An electron can go through both slits; a cat cannot. The phase is not destroyed ── it has only been diluted into the environment
Quantum mechanics' most famous discomfort is that an electron can go through two holes at once while a desk, a cat and the Moon cannot. There must be a boundary somewhere ── but no such boundary is written in the equations. The answer is that there is no boundary; the effect turns on continuously ── and it accelerates explosively with the size of the system. The crucial point is that the phase does not vanish. It is merely diluted into the environment's vast number of degrees of freedom, and in principle it is conserved. It is only that the cost of tracking the diluted phase becomes astronomical, so we do what we did in Episode 1 ── throw it away and write probabilities. That is how "if you look at it coarsely, probability gives the same answer" becomes a theorem with a computable clock.
Facts first. The common sense that "quantum is for small things" is revised by experiment every year.
| System | Scale | What was confirmed |
|---|---|---|
| electron / neutron | 1 particle | interference (textbook) |
| C60 (buckyball) | 60 atoms | double-slit interference (1999) |
| large organic molecules | ~2000 atoms (25,000 amu) | matter-wave interference (2019) |
| current states of a superconducting ring | ~10¹⁰ electrons | superposition of clockwise and counter-clockwise |
| two micro-drums of aluminium | ~10¹³ atoms | entanglement between mechanical oscillators (2021) |
| the mirrors of LIGO | 40 kg (10 kg effective) | cooled near the quantum ground state of motion (2021) |
So there is no wall where "big" stops being quantum. There is only a continuous tendency: bigger is harder to maintain. What makes it harder?
Interference fringes appear only when the phase difference between two paths is definite. If the phase difference is scattered from run to run, the fringes add up flat.
Here is the decisive fact ── once the environment knows which path was taken, interference is gone. No human needs to read it. It is enough for one air molecule to bounce off in a way that carries path information. The instant the information is outside, the fringes vanish as far as the system alone is concerned.
The diagonal entries ("probability it went left," "probability it went right") do not change. Only the off-diagonal entries shrink ── and interference lives nowhere else. When the off-diagonals hit zero, the matrix becomes indistinguishable from an ordinary table of probabilities saying "it went either left or right." That is decoherence. The moment when "probability gives the same answer" becomes true is written down explicitly in the mathematics.
Each nudge from the environment leaks a little phase information. So the leak rate is set by how many nudges per second. A big object has a big cross-section and gets nudged a lot. Let's put numbers in.
Air molecule number density \(n\approx2.5\times10^{25}\ \mathrm{m^{-3}}\), mean speed \(v\approx500\ \mathrm{m/s}\), grain cross-section \(\sigma=\pi r^2=\pi(0.5\times10^{-6})^2\approx8\times10^{-13}\ \mathrm{m^2}\). The collision rate is
$$\text{collision rate}=n\,v\,\sigma\approx (2.5\times10^{25})(500)(8\times10^{-13})\approx 1\times10^{16}\ \text{per second}$$Ten quadrillion times a second. If the grain were in a superposition of two places 1 μm apart, a single collision essentially resolves "which one," so coherence dies on the order of 10⁻¹⁶ seconds. Not remotely observable by a human. That is why dust is always somewhere.
Pump down to laboratory high vacuum (about \(10^{-5}\) Pa) and \(n\) drops by \(10^{10}\), so the rate becomes \(10^{6}\) per second ── coherence time stretches into microseconds. That is why molecular-interference experiments live and die by their vacuum. And even in perfect vacuum, the photons of the cosmic microwave background remain (number density \(4\times10^{8}\ \mathrm{m^{-3}}\)), so it can never be zero.
From this comes the heart of the episode. Make the system bigger and (1) the cross-section grows so it gets nudged more, and (2) the two superposed states become easier to tell apart. Both push the same way, so the decoherence rate rises monotonically with size. Episode 1's \(1/\sqrt{N}\) said "bigger means sharper macroscopic quantities"; this one says "bigger means quantum-ness dies faster." The classical world looks classical for these two reasons together.
On the left of the figure are the interference fringes; on the right is the \(2\times2\) matrix \(\rho\) from the previous section (brighter = larger entry). Move the slider to change the size of the flying object (number of atoms \(N\)).
