Three-Body Recombination of Ultracold Microwave-Shielded Polar Molecules

Abstract

A combined experimental and theoretical study is carried out on the three-body recombination process in a gas of microwave-shielded polar molecules. For ground-state polar molecules dressed with a strong microwave field, field-linked bound states can appear in the intermolecular potential. We model three-body recombination into such bound states using classical trajectory calculations. Our results show that recombination can explain the enhanced loss rates observed at small microwave detunings in trapped samples of bosonic NaCs [Bigagli, $\textit{et al.}$, Nat. Phys. $\textbf{19}$ 1579-1584 (2023)]. Specifically, our calculations reproduce the experimentally measured three-body loss rates across a wide range of microwave Rabi couplings, detunings, and temperatures. This work suggests that for bosonic shielded molecular systems in which the two-body loss is sufficiently suppressed and a field-linked bound state is present, the dominant loss process will be three-body recombination.

Figures (3)

• Figure 1: Semi-classical three-body loss in microwave-shielded molecules. (a) Shielded NaCs molecules are observed to undergo three-body loss when held in an optical dipole trap. (b) Partial energy level diagram for microwave shielding. Molecules are dressed by a $\sigma^+$ field with Rabi coupling $\Omega$ and detuning $\Delta$. Here, $J$ is the total angular momentum of the molecule and $m_J$ is its projection onto the quantization axis defined by the 864 G magnetic field. (c) Calculated potential energy curves for the collision fo two microwave-shielded NaCs molecules for $\Omega / (2 \pi) = 4$ MHz and $\Delta / \Omega = 0.25$. The blue line shows the $L = 0$, $\ket{++}$ shielded potential. Potentials in grey either correspond to higher orbital angular momentum or unshielded states. (d) $L = 0$ adiabatic potential energy curves on the kHz scale for different $\Omega$ and fixed $\Delta / \Omega = 0.25$. The horizontal lines show the bound states in the potentials. The calculation is performed for $\chi = 3(1)$ degree ellipticity of the microwave field (illustrated in inset). (e) Measured three-body loss of molecules in an optical dipole trap for a sample that starts with $8(1) \times 10^3$ molecules at $T = 350(50)$ nK. Molecule number is recorded as a function of hold time and fit to a kinetic model, see Supplemental Material SI, to extract the three-body loss rate coefficient, $L_{\rm 3B}$. Data is shown for $\Omega / (2 \pi) = 8$ MHz and $\Delta / \Omega = 0.25$ MHz. The error bars show the standard error of the mean for two repetitions of the experiment.
• Figure 2: Classical trajectory methods applied to microwave-shielded NaCs molecules. (a) Example of a recombination trajectory for $E=200\,\textrm{nK}$ and $b=0\,a_0$. As a result of the collision, molecules $2$ and $3$ recombine into a bound state and molecule $1$ flies out with excess energy. (b) Calculated opacity function. (right panel) $\Tilde{\mathcal{P}}(b,E,T=300~{\rm nK})\{8,0\}$ as a function of impact parameter, $b$, for a range of collisional energies. There is a maximum impact parameter for a given energy beyond which the opacity function goes to zero. (left panel) $\Tilde{\mathcal{P}}(b, E=500~{\rm nK}, T = 300~{\rm nK})\{8,\Delta\}$ as a function of $b$, for $\Delta$ ranging from 0 to 16 MHz. (c) Calculated thermal dissociation of tetramers. The orange histogram shows all trajectories resulting in bound molecules, plotted as a function of their binding energy. The green histogram shows the relevant trajectories after accounting for thermal dissociation. Calculation is done for $\Omega = 2 \pi \times 8$ MHz and $\Delta/\Omega = 1$ at a collision energy of $k_{\rm B} \times 400$ nK and a sample temperature of $300$ nK. The black dashed line marks the binding energy of the quantum bound state.
• Figure 3: Comparison of measured three-body loss rates and calculated recombination rates. (a) $L_{\rm 3B}$ as a function of $\Delta/\Omega$ for $\Omega / (2 \pi) = 4$, 8, 14, and 31 MHz. Circles are the experimental values with error bars representing the error from the fit. Blue-shaded bands are the result of the trajectory calculation. The width of the bands reflects the experimental uncertainty of the ellipticity, 3(1) degrees. Vertical dashed lines indicate the appearance of a bound state in the collisional potential, grey numbers denote the number of bound states in the potential adiabatic to $L=0$. Solid grey lines are fits to $L_{\rm 3B} = a d_{\rm eff}^x$, yielding $x = 6.8(1.3), \ 5.2(0.3), \ 4.2(1.1), \ 6.3(1.0)$ for $\Omega / (2 \pi) = 4$, 8, 14, and 31 MHz respectively. (b) $L_{\rm 3B}$ as a function of temperature for $\Omega = 2 \pi \times 8$ MHz and $\Delta/\Omega = 1$. Grey solid line shows a fit to $L_{\rm 3B} = a T^y$ yielding $y = -1.7(2)$.