Spontaneous Symmetry Breaking in a Polariton and Photon Laser

We report on the simultaneous observation of spontaneous symmetry breaking and long-range spatial coherence both in the strong- and the weak-coupling regime in a semiconductor microcavity. Under pulsed excitation, the formation of a stochastic order parameter is observed in polariton and photon lasing regimes. Single-shot measurements of the Stokes vector of the emission exhibit the buildup of stochastic polarization. Below threshold, the polarization noise does not exceed 10%, while above threshold we observe a total polarization of up to 50% after each excitation pulse, while the polarization averaged over the ensemble of pulses remains nearly zero. In both polariton and photon lasing regimes, the stochastic polarization buildup is accompanied by the buildup of spatial coherence. We find that the Landau criterion of spontaneous symmetry breaking and Penrose-Onsager criterion of long-range order for Bose-Einstein condensation are met in both polariton and photon lasing regimes.

Figure 1

2D histogram of linear and diagonal components of the pseudo-spin-vector for $P<P_{\mathrm{th}}$ (a) and $P=2.4P_{\mathrm{th}}$ (b) on the $x\mathrm{\text{−}}y$ plane of the Stokes sphere. The mean (c),(f), the standard deviation (d),(g), and the total degree of polarization (e),(h) for linearly (left panel) and circularly (right panel) polarized pump. Above threshold (marked by a red line), we observe the buildup of total degree of polarization, while the average polarization remains close to zero for linearly polarized pump.

Figure 2

Dispersion images for (a) below threshold (linear regime), (b) $P=1.5P_{\mathrm{th}}$ (strong-coupling regime), and (c) $P=4P_{\mathrm{th}}$ (weak-coupling regime). The cyan border marks the regions that are filtered for polariton states (b) and bare cavity photon states (c). The red dashed lines represent the exciton and cavity modes, and the solid red line is the lower polariton mode. (d) The total degree of polarization for circularly polarized (dotted lines) and linearly polarized pump (solid lines) for photon (open circles) and polariton energy states (closed circles). (e),(f) Simulation results for the dependence of the total polarization degree on pump intensity for a linearly polarized (e) and circularly polarized (f) pump. The gray circles show the values for each pulse, and the solid curves show the root mean square.

Figure 3

(a) Normalized intensity on one pixel as a function of phase between the arms of the interferometer. The first-order coherence $g^{(1)}(\mathbf{r},\mathbf{-}\mathbf{r})$ is given by the amplitude. Time-resolved first-order spatial coherence, taken across the autocorrelation point at (b) $P=1.1P_{\mathrm{th}}$ and (c) $P=4P_{\mathrm{th}}$. (d) The profile of the first-order coherence as a function of time in polariton and photon laser regimes, at regions marked in red in (b) and (c).