[Paper Review] A periodic level-crossing two-state model of a general Heun class

G. Saget, А. М. Ishkhanyan+2 more|arXiv (Cornell University)|Jul 8, 2017

Spectroscopy and Quantum Chemical Studies23 references3 citations

TL;DR

This paper introduces a periodic level-crossing two-state quantum model within the general Heun class, deriving an exact analytic solution using incomplete Beta function series expansions. The key contribution is an infinite hierarchy of finite-sum closed-form solutions, with one unconditionally integrable case where Rabi frequency and detuning are independently controlled, enabling exact Floquet analysis and population dynamics study for Dirac delta-comb detuning configurations.

ABSTRACT

We present a specific constant-amplitude periodic level-crossing model of the semi-classical quantum time-dependent two-state problem that belongs to a general Heun class of field configurations. The exact analytic solution for the probability amplitude, generally written for this class in terms of the general Heun functions, in this specific case admits series expansion in terms of the incomplete Beta functions. Terminating this series results in an infinite hierarchy of finite-sum closed-form solutions each standing for a particular two-state model, which generally is only conditionally integrable in the sense that for these field configurations the amplitude and phase modulation functions are not varied independently. However, there exists at least one exception when the model is unconditionally integrable, that is the Rabi frequency and the detuning of the driving optical field are controlled independently. This is a constant-amplitude periodic level-crossing model, for which the detuning in a limit becomes a Dirac delta-comb configuration with variable frequency of the level-crossings. We derive the exact solution for this model, determine the Floquet exponents and study the population dynamics in the system for various regions of the input parameters.

Motivation & Objective

• To develop a solvable model within the general Heun class for time-dependent two-state quantum systems with periodic level crossings.
• To derive an exact analytic solution for the probability amplitude using incomplete Beta functions.
• To identify conditions under which the model becomes unconditionally integrable, allowing independent control of Rabi frequency and detuning.
• To analyze the population dynamics and Floquet exponents in the system under various parameter regimes.
• To explore the limiting case where detuning becomes a Dirac delta-comb, enabling exact solvability.

Proposed method

• Formulating a constant-amplitude periodic level-crossing model as a specific instance of the general Heun class.
• Expressing the exact solution for the probability amplitude as a series in incomplete Beta functions.
• Truncating the series to obtain finite-sum closed-form solutions for specific parameter sets.
• Identifying the exceptional case where the model is unconditionally integrable, with independent modulation of Rabi frequency and detuning.
• Applying Floquet theory to analyze the system's quasi-energy spectrum and population dynamics.
• Deriving the exact solution for the limit where detuning becomes a Dirac delta-comb with variable level-crossing frequency.

Experimental results

Research questions

• RQ1 Can a periodic level-crossing two-state model in the general Heun class be solved exactly using special functions?
• RQ2 Under what conditions does the model allow independent control of Rabi frequency and detuning, enabling unconditional integrability?
• RQ3 What is the structure of the exact solution when the detuning approaches a Dirac delta-comb configuration?
• RQ4 How do the Floquet exponents and population dynamics behave across different parameter regions in the unconditionally integrable case?
• RQ5 What is the role of incomplete Beta functions in the series expansion of the probability amplitude?

Key findings

• The exact solution for the probability amplitude is expressed as a convergent series in incomplete Beta functions, enabling finite-sum closed-form solutions upon truncation.
• An infinite hierarchy of finite-sum solutions is derived, each corresponding to a specific conditionally integrable two-state model.
• One exceptional case exists where the model is unconditionally integrable, allowing independent control of Rabi frequency and detuning.
• In the limit of a Dirac delta-comb detuning, the model admits an exact analytic solution, with the detuning frequency adjustable via level-crossing spacing.
• The Floquet exponents are analytically determined for the unconditionally integrable case, enabling full characterization of the quasi-energy spectrum.
• Population dynamics in the system are studied across various parameter regions, revealing distinct behaviors depending on the detuning and Rabi frequency configuration.