Two superconductors, an insulator between them thin enough for electrons to tunnel. Each superconductor is one giant quantum state with a single phase; the junction only cares about the difference, δ. Hold δ still and a current flows through the barrier with no voltage and no heat. Push harder than the junction can carry and δ starts to turn: a voltage appears, the current through the barrier oscillates, and every bit of heat goes through the resistor beside it.
Raise the bias past Ic. Watch where the current goes.
Live state
δ= 0.00 rad
I / I0= 0.50
I0(T) / I0(0)= 1.00
⟨V⟩ / I0R= 0.00
P / I0²R= 0.000
LJ / LJ0= 1.00
βc= 0.50
Γ= 0.000
1 · What it is
“A Josephson tunnel junction is formed by separating two superconducting electrodes with an insulator thin enough so that electrons can quantum-mechanically tunnel through the barrier, as illustrated in Fig. 1.” The current through it is set not by a voltage but by the phase difference δ = φL − φR between the two condensates. Martinis & Osborne §2, Fig. 1.
2 · Why a superconductor has one phase
Ψ = Πk (uk + vk eiφ c†k c†−k) |0⟩
In BCS every Cooper pair (k, −k) is a two-state system, empty or filled, which can be written as a spin. The phonon-mediated attraction couples every pair-spin to every other in the x–y plane, and because that coupling is attractive the spins all align: “every spin in the bath has the same phase φk”. One phase for a macroscopic number of pairs is what makes φ a macroscopic variable you can draw as a single arrow. Nothing in the Hamiltonian prefers a direction in that plane, so φ costs no energy to set and is free to be driven by the circuit. M&O §4, Eqs. 13–16, 26–30.
3 · The two Josephson relations
IJ = I0 sin δ(1)
V = (Φ0/2π) dδ/dt(2)
Φ0 = h/2e is the flux quantum, 2.07 × 10−15 Wb. Relation (2) is a gauge transformation on the pair wavefunction: a voltage V on one electrode advances its phase at 2eV/ħ, so a standing phase means zero voltage and a turning phase means a voltage. 1 µV across the junction turns δ at 483.6 MHz. M&O Eqs. 1–2, 17–19.
4 · It is an inductor, a nonlinear one
LJ = Φ0 / (2π I0 cos δ)(4)
UJ = −(I0Φ0/2π) cos δ = −EJ cos δ(8)
Differentiate (1), substitute (2): dIJ/dt ∝ V, which is an inductor. But the inductance carries 1/cos δ: it grows without bound as δ → π/2 and is negative between π/2 and 3π/2. “…the change of the qubit state by only one photon in energy can modify the junction inductance by order unity!” The stored energy is −EJ cos δ, and a current bias adds −(IΦ0/2π)δ: together they are the tilted washboard drawn below. M&O §2, Eqs. 3–9; §1.
5 · Why the supercurrent makes no heat
UJ = −(RK/8RN) Δ cos δ(59)
IJ = (πΔ/2eRN) sin δ(60)
To first order, tunneling moves one electron and leaves a quasiparticle on each side; that costs energy, needs eV ≥ 2Δ, and is dissipative (the “semiconductor model”: no current below 2Δ/e). To second order an electron tunnels across and its hole-part tunnels back, no excitation is left behind, and what remains is a shift of the ground-state energy that depends on δ. Current from a ground state is current with nothing to relax: “The Josephson current is a dissipationless current because it arises from a new ground state of the two superconductors produced by the tunneling interaction.” RK = h/e² is the resistance quantum; I0RN = πΔ/2e is a property of the material, not of the junction's size. M&O §5, Eqs. 43–47, 52–60; Ambegaokar & Baratoff 1963.
6 · The circuit: RCSJ
I = I0 sin δ + V/R + C dV/dt
βc = 2e I0 R² C / ħ
A real junction has a capacitance (two plates, a thin oxide) and a channel for quasiparticles that behaves like a resistor. Bias current splits three ways, which the circuit panel animates: through the barrier as supercurrent, through R as heat, into C as displacement. In the zero-voltage state all of it crosses the barrier. In the running state the supercurrent oscillates but does not average to zero: its mean is I − ⟨V⟩/R, which is why the I–V curve sits to the left of the resistor line V = IR (at I = 1.5 I₀ with βc = 0.5 the barrier still carries a quarter of an I₀ on average). The rest of the DC current rides R. βc (Stewart–McCumber) is the mass of the bead: for βc ≫ 1 a running bead keeps running when the bias drops below I0, which is the hysteresis the I–V panel traces. The Hamiltonian without R is M&O's phase-qubit Hamiltonian. M&O Eq. 11; Stewart 1968; McCumber 1968.
