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-Universit´ePSL, CNRS, Universit´ede la Sorbonne et Coll`egede 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 funda- mental 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 ' 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 di- rection. 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 2 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 − y y y y as Sx = (cxcx cycy)=2, Sy = (cxcy + cycx)=2 and δK_ z = γmδKz + γf δIz (4e) y y− − 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 ! E c = ~ , where Vc is the cavity mode volume, ! 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=pN Xa, bility exchange collisions, occurring at rate 1/τ for a ' metastable atom, and 1=T for a ground state atom, with Iz=pN Pa, Ky=pn Xb, Kz=pn Pb, Sy=pnph ' ' ' ' T/τ = Ncell=ncell [16]. Metastability exchange colli- Xc, and Sz=pnph Pc where we have introduced the ' y y 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.