The state that swings like a pendulum

Where the quantum picture hands back the old one does the correspondence argument the usual way: take the quantum numbers large and watch the discreteness become invisible. It works, and it leaves something out, because a stationary state of large quantum number does not behave like a classical oscillator at all.

Its probability density does not move. It is spread over the whole classical range at once, piling up at the turning points where a classical particle is slow, and it stays that way for ever. That is the correct answer to a different question — where an oscillator of definite energy would be found — and a swinging pendulum is not it.

The state that swings has to be a superposition, and finding which superposition is the real content of the classical limit for an oscillator.

One width and no other #

The packet in that figure is the ground state of the well, picked up and moved sideways. Nothing else was done to it — its width was not chosen to be narrow, or to be optimal in any sense; it was chosen to be the width the well’s own lowest state has.

That choice is what makes it rigid. A Gaussian in a harmonic well evolves into a Gaussian, always, and its width obeys an equation of its own that has exactly one fixed point. Start at the fixed point and the width never moves. Start anywhere else and it oscillates about the fixed point.

The doubled frequency is a useful tell, and the figure measures it rather than asserting it. A quantity that is even in the displacement returns to any given value twice per cycle, so a breathing packet is narrowest at both turning points or at neither. Anyone who has watched a squeezed state in an optical experiment has seen the same factor of two, and it is the reason parametric amplifiers are pumped at twice the signal frequency.

The rigidity is special to the harmonic well. In any other potential the restoring force is not proportional to the displacement, different parts of the packet feel different frequencies, and the packet shears out — which is the mechanism the average that obeys Newton shows failing in a quartic well. A harmonic oscillator is the one potential in which a wave packet is permanently a particle.

The minimum it never leaves #

Sharpness has to be paid for gives the floor and says that a Gaussian reaches it. What the figure adds is that reaching it is easy and staying there is not.

Any Gaussian sits at the minimum at the instant it is prepared. A moment later it generally does not, because position and momentum spreads evolve differently and the product opens up. The coherent state is the one whose product is stationary: the two spreads trade in exactly the proportion that keeps their product fixed, for ever, in a well that is doing nothing to help.

That is what makes it usable as a stand-in for a classical particle over long times rather than for one pass. A classical description asserts a definite position and a definite momentum at every future time, and a quantum state can support that assertion, to within $\hbar/2$, only if it does not degrade — which one family of states, and only one, manages.

The zero-point energy that the motion that cannot be stopped accounts for is present here too, and it is exactly the residue. A coherent state’s energy is the classical energy of its swing plus half a quantum, and the half-quantum is the price of the packet having a width at all.

The number of quanta, and how badly it is known #

An oscillator’s energy comes in quanta, so a state with a definite classical amplitude has to say something about how many. A coherent state’s answer is Poisson: the probability of $n$ quanta is $e^{-\bar n}\bar n^n/n!$, with $\bar n$ the square of the amplitude.

That is the distribution of independent arrivals with no memory — the same one that counts raindrops in a bucket, and the reason a coherent beam of light shows shot noise rather than a steady stream. The mean and the variance are equal, which the figure verifies from the drawn distribution, and the equality is a strong statement: it says the quanta neither avoid nor attract one another, which is not true of thermal light and not true of a single-photon source.

The state is not a state of definite energy and does not become one. It is a superposition over every number of quanta, with a phase relationship between the terms that is exactly what makes the density swing. A state of definite energy has no such phases and does not swing; that is the point the essay opened with, restated from the other side.

Why the levels being evenly spaced is the whole trick #

The rigidity in the first three figures and the Poisson weights in the fourth look like two separate facts. They are one fact seen twice, and the connecting statement is about the ladder of energies.

A superposition of stationary states evolves by each term acquiring a phase that turns at a rate set by its own energy. If the energies are unrelated, the terms drift out of step in a complicated way and whatever pattern the sum began with dissolves. If the energies are evenly spaced, every phase turns at a rate that is a whole multiple of one common rate, the whole superposition returns to itself after one period, and any pattern it began with is permanent.

A harmonic oscillator’s levels are evenly spaced — that is the one thing everybody remembers about it — so every superposition of its states is exactly periodic. The packet cannot spread irreversibly because there is nothing for it to spread into: it must come back.

That immediately says which potentials can hold a rigid packet, and the answer is only this one. The box that allows only some energies has levels going as $n^2$, so a packet in a box shears out within a few bounces and only reassembles at a revival time far longer than the bounce. Hydrogen’s levels crowd towards a limit and a Rydberg packet does the same thing. Neither is a failure of the packet; it is the arithmetic of the ladder.

The coherent state is then the particular superposition that starts as narrow as possible. Its weights are the square roots of the Poisson probabilities, with the phases arranged so that all the terms reinforce at one place at one time — and once they do, the even spacing carries that arrangement round the cycle for ever. The Poisson distribution is not an extra property of the state; it is what a maximally narrow superposition on an evenly spaced ladder is forced to weigh its terms by.

Where the classical limit actually is #

Now the correspondence argument can be made properly, and it is a statement about a ratio.

The number of quanta is uncertain by $\sqrt{\bar n}$, always. The energy is uncertain by that many quanta, and the fractional uncertainty is $1/\sqrt{\bar n}$. So a laboratory oscillator — a pendulum, a quartz crystal, a swinging bridge — carrying something like $10^{20}$ quanta has a fractional energy uncertainty of $10^{-10}$, and there is no instrument that would notice.

The phase behaves the same way. The uncertainty in the oscillator’s phase is roughly $1/(2\sqrt{\bar n})$ radians, so the amplitude and phase together become sharp at the same rate. That product — a number uncertainty times a phase uncertainty of order one — is itself a version of the uncertainty relation, and it is the one that matters for a laser, where the linewidth is set by the phase diffusing.

