Sub-shot-noise phase sensitivity with a Bose-Einstein condensate Mach-Zehnder interferometer

Bose-Einstein condensates (BEC), with their coherence properties, have attracted wide interest for their possible application to ultraprecise interferometry and ultraweak force sensors. Since condensates, unlike photons, are interacting, they may permit the realization of specific quantum states needed as input of an interferometer to approach the Heisenberg limit, the supposed lower bound to precision phase measurements. To this end, we study the sensitivity to external weak perturbations of a representative matter-wave Mach-Zehnder interferometer whose input are two Bose-Einstein condensates created by splitting a single condensate in two parts. The interferometric phase sensitivity depends on the specific quantum state created with the two condensates, and, therefore, on the time scale of the splitting process. We identify three different regimes, characterized by a phase sensitivity $\Delta \theta$ scaling with the total number of condensate particles $N$ as (i) the standard quantum limit $\Delta \theta \sim 1∕N^{1∕2}$, (ii) the sub shot-noise $\Delta \theta \sim 1∕N^{3∕4}$, and the (iii) the Heisenberg limit $\Delta \theta \sim 1∕N$. However, in a realistic dynamical BEC splitting, the $1∕N$ limit requires a long adiabaticity time scale, which is hardly reachable experimentally. On the other hand, the sub-shot-noise sensitivity $\Delta \theta \sim 1∕N^{3∕4}$ can be reached in a realistic experimental setting. We also show that the $1∕N^{3∕4}$ scaling is a rigorous upper bound in the limit $N\to \infty$, while keeping constant all different parameters of the bosonic Mach-Zehnder interferometer.

Figure 1

(Color online) Mach-Zehnder phase sensitivity $\Delta \theta$ as a function of $K$ as given by Eq. (23). Here $N=1000$ and $E_c=0.001{\mathrm{ms}}^{-1}$. The green lines are the analytical predictions in three regimes: (i) Rabi Regime, $K⪢N{E}_c$, where $\Delta \theta =1∕\sqrt{N}$, (ii) Josephson Regime, $E_c∕N⪡K⪡N{E}_c$, where $\Delta \theta ={(4K∕E_c)}^{1∕4}1∕N^{3∕4}$, and (iii) Fock Regime, $K⪡E_c∕N$, where $\Delta \theta =\sqrt{2}∕N$. The blue line is given by the variational approach with the wave function (35), which reduced to Eq. (30) in the Rabi and Josephson regimes. Red points represent numerical solutions of Eq. (10).

Figure 2

Comparison between the mean-square fluctuation $\sigma$, as obtained by the variational equations (41, 42) (line) and the one obtained by the exact numerical solution of the Schrödinger equation (11) (points). The parameters are $N=1000$, $E_c=0.001{\mathrm{ms}}^{-1}$, $K\left(0\right)=100{\mathrm{ms}}^{-1}$, and $\tau =10\mathrm{ms}$.

Figure 3

(Color online) Plot of ${\left(\Delta \theta \right)}^2$ as given by Eq. (23) for different values of $t∕\tau$ (in ms). The parameters are $N=1000$, $E_c=0.001{\mathrm{ms}}^{-1}$ and $K\left(0\right)=100{\mathrm{ms}}^{-1}$. The blue line represents the adiabatic behavior.

Figure 4

(Color online) Plot of the scaling parameter $\beta$ with the number of particles [see Eq. (43)]. The calculation has been made with $N=10000$ and $N=1000$, and with fixed parameters $E_c=0.001{\mathrm{ms}}^{-1}$, and $K\left(0\right)=100{\mathrm{ms}}^{-1}$. The blue line represents the adiabatic behavior; the points correspond to the minimum of MZ phase sensitivity occurring for different values of $\tau$ (in ms).