Quasi-one-dimensional atomic gases across wide and narrow confinement-induced resonances

We study quasi-one-dimensional atomic gases across wide and narrow confinement-induced resonances (CIRs). We show using the virial expansion that, by tuning the magnetic field, the repulsive scattering branch initially prepared at low fields can continuously go across CIRs without decay; instead, the decay occurs when the noninteracting limit is approached. The interaction properties essentially rely on the resonance width of the CIR. Universal thermodynamics holds for the scattering branch right at a wide CIR, but is smeared out in a narrow CIR due to the strong energy dependence of the coupling strength. In wide and narrow CIRs, the interaction energy of the scattering branch shows different types of strong asymmetry when the decay is approached from opposite sides of the magnetic field. Finally, we discuss the stability of the repulsive branch for a repulsively interacting Fermi gas in different trapped geometries at low temperatures.

Figure 1

Upper panel: Schematic plot of zero-energy coupling strength $g_{1\mathrm{D}}$ across wide (A1) and narrow (A2) CIRs. The labels $a$–$e$ correspond to $B\ll B_{1\mathrm{D}}\left(a\right);B\to B_{1\mathrm{D}}-0^{+}\left(b\right);B\to B_{1\mathrm{D}}+0^{+}\left(c\right);B>B_{1\mathrm{D}}\left(d\right);B\gg B_{1\mathrm{D}}\left(e\right)$. Lower panel: Phase shift ${\delta }_k$ versus $k$ across wide (B1) and narrow (B2) 1D resonances, with each label $(a,b,c,d,e)$ corresponding to a specific $g_{1\mathrm{D}}$ as marked in (A1) and (A2).

Figure 2

$\Delta b_2$ (a) and ${\epsilon }_{\mathrm{int}}$ (b) for two-species ${}^6$Li fermions across a wide CIR at $T\left(\mu \mathrm{K}\right)=1.5\left(\mathrm{dark}\mathrm{black}\right),3\left(\mathrm{medium}\mathrm{purple}\right),\mathrm{and}6\left(\mathrm{light}\mathrm{pink}\right)$. We consider FR at $B_0=834.1$ G with width $W=-300$ G [22]. Transverse confinements are generated by optical lattices with lattice spacing $a_L=500$ nm and depth $V_0=25E_R$ [$E_R=(1/2m){(\pi /a_L)}^2$], giving ${\omega }_{\perp }=\left(2\pi \right)300$ kHz. CIR occurs at $B_{1\mathrm{D}}=B_0-152.2$ G with $W_{1\mathrm{D}}=-147.9$ G, which satisfies $\delta \mu W_{1\mathrm{D}}\gg 2{\omega }_{\perp }\gg T$. Solid and dashed lines are respectively for the scattering [Eq. (14)] and attractive [Eq. (15)] branches. Dash-dotted lines denote the universal values $-\Delta b_2^{\mathrm{sc}}={\epsilon }_{\mathrm{int}}^{\mathrm{sc}}=1/2$ at the CIR.

Figure 3

Two-body energy levels in the center-of-mass frame for a quasi-1D system confined in a tube ($[-L/2,L/2]$) across wide (a) and narrow (b) CIRs. The orange and black dashed lines denote $\pi /2$ and 0 phase shifts [corresponding to $g\left(E\right)=\infty$ and 0]. $E_0={(2\pi /L)}^2/m$. The dotted lines denote noninteracting energy levels with $E^{\left(0\right)}/E_0={(l+1/2)}^2,l=0,1,2...$.

Figure 4

$\Delta b_2$ (a) and ${\epsilon }_{\mathrm{int}}$ (b) for ${}^{87}$Rb bosons across a narrow CIR at $T\left(\mu \mathrm{K}\right)=1\left(\mathrm{dark}\mathrm{black}\right),2\left(\mathrm{medium}\mathrm{purple}\right),\mathrm{and}3\left(\mathrm{light}\mathrm{pink}\right)$. We consider a FR at $B_0=406.2$ G with width $W=0.4$ mG [22]. The optical lattice is the same as that for ${}^6$Li in Fig. 2, giving ${\omega }_{\perp }=\left(2\pi \right)80$ kHz. CIR occurs at $B_{1\mathrm{D}}=B_0-27.1$ mG with $W_{1\mathrm{D}}=0.5$ mG, which satisfies $\delta \mu W_{1\mathrm{D}}\ll T\ll 2{\omega }_{\perp }$. On decreasing $B$ across $B_{1\mathrm{D}}+W_{1\mathrm{D}}$, the scattering branch (solid lines) continuously evolves to the attractive branch (dashed lines) with a bound state emerging at threshold.