Dynamical many-body phases of the parametrically driven, dissipative Dicke model

R. Chitra and O. Zilberberg

Phys. Rev. A 92, 023815 – Published 10 August 2015

Abstract

Control and manipulation of quantum engineered systems allows for the utilization of time-dependent parametric modulations for accessing novel out-of-equilibrium phenomena. In the absence of such driving, the dissipative Dicke model exhibits a fascinating out-of-equilibrium many-body phase transition as a function of a coupling between a driven photonic cavity and numerous two-level atoms. We study the effect of a parametric modulation of this coupling and discover a rich phase diagram as a function of the modulation strength. We find that in addition to the established normal and super-radiant phases, a new phase with pulsed superradiance, which we term dynamical normal phase, appears when the system is parametrically driven. Employing different methods, we characterize the different phases and the transitions between them. Specific heed is paid to the role of dissipation in determining the phase boundaries. Our analysis paves the road for the experimental study of dynamically stabilized phases of interacting light and matter.

Received 28 January 2015
Revised 1 April 2015

DOI:https://doi.org/10.1103/PhysRevA.92.023815

Authors & Affiliations

R. Chitra and O. Zilberberg

Institute for Theoretical Physics, ETH Zurich, 8093 Zürich, Switzerland

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Vol. 92, Iss. 2 — August 2015

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Figure 1
A sketch of a parametrically driven Dicke model. A single-mode cavity is driven by a laser beam with an oscillating laser-field power $P (t)$ . The cavity is naturally leaky with a dissipation rate $κ$ . The cavity contains a collection of “two-level atoms” arising from either the momentum modes of a Bose-Einstein condensate of an atomic gas [7, 15] or the hyperfine-split lowest-energy states of an ultracold atomic gas [16] that are coupled to the driving laser with coupling $λ (t) \propto \sqrt{P (t)}$ [cf. Eq. (1)]. The atoms are also coupled to an environment with a dissipation rate $η$ . The system is best described by Liouvillian dynamics [cf. Eq. (2)].
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Figure 2
Characteristics of the dynamical phase diagram of the parametrically driven Dicke model as a function of cavity-atoms frequency $ω_{0} = ω_{c} = ω_{a}$ and parametric modulation strength $ε$ for bare coupling $λ_{0} = 0.4 Ω$ and dissipation rates $κ = η = 0.1 Ω$ . We obtain these characteristics from the numerical steady-state solutions of the mean-field equations [cf. Eqs. (6, 7, 8)]. Each point, with a $10^{- 3}$ resolution, is a result of a different numerical integration. Superimposed (striped overlay) is the normal mode stability zone [cf. Eq. (5)]. (a) Density plots of the absolute value of the steady-state time-averaged order parameters. (b) In steady-state the order parameters are oscillating with a complex beat structure; see Appendix pp2. The density plots are of the amplitude of the maximal frequency $ω$ contributing to this oscillation. We see three distinct phases appearing as a function of $ε$ , i.e., extensions of the normal and super-radiant phases (NP and SP), as well as a novel dynamical phase (D-NP). These three phases meet at a shared multicritical point. The properties of each phase are summarized in Table 1.
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Figure 3
Trajectories on the Bloch sphere of the atoms order parameters as a function of time in steady state, $f [x (t), y (t), z (t), t]$ . The trajectories are for $ε = 0.5 Ω, λ_{0} = 0.4 Ω$ , and $κ = η = 0.1 Ω$ . The trajectory that does not encircle the $z$ axis (magenta) is in SP with $ω_{0} = 0.6 Ω$ . The trajectories that encircle the $z$ axis (red and yellow) are for $ω_{0} = 0.65 Ω$ and $ω_{0} = 1.5 Ω$ , respectively.
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Figure 4
Characteristic numerical time-integration plots of Eq. (A8) in different regions of the phase diagram. All curves are with $λ_{0} = 0.4 Ω, ε = 0.5 Ω$ , and $κ = η = 0.1 Ω$ . We see that for (i) $ω_{0} = 0.5 Ω$ , mode $A_{1}$ is stable, whereas $A_{2}$ is not; (ii) $ω_{0} = 0.7 Ω$ , mode $A_{1}$ is unstable, whereas $A_{2}$ is stable; and (iii) $ω_{0} = 0.9 Ω$ , both modes $A_{m}$ are stable.
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Figure 5
The numerical stability diagram of the normal modes $A_{m}$ [cf. Eq. (4)] for $λ_{0} = 0.4 Ω$ and $κ = η = 0.1 Ω$ . As the drive in the Dicke model affects each mode differently, we obtain two “Arnold tongue” stability diagrams that are shifted and scaled with respect to each other. Superposing the zones where both modes are stable leads to the stability of the normal phase (NP) (cf. Fig. 2).
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Figure 6
Characteristic numerical analysis of the mean-field equations [cf. Eqs. (6, 7, 8)] leading to the displayed phase diagram (cf. Fig. 2). All plots are with $λ_{0} = 0.4 Ω, ε = 0.5 Ω$ , and $κ = η = 0.1 Ω$ . (a) and (c) are characteristic numerical time-integration plots of the mean-field equations with $ω_{0} = 0.6 Ω$ and $ω_{0} = 0.65 Ω$ , respectively. (b) and (d) are the corresponding fast fourier transform (FFT). We see that in the super-radiant phase (SP) region [(a) and (b)], all order parameters are oscillating around a nonzero mean with relatively small oscillations amplitudes. In the dynamical normal phase (D-NP) region [(c) and (d)], apart from $z$ , all order parameters have a zero mean value, but their oscillations are large, taking the full length of the central spin in its $x$ and $y$ components.
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