Tunable Molecular Interactions Near an Atomic Feshbach Resonance: Stability and Collapse of a Molecular Bose-Einstein Condensate

Abstract

Understanding and controlling interactions of ultracold molecules is a cornerstone of quantum chemistry. While the laboratory creation of degenerate molecular gases comprised of bosonic atoms has unlocked powerful new platforms for quantum simulation, progress is limited by the absence of a robust theoretical framework for characterizing inter-molecular interactions. This is in stark contrast to the situation for Fermi gases. In this Letter, we present such a framework providing universal expressions for these molecular scattering lengths as functions of experimentally measurable quantities. Our discoveries are crucial for understanding molecular condensate formation. Calculations of the compressibility reveal that a sign change in such molecular scattering lengths is directly correlated with the instability of these condensates. These results offer fresh insight with broad applications for atomic, molecular, and condensed matter physics, as well as quantum chemistry.

Figures (3)

• Figure 1: Scattering due to atomic Feshbach coupling: (a) Leading-order scattering process contributing to the resonant term in the atomic scattering length, $a_s$ [Eq. \ref{['eq:asdef']}]. Two incoming atoms (single blue circles) temporarily combine into a molecule (two attached circles) via Feshbach coupling, and then dissociate into two outgoing free atoms. The orange region represents virtual processes. (b) Corresponding leading-order contribution to the molecular scattering length, $a_{\mathrm{mm}}$, near the resonance. Here, two molecules approach each other and temporarily break up into four free atoms, which then propagate, interact, and recombine into two molecules. Similar to panel (a), this process universally depends on the Feshbach coupling but is of higher order in the coupling strength.
• Figure 2: Phase diagrams and molecular scattering lengths. Ground-state stability phase diagrams for (a) wide and (b) narrow resonances, showing compressibility $\kappa$ [normalized by $\kappa_{\mathrm{bg}}=m_1/(4\pi \hbar^2 a_{\mathrm{bg}})$] as a function of atom number density $n$ and of detuning $\bar{\nu} =\Delta \mu_m (B-B_0)$ (normalized by resonance width $\Delta_{\bar{\nu}}=\Delta \mu_m \Delta B$). Orange indicates stable regions ($\kappa>0$), and blue indicates unstable regions. The atomic condensate is present only in the atomic superfluid (ASF) phase, while the molecular condensate is present in both ASF and molecular superfluid (MSF) phases. Red dashed lines denote the quantum critical point $\bar{\nu}_c(n)$ separating ASF from MSF, while $\bar{\nu}_{c,-}$ and $\bar{\nu}_{c,+}$ mark boundaries between unstable and stable regions. (c),(d) Corresponding two-body molecular scattering length $a_{\mathrm{mm}}$, and its many-body analogue $a_{\mathrm{mm}}^{\mathrm{eff}}(n)$ for $n a_{\mathrm{bg}}^3=1.68\times 10^{-5}$. Parameters for the narrow resonance [panels (b) and (d)] are from the $^{133}$Cs resonance at $B_0 = 19.849$ G Zhang2021Zhang2023Wang2024. For the wide resonance [panels (a) and (c)], $\Delta_{\bar{\nu}}$ is increased by $10^2$ times relative to panels (b) and (d).
• Figure 3: Chemical potential, inverse compressibility and molecular scattering lengths. (a,b) Chemical potential $\mu$ (blue solid and red dashed lines) vs detuning $\bar{\nu}$ at densities $n$ and $n/2$, with $na_{\mathrm{bg}}^3 =1.68\times 10^{-5}$ [same as Figs. \ref{['fig:Fig1']}(c) and \ref{['fig:Fig1']}(d)]. The "atom continuum" threshold is at $\mu=0$. In panel (a), "$-E_b$" represents the dressed molecular energy; in panel (b), it is replaced by its bare value, $\bar{\nu}$. (c), (d) Comparison of the inverse compressibility (blue solid lines) at $na_{\mathrm{bg}}^3 =1.68\times 10^{-5}$ with the atomic scattering length $a_s$ (red dashed lines) and the many-body molecular scattering length $a_{\mathrm{mm}}^{\mathrm{eff}}$ (brown dashed lines with squares), defined in Eq. \ref{['eq:ammeffdef']}. Both $a_{\mathrm{mm}}^{\mathrm{eff}}$ and $a_s$ are plotted in units of $a_{\mathrm{bg}}$.