Pairing of Fermions with Unequal Effective Charges in an Artificial Magnetic Field

Artificial magnetic fields (AMFs) created for ultracold systems depend sensitively on the internal structure of the atoms. In a mixture, each component experiences a different AMF depending on its internal state. This enables the study of Bardeen-Cooper-Schrieffer pairing of fermions with unequal effective charges. In this Letter, we investigate the superconducting (SC) transition of a system formed by such pairs as a function of field strength. We consider a homogeneous two-component Fermi gas of unequal effective charges but equal densities with attractive interactions. We find that the phase diagram is altered drastically compared to the usual balanced charge case. First, for some AMFs there is no SC transition and isolated SC phases are formed, reflecting the discrete Landau level (LL) structure. SC phases become reentrant both in AMF and temperature. For extremely high fields where both components are confined to their lowest LLs, the effect of the charge imbalance is suppressed. Charge asymmetry reduces the critical temperature even in the low-field semiclassical regime. We discuss a pair breaking mechanism due to the unequal Lorentz forces acting on the components of the Cooper pairs to identify the underlying physics.

Figure 1

Phase diagram of the system as a function of dimensionless temperature $\tilde{T}=k_{B}T/\hslash \omega$ and effective density $n_1=N{\pi }^2{\ell }^3$, where $N$ is the real-space density and $\ell =\sqrt{\hslash /m\omega }$. Notice that increasing AMF, $B=m\omega /\sqrt{q_1{q}_2}$, corresponds to lower $n_1$ values. Two frequency ratios ${\omega }_r=1$ and ${\omega }_r=0.95$ are displayed for the same interaction strength ${\tilde{a}}_s=a_s(N{\pi }^2{)}^{1/3}=0.53$. For ${\omega }_r=1$, there is SC transition at any field, the oscillations in $T_C$ (stars) originate from the LL structure. For ${\omega }_r=0.95$, these oscillations evolve into bubble SC regions (shaded areas). The system is not SC even at zero temperature between the bubbles and the transition becomes weakly reentrant in temperature. At the low (many LLs) and high field (only LLL) regimes, $T_C$ is not affected significantly by a small charge imbalance.

Figure 2

Pairing susceptibility (left panel) and respective phase diagrams (right panel) for three frequency ratios ${\omega }_r=1$, 0.95, 0.75. The system is made dimensionless with effective magnetic energy $\hslash \omega$ as in Fig. 1 and decreasing $n_1$ corresponds to increasing AMF at fixed real-space density $N$. The phase diagrams are in linear scale in order to cover $T_C=0$ and plotted for intermediate field strengths, where charge imbalance effects are most prominent. (a) Pairing susceptibility of equal charges diverges at low temperature, guaranteeing SC for any field value. Divergence is more pronounced at LL thresholds. The corresponding phase diagram (b) is obtained by Eq. (7). The phase boundary is also highlighted on the surface. (c) Even a slight asymmetry between the charges, ${\omega }_r=0.95$, lifts the low temperature divergences and the oscillations in (b) turn into bubble SC phases (d). Each LL susceptibility peak is split into smaller peaks, thus, the bubble phases branch into smaller bubbles for weaker interactions (not displayed). (e),(f) The mismatch between LL spectra is greater for smaller ${\omega }_r$, resulting in prominent reentrance with temperature. The maximum reentrance temperature is controlled by $\hslash |{\omega }_1-{\omega }_2|$.