Ballistic quench-induced correlation waves in ultracold gases

We investigate the wave-packet dynamics of a pair of particles that undergoes a rapid change of scattering length. The short-range interactions are modeled in the zero-range limit, where the quench is accomplished by switching the boundary condition of the wave function at vanishing particle separation. This generates a correlation wave that propagates rapidly to nonzero particle separations. We have derived universal, analytic results for this process that lead to a simple phase-space picture of the quench-induced scattering. Intuitively, the strength of the correlation wave relates to the initial contact of the system. We find that, in one spatial dimension, the $k^{-4}$ tail of the momentum distribution contains a ballistic contribution that does not originate from short-range pair correlations, and a similar conclusion can hold in other dimensionalities depending on the quench protocol. We examine the resultant quench-induced transport in an optical lattice in one dimension, and a semiclassical treatment is found to give quantitatively accurate estimates for the transport probabilities.

Figure 1

Ballistic expansion of a bound-state wave function. The black (thin) line represents the initial wave function. The blue (thick, solid) line represents the wave function at $\hslash t/2\mu a_i^2=0.008$, and the magenta (dashed) line is the wave function at $\hslash t/2\mu a_i^2=0.015$.

Figure 2

The Wigner distribution given by Eq. (5) for the ballistic expansion of a 1D bound-state wave function at (a) $t=0$ and (b) $\hslash t/2\mu a_i^2=0.4$. The dashed white line in (b) represents the formula $x=2\hslash kt/m$, which corresponds to the separation of two classical particles that start out on top of each other ($x=0$) and then fly apart with momenta $\pm \hslash k$.

Figure 3

The Wigner distribution given by Eq. (5) for a 1D bound-state wave function quenched to $a_f=2a_i$. The Wigner function is evaluated at the same time as in Fig. 2, and for the same initial scattering length $a_i$. The dashed white line represents the same classical model as shown previously.

Figure 4

Quenching a bound state to $a_f=-a_i/2$. The black (thin) line represents the initial wave function. The blue (thick, solid) line represents the wave function at $\hslash t/2\mu a_i^2=0.008$, and the magenta (dashed) line is the wave function at $\hslash t/2\mu a_i^2=0.015$. Compare with Fig. 1, which depicts a quench to $a_f=\pm \infty$.

Figure 5

Quenching a bound state from $a_i=0.2\ell$ to $a_f=-a_i$ on a single lattice site. The lattice is assumed to have a depth of $10E_R$. (a) The initial two-body probability density $|\mathrm{\Psi }(x_1,x_2){|}^2$ for a bound state on a lattice site $(t=0)$. (b) The same quantity calculated at $\hslash t/m{\ell }^2=0.01$. Note the logarithmic color scale, whose lower limit is a cutoff.

Figure 6

Lattice transport for a bound state that is quenched from $a_i=0.2\ell$ to (a) $a_f=\pm \infty$ and (b) $a_f=-a_i$. The solid blue line denotes the dynamical probability of both atoms occupying the central well of the lattice, $P_{\mathrm{CC}}\left(t\right)$; the solid cyan line is the probability of a single atom occupying the central well, $P_{\mathrm{TC}}\left(t\right)$; and the solid red line is the probability that no atoms occupy the central lattice site, $P_{\mathrm{TT}}\left(t\right)$. The horizontal dashed lines correspond to the semiclassical estimates for these probabilities.

Figure 7

Integration contour in the complex $k^{\prime }$ plane. The crosses denote the poles of the second term in Eq. (A8). The final leg of the integral is along $\arg \left[k^{\prime }\right]=-\pi /4$.