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Physical Review A

Effects of counter-rotating couplings of the Rabi model with frequency modulation

Yiying Yan, Zhiguo Lü, JunYan Luo, Hang Zheng

Published Sep 1, 2017• 23 citations
Publisher

Abstract

We theoretically study the dynamics and resonance shift of the Rabi model with frequency modulation, i.e., the Rabi model driven additionally by a slow longitudinal field, by using the counterrotating-hybridized rotating-wave (CHRW) method, aiming to illustrate the effects of the counterrotating (CR) terms of the transverse field. The CHRW method is based on a unitary transformation and reduces the aperiodic Hamiltonian to an effective periodic Hamiltonian that can be efficiently treated by Floquet theory. The validity of the effective Hamiltonian and widely used rotating-wave approximation (RWA) Hamiltonian is carefully examined compared to the numerically exact results over a wide parameter range. It is found that the effective Hamiltonian gives a correct description, while the RWA breaks down in the strong driving regime. Interestingly, we show that under certain conditions the longitudinal field can be used to modify resonance widths such that resonance widths can be comparable to the magnitude of the Bloch-Siegert (BS) shift, which in turn makes the CR-induced BS shift significant and leads to the complete breakdown of the RWA even in a moderately strong driving regime (in which the RWA holds for the Rabi model without frequency modulation). In addition, by using the effective Hamiltonian, we can efficiently access resonance positions for the bichromatically driven qubit and study how the resonance shifts due to the combined effects of the CR terms and frequency modulation. For a weak longitudinal field, we show that resonance positions can be analytically calculated from the effective Rabi frequency for the effective Hamiltonian, which are in excellent agreement with the numerically exact results.

