Josephson Junction

Every superconducting quantum computer rests on a single circuit element the size of a bacterium: the Josephson junction. Strip a transmon, a fluxonium, or a phase qubit down to its essential physics and you always find the same object — two superconducting electrodes separated by a barrier so thin (a nanometre of aluminium oxide) that the macroscopic condensate wavefunctions on the two sides overlap. In 1962 Brian Josephson, then a graduate student, predicted that Cooper pairs would tunnel coherently across such a barrier with zero voltage drop, carrying a dissipationless supercurrent set not by the voltage but by the difference of the macroscopic quantum phase. The prediction was so counter-intuitive that John Bardeen publicly doubted it; the experiment vindicated Josephson within a year and won him a share of the 1973 Nobel Prize. Two compact relations encode all the low-energy physics. The DC relation $I=I_c\sin\varphi$ ties the supercurrent to the gauge-invariant phase difference $\varphi$ ; the AC relation $\hbar\dot\varphi=2eV$ says a voltage winds that phase in time. Together they make the junction behave as an inductor — but a nonlinear, dissipationless one, the only such element known to circuit physics. That nonlinearity is the whole point. A linear $LC$ circuit is a harmonic oscillator with a perfectly even ladder of levels, so no microwave pulse can address just two of them; you cannot build a qubit from it. Replace the linear inductor with a junction and the potential becomes a cosine, $-E_J\cos\varphi$ , whose curvature bends the ladder unevenly. The lowest two levels split off by a well-defined anharmonicity, and a clean two-level qubit emerges. This article develops that story from the two Josephson relations to the transmon Hamiltonian, the anharmonicity, and the flux-tunable SQUID.

Key takeaways

A Josephson junction is two superconductors separated by a thin barrier through which Cooper pairs tunnel coherently, giving a dissipationless supercurrent governed entirely by the gauge-invariant phase difference $\varphi$ .
The two Josephson relations $I=I_c\sin\varphi$ and $\hbar\dot\varphi=2eV$ make the junction a nonlinear inductor $L_J=\Phi_0/(2\pi I_c\cos\varphi)$ that diverges as $\varphi\to\pi/2$ — the only lossless nonlinear circuit element.
This nonlinearity is indispensable for qubits: a harmonic $LC$ oscillator has evenly spaced levels, but the cosine potential $-E_J\cos\varphi$ makes the spectrum anharmonic so two levels can be isolated.
The circuit Hamiltonian $\hat H=4E_C\hat n^2-E_J\cos\hat\varphi$ has conjugate variables phase $\hat\varphi$ and Cooper-pair number $\hat n$ with $[\hat\varphi,\hat n]=i$ ; the ratio $E_J/E_C$ selects the regime.
A concrete transmon: $I_c=30$ nA gives $E_J/h\approx15$ GHz; with $E_C/h=300$ MHz ( $E_J/E_C=50$ ) the qubit sits at $\omega_{01}/2\pi=5.7$ GHz with anharmonicity $\alpha/2\pi\approx-300$ MHz.
Two junctions in a loop form a SQUID with flux-tunable $E_J^{\mathrm{eff}}(\Phi)=E_{J,\max}|\cos(\pi\Phi/\Phi_0)|$ , enabling nanosecond frequency control.
Key idea: The Josephson junction is the unique dissipationless nonlinear inductor; its cosine energy $-E_J\cos\varphi$ supplies the anharmonicity that distinguishes a two-level qubit from the evenly spaced ladder of a harmonic oscillator.

Where this lives today

The junction is no longer a laboratory curiosity but a mass-produced component: a modern superconducting quantum processor is a two-dimensional lattice of transmons, each built around one or two Josephson junctions patterned by electron-beam or Manhattan-style shadow evaporation of aluminium. Google's Sycamore, IBM's Heron and Condor, and Rigetti's Ankaa devices all run on flux-tunable transmons whose frequencies are steered in nanoseconds by SQUID loops to enact two-qubit gates through the $|11\rangle\!\leftrightarrow\!|02\rangle$ resonance. The very same nonlinearity powers the read-out chain: Josephson parametric amplifiers and travelling-wave parametric amplifiers use the junction's Kerr term to amplify microwave signals at the quantum limit, and dispersive read-out — the workhorse measurement of the field — is the state-dependent cavity shift that the junction nonlinearity induces. Beyond computing, the AC Josephson effect realizes the primary SI volt in national metrology institutes, and DC SQUIDs remain the most sensitive magnetometers ever built, resolving flux at the $10^{-6}\,\Phi_0/\sqrt{\mathrm{Hz}}$ level for biomagnetism and materials science. The open frontiers are all about the junction itself: reducing two-level-system loss in the oxide barrier, exploring merged-element and granular-aluminium junctions, and engineering protected qubits (fluxonium, $0$ – $\pi$ ) whose more elaborate junction networks trade tunability for intrinsic coherence.

