Josephson Effect

Derivation: Josephson Relations from Coupled Superconductors

We derive both Josephson relations using the tunneling Hamiltonian approach. Consider two superconductors with order parameters $\Psi_1 = \sqrt{n_1}\,e^{i\theta_1}$ and$\Psi_2 = \sqrt{n_2}\,e^{i\theta_2}$, weakly coupled through a barrier with tunneling amplitude $K$.

Step 1: The time evolution of each order parameter is governed by coupled Schrodinger-like equations:

where $\mu_1, \mu_2$ are the chemical potentials. With a voltage $V$ across the junction: $\mu_2 - \mu_1 = 2eV$ (for Cooper pairs of charge $2e$).

Step 2: Substituting $\Psi_j = \sqrt{n_j}\,e^{i\theta_j}$and separating real and imaginary parts of the first equation:

Step 3: Since the current $I = 2e\dot{n}_2 = -2e\dot{n}_1$(pair current from 1 to 2), and defining $\phi = \theta_2 - \theta_1$:

This is the first Josephson relation (DC effect).

Step 4: Subtracting the phase evolution equations and using $n_1 \approx n_2$ (large reservoirs):

Therefore $d\phi/dt = -2eV/\hbar = 2eV/\hbar$ (with appropriate sign convention). This is the second Josephson relation (AC effect).

Derivation: RSJ Model I-V Characteristic

We derive the time-averaged voltage in the overdamped RSJ model ($\beta_c \ll 1$).

Step 1: In the overdamped limit ($C = 0$), the RSJ equation reduces to:

Defining normalized variables $\tau = (2eRI_c/\hbar)t$ and $\gamma = I/I_c$:

Step 2: For $\gamma > 1$ (above $I_c$), the phase increases monotonically. The period of one $2\pi$ cycle is:

Step 3: This integral is evaluated using the Weierstrass substitution $t = \tan(\phi/2)$:

Step 4: The time-averaged voltage is:

Therefore $\langle V\rangle = R\sqrt{I^2 - I_c^2}$ for $I > I_c$, and $\langle V\rangle = 0$ for $I < I_c$. This is the characteristic RSJ I-V curve, which approaches Ohm's law $V = IR$ for $I \gg I_c$.

Derivation: DC SQUID Critical Current Modulation

Step 1: A DC SQUID consists of a superconducting loop interrupted by two identical Josephson junctions. The fluxoid quantization around the loop requires:

where $\phi_1, \phi_2$ are the phase differences across the two junctions and$\Phi$ is the total flux through the loop.

Step 2: The total current is the sum of currents through both junctions:

Step 3: Define $\phi_\pm = (\phi_1 \pm \phi_2)/2$. The constraint gives $\phi_- = \pi\Phi/\Phi_0$ (modulo $\pi$), and:

Step 4: The maximum supercurrent (obtained by maximizing over $\phi_+$) is:

This is a two-slit quantum interference pattern for supercurrents, periodic in $\Phi_0$. The transfer function $\partial V/\partial\Phi \sim V_{\text{mod}}/\Phi_0$ is maximized at $\Phi = (n+1/2)\Phi_0$, where the slope of $I_c(\Phi)$ is steepest.

Derivation 2: DC Josephson Effect from Tunneling Hamiltonian

We provide a complete derivation of both Josephson relations using the tunneling Hamiltonian approach, carefully separating real and imaginary parts of the coupled equations.

Step 1: Coupled equations. Consider two superconducting electrodes described by macroscopic wave functions $\psi_1$ and $\psi_2$, coupled through a thin insulating barrier with coupling energy $K$. The time evolution is governed by coupled Schrödinger-like equations:

Here $\mu_1$ and $\mu_2$ are the chemical potentials of the two superconductors, and a voltage $V$ across the junction implies$\mu_2 - \mu_1 = 2eV$ (Cooper pair charge $2e$).

