Supersolid-Based Gravimeter in a Ring Cavity

We propose a novel type of composite light-matter interferometer based on a supersolidlike phase of a driven Bose-Einstein condensate coupled to a pair of degenerate counterpropagating electromagnetic modes of an optical ring cavity. The supersolidlike condensate under the influence of the gravity drags the cavity optical potential with itself, thereby changing the relative phase of the two cavity electromagnetic fields. Monitoring the phase evolution of the cavity output fields thus allows for a nondestructive measurement of the gravitational acceleration. We show that the sensitivity of the proposed gravimeter exhibits Heisenberg-like scaling with respect to the atom number. As the relative phase of the cavity fields is insensitive to photon losses, the gravimeter is robust against these deleterious effects. For state-of-the-art experimental parameters, the relative sensitivity $\mathrm{\Delta }g/g$ of such a gravimeter could be of the order of ${10}^{-10}-{10}^{-8}$ for a condensate of a half a million atoms and interrogation time of the order of a few seconds.

Figure 1

A schematic representation of the system. The cavity is tilted with respect to the horizontal direction by an angle $\theta$. The relative phase $\varphi$ of the two counterpropagating cavity modes ${\widehat{a}}_{\pm }$ is measured in the cavity output to monitor the motion of the BEC along the cavity axis, described by $\widehat{\psi }(x,t)$. The transverse pump laser is indicated by ${\eta }_0$.

Figure 2

The time evolution of the relative phase of the two cavity modes and the center-of-mass motion of the BEC. The solid red line corresponds to the relative phase (left axis) and the dashed red line corresponds to the center-of-mass position of the condensate (right axis), obtained numerically through the mean-field calculation. The inset shows the difference $\delta \varphi$ between the numerical relative phase and the simple heuristic model of Eq. (4) with fitted $\xi =0.167{\omega }_r$ and $\zeta =0.007$. The parameters are set to $({\mathrm{\Delta }}_c,\kappa )=(-8,1){\omega }_r$ and $(\sqrt{N}{\eta }_0,N{U}_0,Mg\sin \theta {\lambda }_c)=(20,-1,10)\hslash {\omega }_r$, where ${\omega }_r\equiv \hslash k_c^2/2M$ is the recoil frequency.

Figure 3

Relative sensitivity for state-of-the-art experimental parameters as a function of time. The red dashed curve presents the sensitivity for a $\kappa =0$ case (solely for the sake of comparison), while the blue solid curve takes into account the effective friction arising from the photon losses, $\kappa ={\omega }_r$. The parameters of the simulation are set to $N=5\times {10}^5$, $n=2.5\times {10}^{11}$, $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$, $k_c=2\pi /780{\mathrm{nm}}^{-1}$, $\xi =0.167{\omega }_r$, and $\zeta =0.007$.