Quantum dynamics of collective spin states in a thermal gas

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Quantum dynamics of collective spin states in a thermal gas

Roy Shaham, Or Katz, and Ofer Firstenberg

Phys. Rev. A 102, 012822 – Published 30 July 2020

Abstract

Ensembles of alkali-metal or noble-gas atoms at room temperature and above are widely applied in quantum optics and metrology owing to their long-lived spins. Their collective spin states maintain nonclassical nonlocal correlations, despite the atomic thermal motion in the bulk and at the boundaries. Here we present a stochastic, fully quantum description of the effect of atomic diffusion in these systems. We employ the Bloch-Heisenberg-Langevin formalism to account for the quantum noise originating from diffusion and from various boundary conditions corresponding to typical wall coatings, thus modeling the dynamics of nonclassical spin states with spatial interatomic correlations. As examples, we apply the model to calculate spin noise spectroscopy, temporal relaxation of squeezed spin states, and the coherent coupling between two spin species in a hybrid system.

Received 7 June 2020
Accepted 7 July 2020

DOI:https://doi.org/10.1103/PhysRevA.102.012822

Physics Subject Headings (PhySH)

Atom or molecule scattering from surfaces Entanglement in quantum gases Lifetimes & widths Quantum description of light-matter interaction Scattering of atoms, molecules, clusters & ions

Atomic ensemble Atomic gases Atoms

Polarization

Stochastic differential equations

Atomic, Molecular & Optical

Authors & Affiliations

Roy Shaham^1,2,*, Or Katz^1,2, and Ofer Firstenberg ¹

¹Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel
²Rafael, Ltd., IL-31021 Haifa, Israel

^*roy.shaham@weizmann.ac.il

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Vol. 102, Iss. 1 — July 2020