Step 2: Substitution. Write each wave function in amplitude-phase form: $\psi_j = \sqrt{n_j}\,e^{i\phi_j}$, where $n_j$ is the Cooper pair density and $\phi_j$ is the macroscopic phase. Substituting into the first equation:

Step 3: Separate real and imaginary parts. Multiply both sides by $e^{-i\phi_1}$ and define the phase difference$\varphi = \phi_2 - \phi_1$. Separating real and imaginary parts yields:

Similarly from the second equation:

Step 4: First Josephson relation (DC effect). The supercurrent is $I = -2e\dot{n}_1 = 2e\dot{n}_2$ (Cooper pairs flowing from 1 to 2). From the imaginary part equations:

where $I_c = 4eK\sqrt{n_1 n_2}/\hbar$ is the critical current. This is the first Josephson relation:$I = I_c\sin(\phi_2 - \phi_1)$.

Step 5: Second Josephson relation (AC effect). Subtract the two real-part equations. For large reservoirs ($n_1 \approx n_2$), the cosine terms cancel:

Therefore $\dot{\varphi} = 2eV/\hbar$ (choosing the conventional sign). This is the second Josephson relation: a constant voltage produces a linearly evolving phase difference, leading to an oscillating supercurrent at frequency$\omega_J = 2eV/\hbar$.

Derivation 3: AC Josephson Effect and Shapiro Steps

We derive the AC Josephson oscillation and show how an applied RF drive produces quantized voltage steps (Shapiro steps) via Bessel function expansion.

Step 1: DC bias. Apply a constant voltage$V_0$ across the junction. The second Josephson relation gives:

Substituting into the first Josephson relation:

where $\omega_J = 2eV_0/\hbar$ is the Josephson frequency. The supercurrent oscillates sinusoidally at this frequency. For $V_0 = 1\;\mu$V, one obtains$f_J = \omega_J/(2\pi) \approx 483.6$ MHz — a microwave-frequency oscillation. The time-averaged DC supercurrent vanishes: $\langle I \rangle = 0$.

Step 2: Add an RF drive. Now apply a combined DC + RF voltage:

Integrating the second Josephson relation:

Step 3: Bessel function expansion. Define the modulation index $a = 2eV_{\text{rf}}/(\hbar\omega_{\text{rf}})$. The current becomes:

Using the Jacobi-Anger expansion$e^{ia\sin\theta} = \sum_{n=-\infty}^{\infty} J_n(a)\,e^{in\theta}$, we expand:

Step 4: Shapiro steps. The time-averaged DC current is nonzero only when $\omega_J + n\omega_{\text{rf}} = 0$, i.e., when$\omega_J = -n\omega_{\text{rf}}$. Since $\omega_J = 2eV_0/\hbar$, the DC current steps appear at quantized voltages:

At each step, $\langle I \rangle = I_c J_{-n}(a)\sin\varphi_0$, so the step amplitude is controlled by the Bessel function $J_n(a)$ and depends on the RF power through $a$. The step height in current is$\Delta I_n = 2I_c|J_n(a)|$.

Physical significance: Shapiro steps provide a fundamental frequency-to-voltage conversion independent of material properties, forming the basis of the Josephson voltage standard. The voltage at each step depends only on fundamental constants and the applied frequency.

Derivation 4: RCSJ Model and Washboard Potential

We derive the resistively and capacitively shunted junction (RCSJ) model and map it to the dynamics of a particle in a tilted washboard potential.

