First order phase transition between two centro-symmetric superradiant crystals

We observe a structural phase transition between two configurations of a superradiant crystal by coupling a Bose-Einstein condensate to an optical cavity and applying imbalanced transverse pump fields. The transition can be interpreted as a transition between two nonpolar, centro-symmetric structures involving a change in polarization. We find that this first order phase transition is accompanied by transient dynamics of the order parameter which we measure in real time. The phase transition and the excitation spectrum can be derived from a microscopic Hamiltonian, in quantitative agreement with our experimental data.

Figure 1

(a) and (c) The free energy $E$ as a function of the phase $\varphi$ of the intracavity light field supports two distinct minima. They correspond to two crystal structures, ${\mathrm{SR}}_1$ and ${\mathrm{SR}}_2$, with different symmetries as sketched in real space in red and green, where dark colors indicate high atomic densities. The black rectangle indicates the primitive cell of the lattice. The colored circles indicate the symmetry centers of the respective structures, and the arrows highlight the shift in position. Dashed gray lines indicate the maxima of the individual standing wave light fields. (b) As the free energy deforms during the experimental sequence, local and global minima swap, and the system undergoes a first order phase transition. The excess energy from the metastable state results in a damped oscillation of the order parameter around the new global minimum. (d) Two imbalanced counterpropagating pump beams ${\mathbf{E}}_{\pm }$ couple the BEC to quadratures $Q$ and $P$ of the intracavity electric field ${\mathbf{E}}_c$. Each quadrature generates a different interference pattern of the electric fields. (e) Using a heterodyne detector, we measure the amplitude $\left|\alpha \right|$ and phase $\varphi$ of the light field leaking from the cavity. From the phase we reconstruct to which quadrature the atoms are coupled and hence which crystal structure they acquire.

Figure 2

(a) and (b) Phase diagram showing the complex order parameter as the expectation value $\langle \widehat{a}\rangle =\left|\alpha \right|e^{i\varphi }$ of intracavity field, measured as a function of the pump lattice depth $V_p$ and the cavity detuning ${\mathrm{\Delta }}_c$. From the amplitude $\left|\alpha \right|$ we extract the intracavity lattice depth $\langle {\widehat{V}}_c\rangle =V_c$, shown in (a). At the phase transition between the two superradiant phases $V_c$ changes abruptly. (b) Phase of the light field mapped to the first quadrant ($\varphi \in [0,\pi /2]$) to highlight the $\pi /2$ phase jump. (c), (d), and (e) Absorption images of the atomic cloud after ballistic expansion, showing the momentum distribution of the atoms. The images are recorded at $V_p=12\left(1\right)E_r$ and ${\mathrm{\Delta }}_c/2\pi =-2,-5,-8$ MHz, as indicated in the phase diagrams. Dark areas show high atomic densities. (c) and (d) correspond to the atomic density distribution of Figs. 1 and 1, respectively. The arrows denote the pump and cavity wave vectors $\hslash {\mathbf{k}}_p$ and $\hslash {\mathbf{k}}_c$.

Figure 3

Phase transition and relaxation of the order parameter. (a) and (b) Cuts of $V_c$ and the $\varphi$ phase diagram [similar to Figs. 2 and 2 but for $\gamma =1.37\left(5\right)$], respectively, for different cavity detunings ${\mathrm{\Delta }}_c/2\pi =-3.75,-2.75,-1.75$ MHz, showing a jump at the phase transition. The transition points $t_{1,2,3}$ correspond, respectively, to $V_p=11\left(1\right)E_r,15\left(1\right)E_r,25\left(2\right)E_r$. (c) The energy landscape [calculated from the parameters of the solid data point in (d)] truncated to the photonic space shows two minima on the different quadrature axes. The system jumps from the local minimum to the global minimum when the phase transition happens. We extract the normal mode frequencies from the curvatures of the energy landscape at the global minimum (see the Supplemental Material), which is plotted as the theoretical prediction in (d). (d) We take the Fourier transform of the time trace of the oscillating phase and extract the dominant frequency. The resulting frequencies are shown as a function of $V_p$ and accordingly changing ${\mathrm{\Delta }}_c$, following the phase boundary between ${\mathrm{SR}}_1$ and ${\mathrm{SR}}_2$, and are compared to the numerical model. The gray shading shows the theoretical prediction including the experimental uncertainties of the imbalance parameter. The Fourier spectrum of the solid data point is shown in the inset. Errors indicate statistical deviation.

Figure 4

Tuning the imbalance parameter $\gamma$. (a) Geometry of the system. We choose the BEC position as the origin of the $y$ axis and define $y_0$ as the coordinate of the beam focus. Moving the lens in the $y$ direction translates the focus and allows tuning of $\gamma$. The mirror has a finite reflectivity, which corresponds to a minor shift in $y$ of the balanced point $\gamma =1$. (b) We extract $\gamma$ from the experimental data by comparing the threshold with numerical calculations. This phase diagram is consistent with an imbalance parameter $\gamma =1.22\left(2\right)$, which gives the calculated phase boundary of ${\mathrm{SR}}_2$ shown as the solid curve. The dashed line is instead the calculated phase boundary for $\gamma =1.28\left(3\right)$ as in Fig. 2. (c) Dependence of the pump lattice depth (blue curve) and imbalance parameter $\gamma$ (black curve) on the beam focus coordinate, calculated from the geometrical model of a retroreflected Gaussian beam. The normalized pump lattice depth ${\overline{V}}_p$ is relative to the maximum measured value of $V_p$. Shaded areas account for the 5% systematic error of the beam waist measurement. The normalized pump lattice depths (blue circles) are measured by performing Raman-Nath diffraction calibration for different $y_0$ values. The imbalance parameters $\gamma$ (black squares) are extracted as in (b). Error bars account for the standard errors of the least squares fits and are smaller than the symbol size for the blue data.