Shifts and widths of Feshbach resonances in atomic waveguides

We develop and analyze a theoretical model which yields the shifts and widths of Feshbach resonances in an atomic waveguide. It is based on a multichannel approach for confinement-induced resonances (CIRs) and atomic transitions in the waveguides in the multimode regime. In this scheme we replace the single-channel scalar interatomic interaction by the four-channel tensorial potential modeling resonances of broad, narrow, and overlapping character according to the two-channel parametrization of Lange et al. [Phys. Rev. A 79, 013622 (2009)]. As an input the experimentally known parameters of Feshbach resonances in the absence of the waveguide are used. We calculate the shifts and widths of $s$-, $d$-, and $g$-wave magnetic Feshbach resonances of Cs atoms emerging in harmonic waveguides as CIRs and resonant enhancement of the transmission at zeros of the free space scattering length. We have found the linear dependence of the width of the resonance on the longitudinal atomic momentum and quadratic dependence on the waveguide width. Our model opens possibilities for quantitative studies of the scattering processes in ultracold atomic gases in waveguides beyond the framework of $s$-wave resonant scattering.

Figure 1

The $s$-wave scattering length $a$ of $|F=3,m_F=3\rangle$ Cesium atoms as a function of the magnetic field $B$. The solid curve shows the analytical result (5) and the dots show the numerical result (see the text).

Figure 2

The transmission coefficient $T$ for the harmonic trap with ${\omega }_{\perp }=14.9$ kHz as a function of the magnetic field $B$ for the case of the two-channel (solid curve) and four-channel (dots) potential (2).

Figure 3

The coupling constant $g_{1\text{D}}$ as a function of the scattering length $a$ in free space in the region near the $s$-wave Feshbach resonance, calculated for the harmonic trap with ${\omega }_{\perp }=14.9$ kHz. Dots indicate the numerical data and the solid curve the analytical results with the formulas derived in the pseudopotential approach [5].

Figure 4

The transmission coefficient $T\left(B\right)$ as a function of the strength of the magnetic field for several trap frequencies ${\omega }_{\perp }$. Here the $s$-wave scattering length $a\left(B\right)$ from Fig. 1 is also given (solid line).

Figure 5

The dependence of the $s$-wave scattering length $a\left(B\right)$ on the width $a_{\perp }$ of the waveguide at the points $B_{\text{min}}$ of the minimum of the transmission coefficient $T(a\left(B\right),a_{\perp })$ (see Fig. 4). The dots, pluses, and circles correspond to the calculated points near the $s$-, $d$-, and $g$-wave magnetic Feshbach resonances, respectively. The solid curve corresponds to the formula $a=a_{\perp }/C$ [5].

Figure 6

(a) The dependence of the width ${\Gamma }_i^{*}$ on the longitudinal momentum $k_0$ near the $d$-wave Feshbach resonance. The solid line has been obtained via Eq. (10); solid circles indicate the widths ${\Gamma }_i^{*}\left(k_0\right)$ extracted from the numerically calculated $T(B,k_0)$. (b) The transmission coefficient $T$ as a function of the magnetic field $B$ calculated for a few $k_0$. For both subfigures ${\omega }_{\perp }$ $=$ 14.9 kHz.

Figure 7

The dependence of the width ${\Gamma }_i^{*}$ on the trap frequency ${\omega }_{\perp }$ near the $d$-wave Feshbach resonance at $k_0=2.0899\times {10}^{-8}\left(a_0^{-1}\right)$. The solid curve has been obtained via Eq. (10); solid circles indicate the widths ${\Gamma }_i^{*}\left({\omega }_{\perp }\right)$ extracted from the numerically calculated $T(B,{\omega }_{\perp })$. The analytical (dashed line) and numerical (squares) results for the right half-width $1/2{\Gamma }_{\text{CIR}}^{\left(i\right)}$ versus ${\omega }_{\perp }$ are also given.

Figure 8

The relative populations $P_i/P_e$ (13) of the molecular states $i=s,d$, and $g$ calculated as a function of $B$ near the $d$-wave Feshbach resonance in Cs for the pair collisions in free space (a) and in the harmonic waveguide with ${\omega }_{\perp }=59.6$ kHz (b). The $s$-wave scattering length $a\left(B\right)$ in free space from Fig. 1 is also given.

Figure 9

Schematic picture (presented in Refs. [6, 17] for a single-channel potential model of interatomic interaction) of the bound state energy $E_b$ of two atoms as a function of $a_{\perp }/a$ in free space (solid line) and in the harmonic waveguide (dashed line). The first excited resonance state energy $E_r$ is also presented.

Figure 10

The relative populations $P_i/P_e$ (13) of the molecular states $i=s,d$, and $g$ calculated as functions of $B$ near the $s$-wave Feshbach resonance for Cs for the pair collisions in free space (a) and in the harmonic waveguide with ${\omega }_{\perp }=59.6$ kHz (b). The $s$-wave scattering length $a\left(B\right)$ from Fig. 1 is also presented.

Figure 11

The relative populations $P_i/P_e$ (13) of the molecular states $i=s,d$, and $g$ calculated as functions of $B$ near the $g$-wave Feshbach resonance in Cs for the pair collisions in free space (a) and in the harmonic waveguide with ${\omega }_{\perp }=59.6$ kHz (b). The $s$-wave scattering length $a\left(B\right)$ from Fig. 1 is also presented.