Nonlinear interferometry with Bose-Einstein condensates

We analyze a proposed experiment [Boixo et al., Phys. Rev. Lett. 101, 040403 (2008)] for achieving sensitivity scaling better than $1/N$ in a nonlinear Ramsey interferometer that uses a two-mode Bose-Einstein condensate (BEC) of $N$ atoms. We present numerical simulations that confirm the analytical predictions for the effect of the spreading of the BEC ground-state wave function on the ideal $1/N^{3/2}$ scaling. Numerical integration of the coupled, time-dependent, two-mode Gross-Pitaevskii equations allows us to study the several simplifying assumptions made in the initial analytic study of the proposal and to explore when they can be justified. In particular, we find that the two modes share the same spatial wave function for a length of time that is sufficient to run the metrology scheme.

Figure 1

$N$ dependence of the inverse volume ${\eta }_N$ in harmonic trap units. The points correspond to the results of the numerical integration of the 3D GP ground-state solution for different trap geometries: circles (blue) correspond to $q=2$, squares (black) to $q=4$, and triangles (red) to $q=10$. The respective Thomas-Fermi predictions for the 1D [Eq. (3.8)] and 3D regimes are the dotted (blue) line for $q=2$, dashed (black) line for $q=4$, and solid (red) line for $q=10$ [15]. The stiffness parameter $k$ of the trapping potential is chosen so that the crossover from 1D to 3D behavior occurs at ${\mathrm{N̄}}_T\simeq 14\hspace{0.5em}000$ for all three values of $q$.

Figure 2

Ramsey fringes for a cigar-shaped ${}^{87}\mathrm{Rb}$ BEC of 1000 atoms (labeling convention as in Fig. 1 [15]). The points represent the numerical results of the integration of the coupled, two-mode, three-dimensional GP equations (3.10) for the different trapping potentials (3.1), whereas the lines are the respective idealized Josephson-approximation predictions (2.10), with the value of ${\eta }_N$ supplied by the numerics of Sec. 3a. The Josephson approximation improves as the trap gets harder.

Figure 3

Ramsey fringes for a cigar-shaped ${}^{87}\mathrm{Rb}$ BEC of 5000 atoms (labeling convention as in Fig. 1 [15]). The points represent the numerical results of the integration of the coupled, two-mode, three-dimensional GP equations (3.10) for the different trapping potentials (3.1), whereas the lines are the respective idealized, Josephson-approximation predictions (2.10). Here we only plot the idealized fringe pattern (2.10) for short times, since it quickly deviates from the simulated nonlinear evolution; the deviation is a consequence of the differentiation of the wave functions of two modes as they evolve separately under the coupled GP equations. (Figure 4 shows a closeup of the first 120 ms.)

Figure 4

Closeup of Fig. 3 for the first 120 ms. The Josephson approximation improves as the trap gets harder.

Figure 5

Same as Fig. 2 (1000 atoms), but here the lines correspond to the improved analytical prediction (4.8) for the three different values of $q$ (labeling convention as in Fig. 1 [15]). The improved model succeeds in predicting a reduction in fringe visibility as time increases, but it does not give a better estimate of the fringe frequency.

Figure 6

Same as Fig. 2 (1000 atoms), but here the lines correspond to the improved analytical overlap (4.8), computed using the numerically computed ${\eta }_N$ from Sec. 3a, as described in the text, for the three different values of $q$ (labeling convention as in Fig. 1 [15]). The improved model, with the ad hoc use of the numerically computed ${\eta }_N$, provides a reasonably good account of the fringe frequency and of the reduction in fringe visibility as time increases, with better agreement for short times and for harder traps.

Figure 7

Same as Fig. 3, but here the lines correspond to the improved analytical overlap (4.8), computed using the numerically computed ${\eta }_N$ from Sec. 3a, for the three different values of $q$ (labeling convention as in Fig. 1 [15]). The improved model, with the ad hoc use of the numerically computed ${\eta }_N$, is surprisingly good even in the third fringe period for $q=10$.