Cavity optomechanical detection of persistent currents and solitons in a bosonic ring condensate

We present numerical simulations of the cavity optomechanical detection of persistent currents and bright solitons in an atomic Bose-Einstein condensate confined in a ring trap. This paper describes a technique that measures condensate rotation in situ, in real time, and with minimal destruction, in contrast to currently used methods, all of which destroy the condensate completely. For weakly repulsive interatomic interactions, the analysis of persistent currents extends our previous few-mode treatment of the condensate [P. Kumar et al. Phys. Rev. Lett. 127, 113601 (2021)] to a stochastic Gross-Pitaevskii simulation. For weakly attractive atomic interactions, we present the first analysis of optomechanical detection of matter-wave soliton motion. We provide optical cavity transmission spectra containing signatures of the condensate rotation, sensitivity as a function of the system response frequency, and atomic density profiles quantifying the effect of the measurement backaction on the condensate. We treat the atoms at a mean-field level and the optical field classically, account for damping and noise in both degrees of freedom, and investigate the linear as well as nonlinear response of the configuration. Our results are consequential for the characterization of rotating matter waves in studies of atomtronics, superfluid hydrodynamics, and matter-wave soliton interferometry.

Figure 1

A schematic setup for the BEC with winding number $L_p$ rotating in a ring trap around the axis of the Fabry-Perot cavity. The red beam represents the Laguerre-Gauss modes with the orbital angular momentum of $\pm \ell \hslash$ used to probe the BEC rotation. The output signal $a_{\text{out}}$ is the field transmitted from the cavity.

Figure 2

(a) Angular profile of the condensate density per particle for a persistent current rotational eigenstate (b) OAM state content of the condensate. Parameters used here are $\mathrm{\Gamma }=0.0001, T=10$ nK $\times k_B/\left(\hslash {\omega }_{\beta }\right), g/\hslash \simeq 2\pi \times 0.02$ Hz, $L_p=1, \ell =10, N={10}^4, \tilde{\mathrm{\Delta }}=-2\pi \times 173$ Hz, $U_0=2\pi \times 212$ Hz, ${\gamma }_0=2\pi \times 2$ MHz, $P_{\text{in}}=0.2$ pW, ${\omega }_c=2\pi \times {10}^{15}$ Hz, $m=23$ amu, and $R=12 \mu \mathrm{m}$ [1, 57].

Figure 3

(a) Power spectra of the output phase quadrature of the cavity field as a function of the system response frequency. The vertical dashed lines (grey and green) correspond to the analytical predictions for the side-mode frequencies of $L_p=1$ and $L_p=2$, respectively, including atomic interactions; see Eq. (A13) [48]. (b) Rotation measurement sensitivity $\zeta$ [Eq. (14)] as a function of the system response frequency $\omega$. Here $G=2\pi \times 7.5$ kHz and $|{\alpha }_s{|}^2=0.096$, which corresponds to $P_{\text{in}}=0.2$ pW. Other parameters are the same as in Fig. 2.

Figure 4

Variation of the fidelity [Eq. (16)] of the ring condensate density for a persistent current with time. The dashes correspond to the fidelity measured for the wave functions having the same phase profile over time. The set of parameters used are the same as mentioned in Fig. 2.

Figure 5

(a) Angular profile of the condensate density per particle for a persistent current superposition [Eq. (17)] with $L_{p1}=1,L_{p2}=3$. (b) OAM state content of the condensate. Here $P_{\text{in}}=0.7$ pW and the other parameters are the same as in Fig. 2.

Figure 6

Persistent current superposition of two combinations of winding numbers: Left column for $L_{p1}=1$ and $L_{p2}=3$ and right column for $L_{p1}=2$ and $L_{p2}=3$. (a) and (b) represent the cavity output spectrum, (c) and (d) show the rotation measurement sensitivities, and (e) and (f) are the temporal evolutions of the fidelity for $L_{p1}=1, L_{p2}=3$ and $L_{p1}=2, L_{p2}=3$, respectively. Vertical dashed lines represent the analytical prediction corresponding to the $L_{p1}\pm 2l, L_{p2}\pm 2l$ modes and their combinations (Eq. (A13) [48, 76]. Here $G=2\pi \times 7.5$ kHz, $|{\alpha }_s{|}^2=0.33$, and the optomechanical measurement time ($t_{\text{meas}}$) is 2.1 ms. All other parameters used are the same as in Fig. 5.

Figure 7

(a) Temporal evolution of the soliton density profile (b) OAM content of the soliton. Here $N=6000, a_s=-27.6a_0$, where $a_0$ is the Bohr radius, $m=7.01$ amu, $L_p=1$, and $P_{\text{in}}=0.4$ pW, and all other parameters are same as in Fig. 2.

Figure 8

(a) Power spectrum of the output phase quadrature of the cavity field as a function of the system response frequency for a soliton. The vertical dashed lines (grey and green) correspond to the analytical predictions for the side modes ($L_p\pm 2\ell$) of $L_p=1$ and $L_p=2$, respectively, see Eq. (A13) [48, 76]. Only the corresponding peaks have been labeled. (b) Variation of rotation measurement sensitivity $\zeta$ with the system response frequency $\omega$ for $L_p=1$. Here $G=2\pi \times 5.8$ kHz and $|{\alpha }_s{|}^2=0.192$, which corresponds to $P_{in}=0.4$ pW. The other set of parameters used here are the same as in Fig. 7.

Figure 9

Fidelity [Eq. (16)] of the soliton density profile versus time. The dashes correspond to the fidelity measured for the wave functions having the same phase profile over time. The set of parameters used is the same as in Fig. 7.

Figure 10

Power spectrum of the output phase quadrature of the cavity field as a function of the system response frequency for different input powers: (a) $P_{\text{in}}=0.5$ pW, (b) $P_{\text{in}}=1$ pW, and (c) $P_{\text{in}}=2$ pW. Vertical dashed lines indicate the analytical predictions of Eq. (A13) [48, 76]. All other parameters used are the same as in Fig. 2.

Figure 11

Location of the peak in the output spectrum (${\omega }_d^{\prime }/2\pi$) [Eq. (A13)] [48, 76] versus input optical power ($P_{\text{in}}$). All other parameters are the same as in Fig. 2. For $P_{\text{in}}\lesssim 0.5\mathrm{pW}$, the location of the ${\omega }_d^{\prime }/2\pi$ peak remains constant, implying the presence of linear response; while for higher power ($P_{\text{in}}\gtrsim 0.5\mathrm{pW}$), the ${\omega }_d^{\prime }/2\pi$ peak begins to deviate from the linear regime, indicating the nonlinear response.

Figure 12

(a)–(c) Temporal evolution of the moving soliton density profile. (d)–(f) OAM states of the condensate occupied by the soliton. (g)–(i) Power spectrum of the phase quadrature of the cavity field versus system response frequency. Vertical dashed lines indicate the analytical predictions of the side modes for $L_p\pm 2\ell$, Eq. (A13) [48, 76]. (j)–(l) Soliton rotation measurement sensitivity as a function of system response frequency for $P_{\text{in}}=0.7$ pW, 1 pW, and 2 pW, respectively. Here $G=2\pi \times 5.8$ kHz and $|{\alpha }_s{|}^2=0.33,0.48,0.96$, respectively, for the above input power values. The remaining parameters are the same as in Fig. 7.