Quantum Phase and Coherence: the variable behind lasers, superconductors and the matter-wave beam

Level 0 · The picture

Hold a wave in your mind. A ripple crossing a pond, a note from a guitar string, the light coming out of a torch. Every wave rises and falls, over and over. Phase is the answer to one small question about it: where is the crest right now?

That is the whole idea. Phase is a position in a cycle, like the position of a second hand on a clock face. One wave's phase, by itself, tells you almost nothing — it depends on where you started counting. What matters, always, is whether two waves agree.

Now the surprising part, and it is the reason this course exists. Everything is a wave. Light is a wave. Sound is a wave. And since 1924 we have known that matter is a wave too — electrons, atoms, whole molecules. Each one carries a phase. Each one can agree, or disagree, with its neighbours.

When many waves agree on phase, physicists call it coherence. Coherence is not a small technical detail. It is the difference between two things you already know.

• A light bulb pours out light with random phases. Every atom in the filament flashes when it feels like it. The result is warm, useful, and hopelessly disordered — you cannot focus it to a point or send it to the Moon.
• A laser pours out light whose waves are all in step. Same atoms, same energy levels, same colour range. The only difference is agreement. That agreement is what lets a laser cut steel, read a disc, carry the internet down a fibre, and measure the length of a four-kilometre tunnel to less than the width of a proton.

The same story runs through matter. Cool certain metals far enough and their electrons pair up and fall into a single shared rhythm; the metal becomes a superconductor, and current runs round a ring for years without fading. Cool a dilute gas of atoms to a hundred-billionth of a degree above absolute zero and the atoms drop into one shared quantum state; the cloud becomes a Bose–Einstein condensate, a lump of matter behaving as one enormous wave. Two condensates released and allowed to overlap show interference fringes — light and dark bands of atoms — photographed for the first time in 1997.

Here is the sentence to carry with you through all six levels:

Phase is where the crest is. Coherence is many waves agreeing on it. And agreement, not energy, is what makes a laser a laser and a superconductor a superconductor.

That raises a question that is now an engineering question rather than a philosophical one. If coherence does that much, can we make it on purpose? With light, we learned how in 1960. With atoms, we learned how in 1995 — by cooling. The frontier this site follows, at Level 5, is a third route: reach in and steer each particle's phase directly, without cooling and without adding energy, using the quantity the vector-potential course is about.

Ways to think about it

• Phase is a clock hand carried by every wave. Coherence is a room full of clocks reading the same time.
• Coherence costs no extra energy. Rowers in time are not rowing harder.
• Almost everything we measure in quantum mechanics is a phase difference, never a phase.

Level 1 · Foundations

Interference: why phase differences are the measurable thing. Send a wave through two narrow slits and let the two halves spread out and overlap on a screen beyond. Where a crest from one slit lands on a crest from the other, they add and the screen is bright. Where a crest lands on a trough, they cancel and the screen is dark. You get stripes: an interference pattern. Thomas Young did it with light around 1801, and the pattern has been the standard tool for reading phase ever since.

Notice what the pattern actually encodes. Move one slit a fraction of a wavelength and the whole striped pattern slides sideways. The stripes do not care about the absolute phase of either wave — only about the difference between them. That is a deep point and it will hold at every level of this course. Absolute phase is a convention, like choosing where a year starts. Phase difference is physics.

Matter is a wave. In his 1924 doctoral thesis Louis de Broglie proposed that a particle with momentum p has a wavelength.

The de Broglie wavelength(1)

$$\lambda =\frac{h}{p}$$

Within three years the prediction was confirmed: electrons fired at a nickel crystal came off in a diffraction pattern, exactly as a wave would. Since then interference has been demonstrated with neutrons, whole atoms, and large molecules. Nothing about being made of matter exempts you from having a phase.

The laser: light put in step, 1960. Theodore Maiman flashed a lamp at a small ruby rod and got a pulse of deep red light at 694 nanometres — the first laser. The trick is stimulated emission: an excited atom nudged by a passing photon emits a new photon into the same wave, with the same direction, the same colour, and the same phase. Put the rod between mirrors so the light passes through again and again, and the in-step component grows while everything else leaks away. What comes out is a beam whose waves agree. The word is an acronym for light amplification by stimulated emission of radiation, but the physics in one word is agreement.

