Feshbach resonances with large background scattering length: Interplay with open-channel resonances

Feshbach resonances are commonly described by a single-resonance Feshbach model, and open-channel resonances are not taken into account explicitly. However, an open-channel resonance near threshold limits the range of validity of this model. Such a situation exists when the background scattering length is much larger than the range of the interatomic potential. The open-channel resonance introduces strong threshold effects not included in the single-resonance description. We derive an easy-to-use analytical model that takes into account both the Feshbach resonance and the open-channel resonance. We apply our model to ${}^{85}\mathrm{Rb}$, which has a large background scattering length, and show that the agreement with coupled-channel calculations is excellent. The model can be readily applied to other atomic systems with a large background scattering length, such as ${}^6\mathrm{Li}$ and ${}^{133}\mathrm{Cs}$. Our approach provides full insight into the underlying physics of the interplay between open-channel (or potential) resonances and Feshbach resonances.

Figure 1

Scattering length as a function of the magnetic field for ${}^{87}\mathrm{Rb}$ in the $\mid f,m_f⟩=\mid 1,1⟩$ hyperfine channel. The horizontal dashed line indicates the direct scattering length $a_{\mathrm{bg}}=100a_0$ (with $a_0$ the Bohr radius), and the vertical dotted line indicates the resonant magnetic field $B_0=1007.4\mathrm{G}$. The field width is given by $\mathrm{\Delta }B=0.2\mathrm{G}$.

Figure 2

(Color online) Left part of figure: schematic illustration of the $P$-channel (dashed) and $Q$-channel (solid) interaction potentials for $B=0$. Right part of figure: the interaction potentials are asymptotically connected to the magnetic-field-dependent two-particle hyperfine-Zeeman eigenenergies. Also shown are the highest $P$-channel vibrational bound state $\left({\nu }_{P,\max }\right)$ and a $Q$-channel vibrational bound state $\left({\nu }_Q\right)$ close to the collision threshold. At the resonant magnetic field $B_0$ the coupling between the $Q$-channel bound state and the $P$-channel scattering state becomes resonant.

Figure 3

Schematic illustration of the poles of the $S$ matrix in the complex $k$ plane. The bound-state pole is located at the positive imaginary $k$ axis. For a less attractive potential, this pole will move down the imaginary $k$ axis, giving rise to a virtual state.

Figure 4

(Color online) Example of an avoided crossing between a $P$-channel and a $Q$-channel bound state. The energy of the bare bound states is indicated by the horizontal and slanted lines. The energies of the dressed states are indicated by the solid lines. One of the dressed states crosses the $P$ threshold at $B_0$. The bare $Q$-channel bound state crosses the $P$ threshold at ${\overline{B}}_0$. Note that the energies are give relative to the $P$-channel collision threshold (horizontal solid line).

Figure 5

(Color online) Effect of virtual state on Feshbach resonance in ${}^{85}\mathrm{Rb}$ in the $\mid 2,-2⟩$ hyperfine channel. Shown are the energy of the unperturbed $Q$-channel bound state (dotted line) and of the (quasi)bound state (solid line) which is “dressed” by the coupling to the $P$ channel. The (quasi)bound state crosses the collision threshold at $B_0$; the unperturbed $Q$-channel bound state crosses the collision threshold at ${\overline{B}}_0$. Note that the energies are give relative to the $P$-channel collision threshold (horizontal solid line).

Figure 6

Schematic illustration of the movement of the poles of the $S$ matrix in the complex $k$ plane (a) and the two-sheeted complex $E$ plane (b). The solid $E$ axes correspond to the first (or physical) Riemann sheet, while the dashed $E$ axes correspond to the second (or nonphysical) Riemann sheet. Both sheets have a branch-cut discontinuity on the real axis running from $0\text{to}\infty$. The two Riemann sheets are connected by the positive real axis, where $\mathrm{Im}\left(k\right)=0$. Initially, the pole is located at the positive imaginary $k$ axis, which corresponds to a bound-state pole on the negative real $E$ axis on the first Riemann sheet. If the interaction potential effectively gets less attractive, the pole moves towards $k=0$, at some point crosses the origin, and turns into a virtual-state pole at the negative imaginary $k$ axis. This corresponds to a virtual-state pole at the negative real $E$ axis on the second Riemann sheet.

