Scattering bright solitons: Quantum versus mean-field behavior

We investigate scattering bright solitons off a potential using both analytical and numerical methods. Our paper focuses on low kinetic energies for which differences between the mean-field description via the Gross-Pitaevskii equation (GPE) and the quantum behavior are particularly large. On the $N$-particle quantum level, adding an additional harmonic confinement leads to a simple signature to distinguish quantum superpositions from statistical mixtures. While the nonlinear character of the GPE does not allow quantum superpositions, the splitting of GPE solitons takes place only partially. When the potential strength is increased, the fraction of the soliton which is transmitted or reflected jumps noncontinuously. We explain these jumps via energy conservation and interpret them as indications for quantum superpositions on the $N$-particle level. On the GPE level, we also investigate the transition from this stepwise behavior to the continuous case.

Figure 1

Energetically (dis)allowed states for $N=100$ particles with $n$ particles on one side of the scattering potential and $N-n$ particles on other side. The parameter regions are displayed as a function of $n$ and center-of-mass kinetic energy $E_{\mathrm{kin}}$. (a) Magenta (gray) region: not energetically allowed as soon as all particles have left the scattering potential; black and white regions: energetically allowed. Only in the white region [which lies above $E_{\mathrm{kin}}\simeq 0.755\left|E_0\right|$ for $N=100$; Eq. (14)] are product states with 50:50 occupation on both side of the barrier energetically allowed. (b) Enhanced lower part of the left panel. In the energy regime below the dashed line, the only energetically allowed states are all particles on the left or all particles on the right of the barrier—and quantum superpositions of both.

Figure 2

Center-of-mass density of a 1D bright quantum-matter-wave soliton in a two-dimensional projection as a function of both time $t$ (in units of the oscillation period $T$) and elongation $x$ (in units of the oscillator length ${\lambda }_{\mathrm{osc}}$). There is additional harmonic confinement ($\propto x^2$); the soliton is modeled within the effective potential approach. Initially, the wave function is centered around $x=-10{\lambda }_{\mathrm{osc}}$. At $t=0.25T$ the soliton scatters off a narrow effective potential modeled by a $\delta$ potential and the center-of-mass wave function splits into two parts, leading to a mesoscopic quantum superposition of all particles being on either the left or right of the scattering potential at $t=0.5T$. (a) If the quantum superposition becomes a statistical mixture due to decoherence at $t\approx 0.5T$, there is a $50\%$ probability to find the particles on either side of the scattering potential at $t=T$. (b) For a quantum superposition, all particles end on the right side with $p_{\mathrm{r}}=98.5\%$ probability at $t=T$.

Figure 3

(a) For the same situation as in Fig. 2, the probability to find all particles on the right side is plotted as a function of the width of the effective potential (19). The cases of the quantum superposition (solid line) and the statistical mixture (dashed line) are clearly distinguishable for repeated measurements. Points: Averaging over an experimentally realistic [48] narrow Gaussian distribution with width 5 centered around $N=100$ and truncated at 90 and 110. (b) Probability $p_r$ to find all particles on the right of the scattering potential as a function of shift $x_s$. From top to bottom: $\ell /{\lambda }_{\mathrm{osc}}=0,0.125,0.5,1.0$. (c) Difference of numeric solution and the computer-algebra-based analytic solution extending (29) to general potentials (19) for the same parameters as in the previous panel. (d) Probability to find all particles on the right-hand side as a function of the shift for $\ell ={\lambda }_{\mathrm{osc}}$. Compared to Eq. (29), the amplitude is lower than one (caused by the wider potential). Furthermore, the fact that the shift of the potential leads to an interference of only parts of the wave packets [discussed below Eq. (29)] is clearly visible.

Figure 4

Reflection of a GPE soliton in a 1D harmonic trap from a narrow Gaussian barrier [used to model the $\delta$ potential in Eq. (19)] as a function of strength of the scattering potential. The initial center-of-mass kinetic energy increases from top to bottom. (a) $E_{\mathrm{kin}}=0.7\left|E_0\right|$: The reflection $R$ remains below 0.22 before jumping to values above 0.77 near $v_0/\left(E_{\mathrm{kin}}{\lambda }_{\mathrm{GPE}}\right)=2.65126$. (b) $E_{\mathrm{kin}}=0.8\left|E_0\right|$: The reflection $R$ remains below 0.36 before jumping to values above 0.64. (c) $E_{\mathrm{kin}}=1.0\left|E_0\right|$: The reflection $R$ is continuous.

Figure 5

Probability of finding $n$ particles on the left side of the scattering potential as a function of time $\tau =Jt/\hslash$ for a total particle number of $N=4$. Initially, the system [modeled by Eq. (30) on 51 lattice sites] is in the $N$-particle ground state (determined via imaginary time evolution) of a one-dimensional harmonic oscillator shifted by 12 lattice sites before removing the shift and switching on the scattering potential. With strong attractive interactions [$NU/J=-4$, panel (a)], the probability of finding four particles on either side of the scattering potential lies above 98$\%$ for $\tau \approx 50$. For weaker interactions [$NU/J=-3$, panel (b)], this value has already dropped to less than $78\%$. Without interaction [$NU/J=0$, panel (c)], the value has dropped to $12.5\%$, the value predicted for product states [see Eq. (A9)]. The values for the scattering potential (${\tilde{v}}_0/J=\{0.27,0.43,0.82\}$) were chosen via bisection; $A=0.001J$.

Figure 6

Scattering a mean-field soliton off a $\delta$ potential in a 1D situation without additional harmonic confinement. (a) The reflection $R_{\min }$ (solid line) and $R_{\max }$ (dotted line) as defined in Eqs. (35) and (36) as a function of initial center-of-mass kinetic energy. As predicted by Eq. (34), both curves lie outside the magenta (gray) region for energetic reasons. (b) Scattering of a GPE soliton with initial center-of-mass kinetic energy $E_{\mathrm{kin}}=0.5\left|E_0\right|$ as a function of the strength of the $\delta$-scattering potential, normalized such that without interaction, there would be 50:50 splitting [cf. Eq. (27)]. From top to bottom: Dotted red (black) line corresponds to the the fraction $p_{\mathrm{soliton}}$ of the final GPE solution which is in solitons (the rest being particle radiation). Solid magenta (gray) line: Total kinetic energy after the scattering as predicted by Eq. (32) using the numerically calculated values for $R$. Dashed blue (black) line: numerically calculated total kinetic energy. The difference between the two lower lines is only large if $p_{\mathrm{soliton}}<1$.