Strong coupling of alkali spins to noble-gas spins with hour-long coherence time
SStrong coupling of alkali spins to noble-gas spins with hour-long coherence time
R. Shaham,
1, 2, ∗ O. Katz,
1, 2, 3, ∗ and O. Firstenberg Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel Rafael Ltd, IL-31021 Haifa, Israel Present address: Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708
Nuclear spins of noble gases can maintain coherence for hours at ambient conditions owing to theirextraordinary isolation by the enclosing, complete electronic shells [1]. This isolation, however, impedes theability to manipulate and control them by optical means or by physical coupling to other spin gases [2–4]. Herewe experimentally achieve strong coherent coupling between noble-gas spins and the optically-accessible spinsof alkali-metal vapor. Stochastic spin-exchange collisions, underlying the coupling, accumulate to a coherentperiodic exchange of spin excitations between the two gases. We obtain a coupling rate 10 times higher thanthe decay rate, observe the resultant avoided crossing in the spectral response of the spins, and demonstratethe external control over the coupling by magnetic fields. These results open a route for efficient and rapidinterfacing with noble-gas spins for applications in quantum sensing and information [5, 6].
INTRODUCTION
Noble gas isotopes with a nonzero nuclear spin, suchas helium-3, feature day-long spin lifetimes and hours-longcoherence times [1, 7]. They are prominent in various fields,from precision sensing [8–11] and medical imaging [12] tosearches of new physics [13–17], and they hold promisefor future quantum information applications such as opticalquantum memories and generation of long-lived entanglement[5, 18–20]. The latter rely on the feasibility of preparing thecollective spin state of the gas and controlling its quantumexcitations [21].Polarized ensembles of alkali-metal spins or noble-gasspins can carry such collective excitations, correspondingclassically to a tilt of the collective spin about the polarizationaxis [22]. These can be modeled as quantum excitationsof a harmonic oscillator. Remarkably, the quantumdescription persists even for gaseous ensembles undergoingrapid diffusion [23, 24] and for overlapping ensembles thatinteract via atomic collisions [21, 25–27]. The collective stateof alkali spins can be addressed and coherently controlled byoptical means [28–30]. The same, however, cannot be donefor the nuclear spins of noble gases, which lack any opticaltransition from the ground levels. Instead, one can accessthe noble-gas spins by collisions with another spin gas, eitherexcited (metastable) helium-3 or alkali vapor, both of whichpossess optically-accessible spins [1, 31, 32]. Alkali atomsexchange spin with noble-gas atoms via a weak electron-nuclear coupling (Fermi contact) during collisions [33]. Theyare normally used for hyperpolarizing the noble gas and forprobing its spin dynamics. The probing relies on the coherentcomponent of the spin-exchange interaction, which is usuallyweak and manifests as a shift in the precession frequenciesof the alkali spins. It is employed for readout of noble-gas-based sensors and for inherent suppression of sensitivityto magnetic fields [2–4, 9]. Recently, we used alkali spinsas off-resonant mediators to couple light to noble-gas spinsbidirectionally [34]. However, these works have been limitedto the regime of detuned or overdamped coupling, with thealkali spin relaxation exceeding the coupling strength to the noble-gas spins.Here we report on strong coherent coupling between thecollective spin states of noble-gas and alkali ensembles.We enter the strong-coupling regime by reaching highpolarizations and densities of the interacting species whileminimizing spin relaxation. We directly probe the dynamicsof both spin ensembles and demonstrate the coherentexchange of excitations between them in the temporal domainas well as the corresponding gap between the normalfrequencies of the coupled system in the spectral domain.These results certify that stochastic spin-exchange collisionsthat are individually weak but altogether frequent enough canaccumulate to form an efficient coherent interface betweentwo spin gases.Consider the bosonic, collective, spin excitations of thealkali and noble-gas spins, represented by the annihilationoperators ˆ a and ˆ b , respectively. The coupling betweenthese excitations relies on the collective enhancement ofthe exchange interaction, due to accumulation of numerouscollisions among the two spin ensembles. The collective,bi-directional, coupling rate J = (˜ ζ/ √ n a p a n b p b thusdepends on the square root of the atomic densities n a , n b and degrees of polarization ≤ p a , p b ≤ [21].The microscopic coupling strength ˜ ζ ( p a ) , incorporating thecollisional cross-section, has a weak dependence on the alkalispin polarization due to the hyperfine structure of the alkaliatoms (see Methods). A simple form of two coupled modescan be used to describe the exchange dynamics, ∂ t (cid:18) ˆ a ˆ b (cid:19) = i (cid:18) ω a + iγ − J − J