The crossing that never happens

Two energy levels depend on some parameter — a magnetic field, a bond length, a gate voltage — and one rises while the other falls. Plotted against the parameter they head for one another and, at the point where the uncoupled energies would be equal, they do not meet.

This happens in atomic spectra, in molecular potential curves, in electronic bands, in coupled oscillators, in coupled optical cavities, and in a pair of pendulums joined by a spring. It has one explanation, it takes two lines of algebra, and the phrase everybody uses for it is misleading.

Two lines of algebra #

Two states, with energies $\pm\varepsilon/2$ depending on some parameter, and a coupling $V$ between them. The Hamiltonian is a two-by-two matrix and its eigenvalues are

$$E_{\pm} = \pm\sqrt{\left(\frac{\varepsilon}{2}\right)^2 + V^2}$$

The expression under the square root is a sum of two squares. It cannot vanish unless both vanish, so the two eigenvalues cannot be equal unless the coupling is zero.

That is the whole of the argument. Nothing repels anything: the gap is an algebraic consequence of adding an off-diagonal element to a matrix, and the size of the gap at closest approach is exactly $2V$ — measured off the drawn curves at 0.10, 0.30 and 0.70 electronvolts for couplings of 0.05, 0.15 and 0.35, which is a check that the drawing and the formula are about the same matrix.

The word repulsion is universal and it is a metaphor. What is true is that the presence of a coupling pushes the eigenvalues apart, in the sense that a matrix with a larger off-diagonal element has more widely separated eigenvalues; there is no force anywhere in the statement, and no energy is required to hold the levels apart.

What actually crosses #

The eigenvalues do not cross. Something does, and identifying it is what makes the phenomenon comprehensible rather than merely surprising.

The right-hand panel of the opening figure plots the character of the upper level: how much of the first basis state it contains. Far to one side it is entirely one state; far to the other it is entirely the other; and across a region whose width is set by the coupling it swaps completely.

So the upper branch arrives as one thing and leaves as another. What the levels avoid is not each other but a label, and the states themselves pass through in the ordinary way. Drawing the diabatic energies — the uncoupled ones, which do cross — alongside the adiabatic ones is the standard way to make this visible, and it is why chemists carry both sets of curves for the same molecule.

There is a way of seeing the exchange that makes it feel inevitable rather than surprising. At the point of closest approach the two diabatic states have equal energy, so neither is preferred, and the eigenstates are their equal mixtures — the sum and the difference. Move away in either direction and the mixture tilts towards whichever diabatic state is lower. Since “lower” swaps as the detuning changes sign, the eigenstates swap with it, and they do so continuously through the equal mixture at the middle. Nothing discontinuous happens anywhere; what changes is which of two labels is the better description, and at the middle neither is.

What actually crosses is the diagonal elements, not the levels. Sweeping the detuning turns the mixture of two basis states through the point where they contribute equally, and at that point the two eigenstates are equal mixtures rather than either basis state — so the labels one started with have swapped, and nothing about the energies has crossed at all. That distinction between what crosses and what does not is the whole subject.

That exchange is also the reason avoided crossings are useful rather than merely decorative. Sweeping a parameter through one is a way to convert one state into another with certainty, and a great deal of quantum control consists of arranging exactly that.

Whether the system follows #

Having established that the levels do not cross, the practical question is whether a system swept through actually follows the branch it is on.

Slowly, and it does: the state stays in the instantaneous eigenstate, follows the gap round, and comes out as the other diabatic state. That is the adiabatic theorem in its most concrete instance.

Quickly, and it does not: the system flies straight through as though the gap were not there, and comes out as what it went in as.

The boundary is the Landau–Zener expression, $P = \exp(-2\pi V^2/\hbar\,\dot\varepsilon)$, and the figure is that expression against a direct integration of the two-state Schrödinger equation through the sweep. The two agree to 0.012 across a hundred-fold range of sweep rates without sharing anything but the physics.

