Nuclear Spin Squeezing in Helium-3 by Continuous Quantum Nondemolition Measurement

Nuclear spin squeezing in Helium-3 by continuous quantum nondemolition measurement

Alan Serafin,1 Matteo Fadel,2 Philipp Treutlein,2 and Alice Sinatra1 1Laboratoire Kastler Brossel, ENS-UniversitéPSL, CNRS, Universitéde la Sorbonne et Collègede France, 24 rue Lhomond, 75231 Paris, France 2Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland (Dated: December 15, 2020) We propose a technique to control the macroscopic collective nuclear spin of a Helium-3 vapor in the quantum regime using light. The scheme relies on metastability exchange collisions to mediate interactions between optically accessible metastable states and the ground-state nuclear spin, giving rise to an effective nuclear spin-light quantum nondemolition interaction of the Faraday form. Our technique enables measurement-based quantum control of nuclear spins, such as the preparation of spin-squeezed states. This, combined with the day-long coherence time of nuclear spin states in Helium-3, opens the possibility for a number of applications in quantum technology.

Introduction. The nuclear spin of Helium-3 atoms in x a room-temperature gas is a very well isolated quantum B y z ϕ system featuring record-long coherence times of up to λ several days [1]. It is nowadays used in a variety of appli- κ 2 cations, such as magnetometry [2], gyroscopes for navi- 1083 nm gation [3], as target in particle physics experiments [1], and even in medicine for magnetic resonance imaging of the human respiratory system [4]. Moreover, Helium-3 gas cells are used for precision measurements in fundamental physics, e.g. in the search for anomalous forces FIG. 1. Illustration of the proposed setup. A Helium-3 va- [5] or violations of fundamental symmetries in nature [6]. por cell is placed inside an asymmetric optical cavity, ensuring that photons leave the cavity at rate κ predominantly through While the exceptional isolation of Helium-3 nuclear the out-coupling mirror. A (switchable) discharge maintains a spins is key to achieving long coherence times, it ren- small fraction of the atoms in a metastable state. The atomic ders measurement and control difficult. Remarkably, no- metastable and nuclear spins are oriented in the x direction ble gas nuclear spins can be polarized by metastability- beforehand by optical pumping. The light polarization, ini- exchange or spin-exchange optical pumping, harnessing tially along x, is rotated by an angle ϕ due to the Faraday collisions between atoms in different states or of different effect, performing a quantum nondemolition measurement of species that transfer the optically induced electronic po- the nuclear spin fluctuations along the light propagation direction. This polarization rotation is continuously monitored larisation to the nuclei [1, 7]. However, the role of quan- via homodyne measurement. tum coherence, quantum noise and many-body quantum correlations in this process is only beginning to be stud- ied [8–10]. Optical quantum control of noble gas nuclear in experiments with alkali vapours [14, 15]. Since our spin ensembles is still in an early stage of development, scheme does not require other atomic species as media- and key concepts of quantum technology such as the gen- tor [10, 11] and the rate constants of metastability ex- eration of non-classical states for quantum metrology [12] change collisions are comparatively high [1], it can oper- or the storage of quantum states of light [13] have not yet ate at room temperature and millibar pressures as com- arXiv:2012.07216v1 [quant-ph] 14 Dec 2020 been demonstrated. monly used in experiments with Helium-3. Moreover, In this paper we propose a technique for the optical the interaction can be switched on and off, by switch- manipulation of Helium-3 nuclear spins in the quantum ing the week discharge that maintains a population in regime. As the nuclear spin state cannot be directly ma- the metastable state. Our scheme will allow to develop nipulated with light, our approach makes use of metasta- quantum-enhanced technologies with Helium-3, such as bility exchange collisions to map optically accessible elec- measurement devices with sensitivity beyond the stan- tronic states into the nuclear state, thereby mediating an dard quantum limit [12]. effective coupling between the light and the nuclear spin. Semiclassical three-mode model. We consider the setup In contrast to earlier ideas put forward by one of us [8, 9], in Fig. 1, where a gas cell containing Ncell Helium-3 atoms −6 the scheme considered here results in a Faraday interac- in the ground state and a small fraction ncell 10 Ncell tion [14] coupling the fluctuations of the light and of the in the metastable state is placed inside an∼ optical cav- nuclear spin. This interaction is nowadays routinely used ity. In the theoretical treatment we assume that the as a powerful and versatile spin-light quantum interface metastable atoms are homogeneously illuminated by the 2

