Out-of-time-order correlators in finite open systems

TL;DR

This paper investigates the behavior of out-of-time-order correlators in finite open quantum systems, revealing exponential saturation in discrete systems and analyzing decay times and saturation values, especially in two-level systems.

Contribution

It provides a microscopic calculation of OTOC decay times and saturation values in open quantum systems, highlighting differences between quantum-chaotic and discrete systems.

Findings

OTOCs saturate exponentially in discrete energy level systems.

Decay times are linked to inelastic transitions and dephasing.

Some OTOCs are immune to dephasing, affecting their decay behavior.

Abstract

We study out-of-time order correlators (OTOCs) of the form $\langle\hat A(t)\hat B(0)\hat C(t)\hat D(0)\rangle$ for a quantum system weakly coupled to a dissipative environment. Such an open system may serve as a model of, e.g., a small region in a disordered interacting medium coupled to the rest of this medium considered as an environment. We demonstrate that for a system with discrete energy levels the OTOC saturates exponentially $\propto \sum a_i e^{-t/\tau_i}+const$ to a constant value at $t\rightarrow\infty$, in contrast with quantum-chaotic systems which exhibit exponential growth of OTOCs. Focussing on the case of a two-level system, we calculate microscopically the decay times $\tau_i$ and the value of the saturation constant. Because some OTOCs are immune to dephasing processes and some are not, such correlators may decay on two sets of parametrically different time scales related to inelastic transitions between the system levels and to pure dephasing processes, respectively. In the case of a classical environment, the evolution of the OTOC can be mapped onto the evolution of the density matrix of two systems coupled to the same dissipative environment.

Equations

$$K(t)=⟨\hat{A}(t)\hat{B}(0)\hat{C}(t)\hat{D}(0)⟩,$$

$$\hat{\mathcal{H}}={\hat{\mathcal{H}}}_0+\hat{V}\hat{X}+{\hat{\mathcal{H}}}_{\text{bath}}(\hat{X}),$$

$$K_{m_1{m}_2,n_1{n}_2}=⟨|n_1⟩⟨m_1|(t)\hat{B}(0)|n_2⟩⟨m_2|(t)\hat{D}(0)⟩,$$

$${\partial }_t{K}_{m_1{m}_2,n_1{n}_2}=$$

$$i(E_{n_1}+E_{n_2}-E_{m_1}-E_{m_2})K_{m_1{m}_2,n_1{n}_2}$$

$$-\sum _{m_1^{\prime },m_2^{\prime },n_1^{\prime },n_2^{\prime }}{\Gamma }_{m_1{m}_2,n_1{n}_2}^{m_1^{\prime }{m}_2^{\prime },n_1^{\prime }{n}_2^{\prime }}{K}_{m_1^{\prime }{m}_2^{\prime },n_1^{\prime }{n}_2^{\prime }}.$$

$$\sum _{m,n}{K}_{nm,nm}=⟨\hat{B}(0)\hat{D}(0)⟩=\text{const}.$$

$$\hat{\mathcal{H}}=\frac{1}{2}B{\hat{\sigma }}_z+\frac{1}{2}\hat{\mathbf{\sigma }}\mathbf{n}\hat{X}+{\hat{\mathcal{H}}}_{bath}(\hat{X}),$$

$$K_{\uparrow \uparrow ,\downarrow \downarrow },K_{\downarrow \downarrow ,\uparrow \uparrow }\propto e^{\mp 2iBt}{e}^{-4{\Gamma }^{\varphi }t},$$

$$K_{\uparrow \downarrow ,\downarrow \downarrow },K_{\downarrow \uparrow ,\uparrow \uparrow },K_{\downarrow \downarrow ,\uparrow \downarrow },\dots \propto e^{-{\Gamma }^{\varphi }t}.$$

$$\displaystyle\partial_{t}\left(\begin{array}[]{c}K_{\downarrow\uparrow,\uparrow\downarrow}\\ K_{\uparrow\downarrow,\downarrow\uparrow}\\ K_{\uparrow\downarrow,\uparrow\downarrow}\\ K_{\downarrow\uparrow,\downarrow\uparrow}\\ K_{\uparrow\uparrow,\uparrow\uparrow}\\ K_{\downarrow\downarrow,\downarrow\downarrow}\end{array}\right)=\left(\begin{array}[]{cccccc}-\Gamma_{\downarrow}-\Gamma_{\uparrow}&0&-\Gamma_{\downarrow}&-\Gamma_{\downarrow}&\Gamma_{\downarrow}&\Gamma_{\downarrow}\\ 0&-\Gamma_{\downarrow}-\Gamma_{\uparrow}&-\Gamma_{\uparrow}&-\Gamma_{\uparrow}&\Gamma_{\uparrow}&\Gamma_{\uparrow}\\ -\Gamma_{\uparrow}&-\Gamma_{\downarrow}&-\Gamma_{\downarrow}-\Gamma_{\uparrow}&0&\Gamma_{\downarrow}&\Gamma_{\uparrow}\\ -\Gamma_{\uparrow}&-\Gamma_{\downarrow}&0&-\Gamma_{\downarrow}-\Gamma_{\uparrow}&\Gamma_{\downarrow}&\Gamma_{\uparrow}\\ \Gamma_{\uparrow}&\Gamma_{\downarrow}&\Gamma_{\uparrow}&\Gamma_{\uparrow}&-2\Gamma_{\downarrow}&0\\ \Gamma_{\uparrow}&\Gamma_{\downarrow}&\Gamma_{\downarrow}&\Gamma_{\downarrow}&0&-2\Gamma_{\uparrow}\end{array}\right)\left(\begin{array}[]{c}K_{\downarrow\uparrow,\uparrow\downarrow}\\ K_{\uparrow\downarrow,\downarrow\uparrow}\\ K_{\uparrow\downarrow,\uparrow\downarrow}\\ K_{\downarrow\uparrow,\downarrow\uparrow}\\ K_{\uparrow\uparrow,\uparrow\uparrow}\\ K_{\downarrow\downarrow,\downarrow\downarrow}\end{array}\right).$$