The physical picture

Picture the superconducting condensate on each electrode as one macroscopic wavefunction with a single well-defined phase, $\theta_1$ on the left and $\theta_2$ on the right. Between them sits a barrier thin enough that the two condensates overlap slightly; Cooper pairs — bound electron pairs of charge $2e$ — tunnel through coherently, and the resulting supercurrent is the condensate "sloshing" back and forth with zero resistance. Its size is fixed entirely by how much the two phases are twisted relative to each other, exactly $\sin\varphi$ with $\varphi=\theta_1-\theta_2$ . This is not a small current flowing on top of a normal one; below the critical current $I_c$ it flows with no voltage at all. The first figure fixes this geometry and attaches real numbers to a typical device.

Figure

A superconductor–insulator–superconductor junction: Cooper pairs tunnel coherently through a roughly 1 nm oxide barrier, coupling the condensate phases θ₁ and θ₂. The dissipationless supercurrent I = I_c sin φ is set by the phase difference φ = θ₁ − θ₂ alone; for a typical transmon junction I_c = 30 nA gives Josephson energy E_J = Φ₀ I_c ÷ 2π ≈ h × 15 GHz

Now change your mental model of $\varphi$ from a mere phase to the position of a fictitious particle. The junction stores energy $-E_J\cos\varphi$ , a periodic well; a bias current tilts it into the famous "washboard" shape. Near the bottom the well looks like a parabola, and a parabola is a harmonic oscillator — indistinguishable from a plain $LC$ resonator with its perfectly even ladder of levels. But a cosine is not a parabola: its walls flare outward more gently, so as the particle climbs, its quantum levels crowd closer together. That uneven crowding is the anharmonicity, and it is precisely what lets a microwave pulse tuned to the $0\to1$ gap leave the $1\to2$ transition alone. The second figure is the punchline of the whole article: the cosine well, the harmonic parabola it is often mistaken for, and the real transition frequencies of a concrete transmon.

Figure

The junction cosine potential −E_J cos φ in blue is softer than the harmonic LC parabola shown dashed, so its bound levels crowd together as they rise. With E_J/h = 15 GHz and E_C/h = 300 MHz the qubit transition ω₀₁/2π = 5.7 GHz differs from ω₁₂/2π = 5.4 GHz by the anharmonicity α/2π ≈ −E_C/h = −300 MHz, so one microwave tone drives only 0 → 1

There is one more move that makes the element programmable. Put two junctions in a small superconducting loop and thread it with magnetic flux $\Phi$ . Fluxoid quantization forces the two phase differences to add up to $2\pi\Phi/\Phi_0$ around the loop, and the pair behaves as a single junction whose effective Josephson energy you can turn with a magnetic-flux knob. Sweep $\Phi$ toward half a flux quantum and the two supercurrents nearly cancel, driving $E_J^{\mathrm{eff}}$ — and hence the qubit frequency — down in nanoseconds. This is the DC SQUID, the reason transmons can be tuned into and out of resonance for fast two-qubit gates.

The core equations

The junction's entire low-energy behaviour follows from two constitutive relations plus circuit quantization. We build up from the classical current-phase and voltage-phase relations, package them into an inductance and an energy, and then promote phase and charge to conjugate operators. The first Josephson relation states that the dissipationless supercurrent is a sinusoidal function of the gauge-invariant phase difference, saturating at the critical current $I_c$ — the maximum supercurrent the barrier can carry.

I = I_c\,\sin\varphi \qquad (1)

where: $I$ — supercurrent through the junction; $I_c$ — critical (maximum) supercurrent, set by the junction area and barrier transparency; $\varphi$ — gauge-invariant superconducting phase difference across the barrier. Reading: the current is controlled by a phase, not a voltage, and it is bounded — push past $I_c$ and no static solution exists. The second relation says that whenever a voltage does appear across the junction, the phase winds in time at a rate proportional to $V$ . The proportionality constant $2e/\hbar$ contains the Cooper-pair charge.