Figure 7

Scattering phase $\delta \left(k\right)$ for ${}^{85}\mathrm{Rb}$ in the $\mid f,m_f⟩=\mid 2,-2⟩$ spin channel. The black dots represent the numerical results. The solid line is obtained from Eq. (21). The dashed line is obtained from the contact potential approximation $\delta \left(k\right)=-\arctan \left[k{a}_{\mathrm{bg}}\right]$. The scattering length of ${}^{85}\mathrm{Rb}$ in this channel is $a_{\mathrm{bg}}=-443a_0$, in agreement with coupled-channel calculations. The background scattering length is $a_{\mathrm{bg}}^P=+119a_0$, and the virtual state is located at $k_{\mathrm{vs}}=-i{\kappa }_{\mathrm{vs}}=-1.78\times {10}^{-3}i[1∕{\mathrm{a}}_0]$, which corresponds to the term $-1∕{\kappa }_{\mathrm{vs}}=-562a_0$ in the expression for $a_{\mathrm{bg}}$.

Figure 8

$S_P$ matrix for negative energies on the nonphysical Riemann sheet for ${}^{85}\mathrm{Rb}$ in the $\mid 2,-2⟩$ hyperfine channel. The pole of the $S_P$ matrix on this sheet is located ${ϵ}_{\text{virtual}}∕k_B=-6.45\mu \mathrm{K}$.

Figure 9

Energy width of the dressed quasibound Feshbach state for ${}^{85}\mathrm{Rb}$ in the $\mid 2,-2⟩$ hyperfine channel. The dotted line shows the energy width according to Eq. (30), where only the virtual state, with ${\kappa }_{\mathrm{vs}}=2.54\times {10}^{-3}{\mathrm{K}}^{1∕2}$ is taken into account, and $A_{\mathrm{vs}}=1.94\times {10}^{-8}{\mathrm{K}}^2$. The solid line shows the energy width according to Eq. (34), where the highest bound state of the $P$ channel is included as well, with ${\kappa }_{\mathrm{bs}}=0.103{\mathrm{K}}^{1∕2}$. In this case, $A_{\mathrm{vs}}=1.92\times {10}^{-8}{\mathrm{K}}^2$ and $A_{\mathrm{bs}}=1.26\times {10}^{-5}{\mathrm{K}}^2$.

Figure 10

(Color online) Energy of the dressed (quasi)bound Feshbach state. The black dots indicate the coupled-channel data. The thick solid line indicates the energy according to our model. The thin solid line is the energy of the unperturbed $Q$-channel bound state, which around threshold is given by the linear expression ${ϵ}_b^Q\left(B\right)=\mathrm{\Delta }{\mu }^{\mathrm{mag}}(B-{\overline{B}}_0)$, with $\mathrm{\Delta }{\mu }^{\mathrm{mag}}=-1.75\times {10}^{-4}K∕G=-3.64\mathrm{MHz}∕G$ and ${\overline{B}}_0=160.1G$.

Figure 11

(Color online) Same as in Fig. 10, but now for energies closer to the $P$-channel threshold.

Figure 12

(Color online) Comparison of coupled-channel calculations of the binding energy (black dots) with the contact model (dotted line), the single-resonance Feshbach model (dash-dotted line), the effective range model (dashed line), and our virtualstate model (solid line). Although for the magnetic fields shown the effective range model and our model give comparable results, our model gives an analytical expression for the binding energy. This is not the case for the effective range model, where afield-dependent parameter $r_0\left(B\right)$ has to be taken into account that has to be calculated numerically.