Ultracold bosons in the vicinity of a narrow resonance: Shallow dimer and recombination

The different resonant regimes that can be achieved by using a magnetic Feshbach resonance are analyzed with a separable two-channel model. Emphasis is put on the case of narrow resonances in a region of intermediate detuning where a shallow dimer exists and an approximate law including the background scattering length for the three-body recombination rate is derived.

Figure 1

Graphical solution of Eq. (24) for $q_{\dim }b\ll 1$ and the ratio $|a_{\mathrm{bg}}|/R^{☆}=2$. The product $q_{\dim }{a}_{\mathrm{bg}}$ is plotted as a function of the energy detuning $\delta \mu (\mathcal{B}-{\mathcal{B}}_0)$. In the regime of negative background scattering length $\left(a_{\mathrm{bg}}<0\right)$ only one shallow dimer exists (solid line). For positive scattering length $(a_{\mathrm{bg}}<0)$ there exists an avoided crossing between the Feshbach dimer and the background dimer respectively (dashed and dotted lines, respectively).

Figure 2

Spectrum of the Feshbach dimer for the resonance at 402 G in ${}^{39}$K. Stars: microscopic computations [21]; continuous line: two-channel model used in this paper where $b=1.2R_{\mathrm{vdW}}$; dashed line: universal law $E_{\dim }={\hslash }^2/\left(m{a}^2\right)$; dashed dotted line: semiclassical prediction from Eq. (41). Inset: for large detunings, the binding energy tends to $-E_{\mathrm{mol}}$.

Figure 3

Spectrum of the shallow dimer for the narrow resonance of ${}^{39}$K at 752 G ($a_{\mathrm{bg}}=-35$ $a_0$, $\Delta \mathcal{B}=-0.4$ G, $\delta \mu =1.5$ ${\mu }_B$, $R_{\mathrm{vdW}}=64.6$ $a_0$). Solid line: binding energy $E_{\dim }$ obtained from the two-channel model in units of $\left|\delta \mu \Delta \mathcal{B}\right|$ as a function of the relative detuning $(\mathcal{B}-{\mathcal{B}}_0)/\left|\Delta \mathcal{B}\right|$; dashed line: approximate law for narrow resonances from Eq. (32). Dotted line: approximation in Eq. (33), derived in the intermediate regime of narrow resonances; dashed dotted line: binding energy from Eq. (45); dashed doubly dotted line: binding energy in the effective range approximation (44).

Figure 4

Three-body recombination constant ${\alpha }_{\mathrm{rec}}$ in units of $\hslash /m$ for the narrow resonance in ${}^{39}K$ in the vicinity of 752 G. Solid line: numerical solution of Eq. (63) for positive value of $a$; dashed line: numerical solution of Eq. (63) for $a_{\mathrm{bg}}<a<0$; dotted line: analytical solution of Ref. [27] valid for large values of the scattering length; dashed-dotted line: approximate law of Eq. (75) valid for large values of the ratio $R^{☆}/\left|a\right|$. To guide the eye, this last curve is also plotted in the region where $R^{☆}/\left|a\right|$ is small.

Figure 5

Three-body recombination constant ${\alpha }_{\mathrm{rec}}$ in the region of small scattering length ($\left|a\right|<|a_{\mathrm{bg}}|$) for the narrow resonance in ${}^{39}K$ in the vicinity of 752 G. Solid line: numerical solution of Eq. (63); dashed line: approximate law of Eq. (75) valid for large values of the ratio $R^{☆}/\left|a\right|$.