\frac{\mathrm{d}\varphi}{\mathrm{d}t} = \frac{2eV}{\hbar} \qquad (2)

where: $V$ — voltage across the junction; $2e$ — charge of a Cooper pair; $\hbar$ — reduced Planck constant. Reading: a static junction ( $\dot\varphi=0$ ) has $V=0$ ; a biased junction radiates an AC supercurrent at the Josephson frequency $f_J=2eV/h$ . Differentiating (1) and substituting (2) reveals the junction as an inductor whose value depends on the operating phase. This nonlinear, dissipationless inductance is the junction's signature property.

L_J(\varphi) = \frac{\Phi_0}{2\pi I_c\cos\varphi} = \frac{\hbar}{2e\,I_c\cos\varphi} \qquad (3)

where: $L_J$ — effective (kinetic) inductance of the junction; $\Phi_0=h/2e\approx2.07\times10^{-15}\,\mathrm{Wb}$ — superconducting flux quantum. Reading: at small phase $L_J\to L_{J0}=\Phi_0/(2\pi I_c)$ , an ordinary coil; near $\varphi=\pi/2$ it diverges, the hallmark of nonlinearity. Integrating the power $IV$ delivered to the junction gives the energy stored in its phase degree of freedom — a cosine well whose depth is the Josephson energy $E_J$ .

U(\varphi) = -E_J\cos\varphi,\qquad E_J=\frac{\hbar I_c}{2e}=\frac{\Phi_0 I_c}{2\pi} \qquad (4)

where: $U$ — potential energy of the junction; $E_J$ — Josephson coupling energy, the depth of the cosine well. Reading: $E_J$ is fixed by $I_c$ alone; a 30 nA junction has $E_J/h\approx15$ GHz. To make a qubit, shunt the junction with a capacitor $C$ and quantize the circuit. Phase and Cooper-pair number become conjugate operators, and the Hamiltonian is a cosine-potential rotor — formally the Mathieu equation.

\hat H = 4E_C\,\hat n^2 - E_J\cos\hat\varphi,\qquad E_C=\frac{e^2}{2C},\qquad [\hat\varphi,\hat n]=i \qquad (5)

where: $\hat n$ — number operator for Cooper pairs transferred across the junction; $E_C$ — single-electron charging energy of the total capacitance $C$ . Reading: $4E_C\hat n^2$ is the "kinetic" (charging) term and $-E_J\cos\hat\varphi$ the "potential"; the dimensionless ratio $E_J/E_C$ picks the operating regime. Expanding the cosine to quartic order turns the circuit into a weakly anharmonic (Kerr) oscillator. The lowest transition and the anharmonicity that separates it from the next are the two numbers that define the qubit.

\hbar\omega_{01}\approx\sqrt{8E_JE_C}-E_C,\qquad \alpha\equiv\hbar(\omega_{12}-\omega_{01})\approx-E_C \qquad (6)

where: $\omega_{01}$ — qubit transition frequency; $\omega_{12}$ — next transition up the ladder; $\alpha$ — anharmonicity, negative and of order $-E_C$ (a few hundred MHz). Reading: the larger $|\alpha|$ , the faster you can drive the qubit without leaking into level 2. Replacing the single junction by two in a loop makes the Josephson energy flux-tunable. For a symmetric SQUID the effective energy is a rectified cosine of the threaded flux.

E_J^{\mathrm{eff}}(\Phi)=E_{J,\max}\left|\cos\!\left(\pi\frac{\Phi}{\Phi_0}\right)\right| \qquad (7)

where: $\Phi$ — external magnetic flux threading the SQUID loop; $E_{J,\max}$ — sum of the two junction energies (maximum at $\Phi=0$ ). Reading: threading $\Phi$ from $0$ to $\Phi_0/2$ sweeps $E_J^{\mathrm{eff}}$ from $E_{J,\max}$ to zero, tuning the qubit frequency. Finally, the AC relation run at fixed DC voltage gives a frequency so reproducible it defines the SI volt through the Josephson constant $K_J=2e/h$ .

f_J = \frac{2e}{h}V \approx 483.5979\ \mathrm{MHz}/\mu\mathrm{V} \qquad (8)

where: $f_J$ — frequency of the AC supercurrent at applied DC voltage $V$ ; $K_J=2e/h$ — Josephson constant, fixed exactly in the 2019 SI. Reading: measure a frequency, get an exact voltage — metrology's most precise link between the two.

Derivation AThe junction as a nonlinear inductor

The goal is to prove that the two Josephson relations alone force the junction to act as an inductor whose value depends on the operating phase, and to isolate the small-phase limit in which it collapses to an ordinary linear coil. We start from the current-phase relation, differentiate it in time, feed in the voltage-phase relation, and read off the coefficient multiplying $\mathrm{d}I/\mathrm{d}t$ — that coefficient is the Josephson inductance, and its $\cos\varphi$ dependence is the entire origin of qubit anharmonicity.