excited state to transfer orientation between the metastable and the 3 ∆ nuclear spins and, as it was shown theoretically, they 2 P0 } can also transfer quantum correlations [8, 9]. Starting from metastabiliy exchange equations for the metastable and nuclear variables [16] plus the Faraday interaction C cavity 8 eld S (1) between K~ and S~, we write a set of nonlinear equa- metastable state tions for the mean values of the collective operators that describe the system dynamics in the semiclassical approx- F =1/2 K 3 imation, i.e. neglecting quantum fluctuations and corre- 2 S1 lations. For x-polarized nuclear and light spins F =3/2 Ncell N nph Ix s = and Sx s = , (2) metastability h i P 2 ≡ 2 h i 2 exchange collisions ground state where [0, 1] is the nuclear polarisation and nph the numberP of ∈ photons in the c cavity mode in steady state 1 x 1 S0 I without atoms, the nonlinear equations of motion admit a stationary solution. In particular, we find FIG. 2. Relevant level scheme of 3He for z quantization axis, 2 which corresponds to the cavity axis. The cavity mode (red) 1 ncell n Kx s = − P2 . (3) addresses the C8 transition between the F = 1/2 metastable h i P 3 + 2 ≡ 2 3 P manifold and the F = 1/2 excited state 2 P0, with detun- 3 ing ∆. The six metastable levels 2 S1 are coupled to the The nonlinear equations of motion can now be linearized 1 purely nuclear 1 S0 ground state by metastability exchange around this stationary solution by setting A = A s + collisions. δA, with A a collective operator and δA ah classicali h fluc-i tuation. By performing an adiabatic elimination of the cavity mode and the magnetic field is zero. Effects of a F = 3/2 metastable manifold, we obtain the reduced set small guiding field and the spatial profile of the cavity of coupled differential equations for the classical fluctua- mode will be discussed at the end of the paper. The tions of the transverse components of three spins relevant level scheme is illustrated in Fig. 2. We in- κ ~ ~ δS˙ z = δSz (4a) troduce the collective spin operators I and K for the − 2 (nuclear) ground state and for the F = 1/2 metastable κ δS˙ y = δSy + χ Sx δKz (4b) manifold, respectively. For the cavity light, propagat- − 2 h is ing in the z-direction and addressing the 23S 23P C 1 0 8 δI˙ z = γf δIz + γmδKz (4c) transition at 1083 nm, we introduce the Stokes− spin op- − δI˙y = γf δIy + γmδKy (4d) erators as a function of the x- and y-polarized modes − † † † † as Sx = (cxcx cycy)/2, Sy = (cxcy + cycx)/2 and δK˙ z = γmδKz + γf δIz (4e) † †− − Sz = (cxcy cycx)/(2i). For a large detuning ∆ and ˙ − 3 δKy = γmδKy + γf δIy + χ Kx s δSz . (4f) in the low-saturation limit, the excited state 2 P0 can be − h i adiabatically eliminated, resulting in the Faraday inter- Here, decay rate and the effective metastability exchange action Hamiltonian [14] rates for the ground state and metastable atoms are 4+P2 1−P2 1 4+P2 1 γ = 2 2 and γ = 2 , respec- H = ~χKzSz (1) f 8−P 3+P T m 8−P τ tively. Note that γm/γf = N/n 1. 2 with coupling strength χ = gc /∆. Here, gc = d8 c/~ and We proceed now with a full quantum treatment of the ω E c = ~ , where Vc is the cavity mode volume, ω the reduced system of three collective spins. 20Vc E Quantum three-mode model. Since S~, K~ and I~ are x- angularq frequency and d8 the dipole matrix element of the chosen transition. polarized and will maintain a large polarization through- The coupling between K~ and I~ is provided by metasta- out the entire protocol, we can perform the Holstein- Primakoff approximation by replacing Iy/√N Xa, bility exchange collisions, occurring at rate 1/τ for a ' metastable atom, and 1/T for a ground state atom, with Iz/√N Pa, Ky/√n Xb, Kz/√n Pb, Sy/√nph ' ' ' ' T/τ = Ncell/ncell [16]. Metastability exchange colli- Xc, and Sz/√nph Pc where we have introduced the ' † † sions can be thought of as an instantaneous exchange bosonic quadratures Xν = (ν +ν )/2, Pν = (ν ν )/(2i), − of the electronic excitation between a ground state and [Xν ,Pν ] = i/2 for ν = a, b, c, that describe the transverse a metastable atom that leaves nuclear and electronic fluctuations of the collective spins. Note that within the spins individually unchanged. They are routinely used Primakoff approximation the mode c cy is associated ' 3 to the y-polarized photons inside the cavity. The Faraday It appears from the adiabatic elimination that Cd is re- Hamiltonian (1) becomes lated to “double jumps” where a photon and a metastable excitation are annihilated at the same time. This process H = ΩP P , (5) ~ b c does not affect the nuclear state vector and it does not with Ω = χ√nnph. In a fully quantum treatment [8], play any role