Realization of the Dicke Model Using Cavity-Assisted Raman Transitions

We realize an open version of the Dicke model by coupling two hyperfine ground states using two cavity-assisted Raman transitions. The interaction due to only one of the couplings is described by the Tavis-Cummings model and we observe a normal mode splitting in the transmission around the dispersively shifted cavity. With both couplings present the dynamics are described by the Dicke model and we measure the onset of superradiant scattering into the cavity above a critical coupling strength.

Figure 1

Schematic representation of the experiment. (a) We trap $N$ rubidium 87 atoms, indicated by the filled ellipses, in a equal mixture of hyperfine states inside a high finesse optical cavity. Two laser beams with Rabi frequencies ${\mathrm{\Omega }}_{\mathrm{r}}$ and ${\mathrm{\Omega }}_{\mathrm{s}}$, indicated by solid arrows, are far detuned from the excited state but near resonant with the cavity-assisted Raman transition, indicated by dashed arrows. (b) With only one beam present, we probe the cavity transmission and observe a normal mode splitting. (c) With both beams present, we observe emission into the cavity above the critical coupling.

Figure 2

Normal-mode splitting in the Tavis-Cummings model. For a single coupling beam present, the cavity transmission spectrum shows a normal mode splitting around the dispersively shifted resonance. We vary the atom number for a fixed ${\lambda }_{\mathrm{r}}$ and $\zeta$, record the cavity transmission $⟨n⟩$ normalized by the empty cavity transmission $⟨n_{\text{empty}}⟩$ and average traces corresponding to dispersive shifts in 10 kHz bins. Gray lines indicate the bare energies ${\omega }_0=-2\pi \times 12.02\mathrm{MHz}-\zeta$ and $\omega ={\omega }_{\mathrm{d}}$. Black lines indicate the eigenenergies of the coupled system for a coupling of ${\lambda }_{\mathrm{r}}=2\pi \times 0.17\mathrm{MHz}$ for a dispersive shift of ${\omega }_{\mathrm{d}}=-2\pi \times 0.5\mathrm{MHz}$. Transmission in excess of 1 is due to shot noise exceeding the amplitude of the Lorentzian used to fit the empty cavity transmission.

Figure 3

Single threshold measurement. We ramp the total power in both Raman coupling beams over 1 ms (diagonal red line) and observe the output of the cavity (blue curve). The critical coupling is inferred from when the output of the cavity reaches $7.8\text{counts}/(5\mu \mathrm{s})$, indicated by the dashed line, which corresponds to an intracavity photon number of 10 and is 10 times higher than the background counts.

Figure 4

Superradiant phase transition in the Dicke model. Above a critical coupling, the system undergoes a phase transition into a superradiant state. Black dots show the observed threshold values. The solid gray line shows the theoretical predictions for a calculated differential stark shift of ${\omega }_{\mathrm{dS}}=-2\pi \times 0.157\mathrm{MHz}$ and ${\omega }_1=-2\pi \times 11.94\mathrm{MHz}$ and $\lambda =2\pi \times 0.173\mathrm{MHz}$ at a dispersive shift of ${\omega }_{\mathrm{d}}=-2\pi \times 0.5\mathrm{MHz}$, as obtained from the normal-mode splitting measurement. The blue shaded area shows the upper and lower bounds inferred from the same measurement where the coupling for the lower bound is increased by a factor of 1.2 and the one for the upper decreased by a factor of 0.89 to account for the varying amount of spontaneous emission. The dashed line is derived from the normal-mode splitting measurement assuming ${\omega }_{\mathrm{dS}}=0$ to illustrate the effect of the varying differential stark shift.