Dissipative quantum chaos unveiled by stochastic quantum trajectories

A new approach to theory and simulation of the non-Markovian dynamics of open quantum systems is presented. It is based on identification of a parameter which is uniformly bounded on wide time intervals: the occupation of the virtual cloud of quanta. By ‘virtual’ we denote those bath excitations which were emitted by the open system, but eventually will be reabsorbed before any measurement of the bath state. A useful property of the virtual cloud is that the number of its quanta is expected to saturate on long times, since physically this cloud is a (retarded) polarization of the bath around the system. Therefore, the joint state of open system and virtual cloud (we call it dressed state) can be accurately represented in a truncated basis of Fock states, on a wide time scale. At the same time, there can be an arbitrarily large number of the observable quanta (which survive up to measurement), especially if the open system is under driving. However, it turns out that the statistics of the bath-measurement outcomes is classical (in a suitable measurement basis): one can employ a Monte Carlo sampling of these outcomes. Therefore, it is possible to efficiently simulate the dynamics of the observable quantum field. In this work we consider the bath measurement with respect to the coherent states, which yields the Husimi function as the positive (quasi)probability distribution of the outcomes. The joint evolution of the dressed state and the corresponding outcome is called the dressed quantum trajectory. The Monte Carlo sampling of these trajectories yields a stochastic simulation method with promising convergence properties on wide time scales.

Dissipative quantum chaos plays a central role in the characterization and control of information scrambling, nonunitary evolution, and thermalization, but it still lacks a precise definition. The Grobe-Haake-Sommers conjecture, which links Ginibre level repulsion to classical chaotic dynamics, was recently shown to fail [Phys. Rev. Lett. 133, 240404 (2024)PRLTAO0031-900710.1103/PhysRevLett.133.240404]. We properly restore the quantum-classical correspondence through a dynamical approach based on the von Neumann entropy (VNE) and out-of-time-order correlators (OTOCs), which reveal signatures of chaos beyond spectral statistics. Focusing on the open anisotropic Dicke model, we identify two distinct regimes: transient chaos, marked by rapid early-time growth of VNE and OTOCs followed by low saturation values, and steady-state chaos, characterized by high long-time values. We introduce a random matrix toy model and show that Ginibre spectral statistics signals short-time chaos rather than steady-state chaos. Our results establish VNE dynamics and OTOCs as reliable diagnostics of dissipative quantum chaos across different timescales.

Information scrambling refers to the phenomenon in which local quantum information in a many-body system becomes dispersed throughout the entire system under unitary evolution. It has been extensively studied in closed quantum systems, where it is quantified by operator size growth, revealing deep connections between condensed matter physics, high-energy physics, and quantum information. However, when extending the study of operator size dynamics to open quantum systems, two different definitions of operator size distributions emerge. These definitions are based on different treatments of the bath. In this work, we aim to establish a unified picture for operator size dynamics in open quantum systems, using the solvable Brownian SYK models at generic system size. In particular, we provide the conditions under which the signature of scrambling transition — discovered using one particular definition — appears in operator size dynamics under the other definition. Additionally, we extend previous studies by exploring finite-size effects that are not captured by the scramblon theory. Our results provide a refined understanding of operator size dynamics in open quantum systems.

Relaxation and decoherence processes are key features of the dynamics of open quantum systems [1]. In the standard approach one tries to develop appropriate master equations for the open system's reduced density matrix ρS which is given by the partial trace taken over the environmental variables coupled to the open system. Invoking the weak-coupling assumption one can formulate in many cases of physical interest a Markovian quantum master equation for the reduced density matrix, expressing the dynamical laws for the irreversible motion of the open system.

This paper discusses several methods for describing the dynamics of open quantum systems, where the environment of the open system is infinite-dimensional. These are purifications, phase space forms, master equation and liouville equation forms. The main contribution is in using Feynman-Kac formalisms to describe the infinite-demsional components.

We present a general analysis of the role of initial correlations between the open system and an environment on quantum dynamics of the open system.

