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Molecular Physics

Phase and temporal average transition probabilities for a multi-level system in a sinusoidal field

J.V. Moloney, William J. Meath

Published May 1, 1976• 55 citations
Publisher

Abstract

A previous exact formal and practical solution for the time-dependent Schrödinger equation for a two-level non-degenerate system in an arbitrarily strong sinusoidal field is transformed for an N-level system to a Floquet form. Using the Floquet form of the solution computationally convenient expressions are obtained for the phase and the phase-long time-averaged induced transition probabilities. The method exploits the time periodicity of the Schrödinger equation and the solution over the initial period of the hamiltonian provides all the information needed to obtain both the unaveraged and the averaged transition probabilities for the system. The method is illustrated by specific examples which are used to discuss some of the temporal features of the multi-photon spectra of two-level systems.

DOI
10.1080/00268977600101211

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quant-pharXiv:2511.03977•Nov 7, 2025•0 citations
A previous exact formal and practical solution for the time-dependent Schrödinger equation for a two-level non-degenerate system in an arbitrarily strong sinusoidal field is transformed for an N-level system to a Floquet form. Using the Floquet form of the solution computationally convenient expressions are obtained for the phase and the phase-long time-averaged induced transition probabilities. The method exploits the time periodicity of the Schrödinger equation and the solution over the initial period of the hamiltonian provides all the information needed to obtain both the unaveraged and the averaged transition probabilities for the system. The method is illustrated by specific examples which are used to discuss some of the temporal features of the multi-photon spectra of two-level systems.
A solution for the time-dependent Schrödinger equation is given for an N-level system interacting with a sinusoidal field of arbitrary amplitude , frequency v and phase δ, in the presence of an applied static field of arbitrary strength . The solution is obtained by a modification of a previous exact method of solution for and retains the essential analytic features of the exact wavefunction for the problem. The Floquet form of the solution yields (a) the location and the widths of allowed transitions and the location of forbidden transitions, by use of characteristic exponent plots, but without evaluating the spectrum, and (b) convenient expressions for the steady-state induced transition probabilities, including the effects of a uniform relaxation mechanism, that only require the solution of the wave equation over the initial period of the hamiltonian. The approach is equally applicable to Stark (or Zeeman) frequency sweep or tuning problems. The ability to incorporate the effects of static fields in the solution is important since a variety of problems (e.g. anti-crossing spectroscopy) involve the application of such fields to a level configuration interacting with a sinusoidal field. As a detailed specific example, the multi-photon steady-state induced transition probabilities and characteristic exponents are studied, as a function of frequency v, for the unperturbed and the perturbed two-level system under saturating conditions. It is also demonstrated, for the coupling strengths of interest in multi-photon calculations, that the method of solution discussed here is a viable unified alternative to dressed-atom techniques.
An exact method of solution to the time-dependent wave equation for a system interacting with a sinusoidal field, which formally treats time and the phase of the field on an equal footing, is used to discuss (1) the importance of averaging the properties of a system over the phase of the applied field and (2) the nature and behavior of the characteristic exponents derived from the Floquet solution (with particular emphasis on the frequency-sweep experiment). The results for the Zeeman tuning experiment are used to resolve recent discrepancies in the literature which relate to the validity of Shirley's important result for the two-level average induced transition probability involving the derivative of the characteristic exponents with respect to the Zeeman splitting parameter ${\ensuremath{\omega}}_{0}$. The explicit calculations included in this work are for the single-photon two-level problem. Important implications can be inferred from them with respect to both the importance of phase averaging and the usefulness of the characteristic exponents as a quantitative means of obtaining resonance frequency shifts and half-widths for multiphoton and multilevel problems.
The phase and the rotational averaging of single and multi-photon transition probabilities for the interaction of an atom or molecule with an applied sinusoidal field is discussed. It is shown that while the correct interval for carrying out the phase average is 2π in general, a phase interval of π is satisfactory under certain specified conditions. Illustrative two-level examples indicate that if these conditions are not satisfied, the two choices of phase interval can lead to significant differences in the phase averaged temporal transition probabilities which are subsequently essentially removed upon long-time averaging. A numerical scheme for evaluating rotational averaged transition probabilities is developed and illustrated for model calculations involving two-level molecular systems. The examples are chosen to illustrate the effects of rotational averaging, both when a molecule possesses non-zero diagonal dipole matrix elements (‘permanent dipole moments’) and when it does not, on single and multi-photon spectra. Often resonance profiles are sharpened, Bloch Siegert shifts and resonance maxima reduced, and the structure due to diagonal dipole matrix elemnts eliminated, upon rotational averaging. A recently developed rotating wave or Rabi type approximation is used to explain the effects of diagonal dipole matrix elements for both the fixed orientation and rotationally averaged spectra.
The interaction of a quantum system with an oscillating field is studied in a formalism which replaces the semiclassical time-dependent Hamiltonian with a time-independent Hamiltonian represented by an infinite matrix. The formalism is developed as a mathematical equivalent to the semiclassical treatment, and interpreted as a classical approximation to the quantum treatment of the field. Combined with a perturbation theory for two nearly degenerate states, the formalism provides a convenient method for determining resonance transition probabilities including frequency shifts and multiple quantum transitions. The theory is illustrated by a detailed study of the simple case of a two-state system excited by a strong oscillating field.
In a famous 1965 paper [Phys. Rev. 138, B979], J. H. Shirley calculates the time-averaged transition probability for an atom in a periodic field in terms of the eigenvectors and eigenvalues of the time-independent Floquet Hamiltonian. It is shown in the present paper that, for the specific case considered by Shirley, a two-state atom in a sinusoidal field, Shirley's solution can be simplified considerably. Also, Shirley's results are generalized to include, for an arbitrary atom-field system, the time-varying components of the transition probability as well.
In this manuscript, we introduce an exact expression for the response of a semi-classical two-level quantum system subject to arbitrary periodic driving. Determining the transition probabilities of a two-level system driven by an arbitrary periodic waveform necessitates numerical calculations through methods such as Floquet theory, requiring the truncation of an infinite matrix. However, such truncation can lead to a loss of significant interference information, hindering quantum sensors or introducing artifacts in quantum control. To alleviate this issue, we use the ⋆\star⋆-resolvent formalism with the path-sum theorem to determine the exact series solution to Schrödinger's equation, therefore providing the exact transition probability. The resulting series solution is generated from a compact kernel expression containing all of the information of the periodic drive and then expanded in a non-harmonic Fourier series basis given by the divided difference of complex exponentials with coefficients corresponding to products of generalized Bessel functions. The present method provides an analytical formulation for quantum sensors and control applications.