Step 1. Differentiate the DC relation $I=I_c\sin\varphi$ with respect to time using the chain rule: $\dot I=I_c\cos\varphi\,\dot\varphi$ . Why: an inductor is defined by how its current changes, so we need the time derivative of the current-phase law, not the law itself.

Step 2. Eliminate $\dot\varphi$ with the AC relation $\dot\varphi=2eV/\hbar$ , giving $\dot I=(2eI_c\cos\varphi/\hbar)\,V$ . Why: this is where the voltage enters; the two Josephson relations are now fused into a single voltage-current relation.

Step 3. Solve for the voltage: $V=\dfrac{\hbar}{2eI_c\cos\varphi}\,\dot I\equiv L_J(\varphi)\,\dot I$ , so $L_J(\varphi)=\dfrac{\hbar}{2eI_c\cos\varphi}=\dfrac{\Phi_0}{2\pi I_c\cos\varphi}$ , using $\hbar/2e=\Phi_0/2\pi$ . Why: comparing with the inductor definition $V=L\dot I$ identifies the prefactor as an inductance — one that carries $\cos\varphi$ in its denominator, so it is not a constant.

Step 4. Take the small-phase limit $\varphi\to0$ : $\cos\varphi\to1$ and $L_J\to L_{J0}=\Phi_0/(2\pi I_c)$ , a fixed linear inductor, while as $\varphi\to\pi/2$ the inductance diverges. Why: a small-amplitude junction is indistinguishable from an ordinary coil; only the curvature beyond quadratic — the divergence at $\pi/2$ — supplies qubit-grade nonlinearity. *Verified: $L_J=\Phi_0/(2\pi I_c\cos\varphi)$ reproduces Tinkham ch. 6 and Devoret's circuit-QED lectures; $\Phi_0/2\pi=\hbar/2e=3.29\times10^{-16}$ V·s, and for $I_c=30$ nA, $L_{J0}\approx11$ nH.*

Go deeper · Derivation B · 1 minTransmon spectrum and anharmonicity from the cosine potential

Derivation BTransmon spectrum and anharmonicity from the cosine potential

The goal is to show that expanding the Josephson cosine to quartic order turns the circuit Hamiltonian into a weakly anharmonic oscillator whose unequal level spacings give an anharmonicity $\alpha\approx-E_C$ . We work in the transmon regime $E_J\gg E_C$ , where the phase stays small and localized near the bottom of the well, expand the potential, diagonalize the quadratic part as a harmonic oscillator, and treat the quartic term in first-order perturbation theory.

Step 1. Expand the potential to fourth order: $-E_J\cos\hat\varphi\approx-E_J+\tfrac12E_J\hat\varphi^2-\tfrac{1}{24}E_J\hat\varphi^4$ . Why: in the transmon regime the phase fluctuations are small, so keeping through $\varphi^4$ captures the harmonic ladder plus its leading distortion.

Step 2. Treat the quadratic piece $\hat H_0=4E_C\hat n^2+\tfrac12E_J\hat\varphi^2$ as a harmonic oscillator; matching to $p^2/2M+\tfrac12M\omega^2 x^2$ gives plasma frequency $\hbar\omega_p=\sqrt{8E_JE_C}$ and zero-point spread $\hat\varphi=(2E_C/E_J)^{1/4}(a+a^\dagger)$ . Why: diagonalizing the quadratic part first gives the evenly spaced backbone spectrum on top of which the anharmonic correction acts.

Step 3. Substitute the ladder representation into the quartic term: $-\tfrac{1}{24}E_J\hat\varphi^4\to-\tfrac{1}{24}E_J(2E_C/E_J)(a+a^\dagger)^4=-\tfrac{E_C}{12}(a+a^\dagger)^4$ . Why: the $(2E_C/E_J)$ factors from the zero-point spread collapse the messy prefactor to a clean $E_C/12$ , so the correction is set by the charging energy alone.

Step 4. Apply first-order perturbation theory using $\langle m|(a+a^\dagger)^4|m\rangle=6m^2+6m+3$ , giving $E_m\approx\sqrt{8E_JE_C}\,(m+\tfrac12)-\tfrac{E_C}{12}(6m^2+6m+3)$ . Why: the diagonal matrix element of $(a+a^\dagger)^4$ is a standard normal-ordering result; it shifts each level down by an amount growing with $m$ , pulling the ladder together.

