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arXiv:2108.00137 · Physical Review Research

Two-photon sideband interaction in a driven quantum Rabi model: Quantitative discussions with derived longitudinal drives and beyond the rotating wave approximation

Byoung-moo Ann, Wouter Kessels, Gary Alexander Steele

Published Jan 3, 2022• 5 citations
arXivHTMLPDF

Abstract

In this paper, we analytically and numerically study the sideband interaction dynamics of the driven quantum Rabi model (QRM). We focus in particular on the conditions when the external transverse drive fields induce first-order sideband interactions. Inducing sideband interactions between two different systems is an essential technique for various physical models, including the QRM. However, despite its importance, a precise analytical study has not been reported yet that successfully explains the sideband interaction rates in a driven QRM applicable for all system parameter configurations. In our paper, we analytically derive the sideband interaction rates based on second-order perturbation theory, not relying on the rotating wave approximation (RWA). Our formula are valid for all ranges of drive frequencies and system's parameters. Our analytical derived formula agrees well with the numerical results in a regime of moderate drive amplitudes. Interestingly, we have found a nontrivial longitudinal drive effect derived from the transverse drive Hamiltonian. This accounts for significant corrections to the sideband interaction rates that are expected without considering the derived longitudinal effect. Using this approach, one can precisely estimate the sideband interaction rates in the QRM not confining themselves within specific parameter regimes for moderate drive amplitudes. This provides important contributions for quantitatively understanding experiments described by the driven QRM.

DOI
10.1103/physrevresearch.4.013005

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In this paper, we analytically and numerically study the sideband interaction dynamics of the driven quantum Rabi model (QRM). We focus in particular on the conditions when the external transverse drive fields induce first-order sideband interactions. Inducing sideband interactions between two different systems is an essential technique for various physical models, including the QRM. However, despite its importance, a precise analytical study has not been reported yet that successfully explains the sideband interaction rates in a driven QRM applicable for all system parameter configurations. In our paper, we analytically derive the sideband interaction rates based on second-order perturbation theory, not relying on the rotating wave approximation (RWA). Our formula are valid for all ranges of drive frequencies and system's parameters. Our analytical derived formula agrees well with the numerical results in a regime of moderate drive amplitudes. Interestingly, we have found a nontrivial longitudinal drive effect derived from the transverse drive Hamiltonian. This accounts for significant corrections to the sideband interaction rates that are expected without considering the derived longitudinal effect. Using this approach, one can precisely estimate the sideband interaction rates in the QRM not confining themselves within specific parameter regimes for moderate drive amplitudes. This provides important contributions for quantitatively understanding experiments described by the driven QRM.
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.
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.
Driving quantum systems periodically in time plays an essential role in the coherent control of quantum states. The rotating-wave approximation (RWA) is a good approximation technique for weak and nearly resonance driven fields. However, these experiments sometimes require large detuning and strong driving fields, for which the RWA may not hold. In this work, we experimentally, numerically, and analytically explore strongly driven two-mode Josephson circuits in the regime of strong driving and large detuning. Specifically, we investigate beam-splitter and two-mode squeezing interaction between the two modes induced by driving a two-photon sideband transition. Using numerical simulations, we observe that the RWA is unable to correctly capture the amplitude of the sideband transition rates. We verify this finding using an analytical model that is based on perturbative corrections. We find that the breakdown of the RWA in the regime studied does not lead to qualitatively different dynamics, but gives the same results as the RWA theory at higher drive strengths, enhancing the coupling rates compared to what one would predict. This is an interesting consequence compared to the carrier transition case, where the breakdown of the RWA results in qualitatively different time evolution of the quantum state. Our work provides an insight into the behavior of time-periodically driven systems beyond the RWA. We also provide a robust theoretical framework for including these findings in the calculation and calibration of quantum protocols in circuit quantum electrodynamics.
Abstract We apply a simple analytical method based on a unitary transformation to calculate the ground state, its excitation spectrum and quantum dynamic evolution of physical quantities for the double-photon quantum Rabi Hamiltonian over the wide coupling-strength range. The concise analytical method possesses the same mathematical simplicity as the approach of the rotating wave approximation (RWA). By quantitative comparison with the numerically exact result obtained by matrix diagonalization, we confirm that our calculated results obtained by transformed rotating-wave method are not only accurate in the weak coupling regime but also correct in intermediate strong-coupling case. In the intermediate ultrastrong-coupling regime, the calculated values of the ground state and lower lying excited states are nearly the same as the exact ones. It turns out that our calculation for the energy spectrum is beyond the ordinary-RWA. Meanwhile, we demonstrate the signatures resulting from the counter-rotating wave terms by monitoring the population, the coherence, the squeezing of the photon under the ultra-strong conditions. In particular, we find that when the frequency of the photon is much larger than the transition frequency of the system, the lineshape of the time evolution becomes complicated with the increase of the coupling strength, which may be verified experimentally.
The extended quantum Rabi models make a significant contribution to understanding of the quantum nature of the atom-light interaction. We transform two kinds of extended quantum Rabi model, the anisotropic Rabi model and the asymmetric Rabi model, into the rotating frame and regard them as periodically driven quantum systems. The analytical solutions of the quasienergy spectrum as well as the Floquet modes for both models are constructed by applying the Floquet theory and the high-frequency expansion, which is applied to the nonstroboscopic dynamics of physical observables such as atomic inversion, transverse magnetization, and atom-field correlation. For the anisotropic Rabi model, the quasienergy fits well with the numerical results even when the rotating-wave coupling is in the deep strong-coupling regime $g\ensuremath{\simeq}\ensuremath{\hbar}\ensuremath{\omega}$ if the counter-rotating term is small enough compared to the driving frequency. Avoided level crossing may occur for quasienergy with the same parity when the positive branch spectrum lines for the total excitation number NNN cross the negative branch lines for N+2N+2N+2, while the high-frequency expansion fails to predict this due to the conservation of the total excitation number. Furthermore, we present analytical and numerical studies of the long-time evolution of population and conclude that the analytical method is credible for the population dynamics. For the asymmetric Rabi model, we find that the external bias field which breaks the parity symmetry of the total excitation number tends to cluster the upper and lower branches into two bundles, and the detuning induced gap in the first temporal Brillouin zone shows a quadratic dependence on the bias. Fourier analysis is applied to extract the frequency composition and two-frequency driving behavior is revealed. Varying the bias strength will change the time-averaged value of the oscillation, which shows how the bias competes with the detuning and atom-field coupling in the driving dynamics. Both models prove that treating the Hamiltonian in the rotating frame by Floquet theory gives an alternative tool in the study of interaction between atom and light.
The Rabi model considers a two-level system (or spin 1/2) coupled to a quantized harmonic oscillator and describes the simplest interaction between matter and light. The recent experimental progress in solid-state circuit quantum electrodynamics has engendered theoretical efforts to quantitatively describe the mathematical and physical aspects of the light-matter interaction beyond the rotating-wave approximation. We develop a stochastic Schr\"odinger equation approach which enables us to access the strong-coupling limit of the Rabi model and study the effects of dissipation and ac drive in an exact manner. We include the effect of Ohmic noise on the non-Markovian spin dynamics, resulting in Kondo-type correlations, as well as cavity losses. We compute the time evolution of spin variables in various conditions. As a consideration for future work, we discuss the possibility of reaching a steady state with one polariton in realistic experimental conditions.