The quadratic in the coupling is what makes this practically important. Doubling the gap makes a system four times harder to shake off the adiabatic path, so a small gap is far more permeable than its size suggests. A gap a tenth as large needs a sweep a hundred times slower to be followed, which is why narrowly avoided crossings behave like crossings for almost all purposes and are drawn as crossings by people who know better.

The same picture, five subjects #

The two-by-two matrix does not know what its states are, so the result appears wherever two things nearly agree in energy and are coupled at all.

The same picture serves five subjects because it is the same matrix. Two coupled oscillators have normal-mode frequencies that are its eigenvalues, and tuning one towards the other produces an avoided crossing in the frequencies with the modes exchanging character — a purely classical statement, with no quantum mechanics in it. Everything peculiar-sounding about level repulsion is already present in two pendulums joined by a spring.

Two coupled pendulums show it classically. Tune one towards the other and the two normal-mode frequencies approach and separate, never meeting, with the modes exchanging which pendulum they are mostly about. The two pendulums that will not stop swapping is that exchange watched in the time domain instead.

There is a nice consequence of that classical case worth stating, because it settles whether any of this is quantum. Nothing in the two-by-two argument mentions Planck’s constant. Avoided crossings occur in coupled classical oscillators, in coupled electrical circuits, in coupled mechanical resonators and in the vibration of a bridge with two nearly-degenerate modes. What quantum mechanics contributes is that energy levels are eigenvalues of a matrix in the first place; once that is granted, the level repulsion is linear algebra and would be equally true of anything else with the same structure.

A molecule shows it as a function of bond length, and the consequences are chemistry. Two electronic states that would cross are coupled, the reaction path follows the lower adiabatic surface round the gap, and the reaction that results is not the one the diabatic states would have given. Where the sweep is fast — a hot collision, a light-driven process — the system jumps the gap instead, and that is a non-adiabatic reaction.

A solid shows it as a function of wavevector. The gap a periodic potential opens at the Brillouin zone boundary is exactly this: two plane waves of equal energy, coupled by the lattice, splitting by twice the relevant Fourier component of the potential. Every band gap in every semiconductor is an avoided crossing, and the curvature it leaves behind sets the effective mass.

An optical waveguide pair shows it as a function of position along the guides, and the exchange is the whole basis of directional couplers: two guides brought close swap their light back and forth, and choosing the interaction length selects any splitting ratio wanted.

A nucleus shows it in the spacing statistics of its levels: the distribution of gaps between neighbouring levels of the same symmetry vanishes at zero, because degeneracy requires a coincidence that coupling forbids. That vanishing is the signature that distinguishes a chaotic spectrum from a regular one, and it is level repulsion applied statistically.

Reading a spectrum for couplings #

The practical value of the two-line result is that it runs backwards: a measured avoided crossing is a measurement of a coupling that may not be accessible any other way.

Sweep a magnetic field across a molecular or atomic spectrum and watch two lines approach. If they cross, the coupling between the two states is zero to within the resolution, which is usually a statement about symmetry. If they do not, the minimum separation is twice the coupling, read straight off the chart with no model in between.

That measurement is used constantly. Spin–orbit coupling constants in molecules are extracted from avoided crossings in field-swept spectra. Tunnel couplings between quantum dots are measured by sweeping a gate voltage and finding the gap at the charge degeneracy point, which is how a double dot is characterised before it is used as a qubit. Cavity–atom couplings are measured from the vacuum Rabi splitting, which is an avoided crossing between an excited atom and a photon in a cavity and is the defining measurement of strong coupling.

In each case the quantity being determined is an off-diagonal matrix element — a number that appears in no energy on its own and is only visible where two energies would otherwise coincide. Two levels far apart are shifted by their coupling only at second order, by $V^2/\varepsilon$, which is small and hard to disentangle from everything else shifting them. At degeneracy the shift is first order and is the whole of the separation. That is why the crossing region is where the information is, and why an experiment will go to some trouble to arrange one.

When two levels really can meet #

The rule that levels of the same symmetry do not cross is stated so often that its condition is easy to lose, and the condition matters.

