We analyze the symmetries of open dissipative models for quantum Liouvillian dynamical systems. Our analysis yields two representation independent symmetry relations. One of the symmetry relations, star-conjugation symmetry, is shown to represent a necessary and sufficient condition for ensuring the reality of observables and the Hermiticity of the system density operator. The other symmetry relation represents a necessary and sufficient condition for the realization of microscopic reversibility. These symmetry relations and the structure of the global dynamics are shown to give rise to a number of generalized symmetry relations for quantities expressible as matrix elements of the retarded, advanced, and global propagators. It is shown that the generalized symmetry relations may be used to facilitate the solution of spectral and temporal problems. This is demonstrated in a concrete way in an application entailing the use of dual Lanczos transformation theory to obtain an exact solution of a prototype model for excited state dynamics. Apart from elucidating the symmetries of open dissipative systems, the reported analysis reveals that there are two different and equally legitimate schemes for performing time-reversal transformations in the matrix formulation of quantum Liouvillian dynamics. The properties and ramifications of these time-reversal transformation schemes are discussed in detail.
We analyze the symmetries of open dissipative models for quantum Liouvillian dynamical systems. Our analysis yields two representation independent symmetry relations. One of the symmetry relations, star-conjugation symmetry, is shown to represent a necessary and sufficient condition for ensuring the reality of observables and the Hermiticity of the system density operator. The other symmetry relation represents a necessary and sufficient condition for the realization of microscopic reversibility. These symmetry relations and the structure of the global dynamics are shown to give rise to a number of generalized symmetry relations for quantities expressible as matrix elements of the retarded, advanced, and global propagators. It is shown that the generalized symmetry relations may be used to facilitate the solution of spectral and temporal problems. This is demonstrated in a concrete way in an application entailing the use of dual Lanczos transformation theory to obtain an exact solution of a prototype model for excited state dynamics. Apart from elucidating the symmetries of open dissipative systems, the reported analysis reveals that there are two different and equally legitimate schemes for performing time-reversal transformations in the matrix formulation of quantum Liouvillian dynamics. The properties and ramifications of these time-reversal transformation schemes are discussed in detail.
We consider the symmetries required of quantum stochastic Liouvillian dynamical models intended to describe fluctuations about equilibrium in a closed system. Our considerations yield symmetry relations that represent necessary and sufficient conditions for the realization of the stationarity of symmetrized, anti-symmetrized, and one-sided equilibrium-averaged time correlation functions. Since stationarity is a physical requirement that must be satisfied by systems in equilibrium, these symmetry relations represent universal constraints on acceptable quantum stochastic Liouvillian dynamical models. In the classical limit, the symmetry relations reduce to the symmetry relation embodying stationarity for closed classical stochastic Liouvillian systems. [W.A. Wassam, Jr., J.A. Balderas-Lopez, and Go. Torres-Vega, J. Math. Phys. 31, 2466 (1990).] The symmetry relations are shown to lead to representation independent formulations of detailed balance and microscopic reversibility. In addition, it is shown that the symmetry relations, along with the basic structure of the global dynamics and star-conjugation symmetry, give rise to a number of generalized symmetry relations for symmetrized, anti-symmetrized, and one-sided equilibrium-averaged time correlation functions and their Fourier and Fourier-Laplace transforms. Apart from elucidating fundamental symmetries of acceptable dynamical models, the reported theory may be used to facilitate the solution of spectral and temporal problems. Also, the theory enables one to construct equilibrium solutions of dynamical models that satisfy the requirements of stationarity. Moreover, the theory may be used as a vehicle for correcting dynamical models that violate the stationarity requirement. These features are illustrated in an analysis of dynamical models for the damped harmonic oscillator and 2-state dissipative atom.
Non-invertible symmetries extend conventional symmetries beyond group-like transformations and play an important role in quantum field theory and many-body physics. However, a general framework of non-invertible symmetries in open quantum systems is still lacking. Here, we formulate a general Liouville-space duality framework that extends Hilbert-space non-invertible dualities to open quantum systems. The resulting defect relates Liouvillians with dual Hamiltonian and dissipative parameters and becomes an exact symmetry at self-duality. We apply this framework to the periodic transverse-field Ising chain with two collective Lindblad channels exchanged by Kramers--Wannier (KW) duality. A strong Z2 symmetry divides Liouville space into four parity sectors, while the defect acts nontrivially only in the even-even sector. At self-duality, the KW defect imposes a momentum-resolved selection rule on Liouvillian Jordan blocks. Modes with different Liouville momenta or KW-defect eigenvalues cannot form a common exceptional Jordan block. For a three-site chain, a global Rényi-2 KW asymmetry dynamically identifies the self-dual conditions. At the same time, the spectral structure exhibits both a semisimple degeneracy and a self-dual exceptional point with a perturbation-selective response. These results extend non-invertible dualities to Markovian quantum dynamics and establish a symmetry-based selection rule for Liouvillian Jordan structures.
