Facilitating spin squeezing generated by collective dynamics with single-particle decoherence

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Facilitating spin squeezing generated by collective dynamics with single-particle decoherence

K. Tucker, D. Barberena, R. J. Lewis-Swan, J. K. Thompson, J. G. Restrepo, and A. M. Rey

Phys. Rev. A 102, 051701(R) – Published 3 November 2020

Abstract

We study the generation of spin squeezing in arrays of long-lived dipoles subject to collective emission, coherent drive, elastic interactions, and single-particle relaxation. It is found that not only does single-particle relaxation not necessarily degrade the squeezing generated in the collective dynamics, but the interplay of single-particle and collective effects can in fact facilitate the generation of squeezing in a specific parameter regime. This latter behavior is connected to the dynamical self-tuning of the system through a dissipative phase transition that is present in the collective system alone. Our findings will be applicable to next-generation quantum sensors with an eye towards atomic clocks in cavity-QED setups and trapped ion systems.

Received 23 December 2019
Accepted 7 October 2020

DOI:https://doi.org/10.1103/PhysRevA.102.051701

Physics Subject Headings (PhySH)

Squeezing of quantum noise

Master equation Monte Carlo methods

Statistical Physics & ThermodynamicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

K. Tucker^1,2, D. Barberena^1,3, R. J. Lewis-Swan^1,3, J. K. Thompson¹, J. G. Restrepo², and A. M. Rey ^1,3

¹JILA, NIST, Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
²Department of Applied Mathematics, University of Colorado, Boulder, Colorado 80309, USA
³Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado 80309, USA

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Vol. 102, Iss. 5 — November 2020

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Images

Figure 1
(a) Steady-state phase diagram of an ensemble of $N$ spin- $1 / 2$ particles subjected to a coherent drive with Rabi frequency $Ω = N Υ / 2$ , collective emission at rate $Γ$ , collective spin-exchange interactions $χ$ , and single-particle relaxation at rate $γ_{s}$ . This system can be engineered using an optical cavity (b) or trapped ion arrays. The spin- $1 / 2$ is encoded in a pair of electronic states, while the collective dissipation and global spin-spin interactions are mediated by spin- $1 / 2$ 's exchanging virtual bosons through a common mode. In the absence of single-particle relaxation, the system undergoes a nonequilibrium phase transition (superradiant to normal) signaled by a change in the total steady-state atomic inversion, which serves as an order parameter. Approaching the transition point from the superradiant phase [points (i) and (ii)], the coherent drive (in the $\hat{x}$ direction) and collective emission combine to generate spin squeezing along $\hat{x}$ , as shown in (c) and (d). In the normal phase no squeezing is observed [point (iii)]. For all three graphs in (c), $N = 2000$ and all spins are initially polarized along $- \hat{x}$ . (d) explicitly displays the Bloch sphere overlaid with a squeezed collective spin distribution of the steady state (pink). Note that this is for illustrative purposes, and that the actual position and orientation of the squeezing can vary with parameters. Introducing finite $γ_{s}$ allows the system to dynamically traverse the phase diagram [red arrow in (a)] and enhances the achievable spin squeezing in the striped region of (a).
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Figure 2
(a) Squeezing vs time for $N = 2000$ , $χ / Γ = 0$ , $Υ / Υ_{c} = 0.9$ , and a range of $γ_{s} / Γ$ . Solid lines indicate squeezing $ξ^{2} (t)$ and dashed lines the corresponding time-dependent effective system size $N^{eff} (t)$ (matching colors). The horizontal black line corresponds to the critical effective particle number $N_{c} = 2 Ω / Υ_{c}$ for which the transition between superradiant and normal phases occurs. (b) Squeezing $ξ^{2} (t)$ (solid) and effective system size $N^{eff} (t)$ (dashed) computed from two individual trajectories of the numerical method with $γ_{s} / Γ = 4$ . For each trajectory, $N^{eff} (t)$ crosses the horizontal line for $N_{c}$ near the point where its corresponding $ξ^{2} (t)$ reaches a minimum. In each panel, all spins are initially polarized along $- \hat{x}$ .
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Figure 3
(a) Minimum transient spin squeezing (see text for clarification) as a function of normalized drive amplitude $Υ / Υ_{c}$ and relaxation rate $γ_{s} / Γ$ with $χ / Γ = 0$ . The strip below the main panel shows a magnified view of the $γ_{s} / Γ = 0$ result for comparison. Note the break in the vertical axis, which is required since attainable simulation times cannot capture the squeezing behavior occurring on timescales of $1 / γ_{s}$ when this value is very large. (b) Minimum transient spin squeezing as a function of the interaction strength $χ / Γ$ with fixed $Υ / Υ_{c} = 0.9$ and $γ_{s} / Γ = 50$ . Inset: Squeezing vs time for a selection of values of $χ / Γ$ and the same $Υ, γ_{s}$ as the main panel. For (a) and (b) we compute the dynamics using a truncated cumulant expansion [47] and $N = 10^{4}$ . Initial conditions in (a) are the coherent spin state (CSS) in the $- \hat{x}$ direction, and in (b) are taken to be the CSS in the direction of the mean-field steady state (for each $χ / Γ$ and $Υ$ ) when $γ_{s} / Γ = 0$ to account for the rotations that result from different values of $χ / Γ$ .
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