Spin-asymmetric Josephson plasma oscillations

The spin-asymmetric Josephson effect is a proposed quantum-coherent tunneling phenomenon where Cooper-paired fermionic spin-$\frac{1}{2}$ particles, which are subjected to spin-dependent potentials across a Josephson junction, undergo frequency-synchronized alternating-current Josephson oscillations with spin-dependent amplitudes. Here, in line with present-day techniques in ultracold Fermi gas setups, we consider the regime of small Josephson oscillations and show that the Josephson plasma oscillation amplitude becomes spin dependent in the presence of spin-dependent potentials, while the Josephson plasma frequency is the same for both spin components. Detecting these spin-dependent Josephson plasma oscillations provides a possible means to establish the yet-unobserved spin-asymmetric Josephson effect with ultracold Fermi gases using existing experimental tools.

Figure 1

Origin of the spin-asymmetric critical Josephson currents. Considering the dynamics of a single Cooper pair across a Josephson junction with spin-dependent potentials, the Josephson current results from a superposition of the paired states ${|\uparrow \downarrow \rangle }_L{|\varnothing \rangle }_R$ and ${|\varnothing \rangle }_L{|\uparrow \downarrow \rangle }_R$. Here we show the tunneling processes that contribute to (a) $I_{\uparrow }^J$ and (b) $I_{\downarrow }^J$. The three processes are the following. First, there is a pair-tunneling contribution via the intermediate state ${|\uparrow \rangle }_L{|\downarrow \rangle }_R$ (left panel). Second, there is another pair-tunneling contribution via the other intermediate state ${|\downarrow \rangle }_L{|\uparrow \rangle }_R$ (middle panel). We have included the loop with label 1 to remind us that these usual Josephson processes describe the interference between the tunneled pair and the initial population (indicated by the label 1) of the state ${|\uparrow \downarrow \rangle }_L{|\varnothing \rangle }_R$. The pair-tunneling processes are the same for both spin components. Finally, we find that there is also a virtual single-particle interference contribution (right panel). The single-particle interference term is different for the spin-$\uparrow$ and spin-$\downarrow$ components due to the presence of the spin-dependent potential that lifts the energy degeneracy of the intermediate states. This causes the spin-asymmetric Josephson effect.

Figure 2

Schematic of a spin-dependent Josephson junction. A superfluid Fermi gas is divided into two reservoirs denoted by $L$ and $R$. The reservoirs are connected via the weak tunneling coupling ${\mathrm{\Omega }}_{\sigma }$, which induces Josephson oscillations across the junction. Additionally, a spin-dependent potential ${\delta }_{\sigma }$ is applied across the junction, which creates a spin asymmetry in the Josephson current.

Figure 3

Asymmetry in the spin-dependent plasma oscillation amplitudes as a function of the difference in the spin-dependent potentials for Josephson frequency ${\omega }_J=0.11E_F$ and interaction strength $k_F{a}_s=-1.0$ [black solid curve, only in (a)], $k_F{a}_s=-1.5$ (blue solid curve), $k_F{a}_s=-2.0$ (red dashed curve), $k_F{a}_s=-2.5$ (yellow dash-dotted curve), and $k_F{a}_s=-3.0$ (purple dotted curve). The temperature is (a) $T=0.05T_F$, (b) $T=0.07T_F$, and (c) $T=0.09T_F$. The temperature regime is the same as in the experiment in [8]. In (a), the $k_F{a}_s=-1.0$ curve is included as the reference line to the unitary Fermi gas regime.

Figure 4

Critical Josephson current $I_{\sigma }^C$ as a function of ${\tilde{\delta }}_{\overline{\sigma }}$ for (a) interaction $k_F{a}_s=-3.0$ (solid curve), $k_F{a}_s=-2.0$ (dashed curve), and $k_F{a}_s=-1.0$ (dotted curve) at temperature $T=0.07T_F$ and (b) temperature $T=0.05T_F$ (solid curve), $T=0.07T_F$ (dashed curve), and $T=0.09T_F$ (dotted curve) for interaction $k_F{a}_s=-1.5$. The divergence of $I_{\sigma }^C$ at ${\tilde{\delta }}_{\overline{\sigma }}=2\mathrm{\Delta }$ is called the Riedel peak.

Figure 5

Asymmetry in the spin-dependent plasma oscillation amplitudes as a function of the difference in the spin-dependent potentials for parameter values obtained in the experiment in [59]. The Josephson frequency is ${\omega }_J=0.08E_F$, the interaction strength is $k_F{a}_s=-4.0$, the temperature is $T=0.06T_F$, and the gap is $\mathrm{\Delta }=0.22E_F$.