The adiabatic theorem and the Berry phase

The light field of the last section oscillates $10^{15}$ times a second — an “extremely fast” disturbance. This section twists the dial the other way: what happens when the Hamiltonian changes extremely slowly?

Start with an experiment within arm’s reach. Put a spin-1/2 particle in a magnetic field; its two levels are the “aligned” and “anti-aligned” states along the field (section 5.3). Now change the field direction gently — sweep it once around a cone, say, taking a few seconds. What happens to a spin in the “aligned” state?

Intuition says: the field turns far more slowly than the spin precesses, so the spin should trail it like a dog on a leash, hugging the field direction throughout and coming home after one full turn. That intuition is essentially right — this is the adiabatic theorem. But in 1984 Michael Berry noticed that “coming home” misses one detail: after the full turn, the wavefunction carries an extra phase — a phase that is neither the dynamical phase of accumulated energy nor removable by redefining phase conventions. It depends only on the solid angle the field direction sweeps out on the sphere — a purely geometric quantity, indifferent to how slowly you turned or how high the levels sat.

Why should something that is “just a phase” cause a fuss? Because phases interfere. This geometric phase turned out to have been hiding all along in chemistry (the sign puzzle of molecular wavefunctions), in optics (polarisation rotation in optical fibres), and in the Aharonov–Bohm effect; today its generalisations (Berry curvature, Chern numbers) are the grammar of the entire field of topological matter.

The adiabatic theorem

Statement: if the Hamiltonian $\hat H(t)$ changes slowly enough with time, and the system starts in an instantaneous eigenstate $\ket{n(0)}$ (where $\hat H(t)\ket{n(t)}=E_n(t)\ket{n(t)}$, and that level never crosses any other), then the system tracks the instantaneous eigenstate $\ket{n(t)}$ throughout, with vanishing probability of jumping to any other level.

“Slowly enough” compared with what? With the energy gap. The quantitative condition is

$$\frac{\hbar\,\bigl|\bra{m}\partial_t\hat H\ket{n}\bigr|}{(E_n-E_m)^2}\ll1 \qquad\text{for all } m\neq n\tag{7.7.1}$$

The numerator is the Hamiltonian’s rate of change (converted to energy per second); the denominator is the gap squared. How to read it: the system has an intrinsic beat $\hbar/(E_n-E_m)$ (the quantum beat period of neighbouring levels); as long as the Hamiltonian’s change within one beat is far smaller than the gap, the system “does not feel” the change happening and has time to adjust itself continuously. An analogy: carry a bowl of water and turn around slowly — the surface stays level, provided the turning period is far longer than the water’s sloshing period.

Tracking along — with two phases in tow

The adiabatic theorem says “the system follows $\ket{n(t)}$”, but quantum states can differ by a phase. What phase does the tracking carry?

The phases of adiabatic evolution: dynamical + geometricadvanced~12 minexpand

Let the Hamiltonian depend on time through a set of external parameters $\vec R(t)$ (field direction, internuclear distance, …): $\hat H(t)=\hat H(\vec R(t))$. Expand the state in instantaneous eigenstates, peeling off the energy phase in advance:

$$\ket{\Psi(t)}=\sum_n c_n(t)\, \exp\!\left[-\frac{\ii}{\hbar}\int_0^tE_n(t')\,\dd t'\right]\ket{n(t)}\tag{7.7.2}$$

The exponential factor is the dynamical phase $\theta_n=-\frac1\hbar\int E_n\dd t$ — the natural generalisation of the stationary phase $-E_nt/\hbar$ to slowly varying energies; nothing new there. The news is in the $c_n$.

Step 1: substitute into the Schrödinger equation.

Completely parallel to section 7.5, the energy terms cancel term by term — but now the basis vectors themselves are moving, and $\partial_t\ket{n(t)}$ contributes new terms. Projecting onto $\bra{m}$:

$$\dot c_m=-c_m\braket{m}{\dot m} -\sum_{n\neq m}c_n\,\frac{\bra{m}\partial_t\hat H\ket{n}}{E_n-E_m}\, \ee^{\ii(\theta_n-\theta_m)}\tag{7.7.3}$$

(The second line uses the identity $\bra{m}\dot n\rangle=\bra{m}\partial_t\hat H\ket{n}/(E_n-E_m)$, obtained by differentiating $\hat H\ket{n}=E_n\ket{n}$.)

