The phase that is only a shape

Polarisation is the direction of the shaking, and a polariser asks about it. This essay is about what happens to the phase of a beam when its polarisation is taken on a journey and brought back.

A circuit on the sphere of polarisations. The Poincaré sphere, on which every polarisation state is a point: linear states around the equator, circular at the poles, and orthogonal states at opposite ends of a diameter. The triangle is a closed circuit — linear at 0°, then linear at 23°, then left circular, and back — taken along geodesics, which is what a sequence of ideal polarisers does. Bringing a state round it returns it to exactly the state it started in, and multiplied by a phase: 0.3927 radians here, against minus half the enclosed solid angle of -0.7854 steradians. The two agree exactly, and neither calculation knows about the other — the phase is the argument of a product of three overlaps between Jones vectors, and the solid angle is spherical geometry. Nothing about the elements used, their thickness, or the wavelength enters. A phase that depends only on the shape of a path is the signature of a geometric phase, and this is the oldest known example of one. — Fig. 1 The Poincaré sphere, on which every polarisation state is a point: linear around the equator, circular at the poles, orthogonal states at opposite ends of a diameter. The triangle is a closed circuit, and taking a state round it returns it to exactly the state it started in — multiplied by a phase of 0.3927 radians, which is minus half the enclosed solid angle of 0.7854 steradians.

The sphere the states live on

A polarisation state is two complex amplitudes with a common phase and a common magnitude removed, which leaves two real parameters. The natural way to hold them is on a sphere.

Linear polarisations sit on the equator, at an angle around it equal to twice their physical angle — so horizontal and vertical, which are orthogonal, are on opposite sides. Circular polarisations sit at the poles. Everything in between is elliptical.

Polarisation is the relative amplitude and phase of a wave’s two transverse components, which is two numbers — and two numbers with an overall phase and normalisation removed is a sphere. That is why the state space is the Poincaré sphere rather than anything more complicated, and why a circuit traced on it has an area: the geometric phase is that area, and it exists because the space is curved.

The factor of two is worth noticing, because it is where the geometry gets interesting: physical rotations by 180° are one full turn of the sphere, so a state’s journey and the physical operations that produce it are not in step.

Malus's law. The fraction of polarised light passing a filter, against the angle between the light's own direction of shaking and the filter's axis. It is the cosine squared: half at 45 degrees, nothing at 90. — Fig. 2 Malus’s law, which is the ordinary content of a polariser: it projects onto a state and transmits the squared overlap. On the sphere, a projection moves the state to a new point along the geodesic between the old point and the polariser’s own state — which is what makes a sequence of polarisers a path.

The phase that a circuit leaves behind

Send a beam through three polarisers at states A, B and C, chosen so that C’s projection returns it to A. The intensity that survives is the product of three squared overlaps, and that is Malus’s law applied three times.

The phase is the argument of the product of the three overlaps themselves:

\gamma = \arg\left(\langle A|B\rangle\langle B|C\rangle\langle C|A\rangle\right)

which is not zero, is not a path length, and is exactly $-\Omega/2$ where $\Omega$ is the solid angle the triangle encloses on the sphere.

The phase is half the area, and nothing else. The phase acquired by a polarisation state taken round a closed circuit on the Poincaré sphere, against the solid angle that circuit encloses, for 6 triangles of quite different shapes and positions. Every point lies on a straight line of slope −1/2, to 1.1e-16 radians. The phase is computed from the states themselves, as the argument of the product of three overlaps ⟨A|B⟩⟨B|C⟩⟨C|A⟩, and the solid angle from the geometry of the triangle they make; the two calculations have nothing in common and agree exactly. What that means is that the phase depends on the shape of the path and not on how the path was traversed — not on the speed, not on the optical elements used, not on any length. It is called a geometric phase for that reason, and it is the same quantity Berry found in 1984 for an adiabatically transported quantum state, of which Pancharatnam's 1956 optical result is a special case that went unnoticed for thirty years. — Fig. 3 The phase acquired against the solid angle enclosed, for six circuits of quite different shapes and positions. Every point lies on the line of slope −1/2, to 10⁻¹⁶ radians. The phase is computed from the states and the solid angle from the geometry, and neither calculation knows about the other.

