Frustrated orbital Feshbach resonances in a Fermi gas

The orbital Feshbach resonance (OFR) is a recently developed scheme for magnetically tuning the interactions in closed-shell fermionic atoms. Remarkably, unlike the Feshbach resonances in alkali-metal atoms, the open and closed channels of the OFR are only very weakly detuned in energy. This leads to a unique effect whereby a medium in the closed channel can Pauli block—or frustrate—the two-body scattering processes. Here, we theoretically investigate the impact of frustration in the few- and many-body limits of the experimentally accessible three-dimensional ${}^{173}\mathrm{Yb}$ system. We find that by adding a closed-channel atom to the two-body problem the binding energy of the ground state is significantly suppressed, and by introducing a closed-channel Fermi sea to the many-body problem we can drive the system towards weaker fermion pairing. These results are potentially relevant to superconductivity in solid-state multiband materials, as well as to the current and continuing exploration of unconventional Fermi-gas superfluids near the OFR.

Figure 1

A depiction of the OFR pairwise interaction for closed-shell fermions: The Zeeman energy separations for the ground $|g\rangle$ and excited $|e\rangle$ orbital levels are ${\delta }_g=g_g\left(\delta m_F\right){\mu }_{B}B$ and ${\delta }_e=g_e\left(\delta m_F\right){\mu }_{B}B$, respectively. Two atoms that begin in the open channel $|g\uparrow ,e\downarrow \rangle$ can scatter into the closed channel $|g\downarrow ,e\uparrow \rangle$ via an interorbital nuclear-spin-exchange process (arrows) which couples the channels together. At zero magnetic field ($B=0$) the scattering channels are degenerate, but when $B>0$ their energies vary by a small amount, $\delta ={\delta }_e-{\delta }_g=\left(\delta \mu \right)B$, where $\delta \mu =\left(\delta g\right)\left(\delta m_F\right){\mu }_B$ is the differential magnetic moment. Here, $\delta g=g_e-g_g$ is the differential Landé $g$ factor, $\delta m_F=m_{F,1}-m_{F,2}$ is the difference in quantum number for the nuclear-spin projection, and the Bohr magneton is ${\mu }_B$ [22, 44]. In ${}^{173}\mathrm{Yb}$, measurements give $\delta \mu /\delta m_F\simeq 2\pi \times 110.7\mathrm{Hz}/\mathrm{G}$ [30].

Figure 2

We have two open-channel atoms ($g\uparrow ,e\downarrow$) and one closed-channel atom ($g\downarrow$) in the harmonic-oscillator levels ${\mathbf{n}}_1,{\mathbf{n}}_2,{\mathbf{n}}_3$. After a collision, as drawn, particle 1 is Pauli blocked from occupying the same state as particle 3.

Figure 3

Energy spectrum for the frustrated few-body system using ultracold ${}^{173}\mathrm{Yb}$: The solution to the two-atom problem was first determined in [28] and corresponds to the green dashed lines ($---$) above. The blue solid lines ($————$) denote the energies when a third atom is added to the closed channel, as found by numerically solving Eq. (19). We focus on the ground-state shift, in orange ($..........$) and highlighted by the box, which is due to Pauli blocking in the closed channel. (Note that all energies are shown relative to the values for a noninteracting system.) For reference, the dashed gray vertical line ($----$) indicates the position of the orbital Feshbach resonance at $\sim 34\mathrm{G}$ [51].

Figure 4

Closed-channel excitation spectrum with empty (unshaded) and filled bands (gray shaded areas): The Zeeman field from Eq. (22), $h>0$, shifts the unfilled band down to negative energies. As a result, we begin occupying quasiparticle levels between $k_{\min }$ and $k_{\max }$ in the zero-temperature ground state. For this range of momenta, the scattering states in the closed channel are Pauli blocked.

Figure 5

The average chemical potential (a) and pairing order parameters (b) for the open and closed channels as functions of the interchannel detuning, $\delta /{\delta }_{\mathrm{res}}$, across the OFR. The density of paired atoms is fixed for all plots at $n_{\mathrm{paired}}={10}^{13}$ atoms per ${\mathrm{cm}}^3$, while the unpaired density in the closed channel is fixed to a different value for each plot: $n_{\mathrm{unpaired}}=0$ (red, $————$), $1\times {10}^{13}{\mathrm{cm}}^{-3}$ (blue, $————$), and $2\times {10}^{13}{\mathrm{cm}}^{-3}$ (green, $————$). To aid comparison, the Fermi energy, $E_F$, and Fermi momentum, $k_F$, are defined in terms of $n_{\mathrm{paired}}$. Consequently, we have ${\delta }_{\mathrm{res}}/E_F\simeq 15$ and $1/\left(k_F{a}_T\right)\simeq 1.5$ (where $a_T$ is the larger of the two fixed scattering lengths [51]). In panel (a), the orange stars ($★$) indicate the limiting behavior of the chemical potential on the BEC side (see Sec. 4c), while a close-up view of the BCS side is provided in the inset.

Figure 6

Atom populations for the three crossover scenarios exhibited in Fig. 5: $n_{\mathrm{unpaired}}/n_{\mathrm{paired}}=0$ (upper panel), 1 (middle panel), and 2 (lower panel). The total density, $n$ (blue solid lines), as well as the densities of paired and unpaired atoms, $n_{\mathrm{paired}}$ and $n_{\mathrm{unpaired}}$, are held fixed in each plot, and $n_{\mathrm{paired}}$ has the same value for all three plots. While the open channel always possesses equal spin populations, $n_{g\uparrow }=n_{e\downarrow }$ (green dashed lines), the closed channel is characterized by a different imbalance, $n_{g\downarrow }-n_{e\uparrow }$, in each panel leading to the shifts in Fig. 5.