Step 2: the adiabatic limit.

The summed terms have slowly varying numerators multiplying the rapidly oscillating phase difference $\ee^{\ii(\theta_n-\theta_m)}$ — a slow envelope times a fast oscillation integrates to approximately zero (this is exactly where the adiabatic condition above comes from). Drop them:

$$\dot c_n=-c_n\braket{n}{\dot n} \quad\Longrightarrow\quad c_n(t)=c_n(0)\,\ee^{\ii\gamma_n(t)}, \qquad \gamma_n(t)=\ii\int_0^t\braket{n}{\dot n}\,\dd t'\tag{7.7.4}$$

($\braket{n}{\dot n}$ is pure imaginary — differentiate $\braket{n}{n}=1$ to see it — so $\gamma_n$ is real and $|c_n|$ is constant: the system really does stay on level $n$, and the adiabatic theorem is proved.)

Step 3: see that it is geometric.

Use the chain rule to trade the time derivative for a parameter derivative: $\ket{\dot n}=\dot{\vec R}\cdot\nabla_{\vec R}\ket{n}$, so

$$\gamma_n=\ii\int_{\vec R(0)}^{\vec R(t)} \braket{n}{\nabla_{\vec R}\,n}\cdot\dd\vec R\tag{7.7.5}$$

Time has vanished. The integral runs only along the path through parameter space: take one second or one year to walk it, $\gamma_n$ comes out identical to the last digit. It is a property of the path, not of the process — hence geometric phase.

Step 4: why it was overlooked for half a century.

The phase of an instantaneous eigenstate was always our arbitrary convention: change conventions, $\ket{n}\to\ee^{\ii\chi(\vec R)}\ket{n}$, and $\gamma_n$ shifts by $\chi(\vec R(0))-\chi(\vec R(t))$. For an open path you can always choose the convention to absorb $\gamma_n$ entirely — it is “not physical”, and textbooks threw it away for fifty years on that basis. But for a closed loop ($\vec R(T)=\vec R(0)$), the endpoint corrections from any change of convention cancel, and what remains,

$$\gamma_n(C)=\ii\oint_C\braket{n}{\nabla_{\vec R}\,n}\cdot\dd\vec R\tag{7.7.6}$$

is independent of the phase convention — a measurable physical quantity. This is the Berry phase. Berry’s contribution was not the integral itself, but seeing that “on a closed loop it cannot be dodged”.

A worked example: spin 1/2 in a slowly turning field

The parameter space is the sphere of field directions. For a spin 1/2 in a field $\vec B=B\hat n$, the Berry connection of the “aligned” state $\ket{+;\hat n}$ can be computed explicitly from the spin states of section 5.3 (write $\ket{+;\hat n}$ in spherical coordinates $\theta,\varphi$, take the gradient in the parameters, do the loop integral). The result is remarkably clean:

$$\gamma_\pm(C)=\mp\frac{\Omega_C}{2}\tag{7.7.9}$$

where $\Omega_C$ is the solid angle enclosed on the direction sphere by the loop $C$ the field direction traces out. In equivalent curvature language: the spin’s Berry curvature is a “monopole field” radiating outward from the centre of the sphere ($B=0$ — the degeneracy point!), $\vec\Omega_\pm=\mp\hat R/2R^2$ — the degeneracy point is a magnetic monopole in parameter space, and the Berry phase is its flux.

Plug in a number: sweep the field once around a cone of half-angle $60^\circ$. The cone encloses a solid angle

$$\Omega=2\pi(1-\cos60^\circ)=\pi\tag{7.7.10}$$

so the “aligned” state collects $\gamma_+=-\pi/2$. To measure it: prepare the spin in a superposition of aligned and anti-aligned; the two branches acquire geometric phases differing by $\Omega=\pi$, so after one turn the superposition’s interference fringes shift bodily by half a period — nuclear magnetic resonance experiments saw this shift in 1987, independent of the turning speed, answering only to the solid angle.

A famous corollary: sweep the field direction around a full great circle (half-angle $90^\circ$): $\Omega=2\pi$, $\gamma=-\pi$ — the spin wavefunction changes sign. A sign flip under a 360° rotation — this is precisely the geometric-phase version of section 5.3’s “a spin 1/2 needs 720° to come back to itself”.

