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arXiv:1602.04413 · Physical Review A

Bias-modulated dynamics of a strongly driven two-level system

Zhiguo Lü, Yiying Yan, Hsi‐Sheng Goan, Hang Zheng

Published Mar 2, 2016• 13 citations
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Abstract

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.

DOI
10.1103/physreva.93.033803

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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 \ensuremathξ\ensuremath{\xi}\ensuremathξ and \ensuremathζ\ensuremath{\zeta}\ensuremathζ, 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 \ensuremathξ\ensuremath{\xi}\ensuremathξ and \ensuremathζ\ensuremath{\zeta}\ensuremathζ, 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 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.
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 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.
We present an interesting dynamical temporal localization of a strongly driven two-level system (TLS), a plateau with quantized oscillation, by an analytical and transparent method, the counter-rotating-hybridized rotating-wave (CHRW) method. This approach, which is based on unitary transformations with a single parameter, treats the rotating and counter-rotating terms on equal footing. In the unitarily transformed representation, we find that it is the multiple-harmonic terms shown in the transformed Hamiltonian that make a crucial contribution to the generation of the exotic plateau phenomenon. By comparing the results of the numerically exact calculation and several other methods, we show that the CHRW results obtained by analytical formalism involving the collective effects of multiple harmonics are in good agreement with the numerical results, which illustrates not only the general tendency of the dynamical evolution of strongly driven TLS, but also the interesting phenomena of plateaus. The developed CHRW method reveals two kinds of plateau patterns: zigzag plateau and armchair plateau, and quantitative analysis is given to comprehensively describe the features of the plateau phenomenon. The plateau phenomenon has a periodical pattern whose frequency is double the driving frequency, and possesses quantized oscillations, the number of which has a certain, precise value. Also, fast oscillation is produced on every plateau that is determined by the relevant driving parameters of the TLS. Our main results are as follows: (i) in the large-amplitude oscillatory case, it turns out that the collective effects of all even harmonics contribute to the generation of zigzag plateau with quantized oscillation, and the general tendency of evolution coincides with the result of the original CHRW method because of a linear trend of the phase function; (ii) in the small-amplitude oscillatory case, the dynamics of the coherent destruction of tunneling under strong driving is exactly exhibited by including the odd-harmonic effect, which offers an attractive dynamical pattern, namely, armchair plateau possessing a two-stair structure rather than the complete destruction. Besides, the characteristics of the plateau (position, frequency, the envelope, and number of quantized oscillations) are revealed by our analytical formalism. Both of the dynamical patterns are of great interest to strongly driven physics in the ongoing research on driven TLS systems.
Single-qubit quantum gates are implemented as the time-controlled unitary evolution of a two-level system under harmonic driving. The rotating-wave approximation (RWA) provides closed-form gate parameters for weak coupling. Larger amplitudes, desirable to speed up the dynamics, make the RWA-designed gates inaccurate because of the neglected counter-rotating term, which also shifts the resonance (Bloch-Siegert shift). Here we study theoretically the infidelity of gates beyond the RWA, using the effective Magnus-Taylor Hamiltonian through second order in the inverse of the driving frequency. By detuning the driving frequency by an amplitude-dependent shift of Bloch-Siegert type and rescaling the operation time, the RWA gate conditions are recovered with corrected parameters. Numerical solutions of the Schrodinger equation show that these corrections lower the gate infidelity by several orders of magnitude, extending the range of usable amplitudes