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.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/0207046Jul 9, 20020 citations

Stationary states of boundary-driven quantum systems: Some exact results

Eric A. Carlen, David A. Huse, Joel L. Lebowitz

We study finite-dimensional open quantum systems in contact with macroscopic equilibrium systems at their boundaries such that the system density matrix $\ensuremath{\rho}$ evolves via a Lindbladian, $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\rho}}=\ensuremath{-}i[H,\ensuremath{\rho}]+\mathcal{D}\ensuremath{\rho}$. Here $H$ is the Hamiltonian of the system and $\mathcal{D}$ is the dissipator. We consider the case where the system consists of two parts, the ``boundary'' $A$ and the ``bulk'' $B$ , and $\mathcal{D}$ acts only on $A$ , so $\mathcal{D}={\mathcal{D}}_{A}\ensuremath{\bigotimes}{\mathcal{I}}_{B}$, where ${\mathcal{I}}_{B}$ is the identity superoperator on part $B$ . Let ${\mathcal{D}}_{A}$ be ergodic, so ${\mathcal{D}}_{A}{\ensuremath{\pi}}_{A}=0$ only for one unique density matrix ${\ensuremath{\pi}}_{A}$. We show that any stationary density matrix $\overline{\ensuremath{\rho}}$ on the full system which commutes with $H$ must be of the product form $\overline{\ensuremath{\rho}}={\ensuremath{\pi}}_{A}\ensuremath{\bigotimes}{\ensuremath{\rho}}_{B}$ for some ${\ensuremath{\rho}}_{B}$. This rules out finding any ${\mathcal{D}}_{A}$ that has the Gibbs measure ${\ensuremath{\rho}}_{\ensuremath{\beta}}={e}^{\ensuremath{-}\ensuremath{\beta}H}/Z(\ensuremath{\beta})$ as a stationary state with $\ensuremath{\beta}\ensuremath{\ne}0$, unless there is no interaction between parts $A$ and $B$ . We give criteria for the uniqueness of the stationary state $\overline{\ensuremath{\rho}}$ for systems with interactions between $A$ and $B$ . Related results for nonergodic cases are also discussed.We study finite-dimensional open quantum systems in contact with macroscopic equilibrium systems at their boundaries such that the system density matrix $\ensuremath{\rho}$ evolves via a Lindbladian, $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\rho}}=\ensuremath{-}i[H,\ensuremath{\rho}]+\mathcal{D}\ensuremath{\rho}$. Here $H$ is the Hamiltonian of the system and $\mathcal{D}$ is the dissipator. We consider the case where the system consists of two parts, the ``boundary'' $A$ and the ``bulk'' $B$, and $\mathcal{D}$ acts only on $A$, so $\mathcal{D}={\mathcal{D}}_{A}\ensuremath{\bigotimes}{\mathcal{I}}_{B}$, where ${\mathcal{I}}_{B}$ is the identity superoperator on part $B$. Let ${\mathcal{D}}_{A}$ be ergodic, so ${\mathcal{D}}_{A}{\ensuremath{\pi}}_{A}=0$ only for one unique density matrix ${\ensuremath{\pi}}_{A}$. We show that any stationary density matrix $\overline{\ensuremath{\rho}}$ on the full system which commutes with $H$ must be of the product form $\overline{\ensuremath{\rho}}={\ensuremath{\pi}}_{A}\ensuremath{\bigotimes}{\ensuremath{\rho}}_{B}$ for some ${\ensuremath{\rho}}_{B}$. This rules out finding any ${\mathcal{D}}_{A}$ that has the Gibbs measure ${\ensuremath{\rho}}_{\ensuremath{\beta}}={e}^{\ensuremath{-}\ensuremath{\beta}H}/Z(\ensuremath{\beta})$ as a stationary state with $\ensuremath{\beta}\ensuremath{\ne}0$, unless there is no interaction between parts $A$ and $B$. We give criteria for the uniqueness of the stationary state $\overline{\ensuremath{\rho}}$ for systems with interactions between $A$ and $B$. Related results for nonergodic cases are also discussed.

arXiv:2408.06887Physical Review AJan 14, 20251 citation

Symmetry of Open Quantum Systems: Classification of Dissipative Quantum Chaos

Kohei Kawabata, Anish Kulkarni, Jiachen Li +2 more

We develop a theory of symmetry in open quantum systems.Using the operator-state mapping, we characterize symmetry of Liouvillian superoperators for the open quantum dynamics by symmetry of operators in the double Hilbert space and apply the 38-fold internal-symmetry classification of non-Hermitian operators.We find rich symmetry classification due to the interplay between symmetry in the corresponding closed quantum systems and symmetry inherent in the construction of the Liouvillian superoperators.As an illustrative example of open quantum bosonic systems, we study symmetry classes of dissipative quantum spin models.For open quantum fermionic systems, we develop the Z 4 classification of fermion parity symmetry and antiunitary symmetry in the double Hilbert space, which contrasts with the Z 8 classification in closed quantum systems.We also develop the symmetry classification of open quantum fermionic many-body systems-a dissipative generalization of the Sachdev-Ye-Kitaev (SYK) model described by the Lindblad master equation.We establish the periodic tables of the SYK Lindbladians and elucidate the difference from the SYK Hamiltonians.Furthermore, from extensive numerical calculations, we study its complex-spectral statistics and demonstrate dissipative quantum chaos enriched by symmetry.We develop a theory of symmetry in open quantum systems.Using the operator-state mapping, we characterize symmetry of Liouvillian superoperators for the open quantum dynamics by symmetry of operators in the double Hilbert space and apply the 38-fold internal-symmetry classification of non-Hermitian operators.We find rich symmetry classification due to the interplay between symmetry in the corresponding closed quantum systems and symmetry inherent in the construction of the Liouvillian superoperators.As an illustrative example of open quantum bosonic systems, we study symmetry classes of dissipative quantum spin models.For open quantum fermionic systems, we develop the Z 4 classification of fermion parity symmetry and antiunitary symmetry in the double Hilbert space, which contrasts with the Z 8 classification in closed quantum systems.We also develop the symmetry classification of open quantum fermionic many-body systems-a dissipative generalization of the Sachdev-Ye-Kitaev (SYK) model described by the Lindblad master equation.We establish the periodic tables of the SYK Lindbladians and elucidate the difference from the SYK Hamiltonians.Furthermore, from extensive numerical calculations, we study its complex-spectral statistics and demonstrate dissipative quantum chaos enriched by symmetry.

