Generator

What a collapse does to a field

One function in the extremes library, called 51 times across 9 essays. Below: what it draws at its defaults, what it draws at every branch an essay asks for, whether the site's own gate puts a claim to it, and everywhere it is called.

At its defaults it draws what a collapse does to a field. A body of radius 700,000 km carrying a field of 0.01 T, collapsing to 10 km. The flux through every comoving loop is fixed, so B goes as 1/R² and the field reaches 4.9·10⁷ T — a compression of 7·10⁴ in radius bought a factor of 4.9·10⁹ in field. The line has slope exactly −2 and that is the only claim being made: what a real object ends up with also depends on how well the flux was held, and on what generated it.

flux-freezing is one function in lib/figures/extremes.js — radiation, cross-sections and self-gravity. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

Drawn even though every essay passes options, because a default nothing exercises is a trap for the next essay to call this with none — which has happened here twice.

A body of radius 700,000 km carrying a field of 0.01 T, collapsing to 10 km. The flux through every comoving loop is fixed, so B goes as 1/R² and the field reaches 4.9·10⁷ T — a compression of 7·10⁴ in radius bought a factor of 4.9·10⁹ in field. The line has slope exactly −2 and that is the only claim being made: what a real object ends up with also depends on how well the flux was held, and on what generated it.

A pendulum with unequal steps

The options are the ones The circuit that forgets its charge passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

A pendulum with unequal steps. The potential −EJ cos φ of the junction against its phase, at EJ/EC = 50, with the lowest 4 levels of the circuit drawn across the well between their classical turning points, in units of the charging energy. The transitions are 18.94, 17.79, 16.50, each smaller than the one below it; a harmonic well of the same curvature would space them all at the square root of 8·EJ·EC, 20.00. The spacing shrinks because the cosine is flatter than a parabola away from its bottom, and the shrinking is what lets a microwave pulse tuned to the lowest transition leave the next one alone.

The potential −EJ cos φ of the junction against its phase, at EJ/EC = 50, with the lowest 4 levels of the circuit drawn across the well between their classical turning points, in units of the charging energy. The transitions are 18.94, 17.79, 16.50, each smaller than the one below it; a harmonic well of the same curvature would space them all at the square root of 8·EJ·EC, 20.00. The spacing shrinks because the cosine is flatter than a parabola away from its bottom, and the shrinking is what lets a microwave pulse tuned to the lowest transition leave the next one alone.

The levels that stop caring about stray charge

The options are the ones The circuit that forgets its charge passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

The levels that stop caring about stray charge. The lowest energy levels of a superconducting island joined to a reservoir through a Josephson junction, against the offset charge on the island in units of a Cooper pair's charge, measured from the lowest point of the ground level and in units of the first transition energy. With the Josephson energy 1 times the charging energy, the ground level wanders by 0.263 of a transition as the offset charge changes; with the Josephson energy 50 times the charging energy, the ground level wanders by 3.0e-8 of a transition as the offset charge changes. Every level repeats with a period of one pair, checked. On the left the levels are parabolas in the offset charge joined where the junction mixes them, and any stray charge moves the transition; on the right they are flat, and the circuit has almost forgotten which charge state it is near.

The lowest energy levels of a superconducting island joined to a reservoir through a Josephson junction, against the offset charge on the island in units of a Cooper pair's charge, measured from the lowest point of the ground level and in units of the first transition energy. With the Josephson energy 1 times the charging energy, the ground level wanders by 0.263 of a transition as the offset charge changes; with the Josephson energy 50 times the charging energy, the ground level wanders by 3.0e-8 of a transition as the offset charge changes. Every level repeats with a period of one pair, checked. On the left the levels are parabolas in the offset charge joined where the junction mixes them, and any stray charge moves the transition; on the right they are flat, and the circuit has almost forgotten which charge state it is near.

Exponentially deaf to charge

The options are the ones The circuit that forgets its charge passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

Exponentially deaf to charge. How much each of the three lowest levels moves as the offset charge sweeps through half a pair, divided by the first transition energy, against the ratio of Josephson to charging energy, both on logarithmic axes. The curves are computed by diagonalising the circuit's Hamiltonian at the two extremes of the offset charge and are checked against the asymptotic result, which falls as the exponential of minus the square root of eight times EJ/EC. At EJ/EC = 50: level 0 3.0e-8, level 1 2.1e-6, level 2 6.5e-5. Each higher level is more sensitive than the one below, because it sits nearer the top of the cosine well where the phase is less confined. Between EJ/EC = 1 and 50 the first excited level becomes 5.4·10⁵ times less sensitive.

How much each of the three lowest levels moves as the offset charge sweeps through half a pair, divided by the first transition energy, against the ratio of Josephson to charging energy, both on logarithmic axes. The curves are computed by diagonalising the circuit's Hamiltonian at the two extremes of the offset charge and are checked against the asymptotic result, which falls as the exponential of minus the square root of eight times EJ/EC. At EJ/EC = 50: level 0 3.0e-8, level 1 2.1e-6, level 2 6.5e-5. Each higher level is more sensitive than the one below, because it sits nearer the top of the cosine well where the phase is less confined. Between EJ/EC = 1 and 50 the first excited level becomes 5.4·10⁵ times less sensitive.

