Chaos in Time: A Dissipative Continuous Quasi Time Crystals

TL;DR

This work addresses how dissipation-driven chaos interacts with time-crystalline order in open quantum systems by studying two coupled driven-dissipative Dicke models and analyzing the phase diagram with mean-field theory and truncated Wigner approximation. It identifies melted, three distinct continuous time crystals (CTC1–CTC3) and a chaotic quasi-time-crystal (CQTC), with chaos emerging at interfaces between CTCs and non-unitary dynamics; CQTC is marked by non-decaying but non-periodic oscillations and corresponds to a 1D sub-manifold of the Liouvillian spectrum. The emission spectra reveal sharp, regulator-like peaks for CTCs and a broad, structureless spectrum for CQTC, reflecting the underlying spectral organization. Overall, the study demonstrates how competing order mechanisms in dissipative quantum systems can produce quasi-periodic or chaotic temporal structure, with potential implications for quantum sensing and non-equilibrium thermodynamics.

Abstract

While a generic open quantum system decays to its steady state, continuous time crystals (CTCs) develop spontaneous oscillation and never converge to a stationary state. Just as crystals develop correlations in space, CTCs do so in time. Here, we introduce a Continuous Quasi Time Crystals (CQTC). Despite being characterized by the presence of non-decaying oscillations, this phase does not retain its long-range order, making it the time analogous of quasi-crystal structures. We investigate the emergence of this phase in a system made of two coupled collective spin sub-systems, each developing a CTC phase upon the action of a strong enough drive. The addition of a coupling enables the emergence of different synchronized phases, where both sub-systems oscillate at the same frequency. In the transition between different CTC orders, the system develops chaotic dynamics with aperiodic oscillations. These chaotic features differ from those of closed quantum systems, as the dynamics is not characterized by a unitary evolution. At the same time, the presence of non-decaying oscillations makes this phenomenon distinct from other form of chaos in open quantum system, where the system decays instead. We investigate the connection between chaos and this quasi-crystalline phase using mean-field techniques, and we confirm these results including quantum fluctuations at the lowest order.

Figures (6)

• Figure 1: Sketch of the model and emergent phases. (a) Two systems, each described by the Dicke model in Eq. \ref{['Eq:Driven_Dissipative_Dicke']}, are coupled and result in Eq. \ref{['eq:ME_spins']}. According to the choice of parameters and the relative strength of the coupling, the collective phase of the system changes between (b) Melted: Irreversible decay towards the steady state. When analyzing the structure of the Liouvillian superoperator [c.f. \ref{['Eq:spectrum']}], all eigenvalues are characterized by Re$(\lambda_j)<0$, except the one associated with the steady state Re$(\lambda_j)=0$. (c) Time Crystal: Undamped, regular oscillations. This results in Re$(\lambda_j)=0$ and equally spaced Im$(\lambda_j)\neq0$ for a portion of eigenvalues. (d) Quasicrystal: Persistent but aperiodic oscillations. Here, Re$(\lambda_j)=0$ and non-commensurate Im$(\lambda_j)$.
• Figure 2: Phase diagram: Characterization of different phases using the largest Lyapunov exponent $\Lambda_L$. Chaotic dynamics, characterized by $\Lambda_L>0$, emerge at the interface of different time crystal phases CTC1, CTC2, and CTC3, described by $\Lambda_L=0$. The melted phase is given by $\Lambda_L<0$. The dashed black and the white dotted-dashed lines separate the CTC from the chaotic CQTC phases, and have been obtained by fixed point analysis.
• Figure 3: Time crystal phases: Time evolution and the emission spectrum of different CTC phases using the mean-field and the truncated Wigner approximation (TWA). Panels (a,d), (b,e) and (c,f) represents the time evolution of $m_z^{A,B}$ of CTC1 ($\Omega/\kappa=1.5,\Gamma/\kappa=0.1$), CTC2 ($\Omega/\kappa=0.1,\Gamma/\kappa=1.5$) and CTC3 ($\Omega/\kappa=0.75,\Gamma/\kappa=0.75$), respectively, for different system sizes. Transient behavior can be observed for finite system sizes. The normalized emission spectra of the corresponding mean-field dynamics is given by (g-i).
• Figure 4: Chaos: Panels (a) and (b) illustrate the time evolution of subsystems $A$ and $B$ using mean-field dynamics and TWA, with parameters $\Omega/\kappa=2$ and $\Gamma/\kappa=1.4$. (c) and (d) The normalized emission spectrum for the dynamics of $m_z^{A}$ shown in (a). The data in (c) have been obtained by the Fourier transform in \ref{['Eq:Observable_time']} over a time $\kappa t = 10^2$, while those in (d) for a time $\kappa t = 10^3$.
• Figure 5: Maximum Lyapunov exponent for different initial states $m^{A,B}_x=\varepsilon,m^{A,B}_y=\delta,m^{A,B}_z=\sqrt{1-\varepsilon^2-\delta^2}$. Positive values of $\lambda_L$ indicate quasi-time crystal phase, $\lambda_L=0$ indicates time-crystal phases, and the system exhibits a time-independent melted phase for $\lambda_L<0$. All the phases are robust to different choices of initial states. The white and black dashed line corresponds to $\Omega/\kappa=\vert (\Gamma/\kappa)^2-1\vert/\sqrt{(\Gamma/\kappa)^2+1}$ and $\Omega/\kappa=\sqrt{(\Gamma/\kappa)^2+1}$ respectively, obtained by fixed point analysis.