Quantum interface for noble-gas spins based on spin-exchange collisions

An ensemble of noble-gas nuclear spins is a unique quantum system that could maintain coherence for many hours at room temperature and above, owing to exceptional isolation from the environment. This isolation, however, is a mixed blessing, limiting the ability of these ensembles to interface with other quantum systems coherently. Here we show that spin-exchange collisions with alkali-metal atoms render a quantum interface for noble-gas spins without impeding their long coherence times. We formulate the many-body theory of the hybrid system and reveal a collective mechanism that strongly couples the macroscopic quantum states of the two spin ensembles. Despite their stochastic and random nature, weak collisions enable entanglement and reversible exchange of nonclassical excitations in an efficient, controllable, and deterministic process. With recent experiments now entering the strong-coupling regime, this interface paves the way towards realizing hour-long quantum memories and entanglement at room-temperature.

Macroscopic systems exhibiting quantum behavior at or above room temperature are of great scientific interest.One such prominent system is a hot vapor of alkali-metal atoms enclosed in a vacuum cell.The collective spin state of these ensembles, consisting of as many as 10 14 atoms, has been used to demonstrate quantum spin squeezing, storage and control of single excitation quanta, and entanglement at room temperature [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18].Despite the rapid thermal motion and atomic collisions, the coherence time of the collective spin in these studies reaches milliseconds and beyond.In some settings, it is unaffected by frequent spin-exchange collisions [18][19][20] and essentially limited by the electron spins' coupling to their surroundings.
Odd isotopes of noble gases, such as 3 He, possess a nonzero nuclear spin.This spin is optically-inaccessible and well protected from the external environment by the enclosing, full, electronic shells and is therefore extremely long-lived.Noblegas spin ensembles have demonstrated lifetimes T 1 exceeding hundreds of hours and coherence times T * 2 of 100 hours at or above room temperature [21][22][23][24].It is to be expected that the collective nuclear spin in these ensembles, similarly to the collective electronic spin in alkali vapor, can be brought to the quantum limit and utilized for quantum optics and sensing applications.Several theory works have proposed to use collisions with optically-excited metastable 3 He atoms for generating and reading-out nonclassical states of ground-level 3 He spins, relying on collisional-exchange of electronic configurations that lead to adiabatic transfer of the quantum state [25][26][27][28].
This paper studies a different mechanism forming a quantum interface to noble-gas spins.The mechanism relies on the stochastic accumulation of weak spin-exchange collisions, directly coupling between the collective spins of polarized atomic gases and thus not limited to adiabatic transfer rates.Recently, we have experimentally demonstrated the coherent coupling between alkali-vapor spins and noble-gas spins, in the regime where the coherent exchange rate dominates the spin relaxation rate [29].Additionally, we have demonstrated * These authors contributed equally to this work.a bi-directional interface between light and noble-gas spins, using the optically-accessible alkali spins as mediators [30].These experiments establish the coherent nature of the collective coupling for classical (coherent) states.If the coupling is indeed quantum, i.e., if it transfers non-classical correlations between the spin ensembles, then it can be used for quantum information applications, such as quantum memories and remote entanglement with long-lived noble-gas spins [31][32][33].
Here we provide a theoretical description of the emergent collective coupling between the spins of polarized alkali and noble-gas atoms.We analyze the effect of weak spin-exchange collisions using a many-body formalism and find that they can efficiently couple the collective quantum excitations of the two ensembles, with negligible quantum noise added due to the collisions' stochasticity.We derive the strong-coupling conditions, where the quantum-coherent exchange dominates over the relaxation.We use numerical many-body simulations and a detailed analytical model to study the controllable periodic exchange of non-classical states between the spin ensembles, further attesting to the quantum nature of the interface.Finally, we outline practical experimental conditions for the efficient exchange of nonclassical states between the alkali and noble-gas ensembles.
System.Noble-gas spins can be accessed via spinexchange collisions with alkali vapor atoms.Consider a gaseous mixture of N b noble-gas atoms with nuclear spin-1/2 and N a N b alkali-metal atoms, all enclosed in a heated spherical cell of volume V and undergoing frequent collisions.A collision between noble-gas atom b and alkali atom a is governed by the Fermi-contact interaction and described by the evolution operator exp(−iφ kb • ŝa ).kb is the noble-gas nuclear spin operator and ŝa is the alkali electronic spin operator of the colliding atoms labeled a, b respectively.φ is the mutual precession angle; see Figs. 1a and 1b [35][36][37].While φ varies between collisions depending on the atoms' random trajectories, its value is always positive [38,39].This is an important property of the isotropic Fermi-contact interaction, leading to a nonzero mean precession φ during collisions.
Between collisions, the electron and nuclear spins of the alkali atoms are altered by their strong hyperfine coupling.Consequently, the slow dynamics of alkali atoms having nuclear a b c d Figure 1.Quantum interface for noble-gas spins via spin-exchange collisions.a, Coherent interaction during a collision between alkalimetal electronic spin ŝa (red) and noble-gas nuclear spin kb (blue).The two spins mutually precess and acquire an angle φ 1 while conserving the total spin, where φ is random and depends on the collision kinematics.b, Stochastic sequence of collisions.Spin exchange occurs over a few picoseconds when the valence electron's wave-function (pink) overlaps with the noble-gas nucleus (blue).c, For polarized ensembles, multiple collisions between different atoms accumulate to a coherent dynamics of bosonic collective-spin excitations, described by local quantum operators â(r) (alkali) and b(r) (noble gas) and a coupling rate J. Incoherent spin dynamics, which enables initialization via spin-exchange optical-pumping, play a minor role for φ ≪ 1. d, Diffusion of the gaseous atoms and the boundary conditions in the cell define nonlocal spin modes.The lowest-order spatial modes â, b govern the coherent evolution of the collective quantum spins (see Methods, [34]).
spin I > 0 is determined by the operator sum fa = ŝa + îa , where îa is the nuclear spin operator of alkali atom a.We shall focus on the conditions of high alkali atomic density and small Zeeman splitting.Under these conditions, the alkali Zeeman states are populated with a spin-temperature distribution, and fa = qŝ a , where q is known as the Larmor slowing-down factor [2I + 1 < q < 2I 2 + I + 1 depending on the degree of polarization, see Eq. ( 12)] [36,40].