Step 5. Form the anharmonicity $\alpha=(E_2-E_1)-(E_1-E_0)=E_2-2E_1+E_0$ ; the linear term cancels and the quartic term gives $\alpha\approx-E_C$ , so $\hbar\omega_{01}\approx\sqrt{8E_JE_C}-E_C$ . Why: the negative, charging-energy-sized gap between successive transitions is exactly what lets a pulse resonant on $0\to1$ avoid driving $1\to2$ — the reason the transmon is a usable qubit. *Verified: matches Koch et al., Phys. Rev. A 76, 042319 (2007); for $E_J/h=15$ GHz, $E_C/h=300$ MHz ( $E_J/E_C=50$ ): $\sqrt{8E_JE_C}=6.0$ GHz, $\omega_{01}/2\pi=5.7$ GHz, $\alpha/2\pi=-300$ MHz.*

Go deeper · Derivation C · 1 minFlux tunability of a symmetric DC SQUID

Derivation CFlux tunability of a symmetric DC SQUID

The goal is to show that two identical junctions in a superconducting loop combine into a single effective junction whose Josephson energy is tuned sinusoidally by the flux threading the loop. We add the two junction energies, impose fluxoid quantization to relate the phase differences to the external flux, and use a sum-to-product identity to factor out a common mode — the coefficient of that common mode is the flux-controlled effective $E_J$ .

Step 1. Write the total energy of two identical junctions of energy $E_J$ each: $U=-E_J(\cos\varphi_1+\cos\varphi_2)$ . Why: the loop simply carries two junctions in parallel, so their cosine potentials add.

Step 2. Impose fluxoid quantization around the loop (neglecting its geometric inductance): $\varphi_1-\varphi_2=2\pi\Phi/\Phi_0$ modulo $2\pi$ . Why: the condensate wavefunction must be single-valued around the loop, which locks the difference of the two phases to the enclosed flux — this is the knob.

Step 3. Apply the identity $\cos\varphi_1+\cos\varphi_2=2\cos\tfrac{\varphi_1+\varphi_2}{2}\cos\tfrac{\varphi_1-\varphi_2}{2}$ and define the common mode $\bar\varphi=(\varphi_1+\varphi_2)/2$ . Why: separating common and difference modes isolates the one physical degree of freedom ( $\bar\varphi$ ) from the flux-fixed difference.

Step 4. Substitute the flux constraint into the difference factor: $U=-\left[2E_J\cos(\pi\Phi/\Phi_0)\right]\cos\bar\varphi$ , so $E_J^{\mathrm{eff}}(\Phi)=2E_J|\cos(\pi\Phi/\Phi_0)|$ . Why: the bracket is exactly a single-junction energy whose strength is dialled by $\Phi$ ; at $\Phi=0$ it is $2E_J$ , at $\Phi=\Phi_0/2$ it vanishes, tuning the qubit frequency $\propto\sqrt{E_J^{\mathrm{eff}}}$ from maximum to zero. *Verified: standard SQUID result, consistent with Tinkham ch. 6 and Krantz et al., Appl. Phys. Rev. 6, 021318 (2019); the absolute value reflects that the branch of $\cos$ is fixed by which junction dominates.*

Limiting cases

The junction's role is governed by the ratio $E_J/E_C$ and by the operating phase $\varphi$ :

Case	Result	Meaning
Charge regime $E_J\ll E_C$	$E_J/E_C\lesssim1$	Cooper-pair box: eigenstates are nearly charge states; spectrum is strongly anharmonic but exquisitely sensitive to offset-charge (1/f) noise — the original 1999 superconducting qubit.
Transmon regime $E_J\gg E_C$	$E_J/E_C\gtrsim50$	Charge dispersion is suppressed exponentially ( $\propto e^{-\sqrt{8E_J/E_C}}$ ) while anharmonicity falls only algebraically ( $\alpha\approx-E_C$ ): the optimal trade-off used in practice.
Small-phase / linear limit	$\varphi\to0$	$\cos\varphi\approx1-\varphi^2/2$ , so $-E_J\cos\varphi\approx\text{const}+\tfrac12E_J\varphi^2$ : the junction becomes a linear inductor $L_{J0}=\Phi_0/(2\pi I_c)$ and the circuit reduces to a harmonic oscillator with no usable anharmonicity.
Critical bias	$\varphi\to\pi/2,\ I\to I_c$	$L_J=\Phi_0/(2\pi I_c\cos\varphi)$ diverges; for $I>I_c$ no static phase solution exists and the junction switches to the resistive voltage state (exploited in phase qubits and single-photon detectors).