What the flatness costs

The options are the ones The circuit that forgets its charge passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

What the flatness costs. The difference between the second and first transition energies, as a fraction of the first, against the ratio of Josephson to charging energy. Zero would be a harmonic oscillator, whose transitions are all the same and whose two lowest levels cannot be addressed alone. The solid curve is computed by diagonalisation; the dashed curve is the result for a weakly anharmonic cosine well, the charging energy divided by the harmonic spacing less one charging energy, with a minus sign. At EJ/EC = 50 the relative anharmonicity is −6.1 per cent; at 1 it is +74.9 per cent. It falls only as one over the square root of the ratio, while the charge dispersion falls exponentially — the asymmetry the design exploits.

The difference between the second and first transition energies, as a fraction of the first, against the ratio of Josephson to charging energy. Zero would be a harmonic oscillator, whose transitions are all the same and whose two lowest levels cannot be addressed alone. The solid curve is computed by diagonalisation; the dashed curve is the result for a weakly anharmonic cosine well, the charging energy divided by the harmonic spacing less one charging energy, with a minus sign. At EJ/EC = 50 the relative anharmonicity is −6.1 per cent; at 1 it is +74.9 per cent. It falls only as one over the square root of the ratio, while the charge dispersion falls exponentially — the asymmetry the design exploits.

Where the design sits

The options are the ones The circuit that forgets its charge passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

Where the design sits. For a qubit whose first transition is fixed at 5 GHz, the two quantities the ratio EJ/EC trades against each other, in hertz on a logarithmic axis: the size of the anharmonicity, which sets how fast the qubit can be driven without leaking into its third level, and the charge dispersion of the first excited level, which sets how much a stray charge moves its frequency. They are equal near EJ/EC = 5.4. At 50 the anharmonicity is 303 MHz and the dispersion 10.3 kHz, with a charging energy of 264 MHz. The notch near EJ/EC of 3 is where the anharmonicity passes through zero and changes sign. Beyond it the anharmonicity declines slowly and the dispersion by many decades.

For a qubit whose first transition is fixed at 5 GHz, the two quantities the ratio EJ/EC trades against each other, in hertz on a logarithmic axis: the size of the anharmonicity, which sets how fast the qubit can be driven without leaking into its third level, and the charge dispersion of the first excited level, which sets how much a stray charge moves its frequency. They are equal near EJ/EC = 5.4. At 50 the anharmonicity is 303 MHz and the dispersion 10.3 kHz, with a charging energy of 264 MHz. The notch near EJ/EC of 3 is where the anharmonicity passes through zero and changes sign. Beyond it the anharmonicity declines slowly and the dispersion by many decades.

What checks it

physicscheck asserts something about flux-freezing that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Electromagnetism

The circuit that forgets its charge

A tiny superconducting island joined to its surroundings through a Josephson junction has discrete energy levels, and two of them make a quantum bit. The first such circuits were ruined by stray charges on nearby surfaces, which moved their levels and scrambled any superposition within a nanosecond. The cure was to make the junction's energy fifty times the charging energy. That makes the levels exponentially insensitive to charge while costing only a power-law loss in the unequal spacing that lets one transition be driven alone.

The field that cannot get out

Squeeze a lump of conducting fluid and its magnetic field comes with it, because the flux through any loop that moves with the material cannot change. Halve the radius and the field goes up fourfold; collapse by a factor of seventy thousand and it goes up by five thousand million.

Electromagnetism

The field that is pushed out

A perfect conductor keeps whatever field was inside it when its resistance vanished. A superconductor expels the field either way, and the difference between remembering and expelling is the experiment that showed superconductivity is a state of matter rather than a very good conductor.

The same force whichever way the surface faces

A magnetic field's stress is usually described as a pressure across the lines with a tension along them, as though it were two effects. It is one. The force per unit area is B²/2μ₀ on every surface however it is oriented, and only the direction changes — the surface normal reflected in the field, so that turning the surface one way turns the force the other. At 45° neither name applies.

The twist that outlives the turbulence

A plasma pinch driven hard enough goes violently unstable, and then settles into the same quiet state however it was started — with the field at its edge pointing backwards. The explanation is that turbulence destroys almost every constraint a perfect conductor obeys and spares one. The magnetic helicity, a measure of how twisted and linked the field is, decays far more slowly than the energy, and a field that has shed all the energy it can at fixed helicity has only one shape available to it.

Electromagnetism

The two in the flux quantum

A superconducting ring cannot hold whatever flux is applied to it. It holds a whole number of quanta and drives a current to make up the difference, and the size of that quantum is Planck's constant divided by twice the electron's charge. The factor of two was measured in 1961, four years after somebody predicted that the carriers are pairs — by an experiment in which no charge is measured at all.

Electromagnetism

The voltage that is a frequency

Two superconductors separated by a barrier a nanometre thick carry a current with no voltage at all, set by the difference of their quantum phases. Push harder and a voltage appears — and a voltage makes that phase difference run, so the current oscillates at 483.6 gigahertz for every millivolt. Shine microwaves on the junction and the voltage locks to exact multiples of the frequency divided by a ratio of fundamental constants, with nothing about the junction in it. That is why a volt is now counted in cycles.

The wave that does not know what the gas is made of

Give the magnetic tension of a bent field line an inertia and it becomes a string. The wave that runs along it goes at the same speed at every wavelength, compresses nothing anywhere, and has a speed containing no temperature, no pressure and no sound speed — it is the only wave in classical physics that is indifferent to what the medium is. Its energy travels along the field whatever direction the wave was sent in.

Electromagnetism

Two lengths, and which one is longer

A superconductor has a depth to which a field leaks in and a distance over which superconductivity itself can be built up. Which of the two is larger decides the sign of the energy of a boundary — and therefore whether the material keeps every field out or fills itself with a lattice of holes.

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