The heating a quantum rotor refuses

Where the quantum picture hands back the old one found that classical behaviour returns from quantum mechanics not because quantum effects die away at large quantum numbers but because they become too fine to resolve: a particle in a box at high energy still has a probability density that oscillates wildly, and only a measurement that averages over the oscillations sees the flat classical distribution. That is the correspondence principle in its mild form, and for most systems it is all that is needed. For systems whose classical motion is chaotic it is not enough. A chaotic system amplifies small differences exponentially, and quantum mechanics has no exponential sensitivity to give it, so the two descriptions must part company — the only question is when, and how badly.

The cleanest place to watch them part is a rotor kicked at regular intervals. It is about as simple as a chaotic system can be: a wheel, or a particle on a ring, turning freely, and every period given a sharp kick by a potential that depends on its angle. Classically it is the standard map, the model that the last curve to go used to find the coupling at which the last barrier to chaotic wandering gives way. Above that coupling, the rotor’s momentum random-walks for ever and its energy grows in proportion to time. The quantum rotor does something else, first found by Giulio Casati, Boris Chirikov, Felix Izrailev and Joseph Ford in 1979, and seen with cold atoms in 1995.

The classical rotor heats without limit #

Classically, a kick of strength $K$ changes the rotor’s momentum by $K\sin\theta$, where $\theta$ is its angle at the moment of the kick, and between kicks the angle advances by the momentum. For small $K$ the motion is regular. For $K$ above about one, the last of the invariant curves that run across the phase space has broken, and an orbit wanders everywhere; for $K = 5$ the angle at successive kicks is effectively random, and the momentum changes by a random amount each time, with a mean square of $K^2/2$. A random walk of such steps makes the mean energy grow by $K^2/4$ per kick, and the first figure’s dashed line, an average over four thousand classical rotors, grows at 6.3 per kick against that estimate of 6.25.

The second figure shows why. With weak kicks, the phase space is threaded by smooth curves that orbits cannot cross, and the momentum is confined between them. With strong kicks those curves have gone, apart from a few small islands, and a single orbit fills the whole space as a cloud of points. That is the classical premise of everything that follows: a system whose motion is chaotic, whose energy grows diffusively, and which has no reason ever to stop.

The diffusion deserves to be taken seriously as diffusion. The equation that only runs forwards, and the walk underneath it found that a random walk of many small independent steps turns, on large scales, into the diffusion equation, whose solutions spread as the square root of time and never return. The kicked rotor’s steps are not random — each is fixed by the angle, which is fixed by every previous step — but chaos makes them as good as random for every statistical purpose, and the classical rotor obeys the diffusion equation in momentum to good accuracy. A deterministic, reversible system behaving irreversibly because its motion is chaotic is one of the standard routes by which the arrow of time is explained. The quantum rotor is deterministic and reversible too. It simply does not become irreversible.

The quantum rotor follows, and then stops #

The quantum rotor obeys the same Hamiltonian with one change: its angular momentum comes in whole multiples of Planck’s constant. In units where the classical momentum is measured in the same way as $K$, the quantum of momentum is an effective Planck constant, $\hbar_{\text{eff}}$, set by the physical parameters — for atoms kicked by a standing wave of light, by the atoms’ recoil and the period between kicks. One period of the quantum motion is a kick, which multiplies the wavefunction by $e^{-i(K/\hbar_{\text{eff}})\cos\theta}$ at every angle, followed by free rotation, which multiplies each momentum state $m$ by $e^{-i\hbar_{\text{eff}}m^2/2}$. The figures compute the motion that way, transforming between angle and momentum with a fast Fourier transform at every step.

The first figure’s two solid curves are the result. For the first few tens of kicks the quantum energy follows the classical line closely; then it bends over and stops, and from then on it only fluctuates. With $\hbar_{\text{eff}} = 1$ it stops near 58, about a tenth of where the classical rotor is after three hundred kicks and falling further behind with every one. With $\hbar_{\text{eff}}$ halved the agreement lasts longer and the energy climbs further before it stops. In the limit $\hbar_{\text{eff}} \to 0$ the classical growth is followed for ever — but only if that limit is taken first. For any fixed $\hbar_{\text{eff}}$, however small, there comes a time after which the quantum rotor stops heating, and the order of the two limits, small Planck’s constant and long time, decides whether classical mechanics comes back at all.

It is worth being clear that the quantum rotor does not violate any of the relations that tie quantum averages to classical ones. The average that obeys Newton found that the expectation values of position and momentum obey Newton’s equations exactly, with the force replaced by its average — Ehrenfest’s theorem. For the kicked rotor that means each kick changes the mean momentum by the mean of $K\sin\theta$ over the quantum distribution of angles, exactly as for a classical ensemble. What it does not say is that the mean of $\sin\theta$ behaves like the sine of a classical angle, and in a chaotic system the two part company almost at once. The theorem holds throughout; it simply stops being useful.

