Simple model of a Feshbach resonance in the strong-coupling regime

We use the dressed potentials obtained in the adiabatic representation of two coupled channels to calculate $s$-wave Feshbach resonances in a three-dimensional spherically symmetric potential with an open channel interacting with a closed channel. Analytic expressions for the $s$-wave scattering length $a$ and number of resonances are obtained for a piecewise constant model with a piecewise constant interaction of the open and closed channels near the origin. We show analytically and numerically that, for strong enough coupling strength, Feshbach resonances can exist even when the closed channel does not have a bound state.

Figure 1

Dressed potential picture of coupled channel model for piecewise constant potential. (a) The level structure of original potentials $U\left(r\right)$. Open channel $U_o\left(r\right)$ and closed channel $U_c\left(r\right)$ (coupling $\tau =0$ and detuning $\nu =0)$. (b) Dressed potentials $U_{+}\left(r\right)$ and $U_{-}\left(r\right)$ for nonvanishing coupling, $\tau >0$, and asymptotic potential energy difference $\nu =0$. (c) Same as (b) but for $\nu \gg 0$. In all panels, dashed lines denote bound states of the $U_o\left(r\right)$ and $U_{-}\left(r\right)$ potentials and dotted lines correspond to $U_c\left(r\right)$ and $U_{+}\left(r\right)$. All length variables are expressed in units of $r_0$ and all energies are given in units of ${\hslash }^2/\left(2m{r}_0^2\right)$.

Figure 2

The depth of the dressed potentials versus the detuning $\nu$, depending on the relative depth of $V_o$ and $V_c$. (a) $V_c<V_o$, and in particular ${\kappa }_o=3,{\kappa }_c=2,\tau =10$. (b) $V_o<V_c$ $\left({\kappa }_o=1.8,{\kappa }_c=7.5,\tau =6.3\right)$, where the avoided crossing is clearly seen. All energies are given in units of ${\hslash }^2/\left(2m{r}_0^2\right)$.

Figure 3

Feshbach resonance for a repulsive potential in the closed channel. (a) Threshold curves (same as in Fig. 6) plotted versus the depth of the closed channel (with ${\kappa }_o=1.8)$. (b) Threshold curves plotted versus the depth of the open channel (for ${\kappa }_c^2=-0.04)$. The dashed blue vertical line refers to the numerical calculations where we keep the depth of the potential fixed $({\kappa }_o=1.8$ and ${\kappa }_c^2=-0.04)$ and increase the value of $\tau$ in the analysis illustrated in Figs. 4 and 5. The purple circle (square) on this line marks the value of $\tau =59.75$ $\left(\tau =60.86\right)$ which is used in the red line in Fig. 4 (black lines in Figs. 4 and 5). Because the circle and square almost coincide on the scale of the figure, we added an appropriate inset.

Figure 4

Scattering length in units of $r_0$ for a repulsive potential in the closed channel as a function of $\nu$ (for ${\kappa }_c^2=-0.04)$. Open channel supports one bound state $\left({\kappa }_o=1.8\right)$. The dashed red curve shows the scattering length for a coupling strength of $\tau =59.75$ (purple circle in Fig. 3) which is below the second threshold at ${\tau }_{\mathrm{th}}=60.06$. The solid black curve is for $\tau =60.86$ (purple square in Fig. 3) which is just above the threshold. See also Fig. 5.

Figure 5

Energy of the bound states of the coupled system in units of ${\hslash }^2/2m{r}_0^2$ for a repulsive potential in the closed channel $\left({\kappa }_c^2=-0.04\right)$ as a function of the energy shift $\nu$ (in the same units as $E_b)$. The coupling strength $\tau =60.86$, is just above threshold (purple square in Fig. 3). The inset shows the small $\nu$ and $E_b$ region and the dashed two-sided arrow depicts its position.

Figure 6

Feshbach resonances in the case of an attractive closed channel potential. The threshold values of the coupling $\tau$ at which successive resonances appear are plotted as a function of ${\kappa }_o$. Each solid curve plotted versus the depth of the closed channel potential (for fixed value of ${\kappa }_c=7.5)$ represents one such threshold. When a solid curve is crossed, a new Feshbach resonance emerges and when a black dashed curve is crossed, one resonance disappears. The dashed blue vertical line refers to our numerical studies, as discussed in the text, where the depth of the potentials is held fixed $\left({\kappa }_o=1.8,{\kappa }_c=7.5\right)$ and the value of $\tau$ is increased. The purple circle (square) on this line marks the value of $\tau =6.3$ $\left(\tau =87.5\right)$ which is used in Figs. 7 and 8 [Figs. 7 and 8].

Figure 7

Scattering length in the units of $r_0$ as a function of energy shift $\nu$ in units of ${\hslash }^2/\left(2m{r}_0^2\right)$ for an attractive closed channel potential with ${\kappa }_c=7.5$. Fixed value of ${\kappa }_o=1.8$ corresponds to the case when the open channel supports one bound state. Coupling strength is equal to (a) $\tau =6.3$ (purple circle in Fig. 6), and (b) $\tau =87.5$ (purple square in Fig. 6). Upon increasing the values of coupling strength from (a) to (b) following the blue line (see Fig. 6) one passes through three threshold curves. Two new resonances are created upon crossing the two solid lines and one resonance disappears upon crossing the dashed line.

Figure 8

(a) Energy of the bound states of the coupled system for an attractive potential in the closed channel as a function of the coupling strength $\tau$ for $\nu =0$. The values of ${\kappa }_o$ and ${\kappa }_c$ are as in Fig. 7. The inset shows the magnification of the top curve near the origin and reveals that the bound state disappears through the threshold and returns. (b) Energy of the bound states for coupling strength $\tau =87.5$ (just above the threshold value; corresponding to purple square in Fig. 6) as a function of the energy shift $\nu$. The inset shows the bound state energy for small values of $\nu$. All energies are given in units of ${\hslash }^2/\left(2m{r}_0^2\right)$.