Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2001.10733 · Physical Review A

Analytical double-unitary-transformation approach for strongly and periodically driven three-level systems

Yingying Han, Xiao‐Qing Luo, Tiefu Li, Wenxian Zhang

Published Feb 18, 2020• 10 citations
arXivHTMLPDF

Abstract

Floquet theory combined with the generalized Van Vleck nearly degenerate perturbation theory has been widely employed for studying various two-level systems that are driven by external fields via time-dependent longitudinal (i.e., diagonal) couplings. However, three-level systems strongly driven by time-dependent transverse (i.e., off-diagonal) couplings have rarely been investigated, due to the breakdown of the traditional rotating wave approximation. Meanwhile, the conventional perturbation theory is not directly applicable, since a small parameter for the perturbed part is no longer apparent. Here we develop a double-unitary-transformation approach to deal with periodically modulated and strongly driven systems, where the time-dependent Hamiltonian has large off-diagonal elements. The first unitary transformation converts strong off-diagonal elements to diagonal ones, and the second enables us to harness the generalized Van Vleck perturbation theory to deal with the transformed Floquet matrix and also allows us to reduce the infinite-dimensional Floquet Hamiltonian to a finite effective one. For a strongly modulated three-level system, with the combination of the Floquet theory and the transformed generalized Van Vleck perturbation theory, we obtain analytical results of the system, which agree well with exact numerical solutions. This method offers a useful tool to analytically study multilevel systems with strong transverse couplings.

DOI
10.1103/physreva.101.022108

Related papers

Floquet dynamics of the Rabi model beyond the counter-rotating hybridized rotating-wave method

Yingying Han, Shuanghao Zhang, Meijuan Zhang +3 more

Monochromatically driven two-level systems (i.e., Rabi models) are ubiquitous in various fields of physics. Though they have been exactly solved, the physical pictures in these exact solutions are not clear. Recently, approximate analytical solutions with neat physics have been obtained by using the counter-rotating hybridized rotating wave (CHRW) method, which has been proven to be effective over a wider range of parameters than the previous analytical solutions. However, the CHRW depends on a parameter $\ensuremath{\xi}$, which has no solution in some regimes. Here, we combine the double-unitary-transformation approach with the generalized Van Vleck nearly degenerate perturbation theory, and present approximate analytical results with clear physics for almost all parameter regimes, which agree well with the numerical solutions and the previous experimental results. Moreover, the dynamic frequencies of the Rabi model are regular, and the frequency with the highest Fourier amplitude changes from the Rabi frequency to $2n\ensuremath{\omega}$ with driving frequency $\ensuremath{\omega}$ and integer $n$, as the driving intensity increases from weak to deep-strong. In addition, we further explore the Floquet dynamics of the dissipative open Rabi model. Remarkably, the dissipations are tunable in the rotating frame, and the approximate analytical results obtained by our method are in good agreement with the numerical results in the strong driving regime. These results pave the way to quantum control using strong and deep-strong driving with applications in quantum technologies.

arXiv:2404.14841•Physical Review A•May 2, 2024•2 citations

Effects of counter-rotating couplings in bichromatically driven three-level V-type systems

Yiying Yan, Zhiguo Lü, Hang Zheng

Abstract We study the resonance and dynamics of a three-level V-type system strongly driven by a quasiperiodic bichromatic field by making use of a unitary-transformation (UT) method. The UT is used to construct a periodic effective Hamiltonian which incorporates the effects of counter-rotating couplings and takes a physically transparent form, and the Floquet theory is used to calculate transition probabilities. By comparing the UT results with those from the rotating-wave approximation (RWA) and generalized Floquet theory, we illustrate that the UT method captures not only accurate resonance structures but also dynamics of the driven V system in a strong-driving regime where the RWA breaks down. We show that there are two dynamical features, namely, peak-dip structures and beat behaviors, which are ascribed to a time-dependent coupling between the two upper levels due to the counter-rotating terms and cannot be described by a RWA Hamiltonian. Moreover, we show that the effective Hamiltonian can be used to calculate higher-order Bloch–Siegert shifts, which is another feature caused by the counter-rotating terms. Present work offers insights into the role of counter-rotating couplings in a bichromatically driven three-level system.

