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arXiv:2502.19137 · SciPost Physics

A perturbation theory for multi-time correlation functions in open quantum systems

Piotr Szańkowski

Published Sep 3, 2025• 1 citation
arXivHTMLPDF

Abstract

Dynamical maps are the principal subject of the open system theory. Formally, the dynamical map of a given open quantum system is a density matrix transformation that takes any initial state and sends it to the state at a later time. Physically, it encapsulates the system’s evolution due to coupling with its environment. Hence, the theory provides a flexible and accurate framework for computing expectation values of open system observables. However, expectation values-or more generally, single-time correlation functions-capture only the simplest aspects of a quantum system’s dynamics. A complete characterization of the dynamics requires access to multi-time correlation functions as well: phenomena like detailed balance, linear and non-linear response, non-equilibrium transport in general, or even sequential measurements of system observables are all described in terms of multi-time correlations. For closed systems, such correlations are well-defined, even though knowledge of the system’s state alone is insufficient to determine them fully. In contrast, the standard dynamical map formalism for open systems does not account for multi-time correlations, as it is fundamentally limited to describing state evolution. Here, we extend the scope of open quantum system theory by developing a systematic perturbation theory for computing multi-time correlation functions.

DOI
10.21468/scipostphys.19.3.066

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Dynamical maps are the principal subject of the open system theory. Formally, the dynamical map of a given open quantum system is a density matrix transformation that takes any initial state and sends it to the state at a later time. Physically, it encapsulates the system’s evolution due to coupling with its environment. Hence, the theory provides a flexible and accurate framework for computing expectation values of open system observables. However, expectation values-or more generally, single-time correlation functions-capture only the simplest aspects of a quantum system’s dynamics. A complete characterization of the dynamics requires access to multi-time correlation functions as well: phenomena like detailed balance, linear and non-linear response, non-equilibrium transport in general, or even sequential measurements of system observables are all described in terms of multi-time correlations. For closed systems, such correlations are well-defined, even though knowledge of the system’s state alone is insufficient to determine them fully. In contrast, the standard dynamical map formalism for open systems does not account for multi-time correlations, as it is fundamentally limited to describing state evolution. Here, we extend the scope of open quantum system theory by developing a systematic perturbation theory for computing multi-time correlation functions.
Standard theoretical descriptions of the dynamics of open quantum systems rely on the assumption that the correlations with the environment can be neglected at some reference (initial) time. While being reasonable in specific instances, such as when the coupling between the system and the environment is weak or when the interaction starts at a distinguished time, the use of initially uncorrelated states is questionable if one wants to deal with general models, taking into account the mutual influence that the open-system and environmental evolutions perform on each other. Here we introduce a perturbative method that can be applied to any microscopic modeling of the system-environment interaction, including fully general initial correlations. Extending the standard technique based on projection operators that single out the relevant part of the global dynamics, we define a family of projections adapted to a convenient decomposition of the initial state, which involves a convex mixture of product operators with proper environmental states. This leads us to characterize the open-system dynamics via an uncoupled system of differential equations, which are homogeneous and whose number is limited by the dimensionality of the open system, for any kind of initial correlations. Our method is further illustrated by means of two case studies, for which it reproduces the expected dynamical behavior in the long-time regime more consistently than the standard projection technique.
We show that the dynamics of any open quantum system that is initially correlated with its environment can be described by a set of d2{d}^{2}d2 (or less) completely positive maps, where ddd is the dimension of the system. Only one such map is required for the special case of no initial correlations. The same maps describe the dynamics of any system-environment state obtained from the initial state by a local operation on the system. The reduction of the system dynamics to a set of completely positive maps allows known numerical and analytic tools for uncorrelated initial states to be applied to the general case of initially correlated states, which we exemplify by solving the qubit dephasing model for such states, and provides a natural approach to quantum Markovianity for this case. We show that this set of completely positive maps can be experimentally characterized using only local operations on the system, via a generalization of noise spectroscopy protocols. As further applications, we first consider the problem of retrodicting the dynamics of an open quantum system which is in an arbitrary state when it becomes accessible to the experimenter and explore the conditions under which retrodiction is possible. We also introduce a related one-sided or limited-access tomography protocol for determining an arbitrary bipartite state, evolving under a sufficiently rich Hamiltonian, via local operations and measurements on just one component. We simulate this protocol for a physical model of particular relevance to nitrogen-vacancy centers and in particular show how to reconstruct the density matrix of a set of three qubits, interacting via dipolar coupling and in the presence of local magnetic fields, by measuring and controlling only one of them.
