We provide model reduction formulas for open quantum systems consisting of a target component which weakly interacts with a strongly dissipative environment. The time-scale separation between the uncoupled dynamics and the interaction allows to employ tools from center manifold theory and geometric singular perturbation theory to eliminate the variables associated to the environment (adiabatic elimination) with high-order accuracy. An important specificity is to preserve the quantum structure: reduced dynamics in (positive) Lindblad form and coordinate mappings in Kraus form. We provide formulas of the reduced dynamics. The main contributions of this paper are (i) to show how the decomposition of the environment into
K components enables its efficient treatment, avoiding the quantum curse of dimension; and (ii) to extend the results to the case where the target component is subject to Hamiltonian evolution at the fast time-scale. We apply our theory to a microwave superconducting quantum resonator subject to material losses, and we show that our reduced-order model can explain the transmission spectrum observed in a recent pump probe experiment.
This thesis develops model reduction methods for open quantum system dynamics, yielding efficient models for bipartite systems, with direct application to circuit QED and broader applicability beyond this platform. First, we consider dissipative dynamics characterized by timescale separation, which translates to a gap on the real axis of the spectrum of the Liouvillian generating it. This allows to compute the reduced evolution for a target subsystem by adiabatically eliminating fast-decaying degrees of freedom. A Sylvester equation integral representation yields closed forms for the reduced generators and explicit reconstruction maps to arbitrary order without a preliminary interaction-picture averaging step, providing a linear center manifold description that separates long-lived from fast-decaying modes by the real parts of the spectrum. The reduced dynamics takes place on the center manifold, that is, the spectral subspace associated with eigenvalues whose real parts are small (long-lived modes), while placing no restriction on imaginary parts, thus allowing for fast unitary dynamics on the target subsystem. We further demonstrate the equivalence of such a geometric approach to adiabatic elimination with the time-convolutionless master equation framework, by showing that the maps defining the reduced model satisfy the same invariance equation in both frameworks. The time-convolutionless formulation of adiabatic elimination extends to cases where the target system undergoes fast unitary dynamics and retains explicit time dependence in the reduced model, hence capturing initial transients that are neglected in time-independent geometric reductions. Second, we apply a unitary chain mapping of microscopic system–environment Hamiltonians to a nearest-neighbor form, enabling efficient time evolution exploiting tensor network time evolution algorithms based on the time dependent variational principle. Accuracy of the tensor network representation, and thus the size of the reduced model, is controlled by entanglement entropy growth, which we quantify numerically. While the methods are general and apply beyond the specific platform considered, we focus on dispersive readout in circuit QED (a qubit dipolarly coupled to a driven, lossy cavity). We quantify the dependence of the relaxation time
T1 of the qubit on intracavity photon number
nˉ (and thus on the drive amplitude) and on the spectral structure of the environment, and compute the emission spectrum directly from environmental observables. Simulations including a Purcell notch filter predict a decrease of
T1 with increasing drive power, a qualitatively different behavior with respect to Born–Markov–secular GKSL predictions but consistent with experimental trends, indicating that explicit filter modeling is required in this case. Together, the results specify when reduced Liouvillian models under a dissipative gap are reliable and when microscopic, non-perturbative simulations are required.
We present an algebraic framework for approximate model reduction of Markovian open quantum dynamics that guarantees complete positivity and trace preservation by construction. First, we show that projecting a Lindblad generator on its center manifold -- the space spanned by eigenoperators with purely imaginary eigenvalue -- yields an asymptotically exact reduced quantum dynamical semigroup whose dynamics is unitary, with exponentially decaying transient error controlled by the generator's spectral gap. Second, for analytic perturbations of a Lindblad generator with a tractable center manifold, we propose a perturbative reduction that keeps the reduced space fixed at the unperturbed center manifold. The resulting generator is shown to remain a valid Lindbladian; explicit finite-time error bounds that quantify leakage from the unperturbed center sector are provided. We further clarify the connection to adiabatic elimination methods, by both showing how the algebraic reduction can be directly related to a first-order adiabatic-elimination and by providing sufficient conditions under which the latter method can be applied while preserving complete positivity. We showcase the usefulness of our techniques in dissipative many-body quantum systems exhibiting non-stationary long-time dynamics.
Consider an open quantum system governed by a Gorini-Kossakowski-Sudarshan-Lindblad master equation with two times-scales: a fast one, exponentially converging towards a linear subspace of quasi-equilibria; a slow one resulting from small decoherence and Hamiltonian dynamics. Usually adiabatic elimination is performed in the Schrödinger picture. We propose here a Heisenberg formulation where the invariant operators attached to the fast decay dynamics towards the quasi-equilibria subspace play a key role. Based on geometric singular perturbations, asymptotic expansions of the Heisenberg slow dynamics and of the fast invariant linear subspaces are proposed. They exploit Carr's approximation lemma from center-manifold and bifurcation theory. Second-order expansions are detailed and shown to ensure preservation, up to second-order terms, of the complete positivity for the slow propagator on a slow timescale. Such expansions can be exploited numerically to derive reduced-order dynamical models.
The so-called “adiabatic elimination” of fast decaying degrees of freedom in open quantum systems can be performed with a series expansion in the timescale separation. The associated computations are significantly more difficult when the remaining degrees of freedom (center manifold) follow fast unitary dynamics instead of just being slow. This paper highlights how a formulation with Sylvester’s equation and with adjoint dynamics leads to systematic, explicit expressions at high orders for settings of physical interest.
Adiabatic elimination methods allow the reduction of the space dimension needed to describe systems whose dynamics exhibit separation of timescale. For open quantum systems, it consists in eliminating the fast part assuming it has almost instantaneously reached its steady state and obtaining an approximation of the evolution of the slow part. These methods can be applied to eliminate a linear subspace within the system Hilbert space or, alternatively, to eliminate a fast subsystem in a bipartite quantum system. In this work, we extend an adiabatic elimination method used for removing fast degrees of freedom within a system [Phys. Rev. A 101, 042102 (2020)] to eliminate a subsystem from an open bipartite quantum system. As an illustration, we apply our technique to a dispersively coupled two-qubit system and in the case of the open Rabi model.