Rotational-state dependence of interactions between polar molecules

The long-range electrostatic interactions between molecules depend strongly on their relative orientation, which manifests as a rotational-state dependence. Interactions between molecules in the same rotational quantum state are well-known attractive rotational van der Waals interactions. Interactions in rotational states that differ by one quantum show resonant dipole-dipole interactions. We show that where molecules are in rotational states that differ by more than one quantum, they exhibit repulsive van der Waals interactions. At temperatures below a millikelvin, this effect can reduce collisional loss by multiple orders of magnitude. These repulsive interactions lead to applications in quantum simulation and impurity physics with ultracold polar molecules.

Figure 1

Schematic representation of rotational energy levels for (a) two molecules in different rotational states (here, $j=1$ and $j=3$) and (b) the combined pair (here, $j+j^{\prime }=1+3$). Different colors represent the various ways of virtually (de-)exciting the molecules via dipole-dipole interactions, creating strong second-order coupling to a state barely below the original state.

Figure 2

Rotational van der Waals $C_6^{\mathrm{rot}}$ coefficients on a logarithmic scale for pairs of bosonic molecules in various rotational states $j_A$ and $j_B$. For each pair, an average is taken over the various ways of coupling both together.

Figure 3

Interaction potential curves for bosonic (a) NaK $0+0$, (b) NaK $0+1$, (c) NaK $0+2$, (d) NaK $1+3$, (e) NaK $1+4$, and (f) KCs $0+2$, with dimer rotational states labeled as $j+j^{\prime }$ where $m_j=m_{j^{\prime }}=0$. All potentials are referenced to the energy of the initial state (solid blue lines) at infinite separation. Due to strong first-order interactions for $0+1$, the energy is scaled by 100 in panel (b). The pure second-order rotational interaction curve is given by the dashed light-blue curve. Addition of an isotropic electronic $C_6^{\mathrm{elec}}$ interaction is given for (c) and (f) by the dash-dotted green curve. Addition of a quadrupole moment for dipole-quadrupole and quadrupole-quadrupole interaction is represent in [(c)–(f)] by the dotted red curve.

Figure 4

Loss rates for various rotational-state pairs $j+j^{\prime }$ with $m_j=m_{j^{\prime }}=0$ for (a) bosonic NaK, (b) bosonic KCs, (c) fermionic NaK, and (d) fermionic KCs. Panel (a) contains loss rates where only the dipole-dipole interaction is taken into account (dashed lines). The full calculations further include contributions from dipole-quadrupole, quadrupole-quadrupole, isotropic electronic $C_6^{\mathrm{elec}}$, and the centrifugal distortion constant.

Figure 5

Spread of loss rates for rotational-state pairs $1+j^{\prime }$ for different $m_j$ and $m_{j^{\prime }}$ for bosonic NaK. The green curve corresponds to the dashed green curve in Fig. 4 and accounts only for the dipole-dipole interaction.

Figure 6

Scaling of loss rates for $j+j^{\prime }=0+2$ of (a) bosonic NaK and (b) bosonic KCs with respect to mass, rotational constant, dipole moment, and quadrupole moment. These results also include an isotropic $C_6^{\mathrm{elec}}$ contribution.

Figure 7

The influence of a static electric field for bosonic and fermionic NaK with $j+j^{\prime }=0+2$ (denoted at zero field) on (a) the loss rate and (b) the induced dipole moment. The universal loss is given for bosonic NaK.

Figure 8

The influence of linearly polarized microwaves for bosonic NaK with rotational $j+j^{\prime }=0+2$ states (denoted at zero field) on (a) the loss rate and (b) the induced dipole moment, both as a function of microwave detuning $\mathrm{\Delta }$ at Rabi frequency $\mathrm{\Omega }=2\pi \times 10$ kHz.

Figure 9

Rotational dimer degeneracy $j+j^{\prime }=2+2\equiv 0+3$ with $m_j=m_{j^{\prime }}=0$ for bosonic NaK. (a),(b) Interaction potentials with centrifugal distortion coefficient taken into account, lifting the degeneracy by $\sim 80\mu \mathrm{K}$, referenced to the prepared states at infinite separation, which are (a) $2+2$ and (b) $0+3$. The main difference is the parity ${(-1)}^{j+j^{\prime }+L}$, which is opposite between these figures. (c),(d) Scaling of the loss rates with respect to the dipole moment, quadrupole moment, and centrifugal distortion constant for the (c) $2+2$ and (d) $0+3$ states.

Figure 10

Collision rates for thermalization of fermions for various rotational-state pairs of NaK as a function of temperature.