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Time-dependent adiabatic elimination in matter-wave optics

Samuel Böhringer, Alexander Bott, Eric P. Glasbrenner

Published May 19, 2026• 0 citations
Publisher

Abstract

We show how the dynamics of a specific subset of states can be separated from the dynamic of the total quantum state via a time-dependent projector-based formalism of adiabatic elimination. Within our formalism, we assume an explicit time dependency in the coupling between both subsystems. Additionally, we do not assume that the elements of the Hamiltonian commute, as in matter-wave optics, this is not given in general. Here, the center-of-mass degrees of freedom frequently need to be taken into account. Assuming non-commutativity also enables the application of this formalism to matrices of arbitrary dimension and the treatment of the adiabatic elimination in high dimensional systems, where the elimination may be needed to reduce the overall complexity. In particular, we apply our formalism to typical mechanisms in matter-wave optics, Raman, and Bragg diffraction.

DOI
10.1116/5.0304521

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We show how the dynamics of a specific subset of states can be separated from the dynamic of the total quantum state via a time-dependent projector-based formalism of adiabatic elimination. Within our formalism, we assume an explicit time dependency in the coupling between both subsystems. Additionally, we do not assume that the elements of the Hamiltonian commute, as in matter-wave optics, this is not given in general. Here, the center-of-mass degrees of freedom frequently need to be taken into account. Assuming non-commutativity also enables the application of this formalism to matrices of arbitrary dimension and the treatment of the adiabatic elimination in high dimensional systems, where the elimination may be needed to reduce the overall complexity. In particular, we apply our formalism to typical mechanisms in matter-wave optics, Raman, and Bragg diffraction.
We show how the dynamics of a specific subset of states can be separated from the dynamic of the total quantum state via a time-dependent projector-based formalism of adiabatic elimination. Within our formalism, we assume explicit time dependency in the coupling between both subsystems. Additionally, we do not assume that the elements of the Hamiltonian commute, as in matter-wave optics this not given in general. Here the center-of-mass degrees of freedom frequently need to be taken into account. Our formalism allows to perform the adiabatic elimination in such a setting.
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.
In open quantum systems theory, reduced models are invaluable for conceptual understanding and computational efficiency. Adiabatic elimination is a useful model reduction method for systems with separated timescales, where a reduced model is derived by discarding rapidly decaying degrees of freedom. So far, adiabatic elimination has been formulated using a geometric approach, which provides a versatile and general framework. This article introduces a reformulation of adiabatic elimination through the framework of the time-convolutionless (TCL) master equation, a widely recognized tool for computing projected time evolution in open quantum systems. We show that the TCL master equation formulation yields results equivalent to those obtained from the geometric formulation. By applying the TCL master equation formulation to typical examples, we demonstrate a practical methodology for performing adiabatic elimination calculation. This study not only bridges two previously independent approaches, thereby making the adiabatic elimination method accessible to a broader audience, but also enables the analysis of complex cases that are challenging within the geometric formulation. Additionally, it reveals a geometric interpretation of the TCL master equation formalism.
Most quantum theorists are familiar with different ways of describing the effective quantum dynamics of a system coupled to external degrees of freedom, such as the Born-Markov master equation or the adiabatic elimination. Understanding the deep connection between these -- sometimes apparently unrelated -- methods can be a powerful tool, allowing us to derive effective dynamics in unconventional systems or regimes. This tutorial aims at providing quantum theorists across multiple fields (e.g., quantum and atom optics, optomechanics, or hybrid quantum systems) with a self-contained practical toolbox to derive effective quantum dynamics, applicable to systems ranging from N -level emitters to mechanical resonators. First, we summarize the projector approach to open quantum systems and the derivation of the fundamental Nakajima-Zwanzig equation. Then, we show how three common effective equations, namely the Brownian master equation, the Born-Markov master equation, and the adiabatic elimination used in atom and molecular optics, can be derived from different perturbative expansions of the Nakajima-Zwanzig equation. We also solve in detail four specific examples using this formalism, namely a harmonic oscillator subject to displacement noise, the effective equations of a mechanical resonator cooled by an optical cavity, the Purcell effect for a qubit coupled to an optical cavity, and the adiabatic elimination in a Lambda system.
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.
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 KKK 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.