© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2606.04835 · Physical Review A

Reconciliation of effective Hamiltonians for intense light-matter interaction

Jakob Nicolai Bruhnke, Jan Marcus Dahlström

quant-phphysics.atom-phphysics.comp-phSubmitted Jun 3, 2026• Announced Jun 4, 2026• 0 citations

Abstract

Essential-state models are central for quantum control and technology in broad regimes of light-matter interaction. The canonical effective Hamiltonian is obtained equivalently from adiabatic elimination, the Markov approximation, and the pole approximation. These approximations are known to break down at high intensities, significantly limiting their applicability to moderate light-matter interaction. We show how this limitation can be addressed by applying quasi-degenerate Rayleigh-Schrödinger perturbation theory (QD-RSPT). We reconcile QD-RSPT with adiabatic elimination and propose a quasi-degenerate extension of adiabatic elimination that is robust when the detuning of the essential states is non-negligible. The accuracy of QD-RSPT is demonstrated in both the low- and high-frequency regime, showing excellent agreement with Floquet calculations at high intensities. The crucial corrections to adiabatic elimination make the eigenvectors of the effective Hamiltonian non-orthogonal. Physically, this allows us to account for the asymmetric strength with which different essential states couple to the non-essential states. We expect that our systematic approach to effective Hamiltonians from QD-RSPT will constitute a new state of the art in intense light-matter interaction and quantum optics with novel forms of strong coupling and quantum control phenomena being conceivable.

Comments: 16 pages, 3 figures

Related papers

Adiabatic elimination in strong-field light-matter coupling

Brian Kaufman, Tamás Rozgonyi, Philipp Marquetand +1 more

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 $N$-photon absorption.

Physical Review A•Dec 23, 2020•18 citations

Commutation Relations in Adiabatic Elimination

Hong Ling Xie, He, Le-Wei, Xiu‐Min Lin

The method of adiabatic elimination has been widely adopted in quantum optics in the past several decades. In the study of cavity-based light-matter interactions, the bad-cavity limit is often encountered, where the damping rate of the cavity is much larger than the interaction strength. The fast-damped cavity will quickly relax to a quasi-stationary state, and one can eliminate the cavity from the equation of motion by setting its time derivative to zero. Elimination of the cavity in the bad-cavity limit can reduce the dimensionality of the equations of motion of the system. However, we find that the adiabatic elimination procedure performed in the quantum Langevin equation leads to an incorrect commutation relation, which was rarely discussed in the former studies, as far as we know. Here, we show the incorrect commutation relation arises from the fact that the high frequency of the vacuum noise should be cut off to perform adiabatic elimination, but the noise with high frequency cutoff is still treated as white noise with infinite bandwidth and delta commutation relation. We also study the correlation function and show that the high frequency part of noise contributes very little when averaged over the bath. Therefore, the adiabatic elimination method can reduce the complexity of the calculations while maintaining physical reliability.

arXiv:2411.10664•Nov 16, 2024•0 citations

Effective Hamiltonians in Cavity and Waveguide QED from Transition-Operator Diagrammatic Perturbation Theory

Mohamed Meguebel, Maxime Federico, Louis Garbe +2 more

We propose an adiabatic-elimination formalism in the dispersive regime based on a transition-centric perturbation theory. The perturbative expansion is recast into a diagrammatic framework, while adiabatic elimination is implemented through controlled projections onto transition subspaces. Our approach applies systematically at arbitrary perturbation order, and is suited to multilevel systems and multiple qubits in both cavity and waveguide quantum electrodynamics. It ultimately enables the explicit construction of effective higher-order Hamiltonians while bypassing important limitations of existing techniques, thereby providing a practical toolbox for multiphoton processes in the dispersive regime.

quant-pharXiv:2605.14100•May 15, 2026•0 citations

Quasi-degenerate perturbation theory using matrix product states

Sandeep Sharma, Guillaume Jeanmairet, Ali Alavi

In this work, we generalize the recently proposed matrix product state perturbation theory (MPSPT) for calculating energies of excited states using quasi-degenerate (QD) perturbation theory. Our formulation uses the Kirtman-Certain-Hirschfelder canonical Van Vleck perturbation theory, which gives Hermitian effective Hamiltonians at each order, and also allows one to make use of Wigner's 2n + 1 rule. Further, our formulation satisfies Granovsky's requirement of model space invariance which is important for obtaining smooth potential energy curves. Thus, when we use MPSPT with the Dyall Hamiltonian, we obtain a model space invariant version of quasi-degenerate n-electron valence state perturbation theory (NEVPT), a property that the usual formulation of QD-NEVPT2 based on a multipartitioning technique lacked. We use our method on the benchmark problems of bond breaking of LiF which shows ionic to covalent curve crossing and the twist around the double bond of ethylene where significant valence-Rydberg mixing occurs in the excited states. In accordance with our previous work, we find that multi-reference linearized coupled cluster theory is more accurate than other multi-reference theories of similar cost.

