Routes to chaos in the balanced two-photon Dicke model with qubit dissipation

Jiahui Li and Stefano Chesi

Phys. Rev. A 109, 053702 – Published 1 May 2024

Abstract

We study the semiclassical limit of the two-photon Dicke model with both cavity decay and qubit dissipation, and extend a recent analysis of its stationary points [Garbe et al., Sci. Rep. 10, 13408 (2020)], where a large unstable region was found. By considering the explicit dynamical evolution, we show that the unstable region actually hosts a rich nonlinear behavior. At variance with other types of Dicke models (without qubit dissipation or with one-photon interactions), the occurrence of chaos does not rely on a large counterrotating interaction. Furthermore, new routes to chaos appear in addition to period-doubling bifurcations, i.e., intermittent chaos and quasi-periodic oscillations. The transition mechanisms under these three distinct routes are investigated in detail through the system's long-time evolution, the optical field power spectrum, Lyapunov exponents, and bifurcation diagrams. Additionally, we provide a comprehensive phase diagram detailing the existence of stable fixed points, limit cycles, and the aforementioned chaos-related dynamics.

Received 18 August 2023
Accepted 20 March 2024

DOI:https://doi.org/10.1103/PhysRevA.109.053702

Physics Subject Headings (PhySH)

Cavity quantum electrodynamics Open quantum systems & decoherence Optical chaos

Quantum cavities Trapped ions

Dicke model

Nonlinear DynamicsAtomic, Molecular & OpticalQuantum Information, Science & Technology

Authors & Affiliations

Jiahui Li^1,2 and Stefano Chesi ^2,3,*

¹School of Future Technology, Henan University, Zhengzhou 450046, China
²Beijing Computational Science Research Center, Beijing 100193, China
³Department of Physics, Beijing Normal University, Beijing 100875, China

^*stefano.chesi@csrc.ac.cn

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Issue

Vol. 109, Iss. 5 — May 2024

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Images

Figure 1
(a1 and a2) Phase diagrams of stable fixed points for different $N$ . The dark blue area represents the $NP$ ; the light blue area represents the $SP$ , and the red area represents the phase without any stable fixed point ( $I$ ). Critical point $g_{t 1}$ (white line) separates $NP$ from other phases. $g_{t 2}$ (red dashed line) is the boundary of $SP$ and phase $I$ , which is obtained numerically. (a2 and a3) Chaotic motion in phase $I$ for $N = 10, ω_{q} / ω_{0} = 0.1, g / ω_{0} = 0.32$ . (b2 and b3) Chaotic motion in phase $I$ for $N = 50, ω_{q} / ω_{0} = 0.1, g / ω_{0} = 0.27$ . Insets: corresponding qubit trajectories in phase space. The other parameters are $κ / ω_{0} = 1$ and $Γ_{↓} / ω_{0} = Γ_{ϕ} / ω_{0} = 0.01$ .
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Figure 2
Evolution from periodic to chaotic motion through period-doubling bifurcation. (a) Bifurcation phase diagram for $ω_{q} / ω_{0} = 0.1$ which plots the oscillation amplitudes of $s_{z} (t) = 2 〈 {\hat{J}}_{z} 〉 (t) / N$ . The insets (from right to left) present the bifurcation of qubit trajectories from the symmetry limit cycle, unsymmetric limit cycle, via period doubling to chaos. The dashed black lines indicate the values of parameters. (b1–b3) Chaotic dynamics ( $g / ω_{0} = 0.384$ ) by qubit evolution $s_{z} (t)$ , perturbations of cavity field $ln (ε_{I_{c}})$ , and power spectrum density of photon field, respectively. (c1–c3) Evolution of period doubling ( $g / ω_{0} = 0.373$ ). Other parameters are the same as Fig. 1.
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Figure 3
Evolution from periodic to fully chaotic motion via IC. By plotting the time evolution of photon field ${\bar{n}}_{c} = 〈 {\hat{a}}^{†} \hat{a} 〉 / N$ , perturbation, and photon field spectrum in the columns from left to right, we present periodic evolution ( $g / ω_{0} = 0.3763$ ), intermittent chaos ( $g / ω_{0} = 0.3795$ ), and fully chaotic dynamics ( $g / ω_{0} = 0.3826$ ) in (a)–(c), respectively. For better insight, we enlarge part of intermittency dynamics (green shaded) in the inset of (b1). Correspondingly, insets in (b2) display system trajectories of chaotic (green) and periodic sectors (purple) in IC. A medium value of qubit frequency $ω_{q} / ω_{0} = 0.15$ is used in the calculation. Other parameters are the same as Fig. 1.
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Figure 4
Evolution from quasi-periodic (left panel: $g / ω_{0} = 0.3716$ ) to chaotic motion (right panel: $g / ω_{0} = 0.3795$ ) for large qubit frequency $ω_{q} / ω_{0} = 0.25$ . The inset in (a1) and (b1) is the blowup of a random prominent peak of $s_{z} (t)$ . In the discrete spectrum (a3), the main frequencies are at integer multiples of $f_{1} = 5.3 \times 10^{- 3} ω_{0}$ , and the distance between auxiliary frequency peaks is $f_{2} = 2.6 \times 10^{- 4} ω_{0}$ . Inset in (b2): chaotic attractor. Insets in (a3) and (b3): evolution of cavity field ${\bar{n}}_{c} = 〈 {\hat{a}}^{†} \hat{a} 〉 / N$ . Other parameters are the same as in Fig. 1.
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Figure 5
(a) Amplitude statistics of qubit oscillation $s_{z} (t)$ during quasi-periodicity to chaos. The dashed lines mark the quasi-periodic and chaotic dynamics presented in Fig. 4. (b) Corresponding values of the LE. The dashed line denotes the zero value. Other parameters are as in Fig. 4.
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Figure 6
Overall phase diagram. It contains stable phases [ $NP$ (dark blue) and $SP$ (light blue)] and oscillatory phases [periodic (green), quasi-periodic (yellow), intermittent chaos (red spots), and full chaos (brown)]. The dashed-dotted lines mark the values of qubit frequencies that we have used in previous figures. The red dashed line is the critical boundary $g_{t 2}$ . Other parameters are the same as in Fig. 1.
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