Bose-Einstein condensation of photons with nonlocal nonlinearity in a dye-doped graded-index microcavity

We consider a microcavity made by a graded-index glass, doped with dye molecules, placed within two planar mirrors and study Bose-Einstein condensation of photons. The presence of the mirrors leads to an effective photon mass, and the index grading provides an effective trapping frequency; the photon gas becomes formally equivalent to a two-dimensional Bose gas trapped in an isotropic harmonic potential. The inclusion of nonlinear effects provides an effective interaction between photons. We discuss, in particular, thermal lensing effects and nonlocal nonlinearity, and quantitatively compare our results with the reported experimental data.

Figure 1

Scheme of the graded-index (GRIN) microcavity; a dye-doped GRIN lens is placed between two planar mirrors $(M_1$ and $M_2)$.

Figure 2

Scheme of the GRIN microcavity in the presence of thermal effects: the massively occupied ground-mode field $\left({\mathrm{TEM}}_{00}\right)$ heats up the medium in the vicinity of the optical axis, causing a nonzero temperature gradient between the center of the cavity and the outer space.

Figure 3

Shift of the condensate photon wavelength as a function of the number of photons in the condensate obtained by a numerical solution of the local 2D GPE [solid (red) line]. The shift in the Thomas-Fermi limit [dotted (green) line] obtained from (32), the shift for small nonlinearity obtained by the perturbative approach [dash-dotted (violet) line], and the shift obtained by a variational calculation [dash-dotted (blue) line] are compared to the numerical solution $\left({\tilde{g}}_N=7.5\times {10}^{-4}\right)$.

Figure 4

Condensate photon wavelength shift as a function of the number of photons in the condensate obtained by the numerical simulation of the nonlocal 2D GPE [solid (red) line] compared to the wavelength shift obtained by perturbation theory [dashed (green) line] and to the shift obtained by the variational approach [dotted (blue) line; see the Appendix]. This figure was obtained with ${\tilde{g}}_N=7.5\times {10}^{-4}$ and $\sigma =0.23$.

Figure 5

Experimental points (courtesy of Jan Klaers and Martin Weitz) of the condensate diameter (FWHM) as a function of the number of photons in the ground mode; data are fitted with the numerical solution of the local 2D GPE [dotted (blue) line] with ${\tilde{g}}_N=7.5\times {10}^{-4}$ and with the numerical solution of the nonlocal 2D GPE with finite nonlocality [solid (red) line] given by Eq. (69) by considering the kernel as in Eq. (70) and with ${\tilde{g}}_N=(7.5\pm 0.1)\times {10}^{-4}$ and $\sigma =0.23\pm 0.02$. A quantitative analysis of the two fits shows that the nonlocal solution provides the best fit. Two nonlocal curves obtained by solving the nonlocal 2D GPE with $\sigma =0.80$ [dashed (green) line] and with $\sigma =1.2$ [dash-dotted (gray) line] are shown as predictions.

Figure 6

Radial temperature profile (in kelvins), as a function of the radial coordinate given in units of $a_{\mathrm{os}}=\sqrt{\hslash /m\Omega }\simeq 7.8\mu \mathrm{m}$ with the given parameters, obtained by the numerical solution of the nonlocal 2D GPE using (71). In the simulation the kernel was normalized to ${\tilde{g}}_N$ (with the fit parameters estimated after the best fit in Fig. 5) and the proportionality constant was taken to be $C=n_0^3{|\partial n_0/\partial T|}^{-1}\hslash \Omega /m{c}^2\simeq 0.22$ K since $\Omega =2.6\times {10}^{11}$ rad/s, $m=6.7\times {10}^{-36}$ kg, and $n_0=1.33,\partial n_0/\partial T=-4.86\times {10}^{-4}$ ${\mathrm{K}}^{-1}$ (methanol [40]) were used in the simulation. The temperature profile is given for four values of the number of photons, corresponding to the occupancies at which the experimental data fitted in Fig. 5 were taken.