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arXiv:2105.14861 · Physical Review Research

Quantization of out-of-time-ordered correlators in non-Hermitian chaotic systems

Wen-Lei Zhao

Published Apr 1, 2022• 20 citations
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Abstract

This paper reports the findings of the late time behavior of the out-of-time-ordered correlators (OTOCs) via quantum kicked rotor and kicked top models with $\mathcal{PT}$-symmetric driving potential. An analytical expression of the OTOCs' power-law growth with time is yielded as $C(t)=G(K){t}^{\ensuremath{\gamma}}$, with $\ensuremath{\gamma}$ being system dependent. Interestingly, the growth rate $G$ features a quantized response to the increase of the kick strength $K$. The physics behind this is the quantized absorption of energy from the non-Hermitian driving potential. This discovery and the ensuing establishment of the quantization mechanism in the dynamics of quantum chaos with non-Hermiticity will provide insights in chaotic dynamics, promising unprecedented observations in updated experiments.

DOI
10.1103/physrevresearch.4.023004

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This paper reports the findings of the late time behavior of the out-of-time-ordered correlators (OTOCs) via quantum kicked rotor and kicked top models with PT\mathcal{PT}PT-symmetric driving potential. An analytical expression of the OTOCs' power-law growth with time is yielded as C(t)=G(K)t\ensuremathγC(t)=G(K){t}^{\ensuremath{\gamma}}C(t)=G(K)t\ensuremathγ, with \ensuremathγ\ensuremath{\gamma}\ensuremathγ being system dependent. Interestingly, the growth rate GGG features a quantized response to the increase of the kick strength KKK. The physics behind this is the quantized absorption of energy from the non-Hermitian driving potential. This discovery and the ensuing establishment of the quantization mechanism in the dynamics of quantum chaos with non-Hermiticity will provide insights in chaotic dynamics, promising unprecedented observations in updated experiments.
We investigate the dynamics of the out-of-time-ordered correlators (OTOCs) via a non-Hermitian extension of the quantum kicked rotor model, where the kicking potential satisfies PT\mathcal{PT}PT-symmetry. The spontaneous PT\cal{PT}PT-symmetry breaking emerges when the strength of the imaginary part of the kicking potential exceeds a threshold value. We find, both analytically and numerically, that in the broken phase of PT\cal{PT}PT symmetry, the OTOCs rapidly saturate with time evolution. Interestingly, the late-time saturation value scales as a pow-law in the system size. The mechanism of such scaling law results from the interplay between the effects of nonlocal operator in OTOCs and the time reversal induced by non-Hermitian driven potential.
We investigate the dynamics of the out-of-time-order correlators (OTOCs) via a non-Hermitian extension of the quantum kicked rotor model, where the kicking potential satisfies PT -symmetry. The spontaneous PT -symmetry breaking emerges when the strength of the imaginary part of the kicking potential exceeds a threshold value. We find, both analytically and numerically, that in the broken phase of PT symmetry, the OTOCs rapidly saturate with time evolution. Interestingly, the late-time saturation value scales as a pow-law in the system size. The mechanism of such scaling law results from the interplay between the effects of the nonlocal operator in OTOCs and the time reversal induced by non-Hermitian-driven potential.
We investigate the quantum entanglement in a non-Hermitian kicking system. In the Hermitian case, the out-of-time ordered correlators (OTOCs) exhibit the unbounded power-law increase with time. Correspondingly, the linear entropy, which is a common measurement of entanglement, rapidly increases from zero to almost unity, indicating the formation of quantum entanglement. For strong enough non-Hermitian driving, both the OTOCs and linear entropy rapidly saturate as time evolves. Interestingly, with the increase of non-Hermitian kicking strength, the long-time averaged value of both OTOCs and linear entropy has the same transition point where they exhibit the sharp decrease from a plateau, demonstrating the disentanglment. We reveal the mechanism of disentanglement with the extension of Floquet theory to non-Hermitian systems.
We investigate both numerically and analytically the dynamics of out-of-time-order correlators (OTOCs) in a non-Hermitian kicked rotor model, addressing the scaling laws of the time dependence of OTOCs at the transition to the spontaneous PT\mathcal{PT}PT-symmetry breaking. In the unbroken phase of PT\mathcal{PT}PT symmetry, the OTOCs increase monotonically and eventually saturate with time, demonstrating the freezing of information scrambling. Just beyond the phase transition points, the OTOCs increase in the power laws of time, with the exponent being larger than 2. Interestingly, the quadratic growth of OTOCs with time emerges when the system is far beyond the phase transition points. The above numerical findings are validated by our theoretical analysis, which provides a general framework with important implications for Floquet engineering and information scrambling in chaotic systems.
It was proposed recently that the out-of-time-ordered four-point correlator (OTOC) may serve as a useful characteristic of quantum-chaotic behavior, because in the semi-classical limit, ℏ→0\hbar \to 0ℏ→0, its rate of exponential growth resembles the classical Lyapunov exponent. Here, we calculate the four-point correlator, C(t)C(t)C(t), for the classical and quantum kicked rotor -- a textbook driven chaotic system -- and compare its growth rate at initial times with the standard definition of the classical Lyapunov exponent. Using both quantum and classical arguments, we show that the OTOC's growth rate and the Lyapunov exponent are in general distinct quantities, corresponding to the logarithm of phase-space averaged divergence rate of classical trajectories and to the phase-space average of the logarithm, respectively. The difference appears to be more pronounced in the regime of low kicking strength, where no classical chaos exists globally. In this case, the Lyapunov exponent vanishes, while the OTOC's growth rate may remain finite corresponding to disconnected local chaotic islands in the phase space. We also show that the quantum correlator as a function of time exhibits a clear singularity at the Ehrenfest time tEt_EtE​: transitioning from a time-independent value of t−1ln⁡C(t)t^{-1} \ln{C(t)}t−1lnC(t) at ttEt t_EttE​. We note that the underlying physics here is the same as in the theory of weak (dynamical) localization [Aleiner and Larkin, Phys. Rev. B 54, 14423 (1996); Tian, Kamenev, and Larkin, Phys. Rev. Lett. 93, 124101 (2004)] and is due to a delay in the onset of quantum interference effects, which occur sharply at a time of the order of the Ehrenfest time.
We investigate both theoretically and numerically the dynamics of out-of-time-ordered correlators (OTOCs) in quantum resonance conditions for a kicked rotor model. We employ various operators to construct OTOCs in order to thoroughly quantify their commutation relation at different times, therefore unveiling the process of quantum scrambling. With the help of quantum resonance condition, we have deduced the exact expressions of quantum states during both forward evolution and time reversal, which enables us to establish the laws governing OTOCs' time dependence. We find interestingly that the OTOCs of different types increase in a quadratic function of time, breaking the freezing of quantum scrambling induced by the dynamical localization under non-resonance condition. The underlying mechanism is discovered, and the possible applications in quantum entanglement are discussed.