That agreement is the whole result. Six circuits, no adjustable constants, agreement to the last bit of a double. And the striking part is what is absent from it: no wavelength, no thickness, no refractive index, no time. Two entirely different optical systems that trace the same circuit produce the same phase.

Turning one plate

The three-polariser version is the clean derivation and a poor demonstration, because polarisers throw most of the light away. The practical version uses one waveplate.

Fig. 4 Circularly polarised light through a half-wave plate whose fast axis is turned. The output state does not change at all — the plate reverses the handedness whatever its angle — and the phase changes by exactly twice the angle turned, measured at 2.000 radians per radian.

A half-wave plate turns right-circular light into left-circular whatever its orientation, so turning it changes nothing about the state that emerges. What it changes is the phase, at exactly two radians per radian of plate rotation, and the factor of two is the sphere’s factor of two: turning the plate by π carries the state twice round a circuit enclosing a full hemisphere.

Nothing about the plate’s thickness enters. A plate that is half-wave at one wavelength and not at another gives the same geometric phase and a different dynamical one, which is the basis of a practical distinction: the geometric part is achromatic and the dynamical part is not.

The index that depends on which way the light is going. The two refractive indices of 3 uniaxial crystals, against the angle between the wave normal and the crystal's optic axis. The flat lines are the ordinary index, which is the same in every direction because the ordinary wave's field is always perpendicular to the axis. The curves are the extraordinary index, which runs from the ordinary value along the axis — where the two waves are identical and the crystal behaves like glass — to its extreme value at right angles to it. calcite (CaCO₃) has n_o = 1.6584 and n_e = 1.4864, so n_e − n_o = -0.1720; quartz (SiO₂) has n_o = 1.5443 and n_e = 1.5534, so n_e − n_o = 0.0091; lithium niobate has n_o = 2.3005 and n_e = 2.2075, so n_e − n_o = -0.0930. The sign of that difference is what makes a crystal positive or negative, and it decides which of the two images in a double-refracting crystal is the one that moves. — Fig. 5 The two refractive indices of a birefringent crystal, which is where a waveplate’s retardation comes from. That retardation is a dynamical phase — thickness times index difference over wavelength — and it is the quantity the geometric phase is being distinguished from.

Why half the area

The factor of one half is not a convention and can be got at without any machinery, which is worth doing because it explains what kind of quantity this is.

Consider a small circuit near the north pole of the sphere — near right-circular polarisation. A state there is $|R\rangle$ plus a small admixture of $|L\rangle$ , and moving around the circuit rotates the phase of that admixture. Going once round a circle of angular radius $\alpha$ about the pole rotates it through $2\pi$ , while the solid angle enclosed is $2\pi(1-\cos\alpha)$ , which for small $\alpha$ is $\pi\alpha^2$ . The phase acquired by the state as a whole is the average over the state of that rotation, weighted by how much of it is in each component — and for a state near the pole the weighting is what produces the half.

The general statement is that the sphere’s natural area form is the curvature of the connection that says how to compare phases at different points, and the phase round a loop is the integral of that curvature over the enclosed area. That is exactly the structure of a magnetic flux: the phase is a “flux” of a fictitious monopole of strength one half sitting at the centre of the sphere.

Interference between two paths is the only way any phase becomes visible, and it is what makes the factor of a half measurable. The monopole analogy is exact enough that the Aharonov–Bohm effect is the same statement with a magnetic flux in place of the solid angle — a phase acquired by going round something, with no local field anywhere along the way.

The monopole picture also explains why the phase is defined only modulo $4\pi$ in the state and $2\pi$ in observables: the strength of the monopole must be a half-integer for the phase to be single-valued, which is the same quantisation condition Dirac derived for magnetic charge.

What “geometric” means

The distinction being drawn is between two kinds of phase, and it is worth stating carefully because the word “geometric” is doing precise work.

