Stable heteronuclear few-atom bound states in mixed dimensions

We study few-body problems in mixed dimensions where two or three heavy atoms are trapped individually in parallel one-dimensional tubes or two-dimensional disks and a single light atom travels freely in three dimensions. Using the Born-Oppenheimer approximation, we find three- and four-body bound states for a broad parameter region. Specifically, the existence of trimer and tetramer states persists to the negative scattering length regime, where no two-body bound state is present. As pointed out by Y. Nishida in an earlier work [Phys. Rev. A 82, 011605(R) (2010)], these few-body bound states are stable against three-body recombination due to geometric separation. In addition, we find that the binding energy of the ground trimer and tetramer state reaches its maximum value when the scattering lengths are comparable to the separation between the low-dimensional traps.

Figure 1

(a) A 1D-1D-3D system with two heavy atoms $A_1$ and $A_2$ confined in two 1D tubes and a light atom $B$ moving freely in 3D. (b) A 2D-2D-3D system with two heavy atoms $A_1$ and $A_2$ confined in two 2D planes and a freely moving light atom $B$.

Figure 2

Regularized effective potential $U(a,z_{12})$ between two heavy atoms $A_{1,2}$ with scattering lengths $a_1=a_2=a=0.2$ [solid (red) line with open circles], 1 [solid (blue) line)], and $\infty$ [dashed (black) line]. We also plot the effective potential $V_{\mathrm{eff}}\left(z_{12}\right)$ for $a_1=a_2=-5$ [dashed-dotted (green) line]. The system of natural units $\hslash =m_B=L=1$ is used throughout this paper.

Figure 3

Depth $D\left(a\right)$ of the regularized part $U(a,z_{12})$ of the effective interaction in the case of $a_1=a_2=a>0$. Note that $D\left(a\right)$ takes a maximum value at $a=1$; i.e., the effective interaction is enhanced when the two-body scattering length is close to the intertube distance.

Figure 4

Binding energy of the ground three-body bound state in a 1D-1D-3D system with a reduced mass of the heavy atoms, $m_{*}=3.33$ and $9.5$. These values correspond to the cases of ($A_1=A_2{=}^{40}$K, $B{=}^6$Li) and ($A_1=A_2{=}^{133}$Cs, $B{=}^7$Li), respectively.

Figure 5

Binding energy $E_{3\mathrm{b}}$ of the ground trimer state as a function of $1/a$ with $a_1=a_2=a$. Reduced masses used in this plot are $m_{*}=3.33$ [solid (black) line], $9.5$ [dashed (blue) line], and $\infty$ [solid (red) line with open circles], respectively.

Figure 6

Binding energy $E_{3\mathrm{b}}$ of the ground trimer state as a function of the reduced mass $m_{*}$ with $a_1=a_2=a=1$ [solid (blue) line with open triangle] and $a_1=a_2=a=\infty$.

Figure 7

Binding energy of the ground three-body bound state in a 1D-1D-3D system with reduced mass $m_{*}=3.33$ and scattering lengths $a_1=a_2=a$. Here, we plot the results given by the BOA [solid (red) line with open circles] and by the exact solution of the Schrödinger equation [33] [solid (blue) line]. Note that the BOA can give reliable results provided that the binding energy of the trimer state is away from the threshold.

Figure 8

Binding energy of the ground three-body bound state in the 2D-2D-3D geometry with reduced masses of the heavy atoms, $m_{*}=3.33$ and $9.5$.

Figure 9

Binding energy of the ground three-body bound state in 2D-2D-3D geometry with reduced mass $m_{*}=3.33$ and scattering lengths $a_1=a_2=a$. Here, we plot the results given by the BOA [solid (red) line with open circles] and the exact solution of the Schrödinger equation [33] [solid (blue) line] and find good agreement, provided that the binding energy is away from the threshold.

Figure 10

A 1D-1D-1D-3D system with the three heavy atoms $A_1,A_2$, and $A_3$ confined in three 1D tubes and the light atom $B$ moving freely in 3D.

Figure 11

Regularized effective potential $U(a;X,Y)$ for two-body scattering length $a_1=a_2=a_3=a=1$ in a 1D-1D-1D-3D system with an equilateral triangle configuration.

Figure 12

Binding energy $E_{4\mathrm{b}}$ of the ground four-body bound state in a 1D-1D-1D-3D system with an equilateral triangle configuration for different values of $a$ with reduced mass $m_{*}=9.5$ [solid (blue) line with circles] and 3.33 [solid (green) line with triangles].

Figure 13

Binding energy $E_{4\mathrm{b}}$ of the ground four-body bound state in a 1D-1D-1D-3D system. Here we take the scattering lengths $a_1=a_2=a_3=1$ and the intertube distances $L=1$ and $L_{1,2}$ defined as in Fig. 10.