Journal of Physics B Atomic Molecular and Optical Physics•Aug 13, 2025•1 citation

Floquet framework for driven polar quantum systems

Viktor Novičenko, Piotr Gładysz, Karolina Słowik +1 more

We present an analytical and numerical Floquet treatment of a driven polar two-level quantum system characterized by both longitudinal and transverse coupling to a periodic field. Analytically, we derive a dressed-frame effective Hamiltonian up to first order in the inverse driving frequency, incorporating the longitudinal coupling nonperturbatively. This yields closed expressions for the effective transverse coupling strength and the effective detuning, both of which are modified by the presence of the longitudinal interaction. In the nonpolar limit, these expressions recover the usual near-resonant Rabi coupling and the Bloch-Siegert shift. As a second main result, we develop a numerical flow-equation framework that yields a time-independent effective Hamiltonian across a broad range of transverse and longitudinal coupling strengths. This dual framework is relevant for a variety of platforms, including driven polar quantum systems, optical lattices, superconducting circuits, and solids subject to surface acoustic waves.

quant-pharXiv:2606.19330•Jun 18, 2026•0 citations

Dissipative two-level system under strong ac driving: A combination of Floquet and Van Vleck perturbation theory

Johannes Hausinger, Milena Grifoni

We study the dissipative dynamics of a two-level system (TLS) exposed to strong ac driving. By combining Floquet theory with Van Vleck perturbation theory in the TLS tunneling matrix element, we diagonalize the time-dependent Hamiltonian and provide corrections to the renormalized Rabi frequency of the TLS, which are valid for both a biased and unbiased TLS and go beyond the known high-frequency and rotating-wave results. In order to mimic environmental influences on the TLS, we couple the system weakly to a thermal bath and solve analytically the corresponding Floquet-Bloch-Redfield master equation. We give a closed expression for the relaxation and dephasing rates of the TLS and discuss their behavior under variation of the driving amplitude. Further, we examine the robustness of coherent destruction of tunneling (CDT) and driving-induced tunneling oscillations (DITO). We show that also for a moderate driving frequency an almost complete suppression of tunneling can be achieved for short times and demonstrate the sensitiveness of DITO to a change of the external parameters.

arXiv:0910.0356•Physical Review A•Feb 22, 2010•104 citations

The dissipative two-level system under strong ac-driving: a Floquet-van-Vleck approach

Johannes Hausinger, Milena Grifoni

We study the dissipative dynamics of a two-level system (TLS) exposed to strong ac driving. By combing Floquet theory with Van Vleck perturbation theory in the TLS tunneling matrix element, we diagonalize the time-dependent Hamiltonian and provide corrections to the renormalized Rabi frequency of the TLS, which are valid for both a biased and unbiased TLS and go beyond the known high-frequency and rotating-wave results. In order to mimic environmental influences on the TLS, we couple the system weakly to a thermal bath and solve analytically the corresponding Floquet-Bloch-Redfield master equation. We give a closed expression for the relaxation and dephasing rates of the TLS and discuss their behavior under variation of the driving amplitude. Further, we examine the robustness of coherent destruction of tunneling (CDT) and driving-induced tunneling oscillations (DITO). We show that also for a moderate driving frequency an almost complete suppression of tunneling can be achieved for short times and demonstrate the sensitiveness of DITO to a change of the external parameters.

•Oct 2, 2009•0 citations

Multiphoton-resonance-induced fluorescence of a strongly driven two-level system under frequency modulation

Yiying Yan, Zhiguo Lü, JunYan Luo +1 more

We study the fluorescence spectrum of a strongly driven two-level system (TLS) with modulated transition frequency, which is a bichromatically driven TLS and has multiple resonance frequencies. We are aiming to provide a reliable description of the fluorescence in a regime that is difficult to tackle with perturbation theory and the rotating-wave approximation (RWA), and illustrate the spectral features of the fluorescence under off- and multiphoton-resonance conditions. To go beyond the RWA, we use a semianalytical counter-rotating-hybridized rotating-wave method that combines a unitary transformation and Floquet theory to calculate the two-mode Floquet states and quasienergies for the bichromatically driven TLS. We then solve the master equation accounting for the spontaneous decay in the bases of the two-mode Floquet states, and derive a physically transparent fluorescence spectrum. In comparison with the numerically exact spectrum from the generalized Floquet-Liouville approach, the present spectrum is found to be applicable in a wide range of the parameters where the RWA and the secular approximation may break down. We find that the counter-rotating (CR) terms of the transverse field omitted in the RWA have non-negligible contributions to the spectrum under certain conditions. Particularly, at the multiphoton resonance the width of which is comparable with the Bloch-Siegert shift, the RWA and non-RWA spectra markedly differ from each other because of the CR-induced shift. We also analyze the symmetry of the spectrum in terms of the transition matrix elements between the two-mode Floquet states. We show that the strict symmetry of the spectrum cannot be expected without the RWA but the almost symmetric spectrum can be obtained at the single-photon resonance that takes the Bloch-Siegert shift into account if the driving is moderately strong and at the multiphoton resonance with a sufficiently weak transverse field.