$$K(t\to \infty )=\frac{1}{\sqrt{2{\Gamma }_{\uparrow }^2+2{\Gamma }_{\downarrow }^2}}$$

$$+{\Gamma }_{\uparrow }{A}_{\uparrow \uparrow }{C}_{\uparrow \uparrow }+{\Gamma }_{\downarrow }{A}_{\downarrow \downarrow }{C}_{\downarrow \downarrow }).$$

$${\rho }_{m_1{m}_2,n_1{n}_2}={⟨{⟨|m_1⟩⟨n_1|(t)⟩}_{{\text{Sys}}_1}{⟨|m_2⟩⟨n_2|(t)⟩}_{{\text{Sys}}_2}⟩}_X$$

$$\hat{\mathcal{H}}={\hat{\mathcal{H}}}_0\otimes \text{mathds}1+\text{mathds}1\otimes {\hat{\mathcal{H}}}_0+\left(\hat{V}\otimes \text{mathds}1+\text{mathds}1\otimes \hat{V}\right)\hat{X}+{\hat{\mathcal{H}}}_{\text{bath}}(\hat{X}),$$

$${\partial }_t{\rho }_{mn}=i(E_{mn}+i{\Gamma }_{mn}){\rho }_{mn},$$

$${\Gamma }_{mn}=-i\int \frac{d\omega }{2\pi }\sum _k\left(\frac{S(\omega )|V_{mk}{|}^2}{\omega -E_{mk}-i0}+\frac{S(-\omega )|V_{nk}{|}^2}{\omega -E_{kn}-i0}\right)+i{V}_{mm}{V}_{nn}\int \frac{d\omega }{2\pi }\frac{S(\omega )+S(-\omega )}{\omega -i0}$$

$${\Gamma }_{mn}=\frac{1}{2}\sum _{k\ne m}{\Gamma }_{m\to k}^{rel}+\frac{1}{2}\sum _{k\ne n}{\Gamma }_{n\to k}^{rel}-i\delta E_m+i\delta E_n+{\Gamma }_{mn}^{deph},$$

$${\Gamma }_{n\to k}^{rel}=|V_{nk}{|}^{2}S(E_n-E_k)$$

$${\Gamma }_{mn}^{deph}=\frac{1}{2}(V_{nn}-V_{mm}{)}^{2}S(0)$$

$$\delta E_m=\sum _{k\ne m}|V_{mk}{|}^2\int \frac{d\omega }{2\pi }\frac{S(\omega )}{E_m-E_k-\omega }$$

$${\partial }_t{\rho }_{nn}=-{\rho }_{nn}\sum _k{\Gamma }_{n\to k}^{rel}+\sum _k{\rho }_{kk}{\Gamma }_{k\to n}^{rel},$$

$${\partial }_t\hat{\rho }=-i[{\hat{\mathcal{H}}}_{\text{eff}},\hat{\rho }]-\frac{1}{2}\sum _{i,j}\left({\hat{\mathcal{L}}}_{ij}^{†}{\hat{\mathcal{L}}}_{ij}\hat{\rho }+\hat{\rho }{\hat{\mathcal{L}}}_{ij}^{†}{\hat{\mathcal{L}}}_{ij}-2{\hat{\mathcal{L}}}_{ij}^{†}\hat{\rho }{\hat{\mathcal{L}}}_{ij}\right),$$

$${\hat{\mathcal{H}}}_{\text{eff}}=\sum _i|i⟩⟨i|(E_i+\delta E_i),$$

$${\hat{\mathcal{L}}}_{ij}=(1-{\delta }_{ij})\sqrt{{\Gamma }_{j\to i}^{rel}}|i⟩⟨j|+{\delta }_{ij}\sqrt{S(0)/N}\sum _l{V}_{ll}|l⟩⟨l|$$

$${\partial }_t{K}_{m_1{m}_2,n_1{n}_2}=i⟨\left[{\hat{\mathcal{H}}}_0+{\hat{\mathcal{H}}}_{\text{coupl}}(t),|n_1⟩⟨m_1|(t)\right]\hat{B}(0)|n_2⟩⟨m_2|(t)\hat{D}(0)⟩$$

$$+i⟨|n_1⟩⟨m_1|(t)\hat{B}(0)\left[{\hat{\mathcal{H}}}_0+{\hat{\mathcal{H}}}_{\text{coupl}}(t),|n_2⟩⟨m_2|(t)\right]\hat{D}(0)⟩,$$

