Suppression of two-body collisional losses in an ultracold gas via the Fano effect

The Fano effect [U. Fano, Phys. Rev. 124, 1866 (1961)] shows that an inelastic scattering process can be suppressed when the output channel (OC) is coupled to an isolated bound state. The Fano effect was originally derived via a first-order perturbation treatment for coupling between the incident channel (IC) and the OC. In this paper, we first generalize the Fano effect to systems with arbitrarily strong IC-OC couplings. We analytically prove that, in a system with one IC and one OC, when the interatomic interaction potentials are real functions of the interatomic distance, the exact $s$-wave inelastic scattering amplitude can always be suppressed to zero by coupling between the IC or the OC (or both of them) and an extra isolated bound state. We further investigate the application of this generalized Fano effect for the suppression of two-body collisional losses of ultracold atoms and show that when the low-energy inelastic collision between two ultracold atoms is suppressed by this effect, the real part of the elastic scattering length between the atoms is still capable of being much larger than the range of interatomic interaction. In addition, when open scattering channels are coupled to two bound states, with the help of the Fano effect, independent control of the elastic and inelastic scattering amplitudes of two ultracold atoms can be achieved. Possible experimental realizations of our scheme are also discussed.

Figure 1

Multichannel models studied in (a–c) Sec. 2 and (d) Sec. 4. Here $r$ is the distance between the two particles. The black curves represent the interaction potentials for the IC $\alpha$ and OC $\beta$ of the inelastic scattering processes, while the blue and orange curves represent the interaction potentials for the closed channels $\eta$ and ${\eta }^{\prime }$, respectively. The red arrows represent interchannel coupling. In Sec. 2, we consider cases where an isolated bound state $|{\mathrm{\Phi }}_b\rangle$ related to the closed channel $\eta$ is coupled to (a) both the IC $\alpha$ and OC $\beta$, (b) only the OC $\beta$, or (c) only the IC $\alpha$. (d) In Sec. 4, we consider the system where the open channels $\alpha$ and $\beta$ are coupled to two isolated bound states $|{\mathrm{\Phi }}_b\rangle$ and $|{\mathrm{\Phi }}_{b^{\prime }}\rangle$ related to the closed channels $\eta$ and $\eta \prime$.

Figure 2

The absolute value $|f_{\beta \alpha }|$ of the inelastic scattering amplitude in the square-well model as a function of the potential energy $U_{\eta \eta }$ of channel $\eta$. In (a)–(c), we show results for cases where $U_{\alpha \eta }=2/b^2, U_{\beta \eta }=3/b^2$, i.e., the cases where the closed channel $\eta$ is coupled to both IC $\alpha$ and OC $\beta$ [the cases in Fig. 1]. In (d)–(f), we show results for cases where $U_{\alpha \eta }=0, U_{\beta \eta }=3/b^2$, i.e., the cases where the closed channel $\eta$ is coupled to only OC $\beta$ [the cases in Fig. 1]. In (g)–(i), we show results for cases where $U_{\alpha \eta }=3/b^2, U_{\beta \eta }=0$, i.e., the cases where the closed channel $\eta$ is coupled to only the IC $\alpha$ [the cases in Fig. 1]. Here we consider systems with potential energies $U_{\alpha \alpha }=-1/b^2, U_{\beta \beta }=-2.5/b^2$; threshold energies $E_{\alpha }=0, E_{\beta }=-0.5/b^2$; interchannel coupling (a, d, g) $U_{\alpha \beta }=1/b^2$, (b, e, h) $3/b^2$, and (c, f, i) $10/b^2$; and incident momentum $k_{\alpha }=0$ (solid black line), $0.3/b$ (dashed blue line), and $1/b$ (dash-dotted red line).

Figure 3

(a)–(c) The absolute value of the inelastic scattering amplitude $f_{\beta \alpha }(E_{\mathrm{s}}=E_{\alpha })$ and the real part of the scattering length as functions of the potential energy $U_{\eta \eta }$ of the closed channel $\eta$ for the square-well model in Sec. 2d. The potential energy $U_{\alpha \alpha }$ of channel $\alpha$ has the values (a) $U_{\alpha \alpha }=-22.5/b^2$, (b) $-21.779/b^2$, and (c) $-21.776/b^2$. In these three cases, when $|f_{\beta \alpha }|$ is suppressed to zero, we have $\mathrm{Re}\left[a\left({\epsilon }_b^{*}\right)\right]=3.4b, 1541b$, and $-931b$, respectively. (d) The scattering length $a\left({\epsilon }_b^{*}\right)$ as a function of $U_{\alpha \alpha }$. Our calculation is done with the parameters $E_{\alpha }=0, E_{\beta }=-0.5/b^2, U_{\beta \beta }=U_{\alpha \beta }=U_{\beta \eta }-3/b^2$, and $U_{\alpha \eta }=2/b^2$.

Figure 4

The elastic scattering length $\overline{a}\left({\epsilon }_{b^{\prime }}\right)\equiv -f_{\alpha \alpha }[E_{\alpha },{\epsilon }_b=\chi \left({\epsilon }_{b^{\prime }}\right),{\epsilon }_{b^{\prime }}]$ for the square-well model in Sec. 4. Here ${\epsilon }_{b^{\prime }}$ is the lowest bound state related to channel $\eta \prime$. The calculation is executed with the parameters $E_{\alpha }=0, E_{\beta }=-0.5/b^2, U_{\alpha \alpha }=-1.5/b^2,U_{\beta \beta }=-3/b^2, U_{\alpha \beta }=U_{\beta \eta }=U_{\alpha {\eta }^{\prime }}=3/b^2$, and $U_{\alpha \eta }=U_{\eta {\eta }^{\prime }}=U_{\beta {\eta }^{\prime }}=0$.