Matter put in step. Photons are bosons — they are perfectly happy to pile into the same state, which is why light is easy to make coherent. Electrons and atoms with an odd count of constituents are fermions, and no two of them may occupy the same state. So matter needs another route. Two have been found.

• Superconductivity (1911, explained 1957). Below a transition temperature the electrons in certain metals bind into pairs. A pair, having an even number of fermions, behaves as a boson. All the pairs then share one quantum state with one phase, spread across the whole piece of metal. That shared phase is what carries a supercurrent, and it is the subject of this site's Josephson-junction course.
• Bose–Einstein condensation (predicted 1924–25, achieved 1995). Cool a dilute gas of bosonic atoms far enough and the atoms crowd into the lowest state together. Eric Cornell and Carl Wieman did it with rubidium at Boulder and Wolfgang Ketterle did it with sodium at MIT, in the same year; the Nobel Prize followed in 2001.

Synchronisation, in one sentence. Oscillators that are even weakly coupled — pendulum clocks on a shared wall, fireflies in a tree, heart-muscle cells, metronomes on a board — spontaneously fall into step once the coupling is strong enough to beat the spread in their natural rates. Yoshiki Kuramoto wrote that down as a model in 1975, and it is the mathematics Charles Chase's slides reach for at Level 5.

Level 2 · The rules

The complex amplitude. Quantum mechanics describes a system by a wavefunction ψ, and the most useful way to write it separates how much from where in the cycle. For a large collection of identical particles with density n and shared phase θ, the standard form is ψ = √n · e^(iθ).

The macroscopic wavefunction(2)

$$\psi =\sqrt{n}{e}^{i\theta }$$

Interference, written down. Bring two waves of intensity I₁ and I₂ together with a phase difference Δφ between them.

Two-wave interference(3)

$$I=I_1+I_2+2\sqrt{I_1{I}_2}\cos \Delta \phi$$

Sweep Δφ and the brightness at a point runs smoothly from four times one beam down to zero and back. Every interferometer ever built — Michelson's, LIGO's, an atom gravimeter, a SQUID — is a machine for turning a phase difference you cannot see into a brightness or a current you can.

Phase and number are a trade. You cannot know exactly how many particles are in a wave and exactly where its crest is at the same time. The two quantities are conjugate, in the way position and momentum are.

Number–phase uncertainty(4)

$$\Delta N\Delta \theta \gtrsim 1$$

Synchronisation, written down. Take N oscillators. Oscillator i has its own natural rate ω_i and its own phase θ_i, and each one is pulled a little towards every other with coupling strength K.

The Kuramoto model(5)

$$\frac{d{\theta }_i}{dt}={\omega }_i+\frac{K}{N}\sum _{j=1}^N\sin ({\theta }_j-{\theta }_i)$$

The critical coupling is the point of the whole model. Synchronisation is not gradual and it is not guaranteed. It is a threshold phenomenon: nudge the coupling past a critical value and order appears across the population all at once. That is a very encouraging fact for anyone trying to engineer coherence, because it means you do not have to coax every oscillator individually. You have to get one number — the coupling — over a line.

Coherence length and coherence time. Real waves are never perfectly pure. A source emits over a small spread of frequencies Δν, and that spread limits how long the wave keeps a predictable phase.

Coherence time and coherence length(6)

$${\tau }_c\simeq \frac{1}{\Delta \nu },{\ell }_c=c{\tau }_c$$

That last clause is the reason cold atoms matter. At room temperature a rubidium atom's matter wavelength is far smaller than the spacing between atoms, so the waves never overlap and nothing coherent can happen. At a hundred nanokelvin the same atom's wavelength is about half a micrometre — comparable to the wavelength of visible light, and much larger than the spacing between atoms in the trap. The waves overlap, and the gas becomes one wave.

