Trap-induced shape resonances in an ultracold few-body system of an atom and static impurities

Hybrid systems of ultracold atoms and trapped ions or Rydberg atoms can be useful for quantum simulation purposes. By tuning the geometric arrangement of the impurities, it is possible to mimic solid-state and molecular systems. Here, we study a single trapped atom interacting with a set of arbitrarily arranged static impurities and show that the problem admits an analytical solution. We analyze in detail the case of two impurities, finding multiple trap-induced resonances which can be used for entanglement generation. Our results serve as a building block for the studies of quantum dynamics of complex systems.

Figure 1

An interaction potential experienced by a trapped atom (red sphere) in the presence of two localized impurities (purple and green spheres), which are localized by external trapping potentials (purple and green). In the vicinity of the static particles, the trapping potential is modified by the atom-impurity interaction; the gray surface is the effective potential experienced by the atom.

Figure 2

The mechanism of the resonance and the characteristic length and energy scales in the system. If the bound states of the atom-impurity potential are at the same level as the bound state in the trap (solid line potential), there is a resonance and the avoided crossing appears (see, e.g., Fig. 3 or 8). The wave function for such state is a mixture of the trap state and the states localized on the impurities [see Figs. 7, 7, and the text for details]. Dashed line potential depicts a situation when the bound states of the atom-impurity potential are at different positions (dashed line potential) than the bound state of the trap, and the system is far from the avoided crossing.

Figure 3

The lowest-energy levels of the atom as a function of the distance between the impurities, equal to $2d$, for different values of the scattering length $a$. The dotted blue and dashed red lines denote even and odd states, respectively. The solid gray lines display the energy levels of the bound states ($E<0$) in absence of the trap.

Figure 4

The dependence of the lowest-energy levels on the distance $2d$ between the symmetrically placed impurities. The results are presented for $a=0.4l_0$ [upper panel (a)] and $a=l_0$ [lower panel (b)]. The color code of the levels is the same as in Fig. 3. Additionally, the dotted-dashed lighter blue and lighter red lines represent the energies of the even and odd bound states, respectively, obtained within variational approach.

Figure 5

Comparison of the results for $a=0.4l_0$ in the vicinity of the lowest avoided crossing obtained within the full method (dotted blue and dashed red), and within the variational approach using three states (lighter blue and lighter red). The inset zooms the vicinity of the avoided crossing. The color code is the same as in Fig. 4.

Figure 6

Comparison of energy levels in the vicinity of avoided crossing. Red dashed line and blue dotted line denote odd and even states of the system with impurities placed symmetrically along the $z$ axis in $z=\pm d$. Gray dotted line denotes the energy levels of the system, where the impurities are placed in $z=-d$ and $z=d+\mathrm{\Delta }d$. Here, $\mathrm{\Delta }d=0.025l_0$.

Figure 7

Cuts along the $z$ axis of the (renormalized) wave functions of the atom for different $d$ and $E$ with the scattering length $\mathrm{a}=0.4l_0$ close to avoided crossings. Gray vertical lines denote the positions of the ions.

Figure 8

The vicinity of the lowest avoided crossing for $a=0.4l_0$. Red dashed line and blue dotted line denote odd and even states of the system with impurities placed symmetrically along the $z$ axis in $z=\pm d$. The points marked with letters (a)–(i) indicate the parameter values for which the eigenstates are studied in Fig. 7.