Step 1: Circuit equation. A real Josephson junction is modeled as the ideal junction element (supercurrent $I_c\sin\varphi$) in parallel with a resistance $R$ (quasiparticle tunneling) and a capacitance $C$(geometric capacitance of the junction electrodes). Applying Kirchhoff's current law with an external bias current $I_{\text{ext}}$:

Step 2: Phase-only equation. Using the second Josephson relation $V = (\hbar/2e)\dot{\varphi}$, substitute to eliminate$V$:

This is the RCSJ equation in $\hbar/(2e)$ units. Dividing by $I_c$ and defining the normalized time $\tau = \omega_p t$ with plasma frequency$\omega_p = \sqrt{2eI_c/(\hbar C)}$:

Step 3: Washboard potential. Multiply the RCSJ equation by $(\hbar/2e)\dot{\varphi}$ to obtain an energy equation. The “particle” of mass $(\hbar/2e)^2 C$ moves in the tilted washboard potential:

where $E_J = \hbar I_c/(2e)$ is the Josephson energy. The first term creates periodic wells (the “washboard”); the second term tilts the potential by an amount proportional to the bias current. For $I_{\text{ext}} < I_c$, local minima exist and the “particle” is trapped (zero voltage, DC Josephson effect). For$I_{\text{ext}} > I_c$, the minima vanish and the particle runs downhill (finite voltage, resistive state).

Step 4: Stewart-McCumber parameter. The dimensionless parameter governing the junction dynamics is:

This is the ratio of the $RC$ time constant to the Josephson oscillation period.

• $\beta_c \ll 1$ (overdamped): The particle experiences strong friction. The junction is non-hysteretic — the I-V curve is single-valued with $\langle V \rangle = R\sqrt{I^2 - I_c^2}$ for$I > I_c$. This is the RSJ limit, preferred for SQUID operation.
• $\beta_c \gg 1$ (underdamped): The particle has large inertia. Once it escapes a well, it gains kinetic energy and continues running even when the tilt is reduced below the critical value. This produces a hysteretic I-V curve: the retrapping current$I_r \approx 4I_c/(\pi\sqrt{\beta_c}) \ll I_c$ is much less than $I_c$. Hysteretic junctions are used for digital logic and qubit applications.

Derivation 5: SQUID Magnetometry

We derive the critical current modulation of a DC SQUID and explain its use as an ultrasensitive magnetometer.

Step 1: Fluxoid quantization. A DC SQUID consists of a superconducting loop interrupted by two Josephson junctions (labeled $a$ and$b$). The single-valuedness of the macroscopic wave function around the loop requires the fluxoid to be quantized:

where $\varphi_a, \varphi_b$ are the gauge-invariant phase differences across each junction, $\Phi$ is the total magnetic flux threading the loop, and$\Phi_0 = h/(2e) \approx 2.068 \times 10^{-15}$ Wb is the flux quantum.

Step 2: Parallel junction currents. The bias current$I$ splits between the two arms. For identical junctions (both with critical current $I_c$):

Step 3: Eliminate one phase. From fluxoid quantization: $\varphi_b = \varphi_a + 2\pi\Phi/\Phi_0$ (setting$n = 0$). Define the average phase$\bar{\varphi} = (\varphi_a + \varphi_b)/2$. Using the sum-to-product identity:

Step 4: Maximum supercurrent. Maximizing over$\bar{\varphi}$ (i.e., $\sin\bar{\varphi} = \pm 1$):

The critical current oscillates between $2I_c$ (at integer flux quanta) and zero (at half-integer flux quanta), with period $\Phi_0$. This is the superconducting analog of a two-slit interference pattern.

Step 5: Voltage-flux characteristic and sensitivity. In practice, the SQUID is current-biased just above the maximum $I_c(\Phi)$. When the flux changes, $I_c$ changes, and since the junction transitions to the resistive state when $I_{\text{bias}} > I_c(\Phi)$, the voltage becomes a periodic function of flux:

The flux sensitivity is characterized by the transfer function:

which is maximized at the steepest slope of $V(\Phi)$, near$\Phi = (n + 1/4)\Phi_0$. With a flux-locked loop (feedback electronics), the SQUID operates at this optimal point, achieving flux noise levels of$\sim 1\;\mu\Phi_0/\sqrt{\text{Hz}}$, corresponding to field sensitivity of $\sim 10^{-15}$ T/$\sqrt{\text{Hz}}$ for typical loop areas.