A dynamical phase is the integral of a rate over time: how fast the state’s energy or optical path accumulates, times how long. Halve the speed and it doubles. It is what almost every phase in optics is.

A geometric phase is a property of the path traced through the space of states, and it is unchanged when the path is traversed at a different speed or by different means. It depends on the shape, not on the schedule. Three polarisers at 0°, 45° and 90° transmit an eighth of the light, which is the intensity version of the same circuit and the part everybody notices first — the phase is the part that needs an interferometer to see.

Pancharatnam found the optical version in 1956, working on interference between beams in different polarisation states, and it was largely ignored. Berry found the general quantum version in 1984, as the phase an adiabatically transported eigenstate acquires when its Hamiltonian is carried round a loop in parameter space — and the two were recognised as the same thing shortly afterwards. That the optical case had been in print for twenty-eight years is a standing example of how a result can be complete, correct, and invisible.

Where it turns into hardware

The geometric phase would be a curiosity if it could not be imposed at will, and the way to impose it is to make the plate’s axis orientation vary across its face.

A plate that is half-wave everywhere but whose fast axis rotates with position imposes a phase $2\theta(x,y)$ on circular light — a phase pattern that is a map of its own orientation pattern. Since any phase pattern can be written that way, any optical element can be made this way: a lens, a prism, a hologram, a vortex generator.

Such devices are called Pancharatnam–Berry optical elements, or geometric-phase optics, and they are what most flat optical components are now made of. They are thin, since the phase comes from orientation rather than from accumulated path; the two circular polarisations get equal and opposite phases, so a geometric-phase lens focuses one handedness and defocuses the other, which is a form of the dependence on handedness a crystal shows made into a design freedom; and the phase profile is the same at every wavelength even though the half-wave condition is not.

A geometric phase is measured by sending half a beam round the circuit and leaving the other half alone, then recombining. That is where it turns into hardware, and it is worth stating because it makes clear the phase is not an abstraction: it is a fringe shift, in an instrument, of a size the solid angle predicts.

The fibre that was coiled

The polarisation sphere is one space a state can be carried round, and the sharpest optical demonstration of the geometric phase uses a different one entirely: the sphere of directions a beam can travel in.

Take a single-mode optical fibre and wind it into a helix. Light travelling along it has its direction of propagation carried round a cone as it goes, and after one turn of the helix that direction has traced a closed circuit on the sphere of directions. The polarisation, which must stay transverse to the propagation, is carried along with it — and it comes back rotated, by an angle equal to the solid angle the circuit enclosed.

That was done in 1986, with a metre or two of fibre wound on formers of different pitch, and the measured rotation matched the solid angle over a range of geometries. Nothing about the glass entered: the fibre’s birefringence was kept negligible, the wavelength did not appear, and the same rotation was obtained from a tightly wound short helix and a loosely wound long one enclosing the same solid angle.

What makes it a good demonstration is that the mechanism is unmistakable. A rotation of polarisation usually means birefringence or a magnetic field, and both were excluded by construction; what was left is that the transverse plane cannot be carried round a closed loop on a sphere and come back the same way it started. That is parallel transport, and its failure to close is the same failure a vector suffers when carried round a triangle on a globe.

It also identifies which sphere a given instance of the effect lives on, which is worth keeping straight. Pancharatnam’s phase is a circuit on the sphere of polarisation states; the coiled fibre’s rotation is a circuit on the sphere of directions. Both give half a solid angle or a solid angle depending on the spin of what is being transported, and they are different quantities in different spaces that happen to be the same shape.

The minus sign a neutron brings back

The quantum version has one prediction so strange that measuring it was worth an experiment on its own, and the measurement is a direct instance of the geometry above.

A spin-half state carried round a closed circuit on its sphere picks up half the solid angle as a phase. Take the circuit that covers the whole sphere once — which is what a physical rotation through $2\pi$ does — and the solid angle is $4\pi$ , so the phase is $2\pi/2 = \pi$ , and the state comes back multiplied by $-1$ .

