© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2607.01030

Analytical connection between exact and approximate solutions of the periodically-driven two-level system starting from the Heun equation

Pietro Follia, Bassano Vacchini

quant-phmath-phSubmitted Jul 1, 2026• Announced Jul 2, 2026• 0 citations

Abstract

We investigate and establish an analytic connection between the exact solutions describing the dynamics of a two-level system driven by periodic external fields, focusing on the cases of linear driving and the so-called rotating-wave approximation, or circular driving. In both cases, the exact solutions can be obtained by mapping the Schrodinger equation onto Heun equations: the confluent Heun equation for linear driving and the Heun equation for the rotating-wave case. In particular, we demonstrate a direct analytic connection between the exact solutions for linear driving and those for the rotating-wave case. This result is obtained by analyzing local solutions expressed in terms of hypergeometric functions, which, in the case of the confluent Heun equation, can be derived by considering path-multiplicative Floquet solutions involving a bilateral series. This series leads to two continued-fraction expansions that can be perturbatively solved by imposing a suitable consistency condition. The connection between the linear-driving and rotating-wave solutions is established through a perturbative procedure that allows us to recover not only the rotating-wave approximation itself, but also the correct Stark and Bloch-Siegert shifts, as well as the so-called high-frequency approximation.

Comments: 15 pages, 1 figure

Related papers

Analytical treatment of the continuous wave driving of a two-level atom without making the rotating wave approximation

Chris O’Brien, Marlan O. Scully

In a straightforward manner, we utilize Floquet theory and adiabatic elimination to derive an analytic expression for a monochromatically driven two-level atom, without making the rotating wave approximation. We show that the counter-rotating terms dropped in the rotating wave approximation are responsible for three major effects. First an ac-Stark phase shift of the driven transition, second increased excited state population from far-detuned driving of the Lorentzian line, and third extra frequencies in the population dynamics that result in “wiggles.” The analytic result agrees well with numerical simultations over a wide range of parameters.

Journal of Modern Optics•Jul 31, 2015•7 citations

Exact solution for quantum dynamics of a periodically driven two-level system

Anirban Gangopadhyay, Maxim Dzero, Victor M. Galitski

We present a family of exact analytic solutions for nonlinear quantum dynamics of a two-level system (TLS) subject to a periodic-in-time external field. In constructing the exactly solvable models, we use a ``reverse engineering'' approach where the form of external perturbation is chosen to preserve an integrability constraint, which yields a single nonlinear differential equation for the ac field. A solution to this equation is expressed in terms of Jacobi elliptic functions with three independent parameters that allows one to choose the frequency, average value, and amplitude of the time-dependent field at will. This form of the ac drive is especially relevant to the problem of dynamics of TLS charge defects that cause dielectric losses in superconducting qubits. We apply our exact results to analyze nonlinear dielectric response of such TLSs and show that the position of the resonance peak in the spectrum of the relevant correlation function is determined by the quantum-mechanical phase accumulated by the TLS wave function over a time evolution cycle. It is shown that in the nonlinear regime, this resonance frequency may be shifted strongly from the value predicted by the canonical TLS model. We also analyze the ``spin'' survival probability in the regime of strong external drive and recover a coherent destruction of tunneling phenomenon within our family of exact solutions, which manifests itself as a strong suppression of ``spin-flip'' processes and suggests that such nonlinear dynamics in LC resonators may lead to lower losses.

arXiv:1005.0652•Physical Review B•Jul 22, 2010•35 citations

Analytic results for the population dynamics of a driven dipolar molecular system

Wen-Jing Zhang, Kang Jin, Luling Jin +1 more

We theoretically investigate the dipolar molecular system driven by monochromatic periodic, linear, parabolic, and ${\mathrm{sech}}^{2}$ forms external fields, respectively. The two-level Hamiltonian model with nonzero diagonal dipole matrix elements is adopted to describe the population dynamics of the driven dipolar molecule, and the corresponding exact solutions are presented in terms of the confluent Heun equations without the generalized rotating-wave approximation. The analytic solutions derived here are valid in the whole parameter space.

