The circuit that forgets its charge

The voltage that is a frequency treated the phase difference across a Josephson junction as a classical quantity. It had a value, the value rotated when a voltage was applied, and microwaves could lock the rotation into steps. That picture is correct for a large junction carrying a large current, and it hides the fact that the phase is a quantum variable with a partner it cannot share certainty with.

The partner is the number of Cooper pairs that have crossed the junction. Phase and pair number are conjugate in the same way as position and momentum, and in a large circuit both can be nearly definite because the uncertainty in each is tiny compared with its size. Make the circuit small — a small island of superconductor, joined to a reservoir through a single junction — and the energy cost of moving one pair onto the island becomes large enough to matter. Then neither the phase nor the charge is definite, the circuit has quantised energy levels, and a pair of them can be used as a quantum bit.

This essay is about the design problem that made such circuits useful, which turned out to be less about making them quantum than about making them indifferent to something.

Two energies and one Hamiltonian #

The circuit has two energy scales. The charging energy $E_C = e^2/2C$, with $C$ the island’s total capacitance, is the cost of adding one electron’s charge. The Josephson energy $E_J$ is the coupling across the junction, proportional to its critical current. The Hamiltonian is

$$H = 4E_C\,(n - n_g)^2 - E_J\cos\varphi,$$

where $n$ is the number of pairs on the island, $\varphi$ the phase difference across the junction, and the factor of four comes from a pair’s charge being $2e$. The quantity $n_g$ is the offset charge, in units of a pair: the charge induced on the island by any voltage or stray charge nearby. It can be set deliberately with a nearby electrode, and it is also changed, uncontrollably, by every charged defect that hops on the surfaces and in the oxides around the circuit.

In the basis of definite pair number, the charging term is diagonal and the junction term moves one pair on or off, so the Hamiltonian is a tridiagonal matrix and its eigenvalues can be found exactly. Everything drawn here comes from that calculation.

On the left, with $E_J = E_C$, the circuit is what was called a Cooper-pair box. Its levels are nearly the parabolas $4E_C(n - n_g)^2$ of definite charge states, joined where the junction mixes neighbouring charges, and they move strongly with the offset charge: the ground level alone wanders by 0.263 of a transition as $n_g$ changes. Every level repeats with a period of exactly one pair in $n_g$, which the calculation checks, because adding a whole pair to the offset is the same as relabelling which charge state is which.

On the right, with $E_J = 50E_C$, the same levels are straight lines. The ground level moves by $3.0\times10^{-8}$ of a transition over the whole range. The circuit has nearly stopped caring what the offset charge is.

Why the Cooper-pair box could not keep a secret #

The first demonstration that a circuit like this could be put into a superposition and kept coherent came in 1999, when Yasunobu Nakamura, Yuri Pashkin and Jaw-Shen Tsai drove a Cooper-pair box between two charge states and watched it oscillate. The oscillations died in about a nanosecond.

The reason is visible in the left panel. A quantum bit stores information in the phase of a superposition, and that phase advances at the transition frequency. If the frequency wanders, the phase wanders with it, and the superposition is scrambled — the qubit dephases. In a Cooper-pair box the transition frequency depends on $n_g$ as steeply as the panel shows, and $n_g$ is never still, because charged two-level defects in the amorphous oxides around the circuit switch at random over every timescale. The effect is a frequency noise with a spectrum rising towards low frequencies, and a circuit this sensitive to it cannot hold a phase for long. Operating at the special point $n_g = 1/2$, where the levels are flat to first order, helped by a factor of several hundred, and was not enough.

There were two possible remedies: remove the fluctuating charges, or build a circuit that does not notice them. The first is a materials problem that has still not been solved completely. The second is a design choice.

A pendulum in a cosine well #

Raising $E_J$ relative to $E_C$ changes which term dominates. When the junction energy is large, the phase prefers to sit near the bottom of the $-E_J\cos\varphi$ well, and the charging term acts as a kinetic energy — the capacitance plays the role of a mass. The circuit becomes a quantum pendulum.

Near the bottom of the well the cosine is a parabola, as every minimum is, and a parabola would give a harmonic oscillator with evenly spaced levels at $\sqrt{8E_JE_C}$, which here is 20 charging energies. The cosine is flatter than its parabola away from the bottom, so the levels crowd together as they rise. The calculated spacings are 18.94, 17.79 and 16.50: the first is lower than the harmonic value by about one charging energy, and each subsequent one by about one charging energy more. The first transition matches $\sqrt{8E_JE_C} - E_C$ to within two per cent at a ratio of eighty, which the figure checks.

It is the quantum version of the period that depends on the swing. A real pendulum’s period lengthens with amplitude because the restoring force grows less than linearly, and a quantum pendulum’s level spacing shrinks with energy for the same reason. That shrinkage is called the anharmonicity, and a qubit cannot work without it.