The accepted formalism for describing the dynamics of the spin ensembles employs the mean-field Bloch equations [11] Here n a = N a /V and n b = N b /V are the atomic densities, ζ = σvφ /q denotes the mean-field interaction strength, where σ is the spin-exchange cross-section, v is the relative thermal velocity, and k se ≡ 1 4 vσφ 2 is known as the binary spin-exchange rate coefficient [11].The first term in Eqs.(1) describes the mutual average precession of the two mean spins.The second term represents an incoherent transfer of spin polarization from one species to another (conserving the total spin) [36]; it is commonly utilized for initially polarizing the noble-gas spins via so-called spin-exchange optical pumping (SEOP) [35].The mean-field Eqs.(1) implicitly assume that any quantum-correlation developed between different atoms due to collisions is rapidly lost.Therefore, this model is insufficient for describing the dynamics of nonclassical, i.e. quantum, spin states.
Dynamics of collective spin states.We describe the macroscopic quantum states of the spin ensembles using collective spin operators [2].Each collective operator is a symmetric superposition of the spins in the cell, where F = Na a=1 fa is the collective alkali spin operator, and K = Nb b=1 kb is the collective noble-gas spin operator.Here we focus on the particular regime of highly-polarized ensembles, with most of the spins pointing downward (−ẑ) [1][2][3].In this regime, the operators Fz and Kz can be approximated by their classical expectation values F z = −p a N a q/2 and K z = −p b N b /2, where p a , p b ≤ 1 are the degrees of spin polarization (p a = p b = 1 for the ideal preparation of fully-polarized ensembles).The quantum state of the collective spins is then fully captured by the ladder operators F± = Fx ± i Fy and K± = Kx ± i Ky .Pictorially, for polarized ensembles, these operators describe a small tilt of the macroscopic spin vector, accompanied by spin uncertainty that scales as √ N a for the alkali ensemble and √ N b for the noble gas.To see how quantum spin excitations are associated with these operators, we apply the Holstein-Primakoff transformation [2], which describes the collective states in terms of excitations of bosonic fields.We define the annihilation operators of the two ensembles as â = F− / 2|F z | and b = K− / 2|K z |.The state |0 a |0 b , with zero spins pointing upwards, is identified as the vacuum, and the creation operators â † and b † flip upwards one alkali or noble-gas spin.We model the quantum dynamics of the collective spins by following the steps illustrated in Fig. 1.We adopt a microscopic picture of a stochastic sequence of random collisions and keep track of the correlations developed between different atoms during collisions (see Methods).With the addition of a magnetic field B ẑ and decoherence for the alkali spins at a rate γ, we find that the dynamics of the coupled quantum systems is well approximated by These equations are written in a rotating frame of the noblegas spins, where ∆ = (g a − g b )B + ∆ c denotes the mismatch of precession frequencies of the two polarized gases, with g a and g b being the gyromagnetic ratios of the alkali and noblegas spins.At zero magnetic field, the detuning is biased by the mean collisional shift ∆ c = ζ(p a n a q − p b n b )/2 due to the difference in the effective magnetic fields induced by one species on the other [39].Fa is a quantum noise operator [41].The decay of the noble-gas spins is omitted here, as we are interested in time scales much shorter than their (hours-long) decoherence time.
A key result of our formalism is the identification of the bidirectional coupling rate J = (ζ/2) √ qp a p b n a n b , which represents the frequency at which the states of the two ensembles are exchanged.Importantly, J is proportional to the squareroot of the atomic densities, which implies that the coupling is collective and benefits from collective enhancement.For the special case of coherent spin states, one can reduce the quantum model [Eqs.Coupling regimes.With the coupling rate J, detuning ∆, and relaxation γ, Eqs.(2) has the canonical form of a coupled two-mode system [2].While J cannot be varied rapidly, ∆ (B) can be controlled efficiently by varying the external magnetic field B along the polarization axis.B alters ∆ by predominantly altering the precession frequency of the alkali spins, owing to the (100 − 1000)-fold difference in the gyromagnetic ratios g a and g b .When the interaction is set offresonance |∆(B)| J, γ, the two collective spins effectively decouple.This decoupling is often used in sensing applications to diminish the effect of the alkali on the noble-gas dynamics [30,[42][43][44].In this regime, the alkali and noble-gas spins precess independently; the alkali spin experiences fast relaxation at a rate γ, while the noble-gas spin maintains its long coherence time.We simulate these dynamics first for coherent spin states, as shown in Fig. 2a.
Conversely, when the magnetic field is tuned to the socalled 'compensation point' ∆(B) = 0 [45], the interaction becomes resonant, and the two spin ensembles hybridize.The magnetic field thus acts as a controllable switch, rapidly coupling or decoupling the two spin ensembles.Romalis and coworkers have demonstrated the alkali-noble-gas hybridiza-tion in the overdamped regime γ J, |∆| [45].In this regime, the noble-gas spins inherit a large fraction of the alkali spins' decoherence rate and thus thermalize before the transfer of excitations is complete, as shown in Fig. 2b.The overdamped regime features a large enhancement in the sensitivity to various external fields and enables the operation as a comagnetometer sensor [45,46].
Here we focus on the recently demonstrated regime J γ, which we identify as strong coupling [29].In this regime, the evolution is governed by the beam-splitter Hamiltonian J(â †b + b † â), which leads to the exchange of quantum states between the spin ensembles.This is illustrated in Fig. 2c for coherent states, demonstrating a coherent transfer of spin excitations, as recently observed [29].One can dynamically tune the exchange rate by varying the magnetic field strength.In particular, maintaining the resonance conditions ∆(B) = 0 for a duration t = π/(2J) and subsequently ramping B up to ∆ J yields a deterministic state transfer between the two ensembles akin to a π pulse.
