Spontaneous collective coherence in driven dissipative cavity arrays

We study an array of dissipative tunnel-coupled cavities, each interacting with an incoherently pumped two-level emitter. For cavities in the lasing regime, we find correlations between the light fields of distant cavities, despite the dissipation and the incoherent nature of the pumping mechanism. These correlations decay faster than any power of the distance for arrays in any dimension but become increasingly long ranged with increasing photon tunneling between adjacent cavities. The interaction-dominated and the tunneling-dominated regimes show markedly different scaling of the correlation length which always remains finite due to the finite photon trapping time. We propose a series of observables to characterize the spontaneous buildup of collective coherence in the system.

Figure 1

(a) The building block of the array, the one-emitter laser, and its main cavity emission properties: (i) cavity population $n_a$ as a function of $P_{\sigma }$ for ${\gamma }_a=0.1g$ and ${\omega }_{\sigma }={\omega }_a$, with the lasing region highlighted in yellow. Below, contour plots of $g^{\left(2\right)}$ as a function of $P_{\sigma }$ and (ii) ${\gamma }_a$ at ${\omega }_{\sigma }={\omega }_a$, or (iii) $\Delta ={\omega }_{\sigma }-{\omega }_a$ at ${\gamma }_a=0.1g$, with $g^{\left(2\right)}>1$ in red, $g^{\left(2\right)}=1$ in white, and $g^{\left(2\right)}<1$ in blue. Also ${\gamma }_{\sigma }=0.01g$ and $J=0$. (b) Scheme of the total system in one dimension: a circular array of $N$ coupled cavities containing single emitters.

Figure 2

(a) Cavity population $n_a$ for ${\omega }_{\sigma }={\omega }_a$ as a function of pump $P_{\sigma }$ for $J=0.5g$ (solid blue) and $J=10g$ (dashed black), with $N=12$, ${\gamma }_a=0.1g$, ${\gamma }_{\sigma }=0.01g$. (b) Corresponding first-order correlations $\mathcal{C}\left(x\right)$ as a function of distance $x$ and emitter frequency ${\omega }_{\sigma }$ at pump rates (1) and (2) in plot (a). Bloch mode resonances are plotted as vertical dashed red lines. (c) Inverse correlation lengths $\lambda$ as obtained from fits (see main text) for $N=108$, $P_{\sigma }=5g$, and $\Delta =0$ (solid), $\Delta =J$ (dotted), or $\Delta =2J$ (dashed).

Figure 3

(a)–(i) Populations of the different modes involved, when sweeping the emitter frequency ${\omega }_{\sigma }$ through the system resonances (vertical red dashed lines): $n_a$ in solid and filled blue, $n_{\sigma }$ in solid pink, the Bloch modes $n_k$ with thin lines and $n_a$ for the case $N=1$ in dashed blue as a reference. (j) Emitter spectrum of emission for $N=1$ and varying ${\omega }_{\sigma }$, showing a Mollow triplet around resonance. In inset, the line shape at resonance. In (k) and (l), the spectra for cases (e) and (f), respectively. We use a temperature color code which goes from blue (0) to red (maximum values). Parameters are $N=4$, 12, 32 and $J=0.5g$, $10g$, $50g$, varying as indicated. Also, $P_{\sigma }=5g$, ${\gamma }_a=0.1g$, ${\gamma }_{\sigma }=0.01g$, ${\Gamma }_d=0.3g$.

Figure 4

Examples for fits of functions $f\left(x\right)=[c_{1}cos\left(\nu x\right)+c_{2}sin\left(\nu x\right)]exp(-\lambda x)$ to the normalized correlations $\mathcal{C}\left(x\right)$ for $N=108$ and the parameters $\Delta$ and $J$ given in the labels of the columns and rows. Other parameters are ${\gamma }_a=0.1g$, ${\gamma }_{\sigma }=0.01g$, $P_{\sigma }=5g$.