Harmonic and subharmonic association of universal dimers in a thermal gas

In a gas of ultracold atoms whose scattering length is controlled by a magnetic Feshbach resonance, atoms can be associated into universal dimers by an oscillating longitudinal magnetic field. In addition to the harmonic resonance with frequency near that determined by the dimer binding energy, there is a subharmonic resonance with half that frequency. If the thermal gas contains dimers, they can be dissociated into unbound atoms by the oscillating magnetic field. We show that the transition rates for association and dissociation can be calculated by treating the oscillating magnetic field as a sinusoidal time-dependent perturbation proportional to the contact operator. Many-body effects are taken into account through transition matrix elements of the contact operator. We calculate both the harmonic and subharmonic transition rates analytically for association in a thermal gas of atoms and for dissociation in a thermal gas of dimers.

Figure 1

Association rate $\mathrm{\Gamma }/N\langle n\rangle$ for a thermal gas of ${}^7\mathrm{Li}$ atoms as a function of the modulation frequency of the magnetic field. The angular frequency $\omega$ is normalized to the angular binding frequency ${\omega }_{\mathrm{D}}$ of the dimer. The bias field is $\overline{B}=734.5$ G, which corresponds to $\overline{a}\approx 1100a_0$. The three sets of curves are for different values of the modulation amplitude $b$ and the temperature $T$: $(b,T)=(0.57\mathrm{G},3\mu \mathrm{K})$ (highest peak value, red), $(0.57\mathrm{G},10\mu \mathrm{K})$ (green), and $(0.14\mathrm{G},3\mu \mathrm{K})$ (lowest peak value, blue). The association rates in the region $\omega <{\omega }_{\mathrm{D}}$ have been multiplied by 10 to make the subharmonic transitions more visible.

Figure 2

Dissociation rate $\mathrm{\Gamma }/N_{\mathrm{D}}$ for a thermal gas of ${}^7\mathrm{Li}$ dimers as a function of the angular frequency $\omega$ of the modulated magnetic field. The angular frequency $\omega$ is normalized to the binding angular frequency ${\omega }_{\mathrm{D}}$ of the dimer. The bias field is $\overline{B}=734.5$ G, which corresponds to $\overline{a}\approx 1100a_0$. The dissociation rate is independent of the temperature. The two sets of curves are for different values of the modulation amplitude $b$: 0.57 G (higher peak value, red) and 0.14 G (lower peak value, blue). The dissociation rates in the region $\omega <{\omega }_{\mathrm{D}}$ have been multiplied by 10 to make the subharmonic transitions more visible.

Figure 3

The Lippmann-Schwinger equation for the transition amplitude $A\left(E_{\mathrm{c}.\mathrm{m}.}\right)$. The blob represents $iA\left(E_{\mathrm{c}.\mathrm{m}.}\right)$. The vertex factor is $-i{\lambda }_0/m$.

Figure 4

Feynman diagrams for the matrix element of ${\varphi }^{†}\left(\mathbf{r}\right)$ between the vacuum and the atom-pair state $|\mathit{K},\mathit{k}\rangle$. The open dot represents the ${\varphi }^{†}$ operator, whose Feynman rule is ${\lambda }_0$. The residue of the pole in the energy in the second diagram can be used to determine the matrix element of ${\varphi }^{†}\left(\mathbf{r}\right)$ between the vacuum and the dimer state $|{\mathit{k}}_{\mathrm{D}}\rangle$.