Asymptotic-bound-state model for Feshbach resonances

We present an asymptotic-bound-state model which can be used to accurately describe all Feshbach resonance positions and widths in a two-body system. With this model we determine the coupled bound states of a particular two-body system. The model is based on analytic properties of the two-body Hamiltonian and on asymptotic properties of uncoupled bound states in the interaction potentials. In its most simple version, the only necessary parameters are the least bound state energies and actual potentials are not used. The complexity of the model can be stepwise increased by introducing threshold effects, multiple vibrational levels, and additional potential parameters. The model is extensively tested on the ${}^6\mathrm{Li}$-${}^{40}\mathrm{K}$ system and additional calculations on the ${}^{40}\mathrm{K}$-${}^{87}\mathrm{Rb}$ system are presented.

Figure 1

Single-atom hyperfine diagrams for ${}^6\mathrm{Li}$ and ${}^{40}\mathrm{K}$. The curves correspond to the eigenvalues of ${\mathcal{H}}^{\mathrm{A}}$ and are labeled by the zero-field quantum numbers $|{\mathit{fm}}_f\rangle$.

Figure 2

ABM calculation for a fictitious homonuclear system with $S=1$ and $I=2$ for an entrance channel with $M_F=M_S+M_I=0$ and $l=0$. The threshold energy of the entrance channel $E_{\alpha \beta }^0$ is shown here explicitly as the dashed line. The energies $E_b$ (solid lines) are labeled by their high-field quantum numbers $|{\mathit{SM}}_S{\mathit{IM}}_I\rangle$. The binding energies of the least bound states in the singlet and triplet potentials are chosen to be ${\epsilon }^0=-10$ and ${\epsilon }^1=-5$. The avoided crossing around $B=0$ is proportional to the hyperfine interaction $a_{\mathrm{hf}}$ and those between the singlet and triplet levels to the wave function overlap ${\eta }_{01}$ and the hyperfine interaction $a_{\mathrm{hf}}$. Four Feshbach resonances occur indicated at the crossings $1$, $2$, and $3$ (double resonance).

Figure 3

Franck-Condon factor ${\eta }_{\nu ,{\nu }^{\prime }}^{01}$ for the least bound states of the ${}^6\mathrm{Li}$-${}^{40}\mathrm{K}$ system, calculated as a function of the triplet binding energy ${\epsilon }^1$ for three different values of ${\epsilon }^0/h=7.16$, 716, and $7.16\times {10}^4$ MHz. ${\eta }_{\nu ,{\nu }^{\prime }}^{01}$ is calculated for the model potential (dashed blue), the $-C_6/r^6$ potential (solid red), and the contact potential, Eq. (17) (dash-dotted green). The black circle indicates the actual value for the least bound state of ${}^6\mathrm{Li}$-${}^{40}\mathrm{K}$ (${\epsilon }^0/h=716$ MHz and ${\epsilon }^1/h=425$ MHz). The nodes in ${\eta }_{\nu ,{\nu }^{\prime }}^{01}$ correspond approximately to the appearance of deeper-lying vibrational states, i.e., for ${\epsilon }^1/h\gtrsim {10}^4\hspace{0.5em}\mathrm{MHz}$, $\nu >1$.

Figure 4

Energies of all the coupled bound states for ${}^6\mathrm{Li}$-${}^{40}\mathrm{K}$ with total spin $M_F=\pm 3$. The black solid line indicates the threshold energy of the entrance channel $|1/2,+1/2\rangle {}_{\mathrm{Li}}\otimes |9/2,+5/2\rangle {}_{\mathrm{K}}$ for $B<0$ and $|1/2,+1/2\rangle {}_{\mathrm{Li}}\otimes |9/2,-7/2\rangle {}_{\mathrm{K}}$ for $B>0$. The gray area represents the scattering continuum and the (colored) lines indicate the coupled bound states. Feshbach resonances occur when a bound state crosses the threshold energy. The color scheme indicates the admixture of singlet and triplet contributions in the bound states obtained from the eigenstates of the Hamiltonian (1). The strong admixture near the threshold crossings at $B\simeq 150$ $\mathrm{G}$ demonstrates the importance of the singlet-triplet mixing in describing Feshbach resonance positions accurately. Since in these calculations the coupled bound states are not coupled to the open channel, they exist even for energies above the threshold.

Figure 5

Bound state spectrum for ${}^{40}\mathrm{K}$-${}^{87}\mathrm{Rb}$ for $M_F=-7/2$ plotted with respect to the threshold energy $E_{\mathrm{K},\mathrm{Rb}}^0$ of the $|9/2,-9/2\rangle {}_{\mathrm{K}}$$+$$|1,1\rangle {}_{\mathrm{Rb}}$ mixture. Solid red lines are ABM calculations and the blue dashed lines are numerical coupled-channel calculations. Good agreement between the two calculations is found, in particular for the weakest bound levels.

Figure 6

Illustration of the threshold behavior in a (fictitious) two-channel version of the dressed ABM. The threshold behavior is determined by the coupling between the least bound level in the open channel in $\mathcal{P}$ space and the resonant bound level in $\mathcal{Q}$ space. The uncoupled levels are shown as the blue (${\epsilon }_P$) and red (${\epsilon }_Q$) dash-dotted lines, with ${\epsilon }_Q$ crossing the threshold at ${\mathrm{B̃}}_0$. The solid black lines represent the dressed levels, with the upper branch crossing the threshold at $B_0$. Near the threshold, the dressed level shows the characteristic quadratic dependence on ($B-B_0$) (see inset). For pure ABM levels (dotted gray) no threshold effects occur and the coupled bound state crosses the threshold at $B_0^{\prime }$.

Figure 7

Dressed bound states for ${}^6\mathrm{Li}$-${}^{40}\mathrm{K}$ for $M_F=-3$ (black lines; see also Table ). The uncoupled $\mathcal{Q}$ and $\mathcal{P}$ bound states (${\mathcal{H}}_{\mathit{PQ}}={\mathcal{H}}_{\mathit{QP}}=0$) are represented by the dot-dashed lines (red and blue, respectively). The gray shaded area is shown in detail in Fig. 8.

Figure 8

A zoom of the dressed ABM as shown in Fig. 7. The dressed molecular states are shown near threshold (black). The field width of a resonance is related to the magnetic field difference between where the dressed and uncoupled $\mathcal{Q}$ bound states cross the threshold.