Bright-soliton quantum superpositions: Signatures of high- and low-fidelity states

Scattering quantum bright solitons off barriers has been predicted to lead to nonlocal quantum superpositions, in particular the NOON state. The focus of this paper lies on signatures of both high- and low-fidelity quantum superposition states. We numerically demonstrate that a one-dimensional geometry with the barrier potential situated in the middle of an additional—experimentally typical—harmonic confinement gives rise to particularly well-observable signatures. In the elastic-scattering regime we investigate signatures of NOON states on the $N$-particle level within an effective potential approach. We show that removing the barrier potential and subsequently recombining both parts of the quantum superposition leads to a high-contrast interference pattern in the center-of-mass coordinate for narrow and broad potential barriers. We demonstrate that the presented signatures can be used to clearly distinguish quantum superposition states from statistical mixtures and are sufficiently robust against experimentally relevant excitations of the center-of-mass wave function to higher-lying oscillator states. For two-particle solitons we extend these considerations to low-fidelity superposition states: even for strong deviations from the two-particle NOON state we find interference patterns with high contrast.

Figure 1

CoM density distribution $|\Psi \left(X,t\right){|}^2$ for scattering a bright soliton off a $\delta$-like effective potential (17), calculated on 6401 lattice points within the effective potential approach (see Sec. 2a). The potential height $v_{0}M{\lambda }_{\mathrm{osc}}/{\hslash }^2=1601$ is chosen such that 50-50 splitting takes place. The initial CoM density distribution corresponds to the shifted oscillator ground state centered around $X_0/{\lambda }_{\mathrm{osc}}=-10$. Variations of the initial particle number are modeled by a truncated Gaussian probability distribution [45] with mean value $N=100$ and standard deviation $\sigma =5$ under consideration of particle numbers $N\in [90,110]$. (a) The scattering potential is switched off at $t/T_{\mathrm{osc}}=0.5$, resulting in an interference of both parts of the superposition at about $t/T_{\mathrm{osc}}\approx 0.75$. The contrast (21) of the interference pattern with fringe spacing $0.37{\lambda }_{\mathrm{osc}}$ is $C\approx 1$ at $t/T_{\mathrm{osc}}\approx 0.75$. It is calculated on the interval $X\in \left[-0.5{\lambda }_{\mathrm{osc}}:0.5{\lambda }_{\mathrm{osc}}\right]$. (b) As (a) but assuming that the quantum superposition is turned into a statistical mixture at $t/T_{\mathrm{osc}}=0.5$.

Figure 2

CoM density distribution $|\Psi \left(X,t\right){|}^2$ for scattering a bright soliton off a barrier potential (15) of width $\ell /{\lambda }_{\mathrm{osc}}=4$, calculated on 9601 lattice points within the effective potential approach from Sec. 2a. The scaled potential height $U_{0}M{\lambda }_{\mathrm{osc}}^2/{\hslash }^2=200.4$ is chosen to ensure 50-50 splitting. The initial CoM density distribution corresponds to the shifted oscillator ground state centered around $X_0/{\lambda }_{\mathrm{osc}}=-20$. Initial variations of the particle number are modeled as in Fig. 1. (a) The scattering potential is switched off at $t/T_{\mathrm{osc}}=0.5$, resulting in an interference of both parts of the superposition around $t/T_{\mathrm{osc}}\approx 0.85$. (b) Zoom into (a) to highlight the resulting interference pattern. (c) As (b) but assuming that the quantum superposition is turned into a statistical mixture at $t/T_{\mathrm{osc}}\approx 0.5$. (d) Density distribution for the quantum superposition at $t/T_{\mathrm{osc}}=0.85$. Blue, thick line: data as in (a). Magenta, thin line: data for $N=100$. The fringe spacing $0.16{\lambda }_{\mathrm{osc}}$ of the interference pattern is not affected by particle number variations. The maximal contrast $C\approx 1$ is reached for $N=100$ (calculated on the interval $\left[-0.5{\lambda }_{\mathrm{osc}}:0.5{\lambda }_{\mathrm{osc}}\right]$), while it is only slightly reduced to $C\approx 0.95$ when particle number variations are included. (e) Same as (d) for the statistical mixture.

Figure 3

Interference patterns after scattering a bright soliton off a barrier potential for 50-50 splitting of the wave function: numerical data (solid black line) and plane-wave approximation (dashed red line). The approximate interference patterns are calculated according to Eq. (28) with reflection and transmission coefficients for the potentials of Eqs. (17) and (29), respectively. The initial CoM wave function is the shifted oscillator ground state centered around $X_0/{\lambda }_{\mathrm{osc}}=-10$, resulting in the mean momentum $\Omega {\lambda }_{\mathrm{osc}}=10$. (a) Scattering at a shifted $\delta$ potential (17) with $X_{\mathrm{s}}/{\lambda }_{\mathrm{osc}}=-0.15$. (b) Scattering at the shifted potential (29) with $\ell /{\lambda }_{\mathrm{osc}}=0.125$ and $X_{\mathrm{s}}/{\lambda }_{\mathrm{osc}}=-0.075$.