Physical Review A•Mar 13, 2018•6 citations
Floquet theory combined with the generalized Van Vleck nearly degenerate perturbation theory has been widely employed for studying various two-level systems that are driven by external fields via time-dependent longitudinal (i.e., diagonal) couplings. However, three-level systems strongly driven by time-dependent transverse (i.e., off-diagonal) couplings have rarely been investigated, due to the breakdown of the traditional rotating wave approximation. Meanwhile, the conventional perturbation theory is not directly applicable, since a small parameter for the perturbed part is no longer apparent. Here we develop a double-unitary-transformation approach to deal with periodically modulated and strongly driven systems, where the time-dependent Hamiltonian has large off-diagonal elements. The first unitary transformation converts strong off-diagonal elements to diagonal ones, and the second enables us to harness the generalized Van Vleck perturbation theory to deal with the transformed Floquet matrix and also allows us to reduce the infinite-dimensional Floquet Hamiltonian to a finite effective one. For a strongly modulated three-level system, with the combination of the Floquet theory and the transformed generalized Van Vleck perturbation theory, we obtain analytical results of the system, which agree well with exact numerical solutions. This method offers a useful tool to analytically study multilevel systems with strong transverse couplings.
Monochromatically driven two-level systems (i.e., Rabi models) are ubiquitous in various fields of physics. Though they have been exactly solved, the physical pictures in these exact solutions are not clear. Recently, approximate analytical solutions with neat physics have been obtained by using the counter-rotating hybridized rotating wave (CHRW) method, which has been proven to be effective over a wider range of parameters than the previous analytical solutions. However, the CHRW depends on a parameter $\ensuremath{\xi}$, which has no solution in some regimes. Here, we combine the double-unitary-transformation approach with the generalized Van Vleck nearly degenerate perturbation theory, and present approximate analytical results with clear physics for almost all parameter regimes, which agree well with the numerical solutions and the previous experimental results. Moreover, the dynamic frequencies of the Rabi model are regular, and the frequency with the highest Fourier amplitude changes from the Rabi frequency to $2n\ensuremath{\omega}$ with driving frequency $\ensuremath{\omega}$ and integer nnn, as the driving intensity increases from weak to deep-strong. In addition, we further explore the Floquet dynamics of the dissipative open Rabi model. Remarkably, the dissipations are tunable in the rotating frame, and the approximate analytical results obtained by our method are in good agreement with the numerical results in the strong driving regime. These results pave the way to quantum control using strong and deep-strong driving with applications in quantum technologies.
Abstract We study the resonance and dynamics of a three-level V-type system strongly driven by a quasiperiodic bichromatic field by making use of a unitary-transformation (UT) method. The UT is used to construct a periodic effective Hamiltonian which incorporates the effects of counter-rotating couplings and takes a physically transparent form, and the Floquet theory is used to calculate transition probabilities. By comparing the UT results with those from the rotating-wave approximation (RWA) and generalized Floquet theory, we illustrate that the UT method captures not only accurate resonance structures but also dynamics of the driven V system in a strong-driving regime where the RWA breaks down. We show that there are two dynamical features, namely, peak-dip structures and beat behaviors, which are ascribed to a time-dependent coupling between the two upper levels due to the counter-rotating terms and cannot be described by a RWA Hamiltonian. Moreover, we show that the effective Hamiltonian can be used to calculate higher-order Bloch–Siegert shifts, which is another feature caused by the counter-rotating terms. Present work offers insights into the role of counter-rotating couplings in a bichromatically driven three-level system.