A rotation through a full turn is supposed to be the identity. For a spin-half it is not: it is minus the identity, and only a rotation through $4\pi$ returns the state unchanged. That is a statement about the mathematics of half-integer spin that had been known since the 1920s and regarded as unobservable, since a global phase of $-1$ changes nothing measurable about a state on its own.

It is observable in an interferometer, because there the sign is relative. Split a beam of neutrons, pass one path through a magnetic field arranged to rotate the spin by a controlled angle, recombine, and count. The fringe pattern repeats with a period of $4\pi$ in the rotation angle rather than $2\pi$ — the beam that has been turned through one full revolution interferes destructively with the one that has not.

Two groups did it independently in 1975 and both saw the $4\pi$ periodicity. It remains one of the few experiments that measures a property of the rotation group directly rather than a consequence of it, and in the language of this essay it is the geometric phase for the largest circuit the sphere has.

The pendulum’s missing angle

The mechanical instance deserves its number, because it makes the “solid angle” concrete in a way the optical cases do not.

A Foucault pendulum’s plane of swing is parallel-transported as the Earth carries it round a circle of latitude. After one rotation of the Earth the plane has turned, relative to the ground, by $2\pi\sin\lambda$ — which at the latitude of Paris is about 271 degrees rather than a full turn.

The missing 89 degrees is the geometry. A circle of latitude on a sphere encloses a cap whose solid angle is $2\pi(1 - \sin\lambda)$ , and that is exactly the deficit. At the pole the circle encloses nothing, the deficit is zero, and the plane turns once per day; at the equator the circle is a great circle, the enclosed cap is a hemisphere, the deficit is the whole turn, and the plane does not rotate at all.

No torque acts on the pendulum’s plane at any point in that argument, and nothing about the pendulum enters — not its length, not its mass, not its period. The rotation is a property of the path the suspension point traced on a sphere, and it would be the same for any object whose orientation is carried along without being twisted.

Which is why the demonstration is usually explained badly. The Coriolis account is correct and computes the rate; the geometric account computes the total round a closed loop and explains why the answer contains only a latitude. Both are right, and only the second says why the number is a solid angle.

The same phase in other systems

In quantum spin. A spin-half carried round a loop in magnetic field direction acquires half the solid angle as a phase, which is the identical formula — the Poincaré sphere and the Bloch sphere are the same sphere with different names. The famous minus sign of a $2\pi$ rotation is that phase for a circuit that covers the whole sphere.

The state space of a spin-half is the same sphere as the state space of a polarisation, so the phase acquired by carrying a spin round a circuit is the same geometric quantity. That is why a spin rotated through 360° comes back with a minus sign — half the solid angle of a full sphere is $2\pi$ , and the phase is $\pi$ . The famous factor of two is a fact about spheres.

In molecules and in solids. The Berry phase around a conical intersection changes the vibrational spectrum of a molecule; the same quantity integrated over a band gives the electric polarisation of a crystal, and its curvature gives the anomalous velocity of an electron in a magnetic field. Whole sections of modern condensed-matter physics are computations of geometric phases.

And in the Foucault pendulum, which is the mechanical version: a pendulum carried round a circle of latitude comes back with its plane rotated by the solid angle its parallel encloses at the pole. The rotation is not caused by any torque and does not depend on how fast the trip was made.

One rotation-without-a-torque is routinely mistaken for another, and the distinction is worth stating because it is the sharpest test of what kind of quantity this is. Faraday rotation also turns a plane of polarisation with nothing mechanical touching it — and it is non-reciprocal: send the light back along the same path and the rotation doubles instead of cancelling, because the magnetic field picks out a direction in space rather than a direction along the beam. A geometric phase does the opposite. Reverse the circuit and the solid angle changes sign, so the phase cancels exactly. Both effects rotate something for free; only one of them can be undone by retracing the path.

The measurement that settles it

A phase that could not be measured would be a bookkeeping choice, so it is worth being specific about how it is seen.

Pancharatnam’s own experiment split a beam, sent one half through a sequence of polarising elements that returned it to its original state, and recombined the two. The fringes shift, and the shift is the phase. Doing it with three polarisers at states whose circuit encloses a known solid angle gives a shift that can be predicted with no free parameters — the derivation has none — and the agreement is what turns the geometry into physics.

