Phase-matching condition for enhanced entanglement of colliding indistinguishable quantum bright solitons in a harmonic trap

We investigate finite number effects in collisions between two states of an initially well defined number of identical bosons with attractive contact interactions, oscillating in the presence of harmonic confinement in one dimension. We investigate two $N/2$ atom bound states, which are initially displaced (symmetrically) from the trap center, and then left to freely evolve. For sufficiently attractive interactions, these bound states are like those found through use of the Bethe ansatz (quantum solitons). However, unlike the free case, the integrability is lost due to confinement, and collisions can cause mixing into different bound-state configurations. We study the system numerically for the simplest case of $N=4$, via an exact diagonalization of the Hamiltonian within a finite basis, investigating left-right number uncertainty as our primary measure of entanglement. We find that for certain interaction strengths, a phase-matching condition leads to resonant transfer to different bound-state configurations with highly non-Poissonian relative number statistics.

Figure 1

Fidelity [left axis, falling line, defined via Eq. (35)] and energy difference [right axis, rising line, defined via Eq. (36)] between the numerical initial condition and analytic initial condition for a range of $\left|g\right|$ (with $g<0$). Initial energy is always overestimated in the numerics as the basis set cannot accurately resolve states on the length scales required. When considering the relative size of the energy discrepancy, we note that the interaction energy for our initial condition scales as $-g^2$ in the harmonic units used.

Figure 2

Minimum left-right number uncertainty $\Delta N_R$ after a single collisions for different values of $x_0$ (given in the legend). Note that the values at $g=0$ should be of the order $exp(-x_0^2)$. For large $\left|g\right|$ the relationship is quadratic.

Figure 3

With $x_0=3$: (a) Frequency of the first three peaks in the Fourier transform of $\Delta N_R\left(t\right)$ (multiplied by $\pi$) compared to the noninteracting values; pluses in the dotted line denote approximate locations of the transfer resonances. (b) Difference between these peak locations and the noninteracting values along with linear fits; equations for fits are shown in the figure inset.

Figure 4

(a) Minimum value taken by $\Delta N_R$ after a given collision for a range of $\left|g\right|$ (with $g<0$). (b) Lines corresponding to near resonant values of $\left|g\right|$. The oscillatory behavior appears to occur with shorter periods at higher $\left|g\right|$, the largest value however does not follow this pattern and is likely an artifact from the numerical breakdown.

Figure 5

(a) Probability of finding $n$ atoms to the right of the trap; (b) right side position expectation value on components of the wave function with exactly $n$ atoms to the right; (c) variance in this value.