Non-Hermitian Hamiltonian Deformations in Quantum Mechanics

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.

28 pages, 6 figures

References in corpus (15)

• Making Sense of Non-Hermitian Hamiltonians
• On space of integrable quantum field theories
• Dynamics of Loschmidt echoes and fidelity decay
• $T\bar{T}$-deformed 2D Quantum Field Theories
• Third quantization: a general method to solve master equations for quadratic open Fermi systems
• Decoherence, Entanglement and Irreversibility in Quantum Dynamical Systems with Few Degrees of Freedom
• Mixed-state evolution in the presence of gain and loss
• Operator growth and Krylov construction in dissipative open quantum systems
• Nonexponential quantum decay under environmental decoherence
• Quantum chaos, scrambling and operator growth in $T\bar{T}$ deformed SYK models
• Integrable nonunitary open quantum circuits
• Decoherence in Conformal Field Theory
• $T\overline{T}$ Deformations of Supersymmetric Quantum Mechanics
• Wormholes and holographic decoherence
• Product of Random States and Spatial (Half-)Wormholes

Cited by in corpus (29)

• Operator growth in open quantum systems: lessons from the dissipative SYK
• Quantum Dynamics in Krylov Space: Methods and Applications
• On Krylov complexity in open systems: an approach via bi-Lanczos algorithm
• Operator dynamics in Lindbladian SYK: a Krylov complexity perspective
• Biorthogonal Dynamical Quantum Phase Transitions in Non-Hermitian Systems
• Diagnosing non-Hermitian Many-Body Localization and Quantum Chaos via Singular Value Decomposition
• Non-Hermitian Stark Many-Body Localization
• Singular-Value Statistics of Non-Hermitian Random Matrices and Open Quantum Systems
• Unitarity breaking in self-averaging spectral form factors
• Proposal for many-body quantum chaos detection
• Quantization of the ModMax Oscillator
• Quantum Chaos and Coherence: Random Parametric Quantum Channels
• Toward a classification of PT-symmetric quantum systems: From dissipative dynamics to topology and wormholes
• Probing quantum chaos through singular-value correlations in sparse non-Hermitian SYK model
• Engineering active motion in quantum matter
• Composite quantum Coriolis forces
• Decoherence rate in random Lindblad dynamics
• Universal Properties of the Spectral Form Factor in Open Quantum Systems
• Reducing dynamical fluctuations and enforcing self-averaging by opening many-body quantum systems
• Non-Hermitian boost deformation
• Fate of non-Hermitian free fermions with Wannier-Stark ladder
• A geometrical description of non-Hermitian dynamics: speed limits in finite rank density operators
• Slow approach to adiabaticity in many-body non-Hermitian systems: the Hatano-Nelson Model
• Embedding of a non-Hermitian Hamiltonian to emulate the von Neumann measurement scheme
• $T\bar{T}$ deformation on non-Hermitian two coupled SYK model
• Double-Bracket Master Equations: Phase-Space Representation and Classical Limit
• Does Cosmology require Hermiticity in Quantum Mechanics?
• Does Gravity Render Probability Quasilocal?
• Krylov Complexity Under Hamiltonian Deformations and Toda Flows