Photon-photon correlation of condensed light in a microcavity

The study of temporal coherence in a Bose-Einstein condensate of photons can be challenging, especially in the presence of correlations between the photonic modes. In this work, we use a microscopic, multimode model of photonic condensation inside a dye-filled microcavity and the quantum regression theorem to derive an analytical expression for the equation of motion of the photon-photon correlation function. This allows us to derive the coherence time of the photonic modes and identify a nonmonotonic dependence of the temporal coherence of the condensed light with the cutoff frequency of the microcavity.

Figure 1

Illustration of a photon BEC setup. (a) A microcavity, formed using a pair of curved and flat mirrors, filled with dye molecules (two-level systems) in a solvent (gray cloud). The figure illustrates different processes such as emission (${\mathcal{E}}_p$), absorption (${\mathcal{A}}_q$), photon loss ($\kappa$), and loss of molecular excitation (${\mathrm{\Gamma }}_{\downarrow }$). The cavity plane can be divided into a grid, with each box indexed as ${\mathbf{r}}_i$ (shaded region), and focused pumping (${\mathrm{\Gamma }}_{\uparrow }^j$) at the center (red region). (b) A one-dimensional portrayal of the BEC setup showing the grids and the probability density $|{\mathrm{\Psi }}_p{|}^2$ for the first six cavity modes. The unexcited (black) and excited (red) are uniformly distributed in the plane (three per grid). The excited molecules are located closer to the pump spot.

Figure 2

The emission spectrum of the first few photonic modes. The figure shows the spectral function $S_p\left(\omega \right)$, normalized with the initial steady-state population $\langle {\widehat{a}}_p^{†}\left(t\right){\widehat{a}}_p\left(t\right)\rangle$ at $t=0$), for the ground state $p=0$ (solid blue line), the first excited $p=1$ (dashed black line), and the second excited $p=2$ (red crossed line) modes. The pump is focused at the center, with a width equal to thrice the oscillator length $l_0$. The cavity cutoff frequency ${\omega }_0\approx 520\text{THz}$, with spacing between two adjacent cavity modes, $\mathrm{\Delta }\omega =1.7\text{THz}$. The rate of loss of cavity photons is equal to $\kappa \approx 0.2\text{THz}$ and the absorption (${\mathcal{A}}_m$) and emission (${\mathcal{E}}_m$) rates are calculated from experimental data [32]. The loss of excitations outside the cavity modes is ${\mathrm{\Gamma }}_{\downarrow }\approx 3\times {10}^{-5}\text{THz}$ and pumping rate ${\mathrm{\Gamma }}_{\uparrow }=0.2\times {\mathrm{\Gamma }}_{\downarrow }$.

Figure 3

Variation of temporal coherence with cavity cutoff frequency. The plots show the coherence time of the lowest energy or ground state mode for different cutoff frequencies of the cavity, and for three different pumping rates ${\mathrm{\Gamma }}_{\uparrow }$, in units of ${\mathrm{\Gamma }}_{\downarrow }$. All other system parameters are the same as in Fig. 2.

Figure 4

Visualization of the term ${\sum }_{i,\forall s_i=0}{\mathrm{\Gamma }}_{\uparrow }^i$ as a square, where each block ${\mathbf{r}}_i$ denotes a position, and black (white) implying the molecule at the position is excited (nonexcited). For a specific $\underline{s}$ there are associated squares with one more excitation, give by a term ${\mathrm{\Gamma }}_{\uparrow }^j$ which creates excitation at a new position ${\mathbf{r}}_j$, with new basis $|\underline{s}+{\underline{s}}_j\rangle \langle \underline{s}+{\underline{s}}_j|$.

Figure 5

Visualization of $|{\underline{s}}^{\prime }\rangle \langle {\underline{s}}^{\prime }|$ as a square, with different contributing terms coming from a set of squares, with one less excitation at position ${\mathbf{r}}_j$ compared to ${\underline{s}}^{\prime }$. Thus, each term contributes ${\mathrm{\Gamma }}_{\uparrow }^j{P}_{\underline{n},{\underline{s}}^{\prime }-{\underline{s}}_j}^{{\underline{k}}_p}$.