A state of an open quantum system is described by a density matrix, whose dynamics is governed by a Liouvillian superoperator. Within a general framework, we explore fundamental properties of both first-order dissipative phase transitions and second-order dissipative phase transitions associated with a symmetry breaking. In the critical region, we determine the general form of the steady-state density matrix and of the Liouvillian eigenmatrix whose eigenvalue defines the Liouvillian spectral gap. We illustrate our exact results by studying some paradigmatic quantum optical models exhibiting critical behavior.
Symmetry transformations are defined by operators in quantum mechanics that preserve the modulus of the scalar product between Hilbert space vectors. According to Wigner's theorem, any such transformation is represented by either a unitary linear operator or an antiunitary (isometric conjugate-linear) operator. Although antiunitary symmetries-most notably time reversal and charge conjugation-are encountered less frequently than unitary ones, they are fundamental to the description of non-conservative and reversible systems. The most frequently treated antiunitary operators are the involutive ones, called conjugations. Any antiunitary operator can be written as a product of a conjugation and a unitary operator. Considering general scattering problems defined by a scattering potential and separating conjugation from the symmetry operator, one can find the role of complex symmetric (in other words self-transpose) unitary operators in physical problems. This approach provides a robust framework for analyzing the role of non-Hermitian symmetries in wave scattering and dissipative dynamics. To demonstrate the practical applicability of these theoretical concepts, we analyze the case of polarized neutron reflectometry (PNR). We show that the scattering potential in PNR, comprising nuclear and magnetic terms, can satisfy the condition of being unitarily equivalent to its transpose, thereby guaranteeing reciprocity under specific orientations of the magnetic field.
We present a full symmetry classification of fermion matter in and out of thermal equilibrium. Our approach starts from first principles, the ten different classes of linear and antilinear state transformations in fermionic Fock spaces, and symmetries defined via invariance properties of the dynamical equation for the density matrix. The object of classification is then the generators of reversible dynamics, dissipation and fluctuations, featuring' in the generally irreversible and interacting dynamical equations. A sharp distinction between the symmetries of equilibrium and out-of-equilibrium dynamics, respectively, arises from the different role played by time in these two cases: In unitary quantum mechanics as well as in microreversible thermal equilibrium, antilinear transformations combined with an inversion of time define time-reversal symmetry. However, out of equilibrium an inversion of time becomes meaningless, while antilinear transformations in Fock space remain physically significant, and hence must be considered in autonomy. The practical consequence of this dichotomy is a novel realization of antilinear symmetries (six out of the ten fundamental classes) in nonequilibrium quantum dynamics that is fundamentally different from the established rules of thermal equilibrium. At large times, the dynamical generators thus symmetry classified determine the steady-state nonequilibrium distributions for arbitrary interacting systems. To illustrate this principle, we consider the fixation of a symmetry protected topological phase in a system of interacting lattice fermions. More generally, we consider the practically important class of mean field interacting systems, represented by Gaussian states. This class is naturally described in the language of non-Hermitian matrices, which allows us to compare to previous classification schemes in the literature.
We present a full symmetry classification of fermion matter in and out of thermal equilibrium.Our approach starts from first principles, the ten different classes of linear and antilinear state transformations in fermionic Fock spaces, and symmetries defined via invariance properties of the dynamical equation for the density matrix.The object of classification is then the generators of reversible dynamics, dissipation and fluctuations, featuring' in the generally irreversible and interacting dynamical equations.A sharp distinction between the symmetries of equilibrium and out-of-equilibrium dynamics, respectively, arises from the different role played by "time" in these two cases: In unitary quantum mechanics as well as in "microreversible" thermal equilibrium, antilinear transformations combined with an inversion of time define time-reversal symmetry.However, out of equilibrium an inversion of time becomes meaningless, while antilinear transformations in Fock space remain physically significant, and hence must be considered in autonomy.The practical consequence of this dichotomy is a novel realization of antilinear symmetries (six out of the ten fundamental classes) in nonequilibrium quantum dynamics that is fundamentally different from the established rules of thermal equilibrium.At large times, the dynamical generators thus symmetry classified determine the steady-state nonequilibrium distributions for arbitrary interacting systems.To illustrate this principle, we consider the fixation of a symmetry protected topological phase in a system of interacting lattice fermions.More generally, we consider the practically important class of mean field interacting systems, represented by Gaussian states.This class is naturally described in the language of non-Hermitian matrices, which allows us to compare to previous classification schemes in the literature.