Spin-Asymmetric Josephson Effect

We propose that with ultracold Fermi gases one can realize a spin-asymmetric Josephson effect in which the two spin components of a Cooper pair are driven asymmetrically—corresponding to driving a Josephson junction of two superconductors with different voltages $V_{\uparrow }$ and $V_{\downarrow }$ for spin up and down electrons, respectively. We predict that the spin up and down components oscillate at the same frequency but with different amplitudes. Furthermore our results reveal that the standard interpretation of the Josephson supercurrent in terms of coherent bosonic pair tunneling is insufficient. We provide an intuitive interpretation of the Josephson supercurrent as interference in Rabi oscillations of pairs and single particles, the latter causing the asymmetry.

Figure 1

Spin-asymmetric Josephson effect setup. (a) The four-component Fermi gas. Particles in internal (e.g. hyperfine) states $|1⟩-|4⟩$ coexist spatially. The components $|1⟩$ and $|2⟩$ as well as $|3⟩$ and $|4⟩$ form Cooper pairs due to the interactions $U_{12}$ and $U_{34}$. The cross-interactions $U_{13}$, $U_{14}$, $U_{23}$, $U_{24}$ are assumed weak, thus not leading to pairing. The tunneling is driven by applying rf fields to couple the states. The Rabi couplings ${\Omega }_{ij}$ correspond to the weak tunneling link between the two superconductors in a Josephson junction. The detunings ${\delta }_{ij}={\nu }_{ij}-{\omega }_{ij}$ play the role of the voltage, $eV$. Remarkably, one can choose ${\delta }_{13}\ne {\delta }_{24}$ and create the analogue of a spin-dependent voltage in a Josephson junction. The states $|1⟩$ and $|3⟩$ ($|2⟩$ and $|4⟩$) must be states of the same atom to allow the rf coupling, but $|1⟩$ and $|2⟩$ can be e.g. ${}^6\mathrm{Li}$ and ${}^{40}\mathrm{K}$. (b) Cooper pairs of a two-component Fermi gas in a spin-dependent double-well potential. This setup is conceptually equivalent to the system of (a) albeit the absence of cross interactions.

Figure 2

Energy level diagram of the four-state model. The basis states $|12⟩$, $|34⟩$, $|14⟩$, $|23⟩$ form a closed subspace under the Rabi oscillation processes caused by the rf coupling. The paired states $|12⟩$ and $|34⟩$ have an energy lower by $U$ than the states of broken pairs $|14⟩$ and $|23⟩$.

Figure 3

Interference processes leading to the Josephson supercurrent. The initial state is $|\varphi ⟩=(|12⟩+|34⟩)/\sqrt{2}$. (a) The second order transition from $|34⟩$ through $|14⟩$ to $|12⟩$ creates an interference with the initial (zeroth order) population in state $|12⟩$. The resulting contribution is of the order ${\Omega }_{13}{\Omega }_{24}$. Also the same process via the state $|23⟩$ contributes. (b) The first order transitions to the excited state $|14⟩$ from $|12⟩$ and $|34⟩$ create an interference proportional to ${\Omega }_{13}{\Omega }_{24}$.