in the homodyne-measurement squeezing one adds appropriate Langevin forces representing quan- scheme we consider [28]. On the contrary we will see tum noise to the semiclassical equations (4). To this ap- that Cs, related to single cavity jumps, is responsible proach however, we prefer here an equivalent formulation for the generation of nuclear spin squeezing at rate Γsq. in terms of a quantum master equation (QME) for the Eqs. (10,11) are one of the main results of our work. density operator ρ describing the three bosonic modes a The factor γf /γm = n/N in Eq. (11), absent in the (nuclear), b (metastable) and c (cavity), squeezing rates obtained for alkali atoms using Faraday interactions, reflects the fact that we optically address n 1 † 1 † ρ˙ = [H, ρ] + CwρC C Cw, ρ . (6) i w − 2{ w } metastable atoms to manipulate N nuclear spins. ~ w=c,m X Quantum non-demolition measurement of the nuclear Besides the interaction Hamiltonian Eq. (5), it includes spin. We now study the evolution of the system in a jump operators for the cavity losses Cc = √κc and single experimental realisation, conditioned on the re- for metastability exchange collisions Cm = √2γmb + sult of a continuous homodyne measurement performed − 2γf a. Initially, the three modes are in the vacuum on the small y-polarized field leaking out of the cavity, state. Due to the Faraday effect caused by quantum the local oscillator phase being chosen to measure Xc pfluctuations of the spin, the polarization of the light is [29]. This is described at the level of the QME by ap- slightly turned and, after a transient time of order 1/κ, propriate jump operators. A density matrix conditioned the number of y-polarized photons in the cavity reaches on the measurement can be reconstructed in the Monte the steady state Carlo wavefunction method by averaging over stochastic realizations with different histories for metastability Ω 2 2γ c†c (t) 1 m . (7) exchange collisions but same history for the homodyne → 2κ − κ + 2(γ + γ ) m f detection. In the limit of a local oscillator with large am- The metastability exchange collisions lead to a hy- plitude, the evolution of the Monte Carlo wavefunction bridization of the nuclear spin and metastable modes. can be approximated by a nonlinear continuous stochas- Their contribution to the three-mode QME is diago- tic evolution [18, 20]. We apply this approach to both nalised introducing the rotated basis the one-mode model and the three-mode model. In the case of the one-mode model Eq. (10), the corre- γm γf α = a + b , (8) sponding stochastic evolution reads γm + γf γm + γf r r γm γf dt 2 β = b a . (9) d φ(t) = ΓsqQ φ(t) + ΓsqdζsQ φ(t) , (12) γm + γf − γm + γf | i − 2 | i | i r r p In practice, as γ γ , α a and β b. In the rotated where Q Pα φ Pα φ and dζs is a real Gaussian ran- m f ≡ − h | | i basis, the system can be reduced≈ to a≈ one-mode model. dom noise of zero mean and variance dt. The stochas- Reduction to a one-mode model. We consider the tic equation (12) describes the evolution of the quantum regime κ γm γf , all being larger than the timescale state of the nuclear spin in a single realization of the ex- of the nuclear spin evolution. During the evolution, the periment. The deterministic term proportional to Γsqdt number of excitations in the “hybridized nuclear” mode α and the random noise proportional to Γsqdζs are is- grows linearly in time, while the “hybridyzed metastable” sued from the jump operator Cs in the original one mode p mode β as well as the cavity mode c will rapidly tend to QME (10) and are physically associated to the measure- a stationary value, allowing their adiabatic elimination. ment process on the nuclear spin [21–23]. For our initial Following a similar procedure as in Ref. [17] within the conditions, the time evolution described by Eq. (12) can Monte-Carlo wavefunction description, we obtain to lead- be solved analytically. For a single realization φ(t) of the ing order in the coupling Ω a one-mode QME describing stochastic evolution, corresponding to a particular his- the slow evolution of the hybridized nuclear mode α tory of homodyne detection, we find that for long times the average Pα φ φ Pα φ stabilizes to a (random) † 1 † h i ≡ h | | i ρ˙α = CwρC [C Cw, ρ] . (10) constant value, and the variance Var (P ) tends to zero w − 2 w φ α w s,d −1 X= as (Γsqt) . Going back to the original three-mode basis, the single realisation variance of the nuclear spin quadra- This QME involves two jump operators, C = Ω2/4κ d I ture P corresponding to I reads with the identity, and C = Γ P with a z I s sq α p γf 2 1 + Γsqt Ω pγf 1 γm Γsq = . (11) Varφ(Pa)(t) = , (13) κ γm 4 1 + Γsqt 4