Nothing in the quantum equation damps the motion. The evolution is unitary: probabilities are conserved, the motion is reversible in principle, and no energy is removed. The kicks keep coming and keep delivering their impulses. The growth stops because the amplitudes for the rotor to reach higher momenta, arriving by many different sequences of kicks, cancel.

The shape of a frozen distribution #

The third figure compares the two momentum distributions after six hundred kicks. The classical one is what a random walk produces, a Gaussian whose width grows as the square root of the number of steps; on the logarithmic scale it is a broad, flat hump, still spreading. The quantum one is a sharp peak at the starting momentum with sides that fall roughly in straight lines on the logarithmic scale — an exponential, $e^{-2|m|/\xi}$, with a localisation length $\xi$ of about fifteen quanta. It has not changed for hundreds of kicks. The rotor’s momentum is localised.

That exponential is the signature of a phenomenon found in a very different place. In 1958 Philip Anderson showed that an electron in a crystal with random impurities can be trapped by interference among the many paths by which it scatters: its wavefunction decays exponentially away from where it started, and the crystal does not conduct. The walk that interference can stop found the same thing for light in a disordered stack of layers. In 1982 Shmuel Fishman, D. R. Grempel and Richard Prange showed that the kicked rotor is mathematically the same problem, transcribed: the momentum states play the part of the sites of a crystal, the kick couples neighbouring sites as hopping does, and the phases $\hbar_{\text{eff}} m^2/2$ that free rotation gives each momentum state, which jump about irregularly from one state to the next, play the part of the random impurity energies. There is no randomness in the kicked rotor’s equations. The disorder is supplied by the irregular values of a quadratic phase, and that pseudo-random sequence is enough to localise the motion. The rotor is an Anderson insulator in momentum space.

Why the break comes when it does #

The time at which the quantum rotor stops following the classical one can be estimated without solving anything. Over $n$ kicks the classical momentum spreads over a range of about $\sqrt{Dn}$, where $D = K^2/2$ is the diffusion constant, which is $\sqrt{Dn}/\hbar_{\text{eff}}$ momentum states. The quantum motion over $n$ kicks can only resolve that many distinct frequencies if it has run for at least as many periods as it has states to distinguish, because its spectrum is discrete. So the classical spreading can continue only until $n$ is comparable with $\sqrt{Dn}/\hbar_{\text{eff}}$, that is until $n^* \sim D/\hbar_{\text{eff}}^2$. By then the momentum has spread over about $D/\hbar_{\text{eff}}^2$ states, and that is the localisation length. The argument, due to Chirikov, Izrailev and Dima Shepelyansky, gives the localisation length and the break time as the same number, both growing as $1/\hbar_{\text{eff}}^2$.

For $K = 5$ and $\hbar_{\text{eff}} = 1$ the estimate gives about twelve kicks and twelve states, against the fifteen states the figure measures; the real numbers fluctuate with $\hbar_{\text{eff}}$ more than the estimate does, because the quantum diffusion constant itself depends on $\hbar_{\text{eff}}$ through corrections of the size of the kick’s phase, and a single run from one starting state has large fluctuations of its own. The scaling is the robust part: the classical behaviour lasts for a time that grows as the inverse square of Planck’s constant, which is very long for any macroscopic rotor and short for an atom.

This break time is much longer than the other well-known time for chaotic systems to go quantum. The error that doubles on a schedule found that a chaotic orbit’s sensitivity grows exponentially, and a quantum wave packet following such an orbit spreads over the whole phase space in a time that grows only as the logarithm of $1/\hbar$, the Ehrenfest time. For the kicked rotor the averages keep following classical diffusion long after the Ehrenfest time, because diffusion is a statistical property that does not depend on tracking any single orbit. It is the slower, algebraic time $1/\hbar_{\text{eff}}^2$ at which even the statistics part company.

Noise brings the heating back #

If interference is what stops the heating, anything that scrambles the phases should restart it, and the fourth figure checks. Varying the strength of each kick at random by a few per cent adds no systematic energy — on average the kicks are the same — but it gives each sequence of kicks a slightly different phase, and the delicate cancellations that held the momentum in place no longer occur. Growth resumes, slower than the classical rate for small noise and approaching it as the noise grows.

This is how the classical world is recovered in practice. Not by making Planck’s constant small — a real rotor’s is fixed — but by coupling the system to an environment that destroys the coherence the quantum behaviour depends on. In 1998, Mark Raizen’s group in Texas and a group in Auckland did exactly this to cold atoms kicked by light, adding spontaneous emission or timing noise, and watched the localised atoms start to heat again at the rate the noise allowed. The return a classical cloud never makes found the opposite kind of fragile coherence, the quantum revival of a wave packet; both are interference effects that the classical description cannot reproduce and that a small amount of noise erases.

