Adiabatic elimination is a model reduction technique commonly used by physicists to eliminate quickly dissipating components from quantum physics equations. We revisit this technique when the target non-dissipating component is driven by Hamiltonian actuation at a fast timescale. Following center manifold theory, we can still write reduced dynamics for the target component, but there may be new conditions to ensure that it takes the standard structure of quantum dynamics, i.e. evolution equations of Lindblad type and coordinate changes in Kraus map form. We here propose various approaches to recover a Lindblad form up to third-order terms: without conditions for finite-dimensional systems, and under some finite set conditions for infinite-dimensional systems.
We consider an open quantum system described by a Lindblad-type master equation with two time scales. The fast time scale is strongly dissipative and drives the system towards a low-dimensional decoherence-free space. To perform the adiabatic elimination of this fast relaxation, we propose a geometric asymptotic expansion based on the small positive parameter describing the time scale separation. This expansion exploits geometric singular perturbation theory and center-manifold techniques. We conjecture that, at any order, it provides an effective slow Lindblad master equation and a completely positive parameterization of the slow invariant sub-manifold associated to the low-dimensional decoherence-free space. By preserving complete positivity and trace, two important structural properties attached to open quantum dynamics, we obtain a reduced-order model that directly conveys a physical interpretation since it relies on effective Lindbladian descriptions of the slow evolution. At the first order, we derive simple formulae for the effective Lindblad master equation. For a specific type of fast dissipation, we show how any Hamiltonian perturbation yields Lindbladian second-order corrections to the first-order slow evolution governed by the Zeno-Hamiltonian. These results are illustrated on a composite system made of a strongly dissipative harmonic oscillator, the ancilla, weakly coupled to another quantum system.
Abstract We consider a composite open quantum system consisting of a fast subsystem coupled to a slow one. Using the time scale separation, we develop an adiabatic elimination technique to derive at any order the reduced model describing the slow subsystem. The method, based on an asymptotic expansion and geometric singular perturbation theory, ensures the physical interpretation of the reduced second-order model by giving the reduced dynamics in a Lindblad form and the state reduction in Kraus map form. We give explicit second-order formulas for Hamiltonian or cascade coupling between the two subsystems. These formulas can be used to engineer, via a careful choice of the fast subsystem, the Hamiltonian and Lindbald operators governing the dissipative dynamics of the slow subsystem.
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.