The return a classical cloud never makes

The state that swings like a pendulum found the one family of quantum states that behaves like a classical oscillator for ever: a packet of the ground state’s width, displaced, sliding back and forth in a harmonic well without changing shape. It also found why the family works only there. The levels of a harmonic oscillator are evenly spaced, so every component of a superposition turns at a whole multiple of one frequency and the whole state repeats after each period.

No real oscillator is exactly harmonic. A pendulum’s period grows with its amplitude, the small lie the textbook pendulum tells; a molecule’s vibrational levels crowd together towards dissociation; a microwave resonator coupled to a Josephson junction has levels whose spacing shrinks by a fixed amount with each photon added. In every one of them the evenly spaced levels pick up a curvature, and the rigid packet stops being rigid.

What happens next is usually summarised as the packet spreads, and the summary is misleading twice. The spreading is not the quantum part. And it is not the end.

A small curvature in the energies #

The model the figures use is the simplest that has the property. In units where the oscillator’s frequency and Planck’s constant are both one, the energy of the state with $n$ quanta is

$$E_n = n + \chi n^2,$$

with $\chi$ small — one two-hundred-and-fortieth here. The spacing between neighbouring levels, $E_{n+1} - E_n = 1 + \chi(2n+1)$, is the frequency at which a packet made of those levels goes round its orbit, and it now depends on $n$: a packet with more quanta, which is a larger swing, orbits slightly faster. That is exactly what a classical oscillator with an amplitude-dependent frequency does, and it is the whole of the anharmonicity.

The same quadratic spectrum turns up in more places than the model suggests. A cavity whose photons interact through a nonlinear element — the Kerr effect, in optics — has it. So does a site in an optical lattice holding $n$ interacting atoms, whose interaction energy grows as $n(n-1)/2$. So, locally, does any spectrum at all, because near a given quantum number every smooth function of $n$ looks like a straight line plus a parabola, and the parabola is where the interesting times come from.

The packet starts as a coherent state with nine quanta on average, which the essay on the pendulum state showed to carry a Poisson spread of three quanta either way. Nine plus or minus three quanta means a spread of orbital frequencies of about $2\chi\sqrt{\bar n}$, and a spread of frequencies is the thing that makes a swing die away.

The collapse is the classical part #

The dashed curve in that figure is not a quantum calculation. It is forty thousand classical oscillators, placed in the position–momentum plane with the same spread the quantum packet has, each going round at the frequency its own energy gives it under the same anharmonic law. Their average position swings and collapses along with the quantum packet’s, and through twice the collapse time the two curves never differ by more than about three per cent of the swing.

That is the first correction to the usual summary. The quantum calculation gives a closed form for the envelope, $\sqrt2\,\alpha\,\exp(-2\bar n\sin^2\chi t)$, which for short times is a Gaussian falling to $1/e$ after $1/(\chi\sqrt{2\bar n})$ — nine periods here. But nothing about that decay needs quantum mechanics. Runners on a track who start together and run at slightly different speeds spread out round it, and after enough laps the pack has no centre; the time that takes is set by the spread of their speeds, and the collapse of the packet is that time. The packet that will not keep its shape spreads for a related reason in free space, where a spread of momenta is a spread of speeds.

So the spreading of a bound packet is dephasing, and dephasing is classical. The average that obeys Newton showed the mean of a spreading packet departing from the classical trajectory of a single particle. The comparison here is with a classical cloud, which is the fairer one, and against that the quantum packet does nothing wrong at all.

Why a cloud of points never regroups #

The cloud in that figure spreads for good, and the reason is worth stating exactly, because the quantum state’s different behaviour follows from the negation of it.

Each member of the classical cloud has an energy that can take any value, so the frequencies in the cloud form a continuum. For the whole cloud to regroup, every member would have to have completed a whole number of extra turns relative to every other at the same instant. With a continuous range of frequencies no such instant exists, and the cloud’s angular order — one for a point, zero for a uniform ring — falls from 0.99 to 0.01 and stays there. A classical ensemble with a smooth spread of frequencies relaxes to a ring, permanently, which is the elementary version of how a gas forgets its initial conditions.

The quantum packet is also a superposition of components with different frequencies, but its frequencies are not a continuum. The state is a sum over whole numbers of quanta, the component with $n$ quanta turns as $e^{-iE_nt}$, and the energies sit on a lattice set by integers. The phase that carries the anharmonicity is $\chi n^2 t$. At $t = 2\pi/\chi$ that phase is $2\pi n^2$ for every $n$ — a whole number of turns, whatever $n$ is, because the square of an integer is an integer. Every component is back in step with every other, and the state is exactly what it was.

