Josephson Junction

Consider superconducting metals I and II separated by a very thin insulating layer, such that that electron wave functions can overlap between the metals (Josephson junction). A battery V is connected across the junction to ensure an average charge neutrality (see Figure 1.1). This situation can be described by means of the coupled Schrodinger equations:

(i)

Here Ψ₁ and Ψ₂ are the probability amplitudes for an electron in I and II, U₂ and U₂ are the electric potential energies in I and II, K is the coupling constant due to the insulating layer, and KΨ_1­Ψ^*₂/Ψ^*₁ and  KΨ_2­Ψ^*₁/Ψ^*₂ describe the battery as a source of electrons.

a) Show that ρ₁ = |Ψ₁|² and ρ₂ = |Ψ₂|²  are constant in time.

Figure 1.1

b) Assuming ρ₁ = ρ₂ = ρ₀ (same metals) and expressing the probability amplitudes in the form

(ii)

find the differential equations for θ₁ and θ₂.

c) Show that the battery current

(iii)

oscillates, and find the frequency of these oscillations.

SOLUTION

a) Take the first of equations (i),

(1)

and its complex conjugate Ψ^*₁ and Ψ₁ multiply them by and respectively:

(2)

(3)

Subtracting (3) from (2) yields

(4)

Similarly, from the second of (i),

(5)

b) Substituting the solutions and  into (1), we obtain the expression for θ₁:

(6)

Taking (6), the analogous expression for θ₂ gives

(7)

(8)

Subtracting (7) from (8), we obtain

(9)

where δ = θ₂ – θ₁. So

(10)

where eV = U₁ – U₂.

c) The battery current

(11)