Dissipative two-level system under strong ac driving: A combination of Floquet and Van Vleck perturbation theory

We study the dissipative dynamics of a two-level system (TLS) exposed to strong ac driving. By combining Floquet theory with Van Vleck perturbation theory in the TLS tunneling matrix element, we diagonalize the time-dependent Hamiltonian and provide corrections to the renormalized Rabi frequency of the TLS, which are valid for both a biased and unbiased TLS and go beyond the known high-frequency and rotating-wave results. In order to mimic environmental influences on the TLS, we couple the system weakly to a thermal bath and solve analytically the corresponding Floquet-Bloch-Redfield master equation. We give a closed expression for the relaxation and dephasing rates of the TLS and discuss their behavior under variation of the driving amplitude. Further, we examine the robustness of coherent destruction of tunneling (CDT) and driving-induced tunneling oscillations (DITO). We show that also for a moderate driving frequency an almost complete suppression of tunneling can be achieved for short times and demonstrate the sensitiveness of DITO to a change of the external parameters.

Figure 1

Quasienergies ${\varepsilon }_{-,n}^{\mathrm{RWA}}$ and ${\varepsilon }_{+,n+m}^{\mathrm{RWA}}$ (triangles) and unperturbed quasienergies ${\varepsilon }_{\uparrow ,n}^0$ and ${\varepsilon }_{\downarrow ,n+m}^0$ (dashed line and dotted-dashed line) for an $m$-photon resonance. The unperturbed quasienergies show an exact crossing at $\varepsilon =m\omega$ according to Eq. (7). The corresponding eigenstates are $|u_{\uparrow ,n}^0\rangle \rangle$ and $|u_{\downarrow ,n+m}^0\rangle \rangle$. For finite $\Delta$ an avoided crossing can be observed. The energy ${\varepsilon }_{-,n}^{\mathrm{RWA}}$ and the corresponding eigenstate $|{\Phi }_{-,n}^{\mathrm{RWA}}\rangle \rangle$ are represented by black downward triangles, whereas ${\varepsilon }_{+,n+m}^{\mathrm{RWA}}$ and $|{\Phi }_{+,n+m}^{\mathrm{RWA}}\rangle \rangle$ are shown by red upward triangles. For $\varepsilon >m\omega$ we find that $|{\Phi }_{-,n}^{\mathrm{RWA}}\rangle \rangle$ approaches $|u_{\uparrow ,n}^0\rangle \rangle$, while $|{\Phi }_{+,n+m}^{\mathrm{RWA}}\rangle \rangle$ becomes $|u_{\downarrow ,n+m}^0\rangle \rangle$ and vice versa for $\varepsilon <m\omega$. The labeling of the perturbed eigenstates and eigenenergies is chosen in a way that ${\varepsilon }_{+,n+m}^{\mathrm{RWA}}⩾{\varepsilon }_{-,n}^{\mathrm{RWA}}$ for all $\varepsilon$.

Figure 2

Quasienergies against static bias $\varepsilon$. The upward triangles result from numerical diagonalization of the Floquet matrix (4), while the solid lines correspond to the analytical formulas (18) and (19). Parameters are $\omega /\Delta =2$, $A/\Delta =3$. At an $m$-photon resonance, we find avoided crossings with a gap distance of ${\Delta }_m$.

Figure 3

Survival probability $P_{\downarrow \to \downarrow }(t)$ close to a 2-photon resonance. The parameters are $\varepsilon /\Delta =4.1$, $\omega /\Delta =2$, $A/\Delta =3$. We compare results obtained from a RWA and the first- and second-order Van Vleck perturbation theory. Furthermore, we show the averaged second-order Van Vleck dynamics, which correspond to Eq. (15) with the Van Vleck frequency [Eq. (20)] and second-order mixing angle [Eq. (22)].

