (0) R ≈ 2π×κ300 MHz estimated at the center of the Gaussian E 1,2 beams. An τ d = 140.37 ns optical delay is introduced by the L ≈ 20 m folded delay line, which is long enough to spatially resolve the counterpropagating nanosecond pulses. To form the counterpropagating E 1,2 pulse pair, we pre-program E 1,2 (t) and E 2,1 (t − τ d ), with a relative delay matching the optical delay line, to ensure the pulse pair with proper carrier frequency ω 1,2 meeting head-on-head in the atomic cloud. To continue multiple SDKs, additional, individually shaped pulses with alternating ω 1,2 can be applied, as in Fig. 2 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 Ω R (t) ∝ E * 1 (t)E 2 (t) (Eq. (2)), therefore, one needs to program the incident E ′ 1(2) pulse according to the retro-reflected 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 τ d . Here, if our goal is to alternate Raman chirps δ 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 δ swp (Sec. II C). As a result, in the balanced chirpalternating 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 ′ -couplings (Eq. (4)). In our experiments, the single-photon detuning ∆ e = 2π × 10 GHz is much larger than δ swp , and we have numerically confirmed that the increasingly chirped waveforms do not significantly affect the Raman dynamics in theŨ (4N ) 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 separatingŨ (4N ) 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 counterpropagating 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.

B. Optimizing Adiabatic SDK
We prepare N A ∼ 10 5 85 Rb atoms in a compressed optical dipole trap at a temperature of T ∼ 200 µK [36]. The atomic sample is optically pumped into the F = 2, |b m ⟩ hyperfine states, elongated along z, with a characteristic radius of σ ≈ 7 µm 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 ±k R , k R = 2k 0 e z are applied to transfer photon momentum by repetitively inverting the atomic population between F = 2, |b m ⟩ and the F = 3, |a m ⟩ hyperfine states. Here k 0 = 2π/λ is the wavenumber of the D1 line SDK pulses at λ = 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 res- , a sequence of two absorption images are taken att1 = 10 µs andt2 =t1 + t tof , separated by τ tof = 160 µs time-of-flight and with a hyperfine repumping pulse along x applied in between (see main text). The center-of-mass position shift δz is retrieved with a typical ±2 µm accuracy by fitting the optical depth (OD). The recoil velocity vn = δz/τ tof is then estimated during typical adiabatic SDK parameter scan as in (b). Here, with n = 25, τc = 60 ns SDK pulses applied, vn is optimized as a function of sweep frequency δswp, and peak laser intensity parametrized by an estimated peak Raman pulse area AR = C (0) R τc/2. Blue, green and orange lines correspond to estimated peak AR of approximately 6π, 9π, 12π respectively. The corresponding peak Rabi frequencies C onant to the D2 line F = 3 − F ′ = 4 hyperfine transition is applied for τ p = 20 µs to record the spatial distribution of atoms in state |a⟩, in the x − z plane, with calibrated absorption imaging [15]. For atoms in |b⟩, this probe is far-detuned and weak enough not to perturb the motion. Next, after a τ tof = 160 µs free-flight time, the 2nd τ p = 20 µs probe pulse is applied to image all the atoms. For the purpose, during the time of flight an additional 50 µs pulse along e x , resonant to F = 2 − F ′ = 3 transition, repumps the |b⟩ atoms to |a⟩ for the 2nd imaging. By comparing atom number N a in state |a⟩ 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⟩ can be measured as a function of the number of SDKs n as ρ 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 = δz/t tof can be retrieved to estimate the photon momentum transfer p n = M v n in unit of ℏk eff .