The accumulated effect of the spin-exchange collisions is coherent only as long as their incoherent contribution is negligible.As described by the incoherent-transfer term in Eq. ( 1), spin-exchange collisions introduce an additional relaxation rate k se n b /q into the alkali spin relaxation γ.Consequently, since n b n a , the strong coupling condition J γ requires that the precession angle φ remains very small φ √ qp a p b n a /n b < 1.Thus, strong coupling of the spin ensembles relies on the weakness of the individual spin-exchange collisions.
Stochastic numerical simulations.To attest to the quantum nature of the interface, we develop a stochastic manybody simulation, which tracks the quantum state of many spins that randomly collide in the strong-coupling regime (see SI).The simulation allows us to witness and visualize the coherent and collective outcome of the stochastic spinexchange interaction as well as the associated relaxation.For Na) via the same process underlying SEOP.b, The localized excitation is incoherently transferred to the noble gas.c, Strikingly, when the collisions are weaker, the exchange fidelities F10 and F01 oscillate with higher contrast and nearly no decay, despite the stochasticity of the process.d, Almost no oscillations are observed for the localized excitation.e, When |ψ0 = |1 a|1 b , the two excitations periodically "bunch" in a superposition of either of the spin ensembles (|2 a|0 b + |0 a|2 b ), manifesting the nonclassical Hong-Ou-Mandel phenomenon and validating the quantum beam-splitter property of the interface.
the sake of simplicity, we assume in the simulations I = 0, i.e. fa ≡ ŝa .We initialize the system with either the symmetric excitation ,d).The exchange between the two ensembles emerges as a collective phenomenon: for the symmetric Fock state, we observe multiple, high-contrast oscillations of the populations of the collective states, whereas, for the localized excitation, the oscillations are negligible.The transfer amplitude accumulates constructively only for the excitation that is symmetrically shared among all spins, maximizing the periodic exchange rate and fidelity.This comparison emphasizes the collective nature of the coupling, which is taken for granted in the mean-field de-scription.
As noted above, the dominance of the coherent exchange over the incoherent transfer and dephasing relies on the collisions being very weak.To exemplify this, we compare between φ = 2•10 −2 (Figs.3a,b) and φ = 10 −5 (Figs.3c,d), the latter corresponding to realistic 3 He-potassium collisions.For φ = 2 • 10 −2 , we observe a dephasing of the alkali spin, with a rate γ ∝ k se n b , reducing the exchange fidelity compared to that with φ = 10 −5 .Finally, we simulate the periodic bunching of two spin excitations, as shown in Fig. 3e.This evolution is analogous to the nonclassical Hong-Ou-Mandel phenomenon, demonstrating that the collisional interface supports the reversible, high fidelity, full exchange Exchange of nonclassical states between alkali and noble-gas spin ensembles.The exchange is calculated for the proposed experimental parameters and assuming the initial excitation is spatially uniform.We compare the two-mode approximation given by Eqs. ( 2) for perfectly-polarized ensembles (black) with an exact solution accounting for higher spatial modes and imperfect polarization (blue, see Methods).At t = 0, the magnetic field is tuned to resonance ∆ (B) = 0.If it is detuned at t = π/(2J), the state transfer is maximal (solid purple, using B = 180 mG).The imperfect polarization is accounted for by initializing the system with an incoherent state and incorporating the associated excess quantum noise (see Methods).a, Probability of populating the uniform (longestlived) diffusion mode of the noble gas with exactly two excitations (Fock state) when the system is initiated with two excitations in alkali's uniform mode.b, Exchange of squeezing between the total alkali spin (proportional to the electron spin Ŝx = a ŝa,x) and the total noble-gas spin ( Ky = b kb,y ).The alkali is initialized with 7 dB squeezing.
of nonclassical states between the two spin ensembles.
To elucidate the physical mechanism that renders a deterministic exchange out of random collisions, we analytically describe the quintessential case of an exchange of a single spin excitation.The noble-gas spins are initialized in the state |0 b with all spins pointing down, and the alkali spins are initialized in the nonclassical Fock state |1 a ≡ N −1/2 a a f+ a |0 a , i.e., a symmetric superposition with one of the spins pointing up.After a short time t, the system wavefunction |1 a |0 b evolves into This evolution, with Jt, 1, is the onset of transfer of the single spin excitation from the alkali to the noble gas via both deterministic and stochastic contributions; The Fock state b manifests the deterministic transfer, while the stochastic wavefunction |δψ represents an incoherent mixture of excited spins (see SI).To quantify the corresponding transition amplitudes Jt and , we assume that any alkali is equally likely to collide with any noble-gas atom at a mean time between collisions τ = 1/(n b σv).We then find that Jt = 1 2 ( φ t/τ ) N a /N b and that, after many collisions, the stochastic variable follows the central limit theorem → φ 2 t/(2τ ).These results show that, at longer times, the transition amplitude of the deterministic term in Eq. ( 3) dominates the transition amplitude of the stochastic term, since Jt .It is the accumulated mean effect of many weak collisions that leads to deterministic transfer scaling linearly with t/τ and to fluctuations that only add up incoherently as t/τ .
Outline for a non-classical demonstration.We now identify relevant experimental parameters for realizing strong coupling in a mixture of potassium-39 and helium-3.To maximize the collective coupling rate J, we consider reasonably high densities of n a = 3 × 10 14 cm −3 (vapor pressure at 215 • C), n b = 2 × 10 20 cm −3 (7.5 atm), and 30 Torr of N 2 for quenching [45].Using standard optical pumping, the alkali spin polarization can be initialized in a spin-temperature distribution with p a ≥ 0.95, for which the slowing down factor is q = 4.1 [2,47].The noble-gas spin can be initialized via SEOP to a moderate yet sufficient polarization of p b 0.75 [48].
