Universality of stochastic control of quantum chaos with measurement and feedback

Abstract

We investigate universal features of measurement-and-feedback control of quantum chaotic dynamics by examining the quantum Arnold cat map, a paradigmatic model of quantum chaos. Inspired by probabilistic control of classical chaos, our protocol stochastically alternates between intrinsic instability and engineered control operations that steer trajectories toward a target point. Simulation of exact quantum dynamics and a semiclassical truncated Wigner approximation reveal universal properties of the cat map's control transition. To further characterize this universality, we introduce the inverted harmonic oscillator as an analytically tractable effective model of instability. By integrating numerical simulations, a semiclassical Fokker-Planck description, and a direct spectral analysis of the stochastic quantum channel, we identify quantum signatures absent in classical limits. The close agreement between quantum simulation, truncated Wigner approximation, and inverted oscillator analysis shows that universal features of the transition are set by uncertainty-limited quantum fluctuations and are insensitive to genuine quantum interference.

Figures (3)

• Figure 1: (a) Order parameter $\bar{\rho}_{00}(p)$ from exact cat-map simulations ($L=\log_2N=4$--$10$, blue) and TWA ($L=10$--$90$, red); inset: TWA collapse near $p=p_c$ with $\nu=1$ and $\beta=0.97$. (b) $\bar{\rho}_{00}(L)$ at $p_c$ for exact (blue) and TWA (red); dashed line: $\bar{\rho}_{00}\propto L^{-\beta}$ with $\beta=0.97$. (c) TWA $\bar{\rho}_{00}$ at $L=90$ vs $p,\theta$, consistent with the phase boundary $p_c(\theta)$ (white dash); $\hbar=\frac{1}{2\pi N}=\frac{1}{2^{L+1}\pi}$.
• Figure 2: (a) Steady-state $\bar{\rho}_{00}(p)$ from the fixed point of $\mathbb{T}$ for $\kappa=\gamma=2\ln\!\left(\tfrac{1+\sqrt{5}}{2}\right)\approx0.42$, with cutoffs $2^{-L-1}<\sigma_\pm<2^{L-1}$ for $L=10,20,30,40,50$; dashed: late-time average over 5000 trajectories; inset: collapse near $p=p_c$ with $\nu=1$ and $\beta=0.96$. (b) $\bar{\rho}_{00}(L)$ at $p_c$ (same cutoffs); dashed: $\bar{\rho}_{00}\propto L^{-\beta}$ with $\beta=0.96$. (c) $\bar{\rho}_{00}(t)$ at $p_c$; dashed: $\bar{\rho}_{00}\propto t^{-1/z}$ with $z=2$.
• Figure 3: Comparison of the power-law form of the distribution $\mathcal{Q}_+(\sigma_+)$ acquired from the FP analysis with the distribution of $\sigma_+$ computed numerically from the fixed point of $\mathbb{T}$ with $\kappa = \gamma = 0.2$ and cutoff $L=50$ for $p=0.6$.