Absence of long-range coherence in the parametric emission of photonic wires

We analytically investigate the spatial coherence properties of the signal emission from one-dimensional optical parametric oscillators. Because of the reduced dimensionality, quantum fluctuations are able to destroy the long-range phase coherence even far above threshold. The spatial decay of coherence is exponential and, for realistic parameters of semiconductor photonic wires in the strong exciton-photon coupling regime, it is predicted to occur on an experimentally accessible length scale.

Figure 1

(Color online) (a) Sketch of a semiconductor photonic wire with an embedded quantum well (QW). Confinement in the $z,y$ directions is provided by, respectively, a pair of distributed Bragg reflectors (DBR) and a lateral etching of the cavity. (b) Linear regime dispersions of signal, pump and idler subbands and sketch of the parametric process under examination. (c) Exciton density in the pump and signal modes as a function of the pump laser intensity for a given $\hslash {\omega }_p=1.398725\mathrm{eV}$. Full (dashed) lines represent linearly stable (unstable) solutions. Cavity parameters inspired by Ref. 18: $\hslash {\omega }_C^o=\hslash {\omega }_X^o=1.4\mathrm{eV}$, $k_z=20\mu {\mathrm{m}}^{-1}$, $\hslash {\Omega }_R=2.5\mathrm{meV}$, $L_y=4\mu \mathrm{m}$, $\hslash g=5\times {10}^{-6}\mathrm{eV}\mu {\mathrm{m}}^2$ and $\hslash {\gamma }_{s,p,i}=0.15\mathrm{meV}$.

Figure 2

Imaginary (a) and real (b) parts of the Bogoliubov excitation spectrum around the mean field steady state. Solid line indicates the Goldstone branch. Pump intensity as at the dotted line in Fig. 1b. Same cavity parameters as in Fig. 1.

Figure 3

Analytical prediction (22) (solid) and QMC data (circles) for the coherence length as a function of the pump power. Same cavity parameters as in Fig. 1.