A coupling rate of J = 1000 s −1 is reached at this temperature, as vσ φ = 2 × 10 −14 cm 3 s −1 (corresponding to φ 2 ≈ φ 2 ≈ 2 × 10 −10 rad 2 ).The resonance condition ∆ = (g a − g b )B + ∆ c = 0 is obtained for a magnetic field B = 94 mG, predominantly compensating for the large collisional shift vσ φ p b n b /2q experienced by the potassium, and yielding the Larmor frequency g b B = g a B + ∆ c = 300 (2π) Hz.The high alkali density and polarization and the relatively small Larmor frequency puts the potassium spins in the 'spin-exchange relaxation-free' (SERF) regime [49][50][51], rendering their relaxation via spin-exchange collisions negligible.The relaxation rate is governed by spin-rotation interaction with 3 He and N 2 and by spin-destruction collisions with other potassium atoms, giving γ = 17.5 s −1 [11].We thus reach the strong-coupling regime with potentially J > 55γ.
The spin state of 3 He in this system can endure for 100 hours, providing that magnetic-field gradients, magnetic impurities in the cell, and alkali-induced dephasing are minimized [23,35,52].The alkali spins can be initialized in a nonclassical state via entanglement-generation schemes [1,15] or by mapping nonclassical light onto the spin orientation moment [53].Calculations of the spin dynamics with these parameters in a 2"-diameter spherical cell are presented in Fig. 4.These calculations account for the spatial dynamics and for the initial imperfect polarization by employing a non-pure density matrix with multiple diffusion modes [34], initialized with incoherent spin excitations corresponding to p a = 0.95 and p b = 0.75.We include the increased quantum noise due to imperfect polarization and calculate expectation values by tracing over the contribution of all diffusion modes (see Methods).These calculations demonstrate the exchange of nonclassical Fock state and squeezed state and their mapping onto the long-living noble-gas spins in a realistic regime of imperfect spin-polarization.
Discussion.We present analytic and numeric quantummechanical models for the hybrid system of alkali-metal and noble-gas spins.The models reveal a collective mechanism that couples the macroscopic quantum states of the two spin ensembles.We highlight feasible experimental parameters for reaching the strong-coupling regime, which enables a faithful quantum-state transfer between the alkali and noble-gas ensembles.
It is intriguing that weak collisions, despite their random nature, allow for an efficient, reversible, and controllable exchange of excitations.It is particularly counter-intuitive that this exchange preserves the unique quantum statistics of nonclassical states.Equations (2) manifest a genuine quantum interface, as they describe the exchange between the operators â and b, which in turn encapsulate the full quantum statistics of the collective spins states.The effect of the randomness of collisions on the quantum statistics is then incorporated in the noise operator Fa .
In stochastic quantum systems, the variance of quantum noise satisfies Fa F † a ≥ 2γ for any relaxation rate γ, where equality is obtained for the case of vacuum noise [41].For perfect spin polarization, we find that the noise due to spinexchange collisions is a vacuum noise, that is, the minimal possible for an open quantum system (see Methods).This result is apparent, for example, in the exchange of a single spin excitation between perfectly-polarized ensembles, where we obtain Fa F † a = 2 /t = 2γ.While it is experimentally possible to approach unity polarization of the alkali atoms p a → 1 [54,55], the highest 3 He polarization demonstrated to date is p b = 0.85 [48].It is therefore important to discuss the consequences of imperfect spin polarization p a , p b < 1.The first and more trivial consequence is a moderate reduction of J ∝ √ p a p b , since now only a fraction p a p b of the atomic collisions contribute to the collective exchange process.The second consequence is an added noise due to initial incoherent population of transverse spin excitations.These incoherent excitations are distributed over a macroscopic number of spatial modes.Therefore, despite having a macroscopic number of depolarized atoms, the number of excitations per mode can be small.Notably, for a finite polarization degree 0 < p ≤ 1, the mean number of incoherent excitations in the uniform spatial mode is nI = (1 − p)/2p.For p close to unity, nI 1 is small.Therefore these excitations might contribute a weak classical background, in principle relevant at all times t > 0, which reduces the fidelity of non-classical states.For example, there is a non-zero probability nI 1 of finding a spin excitation without any stimulation.Finally, the incoherent excitations residing in all other modes form a 'thermal reservoir' of spins that is manifested as an excess quantum noise.In particular, the collisional coupling of the collective spin to this reservoir increases the variance of the quantum noise operators acting on the spin.This noise is accumulated during the dynamics and is thus less relevant at short time scales.Importantly, the excess noise is moderate and, for highly-polarized ensembles, it is on the same order as the vacuum noise.
The quantum interface we study allows for a controllable state-exchange between two spin-gas species, and it is of particular importance when it comes to noble-gas spins, which are extremely long-lived but optically inaccessible.The interface can reach the strong coupling regime and allows for nonadiabatic exchange, thus significantly increasing operations bandwidth.In conjunction with recent experiments demonstrating the coherent, efficient, and bi-directional properties of the collective coupling in the classical regime [29,30], our study thus opens a path to couple light to the transparent spins in the quantum regime.The scenario resembles quantumlogic operations with nuclear ensembles in solids, where a long-lived nuclear spin is accessible via the hyperfine interaction with an electron spin, which is optically manipulated and interrogated [56].The spin-exchange interface therefore paves the way towards wider applications of noble-gas spins in quantum optics, including long-lived quantum memories and long-distance entanglement at ambient conditions [31][32][33], as well as to fundamental research of the limits of quantum theory for entangled macroscopic objects.

