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Physical Review A

Adiabatic elimination in strong-field light-matter coupling

Brian Kaufman, Tamás Rozgonyi, Philipp Marquetand, Thomas C. Weinacht

Published Dec 23, 2020• 18 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.102.063117

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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.
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.
Driven Raman processes — nearly resonant two-photon transitions through an intermediate state that is non-resonantly coupled and does not acquire a sizeable population — are commonly treated with a simplified description in which the intermediate state is removed by adiabatic elimination. While the adiabatic-elimination approximation is reliable when the detuning of the intermediate state is quite large, it cannot be trusted in other situations, and it does not allow one to estimate the population in the eliminated state. We introduce an alternative method that keeps all states in the description, without increasing the complexity by much. An integro-differential equation of Lippmann-Schwinger type generates a hierarchy of approximations, but very accurate results are already obtained in the lowest order.
A nonperturbative analysis is given for an nnn-level atom or molecule dressed by a Stark field and in single-photon or multiphoton (near) resonant interaction with coherent optical field modes, which may be intense. All the field amplitudes and phases may have nonadiabatic time variation. New formulas for probability amplitudes of states with finite lifetimes are given for atom-field levels swept or optical Stark shifted through resonance or into close approach. One of these includes the Landau-Zener formula as a special case. Power broadening is contained in these formulas. They are useful for both weak- and strong-field interaction. Further, it is shown that many familiar single-photon coherent transient phenomena have their analogs in the intense-field and the multiphoton domains. Two new effects, pulse-shape dependence in final transition probability and population inversion due to the nonadiabatic passage of an asymmetric pulse, are demonstrated with numerical results for the process $\mathrm{Na}(3s)+2\ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\rightarrow}\mathrm{Na}(5s)$. Finally, double resonance (single-photon and/or multiphoton) is analyzed, with discussion of adiabatic following and a derivation of a formula for level crossing. Generalization of an earlier procedure for solving the adiabatic eigenvalue problem of the whole atom-field system is given.
We study the general problem of matter-laser interaction within the framework of nonrelativistic quantum electrodynamics, with particular emphasis on strong laser field effects. Consequently, we formulate a well-defined approximation leading in a straightforward manner toward the conventional semiclassical mode of description. Namely, we arrive naturally to two coupled equations of motion: (i) the Schr\"odinger equation which governs the quantum dynamics of an atomic system driven by classically described radiation field (composed of an incoming laser pulse plus radiation emitted from the atom), and (ii) the classical Maxwell wave equation which describes the emission of radiation from the mentioned atomic source. Employing the formalism of adiabatic Floquet theory, we derive a simple criterion of validity of the just described semiclassical approach. It shows that the semiclassical treatment is justified in most situations. On the other hand, it turns out that the semiclassical approximation breaks down completely in certain special but realistic cases, regardless of the fact that the incoming laser pulse contains a huge number of photons. Under such special circumstances, we anticipate new effects arising due to the quantized nature of the radiation field, to be observable, for example, in harmonic generation spectra. Our considerations are illustrated more explicitly using a simple model of a two-level atom strongly driven by a laser. The quantum dynamics of this model problem is resolved within the framework of quantum electrodynamics while adopting well-defined and physically justifiable approximations. As an outcome, analytic formulas are found serving as a quantitative criterion of (non)applicability of the semiclassical approach and demonstrating the breakdown of semiclassical theory under well-defined conditions. An illustrative numerical calculation is provided.
Driven Raman processes – nearly resonant two-photon transitions through an intermediate state that is non-resonantly coupled and does not acquire a sizeable population – are commonly treated with a simplified description in which the intermediate state is removed by adiabatic elimination. While the adiabatic-elimination approximation is reliable when the detuning of the intermediate state is quite large, it cannot be trusted in other situations, and it does not allow one to estimate the population in the eliminated state. We introduce an alternative method that keeps all states in the description, without increasing the complexity by much. An integro-differential equation of Lippmann–Schwinger type generates a hierarchy of approximations, but very accurate results are already obtained in the lowest order.