Discontinuities in Driven Spin-Boson Systems due to Coherent Destruction of Tunneling: Breakdown of the Floquet-Gibbs Distribution

If an open quantum system is periodically driven with high frequency and the driving commutes with the system-bath coupling operator, it is known that the system approaches a Floquet-Gibbs state, a generalization of Gibbs states to periodically driven systems. Here, we investigate the stationary state of an ac-driven system when the driving and dissipation are noncommutative. Then, the resulting stationary state does not obey the Floquet-Gibbs distribution, and the system dynamics is determined by inelastic scattering processes of the driving field. Based on the Floquet-Redfield formalism, we show that the probability distribution can exhibit population inversion and discontinuities, i.e., jumps, for parameters at which coherent destruction of tunneling takes place. These discontinuities can be observed as intensity jumps in the emission into the bath.

Figure 1

(a) The ac-driven spin-boson model. Because of the coherent dynamics, it emits phonons with $\mathrm{\Omega }$, $\mathrm{\Omega }\pm \mathrm{\Delta }$ (unshifted, blueshifted, and redshifted), respectively. The corresponding transitions are depicted in (b).

Figure 2

(a) Quasienergy spectrum as a function of the driving amplitude $h_{z,1}$. (b) and (c) depict the probabilities $p_{\lambda }$ for $\theta =\pi /2$ (${\widehat{\sigma }}_{\pi /2}={\sigma }_x$) and $\theta =\pi /4$ (${\widehat{\sigma }}_{\pi /4}=(1/\sqrt{2}){\sigma }_x+(1/\sqrt{2}){\sigma }_z$), respectively. The Floquet states are approximately given by the eigenstates of ${\sigma }_x$, which we denote by ${|-1⟩}_x$ and ${|+1⟩}_x$ corresponding to eigenvalues $-1$ and 1, respectively. We choose $h_{z,0}=0$ [except for the thin lines in (b), where $h_{z,0}=0.05h_x$], ${\omega }_c=10h_x$, $\mathrm{\Omega }=40h_x$, and $k_{B}T=3h_x$. All results are presented in a $\gamma$-independent way.

Figure 3

(a) The transition coefficients for ${\widehat{\sigma }}_{\pi /2}$. The coefficients for ${\widehat{\sigma }}_{\pi /4}$ are almost equal. (b) Magnification of the jump discontinuity in (a). (c) and (d) depict corresponding intensities of blue- and redshifted emitted phonons for $\theta =\pi /2$ and $\theta =\pi /4$ coupling, respectively. The ratio of the red- and blueshifted emission intensities for $\theta =\pi /4$ is depicted in (e). Parameters are the same as in Fig. 2.

Figure 4

(a) shows the probability of the Floquet state $|{\phi }_0⟩$ in a stationary state of the Hamiltonian Eq. (1). (b) depicts the difference of the blue- and redshifted emission intensities into the bath. Parameters are the same as in Fig. 2.