Level 3 · Undergraduate

The macroscopic wavefunction. The move that makes superconductivity tractable is to promote equation (2) from a description of one particle to a description of an entire condensate. In the Ginzburg–Landau theory of 1950 the order parameter ψ = √n · e^(iθ) is a field defined at every point in the metal, with n the density of pairs and θ a phase that is shared — a single collective variable spread across a macroscopic object. Feynman's Volume III, Chapter 21 derives the consequences from that one assumption in a few pages, without any of the microscopic pairing theory, and gets the Meissner effect and both Josephson relations out of it.

Why a supercurrent is a phase gradient. In an ordinary wire, current flows because an electric field pushes the electrons and resistance fights back. In a superconductor there is no field inside and nothing to fight. The current comes from somewhere else entirely: from the twist in the shared phase from point to point.

The supercurrent as a phase gradient(7)

$${\mathbf{J}}_s\propto \left(\nabla \theta -\frac{2e}{\hslash }\mathbf{A}\right)$$

Read that equation as a definition of the whole subject: a current is a phase gradient. Once you accept that, flux quantisation follows in a line — the phase must come back to itself after one trip around a ring, so its total winding is a whole number of turns, so the enclosed flux is a whole number of flux quanta. The vector-potential course does that derivation at its Level 4; the Josephson-junction course turns the phase difference across a gap into a circuit element. This course is the bridge between them, and its job is the variable itself.

Bose–Einstein condensation, in the laboratory. In June 1995 Anderson, Ensher, Matthews, Wieman and Cornell reported a condensate of about two thousand rubidium-87 atoms below 170 nanokelvin, made by laser cooling followed by evaporative cooling in a magnetic trap. Ketterle's group at MIT reached the same state in sodium a few months later, with far more atoms. Both papers show the same signature: as the cloud is cooled, a sharp narrow peak grows out of the broad thermal background in the velocity distribution. That peak is the condensate — the fraction of the gas that has fallen into one state, with one phase.

The experiment that shows it is really a wave. In 1997 the MIT group did the thing that settles the argument. They made two condensates in a trap split by a sheet of laser light, switched everything off, let the two clouds expand into each other, and photographed the result. What appeared was a set of straight, high-contrast interference fringes running through the overlap region, tens of micrometres apart.

Fringe spacing for two overlapping condensates(8)

$${\lambda }_{\text{fringe}}=\frac{ht}{md}$$

Definitive Bose–Einstein condensation and the interference of two condensates are measured, independently reproduced in laboratories worldwide, and taught in graduate courses. Matter waves interfere; this is not in question anywhere.

The atom laser. If a condensate is a coherent lump of matter, the obvious next device is an output coupler: a way to let a steady stream of atoms out while keeping them in step, the way a partly silvered mirror lets light out of a laser cavity. Mewes and colleagues did it in 1997, tipping atoms out of a trapped sodium condensate with radio-frequency pulses and watching coherent pulses of atoms fall under gravity. Continuous and quasi-continuous versions followed. An atom laser is not a science-fiction weapon; it is a slow, faint, exquisitely controlled beam of matter with a definite phase. But it is the existence proof that matters for Level 5: a coherent beam of matter is a thing that can be built.

Ways to think about it

• One phase shared by a macroscopic number of particles is what "condensate" means. Everything else is consequence.
• Current without a driving field is not mysterious once you see that current is the phase gradient.
• Cooling works because coherence length grows as temperature falls. It is a way of buying phase agreement with refrigeration.

Level 4 · Graduate

The geometric phase. Not all phase comes from a wave ticking along in time. Some of it comes from the shape of the path the system was taken around. Shivaramakrishnan Pancharatnam noticed this for polarised light in 1956: carry a beam's polarisation slowly around a closed circuit of states and it returns with an extra phase equal to half the solid angle the circuit encloses on the sphere of polarisations. In 1984 Michael Berry showed the same thing is true of any quantum system carried adiabatically around a loop in its parameter space.

