Progress towards quantum-enhanced interferometry with harmonically trapped quantum matter-wave bright solitons

We model the dynamics of attractively interacting ultracold bosonic atoms in a quasi-one-dimensional wave guide with additional harmonic trapping. Initially, we prepare the system in its ground state and then shift the zero of the harmonic trap and switch on an additional narrow scattering potential near the center of the trap. After colliding with the barrier twice, we propose to measure the number of atoms opposite the initial condition. Quantum-enhanced interferometry with quantum bright solitons allows us to predict detection of an offset of the scattering potential with considerably increased precision as compared to single-particle experiments. In a future experimental realization this might lead to measurement of weak forces caused, for example, by small horizontal gradients in the gravitational potential—with a resolution of several micrometers given essentially by the size of the solitons. Our numerical simulations are based on the rigorously proved effective potential approach developed in previous papers [Phys. Rev. Lett. 102, 010403 (2009) and Phys. Rev. Lett. 103, 210402 (2009)]. We choose our parameters such that the prerequisite of the proof (that the solitons cannot break apart, for energetic reasons) is always fulfilled, thus exploring a parameter regime inaccessible to the mean-field description via the Gross-Pitaevskii equation due to Schrödinger-cat states occurring in the many-particle quantum dynamics.

Figure 1

As the experimentally accessible signature we suggest to use the probability to find all particles on the side opposite to the initial condition after scattering twice [shown is the two-dimensional projection of the modulus squared of the center-of-mass wave function as a function of both position and time for (a) $N=1$ and (b) a $N=100$ soliton for the parameters of footnote ]. After being prepared in the ground state of a harmonic trap, at $t=0$ the trap is shifted and a narrow repulsive scattering potential switched on at the new center of the trap. After the scattering potential is hit for the first time, Schrödinger-cat states occur; after the second collision all particles are on the side opposite to the initial condition (cf. Ref. [29]). The fact that Schrödinger-cat states occur is a consequence of the fact that slow bright solitons cannot energetically break apart (cf. Ref. [21], footnote ); the interaction remains switched on the entire time.

Figure 2

(a) Our signature that everything is on the side opposite to the initial condition after colliding with the barrier twice, denoted by $T$, as a function of distance $X_{\mathrm{s}}$ between trap minimum and scattering potential for different widths of the $1/{cosh}^2$ potential, introduced in Eq. (16), $X_0=30\mu \mathrm{m}$ and $N=100$ particles. The value $\xi =0.42\mu \mathrm{m}$ corresponds to the parameters from footnote . The limit $\xi \to 0$ corresponds to a delta function potential. (b) Same for varying initial displacements $X_0$ and $\xi =0.42\mu \mathrm{m}$ kept fixed. (c) Displays the analytical result (20) for the same parameters as panel (b). Thus panel (c) confirms that the analytical approximation (20) qualitatively correctly captures the physics displayed in the many-particle quantum dynamics of panel (b).

Figure 3

Our signature that everything is on the side opposite to the initial condition after colliding with the barrier twice as a function of distance between trap minimum and scattering potential [see. Fig. 1 also for parameters used] for (a) a single particle: numerical data [magenta (dark gray) dashed curve] and analytical formula (20) (black solid curve) and (b) repeating the experiment 100 times with single particles [magenta (dark gray) dashed curve] compared to the single-particle result (20) (black solid curve) and replacing this result by a combination of Gaussians [green (light gray) dash-dotted curve]. (c) Shows quantum enhancement by a factor of $1/\sqrt{N}$ for a quantum bright soliton of $N=100$. Shown are formula (20) (black solid curve), numerical data using a delta-function barrier [green (light gray) dash-dotted curves], and the $1/{cosh}^2$ potential corresponding to the parameters of footnote [magenta (dark gray) dashed curve].