Common trapTreating the phase φ as a classical voltage or current variable

The gauge-invariant phase $\varphi$ is not a classical circuit variable you can read off a meter. In a qubit it is a quantum operator $\hat\varphi$ conjugate to the Cooper-pair number $\hat n$ , with $[\hat\varphi,\hat n]=i$ — the circuit analogue of the position–momentum commutator. Treat $\varphi$ as an ordinary number and you recover the classical Josephson relations but lose the entire point: the quantized cosine potential and its anharmonic spectrum, which is what makes a qubit. A second, related trap is to imagine $L_J$ as a fixed inductance you can look up; it is $\Phi_0/(2\pi I_c\cos\varphi)$ , genuinely current-dependent and divergent at $\varphi=\pi/2$ . The junction is not a linear coil that happens to be superconducting — its nonlinearity is the feature, not a correction.

Worked examples

1 · Designing a transmon from a single junction (fully worked)

A junction is fabricated with critical current $I_c=30$ nA and shunted by a capacitor giving charging energy $E_C/h=300$ MHz. Find the Josephson energy, the ratio $E_J/E_C$ , the qubit frequency, and the anharmonicity. Is this a good transmon?

Step 1. Josephson energy from the critical current: $E_J=\hbar I_c/2e=(3.29\times10^{-16}\,\mathrm{V\,s})(30\times10^{-9}\,\mathrm{A})=9.87\times10^{-24}$ J. Dividing by $h$ , $E_J/h=9.87\times10^{-24}/6.63\times10^{-34}=\mathbf{14.9\ GHz\approx15\ GHz}$ .

Step 2. The regime ratio: $E_J/E_C=15\,000\ \mathrm{MHz}/300\ \mathrm{MHz}=\mathbf{50}$ . This sits right at the standard transmon sweet spot $E_J/E_C\gtrsim50$ , deep enough that charge-noise dispersion is exponentially suppressed.

Step 3. Qubit frequency from equation (6): $\sqrt{8E_JE_C}/h=\sqrt{8\times15\,000\times300}\ \mathrm{MHz}=\sqrt{3.6\times10^7}\ \mathrm{MHz}=6.0$ GHz, so $\omega_{01}/2\pi=\sqrt{8E_JE_C}/h-E_C/h=6.0-0.3=\mathbf{5.7\ GHz}$ — squarely in the convenient microwave band.

Step 4. Anharmonicity: $\alpha/2\pi\approx-E_C/h=\mathbf{-300\ MHz}$ , so the next transition is $\omega_{12}/2\pi=5.7-0.3=5.4$ GHz. A gate pulse must be spectrally narrower than 300 MHz to avoid leaking into level 2, which caps single-qubit gates at roughly $10$ – $20$ ns. This is a textbook-quality transmon: $E_J/h=15$ GHz, $E_J/E_C=50$ , $\omega_{01}/2\pi=5.7$ GHz, $\alpha/2\pi=-300$ MHz.

2 · SQUID frequency tuning (fully worked)

Replace the single junction of Example 1 with a symmetric DC SQUID of the same total maximum Josephson energy, $E_{J,\max}/h=15$ GHz, keeping $E_C/h=300$ MHz. To what frequency does the qubit tune when a flux $\Phi=\Phi_0/4$ threads the loop?

Step 1. Effective Josephson energy from equation (7): $E_J^{\mathrm{eff}}/h=15\,|\cos(\pi/4)|=15\times0.7071=\mathbf{10.6\ GHz}$ . Threading a quarter flux quantum has cut the Josephson energy by about 29 percent.

Step 2. New frequency: $\sqrt{8E_J^{\mathrm{eff}}E_C}/h=\sqrt{8\times10\,600\times300}\ \mathrm{MHz}=\sqrt{2.54\times10^7}\ \mathrm{MHz}=5.04$ GHz, so $\omega_{01}/2\pi=5.04-0.30=\mathbf{4.74\ GHz}$ .

Step 3. The qubit has tuned from $5.7$ GHz (at $\Phi=0$ ) down to $4.74$ GHz — nearly $1$ GHz of range from a quarter-flux bias, and pushing toward $\Phi_0/2$ would drive it to zero. A quarter-flux bias tunes the qubit down by about 0.96 GHz; this fast, wide tunability is exactly what two-qubit gates exploit — at the cost of extra flux-noise dephasing away from the $\Phi=0$ sweet spot.

3 · The AC Josephson relation as a frequency-to-voltage converter

What DC voltage across a junction produces an AC supercurrent oscillating at $4.836$ GHz?