The Berry phase(9)

$${\gamma }_n(C)=i{\oint }_C⟨n(\mathbf{R})|{\nabla }_{\mathbf{R}}n(\mathbf{R})⟩\cdot d\mathbf{R}$$

The unifying picture is holonomy, and the vector-potential course develops it at its own Level 4, so here it is one paragraph. Comparing the phase of a quantum state at two nearby points requires a rule, a connection. Transport a state around a closed loop using that rule and it generally comes back rotated; the rotation is the enclosed curvature. The Aharonov–Bohm phase is this statement with the electromagnetic connection and magnetic flux as the curvature. Berry's phase is the same statement for an abstract parameter space. Phase, in other words, is not only a clock reading — it is also a record of where the system has been.

Decoherence: the thing coherence engineers actually fight. A quantum system is never truly alone. Every stray photon, every gas molecule, every vibration in the mount carries away a little information about the system's state, and the moment the environment records which path the system took, the interference term in equation (3) is gone. Wojciech Zurek's 2003 review is the standard account: the environment continuously measures the system, selecting a preferred set of states and washing out the phase relations between them.

Decoherence of the off-diagonal terms(10)

$${\rho }_{12}(t)={\rho }_{12}(0)e^{-t/{\tau }_D}$$

The engineering response has three parts, and every coherent technology uses all three. Isolate: vacuum, magnetic shielding, vibration isolation, optical traps that touch nothing. Cool: fewer thermal photons and phonons to carry information away — dilution refrigerators for superconducting circuits, laser and evaporative cooling for atoms. Correct: encode one logical state across many physical ones so that errors can be detected and reversed faster than they accumulate. Coherence times in superconducting circuits have risen by several orders of magnitude since the first charge qubits, entirely through this work.

Coherent states. What is the quantum state of a laser beam? Roy Glauber answered that in 1963. The coherent states are the eigenstates of the annihilation operator: superpositions of every possible photon number, with amplitudes arranged so that the field's phase is as sharp as equation (4) allows. They have a Poisson spread in photon number, they stay coherent as they evolve, and they are the closest a quantum field comes to a classical wave. Glauber's framework also gave the field its definitions of first and second-order coherence — the correlation functions that let you say precisely how coherent a source is, rather than waving at it.

Arrays that lock themselves. Coherence does not always have to be imposed from outside; sometimes a population finds it. Kurt Wiesenfeld, Pere Colet and Steven Strogatz showed in 1996 that a series array of Josephson junctions maps onto the Kuramoto model of equation (5), with the shared load providing the coupling. That is a striking result: the same threshold mathematics that describes fireflies in a mangrove describes a superconducting circuit on a chip. Strogatz's book Sync tells the story from both ends. It is also the practical reason junction arrays are taken seriously as emitters — they do not need every junction driven in step by hand, because past a critical coupling they do it themselves.

Level 5 · Research frontier

Phase as an engineering variable. The through-line of this course is that phase started as a bookkeeping angle, became a measurable difference, then a controllable one, and is now a design parameter. The volt is realised through a phase difference winding at a rate set by voltage. A SQUID resolves a fraction of a flux quantum by reading a phase difference around a ring. A quantum processor with hundreds of superconducting qubits is hundreds of macroscopic phases held, rotated and read on schedule. None of that is frontier work any more; it is metrology and industry.

The frontier is the next step: making phase agreement happen in systems that do not want it.

The coherent matter-wave beam. In his August 2026 interview with Tim Ventura, Charles Chase — a Lockheed Skunk Works veteran who now runs a laboratory called the UnLab — described the goal directly: putting particles like electrons or atoms in phase together, the way a laser puts light in phase. The obstacle is the one from Level 1: fermions may not share a state, so the laser's own trick is unavailable. Bose–Einstein condensation gets around it by cooling to nanokelvin, which works beautifully and does not scale to a beam you could point at something.

Chase's stated route is different, and it is why this site's vector-potential course exists. The vector potential A shifts a particle's quantum phase without exchanging energy with it — that is the Aharonov–Bohm effect, measured since 1960 and settled by Tonomura's shielded-magnet experiment in 1986. If phase can be steered with no energy exchange, then in principle it can be steered towards agreement. Chase describes the vector potential as more fundamental than the fields and calls this mechanism the secret sauce. The synchronisation mathematics his slides cite is the Kuramoto model of equation (5): the design question becomes whether the coupling A provides can be pushed past the critical value for a population of particles with a spread of natural phases.