Halfway there, at $t = \pi/\chi$, the phase is $\pi n^2$, which is a whole number of turns for even $n$ and half a turn extra for odd $n$ — and multiplying the odd components by $-1$ is precisely the operation that reflects a coherent state to the other side of the well. So the packet returns whole at 120 periods, on the far side, and whole again at 240 periods on the near side. The figure at the top of the page shows both.

That is the second correction. The spreading is not the end, and the reason it is not the end is the one thing about the state that is irreducibly quantum: that the number of quanta is a whole number. A classical cloud can be made to regroup, but only from outside. If at some instant every member’s direction of drift could be reversed — the trick a spin echo plays on the nuclear magnets in a scanner, flipping them so that the fast ones find themselves behind — the cloud would reassemble after as long again as it had spent spreading. Nothing is destroyed by dephasing, classical or quantum; the order is moved into a correlation between where each member is and how fast it goes. What the quantum packet does with no intervention at all is what the echo needs a pulse for, and it can do it because every difference between its frequencies, $1 + \chi(2n+1)$ for neighbouring $n$, is a whole multiple of $2\chi$.

The spectrum is a subtraction makes the point that what a system emits is a difference of discrete levels; the revival is the same discreteness written into the time domain.

Copies in between #

Between the collapse and the return the state is not a featureless smear, and the arithmetic that produced the revival says what it is instead.

At a quarter of the revival time the phase $\chi n^2 t$ is $\pi n^2/2$, which is a whole number of turns for even $n$ and a quarter of a turn out for odd $n$. A state whose even and odd components are multiplied by different constants is a combination of the packet and its reflection, and the arithmetic makes it an equal one: the state at 60 periods is two complete packets on opposite sides of the well, each carrying half the probability, in superposition. At an eighth of the revival time the phases repeat every four values of $n$, and the state is four packets at quarter-turns round the orbit. The figure counts them by the overlap of the state with a packet placed at each point of the orbit, and the counts come out as one, four, two and one, each copy carrying an equal share to within two per cent.

These are fractional revivals, and each of them is a superposition of classical-looking states at macroscopically different places — the thing usually called a Schrödinger cat state. Nobody arranged for them. The oscillator produced them from a single coherent state by doing nothing but evolve under a quadratic spectrum, and it passes through a sequence of them at every rational fraction of the revival time.

The grating that photographs itself meets the same arithmetic in space rather than time: light diffracted by a grating has phases proportional to the square of the diffraction order, so the grating’s image reappears at the Talbot distance and appears in fractional copies in between. The two effects are one theorem about sums whose phases are quadratic in an integer. What the oscillator adds is the comparison with a classical cloud, which has no counterpart for a grating, and which is what turns the revival into a statement about where classical mechanics stops.

The row at an eighth of the revival time shows one further thing. Two of the four copies arrive at the centre of the well at the same moment, one moving each way, and where they overlap the density does not show two bumps added together. It shows fringes. The copies are not a mixture of four possibilities; they are one state, and the fringes are the evidence.

Three clocks, and which one depends on size #

The oscillator now has three timescales, and they separate cleanly. The orbital period is set by the level spacing, about one. The collapse time is set by the spread of spacings across the packet, and since that spread grows as the square root of the number of quanta, the collapse comes sooner for a bigger packet: the figure measures it on the summed state at every size and fits a slope of $-\tfrac12$. The revival time is set by $\chi$ alone and does not know how big the packet is.

The numbers for an ordinary object show how completely both quantum timescales are hidden from it. A pendulum a metre long with a one-kilogram bob is anharmonic in the textbook way — its period grows with the square of its amplitude — and that puts a quadratic term in its energy levels with $\chi = \hbar\omega/16mgL$, about $2\times10^{-36}$. Its revival time is $32\pi mL^2/\hbar$, some $3\times10^{28}$ years. Swinging through a tenth of a radian it holds about $1.5\times10^{32}$ quanta, and the collapse caused by the quantum spread in that number alone would take $2.8\times10^{11}$ years, twenty times the age of the universe. A real pendulum’s swing dies long before, of friction and of the slightly different starting conditions no hand can avoid, which are a classical spread far larger than the quantum one; and the revival, which needs every phase kept for its whole wait, would be erased by the first air molecule to strike the bob.

That ordering contradicts a comfortable reading of where the quantum picture hands back the old one. At fixed anharmonicity, a larger state behaves classically for fewer orbits, not more, before its swing collapses — and in the collapse it is still behaving classically, since the cloud collapses too. The departure from classical behaviour does not arrive gradually as the packet spreads. It arrives with the fractional revivals, when a classical cloud is a ring and the quantum state is a handful of packets.