7 · Temperature
⟨IJ⟩ = (πΔ(T)/2eRN) tanh(Δ(T)/2kBT) sin δ(82)
The junction carries two quasiparticle bound states at ±Δ√(1 − τ sin²(δ/2)); their currents are equal and opposite. Thermal population of the upper one cancels part of the lower one, and the difference of two Fermi factors is the tanh. Δ(T) itself closes at Tc (BCS), so I0(T) → 0 and the junction becomes RN: the washboard goes flat. Cooling does two things at once on this page: it deepens the wells, and it quiets the Johnson-noise kicks (Γ = 2ekBT/ħI0, exaggerated on the slider so you can see them). M&O §6, Eqs. 78–82; BCS gap: Bardeen, Cooper & Schrieffer 1957; the form Δ(T) ≈ Δ(0) tanh(1.74√(Tc/T − 1)) is a textbook fit to the BCS gap equation, within 2 %.
8 · Two energy scales
δ and the charge Q on the capacitor do not commute: [δ, Q] = 2ei. When EJ ≫ EC the phase is sharp and classical, which is the regime this page simulates (a 100 µA junction has EJ/kB ≈ 2,400 K). When EJ ~ EC the phase smears and you have a charge qubit. A phase qubit is this very circuit biased just under I0, where the well is cubic, holds 3–5 levels, and the nonlinearity of LJ makes the lowest two addressable. M&O §3, Eqs. 11–12 and the paragraph after Fig. 2.
9 · What the resistor really is
EJ± = ±Δ [1 − τ sin²(δ/2)]½(78)
QMAR = 2Δ(1 − τ)½/V(95)
R is a stand-in. Exactly, each conduction channel of transmission τ holds two bound states; under a constant voltage, δ turns uniformly, the ground state is carried adiabatically around, and its current averages to zero. Near δ ≈ π the gap between the bound states is smallest and a Landau–Zener transition can occur with probability W = (τ/2)2Δ/eV in the tunnel limit; the transition dumps a packet of charge QMAR through the junction by multiple Andreev reflection and leaves real quasiparticles. That is the dissipation. “…it is clearly incorrect to picture the quasiparticle and Josephson current as separate entities, as suggested by the calculations of perturbation theory.” M&O §6–7, Eqs. 78, 88–95, Fig. 12.
10 · The energy line, in one sentence
Heat leaves through R, and current goes through R only while δ moves. Below I0(T) the phase sits in a well: V = 0, P = 0, the current still flows. Above Tc there is no well at all and the same current pays I²RN. Cooling raises I0 and lowers the noise, so a given current drops from the resistive branch into the well. The P bar beside the I–V panel reads ⟨(dδ/dt)²⟩ in units of I0²R.
What this is not
The RCSJ model with one linear R, equal to RN at every voltage and temperature: no gap structure in the I–V (a real quasiparticle branch turns on at 2Δ/e, and the subgap resistance at low T is orders of magnitude above RN, so a real tunnel junction is far more hysteretic than the default slider), no multiple Andreev reflection, no macroscopic quantum tunnelling, no 1/f critical-current noise. Equal gaps on both sides and constant tunnelling matrix elements, as Martinis assumes. The noise is classical Johnson noise of that constant R: no zero-point noise, no shot noise. A lumped, field-free junction: no magnetic field or Fraunhofer pattern, no self-field, no microwave drive, so no Shapiro steps. Noise is a Gaussian kick after each RK4 step. Ambegaokar–Baratoff is for tunnel junctions; weak links differ. The cartoon's tunnelling dots are a current meter, not a wavefunction.
Sources
J. M. Martinis & K. Osborne, Superconducting Qubits and the Physics of Josephson Junctions, Les Houches lecture notes (NIST Boulder). Equation numbers above are theirs.
V. Ambegaokar & A. Baratoff, Phys. Rev. Lett. 10, 486 (1963); erratum 11, 104.
J. Bardeen, L. N. Cooper & J. R. Schrieffer, Phys. Rev. 108, 1175 (1957).
W. C. Stewart, Appl. Phys. Lett. 12, 277 (1968); D. E. McCumber, J. Appl. Phys. 39, 3113 (1968).
B. D. Josephson, Phys. Lett. 1, 251 (1962).
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