Aharonov–Bohm: a Berry phase twenty-five years early

In 1959 Aharonov and Bohm pointed out something strange: split an electron beam in two and route the branches around the two sides of a long thin solenoid before recombining them. Outside the solenoid the magnetic field is strictly zero; the electrons travel entirely through the region where $\vec B=0$ — as far as classical mechanics is concerned, the solenoid does not exist. Yet when the two branches reunite they carry a phase difference

$$\Delta\phi=\frac{q}{\hbar}\oint\vec A\cdot\dd\vec l=\frac{q\Phi}{\hbar}\tag{7.7.11}$$

where $\Phi$ is the flux inside the tube. The interference fringes shift with $\Phi$ — even though the electrons never touched the field. In 1986 Akira Tonomura ran the experiment with the flux completely shielded inside a superconducting ring; the fringes shifted exactly as predicted.

Seen through this section’s eyes: the electron’s position is the parameter, the vector potential $q\vec A/\hbar$ plays exactly the role of the Berry connection, and the flux is the enclosed “Berry curvature flux”. The AB phase is the geometric phase’s special case in electromagnetism — and it tells us the vector potential is not mere mathematical scaffolding: where the local field is zero, geometry (topology) is still doing work.

∎Key formulas

• Adiabatic condition

  $$\frac{\hbar\,|\bra{m}\partial_t\hat H\ket{n}|}{(E_n-E_m)^2}\ll1$$

  Priced by the gap squared; adiabaticity must fail where the gap closes (Landau-Zener)

• The two phases

  $$\ket{\Psi(t)}=\ee^{\ii\gamma_n}\,\ee^{-\frac{\ii}{\hbar}\int E_n\dd t}\,\ket{n(t)}$$

  The dynamical phase logs energy × time; γₙ logs only the path

• Berry phase

  $$\gamma_n(C)=\ii\oint_C\braket{n}{\nabla_{\vec R}\,n}\cdot\dd\vec R$$

  Gauge-independent and measurable on closed loops; absorbable by convention on open paths

• Spin 1/2

  $$\gamma_\pm=\mp\frac{\Omega_C}{2}$$

  Ω is the solid angle swept by the field direction; degeneracy point = monopole in parameter space

• AB phase

  $$\Delta\phi=\frac{q\Phi}{\hbar}$$

  A geometric phase where the local B=0; the vector potential’s loop integral is measurable

?Self-check4 questions

1. 1.

   The adiabatic theorem’s "slowly enough" — slow compared with what?

   • Compared with the intrinsic beat set by the gap: the Hamiltonian’s change within a time ħ/ΔE must be far smaller than the gap
   • Compared with the speed of light
   • Compared with the response time of the measuring apparatus
   • There is a universal absolute standard of slowness

   Submit
2. 2.

   The essential difference between the Berry phase and the dynamical phase is:

   • The Berry phase depends only on the path through parameter space, not on how long the path took to traverse
   • The Berry phase is smaller
   • The dynamical phase cannot be measured
   • The Berry phase exists only in magnetic-field problems

   Submit
3. 3.

   Why does the geometric phase "not count" on an open path but become a physical quantity on a closed loop?

   • The eigenstates’ phase convention is arbitrary: on an open path γ can be absorbed entirely by convention, while on a closed loop the endpoint corrections cancel and γ is gauge-independent
   • On an open path the system does not return to its origin, so the wavefunction is meaningless
   • On a closed loop the dynamical phase is zero
   • It is a limitation of experimental precision

   Submit
4. 4.

   The field direction is swept slowly once around a cone of half-angle θ = 60°, with the spin in the "aligned" state. Find the magnitude |γ| of the Berry phase it acquires, in units of π. (γ = Ω/2, Ω = 2π(1−cosθ))

   π5% relative tolerance

   Submit

What comes next

The adiabatic approximation holds down the “infinitely slow” end of the scale. The other end — the Hamiltonian switching instantaneously — turns out to be even simpler: the change is so fast that the wavefunction has no time to react at all, and the old wavefunction is “frozen” untouched into the new Hamiltonian. The next section uses this “sudden approximation” on a beautiful real problem: at the instant a tritium nucleus beta-decays, the nuclear charge jumps from 1 to 2 — where does its electron land? It is also the final section of this chapter — after which we face head-on a fact the whole chapter has been dodging: the two electrons of helium are not “electron 1” and “electron 2”; they are indistinguishable identical particles.