What to watch is that the diagonal and off-diagonal have different fates. Raise \(N\) and the top-left and bottom-right ── the probabilities of "went left" and "went right" ── do not budge at all. Only the top-right and bottom-left vanish. And the fringes go. Probability intact, interference dead ── that is what "if you look at it coarsely, probability gives the same answer" really means.
Do try the "Recall the phase with an echo" button. It is the operation of flipping the phase mid-flight and calling it back from the environment (spin echo, or running a quantum circuit backwards). The fact that real experiments can do this is the best possible evidence that the phase was not destroyed but deposited. If the universe really discarded the information, no echo would return.
A natural question arises: "Why go the long way round with decoherence? Why not just write a law that says superpositions break by themselves once things get big?"
People have written exactly that. They are called spontaneous collapse models.
Each particle has a probability \(\lambda\sim10^{-16}\) per second of having its wave function spontaneously collapse to a point. One particle: once in a hundred million years, unnoticeable. But an object of \(10^{23}\) linked particles: \(10^{7}\) times a second ── small things quantum, big things automatically classical. The Penrose–Diósi model goes further and says gravity does the collapsing (a superposition of two different spacetimes cannot be maintained and breaks in \(\tau\sim\hbar/E_G\)).
The idea is attractive and, better, falsifiable. Spontaneous collapse jiggles particles slightly, so it should produce extra heat or radiation ── and you can go hunt for that. And indeed:
So "the universe skips the calculation for big things" is not poetry but a testable physical hypothesis ── and it is currently the side that is losing. Decoherence explains the same phenomena without adding a single new law ── with one caveat.
Established: that interaction with an environment exponentially suppresses the off-diagonal entries of the density matrix; that the rate grows with system size, spatial separation of the superposition and environmental density; that this has been confirmed quantitatively in experiment (controlled decoherence in molecular interferometers, observation of cat-state decay in cavity QED); that spin echo and similar techniques can partially restore coherence; and the experimental records above (molecular interference at the 25,000 amu class, entangled mechanical oscillators, LIGO mirrors near the ground state). The order-of-magnitude estimate is the standard scattering-decoherence evaluation.
Not solved: decoherence explains perfectly why interference becomes invisible, but not "why a single outcome occurs." It gets you to "a density matrix with zero off-diagonals is indistinguishable from a probability table for left-or-right," and no further; what happens beyond that (one outcome actually occurring) remains an interpretational question ── many worlds, Bohm, collapse theories, all still competing and unsettled. The phrasing "the phase is deposited in the environment" takes the standard position that unitary evolution continues globally; if spontaneous collapse is right, the phase really is lost. That side is being whittled away by experiment, which is where we stand. The figure is an order-of-magnitude toy for showing the size dependence, not a reproduction of any particular experiment.
There is no boundary between "electron = quantum" and "cat = classical," only a continuous tendency ── and the records keep climbing: 2000-atom molecules, 10¹⁰ electrons in a superconducting ring, 10¹³-atom drums, 40 kg LIGO mirrors. Interference dies when the environment learns which path. Mathematically, only the off-diagonal entries of the density matrix decay as \(e^{-\Lambda t}\); the diagonal (probability) is untouched. That is why the moment when "probability gives the same answer" can be written down exactly.
The speed can be computed: a 1 μm dust grain in air is nudged \(10^{16}\) times a second and loses coherence in \(10^{-16}\) s. Bigger systems have larger cross-sections and are easier to tell apart, so bigger goes classical faster. But the phase is not destroyed ── the echo proves it. Decoherence is coarse-graining in quantum mechanics. The universe didn't slack off; we gave up tracking. The version where the universe really does slack off (GRW/CSL, Penrose–Diósi) exists as a testable hypothesis ── and it is being whittled away.
Print / make a PDF: ⌘+P (Ctrl+P on Windows). On screen, moving the slider changes the object's size: the fringes vanish and only the off-diagonal entries of the density matrix go dark. "Recall the phase with an echo" shows the recovery. "See the answer" opens each solution.