Three fully polarized fermions close to a $\mathit{p}$-wave Feshbach resonance

We study the three-body problem for three atomic fermions, in the same spin state, experiencing a resonant interaction in the $\mathit{p}$-wave channel via a Feshbach resonance represented by a two-channel model. The rate of inelastic processes due to recombination to deeply bound dimers is then estimated from the three-body solution using a simple prescription. We obtain numerical and analytical predictions for most of the experimentally relevant quantities that can be extracted from the three-body solution: the existence of weakly bound trimers and their lifetime, the low-energy elastic and inelastic scattering properties of an atom on a weakly bound dimer (including the atom-dimer scattering length and scattering volume), and the recombination rates for three colliding atoms towards weakly bound and deeply bound dimers. The effect of ‘‘background’’ nonresonant interactions in the open channel of the two-channel model is also calculated and allows one to determine which three-body quantities are “universal” and which on the contrary depend on the details of the model.

Figure 1

For a square well interaction potential $V\left(r\right)=-\frac{{\hslash }^2{k}_0^2}{m}\theta \left(b-r\right)$, where $\theta$ is the Heaviside function, values of $\alpha$ in units of $1/b$ (dashed line) and ${\mathcal{V}}_s$ in units of $b^3$ (solid line) as functions of $k_{0}b$. Note the divergence of $\alpha$ when ${\mathcal{V}}_s=0$.

Figure 2

Schematic view of a Feshbach resonance configuration: the atoms interact via two potential curves, plotted as a function of the interatomic distance. Solid line: open channel potential curve. Dashed line: closed channel potential curve. When one neglects the coupling $\Lambda$ between the two curves, the closed channel has a molecular state of energy $E_{\text{mol}}$ with respect to the dissociation limit of the open channel. Note that the energy dependence of the two curves is purely indicative, and the spacing between the solid curve and the dashed curve was greatly exaggerated for clarity.

Figure 3

(Color online) For fixed values of the scattering volume ${\mathcal{V}}_s$, parameter $q_{\text{trim}}$ of the trimer (when it exists) as a function of $\alpha b$; $q_{\text{trim}}$ is related to the negative energy $-E_{\text{trim}}$ of the trimer by $E_{\text{trim}}={\hslash }^2{q}_{\text{trim}}^2/m$. (a) Even sector, (b) odd sector, as defined in Sec. 3B. Solid line (black): $\left|{\mathcal{V}}_s\right|/b^3=\infty$. Above the solid line, positive values of ${\mathcal{V}}_s$: short dashed line (blue): ${\mathcal{V}}_s={10}^4{b}^3$; dashed line (red): ${\mathcal{V}}_s={10}^3{b}^3$; dashed-dotted (green): ${\mathcal{V}}_s=100b^3$. Below the solid line, negative values of ${\mathcal{V}}_s$: short dashed line (light blue): ${\mathcal{V}}_s=-{10}^4{b}^3$; dashed line (orange): ${\mathcal{V}}_s=-{10}^3{b}^3$; dashed-dotted (dark green): ${\mathcal{V}}_s=-100b^3$. At the threshold for the existence of the trimer as a true bound state, on the ${\mathcal{V}}_s>0$ side of the resonance, where a dimer exists, the trimer binding energy vanishes, so that the energy of the trimer coincides with the one of the dimer, and $q_{\text{trim}}=q_{\text{dim}}$ (see text).

Figure 4

For ${\mathcal{V}}_s=\infty$, and ${\alpha }_{\text{res}}={\alpha }_{\text{th}}$ [right on the thresholds for the existence of a trimer, see Eqs. (54, 92)], $\mathit{K}$ dependence of the functions (a) $B_{L=1}$ (even sector), (b) $B_{L=0}$ (solid line), $B_{L=2}$ (dashed line) (odd sector), for the trimers. To avoid diverging functions, these functions were multiplied by $\mathit{K}$ in (a) and by $K^2$ in (b). The normalization is arbitrary.

Figure 5

(Color online) Atom-dimer scattering length $a_{\text{ad}}$ as a function of ${\alpha }_{\text{res}}$ for fixed values of the scattering volume ${\mathcal{V}}_s$, ${\mathcal{V}}_s={10}^6{b}^3$ (black solid line), ${\mathcal{V}}_s={10}^3{b}^3$ (red dashed line), ${\mathcal{V}}_s=10b^3$ (green dotted-dashed line). The divergence of $a_{\text{ad}}$ coincides with the threshold of existence of a trimer in the odd sector. In the limit of a broad Feshbach resonance ${\alpha }_{\text{res}}b\to 1/{\pi }^{1/2}$, $a_{\text{ad}}$ tends to $\approx 0.2b$.