Whether two levels can be made to coincide depends on how many independent parameters are available to tune. With one, the answer is generally no — two levels approaching each other in a one-parameter family will repel rather than cross, because making them degenerate requires two conditions to hold at once and there is only one knob. With two parameters the crossings become possible and generically isolated, which is where conical intersections come from.

Making two eigenvalues of a real symmetric two-by-two matrix equal requires two conditions: the diagonal elements must be equal and the off-diagonal element must vanish. With one parameter to vary, two conditions cannot generally be met at once, so the crossing is avoided. With two parameters they can, at isolated points.

Those points are conical intersections, and they are not exotic. In a polyatomic molecule there are always enough vibrational coordinates, and the intersections are where electronic energy is converted into vibrational energy on a femtosecond timescale. That funnel is how a molecule returns to its ground state without emitting light, which is why most molecules do not fluoresce, why sunscreen works, and why vision’s first step is fast enough to beat the competing processes.

Symmetry changes the count. If two states belong to different irreducible representations the coupling between them vanishes identically, so only one condition remains and a crossing occurs with one parameter. That is why crossings in spectra are read as evidence about symmetry — correctly, provided the parameter count is stated.

Where the gap comes from #

Nothing above says what $V$ is, and in each application it is a different physical quantity with a different size.

Where the gap comes from, in a crystal, is a Fourier component. The gap at a zone boundary is twice the relevant component of the periodic potential, which is the two-state result with the two states being the two plane waves that scatter into each other — so a band gap and an avoided crossing are not analogous. They are the same calculation, and the coupling is the lattice.

In a solid it is a Fourier component of the crystal potential. In a molecule it is a matrix element of the nuclear kinetic energy or of a spin–orbit term. In a coupled-cavity system it is the transmission of the mirror between them. In a driven atom it is the Rabi frequency, which is proportional to the field amplitude, so the gap can be turned up and down at will — and that tunability is what makes atomic systems the place where Landau–Zener physics is tested most precisely.

The one universal statement is the relation between the gap and the time it takes to traverse it adiabatically. A gap of $\Delta$ demands a sweep slower than $\Delta^2/\hbar$ per unit of detuning, which by the uncertainty relation between energy and time is a statement that the system needs long enough to resolve the gap. Adiabatic quantum computing lives or dies on that estimate, because the minimum gap along the sweep decides the run time and is generally not known in advance.

The sweep used because it does not need to be tuned #

The Landau–Zener condition is usually presented as a constraint. It is also a technique, and the reason it is used has nothing to do with elegance.

Take the commonest task in magnetic resonance: invert a population of spins. The obvious method is a resonant pulse of exactly the right amplitude and exactly the right duration — a π pulse — which rotates every spin through half a turn. It works, and it works only where the drive amplitude and the resonance frequency are what the pulse was designed for. In a real magnet neither is uniform: the drive field falls off away from the coil, and the resonance frequency varies across the sample. A π pulse calibrated for the middle under-rotates at the edges.

The alternative is to sweep. Ramp the drive frequency through resonance slowly compared with the gap, and the system follows its instantaneous eigenstate round the avoided crossing and comes out inverted.

The virtue is that the outcome does not depend on the details. Provided the sweep is slow enough everywhere, it does not matter what the exact resonance frequency was, or the exact amplitude, or the exact rate — the adiabatic theorem delivers the same final state. A method whose answer is insensitive to its own parameters is worth a great deal in an apparatus whose parameters vary across the sample.

That is why every magnetic resonance imager uses adiabatic pulses where the drive field is inhomogeneous, and why broadband decoupling in spectroscopy is done by sweeping rather than by pulsing.

The same idea is used to build molecules in chosen states. Stimulated Raman adiabatic passage transfers population between two states through a third by following a superposition that never populates the intermediate one — accomplished by applying the two laser pulses in what looks like the wrong order, so that the system is carried along a state whose composition rotates as the pulses overlap. It moves essentially all of the population, it is insensitive to the pulse areas, and it leaves no time for the intermediate state to decay.

