This paper is devoted to the study of behavior of open quantum systems consistently based on the Franke-Gorini-Kossakowski-Lindblad-Sudarshan (FGKLS) equation which covers evolution in situations when decoherence can be distinguished. We focus on the quantum measurement operation which is determined by final stationary states of an open system - so called pointers. We find pointers by applying the FGKLS equation to asymptotically constant density matrix. In seeking pointers, we have been able to propose a perturbative scheme of calculation, if we take the interaction components with an environment to be weak. Thus, the Lindblad operators can be used in some way as expansion parameters for perturbation theory. The scheme we propose is different for the cases of non-degenerate and degenerate Hamiltonian. We illustrate our scheme by particular examples of quantum harmonic oscillator with spin in external magnetic field. The efficiency of the perturbation algorithm is demonstrated by its comparison with the exact solution.
We show that the basic equation of theory of open systems, Gorini-Kossakowski-Sudarshan-Lindblad equation, as well as its linear and nonlinear generalizations have a natural classical probabilistic interpretation - in the framework of prequantum classical statistical field theory. The latter gives an example of a classical probabilistic model (with random fields as subquantum variables) reproducing the basic probabilistic predictions of quantum mechanics.
Derivation of a quantum master equation for a system weakly coupled to a bath which takes into account nonsecular effects, but nevertheless has the mathematically correct Gorini-Kossakowski-Lindblad-Sudarshan form (in particular, it preserves positivity of the density operator) and also satisfies the standard thermodynamic properties is a known long-standing problem in theory of open quantum systems. The nonsecular terms are important when some energy levels of the system or their differences (Bohr frequencies) are nearly degenerate. We provide a fully rigorous derivation of such equation base…
Open quantum systems are, in general, described by a density matrix that is evolving under transformations belonging to a dynamical semigroup. They can obey the Franke-Gorini-Kossakowski-Lindblad-Sudarshan (FGKLS) equation. We exhaustively study the case of a Hilbert space of dimension $2$. First, we find final fixed states (called pointers) of an evolution of an open system, and we then obtain a general solution to the FGKLS equation and confirm that it converges to a pointer. After this, we check that the solution has physical meaning, i.e., it is Hermitian, positive and has trace equal to $…
We investigate departures from the Hamiltonian dynamics such as the quantum decoherence in a supersymmetric quantum system due to its interaction with the environment. The effective description is given by the Franke-Gorini-Kossakowski-Lindblad-Sudarshan (FGKLS) equation. The prescription for supersymmetrization of this equation is elaborated and pairs of related isospectral quantum systems are built. We present some instructive examples of such a construction.
The aims of this paper are two. The first is to give a brief review of the most relevant theoretical results concerning the uniqueness of the steady-state solution of the Lindblad-Gorini-Kossakowski-Sudarshan master equation and the criteria which guarantee relaxingness and irreducibility of dynamical semigroups. In particular, we test and discuss their physical meaning by considering their applicability to the characterisation of the simplest open quantum system \emph{i.e.} a two-level system coupled to a bath of harmonic oscillators at zero temperature. The second aim is to provide a set of …
Gorini-Kossakowski-Sudarshan-Lindblad equation in different bases: application to driven-dissipative two- and multilevel systems
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine（乌克兰国家科学院B.维尔金低温物理与工程研究所）；Center for Quantum Computing, RIKEN（理化学院量子计算中心）；V. N. Karazin Kharkiv National University（V.N.卡拉津哈尔科夫国立大学）；Department of Mathematics, Kyiv School of Economics（基辅经济学院数学系）
We investigate a total thermodynamic entropy production rate of an isolated quantum system. In particular, we consider a quantum model of coupled harmonic oscillators in a star configuration, where a central harmonic oscillator (system) is coupled to a finite number of surrounding harmonic oscillators (bath). In this model, when the initial state of the total system is given by the tensor product of the Gibbs states of the system and the bath, every harmonic oscillator is always in a Gibbs state with a time-dependent temperature. This enables us to define time-dependent thermodynamic entropy f…
We study the non-linear quantum master equation describing a laser under the mean field approximation. The quantum system is formed by a single mode optical cavity and two level atoms, which interact with reservoirs. Namely, we establish the existence and uniqueness of the regular solution to the non-linear operator equation under consideration, as well as we get a probabilistic representation for this solution in terms of a mean field stochastic Schröndiger equation. To this end, we find a regular solution for the non-autonomous linear quantum master equation in Gorini-Kossakowski-Sudarshan-L…
We study the perturbative corrections to the { Gorini-Kossakowski-Sudarshan-Lindblad equation which arises in the weak coupling limit}. The spin-boson model in the rotating wave approximation at zero temperature is considered. We show that the perturbative part of the density matrix satisfies the time-independent Gorini-Kossakowski-Sudarshan-Lindblad equation for arbitrary order of the perturbation theory (if all the moments of the reservoir correlation function are finite). But to reproduce the right asymptotic precision at long times, one should use { an initial condition different} from the…
In this paper we propose a dynamical approach based on the Gorini-Kossakowski-Sudarshan-Lindblad equation for a problem of decision making. More specifically, we consider what was recently called a quantum parliament, asked to approve or not a certain law, and we propose a model of the connections between the various members of the parliament, proposing in particular some special form of the interactions giving rise to a {\em collaborative} or non collaborative behaviour.