arXiv:2212.00605PRX QuantumAug 29, 202373 citations

Extended Hilbert Phase Space and Dissipative Quantum Systems

Tigran Aivazian

We suggest an extension of the Hilbert Phase Space formalism, which appears to be naturally suited for application to the dissipative (open) quantum systems, such as those described by the non-stationary (time-dependent) Hamiltonians $H(x,p,t)$ . A notion of quantum differential is introduced, highlighting the difference between the quantum and classical propagators. The equation of quantum dynamics of the generalised Wigner function in the extended Hilbert phase space is derived, as well as its classical limit, which serves as the generalisation of the classical Liouville equation for the domain of non-stationary Hamiltonian dynamical systems. This classical limit is then studied at some length and it is shown that in the extended phase space the energy plays the role of the coordinate and time that of the conjugated momentum and not the other way around, as might be suggested by the covariant treatment of these quantities arising from the 4-coordinate and 4-momentum in the relativistic context. Furthermore, the canonical form of the equation obtained suggests that that which is perceived as motion in the ordinary phase space is an equilibrium configuration in the extended phase space.We suggest an extension of the Hilbert Phase Space formalism, which appears to be naturally suited for application to the dissipative (open) quantum systems, such as those described by the non-stationary (time-dependent) Hamiltonians $H(x,p,t)$. A notion of quantum differential is introduced, highlighting the difference between the quantum and classical propagators. The equation of quantum dynamics of the generalised Wigner function in the extended Hilbert phase space is derived, as well as its classical limit, which serves as the generalisation of the classical Liouville equation for the domain of non-stationary Hamiltonian dynamical systems. This classical limit is then studied at some length and it is shown that in the extended phase space the energy plays the role of the coordinate and time that of the conjugated momentum and not the other way around, as might be suggested by the covariant treatment of these quantities arising from the 4-coordinate and 4-momentum in the relativistic context. Furthermore, the canonical form of the equation obtained suggests that that which is perceived as motion in the ordinary phase space is an equilibrium configuration in the extended phase space.

arXiv:1702.07746Feb 24, 20170 citations

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

Vladislav Yu. Shishkov, E. S. Andrianov, Alexander Pukhov +2 more

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.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.

arXiv:1803.04083Physical review. EAug 28, 201812 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.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:2108.08316Journal of Physics A Mathematical and TheoreticalApr 8, 202226 citations

H

is the Hamiltonian of the system and

\mathcal{D}

is the dissipator. We consider the case where the system consists of two parts, the ``boundary''

A

and the ``bulk''

B

, and

\mathcal{D}

acts only on

A

, so $\mathcal{D}={\mathcal{D}}_{A}\ensuremath{\bigotimes}{\mathcal{I}}_{B}$, where

{\mathcal{I}}_{B}

is the identity superoperator on part

B

. Let

{\mathcal{D}}_{A}

be ergodic, so ${\mathcal{D}}_{A}{\ensuremath{\pi}}_{A}=0$ only for one unique density matrix ${\ensuremath{\pi}}_{A}$. We show that any stationary density matrix $\overline{\ensuremath{\rho}}$ on the full system which commutes with

H

must be of the product form $\overline{\ensuremath{\rho}}={\ensuremath{\pi}}_{A}\ensuremath{\bigotimes}{\ensuremath{\rho}}_{B}$ for some ${\ensuremath{\rho}}_{B}$. This rules out finding any

{\mathcal{D}}_{A}

that has the Gibbs measure ${\ensuremath{\rho}}_{\ensuremath{\beta}}={e}^{\ensuremath{-}\ensuremath{\beta}H}/Z(\ensuremath{\beta})$ as a stationary state with $\ensuremath{\beta}\ensuremath{\ne}0$, unless there is no interaction between parts

A

and

B

. We give criteria for the uniqueness of the stationary state $\overline{\ensuremath{\rho}}$ for systems with interactions between

A

and

B

. Related results for nonergodic cases are also discussed.

H(x,p,t)

. A notion of quantum differential is introduced, highlighting the difference between the quantum and classical propagators. The equation of quantum dynamics of the generalised Wigner function in the extended Hilbert phase space is derived, as well as its classical limit, which serves as the generalisation of the classical Liouville equation for the domain of non-stationary Hamiltonian dynamical systems. This classical limit is then studied at some length and it is shown that in the extended phase space the energy plays the role of the coordinate and time that of the conjugated momentum and not the other way around, as might be suggested by the covariant treatment of these quantities arising from the 4-coordinate and 4-momentum in the relativistic context. Furthermore, the canonical form of the equation obtained suggests that that which is perceived as motion in the ordinary phase space is an equilibrium configuration in the extended phase space.