$${\partial }_t{K}_{m_1{m}_2,n_1{n}_2}=$$

$$-⟨{\int }_{-\infty }^t\left[{\hat{\mathcal{H}}}_{\text{coupl}}(t^{\prime }),\left[{\hat{\mathcal{H}}}_{\text{coupl}}(t),|n_1⟩⟨m_1|(t)\right]\right]d{t}^{\prime }\hat{B}(0)|n_2⟩⟨m_2|(t)\hat{D}(0)⟩$$

$$-⟨|n_1⟩⟨m_1|(t)\hat{B}(0){\int }_{-\infty }^t\left[{\hat{\mathcal{H}}}_{\text{coupl}}(t^{\prime }),\left[{\hat{\mathcal{H}}}_{\text{coupl}}(t),|n_2⟩⟨m_2|(t)\right]\right]d{t}^{\prime }\hat{D}(0)⟩$$

$$-⟨\left[{\hat{\mathcal{H}}}_{\text{coupl}}(t),|n_1⟩⟨m_1|(t)\right]\hat{B}(0){\int }_{-\infty }^t\left[{\hat{\mathcal{H}}}_{\text{coupl}}(t^{\prime }),|n_2⟩⟨m_2|(t)\right]d{t}^{\prime }\hat{D}(0)⟩$$

$$-⟨{\int }_{-\infty }^t\left[{\hat{\mathcal{H}}}_{\text{coupl}}(t^{\prime }),|n_1⟩⟨m_1|(t)\right]d{t}^{\prime }\hat{B}(0)\left[{\hat{\mathcal{H}}}_{\text{coupl}}(t),|n_2⟩⟨m_2|(t)\right]\hat{D}(0)⟩,$$

$${\partial }_t{K}_{m_1{m}_2,n_1{n}_2}=$$

$$-\frac{1}{2}\left(\sum _{k\ne m_1}{\Gamma }_{m_1\to k}^{rel}+\sum _{k\ne n_1}{\Gamma }_{n_1\to k}^{rel}+\sum _{k\ne m_2}{\Gamma }_{m_2\to k}^{rel}+\sum _{k\ne n_2}{\Gamma }_{n_2\to k}^{rel}\right)K_{m_1{m}_2,n_1{n}_2}$$

$$-{\Gamma }_{n_2\to n_1}^{rel}{K}_{m_1{m}_2,n_2{n}_1}-{\Gamma }_{m_1\to m_2}^{rel}{K}_{m_2{m}_1,n_1{n}_2}-{\Gamma }_{m_1{n}_1,m_2{n}_2}^{\varphi }{K}_{m_1{m}_2,n_1{n}_2},$$

Full text

Quantum information spreading in a quantum system is often described by out-of-time-order correlators (OTOCs) of the form

[TABLE]

where $\hat{A}$, $\hat{B}$, $\hat{C}$ and $\hat{D}$ are Hermitian operators, and $\langle\ldots\rangle$ is the average with respect to the initial state of the system. Correlators of such form have been first introduced by A. Larkin and Y.N. OvchinnikovLarkin and Ovchinnikov (1969) in the context of disordered conductors, where the correlator $\langle[p_{z}(t),p_{z}(0)]^{2}\rangle$ of particle momenta $p_{z}$ has been demonstrated to grow exponentially $\propto e^{2\lambda t}$ for sufficiently long times $t$. The Lyapunov exponent $\lambda$ characterises the rate of divergence of two classical electron trajectories with slightly different initial conditions and serves as a measure of quantum chaotic behaviour in a system.

The concept of OTOC has revivedKitaev (2015) recently in the context of quantum information scrambling and black holes, motivating further studies of such quantities (see, e.g., Refs. Maldacena et al., 2016; Swingle and Chowdhury, 2016; Aleiner et al., 2016; Rozenbaum et al., 2016; Patel and Sachdev, 2016). Despite not being measurable observables111 Because an OTOC involves evolution backwards in time, measuring it requires either using a second copy of the systemYao et al. (2016); Bohrdt et al. (2016); Yunger Halpern et al. (2017) or effectively reverting the sign of the HamiltonianSwingle et al. (2016); Zhu et al. (2016); Gärttner et al. (2016); Danshita et al. (2016); Tsuji et al. (2017); Li et al. (2016), possible, e.g., in spin systems using spin-echo-type techniques or ancilla qubits, OTOCs (1) characterise the spreading of quantum information and the sensitivity of the system to the change of the initial conditions. It is also expected that OTOCs may be usedFan et al. (2016); Chen (2016); Swingle and Chowdhury (2016); Huang et al. (2016); Patel et al. (2017) to distinguish between many-body-localised and many-body-delocalised statesBasko et al. (2006) of disordered interacting systems.

So far the studies of quantum chaos and information scrambling have been focussing on closed quantum systems. In reality, however, each system is coupled to a noisy environment, which leads to decoherence and affects information spreading. Moreover, a sufficiently strongly disordered interacting system may be separated into a small subsystem, of the size of the single-particle localisation length or a region of quasi-localised states, coupled to the rest of the system considered as environment. In this paper we analyse out-of-time order correlators in a quantum system weakly coupled to a dissipative environment.

Phenomenological picture in a strongly disordered material. A system with localised single-particle states and weak short-range interactions exhibits insulating behaviour at low temperaturesBasko et al. (2006). Local physical observables in such a system are strongly correlated only on short length scales, and their properties may be understood by considering a single “localisaton cell”, particle states in a region of space of the size of order of the localisation length $\xi$, which may be considered weakly coupled to the rest of the system.

