First-order phase transition in atom-molecule quantum degenerate mixtures with coherent three-body recombination

TL;DR

This work addresses how coherent three-body recombination (cTBR) alters the atom–molecule Bose–Einstein condensate phase transition. It analyzes a two-mode model with detuning $\Delta$, Feshbach coupling $g_2$, and cTBR coupling $g_3$, forming a mean-field energy density in terms of $\delta=\Delta/(g_2\sqrt{n})$ and $\gamma=g_3 n/g_2$, and corroborates with exact diagonalization, stability analysis, and entanglement measures. The key result is that, unlike the familiar second-order transition driven by Feshbach coupling, dominant cTBR yields a first-order transition accompanied by a double-well energy landscape, bistability, metastability of the molecular condensate, and enhanced atom–molecule entanglement; the transition line and the associated molecular fraction are captured by $\delta_c(\gamma)$ and $2f_M^{(c)}=(1+\gamma)/(3\gamma)$, with sharper spectral avoided crossings as $\gamma$ grows. These findings reveal coherent three-body processes as a tunable knob for quantum-state engineering and control of ultracold chemical reactions, with practical implications for ramp protocols across atom–molecule resonances and potential extensions to finite temperature and confinement.

Abstract

We map the phase diagram of a two-mode atom-molecule Bose-Einstein condensate with Fano-Feshbach and coherent three-body recombination (cTBR) terms. The standard second order phase transition observed as the molecular energy is tuned through the Feshbach resonance, is replaced by a first order transition when cTBR becomes prominent, due to a double-well structure in the free energy landscape. This transition is associated with atom-molecule entanglement, bistability, and molecular metastability. Our results establish cTBR as a powerful knob for quantum state engineering and control of reaction dynamics in ultracold chemistry.

Figures (7)

• Figure 1: First- and second-order phase transitions of atom-molecules with competing Feshbach coupling and cTBR. In the case of pure Feshbach coupling, a second-order transition takes place (dashed line), from a pure molecular state to a mixture of atoms (green line) and molecules (black line). (b) When only three-body coupling is present, the transition becomes first-order, denoted by the vertical dashed line. (c) Molecular fraction, $2\braket{f_M}$, over the $(\delta,\gamma)$ plane. The black-white dashed (orange dash-dotted) curve delineates the first-order (second-order) transition. The box in the upper left corner presents the regime where cTBR prevails. The particle number is $N=800$.
• Figure 2: Phase-space portraits and stability diagram of the atom-molecule phases. (a)-(f) Energy density isolines for $\gamma = 3$ and various $\delta$ (see legends) on the plane defined by the molecular fraction, $2f_M$, and the relative angle between atoms and molecules, $\varphi$. The x, ❍, ✧ symbols mark the energy minima, maxima and saddle points respectively. The $f_M=0.5$ line (mBEC) refers to a single fixed point for which $\varphi$ is ill-defined, while the energy maxima (same fixed point) at $\varphi=0,2\pi$ corresponds to full circles about the $2f_M$ axis. (g) Stability diagram of the molecular condensate. The presence of a real part of the eigenvalues, $\lambda$, pertaining to the linearized Hamiltonian system around $f_M=1/2$ leads to instability within $\left| \delta \right| \leq \delta_s$. In the stable regions, the mBEC can be either a global or a local minimum/maximum, due to the double-well.
• Figure 3: (a)-(c) Distribution of number states, $\left| \braket{n_M | \psi_0} \right|^2$, close to the transition point for different $\gamma$ (see legends). A superposition of a pure molecular condensate and a finite atom-molecule mixture appears at $\gamma = 0.6$ [panel (b)], due to the double-well potential in the mean-field energy [Fig. \ref{['Fig:Energy_contours']}(c)]. The inset in panel (b) presents a distribution profile at the position indicated by the arrow. (d) Maximum of the normalized von Neumann entropy with respect to $\delta$. Its maximum value is attained at $\gamma = 0.6$ (dashed line), in the vicinity of the phase transition character change (from first- to second-order).
• Figure 4: Molecular dissociation dynamics into a finite mixture of atoms (magenta balls) and molecules (balls in the dashed ellipses). A pure mBEC is prepared at large negative detunings, and subsequently $\delta$ is quenched to $-\delta_c(\gamma)+1$ for different $\gamma$ and particle numbers (see legend). A quench for fixed $N$ entails $\gamma$, $g_3$ variation to attain $\delta_c(\gamma)$, while $g_2$ is kept constant. The time-averaged molecular fraction, $2\overline{\langle f_M \rangle}$, reveals that the all-molecule state becomes metastable at large $\gamma$, due to the double-well structure. The inset presents the dissociation dynamics of $N=100$ atoms at $\gamma=2.45$. The vertical dashed line marks the point beyond which the mBEC becomes metastable.
• Figure S1: Energy density profile at $\varphi=\pi$ associated to (a) cTBR [Eq. \ref{['Eq:Energy_density_dim_TBR_Supp']}] and (b) Feshbach coupling [Eq. \ref{['Eq:Energy_density_dim_Supp']} at $\gamma = 0$]. As it can be seen, in the presence of cTBR (panel (a)) for larger negative values of the detuning, e.g. $\delta=-2$, the energy density features a global minimum at $f_M =0.5$ where the molecular condensate is the ground state of the system. However, for decreasing $\delta$ the energy gradually deforms into a double-well structure possessing a global minimum at $f_M \approx 0.15$ (representing the ground state of the atom-molecule mixture) and a local one at $f_M=0.5$ (all molecule state). The local maximum at $f_M \approx 0.4$ refers to the saddle point. In contrast, when solely Feshbach coupling is present (panel (b)) a double-well structure is absent. The energy density is normalized to its minimum, and is presented at different detunings (see legend).