[2211.05437] Non-Hermitian Hamiltonian Deformations in Quantum Mechanics

[Submitted on 10 Nov 2022]

Download PDF

Abstract: The construction of exactly-solvable models has recently been advanced by
considering integrable $T\bar{T}$ deformations and related Hamiltonian
deformations in quantum mechanics. We introduce a broader class of
non-Hermitian Hamiltonian deformations in a nonrelativistic setting, to account
for the description of a large class of open quantum systems, which includes,
e.g., arbitrary Markovian evolutions conditioned to the absence of quantum
jumps. We relate the time evolution operator and the time-evolving density
matrix in the undeformed and deformed theories in terms of integral transforms
with a specific kernel. Non-Hermitian Hamiltonian deformations naturally arise
in the description of energy diffusion that emerges in quantum systems from
time-keeping errors in a real clock used to track time evolution. We show that
the latter can be related to an inverse $T\bar{T}$ deformation with a purely
imaginary deformation parameter. In this case, the integral transforms take a
particularly simple form when the initial state is a coherent Gibbs state or a
thermofield double state, as we illustrate by characterizing the purity,
Rényi entropies, logarithmic negativity, and the spectral form factor. As the
dissipative evolution of a quantum system can be conveniently described in
Liouville space, we further discuss the spectral properties of the
Liouvillians, i.e., the dynamical generators associated with the deformed
theories. As an application, we discuss the interplay between decoherence and
quantum chaos in non-Hermitian deformations of random matrix Hamiltonians and
the Sachdev-Ye-Kitaev model.