The modern versions are cleaner. A half-wave plate on a rotation stage, in one arm of an interferometer, sweeps the phase linearly with the plate angle at two radians per radian, and the fringe count is read off directly. Because the state emerging is unchanged, the fringe contrast does not degrade as the phase is swept, which is the experimental signature that distinguishes this from any dynamical effect: a dynamical phase that varied by many radians would ordinarily come with a change in something else.

A geometric phase shifts every frequency component by the same amount, where a dynamical phase shifts each in proportion to its frequency. That difference is what settles the measurement: send a pulse round the circuit and a dynamical phase delays it while a geometric one does not, so the two can be separated by watching whether the pulse arrives late or merely changed.

That last point is the one with consequences. A dynamical phase from a slab of glass is proportional to frequency, so it delays a pulse; a geometric phase is the same at every frequency, so it changes the phase without delaying anything. Devices exploiting the difference are how broadband phase control is done in ultrafast optics.

What the pictures cannot show

The circuits drawn are geodesic triangles. A circuit made of arbitrary elements is not made of geodesics, and then the phase splits into a geometric part — still minus half the solid angle — and a dynamical part that must be subtracted off. Ideal polarisers happen to move a state along a geodesic, which is why the three-polariser case is the clean one.

The phase of a single beam is not observable. Everything measured here is a relative phase between two beams, one of which went round the circuit. A geometric phase is meaningful for the same reason any phase is: it shows up in interference and nowhere else.

Absorption is ignored. Three polarisers in sequence transmit at most an eighth of the light, and the calculation here is about the phase of what survives. Real geometric-phase devices avoid this by using waveplates rather than polarisers, which is why the practical realisation looks nothing like the derivation.

And the sphere is the space of pure states. Partially polarised light lives inside the sphere, not on it, and the geometric phase of a mixed state is a more delicate object with several competing definitions.

The condition that is usually assumed and rarely stated

One assumption runs through the quantum version and is worth separating out, because the optical case shows it is not needed.

Berry’s derivation is adiabatic: the Hamiltonian is changed slowly enough that the state stays in the same instantaneous eigenstate throughout. That condition does real work there, and it is why the result was thought of for years as a statement about slow processes.

The optical case has no time in it at all. A beam crosses three polarisers in nanoseconds and acquires the full geometric phase; nothing is slow, nothing is adiabatic, and no eigenstate is being followed. What is actually required is only that the state be projected onto a definite state at each step, and the sequence of projections is the path.

That reading came later, in the work of Aharonov and Anandan, and it generalises the phase to any cyclic evolution of a state — adiabatic or not, and produced by any means. It also explains why the effect is so robust in practice: it does not need the slowness that would make it fragile.

Where the condition does still bind, it binds quantitatively rather than as a mood. “Slow” means slow compared with the gap: the rate at which the parameters are driven must be small against the energy separation to the nearest other level, divided by Planck’s constant. A level that is nearly degenerate with its neighbour therefore has almost no adiabatic regime at all, and the closer the approach the slower the circuit has to be traversed to stay in one state — which is why the interesting geometry always sits near a degeneracy and why the interesting geometry is always the hardest place to work.

That is the same degeneracy the solid angle is measured about. The point the circuit encloses on the sphere is the point where the two states coincide, so the quantity that makes the phase nonzero is the quantity that makes the adiabatic condition hardest to meet. The two are not separate problems.

The ladder from here

Later rungs on this anchor: the geometric phase for non-geodesic circuits, and how the dynamical part is separated; the Berry connection and curvature as the differential-geometric objects the phase is an integral of; the non-adiabatic and non-cyclic generalisations, which remove the two conditions the original results assumed; and geometric-phase optics as a design method, where an arbitrary wavefront is written as an orientation pattern.

The neighbouring ladders are the direction of the shaking, which is what a polarisation state is, the crystal that answers twice, which supplies the waveplates, and the rotation a return trip doubles, which is the dynamical rotation this one is most often confused with.