A canonical Hamiltonian for open quantum systems

Abstract

If an open quantum system is initially uncorrelated from its environment, then its dynamics can be written in terms of a Lindblad-form master equation. The master equation is divided into a unitary piece, represented by an effective Hamiltonian, and a dissipative piece, represented by a hermiticity-preserving superoperator; however, the division of open system dynamics into unitary and dissipative pieces is non-unique. For finite-dimensional quantum systems, we resolve this non-uniqueness by specifying a norm on the space of dissipative superoperators and defining the canonical Hamiltonian to be the one whose dissipator is minimal. We show that the canonical Hamiltonian thus defined is equivalent to the Hamiltonian initially defined by Lindblad, and that it is uniquely specified by requiring the dissipator's jump operators to be traceless, extending a uniqueness result known previously in the special case of Markovian master equations. For a system weakly coupled to its environment, we give a recursive formula for computing the canonical effective Hamiltonian to arbitrary orders in perturbation theory, which we can think of as a perturbative scheme for renormalizing the system's bare Hamiltonian.

18 pages; v2 adds some missing citations, v3 fixes a mistyped coefficient in several equations; v4 changes the title, adds some formal theorem statements, and is published in J Phys A

References in corpus (1)

• Foundations and Measures of Quantum Non-Markovianity

Cited by in corpus (18)

• Open-system approach to nonequilibrium quantum thermodynamics at arbitrary coupling
• Initial Correlations in Open Quantum Systems: Constructing Linear Dynamical Maps and Master Equations
• Pontus-Mpemba effects
• Dynamically Emergent Quantum Thermodynamics: Non-Markovian Otto Cycle
• Thermodynamic Roles of Quantum Environments: From Heat Baths to Work Reservoirs
• Strong coupling non-Markovian quantum thermodynamics of a finite-bath system
• Work, Heat and Internal Energy in Open Quantum Systems: A Comparison of Four Approaches from the Autonomous System Framework
• Unveiling coherent dynamics in non-Markovian open quantum systems: exact expression and recursive perturbation expansion
• Local and global approaches to the thermodynamics of pure decoherence processes in open quantum systems
• A constraint on local definitions of quantum internal energy
• Recursive perturbation approach to time-convolutionless master equations: Explicit construction of generalized Lindblad generators for arbitrary open systems
• Energy additivity as a requirement for universal quantum thermodynamical frameworks
• Local energy assignment for two interacting quantum thermal reservoirs
• Lindblad evolution as gradient flow
• Non-Hermitian topological superconductivity with symmetry-enriched spectral and eigenstate features
• From the Choi Formalism in Infinite Dimensions to Unique Decompositions of Generators of Completely Positive Dynamical Semigroups
• Finite-Bath Open Quantum Systems: Exact Dynamics
• The concept of minimal dissipation and the identification of work in autonomous systems: A view from classical statistical physics