Figure 4

Comparison of the main oscillation frequency ${\Omega }_m^{\mathrm{RWA}}$ obtained by the RWA and second-order Van Vleck frequency ${\Omega }_m^{(2)}$ for a fixed static bias, $\varepsilon =4.0\Delta$. The relative mistake performing the RWA is shown against the driving frequency $\omega /\Delta$ and driving amplitude $A/\Delta$. The darkest areas show regions in parameter space of small or no deviations between the two approaches, whereas the lightest areas show a deviation of 15% or more.

Figure 5

Comparison of the main oscillation frequency ${\Omega }_m$ obtained by second-order and third-order Van Vleck perturbation theory for a fixed static bias, $\varepsilon =4.0\Delta$. The relative mistake performing second-order perturbation theory is shown against the driving frequency $\omega /\Delta$ and driving amplitude $A/\Delta$. Color scale is the same as in Fig. 4.

Figure 6

Survival probability $P_{\downarrow \to \downarrow }(t)$ for $\varepsilon /\Delta =4$, $\omega /\Delta =4$, and $A/\Delta =4.1$. Exact numerical results are compared with RWA and second-order Van Vleck results.

Figure 7

Absolute value of the Fourier transform $F(\nu )$ of $P_{\downarrow \to \downarrow }(t)$ in Fig. 6. The oscillation frequency corresponding to ${\Omega }_1=0.45\Delta$ is dominating and is also predicted by the RWA approach. Notice, that there is exact agreement between the second-order Van Vleck and the numerical results.

Figure 8

Survival probability $P_{\downarrow \to \downarrow }$ for $\varepsilon /\Delta =4$, $\omega /\Delta =2.7$, and $A/\Delta =4.1$. Exact numerical results are compared with RWA and second-order Van Vleck predictions.

Figure 9

Absolute value of the Fourier transform of $P_{\downarrow \to \downarrow }(t)$ in Fig. 8. The numerical and the second-order Van Vleck graphs clearly show several dominating oscillation frequencies, while the RWA only shows one of them. Note also the different scale of the $y$ axis compared to Fig. 7.

Figure 10

Fourier coefficient $\left|X_{--,n}\right|$ for various values of $n$ against driving amplitude $A$. We examine the case $\varepsilon =2\omega$. The parameters are $\varepsilon /\Delta =4.0$ and $\omega /\Delta =2.0$. The black triangles show data points from numerical diagonalization of the Floquet matrix; the red (dark gray) dashed curve is obtained from first-order perturbation in $\Delta$ [Eq. (45)], whereas the green (light gray) solid curve is obtained by going to second order in $\Delta$ [Eq. (43)].

Figure 11

Fourier coefficient $\left|X_{-+,n}\right|$ for various values of $n$ against driving amplitude $A$. The parameters are the same as in Fig. 10.

Figure 12

Relaxation (a) and dephasing rate (b) against driving amplitude $A$ for $\omega /\Delta =2.0$, $\varepsilon /\Delta =4.1$, $\hslash \beta \Delta =10$, and $\kappa =0.01$. Results obtained within the second-order Van Vleck perturbation theory are compared with RWA calculations. Notice that the RWA predicts an unphysical vanishing of the relaxation rates at the zeros of ${\Delta }_{-2}$.

Figure 13

Survival probability $P_{\downarrow \to \downarrow }(t)$ close to a 2-photon resonance. The parameters are $\varepsilon /\Delta =4.1$, $\omega /\Delta =2.0$, $A/\Delta =3.0$, $\kappa =0.01$, and $\hslash \beta \Delta =10$. Analytical results obtained by second-order Van Vleck perturbation theory are compared with RWA results. The inset shows the long-time dynamics and visualizes the deviation of the RWA from the Van Vleck dynamics in the asymptotic limit.