Fifty Years of the GKLS Master Equation: Foundations and Early Developments
University of Trieste（的里雅斯特大学）；Istituto Nazionale di Fisica Nucleare, Sezione di Trieste（意大利国家核物理研究所的里雅斯特分部）；Institute of Physics, Nicolaus Copernicus University（尼古拉·哥白尼大学物理研究所）；Department of Physics, University of Bari（巴里大学物理系）；Istituto Nazionale di Fisica Nucleare, Sezione di Bari（意大利国家核物理研究所巴里分部）
Gorini-Kossakowski-Sudarshan-Lindblad equation of Poisson-type for the density matrix is considered. The Poisson jumps are assumed to be unitary operators with generators, which are quadratic in fermionic creation and annihilation operators. The explicit dynamics of the density matrix moments and Markovian multi-time ordered correlation functions is obtained.
We analyze Lindblad-Gorini-Kossakowski-Sudarshan-type generators for selected periodically driven open quantum systems. All these generators can be obtained by temporal coarse-graining procedures, and we compare different coarse-graining schemes. Similar as for undriven systems, we find that a dynamically adapted coarse-graining time, effectively yielding non-Markovian dynamics by interpolating through a series of different but invididually Markovian solutions, gives the best results among the different coarse-graining schemes, albeit at highest computational cost.
A finite-dimensional quantum system is coupled to a bath of oscillators in thermal equilibrium at temperature $T>0$. We show that for fixed, small values of the coupling constant $λ$, the true reduced dynamics of the system is approximated by the completely positive, trace preserving Markovian semigroup generated by the Davies-Lindblad generator. The difference between the true and the Markovian dynamics is $O(|λ|^{1/4})$ for all times, meaning that the solution of the Gorini-Kossakowski-Sudarshan-Lindblad master equation is approximating the true dynamics to accuracy $O(|λ|^{1/4})$ for all ti…
The theory of open quantum system is one of the most essential tools for the development of quantum technologies. Furthermore, the Lindblad (or Gorini-Kossakowski-Sudarshan-Lindblad) Master Equation plays a key role as it is the most general generator of Markovian dynamics in quantum systems. In this paper, we present this equation together with its derivation and methods of resolution. The presentation tries to be as self-contained and straightforward as possible to be useful to readers with no previous knowledge of this field.
The pseudomode approach is discussed with an emphasis to Gorini-Kossakowski-Sudarshan-Lindblad form of this approach. The connection of the pseudomode approach with solutions of both the Friedrichs model and Jaynes-Cummings model with dissipation at zero temperature is shown. The obtained results are applied to non-Markovian phenomena description in the Fenna-Matthews-Olson complexes, estimations based on experimental data are done. A deformation approach to generalization of the pseudomode approach to the finite-temperature case is discussed.
Both quantum information features and irreversible quantum evolution of the models arising in physical systems in one-particle approximation are discussed. It is shown that the calculation of the reduced density matrix and entanglement analysis are considerably simplified in this case. The irreversible quantum evolution described by Gorini--Kossakowski--Sudarshan--Lindblad equations in the one-particle approximation could be defined by a solution of a Shroedinger equation with a dissipative generator. It simplifies the solution of the initial equation on the one side and gives a physical inter…
A longstanding tool to characterize the evolution of open Markovian quantum systems is the GKSL (Gorini-Kossakowski-Sudarshan-Lindblad) master equation. However, in some cases, open quantum systems can be effectively described with non-Hermitian Hamiltonians, which have attracted great interest in the last twenty years due to a number of unconventional properties, such as the appearance of exceptional points. Here, we present a short review of these two different approaches aiming in particular to highlight their relation and illustrate different ways of connecting non-Hermitian Hamiltonian to…
We study the coarse-graining approach to derive a generator for the evolution of an open quantum system over a finite time interval. The approach does not require a secular approximation but nevertheless generally leads to a Lindblad-Gorini-Kossakowski-Sudarshan generator. By combining the formalism with Full Counting Statistics, we can demonstrate a consistent thermodynamic framework, once the switching work required for the coupling and decoupling with the reservoir is included. Particularly, we can write the second law in standard form, with the only difference that heat currents must be de…
Quantum systems interacting with an unknown environment are notoriously difficult to model, especially in presence of non-Markovian and non-perturbative effects. Here we introduce a neural network based approach, which has the mathematical simplicity of the Gorini-Kossakowski-Sudarshan-Lindblad master equation, but is able to model non-Markovian effects in different regimes. This is achieved by using recurrent neural networks for defining Lindblad operators that can keep track of memory effects. Building upon this framework, we also introduce a neural network architecture that is able to repro…
Starting from the quantum stochastic differential equations of Hudson and Parthasarathy (Comm. Math. Phys. 93, 301 (1984)) and exploiting the Wiener-Ito-Segal isomorphism between the Boson Fock reservoir space $Γ(L^2(\mathbb{R}_+)\otimes (\mathbb{C}^{n}\oplus \mathbb{C}^{n}))$ and the Hilbert space $L^2(μ)$, where $μ$ is the Wiener probability measure of a complex $n$-dimensional vector-valued standard Brownian motion $\{\mathbf{B}(t), t\geq 0\}$, we derive a non-linear stochastic Schrodinger equation describing a classical diffusion of states of a quantum system, driven by the Brownian motion…