Figure 6

(Color online) Atom-dimer scattering volume ${\mathcal{V}}_s^{\text{ad}}$ for a total spin $J=1$ (see text) as a function of ${\alpha }_{\text{res}}b$, for a fixed value of the atom-atom scattering volume ${\mathcal{V}}_s/b^3=100$ (dashed-dotted green line), ${\mathcal{V}}_s/b^3={10}^3$ (dashed red line), ${\mathcal{V}}_s/b^3={10}^4$ (solid black line). To reveal the scaling of ${\mathcal{V}}_s^{\text{ad}}$ with ${\mathcal{V}}_s$ close to the Feshbach resonance, ${\mathcal{V}}_s^{\text{ad}}$ is expressed in units of ${\mathcal{V}}_s$. Dotted horizontal line: analytical prediction (66) in the limit ${\mathcal{V}}_s\to +\infty$.

Figure 7

(Color online) Recombination constant ${\mathcal{K}}_{\text{rec}}$ appearing in the expression (81) giving the rate of formation of weakly bound dimers when three low energy atoms are colliding, as a function of ${\alpha }_{\text{res}}$ for a fixed value of the scattering volume. (a) ${\mathcal{K}}_{\text{rec}}$ in units of $\hslash b^8/m$. The scattering volume is, from bottom to top, ${\mathcal{V}}_s=100b^3$ (green line), ${\mathcal{V}}_s=1000b^3$ (red line), and ${\mathcal{V}}_s={10}^4{b}^3$ (black line). Solid lines: numerical solution. Dashed lines: asymptotic formula (90). (b) Ratio of ${\mathcal{K}}_{\text{rec}}$ to the asymptotic formula (90), for ${\mathcal{V}}_s={10}^4{b}^3$. Solid line: numerical solution. Dashed line: analytically predicted Fano profile (93). The inset is exactly the same figure but with a log scale on the vertical axis.

Figure 8

For an infinite scattering volume, quality factor of the trimers, as a function of $q_{\text{trim}}b$, for the even sector (solid line) and for the odd sector (dashed line). For clarity, the quality factor in the even sector was multiplied by 20. The quality factor is defined as the ratio of the trimer binding energy, here equal to $E_{\text{trim}}={\hslash }^2{q}_{\text{trim}}^2/m$ since ${\mathcal{V}}_s=\infty$, and of $\hslash$ times the spontaneous decay rate of the trimer due to the formation of deeply bound dimer and a free atom as estimated by the simple recipe (98).

Figure 9

(Color online) Loss constant $K_{\text{ad}}$ due to formation of deeply bound dimers in the atom-dimer collision, as estimated by the recipe Eq. (98) and the relation Eq. (113), as a function of ${\alpha }_{\text{res}}$ for ${\mathcal{V}}_s={10}^6{b}^3$ (black solid line), ${\mathcal{V}}_s={10}^3{b}^3$ (red dashed line), ${\mathcal{V}}_s=10b^3$ (green dotted-dashed line). $K_{\text{ad}}$ is in units of $\hslash b/m$. A divergence of $K_{\text{ad}}$ occurs at the threshold of odd trimer formation.

Figure 10

(Color online) In presence of open channel interactions, characterized by a fixed value of the background scattering volume ${\mathcal{V}}_s^{\text{bg}}$, parameter $q_{\text{trim}}$ of the trimer (when it exists) (a) in the even sector and (b) in the odd sector, as a function of ${\alpha }_{\text{res}}b$. Here the scattering volume ${\mathcal{V}}_s$ is infinite and ${\alpha }_{\text{res}}$ is the corresponding value of the parameter $\alpha$ for the modified scattering amplitude, see Eq. (132). Solid line (black): ${\mathcal{V}}_s^{\text{bg}}=0$. Dashed line (red): ${\mathcal{V}}_s^{\text{bg}}=-b^3$. Dashed-dotted line (green): ${\mathcal{V}}_s^{\text{bg}}=-10b^3$.

Figure 11

(Color online) Atom-dimer scattering length $a_{\text{ad}}$ as a function of ${\alpha }_{\text{res}}b$ for the model including the open channel interaction, for an infinite scattering volume ${\mathcal{V}}_s$. Solid line (black): ${\mathcal{V}}_s^{\text{bg}}=0$. Dashed line (red): ${\mathcal{V}}_s^{\text{bg}}=-b^3$. Dashed-dotted line (green): ${\mathcal{V}}_s^{\text{bg}}=-10b^3$.

Figure 12

(Color online) In the model including an open channel interaction, atom-dimer scattering volume ${\mathcal{V}}_s^{\text{ad}}$ for a total spin $J=1$ (as detailed in Sec. 4D) as a function of ${\alpha }_{\text{res}}b$, for a fixed value of the atom-atom scattering volume ${\mathcal{V}}_s={10}^4{b}^3$ (solid black line), but for various background scattering volumes in the open channel: ${\mathcal{V}}_s^{\text{bg}}=0$ (black solid line), ${\mathcal{V}}_s^{\text{bg}}=-b^3$ (dashed red line), ${\mathcal{V}}_s^{\text{bg}}=-10b^3$ (dashed-dotted green line). ${\mathcal{V}}_s^{\text{ad}}$ is expressed in units of ${\mathcal{V}}_s$. Dotted horizontal line: analytical prediction Eq. (66) in the limit ${\mathcal{V}}_s\to +\infty$.