Inducing, and enhancing, many-body quantum chaos by continuous monitoring

TL;DR

The paper demonstrates that continuous monitoring combined with a large thermal bath does not universally suppress quantum chaos; instead, in a quenched SYK setup, the steady state is non-thermal and reached via two-stage exponential decay of the retarded Green function. The Lyapunov exponent exhibits rich behavior, including enhancement and re-entrant chaos driven by the monitoring strength κ and bath coupling V, revealing a tunable interplay between decoherence and scrambling. By solving two-time Kadanoff-Baym equations and analyzing unregularized OTOCs, the authors show that monitoring can induce chaotic dynamics even when the bath would suppress it, offering a pathway to control quantum scrambling in many-body systems. These insights have potential implications for designing quantum devices where scrambling properties can be dialed through measurement and environment engineering.

Abstract

It is intuitively expected, and supported by earlier studies, that many-body quantum chaos is suppressed, or even destroyed, by dissipative effects induced by continuous monitoring. We show here that this is not always the case. For this purpose, we study the quenched dynamics of a continuously monitored Sachdev-Ye-Kitaev (SYK) model, described by the Lindblad formalism, coupled to a thermal environment modeled by another SYK maintained at constant temperature. We find that the combined effect of monitoring and the thermal bath drives the system toward a non-thermal steady state independently of the initial conditions. The corresponding retarded Green's function exhibits two stages of exponential decay, with rates that depend non-monotonously on the thermal bath coupling and the monitoring strength. In the limit of weak coupling, the late time decay of the Green's function, computed analytically, is closely related to that of the thermal bath. Strikingly, we identify a range of parameters in which continuous monitoring, despite being a source of decoherence, induces or enhances quantum chaotic dynamics suppressed by the thermal bath. For instance, in the limit of weak coupling to the thermal bath, the Lyapunov exponent increases sharply when monitoring is turned on. For intermediate values of the thermal bath coupling, the Lyapunov exponent exhibits re-entrant behavior: it vanishes at zero or sufficiently weak monitoring strength, and becomes positive again as the monitoring strength is increased. Our results offer intriguing insights on the mechanisms leading to quantum scrambling which paves the way to its experimental control and consequently to a performance enhancement of quantum information devices.

Figures (19)

• Figure 1: Top: Illustration of the model Eq. (\ref{['eq:Lindblad_equation']}) that consists of a Hermitian SYK ($q=4$) with $N$ Majorana fermions coupled to a thermal bath, another SYK ($q_{{\rm{B}}} =4$) with $N_{{\rm{B}}} \gg N$ Majoranas at inverse temperature $\beta_{{\rm{B}}}$ with coupling $V$, $(f_{{\rm{S}}}, f_{{\rm{B}}}) = (1, 3)$ to the Hermitian SYK. The Hermitian SYK is also subjected to continuous measurements described by the jump operators $L_j = \sqrt{\kappa} \psi_j$ ($j=1, \dots, N$). Bottom: Illustration of the Lyapunov exponent $\lambda_{\mathrm{L}}$, that characterizes quantum chaotic dynamics, in the parameter space of the coupling strengths, $\kappa$ (monitoring) and $V$ (thermal bath with bath inverse temperature $\beta_{\rm{B}}=1000$). The values $V_0 \approx 0.94$ and $\kappa_0 \approx 0.33$ indicate the critical values at which $\lambda_{\rm{L}}$ vanishes when the Markovian or thermal bath is absent, respectively. The horizontal axis is shown on a square scale ($x \mapsto x^2$), while the $y$-axis is shown on a square root scale ($y \mapsto \sqrt{y}$). The "Inducing" (dark red and dark blue), "Enhancing" (light blue), and "Weakening" (light red) regions indicate positive Lyapunov exponents $\lambda_{\rm L} > 0$ such that the system is quantum chaotic. By "Inducing" we mean that the Lindblad dynamics induces quantum chaos for $V \geq V_0$. By "Enhancing" ("Weakening") we mean that when $V$ is fixed, the increase of monitoring rate $\kappa$ increases (decreases) the Lyapunov exponent, and hence enhances (weakens) quantum chaos. The "Inducing" region is divided into two parts: for a fixed $V$ with $V \geq V_0$, $\lambda_{\rm L}$ increases as $\kappa$ increases in the dark red region until it reaches a maximum, then it decreases, until it vanishes, in the dark blue region. The "Non-quantum chaotic" (light gray) region corresponds to $\lambda_{\rm L} < 0$.
• Figure 2: The effective temperature $\beta_{\rm eff}$ Eq. (\ref{['eq:betaeff']}) and the energy density $E/N$ Eq. (\ref{['eq:ET']}) as a function of the center-of-mass time $\cal{T}$ obtained from the solution of the KB equations for the monitored dissipative dynamics of the SYK model with parameters $q = 4$, $(f_{{\rm{S}}}, f_{{\rm{B}}}) = (1, 3)$, $V = 0.5$ and $\kappa = 0.1$. The thermal bath is chosen as another SYK with $J_{{\rm{B}}} = J = 1$, $q = 4$ at finite inverse temperature $\beta_{{\rm{B}}} = 100$. The initial conditions are thermal states at various inverse temperatures $\beta_{0} \in \{0, 0.1, 1, 10, 20, 100\}$. The equations are solved in an $8001 \times 8001$ grid with the initial conditions encoded in the third quadruple $(t_1, t_2) \in [-T_{\max}, 0) \times [-T_{\max}, 0)$, with size $4000 \times 4000$. The time separation is set to be $\Delta t = 0.1$ such that $T_{\max} = 400$. We observe that different initial states lead to the same final steady state solution at long time, indicating that the steady state is insensitive to the initial conditions.
• Figure 3: The function $\rho^+/\rho^-$ Eq. (\ref{['eq:rhopm']}) at different center-of-mass time $\mathcal{T}$ for different initial states. Top: Thermal initial state with inverse temperature $\beta_0=0$. Bottom: Thermal initial state with inverse temperature $\beta_0=100$.
• Figure 4: Comparison between the steady state solution of the KB equations and the time-translational invariant SD equations with different $V$, $\beta_{{\rm{B}}}$ and $\kappa$. Top: Comparison for the magnitude of the retarded Green's function Eq. \ref{['eq:GR_def']}. Bottom: Comparison for the spectral function Eq. \ref{['eq:rhopm']}. Other parameters are fixed as $q=q_{{\rm{B}}}=4$, $(f_{{\rm{S}}}, f_{{\rm{B}}}) = (1, 3)$ and $J = J_{{\rm{B}}}=1$, see Eqs. \ref{['eq:Sigma_S']} and \ref{['eq:Sigma_B']}. Excellent agreement is observed in all cases.
• Figure 5: Fitting of $|G^{{\rm{R}}}(t)|$ with Eq. (\ref{['eq:fitting']}) with respect to $t$ for $\kappa = 0.05$ and different values of $V$. The exponential decay occurs in two stages. The early time (short time fitting) decay depends on both the monitoring and the bath strengths while the late time decay (log time fitting) is mostly dependent on the thermal bath only. $1/\gamma$ stands for the typical long-time decay.