We discuss a model of a closed quantum evolution of two-qubits where the joint Hamiltonian is so chosen that one of the qubits acts as a bath and thermalize the other qubit which is acting as the system. The corresponding exact master equation for the system is derived. Interestingly, for a specific choice of parameters the master equation takes the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) form with constant coefficients, representing pumping and damping of a single qubit system. Based on this model we construct an Otto cycle connected to a single qubit bath and study its thermodynamic pro…
Master equations under appropriate assumptions are efficient tools for the study of open quantum systems. For many-body systems, subsystems of which locally couple to thermal baths and weakly interact with each other, the local approach provides a more convenient description than the global approach. However, these local master equations are believed to generate inconsistencies with the laws of thermodynamics when intersubsystem interactions exist. Here we develop an alternative local master equation by virtue of similar approximations used in deriving the traditional Gorini-Kossakowski-Lindbl…
The mathematically sound theory of quantum open systems, formulated in the 70-ties of XX century and highlighted by the discovery of Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) equation, found a wide range of applications in various branches of physics and chemistry, notably in the field of quantum information and quantum thermodynamics. However, it took 40 years before this formalism has been applied to explain correctly the operation principles of long existing energy transducers like photovoltaic, thermoelectric and fuel cells. This long path is briefly reviewed from the personal author's …
In the summer of 2016, physicists gathered in Torun, Poland for the 48th annual Symposium on Mathematical Physics. This Symposium was special; it celebrated the 40th anniversary of the discovery of the Gorini-Kossakowski-Sudarshan-Lindblad master equation, which is widely used in quantum physics and quantum chemistry. This article forms part of a Special Volume of the journal Open Systems & Information Dynamics arising from that conference; and it aims to celebrate a related discovery -- also by Sudarshan -- that of Quantum Maps (which had their 55th anniversary in the same year). Nowadays, mu…
We treat several key stochastic equations for non-Markovian open quantum system dynamics and present a formalism for finding solutions to them via canonical perturbation theory, without making the Born-Markov or rotating wave approximations (RWA). This includes master equations of the (asymptotically) stationary, periodic, and time-nonlocal type. We provide proofs on the validity and meaningfulness of the late-time perturbative master equation and on the preservation of complete positivity despite a general lack of Lindblad form. More specifically, we show how the algebraic generators satisfy …
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 f…
A Fractional Calculus Framework for Open Quantum Dynamics: From Liouville to Lindblad to Memory Kernels
Physical and Computational Sciences Directorate, Pacific Northwest National Laboratory（太平洋西北国家实验室物理与计算科学部）；Theoretical Division, Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室理论部）
The theoretical description of quantum dynamics in an intriguing way does not necessarily imply the underlying dynamics is indeed intriguing. Here we show how a known very interesting master equation with an always negative decay rate [eternal non-Markovianity (ENM)] arises from simple stochastic Schrödinger dynamics (random unitary dynamics). Equivalently, it may be seen as arising from a mixture of Markov (semi-group) open system dynamics. Both these approaches lead to a more general family of CPT maps, characterized by a point within a parameter triangle. Our results show how ENM quantum dy…
We introduce a response theory for open quantum systems within nonequilibrium steady-states subject to a Hamiltonian perturbation. Working in the weak system-bath coupling regime, our results are derived within the Lindblad-Gorini-Kossakowski-Sudarshan formalism. We find that the response of the system to a small perturbation is not simply related to a correlation function within the system, unlike traditional linear response theory in closed systems or expectations from the fluctuation-dissipation theorem. In limiting cases, when the perturbation is small relative to the coupling to the surro…
Theory of Steady States for Lindblad Equations beyond Time-Independence: Classification, Uniqueness and Symmetry
The University of Tokyo（东京大学）；RIKEN Pioneering Research Institute (PRI)（理化学研究所先导研究本部）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理化学研究所交叉理论数学科学中心）
We study open quantum systems whose evolution is governed by a master equation of Kossakowski-Gorini-Sudarshan-Lindblad type and give a characterization of the convex set of steady states of such systems based on the generalized Bloch representation. It is shown that an isolated steady state of the Bloch equation cannot be a center, i.e., that the existence of a unique steady state implies attractivity and global asymptotic stability. Necessary and sufficient conditions for the existence of a unique steady state are derived and applied to different physical models including two- and four-level…
Asymptotic relaxation in quantum Markovian dynamics
Universität Ulm（乌尔姆大学）；Center for Integrated Quantum Science and Technology (IQST), Ulm University（乌尔姆大学集成量子科学与技术中心）；Nicolaus Copernicus University（尼古拉·哥白尼大学）