The energy spectrum of the localisation cell may be probed via response functions of local operators in the cell, e.g., the response function $\chi(\omega)=\sum_{\alpha,\beta}\frac{(f_{\alpha}-f_{\beta})|Q_{\alpha\beta}|^{2}}{E_{\alpha}-E_{\beta}+\omega+i0}$ of the charge $Q$ in a region inside the cell to the voltage in this region, where $E_{\alpha}$ and $E_{\beta}$ are the energies of many-body states and $f_{\alpha}$ is their distribution function. For temperatures smaller than a critical value, quasiparticles in the system have zero decay rate222When studying response functions, a system in the insulating state should be assumed coupled to an external bath, with the value of the coupling sent to zero at the end of the calculationBasko et al. (2006). The quasiparticle decay rate is then given by the bath strength. (“superinsulating” regimeBasko et al. (2006)), and the system thus responds only at a discrete set of frequencies $\omega=E_{i}-E_{j}$, determined by the energy gaps between many-body states, as shown in Fig. 1. The OTOC (1) in this regime oscillates $K(t)\propto\sum_{n}a_{n}e^{i\omega_{n}t}$ with a discrete set of frequencies $\omega_{n}=E_{i_{n}}+E_{i_{n}^{\prime}}-E_{j_{n}}-E_{j_{n}^{\prime}}$.

When the temperature (or the interaction strength at a given temperature) exceeds a critical value, the levels and response functions get broadened (“metallic” phaseBasko et al. (2006)), as illustrated in Fig. 1, becoming smoother with increasing temperature and/or interactions. Near the superinsulator-metal transition the characteristic level width $\Gamma$ is significantly smaller than the gaps between levels, and the localisation cell may be considered as an open system weakly coupled to a dissipative environment. The same model may be applied also to a strongly disordered material with an external bath, such as a system of phonons, which provide a finite level width $\Gamma$ at all finite temperatures. The local operators $\hat{A}$, $\hat{B}$, $\hat{C}$ and $\hat{D}$ in Eq. (1) do not necessarily act on states in one localisation cell, but may involve states in several cells close to each other. These cells may still be considered as a single quantum dot in a noisy environment so long as the level spacing in the dot exceeds the level width. Such a model of an open quantum dot may be also realised directly, e.g., using superconducting qubits or trapped cold atoms.

Summary of the results. We demonstrate that, for a system with discrete non-degenerate levels $E_{n}$, correlator (1) at long times $t$ exponentially saturates to a constant value, $K(t)\propto\sum a_{n}e^{i\omega_{n}t}e^{-t/\tau_{n}}+const$, and calculate microscopically the value of the constant and the relaxation times $\tau_{n}$ as a function of the environment spectral function and the matrix elements of the system-environment coupling. Depending on the choice of the operators $\hat{A}$, $\hat{B}$, $\hat{C}$ and $\hat{D}$, the saturation value may be finite or zero. OTOCs relax due to both inelastic transitions between the system’s levels and pure dephasing processes, which are caused by slow fluctuations of the energies $E_{n}$. While some OTOCs are immune to dephasing processes, a generic correlator has components both sensitive and insensitive to dephasing and thus decays on two sets of parametrically different scales related to dephasing and relaxation respectively, as shown in Fig. 2.

Our results indicate, in particular, that a disordered system of interacting particles cannot exhibit quantum chaotic behaviour if the typical single-particle level splitting $\delta_{\xi}$ in a volume of linear size $\xi$ (localisation length) exceeds the dephasing rate and the rate of inelastic transitions due to interactions and/or phonons. Correlators (1) in this system can only saturate to constant values at $t\rightarrow\infty$, in contrast with quantum-chaotic systems which display exponential growth of OTOCs with time. Our results thus suggest that chaotic behaviour in a disordered interacting system requires either the presence of delocalised single-particle states or sufficiently strong interactions or, e.g., a phonon bath, which would lead to the quasiparticle decay rate exceeding the level spacing $\delta_{\xi}$.

For a classical environment, the evolution of an OTOC (1) in an open system may be mapped onto the evolution of the density matrix of two systems coupled to the same environment, which allows one to measure OTOCs by observing the correlations between two systems in a noisy environment, such as spins in a random time-dependent magnetic field.

Model. We consider a system with discrete non-degenerate energy levels $E_{n}$ coupled to a dissipative environment and described by the Hamiltonian

[TABLE]

where $\hat{\cal H}_{0}=\sum_{n}E_{n}\ket{n}\bra{n}$ is the Hamiltonian of the system, $\hat{\cal H}_{\text{bath}}(\hat{X})$– the Hamiltonian of the environment, and $\hat{V}\hat{X}$ is the coupling between the system and the environment, where the operator $\hat{V}=\sum_{n,m}V_{nm}\ket{n}\bra{m}$ acts on the system degrees of freedom, and $\hat{X}$ is an environment variable which commutes with the system degrees of freedom.