Methods DETAILED MODEL OF THE MANY-BODY PROBLEM
Spin-exchange interaction.Consider a gaseous mixture of N a alkali-metal atoms and N b noble-gas atoms enclosed in a spherical cell.Each alkali atom, labeled by a, has a valence electron with a spin S = 1/2 operator ŝa , in addition to its nuclear spin I > 0 with spin operator îa .Each noble-gas atom, labeled by b, has a nucleus with a K = 1/2 spin, represented by kb .During collisions, the noble-gas spin interacts only with the valence electron.The Hamiltonian of the two spin ensembles is given by H (t) = H 0 + V (t), where is the non-interacting Hamiltonian of the two spin ensembles.a hpf denotes the hyperfine coupling constant in the ground state of the alkali atom [11], and ωa and ωb are the Larmor frequencies of the alkali and noble-gas spins induced by an external magnetic field B = B ẑ.The microscopic manybody interaction Hamiltonian, governed by the Fermi-contact interaction [37,38], is given by This form conserves the total spin of the colliding pairs.The instantaneous interaction strength α ab (t) between atoms a and b is determined by the specific microscopic trajectory of each atom.The spatial degrees of freedom of the thermal atoms are classical.Their coordinates r a (t) and r b (t) follow ballistic trajectories, which are independent of the spin state and governed by the classical Langevin equation.The collisions in the gas can be considered as sudden and binary; the mean collision duration τ c is only a few picoseconds [11], whereas the mean time between collisions for an alkali atom τ is a few nanoseconds at ambient pressure.Since collisions are isolated in time (τ c τ ), the interaction strength can be approximated by a train of instantaneous events α ab (t) = i φ ab denotes the phase φ that spins a and b accumulate during the i th collision, and t (i) ab denotes the time of collision, as determined from the particles trajectories (see SI).
We consider short times τ , typically a few tens of picoseconds, such that τ τ τ c , for which each atom experiences at most a single collision.In other words, we assume that if a collision occurred between an alkali spin a and a noble-gas spin b, then neither a nor b collided with other atoms during τ .Consequently, V (t) has no more than one appearance of each spin operator and thus commutes with itself.Under these conditions, the time-evolution operator is simplified to Here τ i denotes the sum over all collision instances that occur during the short time interval in which t For weak collisions, the mutual precession is small φ (i) ab 1, and the exponential term in Eq. ( 6) can be expanded to leading orders in φ as a Dyson series Here the lowest-order terms are U This simplified form provides for the evolution of any quantum mechanical operator Â after time τ , ∆ Â = U † (t + τ , t) Â (t) U (t + τ , t) − Â (t), where U = e −iH0τ U I is in the Heisenberg picture.Note that e −iH0τ and U I commute, as explained below.Up to second order in φ, the dynamics of Â is given by The first term is the standard Hamiltonian evolution governed by H 0 and independent of φ.The second term describes a unitary evolution during a collision with an effective Hamiltonian which is first-order φ.The third term L (A) is proportional to φ 2 and has the structure of a standard Lindblad term We note however, that this operator is not associated with a decay but is rather a second-order correction to the unitary evolution.
To describe the short-time evolution of the spin ensembles, we derive the time-evolution operator U I in the interaction picture.This operator satisfies i ∂ t U I = V I U I , where V I (t) = e i H0t V (t) e − i H0t is the Hamiltonian in the interaction picture.The collisions are sudden H 0 τ c ≪ 1 (except at strong magnetic fields greater then tens of Tesla), rendering the evolution by H 0 negligible during the short time of collision (typically τ c ≈ 1 psec and B < 1G, such that ωa τ c 10 −7 ).As a result, we can assume that H 0 and V commute, hence V I (t) = V (t).
The evolution of the single-spin operators ŝa and kb in the time interval τ are then derived from Eq. ( 8), yielding This form conserves the total spin of each colliding pair a − b, since ∆ ŝa + kb = 0. Equation ( 11) describes the mutual precession of pairs of spins, as illustrated in Fig. 1a.This evolution is unitary to second order in the precession angle φ, while higher-order contributions are neglected in the truncation of Eq. ( 7).
Between collisions, the nuclear spin of the alkali atoms is altered by the strong hyperfine interaction with the electron.Consequently, the slow dynamics of the alkali atoms should be described in terms of the operator sum fa = ŝa + îa .Here we focus on alkali ensembles in a spin-temperature population-distribution, for which fa = qŝ a , with the slowingdown factor q = q(I, p a ) given by [57] The slow evolution of the spins depends on the cumulative effect of multiple collisions among different atoms.At the macroscopic limit, it is formidable to keep track of the kinematic details of all atoms, given a large set of collision times t Here κ ab (t, τ ) is a Bernoulli process indicating whether a collision between particles a, b has occurred during the short time interval [t, t + τ ], with τ /a hpf .As the phase φ a depends on the kinematic parameters of the collision, such as the impact parameter and the two-body reduced-velocity [35], we treat it as a random variable, with a mean φ and variance var(φ).The operation • denotes an average over the microscopic kinematic parameters.The stochastic nature of φ a manifests the randomness in the interaction strength, while the stochastic nature of κ ab manifests the randomness in pairing the colliding atoms.We derive the statistical properties of κ ab as a function of the microscopic kinematic variables in the SI, yielding κ ab (t, τ )κ cd (t , τ ) = δ ac δ bd τ δ(t − t ) κ ab (t, τ ) and κ ab (t, τ ) = vστ w(r a − r b ).Here the window function w (r) = Θ(l − |r|)/V l represents a control volume V l = 4πl 3 /3, where Θ is the Heaviside function, and l is the coarse-graining scale (larger than the atoms mean free path, see SI).
We are now set to perform spatial coarse-graining.First, we replace the discrete atomic operators with the continuous operators f (r, t) ≡ a fa (t) δ[r − r a (t)] and k (r, t) [2,5]).We then perform the spatial convolutions f(r, t) → f(r, t) * w (r) and k(r, t) → k(r, t) * w (r).The central-limit theorem is valid for this coarse-graining operation as long as V l contains a large number of particles V l n a , V l n b 1.Consequently, f (r, t) and k (r, t) become local symmetric spin operators, and the spatial coordinate r supersedes the specific particle labels.