We present a renormalization group (RG) method which allows for an analytical study of the transient dynamics of open quantum systems on all time scales. Whereas oscillation frequencies and decay rates of exponential time evolution follow from the fixed point positions, the long-time behavior of pre-exponential functions is related to the scaling behavior around the fixed points. We show that certain terms of the RG flow are only cut off by inverse time, which leads to a difference between infrared and ultraviolet scaling. An evaluation for the ohmic spin boson model at weak damping reveals significant deviations from previous predictions in the long-time regime. We propose that weak coupling problems for stationary quantities can in principle turn into strong coupling ones for the determination of the long-time behavior.

We discuss some problems of dissipative chaos for open quantum systems in the\nframework of semiclassical and quantum distributions. For this goal, we propose\na driven nonlinear oscillator with time-dependent coefficients, i.e. with\ntime-dependent Kerr-nonlinearity and time-modulated driving field. This model\nshowing both regular and chaotic dynamics in the classical limit is realized in\nseveral experimental schemes. Quantum dissipative chaos is analyzed on the base\nof numerical method of quantum trajectories. Three quantities are studied: the\nWigner function of oscillatory mode from the point of view of quantum-assemble\ntheory and both semiclassical Poincare section and quantum Poincare section\ncalculated on a single quantum trajectory. The comparatively analysis of these\ndistributions for various operational chaotic regimes of the models is\nperformed, as well as scaling invariance in dissipative chaos and quantum\ninterference effects assisted by chaos are discussed.\n

The dynamical behavior of open quantum systems plays a key role in many applications of quantum mechanics, examples ranging from fundamental problems, such as the environment-induced decay of quantum coherence and relaxation in many-body systems, to applications in condensed matter theory, quantum transport, quantum chemistry and quantum information. In close analogy to a classical Markov process, the interaction of an open quantum system with a noisy environment is often modelled by a dynamical semigroup with a generator in Lindblad form, which describes a memoryless dynamics leading to an irreversible loss of characteristic quantum features. However, in many applications open systems exhibit pronounced memory effects and a revival of genuine quantum properties such as quantum coherence and correlations. Here, recent results on the rich non-Markovian quantum dynamics of open systems are discussed, paying particular attention to the rigorous mathematical definition, to the physical interpretation and classification, as well as to the quantification of memory effects. The general theory is illustrated by a series of examples. The analysis reveals that memory effects of the open system dynamics reflect characteristic features of the environment which opens a new perspective for applications, namely to exploit a small open system as a quantum probe signifying nontrivial features of the environment it is interacting with. This article further explores the various physical sources of non-Markovian quantum dynamics, such as structured spectral densities, nonlocal correlations between environmental degrees of freedom and correlations in the initial system-environment state, in addition to developing schemes for their local detection. Recent experiments on the detection, quantification and control of non-Markovian quantum dynamics are also discussed.

Non-Markovian open quantum systems represent the most general dynamics when the quantum system is coupled with a bath environment. The quantum dynamics arising from many important applications are non-Markovian. Although for special cases, such as Hamiltonian evolution and Lindblad evolution, quantum simulation algorithms have been extensively studied, efficient quantum simulations for the dynamics of non-Markovian open quantum systems remain underexplored. The most immediate obstacle for studying such systems is the lack of a universal succinct description of their dynamics. In this work, we fulfill the gap of studying such dynamics by (1) providing a succinct representation of the dynamics of non-Markovian open quantum systems with quantifiable error, and (2) developing an efficient quantum algorithm for simulating such dynamics with cost $${\mathcal {O}}(t\, \textrm{polylog}(t/\epsilon ))$$ for evolution time t and precision $$\epsilon $$ . Our derivation of the succinct representation is based on stochastic Schrödinger equations, which could lead to new alternatives to deal with open quantum systems as well.

