Spontaneous microcavity-polariton coherence across the parametric threshold: Quantum Monte Carlo studies

We investigate the appearance of spontaneous coherence in the parametric emission from planar semiconductor microcavities in the strong coupling regime. Calculations are performed by means of a quantum Monte Carlo technique based on the Wigner representation of the coupled exciton and cavity-photon fields. The numerical results are interpreted in terms of a nonequilibrium phase transition occurring at the parametric oscillation threshold: below the threshold, the signal emission is incoherent, and both the first and the second-order coherence functions have a finite correlation length which becomes macroscopic as the threshold is approached. Above the threshold, the emission is instead phase coherent over the whole two-dimensional sample and intensity fluctuations are suppressed. Similar calculations for quasi-one-dimensional microcavities show that in this case the phase coherence of the signal emission has a finite extension even above the threshold, while intensity fluctuations are suppressed.

Figure 1

Real (left) and imaginary (right) parts of the Bogoliubov dispersion for different values of the pump frequency $\hslash {\delta }_p=\hslash [{\omega }_p-{\omega }_{LP}\left({\mathbf{k}}_p\right)]=-0.65\mathrm{meV}$ (a), (b), $-0.54\mathrm{meV}$ (c), (d), $-0.49\mathrm{meV}$ (e), (f). At the mean-field threshold $\hslash {\delta }_p=\hslash {\delta }_p^{\left(c\right)}\approx -0.52\mathrm{meV}$, the mean-field energy $\hslash g{\mid {\psi }_X\mid }^2\simeq 0.13\mathrm{meV}$. For the sake of clarity, only the branches corresponding to the LP have been traced, the UP ones being far away in energy. Cavity parameters: $\hslash {\omega }_C^0=\hslash {\omega }_X=1.4\mathrm{eV},2\hslash {\Omega }_R=5\mathrm{meV},\hslash {\gamma }_X=0.15\mathrm{meV},\hslash {\gamma }_C=0.24\mathrm{meV}$. The lower polariton energy at linear regime is $\hslash {\omega }_{LP}\left({\mathbf{k}}_p\right)=1.39935\mathrm{eV}$.

Figure 2

Far-field emission pattern for increasing values of the pump frequency $\hslash {\delta }_p=-0.55\mathrm{meV}$ (a), $-0.53\mathrm{meV}$ (b) below the mean-field threshold, and $-0.52\mathrm{meV}$ (c) close to the mean-field threshold. The pump mode at ${\mathbf{k}}_p$ by far saturates the grey scale, while the idler emission is hardly visible on this scale. The solid rectangles indicate the $\mathbf{k}$-space region which forms the signal emission. Throughout the whole paper, the correlation functions of the selected signal emission are denoted with a bar (e.g., ${\overline{G}}^{\left(2\right)}$ and ${\overline{g}}^{(1,2)}$). (d) Cut for $k_y=0$ of the far-field emission pattern shown in panels (a)–(c). For clarity reasons, only the part of the spectrum corresponding to the signal emission is shown. Same cavity and pump parameters as in Fig. 1, physical size of the system $L_x=L_y=25\mu \mathrm{m}$.

Figure 3

First-order coherence function ${\overline{g}}^{\left(1\right)}\left(\mathbf{x}\right)$ of the selected (see solid rectangle in Fig. 2) signal emission for values of the pump frequency $\hslash {\delta }_p=-0.55\mathrm{meV}$ (dotted), $\hslash {\delta }_p=-0.53\mathrm{meV}$ (dashed), and $\hslash {\delta }_p=-0.52\mathrm{meV}$ (solid), respectively, below and around the mean-field threshold. The cavity and the other pump parameters are the same as in Fig. 1.

Figure 4

Second-order coherence function ${\overline{g}}^{\left(2\right)}\left(\mathbf{x}\right)$ of the selected (see solid rectangle in Fig. 2) signal emission for values of the pump frequency $\hslash {\delta }_p=-0.53\mathrm{meV}$ (dashed) and $-0.52\mathrm{meV}$ (solid), respectively, below and around the mean-field threshold. The cavity and the other pump parameters are the same as in Fig. 1.

Figure 5

(a) Far-field emission pattern for a pump frequency $\hslash {\delta }_p=-0.5\mathrm{meV}$ (above the mean-field threshold). The solid (dashed) rectangles indicate the $\mathbf{k}$-space region which forms the signal (idler) emission. Throughout the whole paper, the correlation functions of the selected signal emission are denoted with a bar (e.g., ${\overline{G}}^{\left(2\right)}$ and ${\overline{g}}^{(1,2)}$). (b) Cut for $k_y=0$ of the far-field emission pattern shown in panel (a). The cavity and the other pump parameters are the same as in Fig. 1.

Figure 6

Intensity in the pump (a) and signal (b) modes as a function of the pump frequency ${\omega }_p$. Solid lines: Wigner–Monte Carlo results. Dotted lines: mean-field prediction. The dashed line in panel (a) gives the intensity in the pump mode for the unstable homogeneous mean-field state of Eqs. (13, 14). The cavity and the other pump parameters are the same as in Fig. 1.

Figure 7

First- (dashed) and second-order (solid) coherence functions ${\overline{g}}^{(1,2)}\left(\mathbf{x}\right)$ of the selected (see solid rectangle in Fig. 2) signal emission for a pump frequency $\hslash {\delta }_p^{\left(c\right)}=-0.5\mathrm{meV}$ (above the mean-field threshold). The cavity and the other pump parameters are the same as in Fig. 1.

Figure 8

Intensity correlation function $G^{\left(2\right)}\left(\mathbf{x}\right)$ of the in-cavity polariton field for values of the pump frequency $\hslash {\delta }_p=-0.55\mathrm{meV}$ (upper panel), $\hslash {\delta }_p=-0.53\mathrm{meV}$ (central panel), and $\hslash {\delta }_p=-0.52\mathrm{meV}$ (lower panel), respectively, below and around the mean-field threshold. The cavity and the other pump parameters are the same as in Fig. 1.

Figure 9

Intensity correlation function $G^{\left(2\right)}\left(\mathbf{x}\right)$ of the in-cavity polariton field for a pump frequency $\hslash {\delta }_p=-0.5\mathrm{meV}$ above the threshold. The cavity and the other pump parameters are the same as in Fig. 1.

Figure 10

Intensity profile ${{\psi }_C\left(\mathbf{x}\right)\mid }^2\mid$ of a single realization for pump frequencies $\hslash {\delta }_p=-0.53\mathrm{meV}$ (upper panel) and $\hslash {\delta }_p=-0.5\mathrm{meV}$ (lower panel), respectively, below and above the mean-field threshold. The cavity and the other pump parameters are the same as in Fig. 1, exception made for the coupling constant $g$ and the driving intensity $I_p$, respectively, increased and decreased by a factor 5 so as to clarify the picture without affecting the mean-field physics.

Figure 11

First- (upper panel) and second-order (lower panel) coherence functions ${\overline{g}}^{(1,2)}\left(\mathbf{x}\right)$ of the selected signal emission for pump frequencies, respectively, $\hslash {\delta }_p=-0.53\mathrm{meV}$ below the mean-field threshold (solid) and $\hslash {\delta }_p=-0.51\mathrm{meV}$ above the mean-field threshold (dotted). The cavity and the other pump parameters are the same as in Fig. 1, exception made for the geometry: the system is here one-dimensional and $L_x=400\mu \mathrm{m}$.