The revival time is a second derivative #

The model’s quadratic spectrum made the revival exact, but the timescales it exposed belong to any discrete spectrum, because any smooth $E(n)$ can be expanded about the quantum number the packet is centred on.

The first derivative $E'(n)$ is the orbital frequency, so the orbital period is $2\pi/E'$. The second derivative is the rate at which that frequency changes across the packet, and the revival time is $4\pi/|E''|$ — the time for the quadratic part of every component’s phase to come back to a whole number of turns. For a particle in a box, with energies proportional to $n^2$, the ratio of the two is $2n$. For hydrogen, with energies proportional to $-1/n^2$, it is $2n/3$. For the oscillator in the other figures it is $1/\chi + 2n$, which is 240 orbits and a little more: the anharmonic term, not the size of the packet, sets it.

The first two are where the correspondence principle lives in this picture. As the quantum number grows the revival recedes, measured in orbits, without limit: a hydrogen atom excited to a packet near $n = 50$ goes round its orbit in 19 picoseconds and revives after 0.63 nanoseconds, thirty-three orbits later, while a packet near $n = 5{,}000$ would wait over three thousand. The classical limit is approached not by the revival disappearing but by its moving out past any time anybody waits. The limit is a statement about timescales, and the quantum answer is always there at the end of the wait.

Revivals of exactly this kind have been seen in each of the systems the figures model. Wavepackets in highly excited atoms were made with short laser pulses from the late 1980s and watched as they spread round their orbits, disappeared, and reassembled — with fractional revivals at the predicted fractions. In 2002 a condensate of rubidium atoms loaded into an optical lattice was used to watch the collapse and revival of its matter-wave interference, driven by the quadratic interaction energy of the atoms sharing each site. And in 2013 a microwave cavity coupled to a superconducting circuit whose nonlinearity is the junction whose voltage is a frequency was watched as a coherent state of its photons spread, split into two, three and four copies, and came back.

Where the model stops #

The spectrum is exactly quadratic. A real spectrum has cubic and higher terms, and each spoils the exact return a little: the revival becomes partial, and at longer times still there are super-revivals set by the third derivative. The hierarchy continues as long as the derivatives do, and a spectrum with no structure at all — a chaotic system’s — has no revivals worth the name.

The spectrum is discrete all the way. A packet whose energy reaches into a continuum, such as an atom excited near its ionisation limit, has part of its weight in components with no fixed frequencies, and that part does not come back.

The classical comparison uses a particular classical Hamiltonian. The cloud’s frequencies were assigned from the same energy function, with the half-quantum of zero-point energy taken into account; a different correspondence rule shifts the collapse slightly but cannot make a cloud with continuous frequencies revive.

And the oscillator is isolated. The revival requires every component’s phase to be preserved for the whole revival time, and anything that disturbs those phases — a stray photon leaving the cavity, one atom hopping out of a lattice site — destroys the revival long before it destroys the collapse, because the collapse needs no coherence and the revival needs all of it.

What the pictures cannot show #

The copies figure draws a probability density, and a density cannot tell a superposition from a mixture. Two packets on opposite sides of the well, half the probability each, draw the same two bumps whether they are one coherent state or a coin tossed to decide which side a single packet is on. Only where copies overlap do the fringes give the difference away, and the pair at a quarter of the revival time never overlaps.

Nor do the envelope figures show how the quantum state keeps the memory the classical cloud loses. The information is in the relative phases of the energy components, which never change their magnitudes and are drawn nowhere. A picture that could show coherence directly would need a representation of the state in the position–momentum plane that keeps those phases — and such a representation exists, at the price of no longer being a probability.

Still open: whether a many-body system can keep coming back #

For a single oscillator the revival is a theorem. For a system of many interacting particles it was expected to be impossible: interactions spread a state’s components over an astronomically large number of energies with no integer structure, and such a system should thermalise and forget its initial state in the way the classical cloud does.

In 2017 a chain of fifty-one Rydberg atoms held in optical tweezers did not. Started in a particular ordered pattern, it oscillated back to that pattern again and again, long after it should have thermalised, while other starting patterns relaxed as expected. The states responsible have been named quantum many-body scars, and they point to a sparse set of energy levels with approximately even spacing hidden inside an otherwise chaotic spectrum. How common such structures are, whether they survive in larger systems, and what they imply for the assumption that isolated quantum systems always thermalise are unsettled.

The habit worth keeping is the one the classical cloud teaches. When a quantum system loses its classical appearance, compare it with a classical ensemble before calling the loss quantum. The collapse of a swing passes that test and is classical. The return does not, and it is the return that follows from the whole-numberness of the quanta.