Speculative The coherent matter-wave beam is a well-posed proposal with a named mechanism, a stated target and a laboratory pursuing it. What would move it up the scale is the first published measurement of induced phase coherence in a beam of fermions that were not cooled into a condensate. That is the experiment to watch.

Junction arrays as phase-locked emitters. The nearest thing to this idea that already runs on a bench is the junction array. Gary Stephenson's 2026 gaser proposal, discussed at length on the channel this site follows, is a phased array of Josephson junctions on a silicon wafer, tuned by the AC Josephson relation and steered by time delay, intended as a coherent emitter. Everything electromagnetic about that design is standard: the tunability is a Josephson relation, the phase locking is the Kuramoto physics of Level 4, and phased-array steering is radar engineering. Stephenson names his own open question, the quantum efficiency of the conversion into gravitational radiation. The Josephson-junction course covers the device in full.

Matter-wave interferometry as a sensor, today. While the beam is a proposal, matter-wave coherence is already a working instrument. Atom interferometers split a cloud of cold atoms into two paths with laser pulses, let them fall, and recombine them; the phase difference reads the local gravitational acceleration. Commercial and field-deployed atom gravimeters now resolve changes in g at the level of parts per billion, which is enough to watch groundwater move underground, monitor magma chambers, and survey for voids. Atom gyroscopes measure rotation the same way. These are Level 3's physics sold in a box.

Definitive Atom interferometry and cold-atom gravimetry are established measurement science with published sensitivities and commercial instruments.

Where the vacuum comes in. One of the summer's most striking results belongs in this course too. A Nature paper published on 19 August 2026 put thin flakes of the superconductor NbSe₂ inside a terahertz split-ring resonator that reshapes the vacuum's own fluctuations, with no drive field, and the superconducting transition temperature rose — about 5% in a six-layer sample and 10% in a bilayer. Read it through this course's lens: reshaping the electromagnetic environment changed how easily a material entered its coherent state. That is coupling to coherence through the environment rather than through cooling, and it is exactly the kind of handle a phase engineer wants.

Open questions

• Can a population of fermions be pushed past a synchronisation threshold by an applied vector potential, and what coupling strength does that need?
• How does the decoherence time of such a beam scale with its intensity and with the residual gas around it?
• What is the honest conversion efficiency from a phase-locked junction array into any non-electromagnetic channel?

What to watch next

1. The UnLab's first published coherence measurement on an electron or atom beam, with the fringe visibility stated and a control run alongside it.
2. A first measurement from a phase-locked junction array built to Stephenson's specification, with the detector and the expected signal level named in advance.
3. Independent groups repeating the cavity-enhanced superconductivity result in their own resonators, and the first attempt to extend it to a higher-temperature material.

The objection

“Quantum phase is a bookkeeping device; only probabilities are real, so 'engineering phase' is a category error”

The answer

Phase differences are measured every working day, in instruments people buy. An interferometer converts a phase difference into a brightness through the cosine term of equation (3); LIGO reads phase differences small enough to detect a passing gravitational wave. A SQUID converts a phase difference around a superconducting ring into a current and resolves a fraction of a flux quantum, which is how the magnetic field of a beating heart is mapped from outside the chest. The international volt is realised through the rate at which a phase difference winds across a Josephson junction, to parts in a billion. And the Aharonov–Bohm effect shows the point most sharply: electrons travelling entirely through field-free space shift an interference pattern in step with a magnetic flux they never touch, first seen by Chambers in 1960 and settled by Tonomura's shielded-magnet experiment in 1986.

The philosophical caution is fair about one thing only: the absolute phase of a single wave is a convention, exactly as the zero of a voltmeter is. Every quantity named above is a phase difference, and differences are gauge-invariant and directly measurable. Engineering them is what a laser already does to light and what a superconductor already does to electron pairs. What to watch next is whether the same control extends to particles that do not condense on their own: the first published measurement of induced coherence in an uncooled fermion beam, and the first junction-array emitter measurement with its expected signal stated in advance.

