Surpassing the standard quantum limit in an atom interferometer with four-mode entanglement produced from four-wave mixing

We theoretically investigate a scheme for atom interferometry that surpasses the standard quantum limit. A four-wave mixing scheme similar to the recent experiment performed by Pertot et al. [Phys. Rev. Lett. 104, 200402 (2010)] is used to generate subshotnoise correlations between two modes. These two modes are then interfered with the remaining two modes in such a way as to surpass the standard quantum limit, whilst utilizing all of the available atoms. Our scheme can be viewed as using two correlated interferometers. That is, the signal from each interferometer when looked at individually is classical, but there are correlations between the two interferometers that allow for the standard quantum limit to be surpassed.

Figure 1

Four-wave mixing. $s$-wave collisions between $|a,0\rangle$ and $|b,k_0\rangle$ scatter atoms into $|a,k_0\rangle$ and $|b,0\rangle$.

Figure 2

(a) Populations of each mode as a function of $t_{\text{fwm}}$. $\langle {\widehat{N}}_{aL}\rangle$ was calculated using the four-mode TW model (blue solid line), and the one-dimensional (1D) multimode TW model (blue solid circles). $\langle {\widehat{N}}_{bL}\rangle$ was calculated using a simple model with no depletion (red “$+$” symbols), the four-mode TW model (red dashed line), and the 1D multimode TW model (red squares). $\langle {\widehat{N}}_{aR}\rangle$ and $\langle {\widehat{N}}_{bR}\rangle$ where omitted, because they are almost identical to $\langle {\widehat{N}}_{bL}\rangle$ and $\langle {\widehat{N}}_{aL}\rangle$, respectively. (b) Normalized number difference variance $v_{i,j}$ for $\{i,j\}=\{aR,bL\}$ (blue “$+$” symbols show simple mode with no depletion, blue solid line shows four-mode TW model, blue solid circles show 1D multimode TW model), $\{aL,bR\}$ (red dashed line shows four-mode TW model, red squares show 1D multimode TW model), and $\{aL,bL\}$ (green dot-dashed line shows four-mode TW model, green triangles shows 1D multimode TW model). $\{aR,bR\}$, $\{aL,aR\}$, and $\{bL,bR\}$ are omitted because they are almost identical to $\{aL,bL\}$. The multimode calculation agrees well with the four-mode calculation for small values of $t_{\text{fwm}}$, but does not reach the same degree of squeezing.

Figure 3

$\langle {\widehat{\psi }}_a^{†}\left(x\right){\widehat{\psi }}_a\left(x\right)\rangle$ (blue solid line) and $\langle {\widehat{\psi }}_b^{†}\left(x\right){\widehat{\psi }}_b\left(x\right)\rangle$ (red dotted line) after 70 ms of separation time for (a) $t_{\text{fwm}}=0.1$ ms and (b) $t_{\text{fwm}}=0.18$ ms. The expectation values were calculated from 1200 trajectories. The trapping frequencies of the harmonic potential at $t=0$ were $\{{\omega }_x,{\omega }_y,{\omega }_z\}=2\pi \{5,1000,1000\}$ Hz. The interaction strength was scaled by the cross-sectional area of the condensate: $U_{1d}=\frac{U}{\pi r_0^2}$ with $r_0=0.55$ $\mu$m. $x_0$ was chosen as the point $x=0.4$ mm, which is roughly halfway between the two wave packets.

Figure 4

Pulse sequence. After the two momentum modes have spatially separated, the interferometry stage is implemented by a sequence of three $\frac{\pi }{2}$ pulses. An appropriate phase shift ${\phi }_{0L}$ is applied to $|b,0\rangle$ and ${\phi }_{0R}$ to $|b,k_0\rangle$ before the first coupling pulse and ${\phi }_{1L}$ and ${\phi }_{1R}$ before the second pulse to ensure that the populations in each mode remain approximately equal after each pulse. These phase shifts may be different, and can be implemented using the ac Stark shift from a localized optical beam. The phase shifts ${\phi }_{2L}$ and ${\phi }_{2R}$ between the second and third pulses is the quantity that the interferometer measures. The population in each mode is measured after the third beam splitter.

Figure 5

(a) $\langle {\widehat{N}}_{aL}-{\widehat{N}}_{bL}\rangle$ (blue solid line shows four-mode TW model, blue “$+$” symbols show 1D multimode TW model), $\langle {\widehat{N}}_{bR}-{\widehat{N}}_{aR}\rangle$ (red dashed line shows four-mode TW model and red squares show 1D multimode TW model), and $\langle \widehat{S}\rangle$ (green dot-dashed line shows four-mode TW model and green circles show 1D multimode TW model) as a function of ${\phi }_2$. The four-mode model has slightly more fringe visibility than the 1D multimode model. The horizontal black dotted lines are a visual guide only. (b) $\frac{V\left(S\right)}{\langle {\widehat{N}}_t\rangle }$ from the four-mode model (solid blue line), and the multimode model (dashed blue line) as a function of ${\phi }_2$. The horizontal black dotted lines indicated the level expected for uncorrelated atoms [${\log }_{10}(V\left(S\right)/\langle {\widehat{N}}_t\rangle )=0$], and squeezing below the uncorrelated level by a factor of 10 [${\log }_{10}(V\left(S\right)/\langle {\widehat{N}}_t\rangle )=-1$]. (c) The phase sensitivity $\Delta \phi$ from the four-mode model (blue solid line) and multimode model (blue dashed line). The minimum of $\Delta \phi \sqrt{N_t}$ for the multimode model is $\Delta \phi \approx 0.41\sqrt{N_t}$—more than twice as sensitive than for uncorrelated atoms.