To compute the OTOC (1), where the operators $\hat{A}$, $\hat{B}$, $\hat{C}$ and $\hat{D}$ act on the system variables, it is convenient to decompose it as $K=K_{m_{1}m_{2},n_{1}n_{2}}A_{n_{1}m_{1}}C_{n_{2}m_{2}}$ (summation over repeated indices implied), where $A_{n_{1}m_{1}}$ and $C_{n_{2}m_{2}}$, are the matrix elements of the operators $\hat{A}$ and $\hat{C}$, and

[TABLE]

where $\langle\ldots\rangle$ is the averaging with respect to both the system and environment states.

In the limit of a vanishing system-environment coupling $\hat{V}$, the correlators (3) oscillate with time, $K_{m_{1}m_{2},n_{1}n_{2}}\propto e^{i(E_{n_{1}}+E_{n_{2}}-E_{m_{1}}-E_{m_{2}})t}$. A finite coupling between the system and the environment leads to dissipation and relaxation processes and thus to the decay of the elements $K_{m_{1}m_{2},n_{1}n_{2}}$. For a weak coupling considered in this paper, the characteristic decay times of the OTOCs significantly exceed the correlation time of the environment degrees of freedom, i.e. of the function $S(t-t^{\prime})=\langle\hat{X}(t)\hat{X}(t^{\prime})\rangle_{\text{env}}$, and the evolution of the elements is described by a system of Markovian Bloch-RedfieldSlichter (1996) equations (see Supplemental Material for the microscopic derivation) of the form

[TABLE]

From the definition of the elements (3) it follows that

[TABLE]

Eq. (5) may be also derived from the microscopic equations of evolution, as shown in Supplemental Material.

Due to the smallness of the decay rates $\Gamma_{m_{1}m_{2},n_{1}n_{2}}^{m_{1}^{\prime}m_{2}^{\prime},n_{1}^{\prime}n_{2}^{\prime}}$ in Eq. (4), the evolution of each element $K_{m_{1}m_{2},n_{1}n_{2}}$ is affected only by the elements $K_{m_{1}^{\prime}m_{2}^{\prime},n_{1}^{\prime}n_{2}^{\prime}}$ with the same oscillation frequency $E_{n_{1}}+E_{n_{2}}-E_{m_{1}}-E_{m_{2}}$ (secular approximation). In this paper we consider systems with sufficiently non-degenerate energy spectra; if two elements oscillate with the same frequency, they may be different only by permutations of indices $n_{1}$ and $n_{2}$ and/or $m_{1}$ and $m_{2}$.

For a generic $N$-level system there are $2N^{2}-N$ elements (3) with zero energy gaps $E_{n_{1}}+E_{n_{2}}-E_{m_{1}}-E_{m_{2}}$ (with $m_{1}=n_{1}$, $m_{2}=n_{2}$ and/or $m_{1}=n_{2}$, $m_{2}=n_{1}$). These elements are immune to dephasing, i.e. to the accumulation of random phases caused by slow fluctuations of the energies $E_{n_{i}}$. Such vanishing of dephasing is similar to that in decoherence-free subspacesZanardi and Rasetti (1997); Wu and Lidar (2002) of multiple-qubit systems. We emphasise, however, that even dephasing-immune correlators in general decay at long times due to the environment-induced inelastic transitions between the levels (relaxation processes).

A generic OTOC (1) includes components both sensitive and insensitive to dephasing, as well as a component independent of time, which exists due to the conservation law (5). For an environment with a smooth spectral function on the scale of the characteristic level splitting, the characteristic decay rate of the dephasing-immune components may be estimated as $1/\tau_{\text{rel}}\sim V_{\bot}^{2}S(\Delta E)$, where $V_{\bot}$ is the typical off-diagonal matrix element of the perturbation $\hat{V}$ and $\Delta E$ is the characteristic level spacing. The other components decay with the characteristic rate $1/\tau_{\text{deph}}+1/\tau_{\text{rel}}$, where $1/\tau_{\text{deph}}\sim V_{\parallel}^{2}S(0)$ is the characteristic dephasing rate, where $V_{\parallel}$ is the typical diagonal matrix element of the perturbation $\hat{V}$. As a result, the decay of the OTOC consist of three stages, corresponding to these characteristic times, as illustrated in Fig. 2.

Two-level system. In order to illustrate the meaning of these time scales and the related phenomena, we focus below on the case of a two-level system, equivalent to a spin-$1/2$ in a random magnetic field (for the microscopic analysis of OTOCs in the generic case of a multi-level system see Supplemental Material), described by the Hamiltonian

[TABLE]

where $\hat{\bm{\sigma}}$ is a vector of Pauli matrices and ${\bf n}$ is a constant unit vector, the direction of the fluctuations of the magnetic field.

The dissipative environment induces transitions $\ket{\uparrow}\rightarrow\ket{\downarrow}$ with the rate $\Gamma_{\downarrow}=\frac{1}{4}(n_{x}^{2}+n_{y}^{2})S(B)$, as well as the opposite transitions $\ket{\downarrow}\rightarrow\ket{\uparrow}$ with the rate $\Gamma_{\uparrow}=\frac{1}{4}(n_{x}^{2}+n_{y}^{2})S(-B)$, where $S(\omega)$ is the environment spectrum, the Fourier-transform of $S(t-t^{\prime})=\langle\hat{X}(t)\hat{X}(t^{\prime})\rangle_{\text{env}}$. Weak fluctuations of the magnetic field in the longitudinal direction lead to dephasing with the rate $\Gamma^{\phi}=\frac{1}{2}n_{z}^{2}S(0)$. We focus below on the long-time dynamics of the system and assume for simplicity that the rate $\Gamma^{\phi}$ of pure dephasing significantly exceeds the rates $\Gamma_{\uparrow}$ and $\Gamma_{\downarrow}$ of inelastic transitions between the levels of the spin; in the opposite case, all OTOC decay rates are of the same order of magnitude.