ω b (cid:19) (cid:18) ˆ a ˆ b (cid:19) + ˆ f . (1)Here ω a and ω b denote the Larmor precession frequenciesof the collective spins of the alkali and noble-gas atoms,respectively. They are set by the external magnetic field B and by the effective magnetic fields exerted by each specieon the other [31]. We tune B to determine the detuning fromresonant couplings ∆ = ω a − ω b . The decoherence rate of thealkali excitations γ is included, while for now we neglect theslow decoherence of the noble-gas spins. Finally, ˆ f denotesthe quantum noise accompanying the relaxation, motion, and a r X i v : . [ qu a n t - ph ] F e b collision processes [21, 23]. In the current study, ˆ f canbe discarded, as we prepare the spin ensembles in coherentspin states and study the evolution of the mean transverseamplitudes (cid:104) ˆ a (cid:105) and (cid:104) ˆ b (cid:105) .When J > γ , Eq. (1) describes the exchange of spins ata rate ˜ J ≈ (cid:112) J + ∆ / and with a contrast ( J/ ˜ J ) , in aframe rotating at ω b . Strong coupling is achieved for J > { γ, ∆ } , which produces exchange oscillations with near-unitycontrast. RESULTSExperimental setup and protocols
We study transverse spin excitations of polarized potassiumvapor and helium-3 gas enclosed in a spherical glass cell, asshown in Fig. 1a. The potassium spins are polarized alongthe axial magnetic field by an optical pumping beam, and thehelium spins are polarized by collisions with the polarizedpotassium (over 10 hours, see Fig. 5). The cell also containsnitrogen for reducing (quenching) the fluorescence from theoptically excited potassium atoms. At a low polarization, thehelium spins exhibit a coherence time of T b2 = 2 hours, asshown in Fig. 1b, and consequently their individual relaxationis henceforth neglected. The exchange experiments start withturning off the pumping beam.We monitor the dynamics of the coupled spin systemfollowing a short, 5- µ s-long pulse of transverse magneticfield B ⊥ , which predominantly excites the collective alkalispin and initializes it at a tilt angle of a few degrees fromthe axial magnetic field B ˆ z . We measure the transversealkali spin using Faraday rotation of an optically-detuned,linearly-polarized probe beam. We perform a tomographic-like reconstruction of the alkali spin in the xy plane byalternating between B ⊥ ˆ y and B ⊥ ˆ x for initialization. Weproperly scale these measurements by the total degree ofpolarization p a ( t ) (measured independently, see Fig. 7) andcalculate the complex amplitude of the collective alkali spin (cid:104) ˆ a ( t ) (cid:105) . To measure the collective noble-gas spin (cid:104) ˆ b ( t ) (cid:105) aftersome exchange duration t , we halt the exchange dynamics at t by rapidly ramping up the axial magnetic field (increasing ∆ )and utilizing the alkali spins as a magnetometer for sensingthe noble-gas spin precession.We realize a maximal coupling rate of J = 78 ± Hz byoperating at high densities of potassium n a = 4 . · / cc (at T = 230 ◦ C ) and helium n b = 6 . · / cc (2.4atm at room temperature) and with relatively high degreesof spin polarization p a (cid:38) . and p b (cid:38) . . At theseconditions, collisions among alkali atoms are frequent enough( > . /µ s ) with respect to the Larmor frequency to keepthe alkali excitations free from spin-exchange relaxation (so-called SERF regime) [31]. The intricate hyperfine manifoldof the alkali atoms maintains a spin-temperature distributiondue to these collisions and manifests as an effective spin-1/2,as shown in Fig. 1c [31]. Remnant spin-relaxation occurring ab pumpingprobe time (hour)0 0.5 1.0 1.5 2.0
0 0.2s 0 0.2s 0 0.2s 0 0.2s 0 0.2s 0 0.2s10-1 t r a n s v e r s e n o b l e - g a s s p i n J potassium-39 He-3 c FIG. 1.
Experimental scheme and coherence-timemeasurements. a.
A glass cell containing optically-pumpedpotassium vapor (alkali spins, red) and helium-3 (noble-gas spins,blue). The polarized ensembles couple via stochastic atomiccollisions that accumulate to a collective spin-exchange interactionat a rate J . An applied magnetic field B ˆ z controls the precessionfrequency difference ∆ = ω a − ω b between the two ensembles. Atransverse excitation of the spins is initialized by a short transversemagnetic field pulse B ⊥ ˆ y and then monitored by Faraday rotation ofan optical probe. b. Precession of the helium-3 spins, measured atlow spin polarizations and normalized to the initial value, featuringa coherence time of T b2 = 2 hours. c. Energy level diagram for thecoupled spins. The spin-polarized alkali atoms, undergoing frequentspin-exchange collisions, can be described as an effective two-levelssystem. during these collisions dominates the decoherence rate of thealkali excitations γ = 6 ± . Hz. We thus achieve J (cid:38) γ .See Methods for a detailed description of the experimentalconditions and analysis procedures. Dynamics of strongly-coupled spins
Under the strong-coupling conditions, the two spingases can coherently exchange collective excitations. To s p i n e xc i t a t i on s / N exchange duration t [ms] FIG. 2.
Exchange of collective spin excitations.
