Bose-enhanced relaxation of driven atom-molecule condensates

Motivated by recent experiments [Zhang et al., Nat. Phys. 19, 1466 (2023)], we study the interconversion between ultracold atomic and molecular condensates, quantifying the resulting oscillations and their slow decay. We find that near equilibrium the dominant damping source is the decay of condensed molecules into noncondensed pairs, with a pair kinetic energy that is resonant with the frequency of the oscillating atom-molecule interconversions. The decay, however, is nonexponential, as strong population of the resonant pairs leads to Bose enhancement. Introducing an oscillating magnetic field, which periodically modulates the molecular binding energy, enhances the oscillations at short times. However, the resulting enhancement of the pair-production process results in an accelerated decay, which rapidly cuts off the initial oscillation growth.

Figure 1

Condensate energy density $E_{\text{cond}}$ corresponding to Eq. (3) for three values of the molecular detuning $\epsilon$. Dashed lines correspond to zero-energy contours. Darker colors represent lower energies. The white spiral in the left figure shows a typical trajectory when the excited atomic states are taken into account, see Sec. 3. Energy and timescales used for dimensionalizing are described in Sec. 5.

Figure 2

(a) Shift of the equilibrium molecular amplitude due to coupling to the atomic continuum: $\mathrm{\Delta }{g}^{★}\left(\alpha \right)=g^{★}\left(\alpha \right)-g^{★}\left(0\right)$. (b) Equilibrium atomic mode occupation $n_k$ weighted by the density of states $\sqrt{{\epsilon }_k}$, Eq. (16), for $\epsilon =0$ and $\alpha \to 0$.

Figure 3

Time evolution of molecule population for two different molecular energies $\epsilon$ without coupling to the atomic continuum, $\alpha =0$. (a) Boundary condition $g\left(0\right)=0$. The dashed horizontal line marks $g_{\text{max}}$ (main text). (b) $g\left(0\right)\lesssim g^{★},\eta \simeq {\eta }^{★}$.

Figure 4

Time evolution of molecule population for boundary condition $g\left(0\right)=0$, continuum coupling $\alpha =0.27$, and three different molecular energies $\epsilon$. The dashed lines mark the equilibrium fixed-point values $g^{★}$.

Figure 5

Numerical demonstration of the resonant production of atomic pairs during small oscillations about the fixed point. (a) Time evolution of weighted atomic density $n_k\left(t\right)\sqrt{{\epsilon }_k}$, obtained by discretizing Eq. (19) on a sufficiently dense momentum grid. Here we used $\epsilon =0,\alpha =0.01$, and initial conditions with the molecular fraction shifted slightly from its equilibrium value. The dashed horizontal line indicates the resonance energy. (b) Linear growth of resonant modes illustrated by plotting $\delta v_k/(1-{\left(u_k^{*}\right)}^2)$ vs time, for $k$ satisfying the resonant condition [dashed white line in panel (a)]. The full numerical solution (red) is contrasted with the approximation (blue) from Eq. (25). (c) Scaled damping rate $\mathrm{\Gamma }$ as function of $\epsilon$, see Eq. (26).

Figure 6

Molecular amplitude $g\left(t\right)$ at $\epsilon =0$ for $\alpha =0.1$ (a) and a smaller value $\alpha =0.01$ (b). In the energy landscape of Fig. 1, the damped oscillation corresponds to a spiral towards the fixed point. The long-time deviation from the ordinary exponential decay, which sets in at $t\gtrsim 1/\left|\delta \mathbf{x}\right|$, is clearly observed in (b).

Figure 7

Molecular amplitude $g\left(t\right)$ subject to off-resonant drive, $\epsilon \left(t\right)=\epsilon +\delta \epsilon cos\left({\omega }_{\epsilon }t\right)$. Here $\epsilon =0$, $\alpha =0.27$ as in the experiment, see Sec. 5, and $\delta \epsilon =0.1$. (a) Driving frequency ${\omega }_{\epsilon }=0.3{\omega }_g$. (b) ${\omega }_{\epsilon }=1.7{\omega }_g$. The analytical approximation is obtained from solving Eq. (28).

Figure 8

Molecular amplitude $g\left(t\right)$ subject to resonant drive, $\epsilon \left(t\right)=\epsilon +\delta \epsilon cos\left({\omega }_{\epsilon }t\right)$, with ${\omega }_g\simeq {\omega }_{\epsilon }$ without coupling to the continuum, $\alpha =0$. Here $\epsilon =0$, $\delta \epsilon =0.1$. The analytical approximation $g_{\text{analyt.}}$, which is valid for short times $t\lesssim 1/\delta \epsilon$ only, is calculated from Eq. (28).

Figure 9

(a) Molecular amplitude $g\left(t\right)$ subject to resonant drive, ${\omega }_g\simeq {\omega }_{\epsilon }$, with coupling to the continuum. Here $\epsilon =0$, $\delta \epsilon =0.1,\alpha =0.27$. At times $t>1/\delta \epsilon$, strong population of the resonant continuum mode sets in, leading to a breakdown of the linearized analytical solution derived from Eq. (28). (b) Time evolution of excited-state population. The horizontal dashed line marks the resonance energy obtained for $\alpha =0$. (c) Time evolution of resonant excited-state mode with ${\mathrm{\Omega }}_k\simeq {\omega }_g$ [see Eq. (23)], compared to the situation without external drive.

Figure 10

Exact vs mean-field dynamics for solitonlike solution with $g\left(0\right)=0^{+}$. (a) $g\left(t\right)$ for fixed total particle number $N$ (mean-field corresponds to $N=\infty$). (b) Scaling of $t_{\text{MF}}$ with $N$ on a log-linear plot.

Figure 11

Exact vs mean-field dynamics for oscillatory solution with $g=g^{★}-0.04$. (a) $g\left(t\right)$ for fixed total particle number $N$, with colors as in Fig. 10. (b) Scaling of $t_{\text{MF}}$ with $N$.