Typical v n measurement results are given in Fig. 3(b). Here, for atoms prepared in |b⟩ states subjected to an n = 25 SDK sequence starting withŨ u (k R )Ũ u (−k R ) (a 2N = 24 double-SDK followed by an additional kick to drive the final Raman transition), the atomic population is largely in |a⟩ while v n is unidirectional along e z . We optimize v n by varying the peak Raman coupling amplitude C (0) R and sweep frequency δ swp of the adia-batic SDK pulses at fixed τ c . As in Fig. 3(b), for a fixed peak Raman pulse area A R , we generally find δ swp to be optimized for efficient photon momentum transfer when it matches C (0) R (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 A R ≈ 9π reaches optimal to ensure the resilience of adiabatic SDK against the up to 50% intensity variation in the setup. Larger A R is accompanied by slow decrease of v n , due to increased probability of spontaneous emission. Here, for τ c = 60 ns, we need to attenuate C (0) R ≈ 2π × 150 MHz to keep the peak A R ≈ 9π. By using the full C (0) R ≈ 2π ×200 MHz available in this work, we are able to reduce τ c to 40 nanoseconds while maintaining nearly identical momentum transfer efficiency at δ swp = 2π × 150 MHz.

C. Inference of fSDK
With the optimal δ swp = 2π × 150 MHz and peak A R ≈ 9π at τ c = 40 ns, we now apply n = 1−25 SDKs to characterize the momentum transfer p n = M v n and normalized population ρ aa,n as a function of kicking number n. Typical results are given in Fig. 4. Here the momentum change p n along e z is again unidirectional along z as in Figs. 4(a). The direction is conveniently reversed by programmingŨ u (−k R ) first in the ±k R alternating sequence, resulting in acceleration of atoms along −e z instead as in Figs. 4(c). In contrast, the hyperfine population ρ 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 chirpalternating sequence n = 4N that combinesŨ u (±k R ) withŨ d (∓k R ) asŨ (4N ) uddu (±k R ) according to Eq. (11), a sequence which will be detailed in Sec. IV for interferometric applications, with p n and ρ aa,n measurement results also given in Fig. 4 to demonstrate similar momentum and population transfer efficiency.
We estimate f SDK by comparing the p n and ρ 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 = |κ| 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 R , a hyperfine Raman population transfer efficiency. The model predicts exponentially reduced increments |p n+1 − p n |/ℏk eff = |ρ aa,n+1 − ρ aa,n | = f R (2f R − 1) n for f R ≈ 1 by each SDK. From measurement data in Fig. 4(a-d), f R ≈ 98.8% can be estimated in bothŨ u (k R ) andŨ u (−k R ) kicks ( Fig. 4(a,c)), slightly less than spontaneous-emission-limited f R ≈ 99.2% predicted by numerical simulation of the experiments, assuming perfect reflection with κ = 1.  We simply attribute the slightly reduced f R from the theoretical value to imperfect retro-reflection (Appendix E). In particular, we numerically find f R reduces with f SDK when 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 ≈ 0.5 in this work and with numerically matched f R , ε leak ≈ 2.5% with ε ∆m ≈ 0.5% and ε sp ≈ 2% can be estimated respectively. We therefore infer from the combined analysis an f SDK = 97.6(3)% for the adiabatic SDK in this experiment, slightly less than the spontaneous-emission-limited f SDK ≈ 98% for the nearly perfect adiabatic SDK. The numbers can be improved further by increasing ∆ e to suppress spontaneous emission as well as the ∆m-leakage. It is important to note that while the imperfect reflection only moderately reduce f SDK , the resulting imperfection of k R ↔ −k R swapping in successiveŨ u (±k R ) control can greatly compromise the cancellation of dynamic phase φ D in double SDK (Eq. (8)) [6,10], a topic to be detailed in Sec. IV D.
The adiabatic SDK demonstrated in this experimental section is the fastest realization to neutral atoms to date [6,10]. 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 retroreflecting mirror (Fig. 2(a)) shifts the SDK diffraction phases (Eq. (5)) [34]. For interferometric measurements with long interrogation time (Sec. IV E), the end mirror for the compact delay line can be vibration-isolated as those in traditional atom interferometry [28][29][30][31][32][33][34].

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 D, 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 ⟩, |b m ⟩} sub-spins (Sec. IV B), or when the non-perfect k R ↔ −k R swapping introduces additional dynamic phases (Sec. IV D).