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Figure 1
(a) Atomic spins in the gas phase, comprising a collective quantum spin $\hat{s} (r, t)$ and undergoing thermal motion. (b) In the diffusive regime, the spins spatially redistribute via frequent velocity-changing collisions. (c) Collisions (local interaction) with the walls of the gas cell may fully or partially depolarize the spin state. (d) Diffusion and wall collisions lead to a multimode evolution, here exemplified for a spin excitation $\hat{a} (r, t) \propto {\hat{s}}_{x} (r, t) - i {\hat{s}}_{y} (r, t)$ with an initial Gaussian-like spatial distribution $〈 \hat{a} (r, t) 〉$ and for destructive wall collisions. In addition to the mode-specific decay $Γ_{n}$ , each spatial mode accumulates mode-specific quantum noise ${\hat{W}}_{n} (t)$ .
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Figure 2
Effect of thermal motion in spin noise spectroscopy. (a) The spins are initially polarized along a magnetic field $B \hat{z}$ , and the spin projection $〈 {\hat{s}}_{x} 〉$ is measured by Faraday rotation using a nondiverging Gaussian probe beam. The calculation assumes a probe waist radius of $w_{0} = 1 mm$ and a cylindrical cell with radius $R = 1$ cm and length $L = 3$ cm. (b) Spin noise spectrum for an uncoated cell with a dense buffer gas ( $D = 1 {cm}^{2} / s, N \leq 1$ ), calculated using Eq. (11). Many spatial (diffusion) modes contribute to the noise, and thus the total signal is a weighted sum of varying Lorentzians, producing a cusp profile. (c) Spin noise spectrum for a cell coated with high-quality paraffin coating ( $D = 3 \times 10^{3} {cm}^{2} / s, N = 10^{6}$ ). The cusp is wider (since $D$ is larger), except for an additional sharp feature associated with the uniform spatial mode, the slow decay of which is governed by wall collisions. The dotted lines in (b) and (c) are a simple Lorentzian with width $Γ_{w} = π^{2} D / w_{0}^{2}$ , provided as reference for a single-mode approximation. (d) Noise content in the vicinity of the resonance (as defined in the main text) for the same two cells and varying probe waists. The narrower the probe beam, the larger its overlap with the high-order, rapidly decaying, diffusion modes, thus leading to weaker signal in the central resonance feature. This effect becomes more pronounced for lower buffer gas pressures.
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Figure 3
Lifetime of spin squeezing. (a) Experimental sequence comprising a short measurement (squeezing) pulse, followed by dephasing in the dark due to thermal motion for duration $t$ , and a verification pulse. The same probe beam is used for both pulses. In the calculations, the Gaussian distribution is initially squeezed by the measurement to spin variance of $〈 {\hat{x}}_{G}^{2} (0) 〉 = 0.05$ (7-dB squeezing). We take the same cell geometry as in Fig. 2 ( $R = 1$ cm, $L = 3$ cm). (b) Degree of spin squeezing vs time in the buffer gas cell, calculated using Eq. (12). Spin squeezing exhibits multiexponential decay associated with multiple diffusion modes. Larger probe beams lead to longer squeezing lifetimes, as the beam overlaps better with lower-order modes. (c) Spin squeezing in the coated cell. In a coated cell, the decay rate of the uniform diffusion mode, dominated by wall coupling, is substantially lower than that of higher-order modes. Therefore, ensuring a significant overlap of the probe beam with the uniform mode is even more important in coated cells for maximizing the squeezing lifetime. The dotted line in (b) and (c) is a single exponential decay $〈 {\hat{x}}^{2} (t) 〉 = 〈 {\hat{x}}^{2} (0) 〉 e^{- 2 Γ_{w} t} + (1 - e^{- 2 Γ_{w} t}) / 4$ , shown for reference with $Γ_{w} = π^{2} D / w_{0}^{2}$ and $w_{0} = 4$ mm; note the difference in timescales between (b) and (c). (d) An extremely squeezed state relies more on higher-order spatial modes and thus loses its squeezing degree rapidly (time is normalized by $T_{w} = R^{2} / π^{2} D$ ). The calculation is initialized with a uniform distribution of squeezing and includes the first 1000 radial modes, required for convergence.
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Figure 4
Excitation exchange between polarized alkali-metal and noble-gas spins. Shown is the exchange fidelity of the doubly excited (Fock) states ${| 2 〉}_{a} {| 0 〉}_{b}$ and ${| 0 〉}_{a} {| 2 〉}_{b}$ . We assume a spherical cell containing potassium and helium-3. The quality $N$ of the wall coating for the alkali metal is varied between no coating ( $N \leq 1$ ) and perfect coating ( $N \to \infty$ ). The noble-gas spins do not couple to the cell walls. The exchange fidelity approaches 1 when $J ≫ Γ_{wall}, Γ_{a}$ , as then the spin exchange is efficient for many diffusion modes; here $Γ_{wall}$ is the contribution of wall collisions to the relaxation rate of the alkali-metal spin (i.e., the typical diffusion rate to the walls), and $Γ_{a}$ is the contribution of atomic collisions. The calculations are performed for a cell radius $R = 5$ mm and with 1 atm of helium-3. The diffusion constants are $D_{a} = 0.35 {cm}^{2} / s$ for the potassium (mean free path $λ_{a} = 50$ nm) so that $Γ_{wall} = π^{2} D_{a} / R^{2} = 1 / (70 ms)$ , and $D_{b} = 0.7 {cm}^{2} / s$ for the helium (mean free path $λ_{b} = 20$ nm). The additional homogeneous decay of the alkali metal is $Γ_{a} \approx 6 s^{- 1}$ [7]. The wall coating plays a significant role, since for $N λ_{a} / R > 1$ (i.e., $N > 10^{5}$ ) the diffusion modes of the potassium and helium spins match.
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Figure 5
Graphical solutions for the Robin boundary condition. Here we solve the 1D equations for a system of length $L = 1 cm$ and mean free path of $λ = 0.5 μ m$ (characteristic of 100 Torr of buffer gas) and for different values of $N$ .
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