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In the previous chapter we built an LC oscillator out of superconducting circuit elements. It has a beautiful, clean problem: it is harmonic. Its energy levels are perfectly evenly spaced, like rungs on a ladder where every step is the same height. That sounds nice, but for a qubit it is fatal. If you send in a microwave pulse tuned to drive the $0 \to 1$ transition, the exact same photon also drives $1 \to 2$, $2 \to 3$, and so on. You cannot isolate a clean two-level system. You need one circuit element that breaks this even spacing, and that element is the Josephson junction.
A junction is almost embarrassingly simple to picture: two superconductors separated by a thin (~1 nm) insulating barrier. Classically nothing should flow. But the superconducting condensate on each side is described by a single macroscopic wavefunction$\psi = \sqrt{n_s},e^{i\theta}$ with a well-defined phase $\theta$. Only the gauge-invariant phase difference
$$ \varphi = \theta_L-\theta_R-\frac{2\pi}{\Phi_0}\int_R^L \mathbf A\cdot d\mathbf l $$
across the barrier matters. This convention takes the positive branch direction to be $R\to L$; positive current and voltage below use the same branch orientation. Reversing the branch sends $\varphi\to-\varphi$ and $I\to-I$ but leaves $U=-E_J\cos\varphi$ unchanged. In a gauge with negligible vector potential across the barrier this reduces to $\theta_L-\theta_R$, and through it Cooper pairs tunnel coherently across the gap. The junction is the one circuit element that is simultaneously nonlinear and non-dissipative, and you cannot build a good qubit without both.
Here is the causal story the rest of the chapter tells, end to end:
flowchart TD
A["Cooper pairs tunnel<br/>(condensate phase phi)"] --> B["First relation<br/>I = I_c sin phi"]
B --> C["Nonlinear<br/>inductance L_J(phi)"]
C --> D["Cosine potential<br/>-E_J cos phi"]
D --> E["Quartic term<br/>flattens the well"]
E --> F["Unequal level spacing<br/>E12-E01 approx -E_C"]
F --> G["Address one<br/>0->1 transition<br/>= a qubit!"]
H["Run at large<br/>E_J/E_C"] --> I["Charge dispersion<br/>~ exp(-sqrt(8 E_J/E_C))"]
I --> J["Dephasing<br/>suppressed<br/>(long T2)"]
G --> K["Transmon<br/>sweet spot"]
J --> K
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The two Josephson relations
Let $\varphi$ be the gauge-invariant phase difference across the junction, and define the branch flux $\Phi=(\Phi_0/2\pi)\varphi$ so that $V=\dot\Phi$. The entire device is governed by two relations:
$$ I = I_c \sin\varphi \qquad\text{(DC / first relation)} $$$$ \frac{d\varphi}{dt} = \frac{2e}{\hbar},V \qquad\text{(AC / second relation)} $$
Where the first relation comes from. Each superconductor carries a condensate with phase $\theta_{L,R}$. Coupling the two through the barrier (Feynman's two-mode tunnelling model) makes the supercurrent depend only on the phase difference, and the simplest such periodic, odd function is $\sin\varphi$. The amplitude is the critical current$I_c$, set by the barrier transparency and the gap. Plain meaning: push a phase across the junction and a dissipationless current flows with no voltage, until you try to exceed $I_c$.
Where the second relation comes from. Each condensate phase evolves as $\dot\theta = -E/\hbar$, where $E$ is the energy of a charge carrier on that side. A voltage $V$ biases the two sides by an energy $2eV$, note the $2e$, because the carriers are Cooper pairs, not single electrons. So the difference winds at $\dot\varphi = 2eV/\hbar$. Integrated, a DC voltage produces an AC supercurrent at the Josephson frequency $f = 2eV/h$, with the universal slope $2e/h \approx 483.6\ \text{MHz}/\mu\text{V}$. That factor of $2e$ is the experimental fingerprint of Cooper pairing.
The nonlinear inductance. Differentiate the first relation and substitute the second:
A linear inductor has constant $L$; here $L_J$ stiffens and softens with $\varphi$, diverges at $\varphi=\pi/2+k\pi$, and has negative incremental sign where $\cos\varphi<0$. That phase-dependence is exactly the ingredient the bare LC circuit was missing.
Pitfall.$L_J$ is only a literal lumped inductor in the small-oscillation linearisation around an operating point. Near $\varphi=\pi/2+k\pi$ it blows up; where $\cos\varphi<0$ it describes negative curvature of the cosine potential, not a stable standalone inductor.