Nevertheless, in this section we demonstrate that the ∆m-leakage and dynamic phases can be efficiently suppressed by the balanced chirp-alternating SDK sequencẽ U (4N ) uddu (k R ) introduced in Sec. II E (Eq. (11)) for faithful implementation of the U (4N ) K (k 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].

A. Average gate fidelity
To evaluate an imperfect double-SDK sequencẽ U (2N ) (k R ; η) 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 ⟩, |b m ⟩} sub-spin space. The average fidelity is written as [53]  For the convenience of related discussions, we define an average ∆m-leakage probability, similar to Eq. (7), as where |ψ (n) m,j ⟩ =Ũ (n) (k R ; η)|ψ m,j ⟩ is defined as the final atomic state after the imperfect control. To exclude spontaneous emission, we simply renormalize to have ⟨ψ   We first consider the chirp-repeating double-SDK se-quencesŨ  simple sequences are widely applied in previous works to drive multiple recoil momentum transfer through both optical [58,69] and Raman excitations [6,10]. In absence of the spin leakage, we expect bothŨ Here, we numerically investigate the ∆m-leakage probability ε   dd (k R ) are effectively identical. The Hamiltonian parameters in the simulation follow the experimental setup described in Sec. III for the 85 Rb D1 line scheme (in particular δ swp = 2π × 150 MHz), except here the reflective coefficient is set as κ = 1 for simplicity, and Γ 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 E 1,2 laser intensities in proportion, parametrized by the Raman pulse area A R in the following. Typical results for 85 Rb are shown in the left panels of Fig. 5, Fig. 6 and Fig. 7.
We first discuss the ∆m-leakage probability ε (n) m,∆m as a function of kick number n for typical A R = 9π presented in Fig. 5(a). Here, for n = 1, the tiny ε  To investigate the laser intensity dependence, the ∆mleakage ε (2N ) m,∆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 A R , for m = 0, ±1, ±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 A R , the spin-leakage probability ε (4N ) m,∆m increases rapidly with 4N in oscillatory fashions. There is hardly any continuous region of laser intensity with ε (4N ) m,∆m < 0.1. It is important to note that although the ∆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, F m,∆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 A R < 3π here) where we instead find F (4N ) m ≈ 0.5, as expected. Clearly, the coherent accumulation of ∆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 [6,10,27,70]. 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.

C. 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 ∆m-leakage. Furthermore, by balancingŨ (2N ) ud withŨ (2N ) du as those in Eq. (11), the dynamic phase can be cancelled inŨ (2N ) uddu 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 ∆m-leakage error can be partly suppressed by the chirp-alternating U du (k R ) sequence prescribed by Eq. (10), we come back to Eq. (4) to better understand the ∆m = ±2 leakage itself. In particular, for atom starting in |a m ⟩ or |b m ⟩ and subjected to a close-to-ideal adiabatic SDK control, with an H u (r, t) Hamiltonian during 0 < t < τ c as prescribed in Sec. II E, the ∆m = ±2 transitions driven by V ′ are often a result of non-adiabatic couplings among sub-spins with nearly equal Stark shifts (for example, the m = ±1 sub-spins in Fig. 1b). By reversing the time-dependence of the Hamiltonian, here with H d (r, t) = H u (r, τ d − t) within τ d < t < τ d + τ 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-alternatingŨ Fig. 5(a), the spin leakage ε (n) m,∆m for all the m = 0, 1, 2 oscillates and is overall efficiently suppressed. Comparing with m = ±1 where the leakage is suppressed for even n, the leakage suppression from the m = 0, ±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 ε (n) m,∆m in 2D vs n and pulse area A 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 = ±1 sub-spins, the ∆m-leakage is suppressed to ∼ 1%, limited by their residual couplings to the |a m ⟩ states with m = ±3 ( Fig. 1(a)). In Appendix F we demonstrate that the spin-leakage suppression becomes perfect in species with nuclear spin I = 1.5 ( 87 Rblike) for the pair of m = ±1 sub-spins. On the other hand, for m = 0, ±2 sub-spins here, there are stripes of A R -region (around A R = 12π and A R = 22π here for example) where the leakage still accumulate with increased n = 4N , even though the suppression still works fairly well (ε m,∆m as in Figs. 6(d)(f) suggests that the difference of dynamic phases among m = 0, ±2 subspins 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 thẽ U (2N ) ud andŨ (2N ) du 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-suppressingŨ We evaluate the average gate fidelity for realizing the quantum gate for U , 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 infi- > 90% span a substantial range of intensity for 4N up to 80, with lower F values for A R > 6π confined around specific A R only (deep-blue area). By moderately increasing ∆ e , in Appendix F we demonstrate F > 99% with more confined low-F region after many kicks. Experimentally, the narrow low-F intensity region should be avoided for faithful multi-Zeeman spinor matterwave control of 85 Rb with the balanced chirp-alternating scheme.
