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arXiv:2310.17881

Signs of the rates in the Lindblad master equations can always be arbitrarily determined

Le Hu, Andrew N. Jordan

Published Oct 27, 2023• 0 citations
arXivHTMLPDF

Abstract

Determining the Markovianity and non-Markovianity of a quantum process is a critical problem in the theory of open quantum systems, as their behaviors differ significantly in terms of complexity. It is well recognized that a quantum process is Markovian if and only if the quantum master equation can be written in the standard Lindblad form with all rates nonnegative for all time. However, here we present a striking result that \textit{any} finite-dimensional open quantum system dynamics can be described by a quantum master equation in the Lindblad form with all rates nonnegative for all time. In fact, it can be shown that one can arbitrarily decide the sign of the rates in any case at any time interval. Note that here we take an unconventional approach where the quantum master equation we construct will in general be state-dependent, which means that the Hamiltonian, jump operators and rates will all depend on the current state of the density matrix $ρ(t)$. Our findings raise serious questions on the current criterion in determining Markovianity and non-Markovianity in open quantum system dynamics.

DOI
10.48550/arxiv.2310.17881

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Determining the Markovianity and non-Markovianity of a quantum process is a critical problem in the theory of open quantum systems, as their behaviors differ significantly in terms of complexity. It is well recognized that a quantum process is Markovian if and only if the quantum master equation can be written in the standard Lindblad form with all rates nonnegative for all time. However, here we present a striking result that \textit{any} finite-dimensional open quantum system dynamics can be described by a quantum master equation in the Lindblad form with all rates nonnegative for all time. In fact, it can be shown that one can arbitrarily decide the sign of the rates in any case at any time interval. Note that here we take an unconventional approach where the quantum master equation we construct will in general be state-dependent, which means that the Hamiltonian, jump operators and rates will all depend on the current state of the density matrix ρ(t)ρ(t)ρ(t). Our findings raise serious questions on the current criterion in determining Markovianity and non-Markovianity in open quantum system dynamics.
We show that all nonrelativistic quantum processes, whether open or closed, are either unitary or probabilistic unitary, i.e., probabilistic combinations of unitary evolutions. This means that, for open quantum systems, their continuous dynamics can always be described by the Lindblad master equation with all jump operators being unitary. We call this formalism the probabilistic unitary formulation of open quantum system dynamics. This formalism is shown to be exact under all cases, and does not rely on any assumptions other than the continuity and differentiability of the density matrix. Moreover, it requires as few as $d\ensuremath{-}1$ jump operators, instead of ${d}^{2}\ensuremath{-}1$, to describe the open dynamics in the most general case, where ddd is the dimension of Hilbert space of the system. Importantly, different from the conventional Lindblad master equation, this formalism is state dependent, meaning that the Hamiltonian, jump operators, and rates in general all depend on the current state of the density matrix. Hence one needs to know the explicit expression of the density matrix in order to write down the probabilistic unitary master equation explicitly. Experimentally, the formalism provides a scheme to control a quantum state to evolve along designed nonunitary quantum trajectories, and can be potentially useful in quantum computing and quantum control scenes since only unitary resources are needed for implementation.
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 system dynamics is efficiently described by the quantum master equation formalism. Therein, quantum master equations in Lindblad form constitute an important subclass describing Markovian dynamics. When an open quantum system is in an out-of-equilibrium state, an exchange of particles between the open system and reservoirs takes place yielding to a non-zero average net current and associated current fluctuations, which can be characterised with the quantum jump formalism for quantum master equations expressed in Lindblad form. However, a large class of quantum master equations cannot be described by Lindblad dynamics. Here we assess the validity and the effectiveness of the quantum jump formalism when the dissipators in the quantum master equation describe a non-additive, open quantum system dynamics. We find that an additive unravelling of the non-additive quantum master equation does not generate a completely-positive dynamics in the scenario of perfect jump detection. Nevertheless, allowing for an imperfect jump detection scenario, we find that an additive unravelling is possible that reproduces the current and the fluctuations obtained via the Landauer-Büttiker formalism.
Abstract If an open quantum system is initially uncorrelated from its environment, then its dynamics can be written in terms of a Lindblad-form master equation. The master equation is divided into a unitary piece, represented by an effective Hamiltonian, and a dissipative piece, represented by a hermiticity-preserving superoperator; however, the division of open system dynamics into unitary and dissipative pieces is non-unique. For finite-dimensional quantum systems, we resolve this non-uniqueness by specifying a norm on the space of dissipative superoperators and defining the canonical Hamiltonian to be the one whose dissipator is minimal. We show that the canonical Hamiltonian thus defined is equivalent to the Hamiltonian initially defined by Lindblad, and that it is uniquely specified by requiring the dissipator’s jump operators to be traceless, extending a uniqueness result known previously in the special case of Markovian master equations. For a system weakly coupled to its environment, we give a recursive formula for computing the canonical effective Hamiltonian to arbitrary orders in perturbation theory, which we can think of as a perturbative scheme for renormalizing the system’s bare Hamiltonian.
By exploiting the peculiarities of a recently introduced formalism for describing open quantum systems (the parametric representation with environmental coherent states) we derive an equation of motion for the reduced density operator of an open quantum system that has the same structure of the celebrated Gorini–Kossakowski–Sudarshan–Lindblad equation, but holds regardless of Markovianity being assumed. The operators in our result have explicit expressions in terms of the Hamiltonian describing the interactions with the environment, and can be computed once a specific model is considered. We find that, instead of a single set of Lindblad operators, in the general (non-Markovian) case there one set of Lindblad-like operators for each and every point of a symplectic manifold associated to the environment. This intricacy disappears under some assumptions (which are related to Markovianity and the classical limit of the environment), under which it is possible to recover the usual master-equation formalism. Finally, we find such Lindblad-like operators for two different models of a qubit in a bosonic environment, and show that in the classical limit of the environment their renown master equations are recovered.
We examine different nonlinear open quantum systems and calculate the non-Markovianity based on the distinguishability between two density matrices. We show that for a single spin (qubit) coupled to a bosonic field the non-Markovianity depends on the spectral density function and can take on any number N with 0 ≤ N ≤ 1, meaning the dynamics can be Markovian or highly non-Markovian. For the main result we consider a system of Ntot identical spins coupled to an environment in the mean-field way. Each spin is coupled to a local and the common reservoir. There are only indirect interactions between the spins through the common reservoir. In limit Ntot → 1 the subsystem consisting of a fixed set of n particles reduces to a factorized state in which each factor is a single spin evolving according to a nonlinear Hartree-Lindblad equation which is exactly solvable. For non-stationary initial spin states the non-Markovianity diverges. In this instance we can never approximate the dynamics of the quantum system by a Markovian master equation and this implies that in the mean field models, memory effects are significant.