High-Precision Quantum-Enhanced Gravimetry with a Bose-Einstein Condensate

We show that the inherently large interatomic interactions of a Bose-Einstein condensate (BEC) can enhance the sensitivity of a high precision cold-atom gravimeter beyond the shot-noise limit (SNL). Through detailed numerical simulation, we demonstrate that our scheme produces spin-squeezed states with variances up to 14 dB below the SNL, and that absolute gravimetry measurement sensitivities between two and five times below the SNL are achievable with BECs between ${10}^4$ and ${10}^6$ in atom number. Our scheme is robust to phase diffusion, imperfect atom counting, and shot-to-shot variations in atom number and laser intensity. Our proposal is immediately achievable in current laboratories, since it needs only a small modification to existing state-of-the-art experiments and does not require additional guiding potentials or optical cavities.

Figure 1

(a) Space-time diagram illustrating SNL gravimetry with a BEC. Unwanted interatomic interactions are reduced by freely expanding the BEC for duration $T_{\exp }$. A $\pi /2-\pi -\pi /2$ Raman pulse sequence then creates a MZ interferometer of interrogation time $T$. The two interferometer modes correspond to internal states $|1⟩$ (red) and $|2⟩$ (blue) with $\hslash k_0$ momentum separation. (b) Quantum-enhanced ultracold-atom gravimetry. During initial expansion duration $T_{\exp }=2T_{\mathrm{OAT}}$, the BEC’s interatomic interactions generate spin squeezing via OAT. (c) Bloch sphere representation of state during quantum-enhanced gravimetry.

Figure 2

Analytic spin squeezing model parameters determined from a GPE simulation of our scheme up to $t=2T_{\mathrm{OAT}}$, with $T_{\mathrm{OAT}}=20\mathrm{ms}$ and an $N={10}^4$ atom BEC initially prepared in a spherical harmonic trap of frequency 50 Hz. (a) Effective squeezing rate $\chi (t)$ (blue, solid) and squeezing degree $\lambda (t)$ (orange, dashed). (b) Mode overlap $|\mathcal{Q}(t)|$. (Bottom) Normalized density slices at radial coordinate $r=0$.

Figure 3

Minimum spin squeezing parameter $\xi$ for $T_{\mathrm{OAT}}=10\mathrm{ms}$ and atom number $N$ [74]. In (a) the BEC is initially prepared in a spherical harmonic trap ($f_r=f_z=50\mathrm{Hz}$), whereas in (b) an initial “pancake” BEC is prepared in a cylindrically symmetric harmonic trap ($f_r=32\mathrm{Hz}$, $f_z=160\mathrm{Hz}$). TW simulations are compared to Eq. (3) with model parameters determined from GPE simulations (“3D GPE”). (c)–(h) Density profiles for $N={10}^6$ at $t=2T_{\mathrm{OAT}}$. The analytic model fails here for the spherical BEC case since spontaneous scattering degrades mode overlap.

Figure 4

One-dimensional TW calculations of sensitivity $\mathrm{\Delta }g$ for an $N={10}^4$ atom BEC initially prepared in a cylindrically symmetric harmonic trap ($f_r=32\mathrm{Hz}$, $f_z=160\mathrm{Hz}$). From top to bottom: (red) MZ with total interrogation time $T+T_{\mathrm{OAT}}$ (no initial period of free expansion), (green) BEC undergoes free expansion for duration $2T_{\mathrm{OAT}}$, followed by MZ of interrogation time $T$ [Fig. 1]; (magenta) quantum-enhanced BEC gravimetry [Fig. 1]; (blue) Eq. (1) with $\xi$ computed via TW. All four cases have the same total duration $2(T_{\mathrm{OAT}}+T)$ with $T=60\mathrm{ms}$. The SNL for an ideal MZ of interrogation time $T$ (dashed) and $T+T_{\mathrm{OAT}}$ (dot dashed) are marked for comparison. Our quantum-enhanced scheme always outperforms MZ schemes, even when phase diffusion is non-negligible.

Figure 5

The effect on our scheme of (a) Gaussian shot-to-shot beam splitting angle fluctuations of variance ${\sigma }_{\theta }^2$, (b) Gaussian shot-to-shot atom-number fluctuations of variance ${\sigma }_N^2$, and (c) imperfect atom detection of resolution $\mathrm{\Delta }n$. Here $N={10}^4$, $T_{\mathrm{OAT}}=10\mathrm{ms}$, with an initial BEC prepared in a cylindrically symmetric harmonic trap ($f_r=32\mathrm{Hz}$, $f_z=160\mathrm{Hz}$). In (a) the sensitivity was obtained via 1D TW simulations of the full interferometer ($T=60\mathrm{ms}$), whereas (b) and (c) computed the spin squeezing parameter from 3D TW simulations.