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arXiv:2603.01826

Time-dependent adiabatic elimination in matter-wave optics

Samuel Böhringer, Alexander Bott, Eric P. Glasbrenner

quant-phSubmitted Mar 2, 2026• Announced Mar 3, 2026• 0 citations
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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 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.

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The following article has been submitted to AVS Quantum Science

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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 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.
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.
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.
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.
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.
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.
We explore the validity of adiabatic elimination in the derivation of an essential-states representation of the time-dependent Schr\"odinger equation in the presence of a strong laser field. We consider the elimination of off-resonant states in generating an effective two-level description of the light-matter interaction, where the initial and final states are two-photon resonant. The treatment is nonperturbative and can be generalized to NNN-photon absorption.