None of this makes the quantum description go away. It makes the disagreement with the classical description shrink below anything measurable, which is a different and weaker claim, and it is the honest form of the correspondence principle: the classical limit is a statement about resolution, not about the state becoming classical.

Where these states come from #

They are not a mathematical curiosity chosen for their properties; they are what ordinary equipment produces.

Drive a harmonic oscillator with a classical force — a current in a coil, a voltage on an electrode, an antenna — starting from the ground state, and the state that results is coherent, exactly, for any driving waveform. That is a theorem rather than an approximation, and it is why a signal generator connected to a resonator produces a coherent state without anybody arranging for it.

A laser well above threshold produces very nearly one too, for a related reason: the gain saturates the amplitude while leaving the phase free, and the steady state of that competition is coherent up to a slow phase drift. The photon statistics of a stable laser are Poisson to good accuracy, and the shot noise on a photodiode illuminated by one is the direct measurement of it. Light from a lamp is not — it is thermal, its number distribution is much wider than Poisson, and the difference is measurable by counting.

Counting is what settles which of the two a source is, and the counting is not difficult. Point a photodiode at a beam, record the arrivals, and look at the variance of the count in a fixed window against its mean. Poisson gives the two equal; thermal light gives a variance larger by a factor of the mean itself, because the photons arrive in clumps. That the quanta are countable at all is light arrives in lumps, and the statistics of the counting is what the fourth figure is a picture of. The measurement was among the first to separate laser light from every other kind, and it did so before anybody could say what a laser was doing differently.

The states that are harder to make are the ones with less noise in one quadrature than a coherent state has, and the ones with a definite number of quanta. Both require a nonlinearity, both are fragile against loss, and the effort spent on making them is the practical measure of how natural the coherent state is. What is easy to produce is the classical-looking state, which is why the world looks classical without anybody arranging for that either.

The disc that stands for the state #

There is a picture that packages all of this and is worth having, because it is the one everybody who works with these states actually thinks in.

Plot position along one axis and momentum along the other. A classical oscillator is a point on a circle in that plane, going round once per period at constant radius, and its energy is set by the radius. A quantum state cannot be a point, and the uncertainty relation says how much of the plane it must occupy: an area of order $\hbar$, and no less.

A coherent state is a disc of exactly that minimum area, centred on the classical point, going round the same circle at the same rate. That single sentence contains all four figures. The disc’s size not changing is the width not breathing; its area being minimal is the uncertainty product sitting on its floor; its radius being the amplitude and its area being fixed is why the relative uncertainty falls as one over the square root of the energy; and the number of quanta being the radius squared, read with a fixed absolute smear, gives the Poisson spread.

It also says immediately what the alternatives are. Squeeze the disc into an ellipse of the same area and one quadrature gets quieter at the other’s expense — that is a squeezed state, and it is what an interferometer wants when it cares about phase and not about amplitude. Smear the disc into an annulus and the amplitude is definite while the phase is not: that is close to a number state. Spread it isotropically and it is thermal.

The area being fixed is the constraint every one of those obeys, and the coherent state is the one that spends it symmetrically. Nothing distinguishes the position direction from the momentum direction in a harmonic well once the units are chosen sensibly, so the symmetric choice is also the natural one — which is another way of saying that the classical-looking state is the one that requires no preference to be expressed.

Where the model stops #

The oscillator is exactly harmonic. Every result above uses the fact that the frequency does not depend on the amplitude. Real oscillators are anharmonic at large amplitude, the number states acquire unequal spacings, the superposition dephases, and the packet spreads after all — slowly, and then all at once. The revival time at which it re-forms is a separate and beautiful subject that the harmonic case cannot even pose.

The state is pure and the oscillator is isolated. Coupling to anything else turns a coherent state into a mixture, and the surprising part is that it is unusually resistant to this: a coherent state losing photons to a cold environment becomes a coherent state of smaller amplitude, rather than a mess. That robustness is why these states survive in a lossy cavity, and it is why the superpositions of two of them decay so much faster than the components do.

The number distribution assumed a single mode. A real beam has many, and a Poisson count in one mode says nothing about correlations between modes, which is where most of the interesting quantum optics lives.

And the classical limit here is a limit of one particular family. A state with $10^{20}$ quanta need not be coherent — it can be squeezed, or thermal, or a superposition of two amplitudes — and none of those looks classical. Large quantum numbers permit classical behaviour; they do not produce it.

What the pictures cannot show #

The propagated figures draw the probability density, and the density is not the state. The phase varies across the packet — it is what carries the momentum — and two of the drawn snapshots have the same density and opposite velocities. A picture of the density alone cannot distinguish a packet moving left from one moving right, which is exactly the information the classical description most needs.

The number-distribution figures draw probabilities and cannot draw the relative phases between the terms, and the phases are what make the state swing. The same distribution with random phases is a thermal-looking state that sits still. Two states with identical figures here would behave completely differently, and the drawing has no room for the difference.

Where the ladder goes next #

The correspondence ladder began with where the quantum picture hands back the old one and continued with the average that obeys Newton, which finds the exact sense in which a mean value follows a classical law and the exact sense in which it does not. This rung asks which state, rather than which limit, and finds a family whose members behave classically at any size. The rungs after it: the revivals of a packet in an anharmonic well, where the classical behaviour is temporary; the phase-space picture in which a coherent state is a disc of fixed area and everything above is a statement about that disc; and decoherence, which explains why superpositions of two coherent states are not found lying about.

The habit worth carrying away is that a limit is usually a statement about a state rather than about a parameter. Large is not the same as classical, and asking which states behave classically produces a sharper answer than asking when a quantum number is big — including the answer that most of them never do.