Measurementof the coherent exchange between the alkali spin (cid:104) ˆ a (cid:105) (red circles)and the noble-gas spin (cid:104) ˆ b (cid:105) (blue triangles) in the strong-couplingregime. A short pulse of transverse magnetic field at t = 0 excites N = |(cid:104) ˆ a (0) (cid:105)| + |(cid:104) ˆ b (0) (cid:105)| = (13 . ± . · spins. Theexperimental conditions at t = 0 are J = 78 ± Hz, γ =6 ± . Hz, and ∆ = − . J (for obtaining maximal extinctionof (cid:104) ˆ a (cid:105) at the minima, see text). Lines present the result of adetailed model using these parameters. Each data-point is averagedover 12 to 20 repetitions of the experimental sequence (shown inFig. 6). Colored errorbars include uncertainties in the spin-projectionmeasurements and the scattering between repetitions. Gray errorbarsalso include the uncertainty in the alkali polarization p a ( t ) , requiredfor converting spin projections to excitations. The bottom panelpresents the same data in terms of |(cid:104) ˆ a (cid:105)| + |(cid:104) ˆ b (cid:105)| , confirming that thetotal number of excitations is conserved by the exchange process, upto an overall decoherence. demonstrate this dynamics, we tune ∆ close to resonanceand generate an initial excitation predominantly of the alkalispin. Fig. 2 presents the measured spin excitations |(cid:104) ˆ a (cid:105)| and |(cid:104) ˆ b (cid:105)| , as they are exchanged back and fourth betweenthe two ensembles. Because the magnetic pulse acts alsoon the noble-gas spin and partially excites it as well,the extinction of |(cid:104) ˆ a ( t ) (cid:105)| at the minima of the observedoscillations is maximized slightly below resonance, at ∆ = − . J ; the presented measurement is taken at this detuning.This detuning is still small in terms of the strong-couplingdynamics, rendering a near-unity ratio between the exchangeand coupling rates ˜ J/J = 1 . .We find that the exchange conserves the total number ofexcitations |(cid:104) ˆ a (cid:105)| + |(cid:104) ˆ b (cid:105)| , which exhibits a roughly monotonicdecay due to spin decoherece. We also directly observethe slowing down of the exchange oscillations, as the spinsgradually decouple due to the dependence of ˜ J on thedecaying alkali polarization p a ( t ) . This gradual decouplingalso leads to residual excitations populating the long-livednoble gas spin. These effects are all captured by a detailedmodel (solid lines), described in Methods, which accountsfor the temporal decrease of J and for small geometricmisalignments. It is instructive at this point to compare the resonant, strong-coupling dynamics to the dynamics off resonance or to anoverdamped dynamics. These are presented in Fig. 3, showingthe measured amplitudes Re (cid:104) ˆ a (cid:105) and Im (cid:104) ˆ a (cid:105) (bottom panel)and total number |(cid:104) ˆ a (cid:105)| (top panel) of collective alkali spinexcitations. For the coherent spin states in our experiment,oscillations of |(cid:104) ˆ a ( t ) (cid:105)| correspond to nutations (tilt) of thecollective alkali spin from the quantization axis ˆ z , whereasoscillations of (cid:104) ˆ a ( t ) (cid:105) also include the Larmor precession inthe xy plane.First, we set ∆ close to resonance ( ∆ = − . J asbefore) and measure the dynamics under the strong-couplingconditions J = 68 ± Hz and γ = 6 . ± Hz (Fig. 3a).As in Fig. 2, we observe oscillations of the number of alkalispin excitations |(cid:104) ˆ a ( t ) (cid:105)| , exchanged back and forth with thenoble-gas spin while gradually decaying. The dynamics far-off resonance is shown for an increased detuning ∆ = 460 Hz ≈ . J (Fig. 3b). In this regime, we observe a decayingprecession of (cid:104) ˆ a ( t ) (cid:105) and an almost monotonic relaxation of |(cid:104) ˆ a ( t ) (cid:105)| at a rate ± Hz, in agreement with the expectedvalue ( γ ). Finally, we repeat the experiments with anincreased relaxation rate γ = 215 Hz ≈ . J (Fig. 3c),implemented by keeping the pumping beam on during themeasurement. No exchange oscillations occur in this regime.The observed relaxation rate is reduced compared to thedecoupled case (Fig. 3c, dotted), since here the proximityto resonance ∆ < γ leads to dissipative hybridization ofthe alkali and noble-gas spins, which increases the coherencetime of the former, as studied by Kornack et al. [3]. Themeasurements in Fig. 3 of the three regimes elucidate thecoherent nature of the exchange interaction under the strong-coupling conditions. Spectral map
A hallmark of strong coupling is the opening of a spectralgap in the response function of the coupled system atresonance. We measure this gap by repeating the experimentpresented in Fig. 3a for different values of ∆ . The spectralmap, shown in Fig. 4a, reveals an avoidance crossing at ∆ = 0 between the normal frequencies, with a wide gap indicating astrong coherent coupling between the two gases. We furthercompare the measurements to calculated spectra. We presentboth a simple model based on Eq. (1) (dashed lines in Fig. 4a)and the results of our detailed model (Fig. 4b). Both modelsreproduce well the main frequency branches. The additionalfeatures in the spectrum, primarily the weak perpendicularbranches and the vanishing amplitude of the horizontal branchat ∆ (cid:38) J (due to reduced sensitivity to magnetic stimulationnear the so-called compensation point [3]) are well capturedby the detailed model. strong coupling a time [ms] -101 00.20.40.60.81 detuned b overdamped c time [ms] -101 FIG. 3.