Finally, it is interesting to note that for the nearly perfect SDK, the gate infidelity 1 − F m,∆m data in Fig. 6 and Fig. 7. This is a result of dynamic phase cancellation in the adiabatic limit for both theŨ (4N ) uu andŨ (4N ) uddu sequences, when the ±k R swapping as detailed next is perfect.

D. Robust cancellation of dynamic phase
The numerical results in Fig. 7 demonstrate that precise dynamic phase cancellation can be achieved by pairingŨ   uddu (k 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 D, the traditional method of dynamic phase cancellation [6] requires perfect k R ↔ −k R swapping for the successive U K (k R ) U K (−k R ) controls. Practically the k-vector swapping is typically accompanied by a modification of 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 κ < 1 factor due to the imperfect reflection, leading to unbalanced dynamic phases φ D associated withŨ (±k R ) to compromise their cancellation in the traditional double-SDK (Eq. (8)) [6,10]. In fact, this systematic exists quite generally in retroreflection setups since the 2-photon shift δ 0 is sensitive to the laser intensities ratios [25,62].
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 δ 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 φ D according to Eq. (9) for theŨ (4) uu (k R ) andŨ (4) uddu (k R ) controls. The f SDK values are given in the same plots, with which we see that the high f SDK in the quasi-adiabatic regime to be hardly affected even by a poor reflectivity r = |κ| 2 = 0.5. On the other hand, in contrast to Fig. 8(a-c) (left) where r ≈ 1 is required for precise suppression of φ D (blue line), in Fig. 8(d-f) (right) the φ D is largely suppressed so long as f SDK ≈ 1. It is worth noting that the residual variation of φ (4) D,m forŨ (4) uddu (k 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 |φ uddu (k R ), to coherently shift any |a m ⟩, |b m ⟩ components of hyperfine spinor matterwave with opposite ±4N ℏk 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 ∆ e /Γ and ∆ e /∆ 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] Here, to demonstrate the utility of the adiabatic SDK sequence for precision measurements, we numerically investigate a simple atom interferometry scheme [3,6,10] where an "enclosed area" A is enhanced by thẽ U (4N ) uddu (±k 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, ∆Φ ∝ A associated with the pathdependent 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 R T 2 is easily evaluated. Here represent regular Raman interferometry controls for R1 = Rφ(π/2) splitter, R2 = Rφ(π) mirror and R3 = Rφ(π/2) combiner of the spinor matterwave respectively with φ = kR · r. The atomic wavefunctions in |b⟩ and |a⟩ states are represented by the blue and red lines [3]. The four thick vertical lines at t = τ1, T − τ1, T + τ1, 2T − τ1 with red curved arrows repre-sentŨ1 =Ũ uddu (kR) sequences respectively. We consider τ1 ≪ T . By properly choosing τ1/T ratio, spurious interference by multiple imperfect controls can be suppressed, and are not included in the simulations. (b-d): The interferometry phase offset δΦ and contrast C as a function of SDK number n = 4N and pulse area AR. Here the single-photon detuning is chosen as ∆e = −3.3 ω hfs,g . The simulations average over m = −2, −1, 0, 1, 2 states, and include Γe = 0.017 ω hfs,e as for the case of 85 Rb. 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 ∆e = −6.6 ω hfs,g .
v R = ℏk 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 [3,34,44].