Physical Review A•Apr 22, 2016•10 citations

Two-state model of a general Heun class with periodic level-crossings

G. Saget, Artur Mikhailovich Ishkhanyan, Claude Leroy +1 more

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 is presented. 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 closedform 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 levelcrossings. The exact solution for this model is derived, the Floquet exponents are determined and study of the population dynamics in the system for various regions of the input parameters is done.

arXiv:1707.02972•Journal of Contemporary Physics (Armenian Academy of Sciences)•Oct 1, 2017•6 citations

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

Qiongtao Xie

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.

Communications in Theoretical Physics•Oct 13, 2025•0 citations

Multi-Directional Periodic Driving of a Two-Level System beyond Floquet Formalism

Michael Warnock, David A. Hague, Vesna F. Mitrovic

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.

quant-pharXiv:2511.03977•Nov 7, 2025•0 citations

We investigate and establish an analytic connection between the exact solutions describing the dynamics of a two-level system driven by periodic external fields, focusing on the cases of linear driving and the so-called rotating-wave approximation, or circular driving. In both cases, the exact solutions can be obtained by mapping the Schrodinger equation onto Heun equations: the confluent Heun equation for linear driving and the Heun equation for the rotating-wave case. In particular, we demonstrate a direct analytic connection between the exact solutions for linear driving and those for the rotating-wave case. This result is obtained by analyzing local solutions expressed in terms of hypergeometric functions, which, in the case of the confluent Heun equation, can be derived by considering path-multiplicative Floquet solutions involving a bilateral series. This series leads to two continued-fraction expansions that can be perturbatively solved by imposing a suitable consistency condition. The connection between the linear-driving and rotating-wave solutions is established through a perturbative procedure that allows us to recover not only the rotating-wave approximation itself, but also the correct Stark and Bloch-Siegert shifts, as well as the so-called high-frequency approximation.

In a straightforward manner, we utilize Floquet theory and adiabatic elimination to derive an analytic expression for a monochromatically driven two-level atom, without making the rotating wave approximation. We show that the counter-rotating terms dropped in the rotating wave approximation are responsible for three major effects. First an ac-Stark phase shift of the driven transition, second increased excited state population from far-detuned driving of the Lorentzian line, and third extra frequencies in the population dynamics that result in “wiggles.” The analytic result agrees well with numerical simultations over a wide range of parameters.

We present a family of exact analytic solutions for nonlinear quantum dynamics of a two-level system (TLS) subject to a periodic-in-time external field. In constructing the exactly solvable models, we use a ``reverse engineering'' approach where the form of external perturbation is chosen to preserve an integrability constraint, which yields a single nonlinear differential equation for the ac field. A solution to this equation is expressed in terms of Jacobi elliptic functions with three independent parameters that allows one to choose the frequency, average value, and amplitude of the time-dependent field at will. This form of the ac drive is especially relevant to the problem of dynamics of TLS charge defects that cause dielectric losses in superconducting qubits. We apply our exact results to analyze nonlinear dielectric response of such TLSs and show that the position of the resonance peak in the spectrum of the relevant correlation function is determined by the quantum-mechanical phase accumulated by the TLS wave function over a time evolution cycle. It is shown that in the nonlinear regime, this resonance frequency may be shifted strongly from the value predicted by the canonical TLS model. We also analyze the ``spin'' survival probability in the regime of strong external drive and recover a coherent destruction of tunneling phenomenon within our family of exact solutions, which manifests itself as a strong suppression of ``spin-flip'' processes and suggests that such nonlinear dynamics in LC resonators may lead to lower losses.

We theoretically investigate the dipolar molecular system driven by monochromatic periodic, linear, parabolic, and

{\mathrm{sech}}^{2}

forms external fields, respectively. The two-level Hamiltonian model with nonzero diagonal dipole matrix elements is adopted to describe the population dynamics of the driven dipolar molecule, and the corresponding exact solutions are presented in terms of the confluent Heun equations without the generalized rotating-wave approximation. The analytic solutions derived here are valid in the whole parameter space.

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 is presented. 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 closedform 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 levelcrossings. The exact solution for this model is derived, the Floquet exponents are determined and study of the population dynamics in the system for various regions of the input parameters is done.

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.

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.