A canonical Hamiltonian for open quantum systems

TL;DR

This paper introduces a unique, canonical Hamiltonian for open quantum systems by minimizing the dissipative part of the dynamics, extending Lindblad's initial Hamiltonian and providing a recursive scheme for perturbative calculations.

Contribution

It defines a canonical Hamiltonian for open quantum systems by fixing the non-uniqueness in the division of dynamics, and extends the concept to perturbative regimes.

Findings

The canonical Hamiltonian is equivalent to Lindblad's original Hamiltonian.

The canonical Hamiltonian is uniquely specified by traceless jump operators.

Provides a recursive formula for perturbative calculation of the effective Hamiltonian.

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.