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arXiv:quant-ph/0609096 · Laser Physics

Non-Hermitian Hamiltonians with real eigenvalues coupled to electric fields: From the time-independent to the time-dependent quantum mechanical formulation

Carla Figueira de Morisson Faria, A. Fring

Published Apr 1, 2007• 48 citations
arXivHTMLPDF

Abstract

We provide a reviewlike introduction to the quantum mechanical formalism related to non-Hermitian Hamiltonian systems with real eigenvalues. Starting with the time-independent framework, we explain how to determine an appropriate domain of a non-Hermitian Hamiltonian and pay particular attention to the role played by PJ symmetry and pseudo-Hermiticity. We discuss the time evolution of such systems having in particular the question in mind of how to couple consistently an electric field to pseudo-Hermitian Hamiltonians. We illustrate the general formalism with three explicit examples: (i) the generalized Swanson Hamiltonians, which constitute non-Hermitian extensions of anharmonic oscillators, (ii) the spiked harmonic oscillator, which exhibits explicit super-symmetry, and (iii) the −x 4-potential, which serves as a toy model for the quantum field theoretical ϕ4-theory.

DOI
10.1134/s1054660x07040196

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We provide a reviewlike introduction to the quantum mechanical formalism related to non-Hermitian Hamiltonian systems with real eigenvalues. Starting with the time-independent framework, we explain how to determine an appropriate domain of a non-Hermitian Hamiltonian and pay particular attention to the role played by PJ symmetry and pseudo-Hermiticity. We discuss the time evolution of such systems having in particular the question in mind of how to couple consistently an electric field to pseudo-Hermitian Hamiltonians. We illustrate the general formalism with three explicit examples: (i) the generalized Swanson Hamiltonians, which constitute non-Hermitian extensions of anharmonic oscillators, (ii) the spiked harmonic oscillator, which exhibits explicit super-symmetry, and (iii) the −x 4-potential, which serves as a toy model for the quantum field theoretical ϕ4-theory.
We provide time-evolution operators, gauge transformations and a perturbative treatment for non-Hermitian Hamiltonian systems, which are explicitly time dependent. We determine various new equivalence pairs for Hermitian and non-Hermitian Hamiltonians, which are therefore pseudo-Hermitian and in addition in some cases also invariant under PT -symmetry. In particular, for the harmonic oscillator perturbed by a cubic non-Hermitian term, we evaluate explicitly various transition amplitudes, for the situation when these systems are exposed to a monochromatic linearly polarized electric field.
We propose a scheme to deal with certain time-dependent non-Hermitian Hamiltonian operators H(t)H(t)H(t) that generate a real phase in their time-evolution. This involves the use of invariant operators IPH(t)I_{PH}(t)IPH​(t) that are pseudo-Hermitian with respect to the time-dependent metric operator and which implies that the dynamics is governed by unitary time evolution. Furthermore, H(t)H(t)H(t) is not generally quasi-Hermitian and does not define an observable of the system but IPH(t)I_{PH}(t)IPH​(t) obeys a quasi-hermiticity transformation as in the completely time-independent Hamiltonian systems case. The harmonic oscillator with a time-dependent frequency under the action of a complex time-dependent linear potential is considered as an illustrative example.
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. This has been driven by the fact that PT\mathcal{PT}PT-symmetric non-Hermitian systems exhibit real energy eigenvalues and unitary time evolution. Historically, the introduction of time into the world of non-Hermitian quantum mechanics has been a conceptually difficult problem to address, as it requires the Hamiltonian to become unobservable. We solve this issue with the introduction of a new observable energy operator and 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\mathcal{PT}PT-symmetry is spontaneously broken. Ordinarily, in the time-independent setting, the energy eigenvalues become complex and the wave functions are asymptotically unbounded. We demonstrate that in the time-dependent setting this broken symmetry can be mended and analysis on the spontaneously broken PT\mathcal{PT}PT regime is indeed possible. We provide many examples of this mending on a wide range of different systems, beginning with a 2×22\times22×2 matrix model and extending to higher dimensional matrix models and coupled harmonic oscillator systems with infinite Hilbert space. Furthermore, we use the framework to perform analysis on time-dependent quasi-exactly solvable models. We present the "eternal life" of entropy 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. We create an elegant framework for Darboux and Darboux/Crum transformations for time-dependent non-Hermitian Hamiltonians.
Explicitly time-dependent pseudo-Hermitian (TDPH) invariants theory systems, with a time-dependent (TD) metric, is developed for a time-dependent non Hermitian (TDNH) quantum systems. We derive a simple relation between the eigenstates of this pseudo-Hermitien (PH) invariant and the solutions of the Schrodinger equation. A physical system is treated in detail: the TD Swanson model, where an explicitly TDPH invariant is derived for this system, the eigenvalues and eigenstates of the invariant are calculated explicitly.
The relevance in physics of non-Hermitian operators with real eigenvalues is being widely recognized not only in quantum mechanics but also in other areas, such as quantum optics, quantum fluid dynamics, and quantum field theory. In this note, a quantum system described by a non-Hermitian Hamiltonian, which is constituted by two types of interacting bosons, is investigated. The real eigenvalues of the Hamiltonian are explicitly determined, as well as complete biorthogonal sets of eigenfunctions of the Hamiltonian and its adjoint. The diagonal representation of H is obtained using pseudo-bosonic operators.
Abstract In the context of non-Hermitian quantum mechanics, many systems are known to possess a pseudo P T symmetry, i.e. the non-Hermitian Hamiltonian H is related to its adjoint H † via the relation, H † = P T H P T . We propose a derivation of pseudo P T symmetry and η -pseudo-Hermiticity simultaneously for the time dependent non-Hermitian Hamiltonians by intoducing a new symmetry operator η ˜ ( t ) = P T η ( t ) that not satisfy the time-dependent quasi-Hermiticity relation but obeys the Heisenberg evolution equation. Here, we solve the SU(1, 1) time-dependent non-Hermitian Hamiltonian and we construct a time-dependent solutions by employing this new symmetry operator and discuss a concrete physical applications of our results. Dedicated to the memory of my mother Djabou Zoulikha and to Djabi Smail