Mixed-State Evolution in the Presence of Gain and Loss

Abstract

A model is proposed that describes the evolution of a mixed state of a quantum system for which gain and loss of energy or amplitude are present. Properties of the model are worked out in detail. In particular, invariant subspaces of the space of density matrices corresponding to the fixed points of the dynamics are identified, and the existence of a transition between the phase in which gain and loss are balanced and the phase in which this balance is lost is illustrated in terms of the time average of observables. The model is extended to include a noise term that results from a uniform random perturbation generated by white noise. Numerical studies of example systems show the emergence of equilibrium states that suppress the phase transition.PACS numbers: 03.65. Ca, 05.30.Rt, 03.65.Yz Over the past decade there have been considerable research interests, both theoretical and experimental, into the static and dynamic properties of classical and quantum systems for which gain and loss are present [1,2]. This is in part motivated by the realisation that when a system is placed in a configuration in which its energy or amplitude is transferred into its environment through one channel, but at the same time is amplified by the same amount through another channel, the resulting dynamics can exhibit features that are similar to those seen in Hamiltonian dynamical systems. The time evolution of such a system can be described by a Hamiltonian that is symmetric under a space-time reflection, that is, invariant under the parity-time (PT) reversal.Interest in the theoretical study of PT symmetry was triggered by the discovery that complex PT-symmetric quantum Hamiltonians can possess entirely real eigenvalues [3]. One distinguishing feature of PT-symmetric quantum systems is the existence of phase transitions associated with the breakdown of the symmetry. That is, depending on the values of the matrix elements of the Hamiltonian, its eigenstates may or may not be symmetric under the parity-time reversal. In the 'unbroken phase' where the eigenstates respect PT symmetry, the eigenvalues are real and there exists a similarity transformation that maps a local PT-symmetric Hamiltonian into a typically nonlocal Hermitian Hamiltonian [4,5]; whereas in the 'broken phase' the eigenvalues constitute complex conjugate pairs. The transition between these phases is similar to second-order phase transitions in statistical mechanics, accompanied by singularities in the covariance matrix of the estimators for the parameters in the Hamiltonian (this can be seen by using methods of information geometry [6]).Experimental realisations of these phenomena are motivated in part by the observation that the presence of a loss, which traditionally has been viewed as an undesirable feature, can positively be manipulated so as to generate unexpected interesting effects (cf. [7]). In particular, PT phase transitions have been predicted or observed in laboratory experiments for a range of systems; most notably in optical waveguides [8][9][10], but ...

Information Geometry of Complex Hamiltonians and Exceptional Points

“…They are also used in the context of open systems as effective Hamiltonians. More recently, complex Hamiltonians that fulfil certain antilinear symmetries have attracted a lot of attention, owing to the facts that such Hamiltonians may possess entirely real eigenvalues and that depending on the parameter values in the Hamiltonian there can be a phase transition where a pair of real eigenvalues degenerates and turns into a complex conjugate pair [15][16][17][18][19][20][21][22][23][24][25][26]. As indicated above, such a critical point is where the associated eigenstates also coalesce, thus constituting an example of an exceptional point.…”

“…In particular, investigations into the properties of complex Hamiltonians have increased significantly over the past decade since the observation of Bender and Boettcher that complex Hamiltonians possessing parity-time (PT) reversal symmetry can possess entirely real eigenvalues [14]. Phase transitions associated with the breakdown of PT symmetry of the eigenfunctions at exceptional points have also been predicted or observed in a range of model systems and experiments [15][16][17][18][19][20][21][22][23][24][25][26], and constitute an interesting and exciting area of application of information geometry. It is our hope that the present paper serves as a concise introduction to the physics of complex Hamiltonians for those who work in the area of information geometry and, at the same time, an introduction to information geometry for those who work in the study of physical systems described by complex Hamiltonians.…”

Embedding quantum systems with a non-conserved probability in classical environments

“…A first general book on the topic has appeared [6]; applications of nonHermitian quantum mechanics involve the study of scattering by complex potentials and quantum transport [7][8][9][10][11][12][13][14][15][16][17], description of metastable states [18][19][20][21][22][23], optical waveguides [24][25][26], multi-photon ionization [27][28][29], and nano-photonic and plasmonic waveguides [30]. The theoretical investigations are also undergoing rapid developments: non-Hermitian quantum mechanics has been investigated within a relativistic framework [31] and it has been adopted by various researchers as a means to describe open quantum systems [32][33][34][35][36][37][38][39][40][41][42]. Moreover, it seems that a few theoretical studies have been dedicated to the statistical mechanics and dynamics of systems with non-Hermitian Hamiltonians [43][44][45][46][47][48][49]...…”

“…Types of noise beyond those arising from Gaussian white noise [41] can then be treated. From a more general perspective, one goal of this work is to develop a numerical formalism (which is complementary to that based on master equations [55]) for studying the dissipative dynamics of, for example, quantum plasmonic metamaterials [56,57] or processes of interest in quantum thermodynamics [58].…”

Non-Hermitian quantum dynamics and entanglement of coupled nonlinear resonators

“…In the present letter, we use a more general von Neumann approach [18][19][20][21] which allows for violation of number conservation due to the non-Hermitian nature of the system. These two approaches are not equivalent to each other and von Neumann approach is proper treatp-1 ment of non-conservative systems with the non-Hermitian models.…”