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Journal of the Optical Society of America B

Further generalization of Landau–Zener calculation

C. E. Carroll, F. T. Hioe

Published Aug 1, 1985• 29 citations
Publisher

Abstract

A three-level model is used for an atom driven by two laser beams of constant amplitude; their detunings are proportional to the time. Simple formulas for the resulting transition probabilities are derived analytically. Special cases of these results are the Landau–Zener result for a two-level model and our earlier result for a symmetrical three-level model.

DOI
10.1364/josab.2.001355

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A three-level model is used for an atom driven by two laser beams of constant amplitude; their detunings are proportional to the time. Simple formulas for the resulting transition probabilities are derived analytically. Special cases of these results are the Landau–Zener result for a two-level model and our earlier result for a symmetrical three-level model.
For an atom or molecule in which two transitions are driven by laser beams, a three-state model is used. Transitions among the three states are caused only by the oscillating electric fields of the laser beams. The amplitudes and detunings of the two laser beams, which are various functions of the time, appear in the Schrödinger equation for the atom or molecule. In certain cases, the Schrödinger equation can be solved analytically, to find transition probabilities and the probability of no transition. This is done by using Clausen’s special function, or by assuming that the sum of the two detunings is zero at all times. Conditions for complete transfer of the population from the ground state to an excited state are obtained from the analytic solutions.
Abstract The Landau–Zener formula describes the diabatic transition probability of a two-level system under linear driving. Its rigorous derivation typically relies on sophisticated mathematical tools, such as special functions, Laplace transforms, or contour integrals. In this work, we present a derivation of the Landau–Zener transition probability using a fundamentally different approach via functional equations. By leveraging integrability, we prove that this transition probability satisfies a functional equation, whose solutions establish the exponential form of the formula. The coefficient in the exponent is then determined through a lowest-order perturbation calculation. This derivation is rigorous and does not involve any sophisticated mathematics. Our work provides insights into the origin of the exponential form of the Landau–Zener transition probability, and shows that the Landau–Zener formula can be viewed as a consequence of integrability, though the two-level Landau–Zener Hamiltonian itself does not satisfy the integrability conditions.
We use a three-state model for an atom or molecule in which two transitions are simultaneously driven by the oscillating electric fields of two laser beams. The amplitudes and detunings of the applied oscillating fields vary during the optical pulse, which can cause transitions from one state to another. For some special cases not previously known, the transition probabilities and probabilities of no transition are obtained analytically. We give conditions for complete transfer of the atomic population from one state to another, and for complete return to the initial state.