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Physical Review A

Interference between driving and dissipation in the spin-boson model: Effect of counter-rotating terms

Chunjian Zhao, Hang Zheng

Published Oct 29, 2010• 15 citations
Publisher

Abstract

The quantum dynamics of the dissipative two-level system in the Ohmic bath with Rabi driving is studied by means of a modified Bloch-Redfield approach based on a unitary transformation. The main purpose is to understand the effect of counter-rotating terms of the system-environment interaction on the dynamics of the two-level system and, in particular, if it is possible to control the effective system-environment coupling by their effect on the interference between driving and dissipation. By calculating the nonequilibrium correlation $P(t)$ for a finite driving it is shown that the counter-rotating terms of the system-environment interaction play an important role in the dynamic evolution with a longer coherent time and a weaker amplitude of the long-time driven oscillation.

DOI
10.1103/physreva.82.043844

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The quantum dynamics of the dissipative two-level system in the Ohmic bath with Rabi driving is studied by means of a modified Bloch-Redfield approach based on a unitary transformation. The main purpose is to understand the effect of counter-rotating terms of the system-environment interaction on the dynamics of the two-level system and, in particular, if it is possible to control the effective system-environment coupling by their effect on the interference between driving and dissipation. By calculating the nonequilibrium correlation P(t)P(t)P(t) for a finite driving it is shown that the counter-rotating terms of the system-environment interaction play an important role in the dynamic evolution with a longer coherent time and a weaker amplitude of the long-time driven oscillation.
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.
The non-Markovian dynamics of the driven spin-boson model at zero and finite temperature is investigated theoretically. Based on a unitary transformation and convolution theorem, the analytical expression of population difference in Laplace space is obtained. This method provides a tool to help us to understand the interference between driving and dissipation within the non-Markovian frame. We study the transient and the steady-state non-Markovian dynamics of population difference P ( t ) in the case of different temperature, driving and spin–bath coupling (SBC) strength. It has been shown that it is necessary to use non-Markovian methods when SBC is strong. At zero temperature, the curve of steady-state amplitude P lt versus Rabi driving amplitude Ω is N-like shaped. While at finite temperature, the curve becomes Λ-like shaped. These curves come from the interference between driving and dissipation. Also, from the interference pattern above, we know that large coherent oscillations can be obtained by optimizing the parameters: Rabi driving amplitude, temperature and SBC strength.
The quantum dynamics of the dissipative two-level system with nonzero bias and sub-Ohmic bath is studied by means of the perturbation approach based on a unitary transformation. It has been shown that for the sub-Ohmic bath it is necessary to use the non-Markovian approach, especially for the short time behavior of the coupled system and environment. The nonequilibrium correlation P(t)P(t)P(t) has been calculated to show that a finite bias may favor the short time coherence. The spectrum of the susceptibility ${\ensuremath{\chi}}^{\ensuremath{''}}(\ensuremath{\omega})$ of the sub-Ohmic case may have a double peak structure in the range of $\ensuremath{\omega}>0$ when the coupling $\ensuremath{\alpha}$ is relatively strong. Besides, the coherence-decoherence transition point ${\ensuremath{\alpha}}_{c}$ is determined for different $0<s\ensuremath{\le}1$ by the condition of ${\ensuremath{\chi}}^{\ensuremath{''}}(\ensuremath{\omega}=0)=\ensuremath{\infty}$ when $\ensuremath{\alpha}={\ensuremath{\alpha}}_{c}$. Finally, we show that Shiba's relation is exactly satisfied in our results.
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.
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.
The dynamics of a driven spin-boson model is studied by means of the perturbation approach based on a unitary transformation. Analytical expressions for the population difference and the coherence of the two state system are obtained. The results show that for weak driving, the population difference displays damping coherent oscillation and/or damping quantum beat, depending on the initial preparation. The coherence exhibits damped oscillation with Rabi frequency. When driving is strong enough, the population difference exhibits undamped large-amplitude coherent oscillation. In addition, our theory leads to correct results in two limiting cases: without dissipation and without driving field.