Resonant and first-order dipolar interactions between ultracold ${}^1\mathrm{\Sigma }$ molecules in static and microwave electric fields

We theoretically study collisions between ultracold polar ${}^1\mathrm{\Sigma }$ molecules that are polarized by microwave or static electric fields. We systematically study the dependence on field strength, microwave polarization, and detuning from rotational transitions. We calculate the loss in two-body collisions that is observable experimentally and compare with the results expected for purely first-order dipolar interactions. For ground-state molecules polarized by a static electric field, the dynamics are accurately described by first-order dipolar interactions. For microwave dressing, instead, resonant dipolar collisions dominate the collision process, in which molecules reorient along the intermolecular axis and interact with the full strength of the transition dipole. For red detuning, reorientation can only be suppressed at extreme Rabi frequencies. For blue detuned microwaves, resonant dipolar interactions dominate even for the highest Rabi frequencies, leading to microwave shielding for circular polarization and structured losses due to resonances for linear polarization. The results are presented numerically for fermionic ${}^{23}\mathrm{Na}{}^{40}\mathrm{K}$ and bosonic ${}^{23}\mathrm{Na}{}^{39}\mathrm{K}$ molecules.

Figure 1

Illustration of the competition of first-order and resonant dipolar interactions explored in this work. Dashed lines indicate diabatic field-dressed interaction potentials, which cross at the Condon point marked $R_c$, whereas the solid lines indicate adiabatic potentials for which the crossing is avoided due to the Rabi coupling. Panel (a) sketches the interaction potentials for a pair of resonantly interacting states occurring above the initial state (in green, indicated by an arrow), as is realizable by dressing with microwaves red detuned from a rotational transition. In the illustration, the attractive branch of the resonant dipolar interaction [in blue (dark gray)] crosses the ground-state interaction potential outside the range of first-order dipolar interactions, where coupling between these potentials can transform the dynamics qualitatively. Panel (b) shows dressing with blue detuning gives access to resonantly interacting states below the initial state, such that the repulsive branch of the resonant dipolar interaction [in orange (light gray)] crosses the initial potential. Repulsive resonant dipolar interactions can reduce short-range loss rates by orders of magnitude even if the interactions become repulsive only at shorter range, as shown in panel (c). The combination of long-range first-order dipolar interactions and repulsive resonant dipolar interactions at shorter range can support quasibound states and lead to scattering resonances.

Figure 2

Level schematic for molecules dressed with microwaves with $\tilde{\mathrm{\Omega }}=2\pi \times 1\mathrm{MHz}$. (a) Single-molecule energy levels for red and blue detuning. (b) Potential curves for resonant dressing with fixed orientation of the intermolecular axis at $\theta ={90}^{\circ }$ relative to the laboratory $z$ axis, for ${\sigma }^{+}$ circular polarization. Adiabatic potential curves are shown as solid lines, the first-order dipolar interactions [Eq. (12)] are shown as dashed lines, and resonant dipole-dipole interactions [Eq. (13)] are shown as dotted lines. Thresholds used for “red detuning” and “blue detuning” are highlighted in color, i.e., the states corresponding to both molecules prepared in the lower or upper-field dressed state, respectively. The detuning $\delta =0$ in both cases.

Figure 3

Potential-energy curves for off-resonant microwave dressing, $\tilde{\mathrm{\Omega }}=2\pi \times 1\mathrm{MHz}$ and $\left|\delta \right|/\mathrm{\Omega }=3$, and at fixed $\theta ={90}^{\circ }$, under the following conditions (a) blue detuning, circular polarization, (b) red detuning, circular polarization, (c) blue detuning, linear polarization, (d) red detuning, linear polarization. The correspondence to molecules in the upper and lower field dressed level is indicated on the right. For blue detuning, the initial state corresponds to two molecules in the upper field-dressed level, $|+\rangle$, and for far off-resonant dressing the lower field-dressed level, $|-\rangle$, is nearly degenerate with the spectator states, $|0\rangle$. For red detuning the initial state corresponds to the lower field-dressed level, $|-\rangle$, and in the case of off-resonant dressing $|+\rangle$ and $|0\rangle$ are nearly degenerate.

Figure 4

Two-body loss rate coefficients for collisions of (a) bosonic and (b) fermionic NaK molecules polarized by a static electric field (lines). Dashed lines show the contributions of different $M_L$. Loss rates for the space-fixed dipole approximation, Eq. (9), are shown as markers. Results shown in black are shown for the physical dipole-dipole interaction, whereas results shown in lighter gray have been obtained with the sign of the interaction reversed.

