Superradiant Quantum Phase Transition in Open Systems: System-Bath Interaction at the Critical Point

Abstract

The occurrence of a second-order quantum phase transition in the Dicke model is a well-established feature. On the contrary, a comprehensive understanding of the corresponding open system, particularly in the proximity of the critical point, remains elusive. When approaching the critical point, the system inevitably enters first the system-bath ultrastrong coupling regime and finally the deepstrong coupling regime, causing the failure of usual approximations adopted to describe open quantum systems. We study the interaction of the Dicke model with bosonic bath fields in the absence of additional approximations, which usually relies on the weakness of the system-bath coupling. We find that the critical point is not affected by the interaction with the environment. Moreover, the interaction with the environment is not able to affect the system ground-state condensates in the superradiant phase, whereas the bath fields are infected by the system and acquire macroscopic occupations. The obtained reflection spectra display lineshapes which become increasingly asymmetric, both in the normal and superradiant phases, when approaching the critical point.

Figures (8)

• Figure 1: Schematic sketch of the system. Two interacting subsystems, A and B, with coupling strength $g$. Each component interacts with its own thermal bath. The interaction of a subsystem with its own reservoir can also be regarded as an input-output port, through which the system can be excited and probed. In this work we will consider explicitly a single tone coherent excitation of subsystem A and calculate the corresponding reflection coefficient $S_{11}$. This framework can be easily generalized to include the interaction with additional thermal baths.
• Figure 2: Excitation energies of the isolated system. Upper and lower excitation energies for the isolated system in the normal (yellow background) and superradiant (cyan background) phases as a function of the (a) normalized coupling and (b) frequency ratio. Parameters: (a)$\omega_b / \omega_a = 1$, (b)$g / \omega_a = 0.18$.
• Figure 3: Excitation energies.(a) Upper and lower excitation energies as a function of the normalized coupling in the closed (blue) and open (red, real parts) Dicke models. (b) Inset zooming near the critical point of the lower polaritons of both the closed and open Dicke models. (c) Negative imaginary parts of the upper (gray) and lower (orange) excitation energies. Parameters used: $\omega_a = \omega_b = 1$, $\gamma_a = 0.3$, $\gamma_b = 0.2$.
• Figure 4: Excitation energies for non-ohmic baths. Comparison of the real (red) and imaginary (blue) parts of the lower excitation energy for ohmic baths (solid lines) with those for (a) subohmic baths (dotted lines) and (b) superohmic baths, plotted as functions of the normalized coupling strength $g/\omega_a$. Parameters used: $\omega_a = \omega_b = 1$, $\gamma_{0a} = 0.3$, $\gamma_{0b} = 0.2$.
• Figure 5: Ohmic reflection spectra.(a-d) 2D ohmic reflection spectra, with an inset near the critical point in (a). The corresponding closed-system excitation energies, $\Omega$, are superimposed on the plots (green lines). (e,f) 1D spectra in the proximity of the critical point, in the (e) normal and (f) superradiant phases. Dashed vertical lines represent the corresponding closed-system lower excitation energies ($\Omega_-$), showing excellent agreement with the minima of the reflection spectra. Parameters used: (a) $\omega_a = \omega_b = 1$, $\gamma_a = \gamma_b = 0.1$; (b) $\omega_a =1.2$, $\omega_b = 1$, $\gamma_a = 0.2$, $\gamma_b = 0.1$; (c) $\omega_a =1.2$, $\omega_b = 1$, $\gamma_a = 0.1$, $\gamma_b = 0.2$; (d) $\omega_a = \omega_b = 1$, $\gamma_a = 0.1$, $\gamma_b = 0.5$; (e,f) $\omega_a = \omega_b = 1$, $\gamma_a = 0.05$, $\gamma_b = 0.075$.