Microwave shielding of ultracold polar molecules on the transition $\boldsymbol{n=1 \rightarrow 2}$

Technical Summary: Microwave Shielding of Ultracold Polar Molecules on the Transition $n = 1 \to 2$

Problem Statement
The production of stable ultracold molecular gases is hindered by strong collisional losses occurring at short range ($R \lesssim 100$ bohr). These losses stem from inelastic collisions, three-body recombination, photoexcitation of collision complexes, and chemical reactions. While static electric fields can suppress these losses in two-dimensional geometries by inducing long-range repulsion, they are ineffective in three-dimensional systems where molecules approach from arbitrary directions. Microwave shielding offers a solution by engineering long-range repulsive barriers through near-degeneracies between molecular pair states. However, a significant drawback of microwave shielding on the standard rotational transition $n = 0 \to 1$ is the formation of two-molecule bound states. These bound states enhance three-body recombination losses, often preventing evaporative cooling to quantum degeneracy. To mitigate this, previous approaches have required "double-field" microwave shielding, utilizing a second field of different polarization to eliminate bound states.

Methodology
The authors employ a coupled-channel formalism to analyze microwave shielding on the rotational transition $n = 1 \to 2$ for strongly dipolar molecules (specifically NaRb, NaCs, and KAg). The theoretical framework includes:

• Hamiltonian Formulation: The interaction between the molecule and a blue-detuned, circularly polarized ($\sigma^-$) microwave field is described. The system is modeled using dressed states defined by the total angular momentum projection $m_{tot} = m_n - N_{ph}$.
• Scattering Calculations: The two-body collision problem is solved using the MOLSCAT package with a fully absorbing boundary condition at short range to model loss. The calculations account for the dipole-dipole interaction as the dominant long-range force, neglecting higher-order multipole terms.
• Universality: The authors utilize reduced units of length and energy to demonstrate that the collision properties are universal for different molecules when expressed in terms of the Rabi frequency ($\Omega$) and detuning ($\Delta$).
• Basis Sets: For the $n=1 \to 2$ transition, the universal basis set involves 36 near-resonant pair states, compared to 10 for the $n=0 \to 1$ transition. Calculations include end-over-end angular momentum up to $L_{max} = 12$.

Key Contributions and Results

1. Effectiveness of $n=1 \to 2$ Shielding: The study demonstrates that microwave shielding on the $n=1 \to 2$ transition is effective in suppressing destructive collisions. The loss rate coefficients can be reduced to levels comparable to those achieved in recent experiments that reached molecular Bose-Einstein condensation (BEC) for NaCs and NaRb.
2. Optimal Initial State: Shielding is most effective when molecules collide in the dressed state with $m_{tot} = 1$, corresponding to a predominant character of $(n, m_n) = (1, 1)$. In this state, the energy separation to lower-lying thresholds is maximized, making nonadiabatic transitions (the primary loss mechanism) slower compared to $m_{tot} = 0$ or $-1$.
3. Absence of Two-Molecule Bound States: A critical finding is that, under the conditions required for effective shielding, the $n=1 \to 2$ transition does not support two-molecule bound states. This is attributed to the significantly weaker long-range attraction compared to the $n=0 \to 1$ transition. The effective dipole moment ($d_{eff}$) for the $m_{tot}=1$ state in the $n=1 \to 2$ transition is a factor of $\sqrt{5}$ smaller than for the $n=0 \to 1$ transition. Consequently, the dipole-dipole forces are 5 times weaker, resulting in a much shallower shielding well that cannot host bound states at experimentally accessible Rabi frequencies (up to $\sim 50$ MHz for NaCs).
4. Elimination of Double-Field Requirement: Because the $n=1 \to 2$ transition naturally avoids the formation of two-molecule bound states, the complex requirement for double-field microwave shielding (using a second field of different polarization to cancel bound states) is obviated. A single circularly polarized microwave field is sufficient.
5. Dependence on Detuning: The loss rate exhibits a strong dependence on the ratio $\Delta/\Omega$. A peak in loss occurs near $\Delta/\Omega \approx 0.35$ due to the crossing of the incoming threshold with a dark state channel. Optimal shielding (minimum loss) is found just above this peak, at $\Delta/\Omega \approx 0.70$.

Significance and Claims
The paper claims that microwave shielding on the $n=1 \to 2$ transition offers a streamlined and effective route to stabilizing ultracold polar molecular gases. Its primary significance lies in the ability to suppress collisional losses without generating the two-molecule bound states that typically plague single-field shielding schemes. By avoiding these bound states, the method prevents the enhancement of three-body recombination, thereby removing the necessity for the more complex double-field shielding protocols previously required to achieve quantum degeneracy. The authors assert that this approach provides a versatile platform for exploring many-body physics with shielded polar molecules, offering collision properties similar to those used in recent BEC achievements but with a simpler experimental setup regarding field configurations.