Measured dynamics of the coupled alkali-noble-gas spin system in three regimes . All measurements begin with a short magneticstimulation of N = (5 . ± . · alkali spin excitations, and the initial alkali-noble-gas coupling rate is J ≈ Hz.
Top:
Collectivespin excitations of the alkali atoms.
Bottom:
Real and imaginary parts of the collective spin amplitude, associated with the two transversespin components in the lab frame, exhibiting Larmor precession in addition to the exchange. a. Strong-coupling , achieved when J exceedsthe alkali relaxation rate γ = 0 . J and close to resonance ∆ = − . J . Collapse and revival of alkali spin excitations provide evidencefor a coherent hybridization with the noble-gas spins. b. Decoupled dynamics , observed when increasing the detuning to ∆ = 6 . J = 69 γ by increasing the magnetic field. The alkali spin, here largely decoupled from the noble-gas spin, undergoes standard Larmor precessionand relaxation. c. Overdamped dynamics , obtained under conditions of weak coupling γ = 3 . J . When near resonance (solid line, ∆ = − . γ ), the long-lived noble-gas spin partially hybridizes with the alkali spin, whose relaxation slows down compared to the non-resonant case (dotted line, ∆ = γ ). DISCUSSION
In summary, we realize a strong coherent coupling betweenthe collective spins of dense alkali vapor and noble gas,relying on stochastic spin-exchange collisions. Each singlecollision is very weak: the mutual precision angle of thealkali (electron) spin around the noble-gas (nuclear) spin ison average only ∼ − radians per collision. Nevertheless,as we show, the collisions can accumulate to an efficientcollective coupling. We demonstrate the coherent exchangeof excitations between the two spin gases and measure asizeable avoided-crossing gap in their spectral response whenscanning the magnetic field across the resonance. Theseobservations manifest the coherent hybridization of the twocollective spins. They attest to the fact that coherence enduresthe randomness and stochasticity of the collisions, providedthat the collisions are individually weak enough, as studiedanalytically and numerically in Ref. [21].The strong-coupling regime J (cid:29) γ offers a fast reversibleinterface to noble-gas spins. While reversible manipulationsare also possible in the overdamped regime J (cid:46) γ , they arelimited to adiabatic operations, slow compared to the couplingrate. We report on J/γ ≈ and estimate that higher valuesare achievable with higher He density and polarization andwith lower densities of the alkali and nitrogen gases. Hepressure exceeding 10 atm was demonstrated [1] as well as85% polarization [35]. A system at ◦ C with 8.2 atm of He polarized to 80% and near unity polarized potassium isexpected to achieve
J/γ > .The hybridization of optically-accessible alkali spinsand long-lived noble-gas spins opens several intriguingpossibilities. One route motivated by quantum informationapplications is using the alkali spins as mediators betweenphotons and noble-gas spins [34]. The strong spin-spincoupling can improve the performance of such applicationsby enhancing the indirect coupling to photons. Noble-gasbased optical quantum memories, for example, would featureenhanced memory bandwidth when operated in the strong-coupling regime [6]. While memories relying on dampedcoupling are expected to have a few-Hz bandwidth [18, 19],the memory bandwidth in the strong-coupling regime isessentially independent of J ; light is first stored on the alkalispins (bandwidth > MHz) and subsequently mapped ontothe noble-gas spins with efficiency e − ( πγ ) / (2 J ) for hours-longstorage [6]. Another example is the generation of long-livedspin-entanglement, which in the strong-coupling regime willbenefit from a significant suppression of the contribution ofalkali projection noise at elevated alkali densities by a factor γ/ ( J T ) , where T is the duration of the entangling pulse[5]. A second potential route is utilizing the strong couplingfor improving noble-gas-based sensors. Alkali-noble-gas co-magnetometers, for example, are used as precise gyrometersand as probes for new physics, as they feature high sensitivityto anomalous fields and to bosonic dark matter [13, 16]. FIG. 4.
Spectral response of the alkali-noble-gas spin system in the strong coupling regime. a.