The charge the pendulum still remembers #

In the pendulum picture it is not obvious why the offset charge should affect the levels at all, since $n_g$ appears only in the kinetic term. The answer is that the phase is an angle. The pendulum’s position $\varphi$ and $\varphi + 2\pi$ are the same configuration, and a wavefunction on a circle can pick up a phase as it goes once round. The offset charge sets that phase: going round the circle once multiplies the wavefunction by $e^{2\pi i n_g}$, in exactly the way a magnetic flux sets the phase an electron acquires going round a path that encloses a flux it never touches.

A pendulum deep in its well almost never goes round. To do so it would have to pass over the top of the potential, a barrier $2E_J$ high, and quantum mechanically that is possible only by tunnelling through a barrier it lacks the energy to climb. The probability amplitude for tunnelling falls exponentially with the barrier’s height and width, and so, therefore, does the sensitivity to offset charge.

The calculation shows the exponential directly, and at a ratio of sixty it agrees with the asymptotic result of Jens Koch and colleagues to within twenty per cent for the two lowest levels, as it should where the asymptotic expansion begins to hold. At a ratio of fifty the ground level’s dispersion is $3.0\times10^{-8}$ of a transition, the first excited level’s $2.1\times10^{-6}$ and the second’s $6.5\times10^{-5}$. Higher levels sit nearer the top of the well, where the barrier is thinner, and tunnel more readily. Between a ratio of one and a ratio of fifty the first excited level becomes a hundred thousand times less sensitive to charge.

That circuit, a junction shunted by a large capacitor to push $E_J/E_C$ to around fifty, was proposed by Koch and colleagues at Yale in 2007 and named the transmon. The large capacitor is usually a pair of metal pads a fraction of a millimetre across, which also couple the qubit to the resonators used to control and read it.

What the flatness costs #

Nothing is free, and what the transmon gives up is anharmonicity. A harmonic oscillator is useless as a qubit because all its transitions coincide: a microwave pulse at the frequency of the lowest one drives every transition at once, and the circuit climbs as far as the pulse pushes it. The qubit needs its first two levels to be addressable on their own, which requires the second transition to differ from the first by more than the frequency spread of the pulse used to drive it.

In the Cooper-pair box regime the anharmonicity is large and positive — at a ratio of one the second transition is 74.9 per cent above the first — because the levels are nearly those of charge states, whose energies grow quadratically. As the ratio rises the anharmonicity passes through zero, where the circuit is briefly as useless as a harmonic oscillator, and settles towards the pendulum’s negative value. At a ratio of fifty it is −6.1 per cent. The pendulum formula, drawn dashed, describes it well once the ratio is above ten or so.

The point of the comparison is the functional form. The anharmonicity falls as one over the square root of $E_J/E_C$. The charge dispersion falls as the exponential of minus the square root. Increasing the ratio from ten to fifty costs roughly a factor of two in anharmonicity and buys several orders of magnitude in charge insensitivity. There are few trades in physics this lopsided, and the transmon is the result of noticing this one.

Where the design sits #

Put in real units, the trade becomes a choice about hertz. Fix the qubit’s first transition at five gigahertz, typical of these devices, and ask what each ratio gives.

At a ratio of fifty the charging energy is 264 megahertz, the anharmonicity 303 megahertz and the charge dispersion of the excited level 10.3 kilohertz. The anharmonicity limits how fast a logic operation can be: a pulse short enough to have a bandwidth comparable with 300 megahertz would drive the second transition too, which puts the natural duration of an operation at some tens of nanoseconds, and pulse shapes designed to cancel the leakage bring it down to around ten. The dispersion limits how long a superposition survives charge noise: ten kilohertz is the full range of frequency shift available to the worst possible offset charge, and the typical fluctuation is much smaller, which makes charge noise a minor contributor to dephasing in a modern transmon rather than the dominant one.

The notch near a ratio of three is where the anharmonicity changes sign and its magnitude briefly vanishes. It is the reason no one builds a qubit there.

Why fifty and not five hundred #

If the charge dispersion falls exponentially and the anharmonicity only as a square root, it might seem that the ratio should be pushed as high as possible. Two things stop it.

The first is the anharmonicity itself. At a ratio of five hundred the relative anharmonicity would be about two per cent, a hundred megahertz for a five-gigahertz qubit, and each operation would have to be three times slower to avoid driving the second transition. Slower operations mean fewer of them within the qubit’s lifetime. Once the charge dispersion is already a few kilohertz, well below the other sources of frequency noise, further suppression buys nothing and the lost anharmonicity is a pure cost. The useful range is where charge noise has stopped being the leading problem, and not much beyond.

The second is the capacitor. The charging energy is lowered by making the capacitance larger, which in practice means larger metal pads. Larger pads spread the qubit’s electric field over more surface, where the lossy oxides are, and couple it more strongly to its neighbours on the chip. Every factor in the ratio has to be paid for in capacitor area, and the area is where the losses live.

