QP Tunneling Rate — Asymmetric Junction

Calculates the quasiparticle (QP) tunneling rate through a Josephson junction with gap asymmetry Δ₂ ≠ Δ₁. The result depends critically on whether the quasiparticles are cold (relaxed to the gap edge) or hot (excited well above the gap).

Scenario A: Cold (Relaxed) Quasiparticles

The QP sits at \(E \approx \Delta_1\) and faces a hard wall: it can only tunnel if a thermal fluctuation overcomes the gap difference \(\delta\Delta = \Delta_2 - \Delta_1\). The rate is the symmetric cold rate, suppressed exponentially by a Boltzmann factor:

\[ \Gamma_{\text{cold,asym}} \approx \frac{16\,E_J}{h\,\Delta_1\,\nu_0\,V} \sqrt{\frac{\pi\Delta_1}{2\,k_B T_{qp}}} \exp\!\left(-\frac{\Delta_2 - \Delta_1}{k_B T_{qp}}\right) \]

Key takeaway: A slight gap asymmetry exponentially suppresses tunneling of relaxed QPs, effectively trapping them on the lower-gap island.

Scenario B: Hot Quasiparticles

The QP has high energy \(E = N\Delta_1\) and easily clears the gap difference \(\Delta_2\). The densities of states on both sides flatten out to roughly 1, and the gap asymmetry becomes mathematically negligible:

\[ \Gamma_{\text{hot,asym}} \approx \frac{8\,E_J}{h\,\Delta_1\,\nu_0\,V} \left(1 + \frac{\Delta_2}{N^2\Delta_1}\right) \]

Key takeaway: Hot quasiparticles punch right through the junction almost as if it were a normal metal, entirely unaffected by the gap asymmetry until they scatter and lose their energy.

Quasiparticle Tunneling Rate — Asymmetric Junction

Scenario A: Cold (Relaxed) Quasiparticles

Scenario B: Hot Quasiparticles