Time crystals in a shaken atom-cavity system

We demonstrate the emergence of a time crystal of atoms in a high-finesse optical cavity driven by a phase-modulated transverse pump field, resulting in a shaken lattice. This shaken system exhibits macroscopic oscillations in the number of cavity photons and order parameters at noninteger multiples of the driving period, which signals the appearance of an incommensurate time crystal. The subharmonic oscillatory motion corresponds to dynamical switching between symmetry-broken states, which are nonequilibrium bond ordered density wave states. Employing a semiclassical phase-space representation for the driven-dissipative quantum dynamics, we confirm the rigidity and persistence of the time crystalline phase. We identify experimentally relevant parameter regimes for which the time crystal phase is long lived, and map out the dynamical phase diagram. We compare and contrast the incommensurate time crystal with the commensurate Dicke time crystal in the amplitude-modulated case.

Figure 1

(a) Schematic representation of the physical system. A gas of ultracold bosons in the density-wave phase is prepared inside a high-finesse cavity with a photon loss rate $\kappa$. The transverse pump beam with intensity ${\epsilon }_p$ and wavelength ${\lambda }_p$ is shaken at a period $T$. (b) Time evolution over two driving cycles of the standing-wave potential produced by the pump. (c) Steady-state response of the cavity mode occupation for nonresonant driving frequencies. The occupation follows the external driving with period $T/2$. (d) Dynamical phase diagram as a function of the maximum pump intensity ${\epsilon }_{\max }$ and unitless shaking amplitude $f_0$ for ${\omega }_d/2\pi =8.8\mathrm{kHz}$. The TC phase is found to be stable for a wide region of the parameter space.

Figure 2

(a) Cavity mode dynamics for ${\omega }_d/2\pi =8.8\mathrm{kHz}$ with fixed ${\epsilon }_{\max }/E_{\mathrm{rec}}=2.24$ and $f_0=0.03$. (b) Dynamics of the ${\mathrm{BDW}}_1$ phase. Insets: each set of inset plots depicts the evolution of the single-particle density over half a period of the driving cycle, i.e., $\rho (y,z,t)$ for $t\in \{t_1,t_1+T/4,t_1+T/2\}$. $\rho (y,z,t)$ around the time when ${\left|\alpha \right|}^2$ is at (top left) a maximum, (bottom) a minimum, and (top right) the next maximum. Circles mark the position of the antinodes of the combined mode functions $\cos [ky+\phi (t\left)\right]\cos \left(kz\right)$. (c) Dynamics for (left axis) the phase difference between the pump and cavity light fields $\mathrm{\Delta }{\theta }_{\alpha }$ and (right axis) the cavity mode occupation. (d) Contour plot for the dynamics of the slice of the SPD at $\rho (y,z=0,t)$. The system oscillates between symmetry broken ${\mathrm{BDW}}_1$ states at a fast time scale corresponding to the driving frequency. The SPD develops an asymmetry on a much longer time scale leading to the TTSB. Oscillating curves correspond to the position of the antinodes of the cavity-pump potential.

Figure 3

Top: phase diagram as a function of ${\omega }_d/2\pi$ (in units of kHz) for fixed values of ${\epsilon }_{\max }/E_{\mathrm{rec}}=2.24$ and $f_0=0.03$. Bottom: exemplary dynamics of the (a)–(f) cavity mode ${\left|\alpha \right|}^2$ and (g)–(l) order parameters ${\mathrm{\Phi }}_{{\mathrm{DW}}_1}$ (blue [dark gray]), ${\mathrm{\Phi }}_{{\mathrm{BDW}}_1}$ (red [gray]), and ${\mathrm{\Phi }}_{{\mathrm{DW}}_2}$ (yellow [light gray]) for the three dynamical phases: ${\mathrm{DW}}_1$, TC, and ${\mathrm{DW}}_2$. (a),(b),(g),(h) ${\omega }_d/2\pi =8.0\mathrm{kHz}$. (c),(d),(i),(j) ${\omega }_d/2\pi =8.8\mathrm{kHz}$. (e),(f),(k),(l) ${\omega }_d/2\pi =9.0\mathrm{kHz}$. Insets: (f) slow decay of the cavity mode occupation in the ${\mathrm{DW}}_4$ phase; (l) slow decay of other order parameters $\left|\mathrm{\Phi }\right|$ with Gaussian filtering except for $|{\mathrm{\Phi }}_{{\mathrm{DW}}_4}|$ in the ${\mathrm{DW}}_4$ phase. The Gaussian filter is chosen to remove the micromotion in $\left|\mathrm{\Phi }\right|$.