The cosine potential
Integrate the energy delivered to the junction, $U = \int I,V,dt$. Using $I = I_c\sin\varphi$ and $V = (\hbar/2e)\dot\varphi$ gives $I,V,dt = I_c\sin\varphi,(\hbar/2e),d\varphi$, and integrating over $\varphi$:
$E_J$ is the Josephson energy, the amplitude of the cosine potential. With the zero chosen as written, the minimum is at $-E_J$ and the zero-bias barrier to the neighboring maximum is $2E_J$. Taylor-expanding,
Using $\Phi=(\Phi_0/2\pi)\varphi$, the $\varphi^2$ term reproduces a harmonic oscillator: $\tfrac12E_J\varphi^2=\Phi^2/(2L_{J0})$, so $L_{J0}=(\Phi_0/2\pi)^2/E_J=\Phi_0/(2\pi I_c)$. The $\varphi^4$ term is the crucial correction: it is negative, so the cosine well is flatter than a parabola away from the bottom. A flatter well means the energy rungs get closer together as you climb, the ladder is no longer evenly spaced.
The cosine (solid) hugs the parabola (the steeper arms) only at the bottom; the levels $E_0, E_1, E_2$ live in the flatter cosine, so $\hbar\omega_{12} < \hbar\omega_{01}$.
Intuition aside. Think of the cosine as a pendulum. Small swings are nearly harmonic and isochronous; but a real pendulum slows for large swings, its period grows with amplitude. That amplitude-dependent frequency is anharmonicity. Tilt the cosine by adding a bias current and you get the tilted washboard: the bob can roll over barriers (phase slips) once $I \to I_c$. A qubit lives in the lowest swings of this superconducting pendulum.
pivot ● tilt with bias current I → /\ /\ /\
|\ / \/ \/ \
| \ angle = phase φ running "washboard":
| \ phase slips when I→I_c
| ● bob gravity well = -E_J cos φ
The Hamiltonian in the charge basis
Pair the cosine potential with the electrostatic charging energy of the island's effective capacitance $C_\Sigma$ (junction plus gate/shunt/network capacitances). The island charge is $Q = 2e(\hat n - n_g)$, so $Q^2/(2C_\Sigma)$ gives the full Hamiltonian:
$$ H = 4E_C,(\hat n - n_g)^2 - E_J\cos\hat\varphi, \qquad E_C = \frac{e^2}{2C_\Sigma}, \qquad [\hat\varphi,\hat n] = i. $$
Here $\hat n$ counts excess Cooper pairs, $n_g$ is the dimensionless offset (gate) charge, and $E_C$ is the charging energy, the single-electron scale $e^2/2C_\Sigma$ that sets the cost of moving charge onto the island (one full Cooper pair costs $4E_C$ at $n_g=0$). The $4E_C(\hat n - n_g)^2$ term is the "kinetic" energy; $-E_J\cos\hat\varphi$ is the "potential." Their ratio $E_J/E_C$ governs everything.
The chapter title promised "in the charge basis," so let's make it concrete. Because $\hat\varphi$ and $\hat n$ are conjugate, $e^{\pm i\hat\varphi}$ shift Cooper-pair number by one. Therefore $\cos\hat\varphi$ is the Hermitian nearest-neighbor hopping operator:
It hops between adjacent charge states $n \leftrightarrow n\pm 1$, literally one Cooper pair tunnelling. In this basis $H$ is tridiagonal: diagonal entries $4E_C(n-n_g)^2$, off-diagonal entries $-E_J/2$. You could type that matrix into NumPy, truncate at $|n|\le 10$, and use eigh to get the spectrum; that is the standard numerical route for plots of the CPB/transmon spectrum. The sketches below are schematic, not generated data.
$E_J/E_C$: the master control knob
Everything about the device follows from one ratio. It interpolates a whole continuum of qubits:
Regime
$E_J/E_C$
Eigenstate character
Charge dispersion
Anharmonicity
Trade-off
Cooper-pair box / charge qubit
$\lesssim 1$
charge (number) states
large (strong $n_g$ sensitivity)
large
very charge-noise sensitive
Intermediate
$\sim 1$-$10$
mixed
moderate
moderate
partially charge-protected, larger anharmonicity
Transmon
$\gg 1$ ($\sim 20$-$100$)
phase-localized
exponentially suppressed
weak (power-law)
charge-insensitive, modest anharmonicity
In the transmon limit $E_J/E_C\gg1$, the charge dispersion scales as $\sim e^{-\sqrt{8E_J/E_C}}$ while the relative anharmonicity scales as $\alpha_r \sim -(8E_J/E_C)^{-1/2}$. The transmon is just one limit of this family, not the only option.