The OTOCs $K_{\uparrow\uparrow,\downarrow\downarrow}$ and $K_{\downarrow\downarrow,\uparrow\uparrow}$ oscillate with frequencies $\pm 2(E_{\downarrow}-E_{\uparrow})=\mp 2B$ and have dephasing rate $4\Gamma^{\phi}$, the same as $\pm 1$-projection states of a spin-$1$ in magnetic field $B$,

[TABLE]

where we have neglected the small relaxation rates $\Gamma_{\uparrow,\downarrow}\ll\Gamma^{\phi}$.

There are 8 elements (3) which correspond to 3 spin indices pointing in one direction and one spin index pointing in the opposite direction. These elements oscillate with frequencies $\pm B$ and have the same dephasing rate as a spin-$1/2$,

[TABLE]

The behaviour of OTOCs at long times $t\gg 1/\Gamma^{\phi}$ is determined by the components with a vanishing frequency $E_{n_{1}}+E_{n_{2}}-E_{m_{1}}-E_{m_{2}}$ of coherent oscillations, because such components are insensitive to dephasing. For a spin-1/2, their evolution is described by the system of equations (as follows from the generic master equations for a multi-level system derived in Supplemental Material)

[TABLE]

The rates of the long-time decay of OTOCs are given by the eigenvalues of the matrix in Eq. (27) (with minus sign) and are shown (except for the zero eigenvalue) in Fig. 3. Such a matrix always has a zero eigenvalue, due to the conservation law (5). The system also has a triply degenerate decay rate $\Gamma_{\uparrow}+\Gamma_{\downarrow}$. The other two decay rates are given by $\frac{1}{2}\left[3\Gamma_{\uparrow}+3\Gamma_{\downarrow}\pm\left(\Gamma_{\uparrow}^{2}+34\Gamma_{\uparrow}\Gamma_{\downarrow}+\Gamma_{\downarrow}^{2}\right)^{\frac{1}{2}}\right]$.

At long times $t\rightarrow\infty$ the correlator (1) saturates to a constant value determined by the projection of the OTOC (1) on the zero-decay-rate mode,

[TABLE]

While we assumed a small inelastic relaxation rate in comparison with the dephasing rate, we emphasise that the result (28) for the saturation value of the OTOC holds for an arbitrary ratio of dephasing and relaxation rates.

Mapping to the evolution of two systems for a classical environment. The evolution of the OTOCs (3) is similar to that of the density-matrix elements

[TABLE]

of a compound system consisting of two identical subsystems (“Sys1” and “Sys2”) coupled to the same dissipative environment, where $m_{1}$ and $n_{1}$ and $m_{2}$ and $n_{2}$ in Eq. (29) are the states of the first and the second subsystems respectively, $\ket{m_{i}}\bra{n_{i}}(t)$ is an operator in the interaction representation, and $\langle\ldots\rangle_{X}$ is the averaging with respect to the environment degrees of freedom. The Hamiltonian of such a compound system is given by

[TABLE]

where $\ldots\otimes\ldots$ is the product of the subsystem subspaces; $\hat{\cal H}_{0}$ and $\hat{V}$ are the Hamiltonian of each subsystem and its coupling to the environment, and the environment variable $\hat{X}$ commutes with all degrees of freedom of subsystems “Sys1” and “Sys2”.

The evolution of the elements (3) and (29) is described by similar Markovian master equations (see Supplemental Material for microscopic derivation). In particular, in the limit of a classical environment ($\left<\hat{X}(t)\hat{X}(t^{\prime})\right>_{\text{env}}=\left<\hat{X}(t^{\prime})\hat{X}(t)\right>_{\text{env}}$), the evolution of OTOCs (3) can be mapped exactly onto that of the density matrix (29) of two systems coupled to this environment, as follows from the definitions of these quantities. The conservation law (5) is mapped then onto the conservation of the trace of the density matrix of a compound system consisting of two subsystems.

In the limit of a classical environment, the spectral function is even, $S(\omega)=S(-\omega)$, the relaxation rate $i\rightarrow j$ for each pair of levels $i$ and $j$ in a system matches the reverse rate $j\rightarrow i$. In particular, in the case of a two-level system $\Gamma_{\uparrow}=\Gamma_{\downarrow}=\Gamma$, and the OTOC has three decay rates at long times $t\gg{\Gamma^{\phi}}^{-1}$: $\Gamma_{1}=6\Gamma$, $\Gamma_{2}=2\Gamma$ (triply degenerate) and $\Gamma_{3}=0$ (doubly degenerate), as shown in Fig. 3. Due to the mapping, these rates match the decay rates of pair-wise correlators of observables in, e.g., an ensemble of spins in a uniform random magnetic field and thus may be conveniently measured in such ensembles.

