© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:1803.04083 · Physical review. E

Zeroth law of thermodynamics for thermalized open quantum systems having constants of motion

Vladislav Yu. Shishkov, E. S. Andrianov, Alexander Pukhov, Alexey P. Vinogradov, Alexander A. Lisyansky

Published Aug 28, 2018• 12 citations

Abstract

We study the evolution of an open quantum system described by a dynamical semigroup having the Lindblad superoperator as a generator. This generator may have an eigenfunction with a zero eigenvalue referred to as a constant of motion (COM). An open quantum system has a unique stationary state if and only if it has no COMs. A system with multiple stationary states has a basis of COMs, any COM of the system is a linear combination of the basis COMs. The basis divides the space of system states into subspaces. In each subspace, its own stationary state is formed, and any stationary state of the system is a linear combination of these states. Usually, neither the basis of COMs nor even their number is known. We demonstrate that finding the stationary state of the system does not require looking for COMs. Instead, one can construct a set of "invariant" subspaces. If the system evolution begins from one of these subspaces, the system will remain in it, arriving at a stationary state independent of evolution in other subspaces. We suggest a direct way of finding the invariant subspaces by studying the evolution of the system. We show that the sets of invariant subspaces and subspaces generated by the basis of COMs are equivalent. A stationary state of the system is a weighted sum of stationary states in each invariant subspace; weighting factors are determined by the initial state of the system.

DOI: 10.1103/physreve.98.022132

Related papers

Analysis of quantum semigroups with GKS–Lindblad generators: II. General

Bernhard Baumgartner, Heide Narnhofer

Semigroups describing the time evolution of open quantum systems in finite-dimensional spaces have generators of a special form, known as Lindblad generators. These generators and the corresponding processes of time evolution are analyzed and characterized as decay, dissipation and dephasing. In relation to these processes the Hilbert space of the system is equipped with a special structure, a decomposition into a sum of mutually orthogonal subspaces. The complete set of all the stationary states and the asymptotic behavior of the evolutions are presented in detail. Some unusual special facts about invariant operators and symmetries are studied and examples are demonstrated. Perturbation theory for the structure and for the stationary states is discussed and performed in case studies.

arXiv:0806.3164•Journal of Physics A Mathematical and Theoretical•Sep 2, 2008•137 citations

Analysis of quantum semigroups with GKS–Lindblad generators: I. Simple generators

Bernhard Baumgartner, Heide Narnhofer, Walter E Thirring

Semigroups describing the time evolution of open quantum systems in finite-dimensional spaces have generators of a special form, known as Lindblad generators. The simple generators, characterized by only one operator, are analyzed. The complete set of all the stationary states is presented in detail, including a formula to calculate a stationary state from the generating operator. Also the opposite task can be fulfilled, to construct an evolution leading to a prescribed stationary state.

arXiv:0710.5385•Journal of Physics A Mathematical and Theoretical•Jan 29, 2008•60 citations

THE STRUCTURES OF STATE SPACE CONCERNING QUANTUM DYNAMICAL SEMIGROUPS

BERNHARD BAUMGARTNER, HEIDE NARNHOFER

Each semigroup describing time evolution of an open quantum system on a finite dimensional Hilbert space is related to a special structure of this space. It is shown how the space can be decomposed into orthogonal subspaces: One part is related to decay, some subspaces of the other subspace are ranges of the stationary states. Specialities are highlighted where the complete positivity of evolutions is actually needed for analysis, mainly for evolution of coherence. Decompositions are done the same way for discrete as for continuous time evolutions, but they may show differences: Only for discrete semigroups there may appear cases of sudden decay and of perpetual oscillation. Concluding the analysis, we identify the relation of the state space structure to the processes of decay, decoherence, dissipation and dephasing.

arXiv:1101.3914•Reviews in Mathematical Physics•Feb 1, 2012•74 citations

Pure stationary states of open quantum systems

Vasily Evgenevich Tarasov

Using Liouville space and superoperator formalism we consider pure stationary states of open and dissipative quantum systems. We discuss stationary states of open quantum systems, which coincide with stationary states of closed quantum systems. Open quantum systems with pure stationary states of linear oscillator are suggested. We consider stationary states for the Lindblad equation. We discuss bifurcations of pure stationary states for open quantum systems which are quantum analogs of classical dynamical bifurcations.

arXiv:quant-ph/0311177•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Nov 19, 2002•45 citations

