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arXiv:2404.01802 · IEEE Conference on Decision and Control (CDC)

Explicit formulas for adiabatic elimination with fast unitary dynamics

Angela Riva, Alain Sarlette, Pierre Rouchon

Published Dec 16, 2024• 0 citations
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Abstract

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.

DOI
10.1109/cdc56724.2024.10886784

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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.
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.
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.
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.
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.
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.
We introduce the prodiabatic elimination, a powerful approximation technique that systematically extends the adiabatic elimination of fast degrees of freedom in light-matter coupled systems. Through a controlled expansion of operators, the prodiabatic elimination incorporates higher-order corrections and consistently includes noise contributions, leading to a significantly improved performance compared to standard adiabatic elimination. Importantly, it retains the simplicity and computational efficiency of the adiabatic elimination, making it convenient for practical applications. We demonstrate the approach on two setups: a driven dissipative Jaynes-Cummings model and a three-level system in a two-mode cavity that performs stimulated Raman adiabatic passage (STIRAP). These examples establish the prodiabatic elimination as a robust and broadly applicable tool for analyzing open quantum systems.