And it is the basis of a whole computing architecture. Adiabatic quantum computing encodes the answer to a problem in the ground state of a final Hamiltonian, starts the system in the easily-prepared ground state of an initial one, and sweeps between them. The run time is set by the smallest gap encountered along the way, and it goes as the inverse square of that gap — which is why the method is easy to state and hard to analyse, since the minimum gap is generally not computable in advance and is the whole question.

Robustness bought with time, and time limited by a number nobody can predict. That is the trade the quadratic in the Landau–Zener exponent imposes, in every one of these applications.

The statistics of the gaps #

Level repulsion between two levels is an algebraic identity. Applied to a spectrum of many levels it becomes a statistical statement, and the statistical statement turns out to be a diagnostic.

Take a system with a great many levels of the same symmetry and ask how the gaps between neighbours are distributed. If the levels are uncorrelated — placed independently, as random points on a line — the spacings follow an exponential distribution, which is largest at zero: close pairs are the commonest outcome, and levels cluster.

If instead every pair of levels repels, close pairs are suppressed and the distribution vanishes at zero. It rises to a maximum near the mean spacing and falls away, and the way it vanishes at the origin is not arbitrary: it is linear, quadratic or quartic according to whether the system has time-reversal symmetry and whether it has spin, which is Dyson’s classification of the three ensembles.

Which of the two shapes a spectrum has turns out to correspond to whether the underlying classical motion is regular or chaotic. A system whose classical trajectories are confined by conserved quantities has levels labelled by those quantities, so levels with different labels do not repel and the spacings are uncorrelated. A system whose classical motion explores its energy surface has no such labels, every level repels every other, and the spacings follow the repelling distribution.

That is a remarkable diagnostic, because it says whether a system is chaotic without solving anything: it requires only a list of measured energies.

It has been tested widely. The original data were the neutron resonances of heavy nuclei, where thousands of levels of the same spin and parity are known and follow the repelling distribution closely. The same has been found in the spectra of rare-earth atoms, in the acoustic resonances of irregularly shaped plates, in the microwave resonances of cavities machined into chaotic billiard shapes — an experiment with no quantum mechanics in it at all, since the wave equation is the same — and in the conductance of small semiconductor devices.

There is one further instance and it is the strangest. The spacings between the zeros of the Riemann zeta function, computed numerically to enormous heights, follow the same distribution as the levels of a chaotic system with no time-reversal symmetry. The correspondence was noticed in conversation in 1973, has been checked to extraordinary precision since, and remains unexplained.

What the pictures cannot show #

Everything is two states. A third state nearby changes the answer, and a whole band of them changes it qualitatively: the sweep then couples to a continuum, and what was a coherent jump becomes an irreversible decay.

The sweep is linear and infinite. Landau and Zener’s result is asymptotic, quoted long before and long after; a real sweep that starts or stops near the crossing leaves the system in a superposition with a phase, and interference between successive traversals produces Stückelberg oscillations that the single-passage formula has no room for.

There is no dissipation. A real system exchanges energy with its surroundings, which populates the lower branch and destroys the coherence, and for a slow enough sweep the relaxation rather than the adiabatic theorem decides the outcome.

The two-by-two matrix is real. A complex off-diagonal element gives the same gap, since only its modulus enters, but a different phase relation between the two branches — and that phase is what a geometric phase accumulates when the crossing is encircled rather than traversed, which is a shape rather than a length.

And the character plot is basis-dependent. Which state is “the first one” is a choice, and the exchange it displays is an exchange between the states chosen. The eigenvalues are not basis-dependent, which is why the gap is the physical statement and the swap is the interpretation.

The ladder from here #

Later rungs on this anchor: Stückelberg interference, where two passages through the same crossing interfere and the phase between them is measurable; conical intersections and the geometric phase a circuit around one acquires; the Landau–Zener problem with dissipation, where the outcome depends on the bath as well as the sweep; and the adiabatic theorem’s failure modes, including the cases where a gap that never closes still does not guarantee adiabaticity.

The neighbouring ladders are what happens when the wells get close, which is this splitting for many states rather than two, the gap a repeat opens, which is the same algebra for two counter-propagating waves, and the mass a curve decides, where what is read off is the curvature the gap leaves behind.