Probing few-particle Laughlin states of photons via correlation measurements

We propose methods to create and observe Laughlin-like states of photons in a strongly nonlinear optical cavity. Such states of strongly interacting photons can be prepared by pumping the cavity with a Laguerre-Gauss beam, which has a well-defined orbital angular momentum per photon. The Laughlin-like states appear as sharp resonances in the particle-number-resolved transmission spectrum. Power spectrum and second-order correlation function measurements yield unambiguous signatures of these few-particle strongly correlated states.

Figure 1

A sketch of the proposed experimental setup. Photons are harmonically trapped inside an optically nonlinear slab with the help of a pair of curved mirrors. A Laguerre-Gauss ${\mathrm{LG}}_0^m$ pump beam is used to inject photons each having an orbital angular momentum of $m\hslash$. Photons emitted from the cavity are collected for measurement.

Figure 2

(a) Ratio $P_2/P_1$ of having two particles to one particle in the steady state as a function of the detuning of pump frequency $\Delta {\omega }_p$ in the presence of a Laguerre-Gauss ${\mathrm{LG}}_0^1$ pump. (b) The overlap $\mathcal{O}$ between the two-photon amplitude and two-particle eigenfunctions as a function of $\Delta {\omega }_p$ (solid blue line for the Laughlin state, dashed red line for the c.m. state). Vertical dashed lines correspond to half the two-particle eigenfrequencies of the isolated system Hamiltonian ${\mathcal{H}}_0$. System and pump parameters: $g_{\text{nl}}/{\ell }^2\omega =0.1$, $\gamma /\omega =0.002$, and $\ell F/\gamma =0.1$.

Figure 3

(a) Ratio $P_3/P_2$ of having three particles to two particles in the steady state as a function of the detuning of pump frequency $\Delta {\omega }_p$ in the presence of a Laguerre-Gauss ${\mathrm{LG}}_0^2$ pump. (b) The overlap $\mathcal{O}$ between the three-photon amplitude and three-particle eigenfunctions as a function of $\Delta {\omega }_p$ (solid blue line for the Laughlin state, dashed red line for the c.m. state). Vertical dashed lines correspond to one third of the three-particle eigenfrequencies of the isolated system Hamiltonian ${\mathcal{H}}_0$. System and pump parameters: $g_{\text{nl}}/{\ell }^2\omega =0.1$, $\gamma /\omega =0.002$, and $\ell F/\gamma =0.1$.

Figure 4

Power spectrum $S(\mathbf{r},{\omega }_s)$ in arbitrary units as a function of frequency ${\omega }_s$ measured with respect to the cavity frequency ${\omega }_c$. (a) and (b) The system is pumped by a Laguerre-Gauss ${\mathrm{LG}}_0^1$ beam at $\Delta {\omega }_p/\omega =2$; $\left|\mathbf{r}\right|=0.05\ell$. (b) A closeup of the original spectrum (solid blue line) together with the spectrum shifted by $-1$ (dashed red line) along the frequency axis. (c) and (d) The system is pumped by a Laguerre-Gauss ${\mathrm{LG}}_0^2$ beam at $\Delta {\omega }_p/\omega =3$; $\left|\mathbf{r}\right|=0.15\ell$. (d) A closeup of the original spectrum (solid blue line) together with the ones shifted by $-1$ (dashed red line) and $-2$ (dash-dotted green line) along the frequency axis. System and pump parameters: $g_{\text{nl}}/{\ell }^2\omega =0.1$, $\gamma /\omega =0.002$, and $\ell F/\gamma =0.1$.

Figure 5

Equal-time second-order correlation function $g^{\left(2\right)}(r_{\circ }=0.5\ell ,\varphi )$ in the steady state for different driving frequencies of the ${\mathrm{LG}}_0^1$ pump. (a) $\Delta {\omega }_p/\omega =2$ (solid blue line), $\Delta {\omega }_p/\omega \approx 2.004$ (dashed red line). (b) $\Delta {\omega }_p/\omega \approx 2.008$. System and pump parameters: $g_{\text{nl}}/{\ell }^2\omega =0.1$, $\gamma /\omega =0.002$, and $\ell F/\gamma =0.1$.

Figure 6

Sketch of the experiment proposed to distinguish between the two-particle Laughlin and c.m. states by measuring the two-particle detection probability $P_2$ as a function of $\beta$.

Figure 7

Absolute value of the power spectrum $\left|S\right(\mathbf{r},{\omega }_s\left)\right|$ in arbitrary units as a function of frequency ${\omega }_s$ measured with respect to the cavity frequency ${\omega }_c$; taking the absolute value smooths out the effect of the finite time window used in the numerical calculation on the $\delta$-shaped pump peak. The inset is a closeup around ${\omega }_s/\omega =1$ showing the double-peak structure. The system is pumped by a Laguerre-Gauss ${\mathrm{LG}}_0^1$ beam at $\Delta {\omega }_p/\omega \approx 2.008$; $\left|\mathbf{r}\right|=0.05\ell$. System and pump parameters: $g_{nl}/{\ell }^2\omega =0.1$, $\gamma /\omega =0.002$, and $\ell F/\gamma =0.1$.

Figure 8

Solid (dashed) line shows $g^{\left(2\right)}(r_{\circ }=0.5\ell ,\varphi )$ for the normalized two-particle Laughlin (c.m.) wave function.

Figure 9

(a) $g^{\left(2\right)}(r_{\circ }=0.5\ell ,\varphi )$ for seven different frequencies of the ${\mathrm{LG}}_0^2$ pump corresponding to the eigenfrequencies of the isolated system. Frequency index increases with increasing energy eigenvalue. (b) $g^{\left(2\right)}(r_{\circ }=0.5\ell ,\varphi )$ for $\Delta {\omega }_p/\omega =3$ (solid blue line) corresponding to the Laughlin state and $\Delta {\omega }_p/\omega \approx 3.00159$ (dashed red line) corresponding to the c.m. state, plotted again for clarity. System and pump parameters: $g_{nl}/{\ell }^2\omega =0.1$, $\gamma /\omega =0.002$, and $\ell F/\gamma =0.1$.