Measured response of the collective alkalispin (cid:104) ˆ a (cid:105) to a weak stimulation, for different detunings between the spins ∆ . The spectrum (cid:104) ˆ a ( ω ) (cid:105) ∝ (cid:82) ∞ (cid:104) ˆ a ( t ) (cid:105) e − iωt dt (normalized separatelyfor each ∆ , see Methods) manifests the eigenvalues of the coupled system; The spectrum maxima correspond to the normal frequencies, andthe spectral widths are indicative of the decay. Dashed lines are the imaginary part of the eigenvalues of Eq. (1). A clear avoided crossing witha sizeable spectral gap at | ∆ | < J indicates the strong, coherent hybridization of the two spin gases. The response at | ∆ | > J corresponds tothe independent precession rates of the alkali and noble-gas spins ω a and ω b respectively. The axes are scaled by the average value J = 47 Hz (rather than the initial value J = 79 Hz) to account for the decrease of J due to alkali depolarization during the 65-ms-long measurement. b. Calculated spectral response from a detailed model. The model includes a misalignment of 3.1 mrad between the magnetic field and thepumping direction; Inset shows the model results without this misalignment.
These sensors however have a relatively slow response time,typically limited by γ/ close to resonance. A strong couplingin these sensors could enhance their bandwidth up to J whilemaintaining their high sensitivity. MATERIALS AND METHODSThe Holstein-Primakoff transformation from spins to bosonicexcitations
The states of the alkali and noble-gas spin ensemblesare characterized by their degree of polarization p a =(2 /N a ) (cid:80) m (cid:104) ˆ s ( m ) z (cid:105) and p b = (2 /N b ) (cid:80) n (cid:104) ˆ k ( n ) z (cid:105) . Here, (cid:80) m ˆs ( m ) j and (cid:80) n ˆk ( n ) j with j = { x, y, z, − , + } are thestandard collective spin operators of the electrons of the alkaliatoms and the nuclei of the noble gas atoms, respectively, and N a = n a V and N b = n b V are the number of atoms inthe volume V . Describing the alkali spins in terms of onlythe electronic spins is possible owing to the frequent alkali-alkali collisions, which constantly drive the alkali atoms toa spin-temperature distribution [36]. In the spin-temperaturedistribution, due to the hyperfine coupling to the alkali nuclearspin, the spin precession around an external magnetic field isslower than that of a bare electron by a factor q ( p a ) , known asthe slowing-down factor; for potassium, q ( p a ) = 2 + 4 / (1 + p ) [36–38].We are interested in the bosonic annihilation operators ˆ a and ˆ b , defined according to the Holstein-Primakofftransformation as ˆ a = (cid:112) q/N a p a (cid:80) m ˆs ( m ) − and ˆ b = (cid:112) /N b p b (cid:80) n ˆk ( n ) − [21, 22]. These are the canonical,normalized version of the collective spin operators transverseto the quantization axis. For the alkali spins, we denote thedepolarization rate by Γ (decay rate of p a = (cid:80) m (cid:104) ˆ s ( m ) z (cid:105) ) andthe transverse relaxation rate by Γ (decay rate of (cid:80) m (cid:104) ˆ s ( m ) x (cid:105) and (cid:80) m (cid:104) ˆ s ( m ) y (cid:105) ). The decoherence rate of the excitations (cid:104) ˆ a (cid:105) is therefore given by γ = Γ − Γ / , neglecting smallvariations of q on short timescales. Apparatus and experimental conditions
We use a spherical cell with diameter (cid:96) = 2 . cm andvolume V = 8 . , made of GE-180 aluminosilicate glass,containing He gas, a droplet of natural abundant potassium,and 50 Torr of nitrogen. The temperature of the cell T =230 ◦ C is maintained using a pair of resistance twisted wireswrapped around an alumina body, which are driven withcurrent oscillating at 320 kHz. The magnetic field is appliedvia three sets of coils: 4-winding double Helmholtz coils forcontrolling B ˆ z and a bird-cage coil for the transverse fields toimprove magnetic uniformity. The coils are placed inside fiveconcentric layers of µ -metal magnetic shields, and the innertwo layers are degaussed.The N a = 4 . · potassium atoms are polarized byoptical pumping using 500 mW of circularly-polarized light at time [hr] FIG. 5.
Spin-exchange optical pumping.
Typical measurementof the pumping process of helium-3 by optically pumped potassiumvapor. Here the potassium density is n a = 4 . · / cc , and thehelium depolarization time is T b1 , act = 3 . hours.
