Pump-probe cavity optomechanics with a rotating atomic superfluid in a ring

Atomic superfluids confined in a ring provide a remarkable paradigm for quantized circulation. Very recently, a technique based on cavity optomechanics has been proposed [Kumar et al., Phys. Rev. Lett. 127, 113601 (2021)] for sensing and manipulating the rotation of a bosonic ring condensate with minimal destruction, in situ and in real time. Here, we theoretically investigate other coherent interference effects that can be supported by the proposed configuration. Specifically, in the presence of a strong control beam, we analyze the influence of atomic rotation on the transmission spectrum of a weak probe laser through a cavity containing a ring condensate. We present a detailed study of the resulting narrow probe transmission profiles and group delay and show that they can be tuned by means of persistent currents. Our results explore a facet of rotating matter waves and are relevant to applications such as atomtronics, sensing, and information processing.

Figure 1

(a) Top: A Fabry-Pérot cavity with partially reflecting mirrors is driven by coherent control beams (red and blue helices on the left) carrying OAM, which interact with a toroidally trapped BEC. A second weak laser (light blue beams) probes the cavity, and the transmission spectrum at the probe frequency is measured at the output. Bottom: Position of the Bragg-scattered side modes on either side of the initial BEC winding number. (b) Illustration of the laser frequencies and the first mechanical sidebands of the control laser.

Figure 2

(a) Bistability plot for $\tilde{g}=0$ (solid blue curve) and $\tilde{g}=12{\tilde{g}}_m$ (dotted black curve). (b) Stable (s) and unstable (u) regions of the system, according to the Routh-Hurwitz criterion mentioned in the text. All plots are obtained at $\tilde{\mathrm{\Delta }}=-{\mathrm{\Omega }}_m$.

Figure 3

(a) Transmission of the probe field for $L_p=0$ obtained at $\tilde{\mathrm{\Delta }}=-{\mathrm{\Omega }}_m$. For this case the critical power is ${\tilde{P}}_{l{c}_{cr}}=28$ fW. (b) Energy levels and transition pathways for the system at $\tilde{\mathrm{\Delta }}=-{\mathrm{\Omega }}_m$.

Figure 4

Transmission $T$ (black, left axis) of the probe field at $\delta ={\mathrm{\Omega }}_m$ and width of the OMIT window ${\mathrm{\Gamma }}_m$ (blue, right axis) as a function of control field power. Here the solid circles represent the numerical values of the transmission and linewidth obtained from Fig. 3. The solid curves represent the analytical solutions from Eqs. (9) and (10). The rest of the parameters are the same as in Fig. 3.

Figure 5

(a) Probe transmission as a function of pump-probe detuning and control field power for $L_p=1$. For this case the critical power is ${\tilde{P}}_{l{c}_{cr}}=33$ fW. (b) Transmission of the probe field as a function of the winding number and pump-probe detuning for $P_{lc}=1$ fW.

Figure 6

Separation between the transparency peaks ($d\delta$) obtained from Fig. 5 as a function of the winding numbers ($L_p$) at control laser detuning $\tilde{\mathrm{\Delta }}=-{\mathrm{\Omega }}_m$ and for different values of OAM. The rest of the parameters are the same as in Fig. 5.

Figure 7

Asymmetric Fano profiles obtained for $L_p=0$ as a function of the pump-probe detuning and cavity detuning. Here, $P_{lc}=1$ fW, and the rest of the parameters are the same as in Fig. 3.

Figure 8

(a) Probe transmission as a function of pump-probe detuning and cavity detuning for $L_p=1$. (b) Transmission of the probe field obtained at $\tilde{\mathrm{\Delta }}=-1.2{\mathrm{\Omega }}_m$ as a function of the winding number and pump-probe detuning. Here, $P_{lc}=1$ fW, and the other parameters are the same as in Fig. 5.

Figure 9

(a) Transmission phase and (b) group delay as a function of pump-probe detuning. Here, $L_p=0$ (dashed blue curve) and $L_p=1$ (solid red curve). The parameters used are the same as in Fig. 5.

Figure 10

Group delay as a function of (a) control power for different $L_p$ and (b) winding number for different power values of the control beam. Here, group delay is obtained at $\delta ={\mathrm{\Omega }}_m$, and the rest of the parameters are the same as in Fig. 9.