The Journal of Chemical Physics•Jan 19, 2016•52 citations

Prodiabatic Elimination: Higher Order Elimination of Fast Variables with Quantum Noise

Jan Neuser, Marcelo Janovitch, Matteo Brunelli +1 more

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.

quant-pharXiv:2602.21896•Feb 26, 2026•0 citations

Phase shifts in nonresonant coherent excitation

Boyan T. Torosov, Nikolay V. Vitanov

Far-off-resonant pulsed laser fields produce negligible excitation between two atomic states but may induce considerable phase shifts. The acquired phases are usually calculated by using the adiabatic-elimination approximation. We analyze the accuracy of this approximation and derive the conditions for its applicability to the calculation of the phases. We account for various sources of imperfections, ranging from higher terms in the adiabatic-elimination expansion and irreversible population loss to couplings to additional states. We find that as far as the phase shifts are concerned, the adiabatic elimination is accurate only for a very large detuning. We show that the adiabatic approximation is a far more accurate method for evaluating the phase shifts, with a vast domain of validity; the accuracy is further enhanced by superadiabatic corrections, which reduce the error well below ${10}^{\ensuremath{-}4}$. Moreover, owing to the effect of adiabatic population return, the adiabatic and superadiabatic approximations allow one to calculate the phase shifts even for a moderately large detuning and even when the peak Rabi frequency is larger than the detuning; in these regimes the adiabatic elimination is completely inapplicable. We also derive several exact expressions for the phases using exactly soluble two-state and three-state analytical models.

arXiv:0911.4812•Physical Review A•Apr 29, 2009•9 citations

Essential-state models are central for quantum control and technology in broad regimes of light-matter interaction. The canonical effective Hamiltonian is obtained equivalently from adiabatic elimination, the Markov approximation, and the pole approximation. These approximations are known to break down at high intensities, significantly limiting their applicability to moderate light-matter interaction. We show how this limitation can be addressed by applying quasi-degenerate Rayleigh-Schrödinger perturbation theory (QD-RSPT). We reconcile QD-RSPT with adiabatic elimination and propose a quasi-degenerate extension of adiabatic elimination that is robust when the detuning of the essential states is non-negligible. The accuracy of QD-RSPT is demonstrated in both the low- and high-frequency regime, showing excellent agreement with Floquet calculations at high intensities. The crucial corrections to adiabatic elimination make the eigenvectors of the effective Hamiltonian non-orthogonal. Physically, this allows us to account for the asymmetric strength with which different essential states couple to the non-essential states. We expect that our systematic approach to effective Hamiltonians from QD-RSPT will constitute a new state of the art in intense light-matter interaction and quantum optics with novel forms of strong coupling and quantum control phenomena being conceivable.

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

N

-photon absorption.

The method of adiabatic elimination has been widely adopted in quantum optics in the past several decades. In the study of cavity-based light-matter interactions, the bad-cavity limit is often encountered, where the damping rate of the cavity is much larger than the interaction strength. The fast-damped cavity will quickly relax to a quasi-stationary state, and one can eliminate the cavity from the equation of motion by setting its time derivative to zero. Elimination of the cavity in the bad-cavity limit can reduce the dimensionality of the equations of motion of the system. However, we find that the adiabatic elimination procedure performed in the quantum Langevin equation leads to an incorrect commutation relation, which was rarely discussed in the former studies, as far as we know. Here, we show the incorrect commutation relation arises from the fact that the high frequency of the vacuum noise should be cut off to perform adiabatic elimination, but the noise with high frequency cutoff is still treated as white noise with infinite bandwidth and delta commutation relation. We also study the correlation function and show that the high frequency part of noise contributes very little when averaged over the bath. Therefore, the adiabatic elimination method can reduce the complexity of the calculations while maintaining physical reliability.

We propose an adiabatic-elimination formalism in the dispersive regime based on a transition-centric perturbation theory. The perturbative expansion is recast into a diagrammatic framework, while adiabatic elimination is implemented through controlled projections onto transition subspaces. Our approach applies systematically at arbitrary perturbation order, and is suited to multilevel systems and multiple qubits in both cavity and waveguide quantum electrodynamics. It ultimately enables the explicit construction of effective higher-order Hamiltonians while bypassing important limitations of existing techniques, thereby providing a practical toolbox for multiphoton processes in the dispersive regime.

In this work, we generalize the recently proposed matrix product state perturbation theory (MPSPT) for calculating energies of excited states using quasi-degenerate (QD) perturbation theory. Our formulation uses the Kirtman-Certain-Hirschfelder canonical Van Vleck perturbation theory, which gives Hermitian effective Hamiltonians at each order, and also allows one to make use of Wigner's 2n + 1 rule. Further, our formulation satisfies Granovsky's requirement of model space invariance which is important for obtaining smooth potential energy curves. Thus, when we use MPSPT with the Dyall Hamiltonian, we obtain a model space invariant version of quasi-degenerate n-electron valence state perturbation theory (NEVPT), a property that the usual formulation of QD-NEVPT2 based on a multipartitioning technique lacked. We use our method on the benchmark problems of bond breaking of LiF which shows ionic to covalent curve crossing and the twist around the double bond of ethylene where significant valence-Rydberg mixing occurs in the excited states. In accordance with our previous work, we find that multi-reference linearized coupled cluster theory is more accurate than other multi-reference theories of similar cost.

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.

Far-off-resonant pulsed laser fields produce negligible excitation between two atomic states but may induce considerable phase shifts. The acquired phases are usually calculated by using the adiabatic-elimination approximation. We analyze the accuracy of this approximation and derive the conditions for its applicability to the calculation of the phases. We account for various sources of imperfections, ranging from higher terms in the adiabatic-elimination expansion and irreversible population loss to couplings to additional states. We find that as far as the phase shifts are concerned, the adiabatic elimination is accurate only for a very large detuning. We show that the adiabatic approximation is a far more accurate method for evaluating the phase shifts, with a vast domain of validity; the accuracy is further enhanced by superadiabatic corrections, which reduce the error well below ${10}^{\ensuremath{-}4}$. Moreover, owing to the effect of adiabatic population return, the adiabatic and superadiabatic approximations allow one to calculate the phase shifts even for a moderately large detuning and even when the peak Rabi frequency is larger than the detuning; in these regimes the adiabatic elimination is completely inapplicable. We also derive several exact expressions for the phases using exactly soluble two-state and three-state analytical models.