Near-threshold bound states of the dipole-dipole interaction

We study the two-body bound states of a model Hamiltonian that describes the interaction between two field-oriented dipole moments. This model has been used extensively in the many-body physics of ultracold polar molecules and magnetic atoms, but its few-body physics has been explored less fully. With a hard-wall short-range boundary condition, the dipole-dipole bound states are universal and exhibit a complicated pattern of avoided crossings between states of different character. For more realistic Lennard–Jones short-range interactions, we consider parameters representative of magnetic atoms and polar molecules. For magnetic atoms, the bound states are dominated by the Lennard–Jones potential, and the perturbative dipole-dipole interaction is suppressed by the special structure of van der Waals bound states. For polar molecules, we find a dense manifold of dipole-dipole bound states with many avoided crossings as a function of induced dipole or applied field, similar to those for hard-wall boundary conditions. This universal pattern of states may be observable spectroscopically for pairs of ultracold polar molecules.

Figure 1

(top) Low-lying adiabats ${\varepsilon }_n^{\mathrm{ad}}\left(r\right)$ of the BCT Hamiltonian (3) for even $L$ and $M_L=0$. (middle and bottom) Lowest adiabat multiplied by $r^3$ and $r^4$, respectively. Short- and long-range approximations to the lowest adiabat, $-2r^{-3}$ and $-4/15r^{-4}$, respectively, are also included.

Figure 2

The total number of states obtained from coupled-channels calculations compared with the semiclassical WKB result. The colored lines show the WKB estimates of the number of bound states supported by the individual adiabats, using the same color coding as in Fig. 1.

Figure 3

Bound states as a function of $r_{\min }^{-1/2}$, which is proportional to the dipole moment for a fixed short-range boundary condition. Bound-state energies are shown as points on an equally spaced grid of values of $r_{\min }$.

Figure 4

Bound states as a function of $r_{\min }^{-1/2}$, both from adiabatic and full coupled-channels calculations.

Figure 5

Bound states as a function of $r_{\min }^{-1/2}$, for odd $L$. Results shown in orange and green correspond to $M_L=0$ and $|M_L|=1$, respectively. Bound-state energies are shown as points on an equally spaced grid of values of $r_{\min }$.

Figure 6

Bound states as a function of $r_{\min }^{-1/2}$, for even and odd $L$ and $M_L=0$ shown in blue and orange, respectively.

Figure 7

Adiabatic potential curves, ${\varepsilon }_n^{\mathrm{ad}}\left(r\right)$, multiplied by $r^3$ for bosons ($L=0$) and fermions ($L=1$) for both $M_L=0$ and 1.

Figure 8

Bound states as a function of the Lennard–Jones well depth. States shown as solid lines are obtained with parameters that are representative of magnetic atoms. Results shown as dashed lines are obtained with the dipole moment set to zero. Blue, orange, and green lines highlight single states on each of the first, second, and third adiabats, which correspond asymptotically to $L=0$, 2, and 4, respectively. The solid and dashed lines for $L=2$ lie almost directly on top of one another.

Figure 9

Bound states, including $L$ assignment, as a function of the Lennard–Jones well depth in the absence of dipole-dipole interactions. Also included is the $s$-wave scattering length, which sets the observed period.

Figure 10

Dipole bound states with Lennard–Jones short-range interactions in the case $R_{\mathrm{dip}}\gg R_6$. (top) On the same scale as Fig. 8. (bottom) The vertical energy scale has been reduced by a factor ${10}^4$. Bound-state energies are shown as points on an equally spaced grid of values of $D_{\mathrm{e}}$.

Figure 11

Dipole bound states with Lennard–Jones short-range interactions in the case $R_{\mathrm{dip}}\gg R_6$, as a function of dipole moment for a fixed well depth. Bound-state energies are shown as points on an equally spaced grid of values of $d$.

Figure 12

Dipole bound states with Lennard–Jones short-range interactions in the case $R_{\mathrm{dip}}\gg R_6$, as a function of dipole moment for fixed well depths. Different colors correspond to different well depths between roughly 800 and $830{\mathrm{cm}}^{-1}$, which scans one cycle of scattering length. Bound-state energies are shown as points on an equally spaced grid of values of $d$.