0.03 (a) (b) 1.0 1.0 (c) (d) 0.02

0.01 0.8 0.8 )/(1/4)

0.00 t

)( 0.6 0.6 a P (

−0.01 ϕ Time Average Var Homodyne Signal 0.4 0.4 −0.02

−0.03 0.2 0.2 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 Γsqt Γsqt Γsqt Γsqt

q FIG. 3. (a) Time evolution of the homodyne signal c + c† (blue) and of the nuclear spin quadrature 2 Γsq hP i (orange) φ κ a φ in a single realization of the experiment where a continuous homodyne measurement of the y-polarized field leaking out of the cavity is performed. (b) Time average of the same quantities. The curves are obtained from the continuous stochastic equation derived from the three-mode QME (6), for a single realization of the stochastic noise describing homodyne detection (the equivalent of dζs of the one-mode model) and averaged over 5 realization of the stochastic noise describing metastability exchange. Parameters: Ω/κ = 1/10, γm/κ = 1/10, γf /κ = 1/100, Γsq/κ = 1/1000. (c) Conditional variance of the nuclear spin quadrature Pa as a function of time. Black: three-mode model with same parameters as (a), Green: analytical prediction (13) of the one-mode model. (d) Effect of decoherence. Black: three-mode model with an additional relaxation rate γ0/κ = 1/1000 in the metastable state, where we now average over 8 realizations of the stochastic noises related to metastability exchange and wall relaxation in the metastable state. Green : one-mode model with the corresponding effective relaxation in the ground 0 state γ0 = Γsq/10. Dashed horizontal line: analytical prediction (16). and the time average of the homodyne signal is propor- this effective relaxation in the one-mode model (10), we tional to the fixed (random) value of Pa of that reali- calculated the squeezing limit in a single realisation in h iφ sation the presence of metastable decoherence for γm γf and 0 Γsq γ0, † t→∞ Γsq c + c φ 2 Pa φ . (14) h i −→ r κ h i 0 t→∞ 1 γ0 t→∞ 1 Γsq Varφ(Pa) and Varφ(Xa) . Note that Varφ(Pa)(t) tends to γf /(4γm) in the t −→ 4 Γ −→ 4 γ0 → ∞ s sq s 0 limit, which is the theoretical spin squeezing limit in- (16) trinsic to this method that uses the metastable state to This kind of scaling, already found for alkali atoms [24], is mediate the interaction. In Fig. 3b-c we compare the further confirmed by our numerical simulations where we analytical predictions (14) and (13) with the numerical introduce an additional jump operator √γ0b in the three- solution of the three-mode model. mode QME (6), see Fig. 3d. An extended theoretical We note that the limit γf /γm 0 of equation (13) → treatment will be published in a separate paper [25]. coincides with the result that one would obtain from a Experimental proposal. We consider a cylindrical va- nuclear spin-light interaction of the quantum nondemo- por cell 20 mm long and 5 mm in diameter, filled with lition or Faraday form N = 2.5 1016 3He atoms at a pressure of p = 2 Torr. cell × n n For a polarization of = 0.4 this gives an effective P 16 Heff = ~Ω PaPc or Heff = ~χ IzSz . (15) number of ground state atoms N = 1.0 10 . We N N n × r take cell = 5 10−6, giving an effective number of Ncell × Effect of decoherence. Due to the long coherence time metastable atoms n = 1.3 1010. From the metasta- of the nuclear spin, we can ignore its decoherence on bility exchange rate coefficient× [1], we determine effec- 6 −1 the time scale of squeezing generation. On the other tive metastability exchange rates γm = 5.2 10 s −1 × hand, decoherence in the metastable state, including and γf = 7.0 s . The cell is placed inside an optical spontanous emission and collisions with the cell walls, cavity to enhance the atom-light interaction [15]. For will affect the performance of the squeezing protocol. a finesse of = 50 and a cavity length of 3 cm, we From analytical calculations we can show that a relax- obtain κ = 2Fπ 1.0 108 Hz. The cavity is laser driven × ation with rate γ0 in the metastable state appears in the on the x-polarization mode so that 5 mW of light exit ground state as an effective relaxation with reduced rate the cavity in this polarization, and we take the light γ0 = γ γf . We thus expect the effect of metastable re- 0 0 γm to be detuned by ∆ = 2π 2.0 GHz from the C8 transi- laxation to become negligible for Γ γ0 . By inserting tion. This results in Ω = 2π 4.1 106 Hz. In steady sq 0 × 5 state, 6.5 105 s−1 y-polarized photons leave the cavity, Dependent Forces Set with a K-3He Comagnetometer, Eq. (7).× The nuclear spin squeezing rate is evaluated Phys. Rev. Lett. 103, 261801 (2009). −1 [6] W. Heil, C. Gemmel, S. Karpuk, Y. Sobolev, K. Tullney, from Eq. (11) to Γsq = 1.4 s . We have assumed that atomic motion averages over spatial inhomogeneities of F. Allmendinger, U. Schmidt, M. Burghoff, W. Kilian, S. Knappe-Grüneberg, A. Schnabel, F. Seifert, and L. the cavity mode, effectively coupling the light homoge- Trahms, Spin clocks: Probing fundamental symmetries neously to all atoms in the cell [30]. From the diffu- in nature, Annalen der Physik 525, 539 (2013). sion coefficient of metastable atoms [26], we estimate the [7] M. Batz, P.-J. Nacher, and G. Tastevin, Fundamentals metastable relaxation rate due to wall collisions to be of metastability exchange optical pumping in helium, J. wall 4 −1 Phys. Conf. Ser 294, 012002 (2011). γ0 = 2.6 10 s [27]. The off-resonant photon scattering rate in× the metastable state, averaged over the cell, [8] A. Dantan, G. Reinaudi, A. Sinatra, F. Laloë,E. Gi- is γscat 2.4 103 s−1 γwall. According to (16), the acobino, and M. Pinard, Long-lived quantum memory 0 0 with nuclear atomic spins Phys. Rev. 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Castin, M. Fadel, P. Treutlein, A. Sinatra, 6

