Theory of parametric resonance for discrete time crystals in fully connected spin-cavity systems

We pinpoint the conditions necessary for discrete time crystal (DTC) formation in fully connected spin-cavity systems from the perspective of parametric resonance by mapping these systems onto oscillatorlike models. We elucidate the role of nonlinearity and dissipation by mapping the periodically driven open Dicke model onto effective linear and nonlinear oscillator models, while we analyze the effect of global symmetry breaking using the Lipkin-Meshkov-Glick model with tunable anisotropy. We show that the system's nonlinearity restrains the dynamics from becoming unbounded when driven resonantly. On the other hand, dissipation keeps the oscillation amplitude of the period-doubling instability fixed, which is a key feature of DTCs. The presence of global symmetry breaking in the absence of driving is found to be crucial in the parametric resonant activation of period-doubling response. We provide analytic predictions for the resonant frequencies and amplitudes leading to DTC formation for both systems using their respective oscillator models.

Figure 1

(a) Sketch of a generic fully connected spin-cavity system, where the $N$ two-level systems are represented by a single collective spin. (b) Sketch of the effective parametric oscillator model of a spin-cavity system. Exemplary dynamics of the LOM, NOM, and DM are shown for (c) $\kappa =0$ and $\{{\omega }_d,A\}=\{0.6\omega ,0.1\}$ and (d) $\kappa =0.5\omega$ and $\{{\omega }_d,A\}=\{0.8\omega ,0.5\}$, with $g_0=0.9g_c$. (e) Sketch of the periodically driven open LMG model. Exemplary dynamics of the open LMG are shown for different (f) spin dissipation strengths and (g) anisotropies. The remaining parameters are set to (f) $\{{\lambda }_0/{\omega }_0,A,{\omega }_d/{\omega }_0,\gamma \}=\{0.8,0.2,1.8,0\}$ and (g) $\{\mathrm{\Gamma }/{\omega }_0,{\lambda }_0/{\omega }_0,A\}=\{0.01,0.8,0.5\}$ with ${\omega }_d\approx 2{\mathrm{\Omega }}_0$.

Figure 2

The ${\omega }_d\text{−}A$ phase diagram of the LOM for (a) $\kappa =0$ and (b) $\kappa =0.5\omega$. Phase diagram of the NOM for (c) $\kappa =0$ and (d) $\kappa =0.5\omega$. The diamond (star) corresponds to the exemplary dynamics of the closed (open) OMs shown in Fig. 1 [Fig. 1]. The remaining parameter is set to $g_0=0.9g_c$.

Figure 3

(a)–(d) Phase diagram of the NOM for increasing values of dissipation strength. The values of $\kappa$ considered are $\kappa =\{0.1\omega ,1.0\omega ,2.0\omega ,20\omega \}$. Trends of the simulated (e) resonant frequency and (f) minimum resonant amplitude are plotted as a function of $\kappa$. In (e) the solid lines correspond to the lower polariton frequency $2{\mathrm{\Omega }}_{-}$, while in (f) they correspond to the analytic expression of $A_r$ derived in Ref. [68]. The phase diagrams in (a)–(d) are identified in (e) and (f) using the gray dashed lines. (g) Real and (h) imaginary components of the eigenmodes of the open LOM as a function of $\kappa$. The black solid line corresponds to ${\kappa }_c^{\prime }$, while the dashed line represents ${\kappa }_c^{\prime \prime }$. (i)-(j) The effective oscillator picture of the polariton modes is sketched for (i) $\kappa <{\kappa }_c^{\prime }$ and (j) $\kappa \gg {\kappa }_c^{\prime \prime }$. The remaining parameter is set to $g_0=0.9g_c$.

Figure 4

Global symmetry breaking in the LMG model in the (a) $\gamma =-1$, (b) $\gamma =0$, and (c) $\gamma >0$ regimes, as presented in the $\mathrm{\Gamma }/{\omega }_0$ vs $\lambda /{\omega }_0$ space. The black region corresponds to the NP characterized by a steady-state solution of $\left|X\right|=0$. The colored region corresponds to the SB phase with steady-state solution of $\left|X\right|\ne 0$. The blue curve represents the critical coupling strength ${\lambda }_c$.

Figure 5

(a) Effective potential of the LMG Hamiltonian in pseudo-position-momentum representation. The potential opening upward (downward) is analogous to the system in the NP (SB phase). (b) Sketch of effective oscillator picture of the driven LMG model. (c) Dynamics of the undriven LMG model at $\{\lambda /{\omega }_0,\mathrm{\Gamma }/{\omega }_0,\gamma \}=\{0.9,0,0\}$ with its (d) corresponding Fourier decomposition, where ${\mathrm{\Omega }}_R$ is the response frequency. We have set $X_0=0$ and $Y_0=5\times {10}^{-2}$ to emphasize the oscillations in $X$.

Figure 6

Bloch sphere dynamics of (a) NP, (b) SB, and (c) $2T$-DTC phases, respectively. The blue (yellow) trajectories mark the dynamics due to an initial state marked by the green (red) dots. These dynamics are observed in the parameter regions (a) $\{\lambda /{\omega }_0,\mathrm{\Gamma }/{\omega }_0\}=\{0.75,0.03\}$, (b) $\{\lambda /{\omega }_0,\mathrm{\Gamma }/{\omega }_0\}=\{1.25,0.03\}$, and (c) $\{{\lambda }_0/{\omega }_0,\mathrm{\Gamma }/{\omega }_0,A,2{\omega }_d/{\omega }_0\}=\{0.8,0.1,0.5,1.7\}$. (d) Phase diagrams of the driven LMG model at varying dissipation and initial interaction strengths. (e) Oscillation envelopes of $X$ at varying dissipation strengths for $\{{\lambda }_0/{\omega }_0,A,2{\omega }_d/{\omega }_0\}=\{0.95,0.15,0.9\}$. All results are for the $\gamma =0$ case.

Figure 7

(a) Phase diagrams for the driven LMG model at varying anisotropy value, where $\mathrm{\Gamma }=0$ and ${\lambda }_0/{\omega }_0=0.8$. Note how the instability region decreases in size as $\gamma$ increases, until eventually disappearing when $\gamma =1$. This behavior is further proven by (b) $A_N/A_T$, the ratio between the area of the NP region and the total area of each phase diagram, where we can see that for different cases of ${\lambda }_0$, the resonance lobe always disappears eventually as $\gamma \to 1$.

Figure 8

Maximum response amplitude of ${\left|b\right|}^2/N$ as a function of $A$, when the system is driven close to resonance. The remaining parameters are set to $\kappa =0.5\omega , g_0=0.9g_c$, and ${\omega }_d=0.8\omega$.

Figure 9

Bloch sphere representation of common non-DTC dynamics in the driven LMG model, i.e., (a) normal beating, (b) symmetry-broken beating, and (c) chaotic behavior, sampled at parameter regions $\{{\lambda }_0/{\omega }_0,A,2{\omega }_d/{\omega }_0\}=\{0.8,0.35,0.5\}, \{0.9,0.9,0.05\}$, and $\{1.1,0.6,1.1\}$, respectively. The blue (yellow) trajectories mark the dynamics due to an initial state marked by the green (red) dots. All results are for the $\gamma =0$ and $\mathrm{\Gamma }/{\omega }_0=0.1$ case.