Mean-field analog of the Hong-Ou-Mandel experiment with bright solitons

In the present work, we theoretically propose and numerically illustrate a mean-field analog of the Hong-Ou-Mandel experiment with bright solitons. More specifically, we scatter two solitons off of each other (in our setup, the bright solitons play the role of a classical analog to the quantum photons of the original experiment), while the role of the beam splitter is played by a repulsive Gaussian barrier. In our classical scenario, distinguishability of the particles yields, as expected, a $0.5$ split mass on either side. Nevertheless, for very slight deviations from the completely symmetric scenario, a near-perfect transmission can be constructed instead, very similarly to the quantum-mechanical output. We demonstrate this as a generic feature under slight variations of the relative soliton speed, or of the relative amplitude in a wide parametric regime. We also explore how variations of the properties of the “beam splitter” (i.e., the Gaussian barrier) affect this phenomenology.

Figure 1

Numerical simulation of the HOM analogous effect for a two (slightly asymmetric) matter-wave bright soliton collision at the Gaussian potential barrier. The parameters are chosen as $q=1,\sigma =0.1,k_1=k_2=1$, and $\Delta =0$. (a) $v_1=1.00,v_2=0.98$, and $x_1/v_1=10$; (b) $v_1=1.00,v_2=1.02$, and $x_1/v_1=10$.

Figure 2

Plots of $E_{+,-}$ [see Eq. (3)] as a function of $v_2/v_1$ varying from 0.9 to 1.1 $\left(v_1=1\right)$. The parameters are chosen as $q=1,\sigma =0.1,k_1=k_2=1,\Delta =0$, and $x_1/v_1=x_2/v_2=20$.

Figure 3

Plots of $E_{+}$ [see Eq. (3)] with $v_2/v_1$ varying from 0.9 to 1.0 for two groups of three values of $v_1$: (a) $v_1=0.5,1.0$, and $2.0$; (b) $v_1=0.1,0.2$, and $0.3$. The parameters are chosen as $q=1,\sigma =0.1,k_1=k_2=1,\Delta =0$, and $x_1/v_1=x_2/v_2=20$.

Figure 4

The top panel shows the variation of the fraction $E_{+}$ as a function of velocity for different relative phases between the bright solitons. The bottom panel shows the shift of the optimal point (of maximal asymmetry) as a function of the relative phase $\Delta$. The other parameters are chosen as $q=1,\sigma =0.1,k_1=k_2=1,v_1=1$, and $x_1/v_1=x_2/v_2=20$.

Figure 5

Plots of $E_{-}$ with $k_2/k_1$ varying from 0.9 to 1.0 for two groups of three values of $k_1$. The parameters are chosen as $q=1,\sigma =0.1,\Delta =0$, and $x_1/v_1=x_2/v_2=20$. (a) $v_1=v_2=1$; (b) $v_1=v_2=0.2$.

Figure 6

(a) Plots of $E_{+}$ with $v_2/v_1$ varying from 0.9 to 1.0 for four values of $\sigma \left(q=1\right)$. (b) Plots of $E_{+}$ with $v_2/v_1$ varying from 0.9 to 1.0 for four values of $q \left(\sigma =0.1\right)$. The parameters are chosen as $k_1=k_2=1,v_1=1,\Delta =0$, and $x_1/v_1=x_2/v_2=20$.

Figure 7

(a) Plots of $E_{+}$ as a function of $v_2/v_1$ for five distinct values of $x_1$. The remaining parameters are $q=1,\sigma =0.1,k_1=k_2=1$, and $v_1=8$. (b) The point of optimal asymmetry quantified by ${(v_2/v_1)}^2$ is shown as a function of $x_1$ together with the corresponding theoretical prediction of our heuristic argument; see the discussion of the last paragraph of Sec. 3. Notice that these panels are constructed for a large value of speed where, as is illustrated, this argument is quantitatively valid.