Where in phase space the quantum state lives #

A different view of the same freezing comes from the phase-space picture of a quantum state. The probability that goes below zero found Wigner’s quasi-probability distribution, a quantum state drawn on the classical plane of angle and momentum, which behaves like a classical ensemble’s density for smooth states and develops fine fringes, some negative, wherever interference matters. A classical ensemble of kicked rotors, drawn that way, spreads ever upward in momentum through the chaotic sea of the second figure. The quantum state’s Wigner distribution starts to do the same and then stops at the localisation length, covered in fringes on the scale of $\hbar_{\text{eff}}$ in both angle and momentum. Those fringes are where the cancellations happen, and they are exactly the structure that an environment’s noise smears out, turning the quasi-probability back into an ordinary positive density that can go on diffusing.

The picture also shows why the break time grows as $1/\hbar_{\text{eff}}^2$ rather than as anything shorter. The quantum state can follow the classical ensemble’s spreading only as long as the region it has to resolve contains more than about one cell of area $\hbar_{\text{eff}}$ per kick it has experienced; with the diffusion filling momentum space at a rate set by $K$, that bookkeeping gives the same estimate as the argument from discrete frequencies.

The other way to fail: resonance #

Localisation is not the only way the quantum rotor departs from the classical one. When $\hbar_{\text{eff}}$ is exactly $4\pi$, the phase $\hbar_{\text{eff}}m^2/2$ that free rotation gives each momentum state is a whole multiple of $2\pi$ for every $m$: free rotation does nothing at all. The kicks then simply accumulate, each adding coherently to the last, and the momentum spreads in proportion to the number of kicks rather than its square root. The fifth figure shows the energy growing as the square of time, sixty-three times faster than the classical diffusion after sixty kicks. Similar quantum resonances occur at every rational multiple of $4\pi$, and with cold atoms they appear as sharp peaks in the energy absorbed against the kicking period.

So quantum mechanics departs from classical chaos in both directions: by interference that cancels the spreading, which is the generic case, and by interference that adds it all up, at special values. The classical diffusion is neither; it is what is left when the phases are averaged.

What was seen with atoms #

The kicked rotor became an experiment in 1995, when Raizen’s group at the University of Texas loaded a few hundred thousand sodium atoms, cooled to a few hundred microkelvin, into a standing wave of laser light switched on in short pulses. The light’s intensity pattern acts on the atoms as a periodic potential, the pulses are the kicks, and the atoms’ momentum along the standing wave changes in units of two photon recoils, the momentum quanta. After a few tens of pulses the atoms’ momentum distribution stopped spreading and settled into the exponential shape of the third figure, which was measured directly by switching off the light and letting the atoms fly apart.

Later experiments with the same kind of system, notably with caesium atoms at Lille, used it to observe the transition between localised and extended states in a three-frequency version of the kicked rotor that mimics a three-dimensional disordered crystal, the Anderson metal–insulator transition itself, and to measure its critical exponents — a quantity that is very hard to measure in a real disordered solid, measured here with atoms and light.

Averages that hide the phases #

The figures compute averages and distributions of momentum; they do not show the wavefunction’s phases, whose cancellations are the whole mechanism. The localised distribution looks static, but the wavefunction is still changing at every kick; only the probabilities have stopped spreading. The quantum runs start from a single momentum state and a single value of the quasi-momentum, so they carry fluctuations that an experiment, averaging over many atoms with slightly different quasi-momenta, would smooth out. And the noise figure models decoherence as random kick strengths, one simple way among many that an environment can scramble phases.

Still open: what classical chaos leaves in a quantum system #

The kicked rotor shows that classical chaos does not survive quantisation as unbounded diffusion. It leaves other marks: the statistics of the quantum energy levels of a classically chaotic system follow random-matrix theory, while those of an integrable system do not, a conjecture of Oriol Bohigas, Marie-Joya Giannoni and Charles Schmit from 1984 that has passed every test and has never been proved. More recently, the spreading of quantum information in many-body systems — how fast a local disturbance becomes scrambled across all the degrees of freedom — has been proposed as the quantum counterpart of a Lyapunov exponent, with a speed limit set by temperature. Whether these different quantum signatures of chaos are aspects of a single thing, and how each connects to the classical exponential sensitivity they replace, is an active question.

The habit worth carrying away is to ask which limit is being taken first. A rotor kicked with strength $K = 5$ heats classically without limit, about 6.3 units per kick; the quantum rotor follows for a break time of order $D/\hbar_{\text{eff}}^2$ and then stops, its momentum frozen into an exponential about fifteen quanta wide at $\hbar_{\text{eff}} = 1$, by the same interference that makes a disordered crystal an insulator. Noise brings the heating back; a special value of Planck’s constant makes it faster than classical; and only when small $\hbar$ is taken before long times does the classical diffusion return.