Performing Hong-Ou-Mandel-type numerical experiments with repulsive condensates: The case of dark and dark-bright solitons

The Hong-Ou-Mandel experiment leads indistinguishable photons simultaneously reaching a 50:50 beam splitter to emerge on the same port through a two-photon interference. Motivated by this phenomenon, we consider numerical experiments of the same flavor for classical wave objects in the setting of repulsive condensates. We examine dark solitons interacting with a repulsive barrier, a case in which we find no significant asymmetries in the emerging waves after the collision, presumably due to their topological nature. We also consider case examples of two-component systems, where the dark solitons trap a bright structure in the second component (dark-bright solitary waves). For these, pronounced asymmetries upon collision are possible for the nontopological bright component. We also show an example of a similar phenomenology for ring dark-bright structures in two dimensions.

Figure 1

Phase diagram of $E^{\pm }$ before (the left two panels for $t=0$) and after (the right two panels for $t=1.6x_1/sin{\phi }_1$) collision. The relevant parameters are $q=0.3,\sigma =0.1,{\psi }_0=1$, and $x_1=15$.

Figure 2

Numerical simulation of a two-dark-soliton collision at center of the impurity. The relevant parameters are $q=0.3,\sigma =0.1,\mu =1,x_1=15$, and ${\phi }_2/{\phi }_1=0.90$. (a) ${\phi }_1=0.35$; (b) ${\phi }_1=0.75$ [recall that the speed of the first soliton is $\propto sin\left({\varphi }_1\right)$]; (c) ${\phi }_1=0.35$ with a parabolic trapping potential $U_{\mathrm{tr}}=\frac{1}{2}{\mathrm{\Omega }}^2{x}^2$ of $\mathrm{\Omega }=0.03$; (d) ${\phi }_1=0.75$ with $\mathrm{\Omega }=0.03$.

Figure 3

Numerical simulation of a two-DB-soliton collision at the center of impurity. The relevant parameters are $q=1,\sigma =0.1,\mu =2,k_1=0.66,k_2=0.46,v_1=v_2=0.5,x_1=10$, and $\mathrm{\Delta }=0$. The lower two panels show the collision with a parabolic trapping potential $U_{\mathrm{tr}}=\frac{1}{2}{\mathrm{\Omega }}^2{x}^2$ of $\mathrm{\Omega }=0.03$.

Figure 4

Plots of $E_{B,D}^{-}$ as a function of $k_2/k_1$ varying from 0.6 to 1.0 ($k_1=0.66$). The relevant parameters are $q=1,\sigma =0.1,\mu =2,v_1=v_2=0.5,x_1=10$, and $\mathrm{\Delta }=0$.

Figure 5

Plots of $E_{B,D}^{-}$ as a function of $k_2/k_1$ varying from 0.6 to 1.0 ($k_1=0.66$). The relevant parameters are $q=1,\sigma =0.1,\mu =2,v_1=v_2=0.5,x_1=10$, and $\mathrm{\Delta }=0$. Left panel: $\mathrm{\Omega }=0,0.01,0.03$, and 0.08, respectively (note that the curves for $\mathrm{\Omega }=0$ and $\mathrm{\Omega }=0.01$ almost overlap); Right panel: comparison between the miscible ($g_{12}=1.0$) and immiscible ($g_{12}=1.1$) examples.

Figure 6

Two-parameter diagram of the propagation asymmetry for the dark (left) and bright (right) components of the DB solitons. The relevant parameters are $q=1,\sigma =0.1,\mu =2,x_1=10$, and $\mathrm{\Delta }=0$.

Figure 7

Plots of $E_{B,D}^{-}$ as a function of $k_2/k_1$ varying from 0.8 to 1.0 ($k_1=0.40$ for the left panel, and $k_1=0.50$ for the right panel). The relevant parameters are $q=-1,\sigma =0.1,\mu =2,x_1=10$, and $\mathrm{\Delta }=0$. These solid and dashed lines in both figures, from upper to lower, correspond to the values of $v$ of 0.7, 0.6, and 0.5, respectively.

Figure 8

Two-parameter diagram of the maximum asymmetry for the DB solitons, again for dark (left) and bright (right) components. The relevant parameters are $q=1,\sigma =0.1,\mu =2,x_1=20$, and $\mathrm{\Delta }=0$.

Figure 9

Two-parameter diagram of the asymmetry for the dark solitons (component). The relevant parameters are $q=-1,\sigma =0.1,\mu =2,x_1=20$, and $\mathrm{\Delta }=0$.

Figure 10

(a) Plots of $E_D^{-}$ (after collision) as a function of $k_2/k_1$. The relevant parameters are $q=1,\sigma =0.1$, $\mu =2,v_1=v_2=0.5,k_1=0.66$, and $\mathrm{\Delta }=0$. $x_1$ is equal to $8,10,12$, and 14, respectively. (b) Plots of $E_D^{-}$ as a function of $k_2/k_1$. The relevant parameters are $q=-1$, $\sigma =0.1,\mu =2,v_1=v_2=0.5,k_1=0.40$, and $\mathrm{\Delta }=0$. $x_1$ is equal to $8,10,12$, and 14, respectively. (c) Plots of $E_D^{-}$ as a function of $v_2/v_1$. The relevant parameters are $q=-1,\sigma =0.1,\mu =2,k_1=k_2=0.3,v_1=0.60$, and $\mathrm{\Delta }=0$. $x_1$ is equal to $18,20,22$, and 24, respectively.

Figure 11

Numerical simulation of a ring DB soliton collision at the center of the ring-shaped impurity. The relevant parameters are $q=1,\sigma =0.1,\mu =2,k_1=k_2=0.60,v_1=0.50,v_2=0.45,r_1=5,r_0=10$, and $\mathrm{\Delta }=0$.

Figure 12

Comparison of ${\phi }_c$ between the effective potential approach (A10) and direct simulation of Eq. (1) ($\sigma =0.1$).