Lellouch-Lüscher relation for ultracold few-atom systems under confinement

Abstract

We derive an analog of the Lellouch-Lüscher (LL) relation for few-body bosonic systems, linking few-body scattering loss rates to the energies and widths of the corresponding harmonically trapped few-body states. Three-body numerical simulations show that the LL relation applies across a broad range of interaction strengths and energies and allows the determination of scattering rates within a single partial wave. Our work establishes a robust theoretical framework for understanding the role of the finite volume effect in few-body observables in optical lattice and tweezer experiments, enabling precise determination of multi-body scattering rates.

Figures (6)

• Figure 1: (a) Schematic illustration of three-body losses in free space and in a harmonic trap. In free space, three atoms in continuum channels $W_{\lambda}(R)$ (solid) undergo inelastic transitions near $R \lesssim R_0$ (shaded) to atom-dimer channels $W_{f}(R)$, characterized by the recombination rate constant $L_3$. In a trap, discrete states $E_q$ (dotted) form in $W_\lambda(R)+V_{\rm ho}(R)$ channels (dashed) and decay to atom-dimer channels with rate $\Gamma_q/\hbar$sm. (b) Trapped-state energies $E_q$ and widths $\Gamma_q$ manifest as Lorentzian resonances in the time delay $\tau_D(E)$ (see text).
• Figure 2: A typical time delay plot for $^{85}$Rb at $\omega_{\rm ho}$=$2\pi\times$252 kHz or, equivalently, $|a|/a_{\rm ho}$=1.11 (solid line) and for $^{87}$Rb at $\omega_{\rm ho}$=$2\pi\times$243 kHz, or $|a|/a_{\rm ho}$=0.24 (dashed line). Both frequencies correspond to $a_{\rm ho}$=$5 r_{\rm vdW}$. The vertical lines indicate the energy levels of three non-interacting atoms, $(2q+3)\hbar\omega_{\rm ho}$. Arrows indicate the shift $\delta E\propto a/a_{\rm ho}$ from the ground state energy $3\hbar\omega_{\rm ho}$ due to interactions. $E_{q}$ denotes the energies for states of the $\lambda=0$ channel and $E_{q}^{(\lambda \neq 0)}$ otherwise.
• Figure 3: Numerical values for the width $\Gamma_q$ of the three-body states (symbols) as a function of their energy $E_q$ for (a) $^{87}$Rb, and (b) for $^{85}$Rb. Solid lines correspond to the results for $\Gamma_q$ from the LL relation (\ref{['gammarM']}), obtained from our free-space numerical simulations of $L_3(E)$. The colored arrows in (b) mark the regime $a<0.25\langle r_q\rangle$, and all data points in (a) fall within this regime. For both Rb isotopes the frequencies correspond to $a_{\rm ho}/r_{\rm vdW}$ = 5, 6, 7, 8, 9 and 10. Symbols in (c) and (d) correspond to $L_3(E_q)$ from the LL relation (\ref{['eq:L3']}) obtained from the numerical calculations for $E_q$ and $\Gamma_q$ shown in (a) and (b). The vertical dashed lines indicate the value of $E_c$=$\hbar^2/(ma^2)$.
• Figure 4: The distribution of the individual contribution, $\gamma_q$, of the trap state to $\Gamma_{\rm th}$ in the threshold regime of $k_BT/E_c=0.1$ (a) and in the unitary regime of $k_BT/E_c=10$ (b). The vertical dashed lines indicate that $E_q=E_c$. (c) shows the temperature dependence behavior of $\Gamma_{\rm th}$ calculated from Eq. (\ref{['gammath']}) and from Eq. (\ref{['gammathX']}) at threshold limit (dotted line) and unitary limit (dashed line). The displayed result is obtained by using Eq. (\ref{['eq:L3toy']}) for $L_3(E)$ with $\alpha=1$, $E_c=0.1 E_{\rm vdW}$ and $\hbar \omega_{\rm ho}=0.001 E_{\rm vdW}$ and Eq. (\ref{['gammarM']}) for $\Gamma_q$, for instance.
• Figure 5: Non-interacting three-body hyperradial wavefunction $F_q(R)$ in a spherical harmonic trap. The solid lines show $F_q^2(R)$ from Eq. (\ref{['fqnon']}), while the dashed lines represent the corresponding Taylor expansion to the leading order. The shaded area indicates the reaction regime $R<R_0$.