More specifically, the enclosed area is defined as A = 2T 0 ∆z(t)dt during a 3-pulse Raman interferometry sequence by integrating the relative displacement ∆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 φ (θ) = cos(θ/2)1 + i sin(θ/2)(e iφ σ + + e −iφ σ − ) for the Raman interferometer, for any spin state within {|a m ⟩, |b m ⟩}. Here φ = k R · r is the local Raman optical phase. Notice the spatial-dependent R φ (θ) 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 φ (π/2), R 2 = R φ (π) and R 3 = R φ (π/2) respectively to manipulate the spin states while imparting the ±ℏk R photon recoil momentum. We now consider enhancing the enclosed area A of the standard 3-pulse Raman interferometer with the chirpalternating SDK sequence. In particular, we consider the Fig. 9(a) scheme with the spinor matterwave diffraction paths marked with solid lines: aŨ 1 =Ũ (4N ) uddu (k R ) is first applied at t = τ 1 to increase the momentum displacement between the two interfering paths from ∆p = ℏk R by R φ (π/2) to ∆p = (2 × 4N + 1)ℏk R with the spindependent kicks. This ∆p-enhancement is followed by an oppositeŨ 2 =Ũ (4N ) uddu (−k R ) at t = T − τ 1 before the R 2 -operation to recover the initial ∆p. To ensure that the interfering paths spatially overlap at t = 2T , an additional pair of opposite momentum boosts,Ũ 3,4 are applied at t = T + τ 1 after the R 2 and t = 2T − τ 1 before the R 3 operation respectively. By properly choosing the τ 1 /T ratio, spurious interference by imperfect R 1,2,3 and U 1,2,3,4 controls can be suppressed [73,74]. With T ≫ τ 1 , the enclosed area of the resulting interfering loop is enhanced to A ′ = (2 × 4N + 1)A.
For the numerical simulation, we consider at t = 0 the atomic state to be initialized at certain |b m ⟩ and subjected to theŨ (4N ) uddu (k R )-enhanced 3-pulse interferometry sequence. The output atomic state, right before the final matterwave combiner R 3 , can then be written as takes into account the free propagation of matterwave between the standard R j and area-enhancingŨ j sequences. We numerically evaluate |ψ m (z, 2T − )⟩ for 1D spinor matterwave between 0 < z < λ/2 (λ is the optical wavelength), as described in Appendix B. To focus on the performance of SDK, we set R 1,3 as perfect π/2 pulses and R 2 as perfect π mirror pulse respectively. A further simplification sets k R = 0 for the idealized R 1,2,3 controls, with which we numerically evaluate ⟨Σ j ⟩ = ⟨ψ m (z, 2T − )|Σ j |ψ m (z, 2T − )⟩ z,m for an initially unpolarized atomic sample.
are summed over all m sub-spins for Pauli matrices with j = x, y, z. Σ 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 τ 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 imperfectŨ 1,2,3,4 diffractions only. Notably, since successive adiabatic SDKs here within each 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(bd) with all spurious interference removed in an efficient manner, we simply apply a digital filter to remove unwanted diffraction orders after eachŨ j sequence.