Figure 14

Absolute value of the Fourier transform $F(\nu )$ of the survival probability for the RWA and second-order Van Vleck perturbation theory. The parameters are the same as in Fig. 13. Next to the relaxation peak at $\nu =0$ the RWA dynamics are governed by a single frequency $\nu ={\Omega }_m^{\mathrm{RWA}}$. The second-order dynamics also exhibit the relaxation peak and a main frequency, which, however, is shifted to $\nu ={\Omega }_m^{(2)}$. Additionally, the higher harmonics of the driving can be seen in the second-order dynamics. For visualization of the $\delta$ peaks, appearing at $\nu =n\omega$, a finite width and height have been artificially introduced. Furthermore, broadened peaks appear at $\nu =n\omega \pm {\Omega }_m^{(2)}$.

Figure 15

Coherent destruction of tunneling for the nondissipative case ($\kappa =0$). The survival probability $P_{\downarrow \to \downarrow }(t)$ is shown at a 3-photon resonance. The parameters are $\varepsilon /\Delta =6.0$, $\omega /\Delta =2.0$, and $A/\Delta =12.7603$. The Van Vleck solution is compared with the RWA and a numerical diagonalization of the Floquet Hamiltonian. Within the RWA, a complete destruction of tunneling can be observed, whereas the analytic Van Vleck solution exhibits driving-induced oscillations. The numerical solution predicts, on the contrary, complete population inversion with low but nonvanishing frequency ${\Omega }_m$. Figure (b) is a blowup from figure (a) for a shorter time scale. There, the numerical and Van Vleck solutions agree well, and one can nicely see the small oscillations resulting from a 3-photon absorption or emission.

Figure 16

Coherent destruction of tunneling for the dissipative case ($\kappa =0.01$, $\hslash \beta \Delta =10$). The remaining parameters are the same as in Fig. 15. A comparison between the numerical solution of the Floquet Hamiltonian and the full master equation (38), the Van Vleck combined with the MRWA approach, and RWA results is provided. The RWA approach predicts a slower relaxation than the numerical one and the Van Vleck solution. Deviations between the numerical and Van Vleck solutions can be seen especially in the long time limit. As in Fig. 15, the numerical result predicts oscillations with a nonvanishing frequency ${\Omega }_m$. For short times, see figure (b), the numerical and Van Vleck results agree well.

Figure 17

Driving-induced tunneling oscillations for the nondissipative case ($\kappa =0$). The survival probability $P_{\downarrow \to \downarrow }(t)$ is shown for $A/\Delta =3.0$, $\omega /\Delta =2.0$, and $\varepsilon /\Delta =5.9011$ (exact 3-photon resonance). Three approaches are compared: a complete numerical solution of the Floquet Hamiltonian, the second-order Van Vleck approach, and the RWA approach. For the first two approaches, complete population inversion is predicted, and for the Van Vleck dynamics we find the main oscillation frequency ${\Omega }_3^{(2)}=\Delta |J_{-3}(A/\omega )|$. Besides, the numerical and Van Vleck approaches exhibit small driving-induced oscillations, see especially figure (b). The RWA predicts a strongly shifted oscillation frequency. Moreover, population inversion is incomplete. For further comparison, the RWA and the Van Vleck approaches are shown for a slightly shifted external frequency, $\omega /\Delta =1.9$. The dynamics in this case are almost completely localized.

Figure 18

Driving-induced tunneling oscillations for the dissipative case ($\kappa =0.01$ and $\hslash \beta \Delta =10$). Remaining parameters are the same as in Fig. 17. The numerical solution of the master equation (38), the analytical Van Vleck–MRWA solution, and the RWA solution for $\omega /\Delta =2.0$ are compared. Good agreement between the numerical and Van Vleck solutions can be observed on both long (a) and short (b) time scales. Within the RWA, not only are the main frequency and amplitude changed and the driving-induced oscillations missed, but also the equilibrium value lies far above the Van Vleck prediction. The RWA and the Van Vleck–MRWA solution for $\omega /\Delta =1.9$ show an almost incoherent decay, and their long time limits differ strongly from the corresponding ones for $\omega /\Delta =2.0$.