Next, we consider the collisional part of the evolution of f(r, t) and k(r, t) at time intervals much longer than τ , The first term in Eqs.(13) represents the average mutual precession of the two symmetric spin operators within the coarsegraining volume, with the local interaction strength given by ζ ≡ σvφ /q.As it describes coherent dynamics, it can be associated with an effective spin-exchange Hamiltonian The second and third terms in Eqs. ( 13) represent incoherent transfer of spin polarization from one specie to the other, while conserving the total spin.Recall that we assume a spin-1/2 noble gas.Here n a = a w(r − r a ) and n b = b w(r − r b ) denote the local densities of the two spin ensembles, and k se ≡ 1 4 vσφ 2 is known as the binary spinexchange rate coefficient [11].In particular, the term k se n b f/q is responsible for hyperpolarization of the noble gas by optically pumped alkali-metal spins, underlying the SEOP technique [11,35].The incoherent SEOP term here replaces the coherent L (A) defined in Eq. (10); it has the same functional form but is now essentially incoherent due to the coarsegraining of the microscopic kinematics.Notably, incoherent effects are second order in φ, and since φ 2 / φ ≪ 1 (typically 10 −5 ), ζ is substantially larger than k se .The fluctuation vector-operator F ex is defined in Eq. (19).
Comparison to mean-field model.We compare Eqs. ( 13) with the existing mean-field theory and associate our model parameters with those obtained from experiments.The meanfield spin operators are related to our formalism by f ≡ d 3 r ψ| f(r, t)|ψ /N a and k ≡ d 3 r ψ| k(r, t)|ψ /N b , where |ψ is the initial many-body wavefunction of the system.Substituting these definitions in Eqs.(13) and using F ex = 0 recover the standard Bloch equations [Eqs.1].
In terms of experimentally measured parameters, the interaction strength ζ is given by ζ = 8πκ 0 g e g n µ B µ n /(3q ), where g e = 2 is the electron g-factor, g n is the g-factor of the noble-gas nucleus, µ B is Bohr magnetron, µ n is the magnetic moment of noble-gas spin, and κ 0 is the enhancement factor over the classical magnetic field due to the attraction of the alkali-metal electron to the noble-gas nucleus during a collision [11,39].For K− 3 He at T = 220 • C, ζ = 4.9 × 10 −15 cm 3 s −1 and k se = 5.5 × 10 −20 cm 3 s −1 [11,39].Roughly estimating a collisional spin-exchange cross-section of σ ≈ 8 × 10 −15 cm 2 from the K− 3 He interatomic potential [58] and using a typical centrifugal potential with an angular momentum 40 yield an estimate of the precession angles φ ≈ ζq/σv = 1.4 × 10 −5 rad and φ 2 ≈ 4k se /σv = 1.6 × 10 −10 rad 2 for highly-polarized alkali vapor.
Dynamics with diffusion and relaxation.To describe the spatial dynamics of a macroscopic ensemble of alkali and noble-gas spins in the presence of relaxation, we write the Heisenberg-Langevin equations for f (r, t) and k (r, t) Here we assume standard noble-gas pressures (10 −1 − 10 4 Torr), of which frequent collisions with the noble-gas atoms render the thermal motion diffusive, as described by the diffusion terms D a ∇ 2 f and D b ∇ 2 k [34].For polarized alkali vapor, the interaction-free Hamiltonian from Eq. ( 4) obtains the simple form H 0 = d 3 r[ω a fz (r, t) + ωb kz (r, t)], and V ex is given in Eq. ( 14).We emphasize that this model can describe the evolution of many-body quantum spin states.
Equations (15) encapsulate the various spin dissipation mechanisms into the relaxation rates γ a and γ b .The relaxation rate of alkali-metal spins is given by consisting of collisional spin-orbit coupling, spin-exchange interaction with the noble-gas nuclei, and spin-destruction via binary collisions of alkali-metal spins (cross-section σ sd , mean atomic velocity v a ) [11].The relaxation rate γ in the main text can then be approximated by γ = γ a + D a π 2 /R 2 , including the leading order of the diffusion relaxation in a spherical cell with radius R (e.g., R = 1" in Fig. 4) [34].
The relaxation rate of the noble-gas spins is given by where T −1 b is the coherence time in the absence of alkali atoms, usually limited by inhomogeneity of the magnetic field [23,35].We note that due to negligible noble-gas-cell-wall coupling, the diffusion-induced decay of the spatially-uniform mode of the noble gas vanishes [34].The incoherent spintransfer terms [third term in Eqs. ( 13)], which have negligible effect on the coherent dynamics, are omitted for brevity.In the experiments considered in the main text, γ −1 a = 50 ms and γ −1 b = 1 − 100 hours, hence γ b ≪ γ a .The Langevin noise operators F a and F b in Eq. ( 15) account for fluctuations and for preserving commutation relations under the relaxations γ a ,γ b and diffusion [41]; their manifestation as scalar operators Fa and Fb after the Holstein-Primakoff approximation is given below.
As described in the main text, we consider highly-polarized ensembles with most spins pointing downward (−ẑ) [1][2][3], approximate f z = −p a n a q/2 and k z = −p b n b /2, and apply the Holstein-Primakoff transformation [2] to represent the collective states as excitations of a bosonic field with local annihilation operators â (r, t) = f− (r, t) / 2|f z | and b (r, t) = k− (r, t) / 2|k z |.The creation operators â † (r, t) and b † (r, t) flip upwards one alkali or noble-gas spin at position r.
When the two gases are polarized, the energy cost of flipping a spin in one species is the sum of Zeeman shift (due to the external magnetic field) and so-called collisional shift (due to effective magnetic field induced by the other species) [39].The altered Larmor frequencies ω a = ωa −ζp b n b /2 and ω b = ωb − ζp a n a q/2 are obtained when rewriting Eqs.(15) in terms of â(r, t) and b(r, t), Importantly, here we obtain the coherent spin-exchange rate J = ζ √ qp a p b n a n b /2, responsible for the local coupling of the two collective spins, as illustrated in Fig. 1c.Numerical solution.We numerically solve the differential Eqs.(18) for two particular cases, using the experimental parameters outlined in the main text.The analytical solution of Eqs.(18) utilizes a decomposition to coupled spatial (diffusion) modes, given in the SI.Our calculation uses the first 100 spherically symmetric least-decaying modes.We assume an alkali polarization of p a = 0.95 and noble-gas spin polarization of p b = 0.75, corresponding to initial â † â = 0.05 and b †b = 0.17 at t = 0. We account for these incoherent excitations, due to the imperfect polarizations, by using a nonpure density-matrix and incorporating the associated (excess) quantum noise.