We emphasise that the mapping between an OTOC and the evolution of two subsystems coupled to the same classical environment holds for an arbitrary system-environment coupling but not only in the limit of a weak coupling considered in this paper. This mapping suggests a way for measuring OTOCs in generic systems in the presence of classical environments through observing correlators $\left<\left<\hat{A}(t)\right>_{\text{Sys}_{1}}\left<\hat{C}(t)\right>_{\text{Sys}_{2}}\right>_{X}$ of observables $\hat{A}$ and $\hat{C}$ between two systems.

Discussion. We computed OTOCs in a system weakly coupled to a dissipative environment and demonstrated that they saturate to a constant value at long times. Because such an open system may serve as a model of a small region in a disordered interacting medium (in the presence or in the absence of a phonon bath), this suggests the absence of a chaotic behaviour in strongly disordered materials. While our result applies to weakly-conducting and insulating materials, for which the system-environment coupling may be considered small, we leave it for a future study whether non-chaotic behaviour persists in systems strongly coupled to the environment (corresponding to an effectively continuous energy spectrum of a localisation cell). For a classical environment, the evolution of an OTOC matches the evolution of correlators of observables between two identical systems coupled to the same environment, which may be used for measuring OTOCs in open systems in classical environments. The possibility to develop a similar measurement method for the case of a quantum environment is another question which deserves further investigsation.

.1 Master equations for the density matrix in an open system

Off-diagonal elements. For a system with non-degenerate energy levels weakly coupled to a dissipative environment, with the Hamiltonian given by Eq. (2), the off-diagonal entries $\rho_{mn}$ of the density matrix satisfy Bloch-Redfield master equations (see, e.g., Ref. Slichter, 1996)

[TABLE]

where $E_{mn}=E_{m}-E_{n}$ is the frequency of coherent oscillations for an isolated system, and the complex quantity

[TABLE]

accounts for the effects of the environment, where $S(\omega)$ is the Fourier-transform of the correlation function $S(t-t^{\prime})=\langle\hat{X}(t)\hat{X}(t^{\prime})\rangle_{env}=\int\frac{d\omega}{2\pi}e^{-i\omega(t-t^{\prime})}S(\omega)$ of the environment degree of freedom $\hat{X}(t)$.

The quantity $\Gamma_{mn}$, given by Eq. (S2), may be decomposed as

[TABLE]

where

[TABLE]

is the rate of environment-induced transitions (relaxation) from level $n$ to level $k$,

[TABLE]

is the pure dephasing rate, and

[TABLE]

is the shift of the energy of the $m$-th level due to the interaction with environment (Lamb shift). The relaxation rate between two levels $n$ and $k$, Eq. (S4), is determined by the environment spectrum $S(\omega)$ at frequency $\omega=E_{nk}$ equal to the energy gap between these levels, while the dephasing rate (S5) is determined by the low-frequency properties of the environment.

Diagonal elements. The dynamics of the diagonal elements of the density matrix is described by the equations

[TABLE]

where the transition rates $\Gamma_{n\rightarrow k}$ are given by Eq. (S4).

Lindblad form. Eqs. (S1) and (S7) for the evolution of the density matrix can be rewritten in the Lindblad form

[TABLE]

where the summation runs over all pairs of indices $i={1,\ldots,N}$ and $j={1,\ldots,N}$ in an $N$-level system; the effective Hamiltonian of coherent evolution is given by

[TABLE]

and the Lindblad operators

[TABLE]

account for the effects of dephasing and dissipation.

.2 Master equations for OTOCs

In what follows we derive microscopically the Bloch-Redfield-type master equations for the out-of-time-order correlator (1), following a procedure similar to the derivation (see, e.g., Ref. Slichter (1996)) of the master equations for the density matrix. Due to the weakness of the system-environment coupling, the OTOCs decay on long times significantly exceeding the characteristic correlation time of the environment.

It follows directly from Eq. (1) that

[TABLE]

where $\hat{\cal H}_{0}$ is the Hamiltonian of the system (without the environment) and $\hat{\cal H}_{\text{coupl}}=\hat{X}\sum_{n,m}V_{nm}\ket{n}\bra{m}$ is the coupling between the system and the environment. By expanding all Heisenberg operators in Eq. (S11) to the first order in the perturbation $\hat{\cal H}_{\text{coupl}}$ and neglecting the change of the density matrix of the system during the characteristic correlation time of the environment, we arrive at the equations for the evolution of the elements $K_{m_{1}m_{2},n_{1}n_{2}}$ in the form

[TABLE]

where only the terms up to the second order in the system-environment coupling have been kept and the lower time integration limit has been extended to $-\infty$ in view of the short correlation time of the environment degrees of freedom, i.e. the correlation time between $\hat{\cal H}_{\text{coupl}}(t^{\prime})\propto\hat{X}(t^{\prime})$ and $\hat{\cal H}_{\text{coupl}}(t)\propto\hat{X}(t)$. Using Eq. (S12), we derive below the master equations for the evolution of the OTOCs in the form (4).

Due to the weakness of the system-environment coupling, the characteristic energy gaps between system levels significantly exceed the decay rates of the OTOCs, which are determined by the last four lines in Eq. (S12); the elements $K_{m_{1}m_{2},n_{1}n_{2}}$ quickly oscillate with frequencies $E_{n_{1}}+E_{n_{2}}-E_{m_{1}}-E_{m_{2}}$ and decay with rates significantly exceeded by these frequencies. Thus, the evolution of each element $K_{m_{1}m_{2},n_{1}n_{2}}$ depends only on other elements corresponding to the same energy splitting $E_{n_{1}}+E_{n_{2}}-E_{m_{1}}-E_{m_{2}}$. Below we consider separately the cases of finite and zero values of the splitting.

