Effective-range approximations for resonant scattering of cold atoms

Studies of cold atom collisions and few-body interactions often require the energy dependence of the scattering phase shift, which is usually expressed in terms of an effective-range expansion. We use accurate coupled-channel calculations on ${}^6$Li, ${}^{39}$K, and ${}^{133}$Cs to explore the behavior of the effective range in the vicinity of both broad and narrow Feshbach resonances. We show that commonly used expressions for the effective range break down dramatically for narrow resonances and near the zero crossings of broad resonances. We present an alternative parametrization of the effective range that is accurate through both the pole and the zero crossing for both broad and narrow resonances. However, the effective-range expansion can still fail at quite low collision energies, particularly around narrow resonances. We demonstrate that an analytical form of an energy and magnetic-field-dependent phase shift, based on multichannel quantum defect theory, gives accurate results for the energy-dependent scattering length.

Figure 1

(a) The field-dependent effective range for the ab channel of ${}^6$Li from coupled-channel calculations (red solid or light gray) and as calculated from Eq. (8) (blue solid or dark gray). The zero-energy $s$-wave scattering length is also shown (green dashed). (b) An expanded view of (a) showing the narrow resonance at $543.40$ G. (c) and (f) The quantity $r_{\mathrm{eff}}{a}_0^2$ (black solid), which is a smoothly varying function of magnetic field through the zero crossing of $a_0$ (green dashed) for both the wide resonance (c) and the narrow resonance (f). (d) The quantity $r_{\mathrm{eff}}{a}_0^2$ (black), which is parabolic as a function of $a_0$ across the width of the wide resonance, except around the narrow resonance. The red dotted line shows the parabola $v+r_0{(a_0-a_{\mathrm{ext}})}^2$ fitted to the coupled-channel results, while the blue dotted (upper) line shows the corresponding parabola from Eq. (9). (e) The function $f\left(r_{\mathrm{eff}}\right)$ of Eq. (11), with parameters appropriate to the wide resonance (cyan). This is constant across the width of the wide resonance in $a_0$ (green dashed) except around the narrow resonance. (g) the quantity $r_{\mathrm{eff}}{a}_0^2$ (black solid), which is parabolic as a function of $a_0$ across the width of the narrow resonance; the red dotted line shows the parabola $v+r_0{(a_0-a_{\mathrm{ext}})}^2$ fitted to the coupled-channel results. (h) $f\left(r_{\mathrm{eff}}\right)$ of Eq. (11), with parameters appropriate to the narrow resonance (cyan). This is constant across the width of the narrow resonance in $a_0$ (green dashed).

Figure 2

Energy dependence of the phase shift $\eta \left(E\right)$ at magnetic fields around narrow resonances in ${}^6$Li, ${}^{39}$K, and ${}^{133}$Cs. Coupled-channel calculations (solid lines) give the actual variation. The effective-range expansions (dotted lines) are given by Eq. (2), except for those at 543.48 G for ${}^6$Li, 744.925 G for ${}^{39}$K, and 226.80 G for ${}^{133}$Cs, which are close to zeroes in $a_0\left(B\right)$ and are therefore calculated with the phase-shift form of Eq. (4b). The effective-range expansions deviate substantially from the coupled-channel results at collision energies on the order of $\mu$K. The MQDT approach of Sec. 4 (dashed lines) gives an accurate representation of $\eta \left(E\right)$ over the full range of collision energies. Top: The resonance at $B_0=543.40$ G in the $\mathrm{ab}$ channel of ${}^6$Li. Center: The resonance at $B_0=744.93$ G in the $\mathrm{aa}$ channel of ${}^{39}$K. Bottom: The resonance at $B_0=226.73$ G in the $\mathrm{aa}$ channel of ${}^{133}$Cs.

Figure 3

Energy dependence of the phase shift, $\eta \left(E\right)$, at magnetic fields around the broad resonance in ${}^6$Li. The results of coupled-channel calculations, the effective-range expansion, and the MQDT approach are indistinguishable on this scale.

Figure 4

The resonance at 226.73 G in the $\mathrm{aa}$ channel of Cs. The two-channel model (dashed lines) includes a bare bound state which crosses the threshold at $B_n$ and a reference potential whose first bound level is at $E_{-1}$. The bound states from the coupled-channel calculations (solid lines) have an avoided crossing with the resonance pole at $B_0$.

Figure 5

The functions $C^{-2}\left(E\right)$ (solid) and $tan\lambda \left(E\right)$ (dashed) for Cs, with a variety of different background scattering lengths. The behavior of these functions is determined by both the $C_6$ coefficient and the background scattering length.

Figure 6

Contour plot of ${sin}^2\eta (E,B)$ for $E>0$ around the resonance at $B_0=226.73$ G in the $\mathrm{aa}$ channel of ${}^{133}$Cs, where $E=0$ is the energy of the two separated atoms. $\eta (E,B)$ is calculated using coupled-channel calculations (top); the effective-range expansion, Eq. (2) (middle); and the MQDT approach, Eq. (16) (bottom).

Figure 7

Contour plot of ${sin}^2\eta (E,B)$ for $E>0$ around the resonance at $B_0=554$ G in the $\mathrm{aa}$ channel of ${}^{133}$Cs, where $E=0$ is the energy of the two separated atoms. $\eta (E,B)$ is calculated using coupled-channel calculations (top) and the MQDT approach, Eq. (16) (bottom). Around the pole of the resonance, where the $a_{\mathrm{bg}}$ fitted by the pole formula is roughly accurate, the MQDT approach works well; toward the edges of the resonance, where the $a_{\mathrm{bg}}$ is shifted, the formula breaks down.