Solvable two dimensional time-dependent non-Hermitian quantum systems with infinite dimensional Hilbert space in the broken PT-regime

Abstract

We provide exact analytical solutions for a two dimensional explicitly time-dependent non-Hermitian quantum system. While the time-independent variant of the model studied is in the broken PT-symmetric phase for the entire range of the model parameters, and has therefore a partially complex energy eigenspectrum, its time-dependent version has real energy expectation values at all times. In our solution procedure we compare the two equivalent approaches of directly solving the time-dependent Dyson equation with one employing the Lewis-Riesenfeld method of invariants. We conclude that the latter approach simplifies the solution procedure due to the fact that the invariants of the non-Hermitian and Hermitian system are related to each other in a pseudo-Hermitian fashion, which in turn does not hold for their corresponding time-dependent Hamiltonians. Thus constructing invariants and subsequently using the pseudo-Hermiticity relation between them allows to compute the Dyson map and to solve the Dyson equation indirectly. In this way one can bypass to solve nonlinear differential equations, such as the dissipative Ermakov-Pinney equation emerging in our and many other systems.

12 pages

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• An introduction to PT-symmetric quantum mechanics -- time-dependent systems
• Exotic entanglement for non-Hermitian Jaynes-Cummings Hamiltonians
• Dynamics of finite dimensional non-hermitian systems with indefinite metric
• Quantum Metric Unveils Defect Freezing in Non-Hermitian Systems
• Spectrally equivalent time-dependent double wells and unstable anharmonic oscillators
• Quasi-exactly solvable quantum systems with explicitly time-dependent Hamiltonians
• Time-independent approximations for time-dependent optical potentials
• Time-dependent metric for the two dimensional, non-Hermitian coupled oscillator
• Pseudo-invariant approach for a particle in a complex time-dependent linear potential
• Generalized gauge transformation with $PT$-symmetric non-unitary operator and classical correspondence of non-Hermitian Hamiltonian for a periodically driven system
• Time-dependent C-operators as Lewis-Riesenfeld invariants in non-Hermitian theories
• Perturbative approach for strong and weakly coupled time-dependent non-Hermitian quantum systems
• Universal scaling at a pre-thermal dark state
• Lewis-Riesenfeld invariants for PT-symmetrically coupled oscillators from two dimensional point transformations and Lie algebraic expansions
• Oscillators with imaginary coupling: spectral functions in quantum mechanics and quantum field theory
• Entanglement of pseudo-Hermitian random states
• Infinite series of time-dependent Dyson maps