Observing Chiral Superfluid Order by Matter-Wave Interference

The breaking of time-reversal symmetry via the spontaneous formation of chiral order is ubiquitous in nature. Here, we present an unambiguous demonstration of this phenomenon for atoms Bose-Einstein condensed in the second Bloch band of an optical lattice. As a key tool, we use a matter-wave interference technique, which lets us directly observe the phase properties of the superfluid order parameter and allows us to reconstruct the spatial geometry of certain low-energy excitations, associated with the formation of domains of different chirality. Our work marks a new era of optical lattices where orbital degrees of freedom play an essential role for the formation of exotic quantum matter, similarly as in electronic systems.

Figure 1

(a) Bipartite lattice geometry. The unit cell with two potential minima denoted by $A$ and $B$ is highlighted. $\lambda =1064\mathrm{nm}$. (b) The second band is plotted across the second Brillouin zone. Two degenerate band minima in the points $X_{\pm }$ arise. (c),(d) The spatial wave functions associated with the Bloch states $|{\psi }_{\pm }⟩$ associated with $X_{\pm }$ are composed of local $s$ and $p$ orbitals with the local phases $\pm 1$ (indicated by red and green) arranged in order to maximize tunneling.

Figure 2

(a) The tubular minima of the lattice potential are split into upper (red) and lower (blue) sections with a cylindrically focused laser beam (light blue). Condensates are simultaneously prepared in the $X_{\pm }$ points of the second band in the two regions of the lattice spatially separated by the light sheet. (b) Band mapping image, showing equally populated condensates at the minima of the second band $X_{\pm }$. (c) Sketch of atomic spatial distribution after a ballistic expansion of 25 ms. The gray area corresponds to the second Brillouin zone. Each of the four Bragg maxima 1–4, corresponding to the $X_{\pm }$ points in Fig. 1, carries an interference grating aligned with the $z$ axis. (d) Absorption image along the line of sight indicated by the vector $\mathbf{r}$ in (c), which lies in the $xy$ plane and encloses an angle of $\alpha =13°$ with the $y$ axis.

Figure 3

(a) The left (I) and right (II) panels show examples of the two classes of observed interference patterns. The upper row shows the original data, which are repeated in the lower row after band-pass filtering and normalization. A data set comprises four density gratings $n_i(\tilde{z})$ with $i\in \{1,2,3,4\}$ corresponding to the four Bragg maxima in Figs. 2 and 2 [see the text for a definition of $n_i(\tilde{z})$]. The black dashed boxes highlight the spatial correlations among the four density gratings. (b) Stacked histogram for the density-density cross correlation ${\rho }_{i,j}$ with both indices odd or even. (c) Stacked histogram for the density-density cross correlation ${\rho }_{i,j}$ with mixed indices. (d) Wave function associated with the state $|\pi /2,+⟩$. The colors parametrize the local phase factor ranging from $+1$ (red) via $+i$ (green) and $-1$ (cyan) to $-i$ (purple). (e) A section of the calculated interference patterns is shown, which exhibits the same spatial correlations as found in (a).

Figure 4

(a) Interference pattern of a slightly excited sample: the inner density gratings (2 and 4) show vertical nodal lines indicated by the arrows at the lower edge. (b) Schematic of the experimental scenario, which explains the observations. The atomic sample in the upper lattice region is divided into two equally sized domains of different chirality indicated by the orange and green areas. The sample in the lower lattice region (green area) carries no excitations. (c) Calculation of the interference based on the scenario illustrated in (b). The nodal lines corresponding to the observations are indicated by black arrows.