Conical Intersections in Laboratory Coordinates with Ultracold Molecules

For two states of opposite parity that cross as a function of an external magnetic field, the addition of an electric field will break the symmetry and induce an avoided crossing. A suitable arrangement of fields may be used to create a conical intersection as a function of external spatial coordinates. We consider the effect of the resulting geometric phase for ultracold polar molecules. For a Bose-Einstein condensate in the mean-field approximation, the geometric phase effect induces stable states of persistent superfluid flow that are characterized by half-integer quantized angular momentum.

Figure 1

Energy levels of KRb as a function of magnetic field $B$. The upper panel shows an overview, with each zero-field level [labeled by $(N,J)$] splitting into $2J+1$ components as a function of magnetic field. The lower panel shows how two levels of different parity cross ($M_J=0$ and $+1$) in the absence of an electric field (solid lines) but avoid one another in the presence of a $5\mathrm{kV}/\mathrm{cm}$ electric field (dashed lines).

Figure 2

Toroidal potentials formed around conical intersections. Left-hand panels: Potential formed when the optical trapping potential (7 nK at $30\mu \mathrm{m}$) is centered at the point of intersection ($\rho =0$), with field gradients $5\mathrm{G}/\mathrm{cm}$ $\widehat{\mathbf{X}}$ and $6.8\mathrm{kV}/{\mathrm{cm}}^2$ $\widehat{\mathbf{Y}}$. Right-hand panels: Potential formed when the trapping potential is offset along $\widehat{\mathbf{X}}$ by $x_0=0.0715\mu \mathrm{m}$, with field gradients $5\mathrm{G}/\mathrm{cm}$ $\widehat{\mathbf{X}}$ and $6.723\mathrm{kV}/{\mathrm{cm}}^2$ $\widehat{\mathbf{Y}}$. The electric field gradient is chosen to minimize the anisotropy in each case. The well depths are given in nK relative to the point of intersection. The lower panels show schematic cuts through the potential at $Y=0$: KRb eigenstates (red dashed lines), optical trapping potential (blue dot-dashed lines), and the resultant toroidal potential (black solid lines).