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arXiv:1806.02103 · Physical Review A

Exactly solvable time-dependent pseudo-Hermitian su(1,1) Hamiltonian models

Roberto Grimaudo, Antonio S. M. de Castro, Marek Kuś, A. Messina

Published Sep 28, 2018• 27 citations
arXivHTMLPDF

Abstract

An exact analytical treatment of the dynamical problem for time-dependent $2\ifmmode\times\else\texttimes\fi{}2$ pseudo-Hermitian $\text{su}(1,1)$ Hamiltonians is reported. A class of exactly solvable and physically transparent scenarios are identified within both classical and quantum contexts. The class is spanned by a positive parameter $\ensuremath{\nu}$ that allows for distinguishing two different dynamical regimes. Our results are usefully employed for exactly solving a classical propagation problem in a guided wave optics scenario. The usefulness of our procedure in a quantum context is illustrated by defining and investigating the $\text{su}(1,1)$ ``Rabi'' scenario, bringing to light analogies and differences with the standard su(2) Rabi model. Our approach, applied to the generalized von Neumann equation describing open quantum systems through non-Hermitian Hamiltonians, succeeds in evidencing that the $\ensuremath{\nu}$-dependent passage from a real to a complex energy spectrum is generally unrelated to the existence of the two dynamical regimes.

DOI
10.1103/physreva.98.033835

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An exact analytical treatment of the dynamical problem for time-dependent 2\ifmmode×\else\texttimes\fi22\ifmmode\times\else\texttimes\fi{}22\ifmmode×\else\texttimes\fi2 pseudo-Hermitian su(1,1)\text{su}(1,1)su(1,1) Hamiltonians is reported. A class of exactly solvable and physically transparent scenarios are identified within both classical and quantum contexts. The class is spanned by a positive parameter \ensuremathν\ensuremath{\nu}\ensuremathν that allows for distinguishing two different dynamical regimes. Our results are usefully employed for exactly solving a classical propagation problem in a guided wave optics scenario. The usefulness of our procedure in a quantum context is illustrated by defining and investigating the su(1,1)\text{su}(1,1)su(1,1) ``Rabi'' scenario, bringing to light analogies and differences with the standard su(2) Rabi model. Our approach, applied to the generalized von Neumann equation describing open quantum systems through non-Hermitian Hamiltonians, succeeds in evidencing that the \ensuremathν\ensuremath{\nu}\ensuremathν-dependent passage from a real to a complex energy spectrum is generally unrelated to the existence of the two dynamical regimes.
In this paper we investigate time-dependent non-Hermitian Hamiltonians, which consist of SU(1,1) and SU(2) generators. The former Hamiltonian is PT symmetric but the latter one is not. A time-dependent non-unitary operator is proposed to construct the non-Hermitian invariant, which is verified as pseudo-Hermitian with real eigenvalues. The exact solutions are obtained in terms of the eigenstates of the pseudo-Hermitian invariant operator for both the SU(1,1) and SU(2) systems in a unified manner. Then, we derive the Lewis–Riesenfeld (LR) phase, which can be separated into the dynamic and the geometrical phases. The analytical results are well consistent with those of the corresponding Hermitian Hamiltonians reported in the literature.
Based on the generalized gauge transformation and invariant methods, we study in this work the rigorous solution for non-Hermitian Hamiltonians consisting of SU(1, 1) and SU(2) generators, which describe the harmonic oscillator and a single spin of arbitrary spin value, respectively, in periodically time-dependent fields. A non-Hermitian Hamiltonian with real eigenvalues is called pseudo-Hermitian. Such a Hamiltonian can be converted to a Hermitian one by a similarity transformation. We extend the concept of pseudo-Hermitian Hamiltonians to the time-dependent case. Under the generalized gauge transformation with a time-dependent Hermitian operator, the Hamiltonian becomes Hermitian and time-independent in the new gauge. The rigorous solution, as well as the Berry phase, is obtained by applying the inverse transformation to return to the original gauge. The time-dependent Schrödinger equation is also solved in terms of non-Hermitian invariant operators, which possess conserved real eigenvalues. We derive the time-dependent solutions and corresponding geometric phases by eigenstate expansion of the invariant operators, combined with the time-evolution operator. It is observed that the non-Hermitian coupling constant indeed has an exceptional point Gc(ω) for the SU(2) non-Hermitian Hamiltonian, where all eigenstates are degenerate with zero eigenvalue and the Berry phase diverges. The eigenvalues become complex beyond Gc(ω) . However, the eigenvalues are real within the entire region of coupling constant for the SU(1, 1) system, different from the common belief that the exceptional point must exist in the non-Hermitian Hamiltonian. The classical counterpart of the SU(1, 1) non-Hermitian Hamiltonian is, in fact, a complex function of position-momentum variables. It can be converted to a real function via the generalized gauge transformation, accompanied by a canonical-variable transformation. This process corresponds exactly to the Hermitization procedure of the pseudo-Hermitian Hamiltonian. We obtain the classical Hannay angle for the periodically time-dependent SU(1, 1) pseudo-Hermitian Hamiltonian. The non-adiabatic Hannay angle and the Berry phase satisfy precisely the quantum-classical correspondence, γn(T)=−(n+12)∆Θ .
We study the time evolution of quantum systems with a time-dependent non-Hermitian Hamiltonian exhibiting a SU(1,1) and SU(2) dynamical symmetry. With a time-dependent metric, the pseudo-Hermitian invariant operator is constructed in the same manner as for both the SU(1,1) and SU(2) systems. The exact common solutions of the Schrödinger equations for both the SU(1,1) and SU(2) systems are obtained in terms of eigenstates of the pseudo-Hermitian invariant operator. Finally some interesting physical applications are suggested and discussed.
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
In the context of non-Hermitian quantum mechanics, many systems are known to possess a pseudo PT symmetry , i.e. the non-Hermitian Hamiltonian H is related to its adjoint H^{†} via the relation, H^{†}=PTHPT . We propose a derivation of pseudo PT symmetry and η -pseudo-Hermiticity simultaneously for the time dependent non-Hermitian Hamiltonians by intoducing a new metric η(t)=PTη(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 metric and discuss a concrete physical applications of our results.