Many transmons also make $E_J$ adjustable. Replacing the single junction with two in parallel forms a small superconducting loop, and the loop’s critical current depends on the magnetic flux through it, oscillating with a period of one flux quantum — the quantum set by the pair’s charge. A flux-tunable transmon can be moved in frequency to bring two qubits into resonance for a two-qubit operation and apart again afterwards. The price is a new sensitivity: the frequency now depends on flux, and fluctuating magnetic moments on the metal surfaces supply flux noise in the same way charged defects supply charge noise. At the flux where the frequency is at its maximum, the dependence is flat to first order, and qubits are parked there when idle, which is the same trick the Cooper-pair box used at its special charge point, applied to the other variable.

Reading the state without disturbing the charge #

A qubit has to be measured, and measuring a transmon’s charge would be exactly the wrong thing to do with a circuit designed to be insensitive to charge. Instead the transmon is read through a microwave resonator coupled to it by a small capacitance.

When the qubit’s transition frequency is far from the resonator’s, the two cannot exchange energy, but each shifts the other’s frequency slightly, and the shift depends on which state the qubit is in. A microwave tone sent through the resonator emerges with a phase that differs between the qubit’s ground and excited states. The measurement records that phase, and it projects the qubit onto one of its two levels without having driven a transition. The size of the frequency shift depends on the coupling, the detuning, and — through the level structure drawn above — on the anharmonicity: a perfectly harmonic oscillator would produce no state-dependent shift at all, because its levels would all push on the resonator equally. The unequal spacing is used twice, once to drive one transition alone and once to tell the states apart.

A pendulum with no bob #

The well figure draws the levels as horizontal lines inside a cosine potential, as though the circuit’s state were a ball at some height in a valley. Nothing on the chip moves along that axis. The phase difference is a property of two superconducting condensates, and the circuit’s state is a wavefunction spread over values of the phase, with no position to point at. The lines are drawn between classical turning points because that is where a classical pendulum of that energy would reverse, a mnemonic for how wide the wavefunction is rather than a boundary it respects: the tunnelling that sets the charge dispersion happens exactly where the drawn lines stop.

The level diagrams have the same limitation in the other variable. Each level is drawn as a sharp line at a definite energy for a definite offset charge, when the offset charge is itself fluctuating and the qubit is coupled to a resonator that shifts every level slightly. The figures show the circuit alone at fixed parameters; the qubit on a chip is never either.

What limits a transmon now #

Losses, not charge. Once charge noise was suppressed, the lifetime of the excited state became the limit, and it is set mainly by energy leaking into defects. Two-level systems in the amorphous oxides on the metal surfaces and at the substrate interface absorb microwave photons, and the transmon’s large pads place much of its electric field in exactly those regions. Improvements in materials — tantalum films, cleaner interfaces, better substrates — have taken the best lifetimes from around a microsecond in the first transmons to hundreds of microseconds.

Quasiparticles. A superconductor at millikelvin temperatures should contain essentially no broken pairs, but stray infrared radiation and even cosmic rays and background radioactivity break some. A quasiparticle tunnelling across the junction changes the island’s charge by one electron — half a pair — which shifts $n_g$ by a half and can also cause the qubit to relax. The transmon’s charge insensitivity makes the first effect small; the second remains, and correlated bursts of quasiparticles from particle impacts affect many qubits on a chip at once.

The cosine is an approximation to the junction. $-E_J\cos\varphi$ is the energy of a tunnel junction with a thin insulating barrier and many weakly transmitting channels. Junctions with a few highly transmitting channels, used in some newer designs, have a potential with higher harmonics, and the level structure changes accordingly.

And the circuit has more than one mode. The single-variable Hamiltonian treats the island as a lump. Real chips have the transmon capacitively coupled to resonators, feed lines and other qubits, and the coupling shifts the levels, mediates interactions and provides extra channels for decay. The Hamiltonian here is the part of the problem that is solved exactly; engineering a processor is mostly the rest.

Still open: how far the materials can be pushed #

The superconducting state has been followed from the field it pushes out, through the pair charge in the flux quantum and the two lengths that sort superconductors into types, to the junction whose voltage is a frequency. Here the junction’s phase becomes a quantum coordinate, and the problem is no longer whether a circuit can be quantum but how to keep it so.

The habit worth carrying away concerns what can be traded against what. When two quantities both depend on the same parameter, compare the forms of the dependence before comparing the values; an exponential against a power law gives a trade that is almost free. The transmon does not remove its sensitivity to charge. It makes the sensitivity an exponentially small tunnelling amplitude while the property it needs, the unequal spacing, declines only as a square root. The same shape of argument decides how narrow a resonance can be for a given lifetime, where one side of the trade is fixed and the design question is which parameter moves the other.

What is not known is where the lifetimes of superconducting circuits will stop improving. Every factor gained so far has come from identifying a loss mechanism — a surface oxide, a substrate defect, a radiation source — and removing it, and each removal has exposed the next. Whether there is a floor set by the physics of the superconductor itself, or only a sequence of materials problems each solvable in turn, decides how large an error-corrected quantum computer built from such circuits has to be, and it is being measured one device generation at a time.