First, the spatially-symmetric (uniform) mode of the alkali spin ensemble is initialized in a Fock state with two spin excitations.The probability of transferring the excitations to the uniform mode of the noble-gas spins is presented in Fig. 4a.Here, by changing ∆ at t = π/(2J), the excitation transfer is complete, and the noble gas becomes decoupled from the alkali and free from collision-induced relaxation.Second, the uniform mode of the alkali spin ensemble is initialized in a squeezed vacuum state with 7 dB of squeezing.The squeezing is transferred to the noble-gas spins with high fidelity, as presented in Fig. 4b.We include 100 modes to ensure convergence of the calculations, with no significant improvement observed with additional modes.

NOISE ASSOCIATED WITH SPIN-EXCHANGE COLLISIONS
The fluctuation vector-operator F ex in Eqs. ( 13) can be defined by (cf.Ref. [59]) Similarly to what we observe in the toy-model simulation (Fig. 3 and SI), the operator F ex describes fluctuations of order φ 2 in the coherent mutual-precession process, manifesting a stochastic superposition of non-symmetric local spinoperators.Fluctuation in the incoherent terms are of order φ 4 and thus negligible.We now examine the statistical properties of the fluctuation operator F ex .It is zero on average F ex (r, t) = 0, and its correlations satisfy where Lij = δ ij 1 − 2{ ki , ŝj }/(n a n b ) + i ijm km /n b + ŝm /n a .Here i, j, m ∈ {x, y, z} and the symbol ijm is the Levi-Civita tensor.We interpret F ex as temporally and spatially white, since its correlations are proportional to δ (t − t ) and to the coarse-grained delta function w(r − r ).Furthermore, the coarse-grained commutation relations ŝi (r, t), ŝj (r , t) = iw(r − r ) ijm ŝm (r, t) and ki (r, t), kj (r , t) = iw(r − r ) ijm km (r, t) are preserved.Indeed, the relaxation of the commutation relations after dt due to the loss terms in Eqs. ( 13) is exactly balanced by the fluctuations Fex,i (r, t) Fex,j (r , t)dt 2 .We therefore formally identify F ex as a quantum white noise operator [41] originating from the randomness of the collisional interaction.
For fully-polarized spin ensembles, the corresponding noise correlations are found to have the standard form of a vacuum noise [41], satisfying The contribution of F ex to the alkali noise operator Fa in Eqs. ( 2) and ( 18) is given by Fa = F − ex / 2|f z |.Therefore, the variance is Fa (r, t) F † a (r , t ) = 2γ ex δ (t − t ) w(r−r ), where γ ex = n b k se /q is the spin-exchange relaxation for the alkali spins.We thus conclude that, for polarized ensembles, the spin-exchange noise appears as a vacuum noise, which is the minimal possible noise.
For general spin polarizations p a , p b ≤ 1, the second-order moments of the noise are given by Importantly, for highly-polarized ensembles (1 − p a 1, 1 − p b 1), the excess noise is small, since 2−pa−pb 4pa 1 and 2+pa+pb 4pa − 1 1.It is interesting to note that imperfect polarization contributes to other quantum noise terms as well, and this contribution is quantitatively similar to that in F ex and Fa .It follows that the coherent interface based on spin-exchange collisions is not much more sensitive to imperfect polarization compared to other quantum effects in singlespecie spin systems.
In practice, the relaxation due to spin exchange, including the effect of F ex , is small compared to that originating from other sources.For example, the relaxation rate of the alkali electron spin due to spin exchange is n b k se , whereas the relaxation rate due the spin-orbit coupling during collisions is n b σ sr v, where σ sr is the spin-rotation cross-section.The relative importance of these two mechanisms is characterized by the parameter η = k se /(σ sr v), where η = 0.34 for potassium-helium, η = 0.024 for rubidium-helium, and η < 0.01 for cesium-helium at 215 • C [11].The relaxation of the noble-gas spins due to spin exchange with alkali spins is negligible for t (n a k se ) −1 ≈ 17 hours when operating with n a = 3 × 10 14 cm −3 .When perfectly-polarized ensembles are initialized, quantum noises other than spin exchange also behave as vacuum noises.In this case, the noise terms Fa = ( Fa,x +i Fa,y )/ 2|f z | and Fb = ( Fb,x +i Fb,y )/ 2|k z | )]δ (t − t ) for χ ∈ {a, b}, including the fluctuations induced by the spinexchange interaction.The function C χ (r, r ) is the diffusion component of the noise correlation function [34], independent of the spin-exchange interaction or of the other relaxation mechanisms incorporated in γ χ .
Finally, the incoherent excitations due to imperfect polarization also add a classical (mix state) background to the (otherwise pure) expectation values, which reduces the significance or fidelity of non-classical phenomena.For example, the probability of finding a collective excitation in an otherwise unexcited ensemble is 2p where θ v ab ∈ [0, π] is the relative angle between r ab and v ab .Therefore t ab exists only if sin 2 (θ v ab ) ≤ 2 /r 2 ab (t).Since is about a few angstroms, collisions occur only at small angles θ v ab 1. Neglecting the collision duration τ c 2 /v ab τ , we can approximate the collision time as the average of the two solutions, yielding t (i) ab = t + r ab (t) /v ab , if θ v ab ≤ /r ab (t).We can then write the expression of κ ab (t, τ ) as which determines if a pair has collided given the relative location and velocities.
