Generalized Gauge Transformation with PT$PT$‐Symmetric Non‐Unitary Operator and Classical Correspondence of Non‐Hermitian Hamiltonian for a Periodically Driven System

Abstract

This paper demonstrates that the parity‐time (PT$PT$)‐symmetric non‐Hermitian Hamiltonian for a periodically driven system can be generated from a kernel Hamiltonian by a generalized gauge transformation. The kernel Hamiltonian is Hermitian and static, while the time‐dependent transformation operator has to be PT$PT$ symmetric and non‐unitary in general. Biorthogonal sets of eigenstates appear necessarily as a consequence of the non‐Hermitian Hamiltonian. The wave functions and associated non‐adiabatic Berry phase γn$\gamma _{n}$ for the nth eigenstate are obtained analytically. The classical version of the non‐Hermitian Hamiltonian becomes a complex function of canonical variables and time. The corresponding kernel Hamiltonian is derived with PT$PT$ symmetric canonical‐variable transfer in the classical gauge transformation. Moreover, with the change of position‐momentum to angle‐action variables it is revealed that the non‐adiabatic Hannay's angle normalΔθH$\Delta \theta _{\text{H}}$ and Berry phase satisfy precisely the quantum‐classical correspondence, γn=(n+1/2)normalΔθH$\gamma _{n}= (n+1/2)\Delta \theta _{\text{H}}$.

Non-Hermitian Hamiltonian beyond PT-symmetry for time-dependant SU(1,1) and SU(2) systems -- exact solution and geometric phase in pseudo-invariant theory

“…The exact solution of a PT -symmetric non-Hermitian Hamiltonian was presented recently for the periodically driven SU(1, 1) generators [18,19]. We emphasize that the spectrum reality of a non-Hermitian Hamiltonian is not confined to the PT -symmetry.…”

How “Berry Phase” Analysis of Non-adiabatic Non Hermitian Systems Reflects Their Geometry

“…The latter fits a storytelling approach better, although one always must give the description somewhere! As Gu et al [1] point out: "when a classical or quantum system undergoes a cyclic evolution governed by slow change in its parameter space, it acquires a topological phase factor known as the geometric or Berry phase, which reveals the gauge structure in quantum mechanics". "Hannay's angle" is the classical counterpart of this additional quantum phase [2] as is clear from the elegant treatment of a spinning top [3].…”

“…Again, Gu et al [1] point out that "in standard quantum theory, observable quantities are associated with Hermitian operators, and time evolution is generated by a Hermitian Hamiltonian, where the Hermiticity (or more precisely self-adjointness) of the Hamiltonian ensures both the real-valuedness of the energy spectrum and more importantly the unitarity" (that is, losslessness, or lack of dissipation) of time evolution. It therefore came as a surprise when Bender & Boettcher showed in 1998 [15] that non-Hermitian Hamiltonians can still possess real and positive eigenvalues.…”

“…It therefore came as a surprise when Bender & Boettcher showed in 1998 [15] that non-Hermitian Hamiltonians can still possess real and positive eigenvalues. They claimed that the realvaluedness of the spectrum is due to parity-time (PT) symmetry of the Hamiltonian: Gu et al [1] also studied PT-symmetric (or near-PT-symmetric) systems where the Hamiltonian is not Hermitian and where the Berry phases are non-adiabatic.…”

“…Gu et al [1] propose a gauge transformation method to solve the time-dependent (periodically driven) Schrödinger equation whose Hamiltonian is a non-Hermitian (non-unitary) but PT-symmetric operator. Quantum wave functions are obtained analytically along with the non-adiabatic Berry phase for the system.…”