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Structure-preserving adiabatic elimination for open bipartite quantum systems

Rémi Azouit, Francesca Carlotta Chittaro, Alain Sarlette, Pierre Rouchon

Published Jul 1, 2017• 4 citations
Publisher

Abstract

We consider a quantum system composed of a fast quantum subsystem coupled to a slow one. We provide expressions of the approximate reduced model describing the slow subsystem (adiabatic elimination), based on an asymptotic expansion that we solve up to second order. The specificity of our expressions is to preserve the quantum structure in the reduced model: we provide the reduced dynamics in Lindblad form, and the mapping defining the slow manifold as a completely positive map in Kraus form.

DOI
10.1016/j.ifacol.2017.08.2000

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We consider a quantum system composed of a fast quantum subsystem coupled to a slow one. We provide expressions of the approximate reduced model describing the slow subsystem (adiabatic elimination), based on an asymptotic expansion that we solve up to second order. The specificity of our expressions is to preserve the quantum structure in the reduced model: we provide the reduced dynamics in Lindblad form, and the mapping defining the slow manifold as a completely positive map in Kraus form.
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.
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.
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.
This paper discusses quantum adiabatic elimination, which is a model reduction technique for a composite Lindblad system consisting of a fast decaying subsystem coupled to another subsystem with a much slower timescale. Such a system features an invariant manifold that is close to the slow subsystem. This invariant manifold is reached subsequent to the decay of the fast degrees of freedom, after which the slow dynamics follow on it. By parametrizing the invariant manifold, the slow dynamics can be simulated via a reduced model. To find the evolution of the reduced state, we perform an asymptotic expansion with respect to the timescale separation. So far, the second-order expansion has mostly been considered. It has then been revealed that the second-order expansion of the reduced dynamics is generally given by a Lindblad equation, which ensures complete positivity of the time evolution. In this paper, we present two examples where complete positivity of the reduced dynamics is violated with higher-order contributions. In the first example, the violation is detected for the evolution of the partial trace without truncation of the asymptotic expansion. The partial trace is not the only way to parametrize the slow dynamics. Concerning this nonuniqueness, it was conjectured in [Quantum Sci. Technol. 2, 044011 (2017)] that there exists a parameter choice ensuring complete positivity. With the second example, however, we refute this conjecture by showing that complete positivity cannot be restored in any choice of parametrization. We discuss these results in terms of the invariant slow manifold consisting of quantum correlated states.
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.