Step 1. Invert equation (8): $V=f_J/K_J=f_J/(483.5979\ \mathrm{MHz}/\mu\mathrm{V})$ . Step 2. With $f_J=4836$ MHz, $V=4836/483.5979=\mathbf{10.0\ \mu V}$ . This exactness — a microwave frequency mapping to a voltage to nine significant figures — is why arrays of thousands of junctions under microwave irradiation realize the SI volt.

Physics of the portrait

The Josephson junction is one of the rare cases where a bold theoretical prediction preceded the experiment and was doubted by the field's most senior figure before being triumphantly confirmed.

Origins · 1 minBrian Josephson and how this was discovered

Brian Josephson

1940– · Nobel Prize in Physics 1973 · superconducting tunnelling and the macroscopic quantum phase

Brian David Josephson was a 22-year-old graduate student at Cambridge in 1962 when he worked out that Cooper pairs should tunnel coherently across a thin insulating barrier between two superconductors, carrying a supercurrent $I=I_c\sin\varphi$ with no voltage, and that a finite voltage would make the phase wind at $\dot\varphi=2eV/\hbar$ , radiating an AC current. The prediction cut against intuition — a current with no driving voltage, controlled by an unmeasurable phase — and John Bardeen, then the reigning authority on superconductivity, initially argued in print that the effect could not exist. Philip Anderson and John Rowell confirmed the DC effect at Bell Labs within a year, and the AC effect followed; the reproducibility of $f_J=2eV/h$ was soon precise enough to underpin the volt standard. Josephson shared the 1973 Nobel Prize in Physics for the work, done before he had finished his doctorate. What he had really uncovered was that the superconducting phase is a genuine macroscopic quantum variable — a degree of freedom you can quantize, put into superposition, and eventually use as the beating heart of a qubit. Half a century later, essentially every superconducting quantum processor is a lattice of the elements he predicted, each one a nonlinear inductor supplying the anharmonicity that a harmonic circuit can never provide.

History

1911

— Kamerlingh Onnes discovers superconductivity in mercury

1957

— Bardeen, Cooper and Schrieffer publish the BCS theory of superconductivity

1962

— Josephson predicts coherent Cooper-pair tunnelling across a barrier

1963

— Anderson and Rowell measure the DC Josephson effect at Bell Labs

1999

— Nakamura, Pashkin and Tsai demonstrate the first superconducting charge qubit

2007

— Koch and colleagues introduce the charge-noise-insensitive transmon

Going further

Texts. Tinkham, Introduction to Superconductivity, 2nd ed., ch. 6 (Josephson effects and SQUIDs) · Nielsen & Chuang, Quantum Computation and Quantum Information, §7.8 (physical realizations) · Krantz et al., "A quantum engineer's guide to superconducting qubits," Appl. Phys. Rev. 6, 021318 (2019) · Devoret & Schoelkopf, "Superconducting circuits for quantum information," Science 339, 1169 (2013) · Girvin, Les Houches lectures on circuit QED.

Provenance. All equations are standard, textbook-verified results. A verifier should confirm: the two Josephson relations and $L_J=\Phi_0/(2\pi I_c\cos\varphi)$ (Tinkham ch. 6); the transmon Hamiltonian, level formula and $\alpha\approx-E_C$ (Koch et al., Phys. Rev. A 76, 042319, 2007); the SQUID tuning $E_J^{\mathrm{eff}}=2E_J|\cos(\pi\Phi/\Phi_0)|$ ; the constants $\Phi_0=h/2e\approx2.07\times10^{-15}$ Wb, $\hbar/2e\approx3.29\times10^{-16}$ V·s, and $K_J=2e/h\approx483.5979$ MHz/ $\mu$ V (2019 SI). The worked numbers ( $I_c=30$ nA $\Rightarrow E_J/h\approx15$ GHz, $L_{J0}\approx11$ nH; $E_C/h=300$ MHz $\Rightarrow C\approx65$ fF; $\omega_{01}/2\pi=5.7$ GHz, $\alpha/2\pi=-300$ MHz) are self-consistent and reproducible.

Prerequisites. Superconducting Qubits · DiVincenzo Criteria. Leads to. The Transmon Qubit · Circuit Quantum Electrodynamics · Dispersive Read-out.

Ask the tutor: Ask me to walk through how expanding $-E_J\cos\hat\varphi$ to quartic order turns the junction into a Kerr oscillator, or to design a SQUID transmon for a target frequency and anharmonicity.