Figure 5

Two-body loss rate coefficients for NaK molecules polarized by microwave fields with (a), (d) $\tilde{\mathrm{\Omega }}=2\pi \times 1\mathrm{kHz}$, (b), (e) $\tilde{\mathrm{\Omega }}=2\pi \times 1\mathrm{MHz}$, (c), (f) $\tilde{\mathrm{\Omega }}=2\pi \times 100\mathrm{MHz}$. Panels (a)–(c) correspond to fermionic molecules and (d)–(f) to bosonic molecules. Red and pink lines show results at red detuning for linear and circular polarization, respectively. Blue and light-blue lines show results at blue detuning for linear and circular polarization, respectively. For comparison, this graph includes the static field loss rates for the physical (sign-reversed) dipole-dipole interaction in black (gray). For clarity, some of the nearly overlapping lines are instead rendered as markers adhering to the same color coding. In gray scale, results for red detuning are rendered as markers in panels (a), (c), (d), and (f) and the top two lines in panels (b) and (e), whereas blue detunings are rendered as the corresponding lines in panels (a) and (c), the lowest two lines in panels (b) and (e), and the structured line in panels (c) and (f), and in both cases linear and circular polarization are rendered as dark and light gray, respectively. Sharp structure in the observable loss rate stems from stable scattering resonances supported by the long-range interactions, protected from universal short-range loss by repulsive resonant dipolar interactions, see Fig. 1. In panels (c) and (f), the dotted blue line shows the same loss rate averaged over 1 dB uncertainty in the microwave power.

Figure 6

Two-body loss rate coefficients for fermionic and bosonic molecules for $\tilde{\mathrm{\Omega }}=2\pi \times 1\mathrm{kHz}$, from c.c. calculations. At this Rabi frequency, $\hslash \mathrm{\Omega }\ll k_{B}T$ at 500 nK, such that the interaction is determined solely by the resonant dipole-dipole interactions, not the space-fixed induced dipole moment. Hence, loss curves for all polarizations and signs of the detuning collapse onto one another as a function of $\delta /\mathrm{\Omega }$, and the width of the plotted line indicates the variation of the loss rate with sign of the detuning and microwave polarization. For fermionic molecules, the Condon approximation is also shown, where the width indicates the effect of the difference in Condon point for red and blue detuning. The Condon approximation is accurate for large detuning, but misses the background universal loss rate which causes an offset for bosonic molecules.

Figure 7

Two-body loss rate coefficients for fermionic NaK molecules polarized by microwave fields with $\tilde{\mathrm{\Omega }}=2\pi \times 1\mathrm{MHz}$, on a doubly logarithmic scale. The data shown are identical to that in Fig. 5, but the logarithmic scale emphasizes off-resonant dressing, which results in small induced dipole moments in individual molecules, yet causes resonant loss an order of magnitude faster than in the static case. Results for red and blue detuning are shown in corresponding colors (top two and bottom two lines, respectively), whereas linear and circular polarization are rendered as dark and light colors, respectively. For comparison, this graph includes the static field loss rates for the physical (sign-reversed) dipole-dipole interaction in black (gray).

Figure 8

Two-body loss rate coefficients for fermionic NaK molecules induced by microwave dressing with $\tilde{\mathrm{\Omega }}=2\pi \times 1\mathrm{MHz}$ and (a) red detuning and $\pi$ polarization, (b) red detuning and ${\sigma }^{+}$ polarization, (c) blue detuning and $\pi$ polarization, and (d) blue detuning and ${\sigma }^{+}$ polarization. Losses are decomposed into contributions of $M_L=0$ and $\pm 1$ (lines), and these into short-range and inelastic losses, shown as correspondingly colored markers.

Figure 9

Two-body loss rate coefficients for bosonic NaK molecules induced by microwave dressing with $\tilde{\mathrm{\Omega }}=1\mathrm{MHz}$ and (a) red detuning and $\pi$ polarization, (b) red detuning and ${\sigma }^{+}$ polarization, (c) blue detuning and $\pi$ polarization, and (d) blue detuning and ${\sigma }^{+}$ polarization. Losses are decomposed into contributions of $M_L=0, \pm 1$, and $\pm 2$ (lines), and these into short-range and inelastic losses, shown as correspondingly colored markers.