Figure 4

Long-time averaged (a) ${\mathrm{\Phi }}_{{\mathrm{DW}}_1}$ and (b) $|{\mathrm{\Phi }}_{{\mathrm{BDW}}_1}|$ as a function of ${\omega }_d$ and $f_0$ for fixed ${\epsilon }_{\max }/E_{\mathrm{rec}}=2.24$.

Figure 5

Comparison between (a),(b) the dynamical bond-density wave phase and (c),(d) the dynamical normal phase. Dependence of the ratio (left) ${\omega }_{{\mathrm{BDW}}_1}/{\omega }_d$ and (right) ${\omega }_{{\mathrm{DW}}_1}/{\omega }_d$ for varying pump intensity and driving strength. For the dynamical ${\mathrm{BDW}}_1$ phase, we use ${\omega }_d/2\pi =8.8\mathrm{kHz}$. For the dynamical normal phase, the amplitude of the pump field is modulated by ${\omega }_d/2\pi =8.0\mathrm{kHz}$.

Figure 6

(a) Stroboscopic order parameter portraits of ${\mathrm{\Phi }}_{{\mathrm{DW}}_1}$ and ${\mathrm{\Phi }}_{{\mathrm{BDW}}_1}$ for three phases: ${\mathrm{DW}}_1, {\mathrm{BDW}}_1$, and DNP. For the portraits, the relevant order parameters are recorded every $mT$ for integer $m$ in the long-time limit. Representative long-time dynamics of ${\mathrm{\Phi }}_{{\mathrm{DW}}_1}$ (blue [dark gray]) and ${\mathrm{\Phi }}_{{\mathrm{BDW}}_1}$ (red [gray]) for (b) the dynamical ${\mathrm{BDW}}_1$ phase and (c) the DNP.

Figure 7

Rigidity of TTSB to quantum fluctuations. MF (blue [dark gray]) and TWA (red [gray]) results for the cavity dynamics at (a) short and (b) long times. (c) Frequency spectrum for the cavity dynamics. The parameters are ${\epsilon }_{\max }/E_{\mathrm{rec}}=2.24, f_0=0.03$, and ${\omega }_d/2\pi =8.8\mathrm{kHz}$.

Figure 8

Persistence of TTSB for two sample trajectories in TWA. Cavity dynamics for (a) short and (b) long time scales. (c) Fourier spectrum as in Fig. 7. The parameters are the same as Fig. 7.

Figure 9

Persistence of TTSB quantified by the relative shift in the frequency response in the presence of perturbation $\delta /f_0$ in the second half of the driving. The parameters are the same as Fig. 7.

Figure 10

Dynamics of the five largest eigenvalues (from light to dark gray) of the SPDM for the same parameters as in Fig. 7. Only two eigenmodes of the SPDM are highly occupied in the TC phase.

Figure 11

(a) MF and (b) TWA results for dynamics of the cavity mode in the presence of collisional atom-atom interaction quantified by the interaction energy $E_{\mathrm{int}}$ (from black to yellow [light gray] for increasing $E_{\mathrm{int}}$). We filter out frequency faster $\pi /T_{\mathrm{B}}$ using a Gaussian filter. The shaking amplitude is fixed to $f_0=0.03$, while the pump intensity and the driving frequency are adjusted to match the initial ${\left|\alpha \right|}^2$ and $T_B$ for different $E_{\mathrm{int}}$.

Figure 12

Comparison of the cavity mode dynamics in the TC phase between the MF results for the thermodynamic limit $N_a\to \infty$ and the TWA results for finite $N_a$ (from yellow [light gray] to black for increasing $N_a$). Similar to Fig. 11, a Gaussian filter is also applied here to remove fast oscillations. The parameters are the same as Fig. 7 except for ${\mathrm{\Delta }}_0$, which is adjusted such that $N_a{\mathrm{\Delta }}_0=-2\pi \times 21.6\mathrm{kHz}$ for varying $N_a$.