Analytical determination of multi-time correlation functions in quantum chaotic systems

Abstract

The time-dependence of multi-point observable correlation functions are essential quantities in analysis and simulation of quantum dynamics. Open quantum systems approaches utilize two-point correlations to describe the influence of an environment on a system of interest, and in studies of chaotic quantum system, the out-of-time-ordered correlator (OTOC) is used to probe chaoticity of dynamics. In this work we analytically derive the time dependence of multi-point observable correlation functions in quantum systems from a random matrix theoretic approach, with the highest order function of interest being the OTOC. We find in each case that dynamical contributions are related to a simple function, related to the Fourier transform of coarse-grained wave-functions. We compare the predicted dynamics to exact numerical experiments in a spin chain for various physical observables. We comment on implications towards the emergence of Markovianity and quantum regression in closed quantum systems, as well as relate our results to known bounds on chaotic dynamics.

Figures (9)

• Figure 1: Coarse-grained mid-energy chaotic eigenstates $\Lambda(\mu, \alpha)$ (see Eq. \ref{['eq:Lambda_def']}) of the spin-chain model used in Sec. \ref{['ED']}, with Lorentizian and Gaussian fits in weak coupling (a) and strong coupling (b) limits. Parameters: $J_x^{\rm i} =$ 0.1 (a), 0.8 (b), each have $B_z^{\rm s}=B_x^{\rm s}=0.4$, $B_x^{\rm b} = 0.3$, $J_x^{\rm b}=0.7$, $J_z^{\rm i}=0.2$, $r_1=5$, $r_2=10$, $N=12$.
• Figure 2: Observable correlation functions in weak coupling regime, $J_x^{\rm i}=0.1$. (a) and (b) One-point correlation functions, analytic results given by \ref{['1-point-weak']} and \ref{['1-point-strong']}. (c) and (d) The real part of two-point observable correlation functions, analytic results given by \ref{['2-point-weak']} and \ref{['2-point-strong']}. The initial state is $|\Psi_0\rangle=|\phi_\alpha\rangle$, where $\alpha=2041$. The system consists of 12 spins and the other parameters are set to $B_z^{\rm s}=B_x^{\rm s}=0.4$, $B_x^{\rm b} = 0.3$, $J_x^{\rm b}=0.7$, $J_z^{\rm i}=0.2$, $r_1=5$, $r_2=10$. We work with $\Gamma=0.087$ (Lorentzian $\Lambda)$ and $K=0.005$ (Gaussian $\Lambda$).
• Figure 3: Observable correlation functions in strong coupling regime, $J_x^{\rm i}=0.8$. (a) and (b) One-point correlation functions, analytic results given by \ref{['1-point-weak']} and \ref{['1-point-strong']}. (c) and (d) The real part of two-point observable correlation functions, analytic results given by \ref{['2-point-weak']} and \ref{['2-point-strong']}. We work with $\Gamma=0.79$ (Lorentzian $\Lambda)$ and $K=0.31$ (Gaussian $\Lambda$).
• Figure 4: Out-of-time-ordered correlators defined as $F_{xz}(t)=\langle\sigma_x(t)\sigma_z\sigma_x(t)\sigma_z\rangle$ and $F_{xx}(t)=\langle\sigma_x(t)\sigma_x\sigma_x(t)\sigma_x\rangle$. (a) Weak coupling regime, $J_x^{\rm i}=0.1$. (b) Strong coupling regime $J_x^{\rm i}=0.8$. The analytic results are given by \ref{['4-point-weak']} and \ref{['4-point-strong']}.
• Figure 5: Time dependence of the squared commutator in (a) weak ($J_x^{\rm i}=0.1$) and (b) strong ($J_x^{\rm i}=0.8$) coupling regime. The analytic results follow Eq. \ref{['commutator_ED']}.