A canonical Hamiltonian for open quantum systems

Patrick M. Hayden, Jonathan Sorce

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.

arXiv:2108.08316•Journal of Physics A Mathematical and Theoretical•Apr 8, 2022•26 citations

Dynamical invariants of open quantum systems

S. L. Wu, X. Y. Zhang, X. X. Yi

For a closed quantum system, a dynamical invariant is defined as an operator whose expectation value is a constant. In this paper, we extend the concept of dynamical invariants from closed systems to open systems. A dynamical equation for invariants (the dynamical invariant condition) is derived for Markovian dynamics. Different from dynamical invariants of closed quantum systems, the time evolution of dynamical invariants of open quantum systems is no longer unitary, and eigenvalues of any invariant are time dependent in general. Since any Hermitian operator which can fulfill the dynamical invariant condition is a dynamical invariant, we construct a type of special dynamical invariant of which a part of the eigenvalues is still constant. The dynamical invariants in the subspace spanned by these eigenstates thus evolve unitarily.

arXiv:1510.01038•Physical Review A•Dec 15, 2015•12 citations

We study the evolution of an open quantum system described by a dynamical semigroup having the Lindblad superoperator as a generator. This generator may have an eigenfunction with a zero eigenvalue referred to as a constant of motion (COM). An open quantum system has a unique stationary state if and only if it has no COMs. A system with multiple stationary states has a basis of COMs, any COM of the system is a linear combination of the basis COMs. The basis divides the space of system states into subspaces. In each subspace, its own stationary state is formed, and any stationary state of the system is a linear combination of these states. Usually, neither the basis of COMs nor even their number is known. We demonstrate that finding the stationary state of the system does not require looking for COMs. Instead, one can construct a set of "invariant" subspaces. If the system evolution begins from one of these subspaces, the system will remain in it, arriving at a stationary state independent of evolution in other subspaces. We suggest a direct way of finding the invariant subspaces by studying the evolution of the system. We show that the sets of invariant subspaces and subspaces generated by the basis of COMs are equivalent. A stationary state of the system is a weighted sum of stationary states in each invariant subspace; weighting factors are determined by the initial state of the system.

Semigroups describing the time evolution of open quantum systems in finite-dimensional spaces have generators of a special form, known as Lindblad generators. These generators and the corresponding processes of time evolution are analyzed and characterized as decay, dissipation and dephasing. In relation to these processes the Hilbert space of the system is equipped with a special structure, a decomposition into a sum of mutually orthogonal subspaces. The complete set of all the stationary states and the asymptotic behavior of the evolutions are presented in detail. Some unusual special facts about invariant operators and symmetries are studied and examples are demonstrated. Perturbation theory for the structure and for the stationary states is discussed and performed in case studies.

Each semigroup describing time evolution of an open quantum system on a finite dimensional Hilbert space is related to a special structure of this space. It is shown how the space can be decomposed into orthogonal subspaces: One part is related to decay, some subspaces of the other subspace are ranges of the stationary states. Specialities are highlighted where the complete positivity of evolutions is actually needed for analysis, mainly for evolution of coherence. Decompositions are done the same way for discrete as for continuous time evolutions, but they may show differences: Only for discrete semigroups there may appear cases of sudden decay and of perpetual oscillation. Concluding the analysis, we identify the relation of the state space structure to the processes of decay, decoherence, dissipation and dephasing.

Using Liouville space and superoperator formalism we consider pure stationary states of open and dissipative quantum systems. We discuss stationary states of open quantum systems, which coincide with stationary states of closed quantum systems. Open quantum systems with pure stationary states of linear oscillator are suggested. We consider stationary states for the Lindblad equation. We discuss bifurcations of pure stationary states for open quantum systems which are quantum analogs of classical dynamical bifurcations.

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.

For a closed quantum system, a dynamical invariant is defined as an operator whose expectation value is a constant. In this paper, we extend the concept of dynamical invariants from closed systems to open systems. A dynamical equation for invariants (the dynamical invariant condition) is derived for Markovian dynamics. Different from dynamical invariants of closed quantum systems, the time evolution of dynamical invariants of open quantum systems is no longer unitary, and eigenvalues of any invariant are time dependent in general. Since any Hermitian operator which can fulfill the dynamical invariant condition is a dynamical invariant, we construct a type of special dynamical invariant of which a part of the eigenvalues is still constant. The dynamical invariants in the subspace spanned by these eigenstates thus evolve unitarily.