Phase transition in the periodically pulsed Dicke model

We study the effect of pulsed driving and kicked driving of the interaction term on the nonequilibrium phase transition in the Dicke model. Within the framework of Floquet theory, we observe the emergence of new nontrivial phases on impingement by such periodic pulses. Notably, our study reveals that a greater control over the dynamical quantum criticality is possible through the variation of multiple parameters related to the pulse, as opposed to a single parameter control in a monochromatic drive. Furthermore, the probability of the system remaining trapped in a metastable state during the observed first order transition from the superradiant to normal phase is found to be higher for small number of kicks (or pulses) in comparison to the sinusoidal perturbation.

Figure 1

This figure shows the unstable regions in the parameter space for the maps derived from Eq. (15) and its counterpart for the $q_{+}$ mode combined. The black background (dark shaded) region represents the NP, the red background (light shaded) represents the SR phase, and the yellow striations (in the dark-shaded region) represent the unstable zones. We expand and perform the energy surface calculations around ${\epsilon }_{\pm }=m\mathrm{\Omega }/2$, for $m=0$, which is the region obtained at a low ${\lambda }_0$.

Figure 2

The quasienergy surface $E_g(X,Y)$ is shown for the renormalized kick strength: $k{\lambda }_1=0$ (a), $k{\lambda }_1=6$ (b), $k{\lambda }_1=10.66$ (c), and $k{\lambda }_1=14$ (d). There is a transition from a scenario with two global minima in (a) representing an SR phase to a scenario with a single global minimum (d) representing the NP. The critical point lies at $k{\lambda }_1=10.66$ when a first order phase transition takes place as the values of the order parameter $(X,Y)$ undergo a finite jump to $(0,0)$.

Figure 3

We show above the cross sections corresponding to the quasienergy contours in Fig. 2. The first order phase transition and the point of coexistence of the SR and normal phase is clearly seen in panel (c). The behavior of the LQE observed above is the same as the one reported in Ref. [28]. The difference lies in the fact that unlike the single parameter control of the LQE in the monochromatic case we can now use both the kick strength and the number of kicking cycles to modify the LQE.

Figure 4

The energy surface plots for the first order phase transition obtained keeping the number of pulses $k=1$ constant. The panels represent the LQE for various values of the pulse strength (a) ${\lambda }_1=0.0$, (b) ${\lambda }_1=12.0$, (c) ${\lambda }_1=15.02$, and (d) ${\lambda }_1=24.0$; we can see a distinct crossover of the nature from SR to NP at the critical point represent in panel (c).

Figure 5

The energy surface cross section corresponding to Fig. 4. We can clearly make out the transition and the coexistence point represented in panel (c).

Figure 6

The energy surface plots for the first order phase transition keeping the pulse strength ${\lambda }_1=1$ constant. The panels represent the LQE for various values of the number of pulses $k$ (a) $k=0$, (b) $k=12$, (c) $k=17$, and (d) $k=24$, we can see a distinct crossover of the nature from SR to NP at the critical point represent in panel (c).

Figure 7

The energy surface cross section corresponding to Fig. 6. We can clearly make out the transition and the coexistence point represented in panel (c).

Figure 8

This figure reflects the role of the asymmetry in the pulse width (in time). The parameters are fixed at ${\lambda }_1=2$ and $k=6$. Panels (a) and (c) represent the LQE and its cross section for $\mathrm{\Delta }=T/2$, and panels (b) and (d) represent the LQE and its cross section for $\mathrm{\Delta }=T/5$. We can see that the case in which the pulse width is broader there is a phase transition from SR to NP, while in the case of the shorter pulse width the system remains in the SR phase.