Contrast the two oscillators a newcomer might confuse:
Property
Linear LC oscillator
Josephson (transmon)
Potential shape
$\tfrac12 L^{-1}\Phi^2$ parabola
$-E_J\cos\varphi$ cosine
Inductance
constant $L$
$L_J(\varphi)=\Phi_0/(2\pi I_c\cos\varphi)$
Level spacing
equal ($\hbar\omega$)
unequal (shrinks with $m$)
Usable as a qubit?
No (cannot address one transition)
Yes
Key parameter
$\omega = 1/\sqrt{LC}$
ratio $E_J/E_C$
Dissipation
non-dissipative if built from ideal superconducting $L,C$
non-dissipative in the ideal junction model
Energy levels and anharmonicity
In the transmon regime $\varphi$ is localized near $0$, so expand the cosine to quartic order and treat the $\varphi^4$ term in first-order perturbation theory using harmonic-oscillator matrix elements. The result (Koch et al. 2007):
The harmonic part is $4E_C\hat n^2+\tfrac12E_J\hat\varphi^2$. With $[\hat\varphi,\hat n]=i$, choose $\hat\varphi=\varphi_{\rm zpf}(a+a^\dagger)$ and $\hat n=\tfrac{i}{2\varphi_{\rm zpf}}(a^\dagger-a)$; balancing the two quadratures gives $\varphi_{\rm zpf}=(2E_C/E_J)^{1/4}$ and the plasma frequency$\hbar\omega_p=\sqrt{8E_JE_C}$. The $m^2$ term bends the ladder. Subtracting levels:
So the energy anharmonicity is $\alpha_E\approx -E_C$; quoted as a cyclic frequency, $\alpha_E/h$ is typically a few hundred MHz, and the relative anharmonicity shrinks only as a weak power law of $E_J/E_C$. Why does this finally give a qubit? The $0\to1$ and $1\to2$ transitions now sit at different frequencies, detuned by $\alpha_\omega$. A pulse resonant with $\omega_{01}$ is off-resonant from $\omega_{12}$, so it leaves higher levels alone, provided its bandwidth $\sim 1/\tau$ stays below $|\alpha_\omega|/2\pi$. That is a hard gate-speed limit: too-fast pulses leak population into level $2$. Pulse-shaping (e.g. DRAG) is the practical fix.
Pitfall. A transmon is not a true two-level system, it is a weakly anharmonic multi-level oscillator. Levels $2, 3, \dots$ are always there, which is exactly why $|\alpha_\omega|$ and DRAG matter. And the nonlinearity comes entirely from the cosine; $E_C$ only sets the oscillator scale and the size of the anharmonicity, the charging term itself is not nonlinear.
Charge dispersion: the whole point of the transmon
Each level's energy depends on the gate charge $n_g$. Define the signed charge dispersion$\epsilon_m \equiv E_m(n_g=1/2)-E_m(n_g=0)$; the peak-to-peak width is $|\epsilon_m|$, and it is suppressed exponentially (Koch et al. 2007):
The physics: making the band depend on $n_g$ requires the phase to tunnel between adjacent cosine wells ($\varphi\to\varphi+2\pi$). A WKB/instanton estimate of that tunnelling action gives the $e^{-\sqrt{8E_J/E_C}}$ factor (the exact problem maps onto Mathieu's equation, which supplies the algebraic prefactor). Flat bands mean $df_q/dn_g \approx 0$: stray $1/f$ charge noise barely shifts the qubit frequency, so dephasing is suppressed and $T_2$ is long. This is the transmon's defining payoff, and it costs only a weak power-law penalty in anharmonicity.
Pitfall. "Charge-insensitive" does not mean $n_g$ vanishes from $H$, it still appears. The point is the resulting bands are exponentially flat. Likewise "non-dissipative" is an ideal-element statement; real junctions still suffer quasiparticle, dielectric, and radiative losses.
A fully worked transmon (illustrative numbers)
Pick two energy scales (round teaching values, illustrative, not any real device): $E_C/h = 250\ \text{MHz}$ and $E_J/E_C = 50$, so $E_J/h = 12.5\ \text{GHz}$. All energies are quoted as frequencies by dividing by $h$.
Step 1: back out the hardware.
Total island capacitance: $C_\Sigma = e^2/(2E_C) = e^2/(2h\cdot 250,\text{MHz}) \approx 78\ \text{fF}$ (a large pad, how you reach big $E_J/E_C$).
