[Paper Review] Lindblad master equations for quantum systems coupled to dissipative bosonic modes

Simon B. Jäger, Tom Schmit+3 more|arXiv (Cornell University)|Mar 7, 2022

Cold Atom Physics and Bose-Einstein Condensates63 references53 citations

TL;DR

This paper presents a general method to derive Lindblad master equations for quantum systems coupled to dissipative bosonic modes by employing a Schrieffer-Wolff-type transformation that eliminates the bosonic degrees of freedom while preserving complete positivity. The approach self-consistently determines effective fields and yields a valid Lindblad-form master equation that accurately captures the dissipative Dicke model's phase transition, steady state, and quantum metastability beyond mean-field theory, validated via exact diagonalization and stochastic simulations.

ABSTRACT

We present a general approach to derive Lindblad master equations for a subsystem whose dynamics is coupled to dissipative bosonic modes. The derivation relies on a Schrieffer-Wolff transformation which allows to eliminate the bosonic degrees of freedom after self-consistently determining their state as a function of the coupled quantum system. We apply this formalism to the dissipative Dicke model and derive a Lindblad master equation for the atomic spins, which includes the coherent and dissipative interactions mediated by the bosonic mode. This master equation accurately predicts the Dicke phase transition and gives the correct steady state. In addition, we compare the dynamics using exact diagonalization and numerical integration of the master equation with the predictions of semiclassical trajectories. We finally test the performance of our formalism by studying the relaxation of a NOON state and show that the dynamics captures quantum metastability beyond the mean-field approximation.

Motivation & Objective

• To develop a general, systematic method for deriving effective Lindblad master equations for open quantum systems coupled to dissipative bosonic modes.
• To ensure the derived master equations preserve complete positivity, a key requirement for physical consistency.
• To accurately describe non-equilibrium dynamics, including critical phenomena and metastable states, beyond mean-field approximations.
• To benchmark the formalism against exact diagonalization and stochastic simulations in the context of the dissipative Dicke model.

Proposed method

• Utilizes a displacement transformation via the operator ˆD = exp(∑k(ˆa†kˆαk − ˆα†kˆak)) to displace the bosonic modes based on the system's eigenstates and energies.
• Applies perturbation theory up to third order in ϵ = ∥ˆαk∥ ≪ 1 to systematically eliminate bosonic degrees of freedom while preserving the system's dynamics.
• Derives an effective master equation in the form ∂ˆρsys/∂t = Leffˆρsys = −i[ˆHeff, ˆρsys] + ∑k κk D[ˆαk]ˆρsys, with ˆHeff = ˆHS + ½∑k (ˆα†k ˆSk + ˆS†k ˆαk).
• Solves for the effective fields ˆαk via the equation ∂ˆαk/∂t = −i[ˆHS, ˆαk] −i∑k′ ˆΩk,k′S ˆαk′ −i ˆSk −κk ˆαk, which includes retardation effects from the system's evolution.
• Validates the formalism by comparing dynamics from the effective master equation with exact diagonalization and stochastic trajectory simulations.

Experimental results

Research questions

• RQ1 Can a general, physically consistent Lindblad master equation be derived for a quantum system coupled to dissipative bosonic modes?
• RQ2 Does the derived master equation correctly capture the Dicke phase transition and the system's steady state?
• RQ3 Can the formalism describe quantum metastability beyond the mean-field approximation?
• RQ4 How does the inclusion of retardation effects in the effective fields influence the dynamics and critical behavior?

Key findings

• The derived effective master equation is in Lindblad form, ensuring complete positivity and physical consistency for bounded effective fields.
• The formalism accurately predicts the Dicke phase transition and the correct steady-state population distribution in the dissipative Dicke model.
• The dynamics from the effective master equation agree quantitatively with exact diagonalization results, validating its accuracy.
• The stochastic simulation based on the effective dynamics reproduces the correct moments of spin operators, including the Z-polarized and NOON state initializations.
• The formalism captures quantum metastability in the NOON state dynamics, including fast oscillations not accessible via mean-field theory.
• The mean-field approximation derived from the effective master equation yields the correct dynamical critical exponent ν = 1 for the dissipative Dicke model, with ν = 0.5 in the non-dissipative limit.