in preparation. the spatial variations of the cavity mode, ensuring the [26] W. A. Fitzsimmons,N. F. Lane, and G. K. Walters, Diffu- validity of a description in terms of collective interactions. 3 3 1 sion of He(2 S1) in Helium Gas; 2 S1 − 1 S0 Interaction [31] We consider that the effect of a magnetic field B over Potentials at Long Range, Phys. Rev. 174 193 (1968). a time t is negligible if the precession of the noise el- [27] W. Franzen, Spin Relaxation of Optically Aligned Ru- lipse of a 10 dB squeezed state degrades the squeezed bidium Vapor, Phys. Rev. 115, 850 (1959) variance by less than 10% (this corresponds to an an- [28] This is because the produced photon is in this case in- gle of 1.8 degrees). Given that the Larmor frequency in coherent with the pump and does not contibute to the the ground state is 3.24 kHz/G, we obtain the condition homodyne signal. It would on the contrary play a role in B[G] × t[s] ≤ 1.5 × 10−6. Although the Larmor frequency a scheme based on photon counting as in [19]. in the metastable state is much larger, 1.87 MHz/G, the [29] Being the conjugate quadrature to Pc, Xc carries the precession in this state is negligible for magnetic fields information about Pα (see Eqs. (5) and (8)-(9)). up to ∼ 10 mG since the rotation in the zy plane occurs [30] The squeezing time scale 1/Γsq is long compared to the only during the short time 1/γm between two metasta- wall time scale 1/γ0 for atomic motion between cell walls, bility exchange collisions, corresponding to an angle of wall 4 γ0 /Γsq ∼ 10 . The atomic motion thus averages over order 1 degree.