Illustrated Physics
physics, drawn
Concept

Avoided crossing — where it appears

Two energy levels approaching as a parameter is varied and separating without meeting, by exactly twice the coupling between them. Nothing repels: the eigenvalues of a matrix with an off-diagonal element cannot coincide, and what actually crosses is the identity of the two states, which swap across the gap.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

The response of a machine with a 10% absorber bolted to it. The amplitude of the driven mass, in units of the deflection the same force would cause if applied slowly, against drive frequency in units of the machine's own natural frequency. Dashed: the machine alone, with its single resonance. Solid: the same machine with a second mass and spring attached, weighing 10 per cent of it and tuned to the same frequency. At that frequency the driven mass does not move at all — the amplitude is zero rather than small, and the absorber is moving 10.0 static deflections to make it so. What the device costs is the two new resonances it creates, at 0.854 and 1.171 times the original frequency, which are infinite in this undamped calculation and straddle the frequency the machine was protected at.

The mass that makes another stand still

Bolt a small mass on a spring to a machine that is shaking itself apart, tune it to the frequency that is doing the damage, and the machine stops moving. Not moves less — stops, exactly, at that one frequency. The price is two new resonances either side of it, and the whole device is a bet that the drive stays where it was put.

waves · Resonance Two levels that refuse to cross. On the left, the energies of a two-state system as one state is swept past the other, for couplings of 0, 0.05, 0.15, 0.35 electronvolts. With no coupling at all the two levels cross, which is the dotted pair. With any coupling whatever they do not: the eigenvalues are plus and minus half the square root of the squared detuning plus four times the squared coupling, so the closest approach is exactly twice the coupling — measured off the drawn curves as 0.000, 0.100, 0.300, 0.700 eV against 2V, agreeing to 0.0e+0. Far from the crossing the curves rejoin the uncoupled lines, so the repulsion is local: it is largest where the two states are degenerate and dies away as the square of the coupling divided by the detuning. On the right is what makes this more than a picture of two hyperbolas — the character of the upper state, meaning how much of the first basis state is in it. It swaps completely across a region whose width is set by the coupling, so the level that arrives as one thing leaves as the other. That exchange of identity is why the phrase avoided crossing is misleading: nothing is avoided, the labels are.

The crossing that never happens

Two energy levels swept past one another do not cross. Any coupling between them, however small, opens a gap of exactly twice the coupling — and the two levels exchange their identities across it, so the state that arrives as one thing leaves as the other while the labels are what avoided anything.

quantum · Bands The light follows one supermode through the crossing. Inside a tapered coupler 800 µm long at 1550 nm. Above: the fraction of the light in the first guide along the coupler, from integrating the coupled-mode equations, and dashed, the share of the first guide in the local supermode the light was launched into. They agree to within 8.3 percentage points along the whole length: the light does not beat between the guides but follows the supermode as that supermode changes from being in the first guide to being in the second. Below: the two supermodes' propagation constants relative to their average, which approach each other and repel across a gap of twice the coupling, 0.084 per micrometre, at the point where the guides are equally wide; dashed, the two single guides' constants, which cross.

The coupler that does not care about the colour

Two identical guides side by side swap their light back and forth, and a coupler cut to the length of one swap works perfectly at one wavelength and badly at every other. Make the guides unequal, and sweep the inequality from one sign to the other along their length, and the light no longer swaps — it follows a single mode of the pair as that mode moves from one guide to the other. The transfer is then nearly complete across hundreds of nanometres of wavelength and immune to the widths being made wrong, and it costs length.

waves · Guided waves

Named alongside it

The objects these essays reach for when they reach for this one.

Landau zenerAdiabatic followingAdiabatic theoremAmplitudeBand gapCoupled mode theoryCoupled oscillatorsCouplingDampingDiabaticDirectional couplerDissipation

All concepts