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

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.

Figure 1

$\langle {\sigma }_z\left(t\right)\rangle$ as a function of $t$ is shown for $A/{\omega }_L=0.1$, $\alpha =0.01$, and (a) ${\omega }_L=\Delta$ and (b) ${\omega }_L=4\Delta$.

Figure 2

$\langle {\sigma }_z\left(t\right)\rangle$ as a function of $t$ is shown for $A/{\omega }_L=1$, $\alpha =0.01$, and (a) ${\omega }_L=\Delta$ and (b) ${\omega }_L=4\Delta$.

Figure 3

[(a), (c)] $\langle {\sigma }_z\left(t\right)\rangle$ and [(b), (d)] $\langle {\sigma }_x\left(t\right)\rangle$ as functions of $t$ are shown for $A/{\omega }_L=2.5$, $\alpha =0.01$, and driving frequencies [(a), (b)] ${\omega }_L=\Delta$ and [(c), (d)] ${\omega }_L=4\Delta$.

Figure 4

[(a), (c)] $\langle {\sigma }_z\left(t\right)\rangle$ and [(b), (d)] $\langle {\sigma }_x\left(t\right)\rangle$ as functions of $t$ are shown for $A/{\omega }_L=1$, $\alpha =0.01$, and driving frequencies [(a), (b)] ${\omega }_L=\Delta$ and [(c), (d)] ${\omega }_L=4\Delta$.

Figure 5

(a) The relaxation rate and (b) the dephasing rate as functions of $A$ are shown for $\alpha =0.01$ and ${\omega }_L=\Delta$. ${\gamma }_0=\pi G\left(\Delta \right)/4$ is set as units.

Figure 6

(a) The steady oscillation amplitude $P_{\infty }$ and (b) ${\langle {\tilde{\sigma }}_z\rangle }_{ss}$ as functions of $A$ are shown for $\alpha =0.01$ and driving frequency ${\omega }_L=\Delta$.

Figure 7

(a) The steady oscillation amplitude $P_{\infty }$ and (b) the modified driving strength $\tilde{A}$ as functions of ${\omega }_L$ are shown for $A=0.5\Delta$ and $\alpha =0.01$.

Figure 8

(a) The static value of coherence in the long-time limit as a function of $A$ is shown for $\alpha =0.01$ and ${\omega }_L=\Delta$. (b) The static value of coherence as a function of ${\omega }_L$ is shown for $\alpha =0.01$ and $A=0.5\Delta$.

Figure 9

(a) The steady oscillation amplitude of the coherence as a function of $A$ is shown for $\alpha =0.01$ and ${\omega }_L=\Delta$. (b) The steady oscillation amplitude of the coherence as a function of ${\omega }_L$ is shown for $\alpha =0.01$ and $A=0.5\Delta$.

Figure 10

The probability of the TLS remains in its initial state as a function of $t$ is shown for $\alpha =0.01$, ${\omega }_L=10\Delta$, and $A=24.0483\Delta$.