Abstract We investigate the dynamics of open quantum systems which are initially correlated with their environment. The strategy of our approach is to analyze how given, fixed initial correlations modify the evolution of the open system with respect to the corresponding uncorrelated dynamical behavior with the same fixed initial environmental state, described by a completely positive dynamical map. We show that, for any predetermined initial correlations, one can introduce a linear dynamical map on the space of operators of the open system which acts like the proper dynamical map on the set of physical states and represents its unique linear extension. Furthermore, we demonstrate that this construction leads to a linear, time-local quantum master equation with generalized Lindblad structure involving time-dependent, possibly negative transition rates. Thus, the general non-Markovian dynamics of an open quantum system can be described by means of a time-local master equation even in the case of arbitrary, fixed initial system–environment correlations. We present some illustrative examples and explain the relation of our approach to several other approaches proposed in the literature.
Open quantum systems have drawn attention over decades due to its applicability in the foundation of theoretical physics, e.g. statistical mechanics, quantum mechanics and condensed matter physics. The dynamics of open quantum systems has been described as separate entities from their surrounding environment that consists of a very large number of modes, somehow coupled to the mode of the system. Even though the exact solution of the dynamical behavior of the system is impossible to calculate, we obtain a tentative solution using the crucial Markov approximation. The input-output formalism of the quantum Langevin equation (QLE) has been considered as a useful tool which provides a semi-classical description of the dynamics of the system, whereas the master equation provides a complete picture of the dynamics of the system expressed in terms of density matrix. While studying the dynamics of nonlinear system/environment coupling using QLE, we see, for a small value of external field, that the steady state system field does not change much from the steady state field obtained in the absence of nonlinear dissipation. However, in a case where a stronger external field is applied, we see that the deviation becomes substantial from the solution of linear system. We also see that the nonlinear coupling introduces significant difference in the cavity fluctuation spectrum. The description, therefore, provides a potential explanation of parametric effects in terms of nonlinear dissipation phenomena associated with the nonlinear coupling. Even though the theories developed in the context of open quantum systems have proven to be powerful tools, they do not provide a satisfactory platform to be implemented on non-linear Hamiltonians. We often approximate it by linearizing over nonlinear steady state field amplitude, and therefore, the interesting effects are often overlooked. The limitation of the analytics provokes us to simulate open quantum dynamics numerically. The numerical method consists of transformation of the environmental degrees of freedom to a one-dimensional many-body chain, and the computational technique includes numerical diagonalization and renormalization process. The time-adaptive density matrix renormalisation group (t-DMRG) is known as one of the most powerful techniques for the simulation of strongly-correlated many-body quantum systems. In this thesis, along with the theoretical modeling, we implement DMRG numerical scheme for the simulation of canonical S/B model by mapping it to one-dimensional harmonic chain with nearest neighbor interactions, and use the method to investigate the dynamics of the free dissipative system. The thermalization of open quantum systems is also studied by generating minimally entangled typical thermal states (METTS) through imaginary time evolution, and real-time evolving an empty system in the presence of the thermal bath. Further, we simulate coherently driven free dissipative Kerr nonlinear system numerically using Euler's method by solving Heisenberg equation of motion and t-DMRG algorithm, and demonstrate how the numerical results are analogous to classical bistability. By comparing with analytics, we see that the DMRG numerics is analogous to the quantum-mechanical exact solution obtained by mapping the equation of motion of the density matrix of the system to a Fokker-Plank equation. The comparison between two different numerical techniques shows that the semi-classical Euler's method determines the dynamics of the system field of one among two coherent branches, whereas DMRG numerics gives the superposition of both of them. Hence, DMRG-determined time dynamics undergoes generating non-classical states. Our approach of dealing with nonlinearity represents an important contribution in the developments of technique to study the dynamical and steady-state behavior of open quantum systems, which is a fundamental aspect of quantum physics.
Remarkable experimental progress and theoretical advancements in the study of strongly corre- lated systems have opened up new doors towards a better understanding of interesting states of matter. Such states could possibly be used to develop novel technological applications based on the principles of quantum mechanics. In recent years, the investigation of both complex solid state and ultracold atom systems has lead to a number of fascinating discoveries with regards to their non-equilibrium properties. One route to probe non-equilibrium physics is to consider the evolution of open systems defined by the coupling to an external environment introducing dissipation. Engineering the environment according to predefined requirements can then be used to dynamically prepare states which are difficult to access otherwise. This thesis explores the dynamics of many-body open quantum systems aiming at under- standing the effects resulting from the interplay of kinetic motion, interactions and dissipation in different physical settings. First, we build upon recent developments concerning the coupling of ultracold atoms to optical resonators which are subject to photon losses. We show how the dissipation can trigger an attractor dynamics towards a topologically non-trivial steady state that is robust against perturbations. The numerically obtained full open system evolution en- ables us to not only characterize in detail the steady state but also to analyze the features of the approaching dynamics. Second, we study the propagation of two-time correlation functions in interacting quantum spin-1/2 chains in contact with an environment causing dephasing. Pro- vided with the numerically exact full time-evolution, we identify an algebraic scaling regime where time-translation invariance breaks down and uncover the emergence of aging dynam- ics. Finally, we study a system of interacting fermionic atoms in a one-dimensional lattice with particle loss at the central site. There we report on the dissipation-induced deceleration of the non-equilibrium dynamics known as the quantum Zeno effect. As in the majority of cases quantum many-body systems are not solvable analytically, another central aspect of this thesis deals with adapting numerical methods based on matrix product state algorithms to investigate open quantum systems dynamics.