770 nm. This pumping light is generated using a free-runningdiode laser followed by a tapered amplifier. We tune the lasernear the optical D1 transition, which in our setup appears asa single absorption line with a full width of 32 GHz due topressure broadening, producing an on-resonance optical depthof n a σ abs (cid:96) ≈
220 ( σ abs = 1 . · − cm is the absorptioncross-section of the 32-GHz-wide line). The pumping beamis Gaussian with a 25-mm waist diameter. We detune it fromresonance to reduce its depletion and by that achieve the highdegree of spin polarization p a ≥ . .The depolarization rate of the potassium spins in the dark Γ = 8 . Hz is dominated by spin-destruction collisionsamong potassium atoms and by spin-rotation interaction ofpotassium atoms with the buffer gas [39]. Rapid spin-exchange collisions among potassium atoms at a rate R se =86 kHz and the operation at low Larmor precession rates | ω a | (cid:28) √ R se Γ puts the potassium in the so-called spin-exchange relaxation-free (SERF) regime [37, 40], renderingthe relaxation induced by spin-exchange collisions negligible.Consequently, the transverse spin relaxation rate Γ = 10 . Hz is dominated by the depolarization processes, with minorcontribution from magnetic inhomogeneity. These lead to adecoherence rate of γ = Γ − Γ / ≈ Hz for the bosonicexcitations of the potassium spins.The N b = 5 . · helium atoms are hyperpolarizedusing spin-exchange optical pumping (SEOP) [36] at a rate (3 . ± . · − Hz in the presence of an axial magneticfield B = 400 mG. A typical SEOP measurement settlingat p b ≥ . is presented in Fig. 5. In our system, at lowtemperature, the measured depolarization and decoherencetimes of the helium spins T b1 = 22 hours and T b2 = 2 hours are limited by magnetic field inhomogeneity within the cellvolume. At elevated temperature and polarizations, wemeasure T b1 , act = 3 . (see Fig. 5) due to inhomogeneityof the magnetizations of the two ensembles in the cell,which slightly deviates from an ideal sphere [41]. Tomoderate the helium depolarization during the experiments,we intermittently turn on the SEOP in between measurements.The polarized spin ensembles exert an equivalentmagnetic field (EMF) on each other, via collisions andvia the macroscopic magnetic fields generated by their magnetization. While the EMF experienced by the helium B a → b = − . mG (for p a = 0 . ) is small, the EMFexperienced by the potassium B b → a = − . mG (for p b = 0 . ) is considerable. The detuning from resonantcoupling ∆ is thus quite sensitive to p b , which we monitorduring the experiment. We do so by applying a constantmagnetic field − B b → a + 1 . mG, and monitoring theprecession frequency of the decoupled alkali spins followinga small transverse magnetic pulse.In the experiments presented in Fig. 2, Fig. 3, and Fig. 4,we use a transverse magnetic pulse to tilt the alkali spins by θ a = 9 . ± . ◦ , θ a = 6 . ± . ◦ , and θ a = 0 . ± . ◦ ,respectively. In terms of the number of excitations |(cid:104) ˆ a (cid:105)| = qN a p a θ / , these correspond to |(cid:104) ˆ a (cid:105)| = (12 . ± . · , |(cid:104) ˆ a (cid:105)| = (5 . ± . · , and |(cid:104) ˆ a (cid:105)| = (6 . ± . · .In all experiments, we measure the transverse spincomponent of the alkali atoms along the ˆ x axis using Faradayrotation of a linearly-polarized probe beam. The 5-mmdiameter, 260 µ W, probe beam is detuned by ∼ GHzabove the D1 transition and its polarization is measured afterthe cell using balanced photodetection method [42]. Wesubtract from all measurements a background signal takenwithout the magnetic pulse. This background signal is smalland is dominated by excitations of transverse spins duringthe fast variation of B ˆ z (when setting ∆ ), due to imperfectalignment between the optical and magnetic axes. Reconstruction and scaling of (cid:104) ˆ a (cid:105) and (cid:104) ˆ b (cid:105) We use optical Faraday rotation to measure (cid:104) ˆ a (cid:105) and (cid:104) ˆ b (cid:105) .For the optically-broadened line and the far-detuned probein our setup, and as long as the Faraday-rotation angle issmall, the balanced-detection readout is proportional to the ˆ x axis of the collective alkali spin (cid:104) (cid:80) m ˆs ( m ) x (cid:105) , i.e. , to theelectron spin projection along the probing axis [43]. Fromthese measurements we extract the normalized transversespin component ¯ S x ( t ) = (cid:104) (cid:80) m ˆs ( m ) x ( t ) (cid:105) / [ N a p a (0) / . Thenormalization factor is calibrated separately by tilting theinitial spin [ N a p a (0) / z all the way to the ˆ x direction(equivalent to θ a = 90 ◦ ) and measuring the maximal Faradayrotation angle ( ∼ |(cid:104) ˆ a ( t ) (cid:105)| and |(cid:104) ˆ b ( t ) (cid:105)| presented inFig. 2 are done according to the experimental sequence shownin Fig. 6a. The sequence starts by initializing the spins witha small transverse component under conditions of small ∆ .After some evolution and partial decay in the dark, at time t , we increase ∆ by an order of magnitude (by increasing B + B b → a to . mG), thus largely decoupling the alkali andnoble-gas spins. We continue to monitor the alkali spins anduse them as a magnetometer for sensing the noble-gas spins.During the experiment, when the pumping light is off, thepolarization of the alkali spin decays p a ( t ) ≤ p a (0) . Thisdecay, which is to leading order exponential with decay rate Γ , changes the slowing-down factor q ( t ) = q [ p a ( t )] and time [ms] -0.0100.01 s p i n m ea s u r e m en t f o r de c oup li ng a t t = m s measurement s x low pass filtered k x two-species precession fitexchangeduration t ab detuning excitationpumping FIG. 6.