.2.1 Finite energy splitting

For each combination of different $m_{1}$, $m_{2}$, $n_{1}$ and $n_{2}$ there are four elements $K$ which correspond to the same energy splitting and differ from each other by permutations of indices. We assume for simplicity that there is no additional degeneracy of the quantities $E_{n_{1}}+E_{n_{2}}-E_{m_{1}}-E_{m_{2}}$ when all of the indices $m_{1}$, $m_{2}$, $n_{1}$ and $n_{2}$ are different. Eq. (S12) in that case gives

[TABLE]

where the transition rates $\Gamma_{i\rightarrow j}^{rel}$ are given by Eq. (S4); $\delta E_{i}$ is the renormalisation of the $i$-th level by environment, given by Eq. (S6); and

[TABLE]

is the dephasing rate in a compound system consisting of two copies of the original system coupled to the same bath.

.2.2 Zero energy splitting

Elements $K_{m_{1}m_{2},n_{1}n_{2}}$ with zero splitting $E_{n_{1}}+E_{n_{2}}-E_{m_{1}}-E_{m_{2}}$ have a greater degeneracy and require separate analyses.

“Diagonal” elements. Let us first consider the elements with $n_{1}=m_{1}$ and $n_{2}=m_{2}$. These elements satisfy the same equations of evolution as the diagonal elements of the density matrix of a compound system consisting of two copies of the original system. For $n_{1}=n_{2}=n\neq m=m_{1}=m_{2}$ we obtain from Eq. (S12)

[TABLE]

In the case $n=m$ Eq. (S12) gives

[TABLE]

From Eqs. (S15) and (S16) it follows immediately that

[TABLE]

which corresponds to the conservation of the sum of the diagonal elements of the density matrix of a compound system.

“Non-diagonal” elements. The other set of elements with zero energy splitting, different from the “diagonal” elements, correspond to $m_{1}=n_{2}$ and $m_{2}=n_{1}$. Their evolution is described by the equations

[TABLE]

.3 Master equation for the density matrix for two copies of a system coupled to the same environment

The equations for the evolution of the elements $K_{m_{1}m_{2},n_{1}n_{2}}$ are similar to the equations of evolution of the density-matrix elements $\rho_{m_{1}m_{2},n_{1}n_{2}}=\left<\ket{n_{1}n_{2}}\bra{m_{1}m_{2}}(t)\right>$ of a compound system consisting of two copies of the original system coupled to the same environment, where $n_{i}$ and $m_{i}$ label the states of the $i$-th subsystem; $i=1,2$. The Hamiltonian of such a compound system is given by Eq. (30). To the second order in the system-environment coupling $\hat{V}$ the evolution of the density matrix elements is described by the equation

[TABLE]

The form of the coupling $\hat{V}=\sum_{n,m}V_{nm}\ket{n}\bra{m}$ and Eq. (S19) give, when all of the indices $n_{1}$, $n_{2}$, $m_{1}$ and $m_{2}$ are different,

[TABLE]

where the quantity

[TABLE]

gives the rate of the flip-flop processes, i.e. the rate of the coherent interchange $n_{1}\leftrightarrow n_{2}$, and the dephasing rate $\Gamma^{\phi}_{m_{1}n_{1},m_{2}n_{2}}$ is defined by Eq. (S14).

Lindblad form. The master equations for the evolution of the density matrix of two systems in the same environment may may be also rewritten in the Lindblad form (S8) with the effective Hamiltonian

[TABLE]

and the Lindblad operators

[TABLE]

Mapping between OTOCs and two-system density matrix. Eq. (S13), which described the evolution of OTOCs for an open system in a dissipative environment, resembles Eq. (S20), which describes the evolution of the density matrix elements for two copies of the system coupled to this environment. Indeed, both equations have the same diagonal part, i.e. the part which relates the evolution of the element $\rho_{m_{1}m_{2},n_{1}n_{2}}$ or $K_{m_{1}m_{2},n_{1}n_{2}}$ to itself. Both equations also have terms with interchanged indices $n_{1}\leftrightarrow n_{2}$ or $m_{1}\leftrightarrow m_{2}$. While two systems coupled to an environment allow for a coherent (“flip-flop”) as well as inelastic interchange, the respective processes for OTOCs are purely inelastic.

As discussed in the main text, in the limit of a classical environment the evolutions of the OTOC and two systems coupled to this environment may be mapped onto each other. Classical environment corresponds to the odd spectrum $S(\omega)=S(-\omega)$, which leads to the vanishing of the flip-flop rates (S21) and identical relaxation rates $\Gamma_{n_{1}\rightarrow n_{2}}^{rel}=\Gamma_{n_{2}\rightarrow n_{1}}^{rel}$ of the transitions $n_{1}\rightarrow n_{2}$ and $n_{2}\rightarrow n_{1}$ for each pair of states $n_{1}$ and $n_{2}$. The equations (S13) and (S20) for the evolution of the OTOC and the two systems become identical in this limit.