We are particularly interested in the interferometry contrast C and diffraction phase offset δΦ. The contrast C sets the quality of the final matterwave interference fringes. The phase offset δΦ, stemming from the unbalanced dynamic phase by the fourŨ uddu sequences, enters the interferometry readout as systematic bias against any precision measurements or controls. We evaluate the in- 3 ω hfs,g so both ε sp and ε ∆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 C ≈ 0.5 (which is still highly useful [34]), with δΦ < 0.01 to be experimentally calibrated when necessary. Here, to avoid excessive spontaneous emission and coherent leakage ( Fig. 6(d-f), the peak A R ≈ 6−8π needs to be chosen ( Fig. 9(b-d)). On the other hand, by doubling the single photon detuning to ∆ e = −6.6 ω hfs,g (with laser intensity increased in proportion to maintain the Raman Rabi frequency), ε sp are halved, while the impact of ε ∆m leakage are dramatically suppressed in theŨ uddu sequence (Figs. 9(e-g), also see Appendix F) [77]. The further detunedŨ uddu sequence should thus support up to 50-fold enhancement of interferometry enclosed area, with spontaneous-emission-limited 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 [3,6,10] and for quantum information processing with trapped ions [5,[11][12][13]17]. Practically, unlike interrogating microscopically confined single ions where Raman excitation with multiple-THz single-photon detuning is feasible [8,39], addressing larger samples prefers efficient excitations at moderate single-photon detunings. The seemingly unavoidable imperfections associated with spinleakages 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 ∆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 ∼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 com-paring 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 [14,[84][85][86][87], our method may drastically enhance the practical benefits of large-momentum beamsplitting in Raman atom interferometry [3,6,10], and to support ultra-precise matterwave control for quantum enhanced technologies [18][19][20][21].
The full H eff in Eq. (A1) 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 with the convention of summing over repeated m, n, l indices [51]. The single photon detuning is defined as ∆ e = ω 1 − ω ea = ω 2 − ω eb . The vector k R = k 2 − k 1 is the k-vector associated with the Raman transition driven by the counter-propagating pulses. The Raman coupling associated with σ bnam in line 2 of Eq. (A3) is accompanied by a ±ℏk R momentum transfer to the spinor matterwave. Similarly, the "counterrotating" term in line 3 of Eq. (A3) leads to an opposite, ∓ℏk R momentum transfer. For smooth laser pulses with bandwidth δω ≪ ω 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 |ψ(r, t)⟩ for atom at location r under the non-Hermitian Hamiltonian H eff (see Eq. (A1) with non-Hermitian part ℏ e iΓ e σ e l e l /2 in the first line). Here Γ 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 |ψ(r, t)⟩ densely over a uniform grid within 0 < z < λ/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 τ c for single SDK, the internal state dynamics is evaluated within a single step with frozen external motion under a local |ψ(r)⟩ basis, with atomic position r treated as a parameter of H eff . The evaluation of observables later is normalized by N = r ⟨ψ(r, t = 0)|ψ(r, t = 0)⟩, with corrections from stochastic contributions to be discussed shortly. Between SDKs, a Fourier transform along 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 ⟨c m |ψ(r, t)⟩ for any specific spin state |c m ⟩.
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⟩, |b⟩} 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, ε sp = 1 − 1 N r ⟨ψ(r, τ tot )|ψ(r, τ tot )⟩, after a total evolution time τ 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 fSDK 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 "momentumlattice" [15]. Our numerical simulation suggests that with fair efficiency of single adiabatic pulse to achieve hyperfine transfer efficiency of f R > 95%, the resulting average momentum p n and population ρ 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 ρ aa − ρ 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 ℏk R ) and population contrast C n ≡ |ρ aa,n − ρ bb,n |, then the next kick will impart momentum as 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 ℏk R momentum on average.
