Entropy and entanglement in a bipartite quasi-Hermitian system and its Hermitian counterparts

We consider a quantum oscillator coupled to a bath of $N$ other oscillators. The total system evolves with a quasi-Hermitian Hamiltonian. Associated to it is a family of Hermitian systems, parameterized by a unitary map $W$. Our main goal is to find the influence of $W$ on the entropy and the entanglement in the Hermitian systems. We calculate explicitly the reduced density matrix of the single oscillator for all Hermitian systems and show that, regardless of $W$, their von Neumann entropy oscillates with a common period which is twice that of the non-Hermitian system. We show that generically, the oscillator and the bath are entangled for almost all times. While the amount of entanglement depends on the choice of $W$, it is independent of $W$ when averaged over a period. These results describe some universality in the physical properties of all Hermitian systems associated to a given non-Hermitian one

Time-dependence in non-Hermitian quantum systems

In this thesis we present a coherent and consistent framework for explicit time-dependence in non-Hermitian quantum mechanics. The area of non-Hermitian quantum mechanics has been growing rapidly over the past twenty years [2]. This has been driven by the fact that PT -symmetric non-Hermitian systems exhibit real energy eigenvalues and unitary time evolution [1, 3, 4]. Historically, the introduction of time into the world of non-Hermitian quantum mechanics has been a conceptually difficult problem to address [5, 6], as it requires the Hamiltonian to become unobservable. However, we solve this issue with the introduction of a new observable energy operator [7]. We explain why its instigation is a necessary and natural progression in this setting. For the first time, the introduction of time has allowed us to make sense of the parameter regime in which the PT -symmetry is spontaneously broken. Ordinarily, in the time-independent setting, the energy eigenvalues become complex and the wave functions are asymptotically unbounded. However, we demonstrate that in the time-independent setting this broken symmetry can be mended and analysis on the spontaneously broken PT regime is indeed possible. We provide many examples of this mending on a wide range of different systems, beginning with a 2 x 2 matrix model [8] and extending to higher dimensional matrix models [9] and coupled harmonic oscillator systems with infinite Hilbert space [10, 11]. Furthermore, we use the framework to perform analysis on time-dependent quasi-exactly solvable models [12]. The ability to make sense of the spontaneously broken PT regime has revealed a vast array of new and exotic effects. We present the "eternal life" of entropy [13] in this thesis. Ordinarily, for entangled quantum systems coupled to the environments, the entropy decays rapidly to zero. However, in the spontaneously broken regime, we find the entropy decays asymptotically to a non-zero value. Finally, we create an elegant framework for Darboux and Darboux/Crum transformations for time-dependent non-Hermitian Hamiltonians [14]. This combines the area of non-Hermitian quantum mechanics with non linear differential equations and solitons

Quantum chimera states

We study a theoretical model of closed quasi-hermitian chain of spins which exhibits quantum analogues of chimera states, i.e. long life classical states for which a part of an oscillator chain presents an ordered dynamics whereas another part presents a disordered chaotic dynamics. For the quantum analogue, the chimera behavior deals with the entanglement between the spins of the chain. We discuss the entanglement properties, quantum chaos, quantum disorder and semi-classical similarity of our quantum chimera system. The quantum chimera concept is novel and induces new perspectives concerning the entanglement of multipartite systems

Non-Hermitian quantum dynamics and entanglement of coupled nonlinear resonators

We consider a generalization of recently proposed non-Hermitian model for resonant cavities coupled by a chiral mirror by taking into account number non-conservation and nonlinear interactions. We analyze non-Hermitian quantum dynamics of populations and entanglement of the cavity modes. We find that an interplay of initial coherence and non-Hermitian coupling leads to a counterintuitive population transfer. While an initially coherent cavity mode is depleted, the other empty cavity can be populated more or less than the initially filled one. Moreover, presence of nonlinearity yields population collapse and revival as well as bipartite entanglement of the cavity modes. In addition to coupled cavities, we point out that similar models can be found in $\mathcal{PT}$ symmetric Bose-Hubbard dimers of Bose-Einstein condensates or in coupled soliton-plasmon waveguides. We specifically illustrate quantum dynamics of populations and entanglement in a heuristic model that we propose for a soliton-plasmon system with soliton amplitude dependent asymmetric interaction. Degree of asymmetry, nonlinearity and coherence are examined to control plasmon excitations and soliton-plasmon entanglement. Relations to $\mathcal{PT}$ symmetric lasers and Jahn-Teller systems are pointed out

Dissipative quantum theory: Implications for quantum entanglement

Three inter-related topics are discussed here. One, the Lindblad dynamics of quantum dissipative systems; two, quantum entanglement in composite systems and its quantification based on the Tsallis entropy; and three, robustness of entanglement under dissipation. After a brief review of the Lindblad theory of quantum dissipative systems and the idea of quantum entanglement in composite quantum systems illustrated by describing the three particle systems, the behavior of entanglement under the influence of dissipative processes is discussed. These issues are of importance in the discussion of quantum nanometric systems of current research.Comment: 12 pages, 1 Tabl