Non-Hermitian Hamiltonian deformations in quantum mechanics

Abstract

The construction of exactly-solvable models has recently been advanced by considering integrable $$ T\overline{T} $$ T 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 $$ T\overline{T} $$ T 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 study 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.

On Krylov complexity in open systems: an approach via bi-Lanczos algorithm

“…However, in the case of the open systems, the evolution is non-unitary and hence the probability is expected to decay due to the dissipative effects. This decay is expected to be very similar to the deacy due to decoherence discussed in the equation (5.9) of [25]. Similarly, we can try to (naively) define the K-complexity in the bi-Lanczos basis as…”

“…For example, it should apply equally well in the studies of the spread complexity in the Schrödinger picture given the Hamiltonian under which the evolution happens, is non-Hermitian (see Appendix C). A few of such scenarios, where a non-Hermitian Hamiltonian evolution occurs, are i) the evolution of mixed state density matrix in terms of an effective non-Hermitian Hamiltonian [25,55] derived from the Lindblad master equation and ii) the evolution of quantum states under projective measurements after regular intervals (also known as the first passage problem) [56]. We hope to address some of them in future studies.…”

“…This direction of the study was initiated in our previous exploration [1] and continued in other following works [2,18] (see [19][20][21][22][23][24][25][26] for some recent works on op-erator growth and chaos on non-Hermitian and open quantum systems). Unlike the closed systems, the open system operator evolution (under the realm of Markovian dynamics [27]) is characterized by a non-unitary evolution through the exponentiated non-Hermitian Lindbladian L o ⋅ = [H, ⋅ ] + iT , where the second term represents the non-Hermitian part [28,29].…”

Composite quantum Coriolis forces

“…also a more extensive commentary in [23]). Incidentally, such an observation simplifies transition to a statistical version of QHQM where the states become represented by the non-Hermitian and time-dependent version of the density matrix (cf., e.g., [24]),…”

Embedding of a non-Hermitian Hamiltonian to emulate the von Neumann measurement scheme

“…There have also been attempts to describe open quantum systems effectively in terms of non-Hermitian Hamiltonians [25][26][27][28][29][30][31][32][33]. Specifically, the link between the Lindblad master equation and non-Hermitian dynamics has been studied before [34][35][36].…”