Ultracold bosonic scattering dynamics off a repulsive barrier: Coherence loss at the dimensional crossover

We explore the impact of dimensionality on the scattering of a small bosonic ensemble in an elongated harmonic trap off a centered repulsive barrier, thereby taking particle correlations into account. The loss of coherence as well as the oscillation of the center of mass are studied and we analyze the influence of both particle and spatial correlations. Two different mechanisms of coherence losses in dependence of the aspect ratio are found. For small aspect ratios, loss of coherence between the region close to the barrier and outer regions occurs, due to spatial correlations, and for large aspect ratios, incoherence between the two density fragments of the left and right side of the barrier arises due to particle correlations. Apart from the decay of the center-of-mass motion induced by the reflection and transmission, further effects due to the particle and spatial correlations are explored. For tight transversal traps, the amplitude of the center-of-mass oscillation experiences a weaker damping, which can be traced back to the population of a second natural orbital, and for a weaker transversal confinement, we detect a strong decay due to the possibility of transferring energy to transversal excited modes. These effects are enhanced if the aspect ratio is integer valued.

Figure 1

Time evolution of the numerically obtained one-body density (first row) for $N=5$ interacting atoms with $h_{\text{1D}}=0.1293$, $\sigma =0.1$, and barrier $H_{\text{1D}}=0.4$, $S=0.2$. The color scales are normalized with respect to the maximal value of the density. (b)–(d) The numerically exact density profile (green line) is compared with the approximation (10) (blue line) for three instants in time. In panel (e), the corresponding expectation value of the CM is compared with Eq. (9). Vertical dashed lines mark the three instants in time used in panels (b)–(d).

Figure 2

Shown is $v_{12}\left(t\right)$ (gray solid line), $G\left(t\right)$ (black solid line), and $\tilde{G}$ (black dashed line) versus time $t$, with $c=0.01$. $c$ and $v_{12}$ are obtained by using the parameters $\mathrm{\Omega }=1$, $H_{\text{1D}}=0.38$, $S=0.2$, and $b=3$.

Figure 3

Temporal evolution of the density both within (a) the one-dimensional GPE and (b) when taking particle correlations into account, for $N=5$ bosons in a harmonic trap with barrier $H_{\text{1D}}=1.54$, $S=0.2$ and interaction $h_{\text{1D}}=2.01$, $\sigma =0.1$. Panels (c) and (d) show the first and second natural orbitals ${\alpha }_1(z,t)$ and ${\alpha }_2(z,t)$ weighted with their natural populations $a_1$ and $a_2$, which are shown as white solid lines. The color scales are normalized with respect to the maximal value of the density.

Figure 4

Panel (a) shows the CM oscillation and panel (b) shows the probability to find two bosons on the same side of the trap; see Eq. (12). The same parameters as in Fig. 3 are used. For both figures, the dark blue line represents the mean-field result (3) and the green light line denotes a ML-MCTDHB result, where particle correlations are taken into account. The magenta line represents the expectation value of ${\delta }_{\mathrm{BMF}}=\langle z\rangle$ evaluated with respect to the second natural orbital and weighted with its population.

Figure 5

Absolute value of the first-order correlation function for three different times $t=\{9\pi ,9.5\pi ,10\pi \}$. The black lines indicate the density profiles at the corresponding time instants and the dashed lines mark the position of the barrier. Areas of density smaller than ${10}^{-6}$ are colored in black.

Figure 6

Panels (a) and (b) show the longitudinal and transversal density profiles, respectively. Parameters are barrier height $H=5$, width $S=0.2$, displacement $b=3$, and aspect ratio $\eta =2$. The color scales are normalized with respect to the respective maximal value of the density.

Figure 7

(a) Populations of $d_{0,0}$ (dark blue), $d_2={\sum }_{m=0}^{\infty }{d}_{2,m}$ (red), $d_0={\sum }_{m=1}^{\infty }{d}_{0,m}$ (light green) for the aspect ratio $\eta =3$ (solid lines) and for $\eta =2.5$ (dashed lines). (b) Normalized transversal energy $E_{\perp }\left(t\right)/E_{\perp }\left(0\right)$ for the integer aspect ratio $\eta =3$ (dark blue solid line) and the half-integer aspect ratio (light green solid line) for the first oscillation periods, continued in panel (c) for longer times. Black dotted lines mark times, when the atom scatters with the barrier. All other parameters are as in Fig. 6.

Figure 8

(a) The oscillation of the CM for $\eta =2$ (dark blue line), $\eta =2.5$ (light green line), and $\eta =3$ (magenta dashed line). Panel (b) depicts the fitted decay coefficient $c$ of model (9) for various aspect ratios. All other parameters coincide with those of Fig. 6.

Figure 9

First-order correlation function $|g_1(z,z^{\prime })|$ for different aspect ratios $\eta$ at the time instant $t=9\pi$. Areas of density smaller than ${10}^{-6}$ are colored in black. The black-white dashed line marks the position of the barrier and the black solid line is the longitudinal density profile. Same physical parameters are used as in Fig. 6.

Figure 10

Time-averaged depletions (for a definition see text) for integer (half integer) aspect ratios in the main figure (inset), measuring particle ($D^{\left(\text{3D}\right)}$, dark blue), spatial transversal ($D^{(s=1,2)}$, light green), and longitudinal ($D^{(s=3)}$, magenta) correlations. Circles denote MF simulations, and stars represent BMF simulations. Physical parameters are $N=5$, $H=9$, $S=0.2$, $h=91.125$, and $\sigma =0.1$. Data points are connected by a line to guide the eye.

Figure 11

(a) The CM dynamics is shown for the different aspect ratios $\eta =2,2.5,3$. Panel (b) shows the fitted decay constant $c$ of the model (9) as a function of $\eta$. The parameters are the same as in Fig. 10.

Figure 12

First-order correlation function $|g_1(z,z^{\prime })|$ for different aspect ratios $\eta$ at the time instant $t=9\pi$. Areas of density smaller than ${10}^{-6}$ are colored in black. The black-white dashed line marks the position of the barrier and the black solid line is the longitudinal density profile. The physical parameters are the same as used in Fig. 10.