Limits to phase resolution in matter-wave interferometry

We study the quantum dynamics of a two-mode Bose-Einstein condensate in a time-dependent symmetric double-well potential using analytical and numerical methods. The effects of internal degrees of freedom on the visibility of interference fringes during a stage of ballistic expansion are investigated varying particle number, nonlinear interaction sign and strength, as well as tunneling coupling. Expressions for the phase resolution are derived and the possible enhancement due to squeezing is discussed. In particular, the role of the superfluid–Mott insulator crossover and its analog for attractive interactions is recognized.

Figure 1

Particle density bounded by density noise $⟨G_1⟩\pm \Delta G_1$ after ballistic expansion of a two-mode condensate with $N=100$ in the ground state in the “superfluid” regime, $G={10}^{-3}$, where the visibility is high. The dashed lines show the density of the mode functions. Transverse distance is measured in units of the oscillator width $\Delta y$ and the initial condensates were centered around $d=\pm 8\Delta y$. The modes were allowed to expand ballistically until $\omega t=20$.

Figure 2

Ground-state distributions as function of the dimensionless parameter $G$ for attractive interaction. In (a) magnitudes of ground-state components $\mid c_m\left(G\right)\mid$ are shown for attractive interaction, and in (b) the distribution of relative phase components $\mid c_{{\theta }_m}\left(G\right)\mid$ in Eq. (26).

Figure 3

Ground-state distributions as a function of the dimensionless parameter $G$ for repulsive interaction. In (a) magnitudes of ground-state components $\mid c_m\left(G\right)\mid$ are shown for attractive interaction, and in (b) the distribution of relative phase components $\mid c_{{\theta }_m}\left(G\right)\mid$ in Eq. (26).

Figure 4

The uncertainties $\Delta {\widehat{J}}_y^2$, $\Delta {\widehat{J}}_z^2$ and the product $\Delta {\widehat{J}}_y\Delta {\widehat{J}}_z$ as functions of $G$ for $N=100$. The phase uncertainty $\Delta {\widehat{J}}_y^2$ goes through a global minimum at $G=-1$, the border between attractive superfluid and superposition states, and increases in the repulsive regime until it saturates to a constant value as the system passes through the Mott-insulator transition.

Figure 5

Visibility of interference fringes for a ballistically expanding two-mode Bose-Einstein condensate as a function of $G$, the ratio between the mean-field and tunneling energies, for particle numbers ranging from 10 to 200. For attractive interaction the visibility decreases abruptly for $G<-1$. In the case of repulsive interaction the behavior is qualitatively and quantitatively different for even (solid) and odd (dashed) atom numbers, as explained in the text.

Figure 6

Squeezing of the phase variance relative to the standard quantum limit for the two-mode ground state. For attractive interaction the phase squeezing is minimal just below $G=-1$. The crossover from Poissonian fluctuations in the superfluid regime $G<-1$ to the superposition state $G>-1$ becomes sharper with increased number of particles. For repulsive interaction there is a smooth transition between the superfluid regime of low $G$ values to the Mott-insulator state. In the asymptotic limit $G\to \infty$, the squeezing approaches a plateau, which is different for odd and even atom numbers.

Figure 7

Phase variance in the ground state of the double-well system at the “critical point” $G=-1$ for attractive condensates, and at $G=N^2∕2$ for repulsive condensates, as functions of particle number. The dashed lines show the standard quantum limit (SQL) and Heisenberg limit (HL) for comparison.

Figure 8

Phase squeezing for $g<0$ when $\Delta E\left(t\right)$ is taken to vary according to Eq. (42), for several values of the $\tau$. The dashed lines show squeezing of the instantaneous ground state for comparison.

Figure 9

Squeezing of the relative phase $\Delta \Theta$ for a double well split in time according to Eq. (42).