Observation of a Space-Time Crystal in a Superfluid Quantum Gas

Time crystals are a phase of matter, for which the discrete time symmetry of the driving Hamiltonian is spontaneously broken. The breaking of discrete time symmetry has been observed in several experiments in driven spin systems. Here, we show the observation of a space-time crystal using ultracold atoms, where the periodic structure in both space and time is directly visible in the experimental images. The underlying physics in our superfluid can be described ab initio and allows for a clear identification of the mechanism that causes the spontaneous symmetry breaking. Our results pave the way for the usage of space-time crystals for the discovery of novel nonequilibrium phases of matter.

Figure 1

Schematic representation of the experimental setup, the timeline, and the imaging sequence. (a) Schematic view of the imaging system and atomic cloud. (b) Experimental sequence and schematic representation of the driving. (c) Imaging sequence showing the first six and the last image of a selected run. In total, 50 images are taken each run with 3.28 ms between images.

Figure 2

Line density $n_{\ell }$ as a function of the time and position starting 500 ms after the onset of the drive. Time slices are taken from a single experimental run. In both space and time, a recurring pattern is observed. The (temporal) period of the pattern corresponds to twice the breathing period $T_D$. The diagonal streaks in the image are caused by a correction for the uncoupled axial center-of-mass motion and darker areas in the imaging. The decrease of the signal is attributed to a slight particle loss (3% per shot) due to interactions of imaging light with the atoms.

Figure 3

Fourier analysis and comparison of the experiment with the simulation. (a) Line density at the center of the cloud before the onset of the space-time crystalline phase, directly after the excitation. (b) Fourier transform of the data in (a). Peaks at $f/f_D=\pm 1$ are associated with a weakly excited scissor mode. The signal around the origin is associated with the equilibrium profile of the condensate. (c) Take-out of Fig. 2. Line density at the center of the cloud after the transition to the space-time crystalline phase, after a driving time of 500 ms. A lattice has formed. (d) Fourier transform of the data in (c), with the appearance of four additional peaks due to the space-time crystal at $(k/k_c,f/f_D)=(\pm 1,\pm 0.5)$, where $k_c$ is the center wavelength [25]. (e) Simulated line density for a modulation depth of 0.02, after a wait time of $25T_D$. (f) Fourier transform of the data in (e). Notice that in the simulation only the equilibrium profile is visible. (g) Simulated line density for a modulation depth of 0.2 after a wait time of $25T_D$. A pattern similar to the experimental data of (c) is observed. (h) Fourier transform of (g). Note the appearance of the four additional peaks at $(k/k_c,f/f_D)=(\pm 1,\pm 0.5)$ attributed to the space-time crystal. The line density in (a), (c), (e), and (g) is in units of $10^{11}\mathrm{atoms}/\mathrm{m}$. Fourier images in (b), (d), (f), and (h) are truncated and normalized to 1 for the experimental data.

Figure 4

Long-term behavior of the amplitudes of the drive and crystal fraction. (a) Relative amplitude of the radial breathing mode derived by fitting a two-dimensional profile to the data. (b) Crystal fraction determined from each measurement run. The dashed line indicates the background signal from shot-to-shot noise. Notice that the crystalline phase appears a certain time after the driving mode revives [25].

Figure 5

Minimum required driving amplitude $⟨\delta A_D/{\overline{A}}_D⟩$ as a function of the driving frequency $f_D$ in a linear response analysis [36]. (a) Without damping (solid line) a mode $j$ can be driven with an arbitrary small amplitude, if the driving frequency coincides with twice the mode frequency $f_j$, whereas for small damping (dashed line) there is for any drive frequency a threshold, below which the mode is not excited. (b) Under our experimental conditions (indicated by the red star), the damping is larger and the threshold for exciting modes becomes larger. In the case of linear response, the modes $j=16$, 17, and 18 can be excited, and, depending on the competition between these modes, one of them dominates the pattern [36].