DOI
10.1103/physreva.96.033802

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Journal of Physics B Atomic Molecular and Optical Physics•Aug 13, 2025•1 citation
We theoretically study the dynamics and resonance shift of the Rabi model with frequency modulation, i.e., the Rabi model driven additionally by a slow longitudinal field, by using the counterrotating-hybridized rotating-wave (CHRW) method, aiming to illustrate the effects of the counterrotating (CR) terms of the transverse field. The CHRW method is based on a unitary transformation and reduces the aperiodic Hamiltonian to an effective periodic Hamiltonian that can be efficiently treated by Floquet theory. The validity of the effective Hamiltonian and widely used rotating-wave approximation (RWA) Hamiltonian is carefully examined compared to the numerically exact results over a wide parameter range. It is found that the effective Hamiltonian gives a correct description, while the RWA breaks down in the strong driving regime. Interestingly, we show that under certain conditions the longitudinal field can be used to modify resonance widths such that resonance widths can be comparable to the magnitude of the Bloch-Siegert (BS) shift, which in turn makes the CR-induced BS shift significant and leads to the complete breakdown of the RWA even in a moderately strong driving regime (in which the RWA holds for the Rabi model without frequency modulation). In addition, by using the effective Hamiltonian, we can efficiently access resonance positions for the bichromatically driven qubit and study how the resonance shifts due to the combined effects of the CR terms and frequency modulation. For a weak longitudinal field, we show that resonance positions can be analytically calculated from the effective Rabi frequency for the effective Hamiltonian, which are in excellent agreement with the numerically exact results.
Abstract We investigate the dynamics of a two-level system driven by a deep-strong slow sinusoidal field using a novel counterrotating-hybridized rotating-wave (CHRW) method. By two successive unitary transformations, we obtain a transformed Hamiltonian which possesses a simple rotating-wave interaction with a renormalized driving strength and a modified tunneling strength. The modified CHRW method allows us to directly calculate the analytical results of the dynamics and Bloch-Siegert shift. We show that the results given by the modified CHRW method are in good agreement with the numerical solutions of the Rabi model with a moderate strong driving strength in the off-resonance case. It turns out that the modified CHRW method is beyond the traditional rotating-wave approximation in some interesting parameter regions. As the modified CHRW method is mathematically simple and physically clear, it may be applied to driven dissipative systems which are difficult to solve numerically.
We investigate the dynamics of a driven two-level system (classical Rabi model) using the counter-rotating-hybridized rotating-wave method (CHRW), which is a simple method based on a unitary transformation with a parameter $\ensuremath{\xi}$. This approach is beyond the traditional rotating-wave approximation (Rabi-RWA) and more importantly, remains the RWA form with a renormalized tunneling strength and a modified driving strength. The reformulated rotating-wave method not only possesses the same mathematical simplicity as the Rabi-RWA but also allows us to explore the effects of counter-rotating (CR) components. We focus on the properties of off-resonance cases for which the Rabi-RWA method breaks down. After comparing the results of different RWA schemes and those of the numerically exact method in a wide range of parameter regime, we show that the CHRW method gives the accurate driven dynamics which is in good agreement with the numerical method. Moreover, the other RWA methods appear as various limiting cases of the CHRW method. The CHRW method reveals the effects of the CR terms clearly by means of coherent destruction of tunneling and Bloch-Siegert shift. Our main results are as follows: (i) the dynamics of the coherent destruction of tunneling is explicitly given and its dependence on $\ensuremath{\Delta}$ is clarified, which is quantitatively in good agreement with the exact results; (ii) the CR modulated Rabi frequency and the Bloch-Siegert shift are analytically calculated, which is the same as the exact results up to fourth order; (iii) the validity of parameter regions of different RWA methods are given and the comparison of dynamics of these methods are shown. Since the CHRW approach is mathematically simple as well as tractable and physically clear, it may be extended to some complicated problems where it is difficult to do a numerical study.
We investigate the bias-modulated dynamics of a strongly driven two-level system using the counterrotating-hybridized rotating-wave (CHRW) method. This CHRW method treats the driving field and the bias on equal footing by a unitary transformation with two parameters $\ensuremath{\xi}$ and $\ensuremath{\zeta}$, and is nonperturbative in driving strength, tunneling amplitude, or bias. In addition, this CHRW method is beyond the traditional rotating-wave approximation (Rabi-RWA) and yet by properly choosing the two parameters $\ensuremath{\xi}$ and $\ensuremath{\zeta}$, the transformed Hamiltonian takes the RWA form with a renormalized energy splitting and a renormalized driving strength. The reformulated CHRW method possesses the same mathematical simplicity as the Rabi-RWA approach and thus allows us to calculate analytically the dynamics and explore explicitly the effect of the bias. We show that the CHRW method gives the accurate driven dynamics for a wide range of parameters as compared to the numerically exact results. When energy scales of the driving are comparable to the intrinsic energy scale of the two-level systems, the counterrotating interactions and static bias profoundly influence the generalized Rabi frequency. In this regime, where ordinary perturbation approaches fail, the CHRW works very well and efficiently. We also demonstrate the dynamics of the system in the strong-driving and off-resonance cases for which the Rabi-RWA method breaks down but the CHRW method remains valid. We obtain analytical expressions for the generalized Rabi frequency and bias-modulated Bloch-Siegert shift as functions of the bias, tunneling, and driving field parameters. The CHRW approach is a mathematically simple and physically clear method. It can be applied to treat some complicated problems for which a numerical study is difficult to perform.
Abstract The resonance and dynamics of a qubit exposed to a strong aperiodic bichromatic field are studied by using a periodic counter‐rotating hybridized rotating wave (CHRW) Hamiltonian, which is derived from the original Hamiltonian with the unitary transformations under a reasonable approximation and enables the application of the Floquet theory. It is found that the consistency between the CHRW results and numerically exact generalized‐Floquet‐theory (GFT) results in the valid regime of the former while the widely used rotating‐wave approximation breaks down. It is illustrated that the resonance exhibits band structure and the Bloch–Siegert shifts induced by the counter‐rotating couplings of the bichromatic field become notable at the multiphoton resonance band. In addition, the CHRW method is found to have a great advantage of efficiency over the GFT approach particularly in the low beat‐frequency case where the latter converges very slowly. The present CHRW method provides a highly efficient way to calculate the resonance frequency incorporating the Bloch–Siegert shift and provides insights into the effects of the counter‐rotating couplings of the bichromatic field in the strong‐driving regimes.
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.