We present an analytical and numerical Floquet treatment of a driven polar two-level quantum system characterized by both longitudinal and transverse coupling to a periodic field. Analytically, we derive a dressed-frame effective Hamiltonian up to first order in the inverse driving frequency, incorporating the longitudinal coupling nonperturbatively. This yields closed expressions for the effective transverse coupling strength and the effective detuning, both of which are modified by the presence of the longitudinal interaction. In the nonpolar limit, these expressions recover the usual near-resonant Rabi coupling and the Bloch-Siegert shift. As a second main result, we develop a numerical flow-equation framework that yields a time-independent effective Hamiltonian across a broad range of transverse and longitudinal coupling strengths. This dual framework is relevant for a variety of platforms, including driven polar quantum systems, optical lattices, superconducting circuits, and solids subject to surface acoustic waves.
We study the dissipative dynamics of a two-level system (TLS) exposed to strong ac driving. By combining Floquet theory with Van Vleck perturbation theory in the TLS tunneling matrix element, we diagonalize the time-dependent Hamiltonian and provide corrections to the renormalized Rabi frequency of the TLS, which are valid for both a biased and unbiased TLS and go beyond the known high-frequency and rotating-wave results. In order to mimic environmental influences on the TLS, we couple the system weakly to a thermal bath and solve analytically the corresponding Floquet-Bloch-Redfield master equation. We give a closed expression for the relaxation and dephasing rates of the TLS and discuss their behavior under variation of the driving amplitude. Further, we examine the robustness of coherent destruction of tunneling (CDT) and driving-induced tunneling oscillations (DITO). We show that also for a moderate driving frequency an almost complete suppression of tunneling can be achieved for short times and demonstrate the sensitiveness of DITO to a change of the external parameters.
We study the dissipative dynamics of a two-level system (TLS) exposed to strong ac driving. By combing Floquet theory with Van Vleck perturbation theory in the TLS tunneling matrix element, we diagonalize the time-dependent Hamiltonian and provide corrections to the renormalized Rabi frequency of the TLS, which are valid for both a biased and unbiased TLS and go beyond the known high-frequency and rotating-wave results. In order to mimic environmental influences on the TLS, we couple the system weakly to a thermal bath and solve analytically the corresponding Floquet-Bloch-Redfield master equation. We give a closed expression for the relaxation and dephasing rates of the TLS and discuss their behavior under variation of the driving amplitude. Further, we examine the robustness of coherent destruction of tunneling (CDT) and driving-induced tunneling oscillations (DITO). We show that also for a moderate driving frequency an almost complete suppression of tunneling can be achieved for short times and demonstrate the sensitiveness of DITO to a change of the external parameters.
We study the fluorescence spectrum of a strongly driven two-level system (TLS) with modulated transition frequency, which is a bichromatically driven TLS and has multiple resonance frequencies. We are aiming to provide a reliable description of the fluorescence in a regime that is difficult to tackle with perturbation theory and the rotating-wave approximation (RWA), and illustrate the spectral features of the fluorescence under off- and multiphoton-resonance conditions. To go beyond the RWA, we use a semianalytical counter-rotating-hybridized rotating-wave method that combines a unitary transformation and Floquet theory to calculate the two-mode Floquet states and quasienergies for the bichromatically driven TLS. We then solve the master equation accounting for the spontaneous decay in the bases of the two-mode Floquet states, and derive a physically transparent fluorescence spectrum. In comparison with the numerically exact spectrum from the generalized Floquet-Liouville approach, the present spectrum is found to be applicable in a wide range of the parameters where the RWA and the secular approximation may break down. We find that the counter-rotating (CR) terms of the transverse field omitted in the RWA have non-negligible contributions to the spectrum under certain conditions. Particularly, at the multiphoton resonance the width of which is comparable with the Bloch-Siegert shift, the RWA and non-RWA spectra markedly differ from each other because of the CR-induced shift. We also analyze the symmetry of the spectrum in terms of the transition matrix elements between the two-mode Floquet states. We show that the strict symmetry of the spectrum cannot be expected without the RWA but the almost symmetric spectrum can be obtained at the single-photon resonance that takes the Bloch-Siegert shift into account if the driving is moderately strong and at the multiphoton resonance with a sufficiently weak transverse field.