Generation and spectroscopic signatures of a fractional quantum Hall liquid of photons in an incoherently pumped optical cavity

We investigate theoretically a driven dissipative model of strongly interacting photons in a nonlinear optical cavity in the presence of a synthetic magnetic field. We show the possibility of using a frequency-dependent incoherent pump to create a strongly correlated $\nu =1/2$ bosonic Laughlin state of light: Due to the incompressibility of the Laughlin state, fluctuations in the total particle number and excitation of edge modes can be tamed by imposing a suitable external potential profile for photons. We further propose angular-momentum-selective spectroscopy of the emitted light as a tool to obtain unambiguous signatures of the microscopic physics of the quantum Hall liquid of light.

Figure 1

Density profile $\rho \left(r\right)$ of the Laughlin state with $N=2$ (solid blue curve) and $N=3$ (red dashed curve) particles as a function of the radial coordinate $r$, $\ell$ being the magnetic length. Also plotted by a black line is a cylindrical step potential with radius $R$ and strength $V$ (plotted in arbitrary units). As the three-particle state occupies more space than the two-particle state due to the incompressible nature of the Laughlin state, it experiences the effect of the external potential more and is therefore shifted upward in energy.

Figure 2

Schematic energy levels of the isolated system. (a) In the presence of a magnetic field, noninteracting particles occupy the lowest Landau level. The energy separation between each successive $N$-particle manifold is $\hslash {\omega }_0$. (b) When interactions are included, the ground state for each $N$ is massively degenerate and contains the $\nu =1/2$ bosonic Laughlin state and its zero-energy edge and quasihole excitations (shown in red for $N=2$, purple for $N=3$). This Laughlin manifold is separated from all other excitations by a gap $\sim \mathrm{\Delta }$. (c) Low-energy states in the presence of an external potential with matrix elements $V_{\text{ext}}^{\overline{m}\overline{m}}$. Degeneracies are lifted and for a suitably designed potential profile it is possible to appreciably blueshift all the excitations (shown in orange) of a target $N$-particle Laughlin state and all energies in the $(N+1)$-particle Laughlin manifold (shown in light purple). In both these cases, states above the Laughlin gap $\mathrm{\Delta }$ are marked in black. If the transition frequency ${\omega }_{\text{at}}$ of two-level emitters matches ${\omega }_0$, the target Laughlin state (here $N=2$) can be sizably populated provided the energy shifts ${\delta }_1,{\delta }_2,...$ are larger than the pump linewidth ${\mathrm{\Gamma }}_p$.

Figure 3

Low-lying energy levels $E_N^{\prime }=E_N-N\hslash {\omega }_0$ of the isolated system in the presence of the step potential with $V/\mathrm{\Delta }=50\pi$ and $R/\ell =3.7$, versus total angular momentum $L_z$: (a) $N=3$, (b) $N=2$, and (c) $N=1$. (d) Scaled Fourier amplitudes ${\mathcal{F}}_{i\to f}^{\left(m\right)}$ of transitions as a function of the difference between the lost-photon frequency ${\omega }_{if}={\omega }_i-{\omega }_f$ and the reference frequency ${\omega }_0$. Different colors designate the angular momentum $m\hslash$ of the photon lost from the cavity. Vertical dashed lines show the position of allowed transitions from the $N=2$ Laughlin state [inside the red circle in (b)] to the three states in the $N=1$ manifold of (c), at $\hslash ({\omega }_{if}-{\omega }_0)/\mathrm{\Delta }\sim -0.007,0.01,0.012$ with $m=0,1,2$, respectively. The loss and incoherent-pumping parameters are $\hslash {\mathrm{\Gamma }}_l/\mathrm{\Delta }=\pi \times {10}^{-5}$, $\hslash {\mathrm{\Gamma }}_e/\mathrm{\Delta }=5\pi \times {10}^{-4}$, $\hslash {\mathrm{\Gamma }}_p/\mathrm{\Delta }=5\pi \times {10}^{-3}$, and ${\omega }_{at}={\omega }_0$.

Figure 4

Low-lying energy levels $E_N^{\prime }=E_N-N\hslash {\omega }_0$ of the isolated system in the presence of the step potential with $V/\mathrm{\Delta }=50\pi$ and $R/\ell =4.15$, versus total angular momentum $L_z$: (a) $N=4$, (b) $N=3$, and (c) $N=2$. (d) Scaled Fourier amplitudes ${\mathcal{F}}_{i\to f}^{\left(m\right)}$ of transitions as a function of the difference between the lost-photon frequency ${\omega }_{if}={\omega }_i-{\omega }_f$ and the reference frequency ${\omega }_0$. Different colors designate the angular momentum $m\hslash$ of the photon lost from the cavity. Vertical dashed lines show the position of allowed transitions from the $N=3$ Laughlin state [inside the red circle in (b)] to the six lowest-energy states in the $N=2$ manifold of (c), at $\hslash ({\omega }_{if}-{\omega }_0)/\mathrm{\Delta }\sim -0.008,-0.005,-0.003$, 0.009, 0.01, 0.013 with $m=2$, 1, 0, 3, 2, 4, respectively. Also shown inside a red rectangle are the two states degenerate with the $N=2$ Laughlin state in the absence of an external potential. Transitions to these two states with $m=2$ are further indicated by arrows in (d). The loss and pump parameters are the same as in Fig. 3.

