Faster State Preparation across Quantum Phase Transition Assisted by Reinforcement Learning

Shuai-Feng Guo, Feng Chen, Qi Liu, Ming Xue, Jun-Jie Chen, Jia-Hao Cao, Tian-Wei Mao, Meng Khoon Tey, and Li You

Phys. Rev. Lett. 126, 060401 – Published 9 February 2021

Abstract

An energy gap develops near quantum critical point of quantum phase transition in a finite many-body (MB) system, facilitating the ground state transformation by adiabatic parameter change. In real application scenarios, however, the efficacy for such a protocol is compromised by the need to balance finite system lifetime with adiabaticity, as exemplified in a recent experiment that prepares three-mode balanced Dicke state near deterministically [Y.-Q. Zou et al., Proc. Natl. Acad. Sci. U.S.A. 115, 6381 (2018)]. Instead of tracking the instantaneous ground state as unanimously required for most adiabatic crossing, this work reports a faster sweeping policy taking advantage of excited level dynamics. It is obtained based on deep reinforcement learning (DRL) from a multistep training scheme we develop. In the absence of loss, a fidelity $\geq 99 %$ between prepared and the target Dicke state is achieved over a small fraction of the adiabatically required time. When loss is included, training is carried out according to an operational benchmark, the interferometric sensitivity of the prepared state instead of fidelity, leading to better sensitivity in about half of the previously reported time. Implemented in a Bose-Einstein condensate of $\sim 10^{4}$ ${}^{87}{Rb}$ atoms, the balanced three-mode Dicke state exhibiting an improved number squeezing of $13.02 \pm 0.20 dB$ is observed within 766 ms, highlighting the potential of DRL for quantum dynamics control and quantum state preparation in interacting MB systems.

Received 19 September 2020
Accepted 12 January 2021

DOI:https://doi.org/10.1103/PhysRevLett.126.060401

Physics Subject Headings (PhySH)

Entanglement production Quantum control Quantum phase transitions Quantum state engineering Spinor Bose-Einstein condensates

General PhysicsAtomic, Molecular & Optical

Authors & Affiliations

Shuai-Feng Guo ^1,*, Feng Chen ^1,*, Qi Liu ¹, Ming Xue ¹, Jun-Jie Chen¹, Jia-Hao Cao¹, Tian-Wei Mao¹, Meng Khoon Tey^1,2,†, and Li You ^1,2,‡

¹State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, China
²Frontier Science Center for Quantum Information, Beijing, China

^*These authors contributed equally to this work.
^†mengkhoon_tey@tsinghua.edu.cn
^‡lyou@tsinghua.edu.cn

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Vol. 126, Iss. 6 — 12 February 2021

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Images

Figure 1
Dicke state preparation assisted by DRL. (a) The learning loop between the DRL agent and the simulated system. (b) Illustration of our multistep training scheme for successively increased system size from $N = 10$ to $N = 1000$ . Dependence of the mean values for the optimized policy $π^{*} (A | S)$ on mean-field state space ${ρ_{0}, 0, ρ_{0} (1 - ρ_{0}) / 2, θ_{s}}$ (mean-field policy) are displayed as density plots in (b1),(b2), and (b3), respectively, for the corresponding $N$ . Their close resemblance reflects nice generalization ability of the policy. Red dashed lines denote evolution trajectories starting from the polar state.
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Figure 2
Results from DRL policy without atom loss for $N = 2000$ . (a) The sweeping profile (in three line segments of black, blue, and red) and corresponding simulated fidelity (dot-dashed line referred to right axis); (b) Density plot of probability distribution $W_{n} = | ⟨ ψ_{n} (t) | ψ (t) ⟩ |^{2}$ for simulated state in the instantaneous eigenstate basis $| ψ_{n} (t) ⟩$ (vertical axis); Insets show enlargements of distributions on the boundaries $| c_{2} | t = 2.6$ and 10 between stages; (c1),(c2), and (c3), respectively, show evolution of state expanded in the Fock state basis $| k, N - 2 k, k ⟩$ over the three stages; (d1) their corresponding mean energies $⟨ ψ (t) | H (q) | ψ (t) ⟩ - E_{g} (q)$ shown by the same colored lines as in (a) with 1 standard deviation in shaded regions, and (d2) for an expanded view around the QCP. Arrows indicate direction of increasing time and gray dashed lines denote the CGC.
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Figure 3
Results (including loss) for $N ≃ 11800$ . (a) Comparison of $q (t)$ profile from DRL (red solid line) and the empirically optimized one from Ref. [54] (black dot-dashed line). The shaded region paints the BA phase. (b) The corresponding evolutions for fractional populations $ρ_{m_{F}} = N_{m_{F}} / N$ are compared in the same colors as in (a), with lines (gray shaded regions) denoting simulated mean values (uncertainties). Data points from ten experimental runs are marked by circles or triangles, respectively, for $m_{F} = 0$ , $\pm 1$ components. The results for $m_{F} = \pm 1$ overlay almost completely due to their correlated generation. (c1) The measured histograms of $L_{z}$ for the prepared Dicke state obtained from 1000 continuous experiment runs. The solid line denotes a Gaussian fit, whereas the dashed line is the theoretical envelope for a coherent state with same $N$ . (c2) The histograms of $L_{z} / N$ for the prepared state after a $π / 2$ rotation about the $x$ axis from 1057 continuous experiment runs. The solid line represents the theoretical results for an ideal balanced Dicke state. The inset spheres in (c) denote the generalized Bloch sphere representations for the corresponding states.
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