Pulse sequence and typical results of an excitation-exchange measurement. a.
First, we turn off the pumping and bringthe two species to strong coupling with a small detuning ∆ . We thengenerate a transverse excitation with a pulse of transverse magneticfield. At a later time t , we halt the exchange by increasing the axialmagnetic field and setting a large ∆ . b. Example of a measuredsignal with exchange duration t = 11 ms, with ∆ = − . J before t , and ∆ = 790 Hz (cid:29) J after t . We measure the alkali electron spin(red) which, once ∆ is increased, can be used as a magnetometerthat senses the noble-gas spin. The fast oscillations of the signalcorrespond to the Larmor precession of the alkali spin, and the slowmodulation correspond to the noble-gas precession. The latter ishighlighted by the blue line (generated by low-pass filtering of thesignal for illustrative purposes). We fit the signal to the model fromEq. (3) (dashed black line) and find the amplitudes of the alkali andnoble-gas components at time t , which are used to estimate (cid:104) ˆ a ( t ) (cid:105) and (cid:104) ˆ b ( t ) (cid:105) , respectively. The same fit also provides p a ( t ) . thus shifts the Larmor precession frequency of the alkali spin.Denoting τ as the time elapsed from the decoupling time t , theinstantaneous precession frequency of the alkali spin is givenby ω a ( ω , p a ( t ) , Γ ; τ ) = 2 ω / [1 + p ( t ) e − τ ] , (2)where ω = ω a ( p a = 1; τ = 0) . To each measured signal ¯ S x ,we therefore fit the model ¯ S x ( t + τ ) = Re (cid:104) σ a ( t ) e i (cid:82) τ ω a ( ω ,p a ( t ) , Γ ; τ (cid:48) ) dτ (cid:48) − γ a τ + σ b ( t ) e ( iω b − γ b ) τ (cid:105) . (3)Here σ a ( t ) , σ b ( t ) are complex fitting parameters,corresponding to the amplitudes of the two frequencycomponents, and γ a , γ b , ω , ω b , p a ( t ) are real fittingparameters. One such fit is demonstrated in Fig. 6b, and theextracted p a ( t ) , ω a ( t, τ = 0) , | σ a ( t ) | , and | σ b ( t ) | are shownin Fig. 7 [note the factor | σ b /σ a | ≈ ( J/ ∆) < / ].The value Γ = 8 . Hz is set in Eq. (3) for all fits, and itis (self) consistent with Γ = 8 . ± . Hz extracted fromFig. 7a. Deviations from a pure exponential decay in Fig. 7bat t > ms can be attributed to multi-mode spatial dynamics[23], to SEOP of the alkali by the noble gas, and to lowsignal-to-noise ratios. FIG. 7.
Variables extracted from fitting Eq. (3) to the measuredsignals for each exchange time t , as exemplified in Fig. 6. a. The change in alkali precession frequency ω a ( t, τ = 0) [seeEq. (2)] manifests the change in the slowing-down factor due toalkali depolarization. b. The degree of alkali polarization p a ( t ) (in semi-log scale). In a and b , dashed black lines correspond toan exponential decay of p a ( t ) at the rate Γ = 8 . Hz. Lessreliable data, extracted when the excitations reside predominantlyin the noble-gas spins, are marked in gray. c. The two frequencycomponents (amplitude squared) of the normalized Faraday rotationsignal ¯ S x ( t + τ ) . Note the factor of (∆ /J ) (cid:38) between them.Each data-point is averaged over 12 to 20 repetitions of the sequence. With p a ( t ) at hand, we obtain the factor η ( t ) = N a q [ p a ( t )] p (0)4 p a ( t ) between the number of alkali excitations and ¯ S x . The alkali and noble-gas excitations presented in Fig. 2are then given by |(cid:104) ˆ a ( t ) (cid:105)| = η ( t ) (cid:12)(cid:12) σ a ( t ) + σ b ( t ) (cid:12)(cid:12) and |(cid:104) ˆ b ( t ) (cid:105)| = η ( t ) (cid:12)(cid:12) ∆( t ) J ( t ) σ b ( t ) − J ( t )∆( t ) σ a ( t ) (cid:12)(cid:12) , where ∆( t ) = ω a ( t, τ = 0) − ω b and J ( t ) = (cid:113) p a ( t ) p a (0) q a (0) q a ( t ) J ( t = 0) . Theseexpression neglect terms of order ( J/ ∆) and higher. Forthe experiments presented in Figs. 3 and 4, we reconstructthe complex-valued (cid:104) ˆ a ( t ) (cid:105) = (cid:112) η ( t )[ ¯ S x ( t ) − i ¯ S y ( t )] . Thetwo normalized projections ¯ S x ( t ) and ¯ S y ( t ) are measured intwo consecutive experiments that differ in the direction of theinitial pulsed excitation (alternating between B ⊥ ˆ y and B ⊥ ˆ x ).We use the p a ( t ) = p a (0) e − Γ t for Figs. 3a and 3b andestimate p a = p a ( t = 0) = 0 . for Fig. 3c. Finally, in Fig. 4we present the normalized Fourier amplitudes |(cid:104) ˆ a ( ω ) (cid:105)| = | (cid:82) ∞ (cid:104) ˆ a ( t ) (cid:105) dt | / (cid:113) T (cid:82) ∞ |(cid:104) ˆ a ( t ) (cid:105)| dt , where T = 65 ms is thesequence duration. Detailed model