Figure 5

Laughlin state population $P_{\mathrm{FQH}}^{\left(N\right)}$ as a function of the strength $V$ and the radius $R$ of the cylindrically symmetric step potential $V_{\mathrm{ext}}\left(r\right)=V\mathrm{\Theta }(r-R)$ for different target particle numbers (a) $N=2$ and (b) $N=3$. Red dots correspond to the $V$ and $R$ values used for subsequent results. The loss and pump parameters are the same as in Fig. 3.

Figure 6

Total population $P^{\left(N\right)}$ of the different $N$-particle states as a function of the detuning ${\omega }_{at}-{\omega }_0$ for step-potential parameters $V/\mathrm{\Delta }=50\pi$ and (a) $R/\ell =3.7$ and (b) $R/\ell =4.15$ chosen to obtain an $N=2$ or 3 target Laughlin state. The loss and pump rates are the same as in Fig. 3.

Figure 7

$N$-particle Laughlin state populations $P_{\mathrm{FQH}}^{\left(N\right)}$ as a function of the detuning ${\omega }_{at}-{\omega }_0$ for step-potential parameters $V/\mathrm{\Delta }=50\pi$ and (a) $R/\ell =3.7$ and (b) $R/\ell =4.15$ chosen to obtain an $N=2$ or 3 target Laughlin state. The loss and pump rates are the same as in Fig. 3.

Figure 8

Population $P_{\mathrm{FQH}}^{\left(3\right)}$ of the $N=3$ Laughlin state for $V/\mathrm{\Delta }=50\pi$, $R/\ell =4.15$, and ${\omega }_{\text{at}}={\omega }_0$ as a function of pump ${\mathrm{\Gamma }}_p$ and emission ${\mathrm{\Gamma }}_e$ rates for a fixed loss rate $\hslash {\mathrm{\Gamma }}_l/\mathrm{\Delta }=\pi \times {10}^{-5}$. Black lines correspond to several fixed ${\mathrm{\Gamma }}_p/{\mathrm{\Gamma }}_e=2$, 3, 4, 5, 10 ratios and the red dot locates the values $\hslash {\mathrm{\Gamma }}_e/\mathrm{\Delta }=5\pi \times {10}^{-4}$ and $\hslash {\mathrm{\Gamma }}_p/\mathrm{\Delta }=5\pi \times {10}^{-3}$ used in the main text.

Figure 9

(a) Matrix elements $V_{\mathrm{ext}}^{mm}$ of the cylindrically symmetric step potential with $V/\mathrm{\Delta }=50\pi$ in the angular momentum basis, for $R/\ell =3.7$ (blue solid line) and $R/\ell =4.15$ (red dashed line). (b) Matrix elements of two external potential profiles given in (c), one yielding $P_{\mathrm{FQH}}^{\left(2\right)}\approx 0.91$ (blue solid line) and the other giving $P_{\mathrm{FQH}}^{\left(3\right)}\approx 0.76$ (red dashed line) for the same loss and pump parameters as in Fig. 3. Note that both potentials keep monotonically increasing for large $r$.

Figure 10

Scaled Fourier amplitudes ${\mathcal{F}}_{i\to f}^{\left(m\right)}$ of transitions as a function of the difference between the lost-photon frequency ${\omega }_{if}={\omega }_i-{\omega }_f$ and the reference frequency ${\omega }_0$ for the potential profiles in Fig. 9 shown by (a) a blue solid line and (b) a red dashed line. Different colors designate the angular momentum $m\hslash$ of the photon lost from the cavity. Vertical dashed lines show the position of allowed transitions at $\hslash ({\omega }_{if}-{\omega }_0)/\mathrm{\Delta }\sim 0.0210$, 0.0212, 0.0215 with $m=2,0,1$, respectively in (a) and at $\hslash ({\omega }_{if}-{\omega }_0)/\mathrm{\Delta }\sim -0.0016$, 0.0046, 0.0081, 0.0083, 0.0130, 0.0142 with $m=2$, 1, 4, 2, 0, 3, respectively in (b). Transitions to two states involving the loss of $2\hslash$ angular momenta are further indicated by arrows in (b). The loss and pump parameters are the same as in Fig. 3.