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Communications in Theoretical Physics

Exact analytical results for a two-level system driven by a pulse with Lorentzian frequency and sub-Lorentzian amplitude modulations

Qiongtao Xie

Published Oct 13, 2025• 0 citations
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Abstract

Abstract We investigate the dynamics of a two-level quantum system driven by a laser pulse characterized by Lorentzian frequency and sub-Lorentzian amplitude modulations. Complete analytical solutions, expressed via confluent Heun functions, are derived. Our analysis reveals that explicit exact analytical solutions exist under infinite sets of specific parameter conditions. The effects of modulation parameters and initial conditions on the final transition probabilities are examined analytically and numerically. Furthermore, the method is demonstrated to be directly applicable to two closely related models with Lorentzian pulses.

DOI
10.1088/1572-9494/ae120a

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Abstract We investigate the dynamics of a two-level quantum system driven by a laser pulse characterized by Lorentzian frequency and sub-Lorentzian amplitude modulations. Complete analytical solutions, expressed via confluent Heun functions, are derived. Our analysis reveals that explicit exact analytical solutions exist under infinite sets of specific parameter conditions. The effects of modulation parameters and initial conditions on the final transition probabilities are examined analytically and numerically. Furthermore, the method is demonstrated to be directly applicable to two closely related models with Lorentzian pulses.
This work presents an analytic description of the coherent excitation of a two-state quantum system by an external field with a Lorentzian temporal shape and a constant frequency. An exact analytical solution for the differential equation of the model is derived in terms of Confluent Heun functions. Also a very accurate analytic approximation to the transition probability is derived by using the Dykhne-Davis-Pechukas approach. This approximation provides analytic expressions for the frequency and amplitude of the probability oscillations.
We study the coherent excitation of a two-state quantum system by a field with a Lorentzian temporal envelope and constant carrier frequency. The associated differential equation admits an exact local representation in terms of confluent Heun functions. For the transition probability we develop a Dykhne-Davis-Pechukas (DDP) description based on the two relevant complex transition points and their interference. The DDP action is analyzed directly in the weak- and strong-coupling limits, yielding explicit asymptotic expressions for the oscillation phase, oscillation envelope, far-detuned line shape, and near-resonant behavior. In particular, the strong-coupling asymptotics imply a linewidth proportional to the inverse peak Rabi frequency for resonant odd-πππ pulses, thereby exhibiting the power-narrowing characteristic of the Lorentzian pulse.
We develop the group-theoretical approach for analytically solving the time-dependent Schrodinger equations for multilevel atoms driven by amplitude- and/or frequency-modulated concurrent laser fields. The time evolution of externally driven N-level atoms is calculated in an explicit matrix form. The exact analytical conditions for complete population transfer and its return are simply connected with the parameters of the symmetry. We consider in detail the case of a four-level system driven by concurrent laser pulses of different forms.
We have studied a special class of time-symmetric pulses interacting with a two-level system. These pulses are built out of a class of asymmetric functions for which the analytic description of the interaction was possible. The time behavior of these pulses is given. We have evaluated the transition probability in a closed form. These pulses can be used to simulate the familiar Lorentzian and Gaussian pulse shapes remarkably well. Therefore, they can be used in computing the overall features of the interaction process. Comparisons of the transition amplitudes are made between the closed-form solution and numerically calculated Lorentzian and Gaussian cases.
Analytic solutions for a class of pulses that includes an infinite variety of amplitude and frequency modulations which we previously obtained are applied specifically to the square, Gaussian, Lorentzian, hyperbolic-secant, and the exponential pulses, among others. For all the different types of amplitude- and its corresponding frequency-modulated functions, the outcome of the atomic excitation is given by the same simple analytic formulas. Our analytic solutions apply equally to two-state collision problems.