Similarly, the population distribution can be written as We define Raman transfer efficiency as f R = f 0 (1 − ε sp /2). When both f 0 and 1 − ε sp are close to unity, Eqs. (C1)(C3) can be approximated as With the recursion relations by Eq. (C4), we arrive at Finally, we remark that for the Raman SDK, there is a slight difference of spontaneous emission loss for single kicks between the a → b and b → a process. As we consider repetitive SDK with n up to a quite large number (e.g. n max = 25 in our experiment), we effectively set the same ε sp parameter for the opposite population transfer processes. |κ| 2 of the retro-reflection mirror (Fig. 2) affects f SDK at the 1%-level. To investigate the effect, we sample 0 < r < 1 during the simulation, and calculate Raman transfer efficiency f R and SDK fidelity f SDK with peak Raman pulse area A 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 ⟩. For all the simulation, we evaluate peak Rabi frequency Ω 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 ⟨f R ⟩ and ⟨f SDK ⟩ at various reflectivity r in Fig. A1(b). On the plot, the measured Raman transfer efficiency f R suggest r ∼ 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 85 Rb atoms featuring m = 0, ±1, ±2 weaklycoupled sub-spins (Fig. 1). In particular, in Sec. IV we have shown that within the experimentally explored parameter regime, a balanced chirp-alternating sequencẽ U (4N ) uddu (k R ) (Eq. (11)) quite efficiently suppress the leakage among the five sub-spins to achieve phase gate fidelity F (4N ) m ≈ 90 ∼ 98% at large N , for most of light intensities (Fig. 7). As clarified in Sec. IV C, 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. A2, Fig. A3,  Fig. A4 respectively. To compare with the Fig. 7 results, we set the hyperfine splittings ω hfs,g , ω hfs,e to be identical to 85 Rb for all the simulations. In addition, we choose ∆ e = −6.6ω hfs,g , twice as those for Fig. 7, to demonstrate the substantially enhanced spin-leakage suppression. Figure A2 compares the SDK phase gate fidelity by the traditionalŨ (right) sequences, for the 87 Rb-like atom. With F b = 1 for I = 1.5 ( Fig. 1(a)), there are three sub-spins with m = 0, ±1. The m = 0 sub-spin is only coupled to single |a m ⟩ states with m = ±2 (similar to the m = ±3 end-couplings in Fig. 1(a)). The gate infidelity forŨ uddu is still preferred practically, not only because it supports the m = ±1 sub-spin performance to be discussed next, but also due to the robust dynamic phase cancellation (Sec. IV D).
In contrast, the resonant leakages between the degenerate m = ±1 sub-spins is substantial which strongly affect the gate fidelity forŨ (4N ) uu . This is shown in Fig. A2(b) similar to the Fig. 7(b) results. Interestingly, in absence of the m = ±3 end-couplings for 85 Rb (Fig. 1(a)), here the leakage-suppression by chirp-alternating theŨ (4N ) uddu sequence is essentially perfect for the 87 Rb-like atom. As in Fig. A2(e), the F (4N ) m reaches 99.99% level at large N , which appears to be only limited by the SU(2) nonadiabaticity of single adiabatic SDK [59], as suggested by f ′ SDK in Fig. A2(c,f) evaluated according to Eq. (6), after setting ω hfs,e = 0 to remove the m-changing couplings (Eq. (1)).
Next, Figure A3 investigates SDK phase gate fidelity for 85 Rb featuring five nearly degenerate sub-spins with m = 0, ±1, ±2, as in the main text. Here the single- photon detuning ∆ e = −6.6ω hfs,g is doubled to be compared with the Fig. 7 results. We see that for bothŨ (4N ) uu andŨ (4N ) uddu sequences, at the larger ∆ e the intensitydependent F (4N ) m oscillatory features due to the spinleakages are slowed (with respect to the increasing n = 4N ) and sharpened. With the Ω ±2 couplings halved (Eq. (1)) relative to the Fig. 7 case, the infidelity for theŨ (4N ) uddu sequence is now below 1% at the large N , for most of large laser intensities. The low-F region are now narrowed within 11 < A R /π < 13. Finally, Figure A4 investigates SDK phase gate fidelity for 133 Cs-like atom featuring seven nearly degenerate sub-spins with m = 0, ±1, ±2, ±3. With the large number of sub-spins involved, the spin-leakage dynamics are more complex. Nevertheless, comparing with the gate fidelity forŨ  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 , except for ∆ = −6.6ω hfs,g here and for a fictitious 85 Rb atom with nuclear spin I = 3.5. The corresponding 2-photon detuning profiles are illustrated on top of (a,e).
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].