Figure 4

Same protocol as in Fig. 1 for $N=100$. Statistically averaged CoM probability density (31) for $T=450$ pK, $\omega =2\pi \times 10$ Hz, $N=100$ ${}^7$Li atoms, and $M=2$. (a) $\delta$-like barrier potential (17). The contrast at $t/T_{\mathrm{osc}}=0.75$ is reduced from $C\approx 1$ to $C=0.987$. (b) Scattering potential (15) with $\ell =1.5{\lambda }_{\mathrm{osc}}$. The contrast at $t/T_{\mathrm{osc}}=0.8$ is reduced from $C\approx 1$ to $C=0.574$. (c) CoM density from (a) at $t/T_{\mathrm{osc}}=0.75$. (d) CoM density from (b) at $t/T_{\mathrm{osc}}=0.8$.

Figure 5

CoM probability density $|\Psi \left(X,t\right){|}^2$ for scattering twice off a $\delta$ potential (17). (a) Initial state: ground state. The probability to find the particles to the right side of the barrier potential at $t/T_{\mathrm{osc}}=1$ is $p_{\mathrm{right}}\left(T_{\mathrm{osc}}\right)=0.985$. (b) Initial state: first excited state. (c) Initial state: second excited state. (d) Statistically averaged CoM probability density (31) for the same parameters as in Fig. 4. The probability to find the particles to the right side of the barrier potential stays about the same: $p_{\mathrm{right}}\left(T_{\mathrm{osc}}\right)=0.984$. (e) Same as (d) but assuming that the quantum superposition is turned into a statistical mixture at $t/T_{\mathrm{osc}}\approx 0.5$.

Figure 6

Time evolution for a two-particle soliton being scattered off a $\delta$ potential in an additional slight harmonic confinement with $A/J=9\times {10}^{-6}$, calculated on 201 lattice sites. The wave function is initially centered around $X/{\lambda }_{\mathrm{osc},2}=-4.61$. The initial state is determined by imaginary time evolution [25]. CoM density distribution ${\rho }_{\mathrm{CoM}}\left(X\right)$ (solid, black line) and single-particle density $\rho \left(x\right)$ (dash-dotted red line) are shown vs the spatial coordinate at exemplary times for interaction strengths $U/J=-1$ [(a)–(c)] and $U/J=-0.2$ [(d)–(f)]. (a) Initial state. (b) High-fidelity quantum superposition with occupation probabilities ${\tilde{p}}_0\approx {\tilde{p}}_2\approx 0.5$ and ${\tilde{p}}_1\approx 0$ at $t/T_{\mathrm{osc}}=0.5$. (c) Interference patterns at $t/T_{\mathrm{osc}}=0.77$. The contrast (21) (calculated on the interval $-1.38<X/{\lambda }_{\mathrm{osc},2}<1.38$) has a value of 1 for the CoM density distribution and is reduced to 0.71 in the single-particle density. (d)–(f) Same for $U/J=-0.2$. (e) Resulting low-fidelity quantum superposition with ${\tilde{p}}_0\approx 0.34$, ${\tilde{p}}_1\approx 0.32$, and ${\tilde{p}}_2\approx 0.34$ at $t/T_{\mathrm{osc}}=0.5$. (f) Interference pattern at $t/T_{\mathrm{osc}}=0.77$ with contrast $C=0.83$ for the CoM density and $C=0.34$ for the single-particle density.

Figure 7

Same setup as in Fig. 6. Occupation probabilities ${\tilde{p}}_n$ as defined below Eq. (33) vs dimensionless time $t/T_{\mathrm{osc}}$ for (a) $U/J=-1$ and (c) $U/J=-0.2$. Dash-dotted black line: ${\tilde{p}}_2$. Solid blue line: ${\tilde{p}}_1$. Dashed red line: ${\tilde{p}}_0$. (b) Interference pattern in CoM density ${\rho }_{\mathrm{CoM}}\left(X\right)$ for $U/J=-1$ at $t/T_{\mathrm{osc}}\approx 0.77$ with contrast $C_{\mathrm{CoM}}=1.0$. There is no contribution from $|1,1\rangle$. (d) Interference pattern in CoM density (solid black line) with $C_{\mathrm{CoM}}=0.83$ and CoM density minus contribution from $|1,1\rangle$ (dashed red line) with $C_{\mathrm{CoM},\mathrm{corr}}=0.90$ at $t/T_{\mathrm{osc}}=0.77$ for $U/J=-0.2$.