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

Journal of Modern Optics

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

Chris O’Brien, Marlan O. Scully

Published Jul 31, 2015• 7 citations
Publisher

Abstract

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.

DOI
10.1080/09500340.2015.1066457

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

Dynamical effects of counter-rotating couplings on interference between driving and dissipation

Yiying Yan, Zhiguo Lü, Hang Zheng

We use an analytical method to study the dissipative dynamics of a two-level system (TLS) under a harmonic driving. The method is based on a combination of the unitary transformation and Born--Markov master-equation approach. Our main aim is to clarify the effects of counter-rotating (CR) terms of both the driving and TLS-bath (dissipative) coupling on the dynamics, in comparison with the rotating-wave results of different schemes, i.e., the well-known traditional rotating-wave approximation method, and two particular methods: one just takes into account the effects of the driving CR terms and the other the effects of the dissipative CR terms, which are derived from our general treatment. Our main results are as follows: (i) by calculating the time-dependent population difference and coherence, in the case of resonant strong driving, we demonstrate that the CR terms of both the driving and dissipative coupling play an important role in the relaxation and dephasing processes, and also the properties of the steady state; (ii) in the case of largely detuned driving, we find that the CR terms of the dissipative coupling become negligible while those of the driving contribute dominant modifications to the time evolution and steady state. Moreover, we examine the influence of the dissipation on coherent destruction of tunneling under a largely detuned strong driving. We show that an almost complete suppression of the tunneling can be achieved for a relatively long time; (iii) under certain conditions, we find numerical equivalence between one of two particular methods and the Floquet--Born--Markov approach based on exact numerical treatment of the Floquet Hamiltonian. It turns out that our method is more simple and efficient than the Floquet--Born--Markov approach for both analytical and numerical calculations. By the general comparison of different treatments we demonstrate the dynamical effects of CR terms of both the driving and the dissipative coupling on the coherence and population difference.

Physical Review A•Nov 26, 2014•8 citations

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

Pietro Follia, Bassano Vacchini

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.

quant-pharXiv:2607.01030•Jul 2, 2026•0 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

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

Effects of counter-rotating-wave terms of the driving field on the spectrum of resonance fluorescence

Yiying Yan, Zhiguo Lü, Hang Zheng

We investigate the fluorescence spectrum of a two-level system driven by a monochromatic classical field by the Born-Markovian master equation based on a unitary transformation. The main purpose is to understand the effects of counter-rotating-wave terms of the driving on spectral features of the fluorescence. We have derived an analytical expression for the fluorescence spectrum, which is different from Mollow's theory, while Mollow's result on resonance is the limiting case of ours in moderately weak driving regimes. Our results demonstrate precisely that the counter-rotating-wave terms of the driving play an important role in the fluorescence spectrum for intense driving: (i) the counter-rotating coupling suppresses the red sideband in the Mollow triplet and it enhances the blue one in explicitly contrast to the well-known equal intensity of the sideband in Mollow's theory, (ii) the higher-order Mollow triplets appear as a characteristic spectral feature arising from counter-rotating-wave terms of the driving, and (iii) a significant frequency shift of the sidebands is observed, which depends on both the detuning and driving strength.

Physical Review A•Nov 15, 2013•21 citations
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.
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.
We use an analytical method to study the dissipative dynamics of a two-level system (TLS) under a harmonic driving. The method is based on a combination of the unitary transformation and Born--Markov master-equation approach. Our main aim is to clarify the effects of counter-rotating (CR) terms of both the driving and TLS-bath (dissipative) coupling on the dynamics, in comparison with the rotating-wave results of different schemes, i.e., the well-known traditional rotating-wave approximation method, and two particular methods: one just takes into account the effects of the driving CR terms and the other the effects of the dissipative CR terms, which are derived from our general treatment. Our main results are as follows: (i) by calculating the time-dependent population difference and coherence, in the case of resonant strong driving, we demonstrate that the CR terms of both the driving and dissipative coupling play an important role in the relaxation and dephasing processes, and also the properties of the steady state; (ii) in the case of largely detuned driving, we find that the CR terms of the dissipative coupling become negligible while those of the driving contribute dominant modifications to the time evolution and steady state. Moreover, we examine the influence of the dissipation on coherent destruction of tunneling under a largely detuned strong driving. We show that an almost complete suppression of the tunneling can be achieved for a relatively long time; (iii) under certain conditions, we find numerical equivalence between one of two particular methods and the Floquet--Born--Markov approach based on exact numerical treatment of the Floquet Hamiltonian. It turns out that our method is more simple and efficient than the Floquet--Born--Markov approach for both analytical and numerical calculations. By the general comparison of different treatments we demonstrate the dynamical effects of CR terms of both the driving and the dissipative coupling on the coherence and population difference.
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.
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 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.
We investigate the fluorescence spectrum of a two-level system driven by a monochromatic classical field by the Born-Markovian master equation based on a unitary transformation. The main purpose is to understand the effects of counter-rotating-wave terms of the driving on spectral features of the fluorescence. We have derived an analytical expression for the fluorescence spectrum, which is different from Mollow's theory, while Mollow's result on resonance is the limiting case of ours in moderately weak driving regimes. Our results demonstrate precisely that the counter-rotating-wave terms of the driving play an important role in the fluorescence spectrum for intense driving: (i) the counter-rotating coupling suppresses the red sideband in the Mollow triplet and it enhances the blue one in explicitly contrast to the well-known equal intensity of the sideband in Mollow's theory, (ii) the higher-order Mollow triplets appear as a characteristic spectral feature arising from counter-rotating-wave terms of the driving, and (iii) a significant frequency shift of the sidebands is observed, which depends on both the detuning and driving strength.