Many-body theory of reactive loss, rethermalization, and deep-degenerate dynamics in a Fermi gas of polar molecules

The 2019 realization of a quantum-degenerate Fermi gas of fermionic potassium-rubidium ($^{40}$K$^{87}$Rb) polar molecules showed that chemical reactions are suppressed as the gas enters degeneracy, and posed four linked open questions: (1) a complete many-body theory connecting microscopically defined two-body reaction probabilities to the observed, suppressed bulk loss rates, beyond the suggestive density-fluctuation picture; (2) the degree and timescale of rethermalization of the gas when reactive loss preferentially removes low-energy particles; (3) the behavior of reaction and collision energetics well below $T/T_F\sim 0.3$, where collisions involve molecules near the Fermi surface and the mean relative collision energy departs from classical equipartition toward the quantum value $\frac{3}{4}k_B T_F$ per particle; and (4) the physics of degenerate polar molecules under strong dipole-dipole interaction in applied electric fields, including Fermi-surface deformation and novel correlated many-body phases. Later literature has achieved deep degeneracy by collisional shielding, observed Fermi-surface deformation in a microwave-shielded degenerate NaK gas, measured Pauli suppression of elastic molecular collisions, and produced two competing, mutually inconsistent theoretical accounts of the loss suppression (p-wave contact/Fermi-liquid theory versus beyond-two-body complex-molecule interactions). The common objective -- a quantitative, mechanistic, and experimentally validated theory of reactive loss and rethermalization in a degenerate molecular Fermi gas, and the exploitation of degenerate dipolar interactions for correlated phases -- remains open.

Background

Ultracold polar molecules, diatomic molecules prepared in their rovibrational ground state with a permanent electric dipole moment, are a leading platform for quantum simulation and controlled quantum chemistry because their dipole-dipole interaction is long-ranged, anisotropic, and electric-field-tunable. Reaching quantum degeneracy for molecules proved far harder than for atoms: in a reactive molecule such as fermionic $^{40}$K$^{87}$Rb, two ground-state molecules can undergo the exothermic exchange reaction $\mathrm{KRb}+\mathrm{KRb}\to\mathrm{K}_2+\mathrm{Rb}_2$ even at ultracold temperatures, so the gas is continuously lost. In 2019, De Marco, Valtolina, Matsuda, Tobias, Covey, and Ye produced a Fermi-degenerate gas of spin-polarized $^{40}$K$^{87}$Rb at $T/T_F$ as low as about 0.3 ($T_F$ is the Fermi temperature, the temperature scale below which Fermi-Dirac statistics dominate a fermionic gas) by associating molecules from an already-degenerate K-Rb atomic mixture, and observed that degeneracy suppresses the bulk chemical loss rate.

For a single-component (spin-polarized) Fermi gas, the Pauli exclusion principle forbids s-wave (zero-angular-momentum) collisions, so both elastic scattering and inelastic reactive loss proceed through p-wave (one-unit angular momentum) channels. At ultracold energies the two-body p-wave reactive rate follows the Bethe-Wigner threshold law (the universal low-energy scaling of two-body collision rates), growing linearly with temperature; the relevant microscopic input is the complex p-wave scattering volume, calculable from multi-channel quantum defect theory (MQDT), a framework that parametrizes short-range collision physics by a few quantum defects. The experiment modeled the measured density decay with a two-body rate equation (Eq. 1 of that paper) and found that the temperature-normalized loss coefficient $\beta/T$ drops sharply for $T/T_F$ below about 0.6, in qualitative agreement with the reduction of the average relative density fluctuation $\langle\delta n^2(\mathbf{r})\rangle/\langle n(\mathbf{r})\rangle$ -- a measure of the suppression of local particle-number fluctuations by Fermi statistics, related to the two-particle correlation at zero separation.

The authors were explicit that this qualitative agreement is not a theory, and four questions follow from their discussion. First, a complete theory must capture the higher-order effects causing deviations from the rate-equation fit and quantitatively connect microscopic two-body reaction probabilities to the suppressed bulk loss; in particular, the p-wave contact (the short-distance pair-correlation amplitude conjugate to the p-wave scattering volume, which fixes loss in a weakly interacting gas) and possible beyond-two-body physics involving the transient four-atom collision complex ($\mathrm{K}_2\mathrm{Rb}_2$) formed before reaction must both be accounted for. Second, reactive loss preferentially removes molecules in low-energy collisions, and the degree and timescale on which the remaining gas rethermalizes is unknown. Third, for $T/T_F$ well below 0.3, collisions are dominated by molecules near the Fermi surface (the boundary of occupied momenta in momentum space), and the mean relative collision energy is expected to leave the classical equipartition value $3k_B T/2$ per particle pair and approach the quantum value $\frac{3}{4}k_B T_F$ per particle, so the loss energetics require new theoretical and experimental description. Fourth, applying electric fields to degenerate polar molecules turns on strong dipole-dipole interactions, expected to produce interaction-induced deformation of the Fermi surface (observed earlier only in dipolar atomic gases such as erbium) and eventually novel correlated many-body phases such as topological p-wave superfluidity, a superfluid phase of identical fermions paired by an attractive p-wave interaction that supports topologically protected edge states.

Problem Statement

For a single-component (spin-polarized) degenerate Fermi gas of ground-state $^{40}$K$^{87}$Rb molecules (and, where shielding or other species are involved, degenerate molecular Fermi gases generally), quantify and explain, from microscopic two-body physics, the reactive loss and dynamics in the quantum-degenerate regime, and exploit the resulting stability for dipolar many-body physics. Concretely, all four of the following questions posed by the source experiment are required and independent, so that resolving one leaves the others open: (1) Construct a complete many-body theoretical treatment of chemical reactions inside the degenerate gas that quantitatively connects microscopically defined two-body reaction probabilities (e.g., the complex p-wave scattering volume, the p-wave contact, and intermediate four-body reaction complexes) to the observed, suppressed bulk loss rates, resolving whether the suppression of the temperature-normalized loss coefficient $\beta/T$ for $T/T_F$ below roughly 0.6 is explained by two-body p-wave physics with Fermi statistics and trap inhomogeneity, by reduced density fluctuations, or by beyond-two-body (complex-molecule, medium, or light-assisted) processes. Quantify how the microscopic two-body inputs (the complex p-wave scattering volume, the p-wave contact, intermediate four-body complexes) and the many-body effects (Fermi statistics, pair correlations, trap inhomogeneity) produce the observed suppression of $\beta/T$ below about 0.6, and compare the predicted temperature dependence of $\beta/T$ quantitatively with the measured loss data. (2) Determine the degree and timescale on which the remaining molecules rethermalize when reactive loss preferentially removes low-energy particles, experimentally and theoretically: the rethermalization rate or timescale and the degree of re-equilibration of the loss-perturbed gas, together with the measurement that would determine them. (3) Determine the reaction and collision energetics for $T/T_F$ well below 0.3, where collisions are dominated by molecules near the Fermi surface and the mean relative collision energy departs from classical equipartition and approaches the quantum value of $\frac{3}{4}k_B T_F$ per particle, obtaining an explicit value or scaling for the collision energy in this deep-degenerate regime and for the residual two-body loss rate as $T\to 0$, tied to microscopic scattering parameters. (4) Explore degenerate polar molecules under applied electric fields, where the dipole-dipole interaction dominates: characterize interaction-induced Fermi-surface deformation quantitatively (an explicit magnitude or functional form as a function of interaction strength), and independently identify and realize the novel many-body correlated phases expected in this strongly dipolar degenerate regime, including the conditions under which a specific such phase can be realized.

References

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