Equation (1) describes idealized dynamics of the spin gases.For the calculations presented in Figs. 2 and 4b, we usea detailed model, which includes the decay of the alkalipolarization p a = p a ( t ) during the experimental sequence,the dependence of ∆ on p a ( t ) via the slowing-down factor q [ p a ( t )] , misalignment of the optical and magnetic axes, andresidual transverse magnetic fields.The model assumes both spin ensembles are polarizedalong − ˆ z . It follows Refs. [21, 36] and describes the dynamicsof the collective spin excitations S − = (cid:80) m (cid:104) ˆ s ( m ) − (cid:105) and K − = (cid:80) n (cid:104) ˆ k ( n ) − (cid:105) , coupled by the Fermi-contact interactionoccurring during stochastic collisions. In the presence ofaxial magnetic field B ˆ z and transverse magnetic field B − = B x − iB y , the coupled spin equations are given by ∂ t S − = i ( ω a + i Γ ) S − − i n a qn b J a K − + i g e q N a p a B − ,∂ t K − = − i qn a n b J b S − + iω b K − + ig b N b p b B − . (4)Here J a = √ q ˜ ζn b p a / and J b = ˜ ζn a p b / √ q are theuni-directional coupling rates, eventually composing the bi-directional rate J = √ J a J b , with ˜ ζ = (2 · −
14 cm / s ) / √ q .The gyromagentic ratios of the electron and helium-3 spinsare g e = 2 . · Hz/G and g b = − . · Hz/G, and theprecession frequencies are ω a = g e B/q + ˜ ζn b p b / √ q and ω b = g b B + √ q ˜ ζn a p a / .We simulate the experimental sequences bynumerically solving these equations. From thesimulation results, we calculate the expectation values (cid:104) ˆ a (cid:105) = (cid:112) q [ p a ( t )] /N a p a ( t ) S − ( t ) and (cid:104) ˆ b (cid:105) = (cid:112) /N b p b K − ( t ) .For the model parameters, we use known constants or themeasured values from the calibration experiments. For thealkali polarization, we use p a ( t ) = p a (0) e − Γ t with Γ = 8 . Hz and p a (0) = R p / ( R p + Γ ) = 0 . , where R p = 430 Hzis the optical pumping rate. For the noble-gas polarization,we set p b = n a k se p a (0) T b1 , act = 0 . due to SEOP, where k se = 5 . · − cm / s is the SEOP rate.The model can account for various geometricmisalignments and other experimental imperfections: (1)Misalignment of the pumping beam from the ˆ z axis generatesan initial transverse spin component. If the pumping beampoints towards η x e x + η y e y + e z (given η x,y (cid:28) ), the initialvalue of S − is N a p a ( t = 0)( η x − iη y ) / . (2) A residualmagnetic field pointing towards β x e x + β y e y + e z (given β x,y (cid:28) ) during the SEOP process would turn the initialvalue of K − to N b p b ( β x − iβ y ) / . (3) A non-vanishingtransverse magnetic field during the sequences can beaccounted for by a constant offset of B − . When varying ∆ during the sequence, these misalignments could tilt thespins and introduce spurious (background) excitations. (4)A misalignment of the probe field can be accounted for byextracting the signal S = Re(1 + iε (cid:107) ) S − + ε ⊥ N a p a / for ε (cid:107) , ⊥ (cid:28) (rather than simply S = S − ) from the simulationresults.We calculate the spectral map presented in Fig. 4b byrepeating the calculation for . < B < . (corresponding to − < ∆ /J < ). The excitation issimulated by applying B ⊥ ( t ) = 2 . × exp[ − t / (2 . µ s) ] ( θ a = 3 ◦ ). We calculate the Fourier transform of (cid:104) ˆ a (cid:105) and the normalized amplitude |(cid:104) ˆ a ( ω ) (cid:105)| , as done for theexperimental data. We reproduce the imperfection generatingthe perpendicular frequency branch by introducing a minutemisalignment β x = β y = 2 . mrad (and η x = η y = ε (cid:107) = ε ⊥ = 0 ). The inset of Fig. 4b is calculated with β x = β y = 0 . The calculations for Fig. 2a (solid lines)are done with B = 11 .
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