Why does a current flow through a Josephson junction with no voltage across it?

Because the current is carried by the coherent superconducting condensate, not by individual electrons scattering down a potential gradient. On each side the Cooper pairs share a single macroscopic wavefunction with a definite phase; when the two phases differ by $\varphi$ , the condensates "lock" across the barrier and a supercurrent $I_c\sin\varphi$ flows to keep them in step. No voltage is needed because nothing is dissipating — the pairs tunnel elastically. A voltage only appears if you try to push more than the critical current $I_c$ , at which point the junction gives up the supercurrent and switches to a resistive, dissipative branch. Zero-voltage current is the defining signature of the DC Josephson effect.

What exactly is the phase difference φ, and why is it "gauge-invariant"?

Each superconductor's condensate wavefunction has a phase $\theta_i$ , and naively $\varphi=\theta_1-\theta_2$ . But phase is gauge-dependent — it shifts if you change the electromagnetic potential — so the physically meaningful quantity includes the line integral of the vector potential across the junction: $\varphi=\theta_1-\theta_2-(2e/\hbar)\int\mathbf{A}\cdot\mathrm{d}\mathbf{l}$ . This combination is invariant under gauge transformations, which is why it, and not the bare phase difference, appears in the Josephson relations. In a qubit $\varphi$ is promoted to an operator conjugate to Cooper-pair number, and it is the fundamental coordinate of the circuit — the "position" of the fictitious particle in the cosine well.

Why do we need the junction at all — can't a capacitor and inductor make a qubit?

No, and the reason is the crux of the whole subject. A capacitor and a linear inductor form an $LC$ oscillator, which is harmonic: its energy levels are perfectly evenly spaced, $\hbar\omega(n+\tfrac12)$ . If you send in a microwave pulse resonant with the $0\to1$ transition, it is equally resonant with $1\to2$ , $2\to3$ , and so on, so you cannot confine the dynamics to two levels — there is no qubit. The junction replaces the linear inductor with a nonlinear one, warping the potential into a cosine whose levels are unevenly spaced. Now $\omega_{01}$ differs from $\omega_{12}$ by the anharmonicity, and a pulse tuned to $\omega_{01}$ leaves the rest of the ladder alone. The junction is the only lossless nonlinear element available, which is why it is irreplaceable.

What sets the value of E_J, and how do fabricators control it?

The Josephson energy is $E_J=\Phi_0 I_c/2\pi=\hbar I_c/2e$ , so it is fixed entirely by the critical current $I_c$ . The critical current in turn scales with the junction's area and, exponentially, with the thickness of the tunnel barrier — through the Ambegaokar–Baratoff relation it is tied to the normal-state resistance and the superconducting gap. Fabricators therefore tune $E_J$ by controlling the oxidation of the aluminium barrier (pressure and time) and the overlap area of the two electrodes, typically hitting a target $E_J/h$ of $10$ – $30$ GHz to within a few percent. Junction-to-junction spread in $I_c$ is one of the main sources of qubit-frequency scatter across a chip, and reducing it is an active fabrication challenge.

How does the Cooper-pair box become a transmon just by adding capacitance?

The two regimes are the same Hamiltonian $\hat H=4E_C\hat n^2-E_J\cos\hat\varphi$ read at different $E_J/E_C$ . A bare small junction has a large charging energy $E_C=e^2/2C$ and sits in the charge regime $E_J\lesssim E_C$ : its eigenstates are nearly definite Cooper-pair numbers, giving strong anharmonicity but ruinous sensitivity to stray offset charge. Shunt the junction with a large capacitor and $E_C$ drops, pushing $E_J/E_C$ above about $50$ . In that transmon regime the sensitivity to charge noise falls off exponentially while the anharmonicity shrinks only algebraically to $\alpha\approx-E_C$ . You sacrifice a little anharmonicity for a vast gain in coherence — the trade that made superconducting qubits practical.

Is a real Josephson junction truly dissipationless?

The supercurrent branch is, in principle, lossless: Cooper pairs tunnel elastically with zero voltage, so no energy is dissipated as long as $I<I_c$ . In a real device, though, small imperfections add loss. The amorphous oxide barrier hosts two-level-system defects that absorb microwave energy, quasiparticles (unpaired electrons) can tunnel and cause dissipation and dephasing, and the surrounding materials add dielectric loss. These are what limit transmon coherence times to the hundreds-of-microseconds range rather than infinity. So the ideal junction is dissipationless — that is the theoretical basis of the whole platform — but engineering the real barrier and its environment to approach that ideal is the central materials problem of the field.