Teaching aids

Three demonstrations you can run

1. Metronomes on a board. Stand four or five wind-up metronomes on a light board resting across two empty soda cans, set them all to the same tempo, and start them at random. Within a minute or two they are swinging as one. The board is the coupling: each metronome's swing rocks it a little, and the rocking nudges the others. Ask the class what changed. Nothing was tuned, no energy was added, and no metronome was told what to do — the coupling simply crossed the Kuramoto threshold of equation (5). Then take the board off the cans and put it flat on the table. The coupling drops to almost nothing and the metronomes drift apart again.
2. Two slits with a laser pointer. In a darkened room, shine a cheap laser pointer through a single strand of hair held in a paperclip, or through two fine slits cut a hair's width apart in kitchen foil with a craft knife, and let the light land on a wall two or three metres away. You will see a row of bright and dark bands. Move the wall further away and the bands spread. This is equation (3) on a wall, and it costs nothing. Then try it with a torch instead of the laser: the bands vanish, because the torch's coherence length is a few micrometres and its waves cannot agree over the path difference.
3. A human Kuramoto. Put twenty or thirty people in a room and ask each of them to clap at a steady rate of their own choosing. It will be noise. Now give one instruction: keep clapping, but try to match the people either side of you. Within about twenty seconds the room is clapping in unison. Discuss what happened — nobody led, nobody counted, and there was no shared clock. Local coupling was enough. Then ask half the room to face away and cover their ears, and watch the order fall apart. That is the coupling dropping below critical.

Self-check (answers below)

1. What does the phase of a wave actually tell you, and why is a single wave's phase not measurable on its own?
2. A light bulb and a laser can emit the same colour with the same power. What is physically different about the laser's light?
3. Two beams of equal intensity I₀ meet with a phase difference of zero. What is the brightness where they meet, and where did the missing darkness go?
4. Why does cooling a gas of atoms to nanokelvin make it possible for the atoms to interfere with one another?
5. In one sentence: why can photons be made coherent easily, and what two routes exist for making matter coherent?

Answers. (1) It tells you where the wave is in its cycle right now. Absolute phase depends on where you start counting, so only the difference between two waves is physical — which is why every measurement is an interference measurement. (2) The laser's waves agree on phase, so they can be focused, sent enormous distances, and interfered; the bulb's phases are random. (3) Four times I₀ — and nothing is missing, because at other points on the screen the same two beams cancel to darkness; interference redistributes light rather than creating or destroying it. (4) The de Broglie wavelength grows as momentum falls, and at nanokelvin the atoms' matter waves become larger than the spacing between the atoms, so the waves overlap and one shared state becomes possible. (5) Photons are bosons and may share a state, so stimulated emission puts them in step by itself; matter needs either condensation, by cooling until the waves overlap, or direct phase steering with the vector potential, which is the route the UnLab proposes.

One-page summary for the wall

• Phase is where the crest is. Coherence is many waves agreeing on it. Agreement, not extra energy, is what a laser adds to light.
• Only phase differences are measurable — and they are measured constantly: interferometers, SQUIDs, the volt, the Aharonov–Bohm effect.
• The complex amplitude ψ = √n · e^(iθ) holds both halves: how much, and where in the cycle. Number and phase trade off against each other.
• Interference brightness is I₁ + I₂ + 2√(I₁I₂) cos Δφ. That third term is the whole of interferometry.
• Coupled oscillators lock past a critical coupling — the Kuramoto threshold. Fireflies, metronomes and junction arrays all do it.
• In a superconductor the current is a phase gradient. In a condensate, matter interferes with matter — photographed in 1997.
• Decoherence is the enemy; isolation, cold and error correction are the reply, and coherence times keep rising.
• The frontier: steer phase directly with the vector potential to bring uncooled particles into step, and drive arrays of junctions past their locking threshold as coherent emitters.