[Paper Review] Quantum Evolution of the Time-Dependent Non-Hermitian Hamiltonians: Real Phases

Mustapha Maamache, Oum Kaltoum Djeghiour+2 more|arXiv (Cornell University)|May 17, 2017

Quantum Mechanics and Non-Hermitian Physics6 references3 citations

TL;DR

This paper develops a time-dependent pseudo-Hermitian (TDPH) invariant theory for non-Hermitian Hamiltonians with explicitly time-dependent metrics, enabling exact solutions to the time-dependent Schrödinger equation. It applies the Lewis-Riesenfeld invariants method to the generalized Swanson model with time-dependent coefficients, deriving real-valued dynamical phases and exact eigenstates, thus ensuring unitary time evolution despite non-Hermiticity through a time-dependent metric and quasi-Hermitian structure.

ABSTRACT

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.

Motivation & Objective

• To establish a consistent framework for unitary time evolution in time-dependent non-Hermitian (TDNH) quantum systems with explicitly time-dependent metrics.
• To resolve conceptual challenges in extending quasi-Hermiticity and unitarity to TDNH systems where the metric operator evolves in time.
• To develop a time-dependent pseudo-Hermitian invariant theory that generalizes the standard invariants method for TD Hamiltonians.
• To apply the formalism to the generalized Swanson model with time-dependent complex coefficients, providing exact solutions.
• To ensure the reality of the dynamical phase factor in the time evolution, which is essential for probabilistic interpretation and unitarity.

Proposed method

• Introduces a time-dependent pseudo-Hermitian (TDPH) invariant operator $ I^{PH}(t) $, related to a Hermitian invariant $ I^h(t) $ via a time-dependent similarity transformation $ I^h(t) = \rho(t) I^{PH}(t) \rho^{-1}(t) $.
• Applies the Lewis-Riesenfeld invariants method to solve the time-dependent Schrödinger equation for TDNH Hamiltonians by expressing solutions as eigenstates of $ I^{PH}(t) $ with a time-dependent global phase $ \gamma_n(t) $.
• Derives the general form of the TDPH invariant for the generalized Swanson Hamiltonian with time-dependent coefficients $ \omega(t), \alpha(t), \beta(t) $, using a time-dependent canonical transformation.
• Constructs the eigenstates $ \phi_n^H(x,t) $ of the invariant in terms of Hermite polynomials and a time-dependent weight factor $ \eta(t) $, ensuring orthonormality under the $ \eta $-inner product.
• Imposes conditions on the time-dependent frequency $ W(t) $ to ensure the dynamical phase $ \gamma_n(t) $ is real, which is crucial for unitarity and physical consistency.
• Uses the relation $ \left| \Psi^h(t) \right\rangle = \rho(t) \left| \Phi^H(t) \right\rangle $ to map solutions of the Hermitian system to the non-Hermitian one, preserving unitarity through the time-dependent metric.

Experimental results

Research questions

• RQ1 Can a consistent time-dependent pseudo-Hermitian invariant theory be formulated for non-Hermitian Hamiltonians with explicitly time-dependent metrics?
• RQ2 Does the Lewis-Riesenfeld method remain applicable to TDNH systems when the metric is time-dependent, and can it yield unitary time evolution?
• RQ3 What conditions ensure the reality of the dynamical phase $ \gamma_n(t) $ in the time evolution of non-Hermitian systems with time-dependent metrics?
• RQ4 How can the generalized Swanson Hamiltonian with time-dependent complex coefficients be solved exactly using this invariant framework?
• RQ5 What is the explicit form of the eigenstates and eigenvalues of the TDPH invariant in the generalized Swanson model?

Key findings

• The time-dependent pseudo-Hermitian invariant $ I^{PH}(t) $ is explicitly constructed for the generalized Swanson Hamiltonian with time-dependent coefficients, expressed in terms of position and momentum operators with time-varying coefficients.
• The eigenfunctions $ \phi_n^H(x,t) $ are derived as harmonic-oscillator-like states involving Hermite polynomials and a time-dependent Gaussian weight factor $ \eta(t) = \exp\left[ \frac{\Phi(\chi-1)}{(\Phi - \chi)(1 - \Phi)} x^2 \right] $, ensuring orthonormality under the $ \eta $-inner product.
• The dynamical phase $ \gamma_n(t) $ is shown to be real when the frequency $ W(t) $ is real, which is achieved by imposing specific time-dependent constraints on the parameters $ \omega(t), \alpha(t), \beta(t) $, ensuring unitary evolution.
• The exact solution to the time-dependent Schrödinger equation is given by $ \left| \Phi^H(t) \right\rangle = \sum_n C_n(0) \exp\left( i k_n \int_0^t \frac{2}{\vartheta_0} \left[ |\omega|(\Phi^2 + \chi)\cos\varphi_\omega - 4\Phi|\alpha|\cos\varphi_\alpha \right] dt' \right) \left| \phi_n^H(t) \right\rangle $, with real phase evolution.
• The eigenvalues of the invariant are $ 2k_n = n + \frac{1}{2} $, confirming the harmonic-oscillator spectrum structure under the time-dependent transformation.
• The formalism confirms that unitary time evolution is possible for TDNH systems when the metric is time-dependent, provided the invariant is pseudo-Hermitian and the phase remains real, resolving prior inconsistencies in the literature.