The pseudomode model effectively captures the nonperturbative dynamics of open quantum systems with significantly reduced degrees of freedom. However, it is limited by the exponential growth of the Hilbert space dimension. To overcome the computational challenges, we propose a novel method that combines the multiple Davydov Ansatz with the Choi-Jamiolkowski isomorphism. Within this framework, the Lindblad equation is transformed into the non-Hermitian Schrödinger equation in a double Hilbert space, with its dynamics determined using the time-dependent variational principle. Three cases are calculated to demonstrate the effectiveness of the proposed method. We first discuss how the Davydov Ansatz works for the model with a single pseudomode. Extending the method to multiple pseudomodes, we show that the Ansatz effectively circumvents the exponential growth of the Hilbert space. Additionally, the method is also capable of addressing potential intersections that emerge in multibath scenarios. This approach offers potential applicability to various types of pseudomode models and other dissipative systems, providing a promising tool for the studies of open quantum dynamics.

A systematic approach to the non-Markovian quantum dynamics of open systems is given by the projection operator techniques of nonequilibrium statistical mechanics. Combining these methods with concepts from quantum information theory and from the theory of positive maps, we derive a class of correlated projection superoperators that take into account in an efficient way statistical correlations between the open system and its environment. The result is used to develop a generalization of the Lindblad theory to the regime of highly non-Markovian quantum processes in structured environments.

The dynamics of Markovian open quantum systems are described by Lindblad master equations. For fermionic and bosonic systems that are quasi-free, i.e. with Hamiltonians that are quadratic in the ladder operators and Lindblad operators that are linear in the ladder operators, we derive the equation of motion for the covariance matrix. This determines the evolution of Gaussian initial states and the steady states, which are also Gaussian. Using ladder super-operators (a.k.a. third quantization), we show how the Liouvillian can be transformed to a many-body Jordan normal form which also reveals the full many-body spectrum. Extending previous work by Prosen and Seligman, we treat fermionic and bosonic systems on equal footing with Majorana operators, shorten and complete some derivations, also address the odd-parity sector for fermions, give a criterion for the existence of bosonic steady states, cover non-diagonalizable Liouvillians also for bosons, and include quadratic systems. In extension of the quasi-free open systems, quadratic open systems comprise additional Hermitian Lindblad operators that are quadratic in the ladder operators. While Gaussian states may then evolve into non-Gaussian states, the Liouvillian can still be transformed to a useful block-triangular form, and the equations of motion for k-point Green’s functions form a closed hierarchy. Based on this formalism, results on criticality and dissipative phase transitions in such models are discussed in a companion paper.

This paper proposes the concept of subsystem controllability for both closed and open quantum dynamical systems, and the existence conditions of controllable subsystems are derived accordingly. It is demonstrated that a controllable subsystem may exist for a closed quantum dynamical system even if the overall system is not controllable. Furthermore, we also demonstrate that, for a class of decoherence-free subsystems of Markovian open quantum dynamical systems, the subsystems are controllable under open-loop coherent control if the initial states of the open systems are factorized. To a certain extent, this paper gives an alternative control theoretical interpretation on why decoherence-free subsystems are useful for quantum computation.

The repeated interaction model provides a framework for emulating and analyzing the dynamics of open quantum systems. We explore here the dynamics generated by this protocol in a system that is simultaneously coupled to two baths through noncommuting system operators. One bath is made to couple to nondiagonal elements of the system, thus it induces dissipative dynamics, while the other couples to diagonal elements, and by itself it generates pure dephasing. By solving the problem analytically exactly, we show that when both baths act concurrently, a strong systembath coupling gives rise to nonadditive effects in the dynamics. A prominent signature of this nonadditivity is the characteristic slowing down of population relaxation, driven by the influence of the dephasing bath. Beyond dynamics, we investigate the thermodynamic behavior of the model. Previous studies, using quantum master equations, showed that strong system-bath coupling created bath-cooperativity in this model, allowing heat exchange to the dephasing (diagonally coupled) bath. We find instead that, under the repeated interaction scheme, heat flows exclusively to the dissipative bath (coupled through nondiagonal elements). Our results highlight the need for a deeper understanding of the types of open quantum system dynamics and steady-state phenomena that emerge within the repeated interaction framework and the relation of this protocol to other common open quantum system techniques.