Step 3: anharmonicity. Angular-frequency anharmonicity: $\alpha_\omega/2\pi \approx -E_C/h = -250\ \text{MHz}$, so $\omega_{12}/2\pi \approx 4.50\ \text{GHz}$, the $1\to2$ transition sits $250\ \text{MHz}$below the $0\to1$. Relative: $\alpha_r \approx -(8\cdot 50)^{-1/2} = -1/\sqrt{400} = -0.05$, i.e. $-5%$.
Step 4: charge dispersion (the payoff). Exponent $\sqrt{8\cdot 50} = 20$, so $e^{-20}\approx 2.06\times10^{-9}$. The ground level barely moves: $\epsilon_0/h \approx (250,\text{MHz})\cdot 32\cdot 0.798\cdot 25^{0.75}\cdot 2.06\times10^{-9} \approx 150\ \text{Hz}$. But the quantity you actually measure is the $0\to1$ transition, and its dispersion is set by the upper level: $\epsilon_m$ grows fast with $m$ (the $2^{4m+5}$ prefactor), so evaluating the formula at $m=1$ gives $|\epsilon_1|/h \approx 11.8\ \text{kHz}$, about $80\times$ larger than $\epsilon_0$. Because $\epsilon_0$ and $\epsilon_1$ alternate in sign (the $(-1)^m$), the $0\to1$ frequency wiggles peak-to-peak by $|\epsilon_1-\epsilon_0|/h \approx |\epsilon_0|/h+|\epsilon_1|/h \approx 11.9\ \text{kHz}$ across a full $n_g$ period, about one part in $5\times10^5$ of $4.75\ \text{GHz}$. That is why the transmon barely feels charge noise. (Diagonalizing the Mathieu problem directly confirms the $\sim 12\ \text{kHz}$ transition swing.)
Step 5: gate-speed sanity check. With $|\alpha_\omega|/2\pi = 250\ \text{MHz}$, a plain resonant pulse must keep its spectral width $\sim 1/\tau$ well under $250\ \text{MHz}$, so $\tau\gtrsim$ a few ns; DRAG relaxes this.
It's $2e$, not $e$. Both Josephson relations and $\Phi_0 = h/2e$ carry the Cooper-pair charge. The factor-of-two error is everywhere.
Bigger $E_J/E_C$ is not strictly better. It suppresses charge dispersion exponentially but preserves anharmonicity only as a power law; too little $|\alpha_\omega|$ means leakage. The transmon is a deliberate compromise.
Two anharmonicities.$\alpha_E$ is an energy ($\approx -E_C$), while $\alpha_\omega=\alpha_E/\hbar$ is angular frequency and $\alpha_E/h$ is the value quoted in Hz; $\alpha_r$ (relative, $\approx -5%$, dimensionless) measures fractional unevenness.
The cosine is flatter, not steeper. The negative quartic term lowers upper levels and shrinks spacings, that is what "negative anharmonicity" means.
Key takeaways
The Josephson junction supplies the standard nonlinear, ideally non-dissipative circuit element; a bare superconducting LC oscillator can be low-loss, but it remains harmonic and therefore unusable as an addressable qubit.
Two relations: $I = I_c\sin\varphi$ and $\dot\varphi = 2eV/\hbar$, giving the phase-dependent inductance $L_J(\varphi) = \Phi_0/(2\pi I_c\cos\varphi)$.
The junction energy is the cosine potential$-E_J\cos\varphi$; its quartic term flattens the well and unevenly spaces the ladder.
$H = 4E_C(\hat n - n_g)^2 - E_J\cos\hat\varphi$ is tridiagonal in the charge basis and governed by $E_J/E_C$, a continuum from charge qubit to transmon.
Anharmonicity $\alpha_\omega\approx -E_C/\hbar$ lets a selective pulse address one $0$-$1$ transition (gate-speed limited by $|\alpha_\omega|$); charge dispersion $\sim e^{-\sqrt{8E_J/E_C}}$ is the exponential charge-noise immunity that defines the transmon.
Go deeper
Koch et al., "Charge-insensitive qubit design derived from the Cooper pair box," Phys. Rev. A 76, 042319 (2007), the foundational transmon paper and source of the $E_m$, $\omega_q$, $\alpha_r$, and charge-dispersion formulas above. arXiv:cond-mat/0703002.
Krantz et al., "A Quantum Engineer's Guide to Superconducting Qubits," Appl. Phys. Rev. 6, 021318 (2019), the standard modern review. arXiv:1904.06560.
Blais et al., "Circuit Quantum Electrodynamics," Rev. Mod. Phys. 93, 025005 (2021), authoritative derivations of the charge-basis Hamiltonian and charge dispersion. arXiv:2005.12667.