To derive the statistical properties of κ ab , we first average over the pairs velocities.We assume a Maxwell-Boltzmann distribution for the velocity v where v T stands for the thermal relative velocity of the pair.The velocity-average collision probability is then given by such that two particles collide, on average, depending on their relative solid angle σ/(4πr 2 ab (t)), provided that their spatial separation is small r ab < τ v T .Our model relies on the motion of the particles being ballistic, which is valid for short intervals v T τ 1/(n b σ).We are interested in the spatially coarse-grained dynamics on the length-scale l 1/(n b σ).Using the radial window function w(r), we obtain κ ab (t, τ ) v * w(r) = where in the second line we used v T τ ≪ l.This expression can be used to estimate standard kinematic relations, such as the mean collision times.The probability that two spins a − b would collide in time interval τ is given by p ab (t, τ ) = κ ab (t, τ ) v * w(r).We can now find the mean time that particle a has collided with any other noble gas atom.This probability is given by We calculate the second moment of κ ab assuming that different collisions are statistically independent, where in the second line we assumed that the times t, t are sampled with intervals dt, dt τ to include multiple collisions.Spin exchange interactions, experienced during binary collisions as considered so far, lead to phase accumulation of the colliding spins.The spin dynamics are determined by the statistics of the collisions and are governed by Moreover, κ ab (t, τ ) is defined such that when the spins a, b do not approach each other closer then the collision distance , then φ ab (t) = 0, and φ ab (t) = 0 only when κ ab (t, τ ) = 1.Therefore φ ab (t) |κ ab (t, τ ) = 1 = φ ab (t) = φ , where φ is averaged over all impact velocities and impact parameters.The value of φ and the dependence of φ ab (t) on the collision trajectory (velocity and impact parameter) were discussed in Eqs.(S12) and (S15) are used to derive the properties of ζ in Eq. ( 13) in the methods section, and in the identity of the spinexchange noise F ex (r, t) = 0. Eqs.(S13) and (S16) are used in deriving the properties of the incoherent exchange terms in Eq. ( 13) in the methods section and in the derivation of the variance identities of the spin-exchange noise in Eq. (19).

Figure 2 .
Figure 2. Evolution of spin excitations of interacting alkali and noble-gas ensembles in different coupling regimes.Both ensembles are initialized in coherent spin states, with the alkali spins initially having â † â = 1000 excitations and the noble-gas spins in the vacuum state b † b = 0. a, Decoupled modes.At large detunings ∆ = 10J = 600γ, the spin ensembles are decoupled, and the alkali excitations decay at a rate γ.The noble-gas spins exhibit negligible exchange with the alkali (inset).b, Overdamped coupling.At ∆ = 0, the alkali and noble-gas spins hybridize and decay, here at a rate γ = 10J, exhibiting partial transfer of the excitations (inset).c, Strong coupling.The periodic exchange for J = 57γ at ∆ = 0 allows for coherent transfer of spin excitations.Application of a large magnetic field at t = π/(2J) decouples the spin ensembles and 'stores' the excitations in the noble-gas spin state (dashed line).

( 2 )
] to the mean-field model [Eqs.(1)], by employing the transformations â = N a /(qp a ) f− and b = N b /p b k− , which associate the collective displacements with the mean spin of the ensembles.

Figure 3 .
Figure 3. Stochastic simulation of the collisional interface.We solve the unitary evolution of the quantum state of Na = 100 electron spins and N b = 10 4 noble-gas spins (in e, Na = 30, N b = 300), initialized in the state |ψ0 .Each electron spin undergoes a spin-exchange collision of random strength (φrandom 1) with a randomly chosen noble-gas spin every simulation time-step τ = 1; see SI for more details.The collisions are either very weak φ = 10 −5 (c-e) or more moderate φ = 2 • 10 −2 (a,b).The electron spins can be initialized with all pointing downwards |0 a = | ↓ . . .↓ a ("vacuum"), or with one arbitrary spin pointing upwards f+ a=1 |0 a = | ↑↓ . . .↓ a ("localized excitation"), or in the symmetric state with a single collective excitation |1 a = N −1/2 a a f+ a |0 a = N −1/2 a i | ↓ . . .↓↑i↓ . . .↓ a ("collective excitation").The nuclear spins are initialized in either |0 b or |1 b .a, An initial symmetric excitation |ψ0 = |1 a|0 b is coherently exchanged between the two spin ensembles.The inset highlights the stochasticity of the process.The exchange is accompanied by dephasing due to incoherent transfer of the excitation to the large noble-gas ensemble (N bNa) via the same process underlying SEOP.b, The localized excitation is incoherently transferred to the noble gas.c, Strikingly, when the collisions are weaker, the exchange fidelities F10 and F01 oscillate with higher contrast and nearly no decay, despite the stochasticity of the process.d, Almost no oscillations are observed for the localized excitation.e, When |ψ0 = |1 a|1 b , the two excitations periodically "bunch" in a superposition of either of the spin ensembles (|2 a|0 b + |0 a|2 b ), manifesting the nonclassical Hong-Ou-Mandel phenomenon and validating the quantum beam-splitter property of the interface.

Figure 4 .
Figure 4. Exchange of nonclassical states between alkali and noble-gas spin ensembles.The exchange is calculated for the proposed experimental parameters and assuming the initial excitation is spatially uniform.We compare the two-mode approximation given by Eqs.(2) for perfectly-polarized ensembles (black) with an exact solution accounting for higher spatial modes and imperfect polarization (blue, see Methods).At t = 0, the magnetic field is tuned to resonance ∆ (B) = 0.If it is detuned at t = π/(2J), the state transfer is maximal (solid purple, using B = 180 mG).The imperfect polarization is accounted for by initializing the system with an incoherent state and incorporating the associated excess quantum noise (see Methods).a, Probability of populating the uniform (longestlived) diffusion mode of the noble gas with exactly two excitations (Fock state) when the system is initiated with two excitations in alkali's uniform mode.b, Exchange of squeezing between the total alkali spin (proportional to the electron spin Ŝx = a ŝa,x) and the total noble-gas spin ( Ky = b kb,y ).The alkali is initialized with 7 dB squeezing.
ab and strengths φ (i) ab .Instead, we represent the exact values of t (i) ab and φ