Quantum Physics Vol. 1 Pixel Sticker Sheet

12 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Quantum Physics Vol. 1. Every character below has its own sticker page with the full physics panel.

Berryphase

Carry a quantum system slowly round a closed loop of its settings and it returns to its starting state, apart from a phase. Michael Berry showed in 1984 that part of this phase depends only on the geometry of the loop, not on how long it takes (Proc. R. Soc. A 392, 45). For a spin-½ in the state aligned with a magnetic field (m = +½), with the field direction swept slowly round a closed loop, the geometric phase is minus half the solid angle that loop encloses: γ = −mΩ = −Ω/2. The anti-aligned state (m = −½) picks up +Ω/2.

γ = −Ω/2

γ: geometric (Berry) phase of the aligned state (rad); Ω: solid angle enclosed by the loop traced by the field direction, seen from the origin (sr); m: spin projection along the field in units of ħ (dimensionless; +½ aligned, −½ anti-aligned)

Dehaasa

Magnetise an iron rod and it starts to turn. In 1915 Albert Einstein and Wander de Haas hung an iron cylinder on a fine fibre inside a coil (Verh. Dtsch. Phys. Ges. 17, 152). Reversing the magnetisation flips the tiny magnets inside the iron, and their angular momentum flips with them. Angular momentum is conserved, so the whole rod twists the other way. The ratio of magnetic moment to angular momentum sets the g-factor.

L = 2mμ/(ge)

L: angular momentum of the carriers (J·s); μ: magnetic moment (A·m²); m: electron mass (kg); e: elementary charge (C); g: gyromagnetic g-factor

Homestaka

Raymond Davis Jr.'s experiment in the Homestake gold mine, South Dakota, 1478 m underground, used a tank of about 615 tonnes of perchloroethylene (tetrachloroethylene, C₂Cl₄). A solar electron neutrino can turn a chlorine-37 nucleus into argon-37 (threshold 0.814 MeV), and the few argon atoms were extracted and counted. From 1970 to 1994 it measured about a third of the rate predicted by John Bahcall's standard solar model.

measured 2.56 ± 0.16 ± 0.16 SNU vs predicted 7.6⁺¹·³_−1.1 SNU

SNU: solar neutrino unit: 10⁻³⁶ captures per target atom per second; 2.56: final Homestake result (Cleveland et al., ApJ 496, 505, 1998), statistical and systematic errors; 7.6: chlorine prediction of the BP2000 standard solar model (Bahcall, Pinsonneault and Basu, 2001)

Helicita

In 1958 Maurice Goldhaber, Lee Grodzins and Andrew Sunyar measured the handedness of the neutrino (“Helicity of Neutrinos”, Phys. Rev. 109, 1015). Europium-152m captures an electron and emits a neutrino; the daughter nucleus recoils and then emits a gamma ray. Selecting gammas emitted against the recoil and measuring their circular polarisation with magnetised iron tied the neutrino's spin to its motion. Neutrinos turned out left-handed: spin opposite to motion.

h = −1

h: helicity, the spin component along the direction of motion, in units of the maximum

Expulsieur

A superconductor expels magnetic field from its interior, which a merely perfect conductor would not do. The field only penetrates a thin layer, dying away exponentially over the London penetration depth.

B_inside = 0

B_(inside): magnetic field in the bulk of the superconductor (applied field below the critical field) (T); B(x): field at depth x below the surface (T); B₀: field at the surface (T); x: depth below the surface (m); λ_L: London penetration depth (m); m: electron mass (kg)

Landaura

In a narrow ballistic conductor, such as a quantum point contact in a two-dimensional electron gas, electrons travel in a few transverse modes, like lanes on a road. Each fully open lane carries one conductance quantum, counting both spin states. Widening the constriction opens one lane at a time, so the conductance rises in a staircase. This was first seen in 1988 in GaAs point contacts (van Wees et al.; Wharam et al.).

G = (2e²/h) Σ_n T_n, G₀ = 2e²/h ≈ 7.748 091 729 × 10⁻⁵ S

G: conductance (siemens, S); e: elementary charge, exact in the 2019 SI; h: Planck constant, exact in the 2019 SI; T_n: transmission probability of mode n (0 to 1); 2: spin degeneracy, two spin states per lane; G₀: conductance quantum, exact (CODATA 2022)

Squidra

A dc SQUID is a superconducting loop broken by two Josephson junctions. The current waves through the two junctions interfere, so the voltage across the device swings periodically as the magnetic flux through the loop changes. One full period is one flux quantum. R. C. Jaklevic, J. Lambe, A. H. Silver and J. E. Mercereau first showed this interference (“Quantum Interference Effects in Josephson Tunneling”, Phys. Rev. Lett. 12, 159, 1964).

Φ₀ = h/2e ≈ 2.067833848 × 10⁻¹⁵ Wb

Φ₀: magnetic flux quantum (weber, Wb); h: Planck constant; 2e: charge of a Cooper pair

Stairohm

In a thin conductor at low temperature and strong magnetic field, the Hall resistance locks onto exact plateaus at h/e² divided by a whole number. These plateaus are so precise and reproducible that they now serve as the standard of electrical resistance.

R_xy = h/(νe²) = R_K/ν

R_(xy): Hall resistance: transverse voltage divided by current (Ω); h: Planck constant (exact in the SI since 2019) (J·s); e: elementary charge (exact in the SI since 2019) (C); ν: filling factor: an integer on each plateau; R_K: von Klitzing constant, h/e² (Ω); Rₓₓ: longitudinal resistance along the sample (Ω)

Tesora

A transition-edge sensor is a thin superconducting film held right at the edge of its superconducting transition, where its resistance rises from zero to normal within a tiny temperature range. Absorb a photon or particle and the film warms very slightly, its resistance jumps, and the current through it changes measurably. K. D. Irwin showed how to hold it stable on the edge with voltage bias and electrothermal feedback (Appl. Phys. Lett. 66, 1998, 1995). The steepness is measured by a logarithmic sensitivity.

α = d ln R / d ln T

α: logarithmic temperature sensitivity, dimensionless; R: film resistance (Ω); T: film temperature (K)

Echoa

In a magnetic field, nuclear spins precess at slightly different rates because the field is not perfectly uniform, so after a pulse knocks them over they fan out and the signal fades. Erwin Hahn found in 1950 (“Spin Echoes”, Phys. Rev. 80, 580) that a second pulse makes them run back together, and a signal reappears: the spin echo. In the version by H. Y. Carr and E. M. Purcell (Phys. Rev. 94, 630, 1954), a π/2 pulse is followed after a time τ by a π pulse that flips the fan over. The fast spins are now behind the slow ones, catch up, and all line up again at 2τ.

t_echo = 2τ

τ: time between the first and second pulse (s); t_echo: time of the echo after the first pulse (s)

Purcella

An excited atom's spontaneous emission rate is not fixed: it depends on the electromagnetic modes around it. Put the atom in a small, high-quality resonant cavity tuned to its transition and it can emit much faster than in free space. E. M. Purcell pointed this out for nuclear magnetic moments at radio frequencies in a short abstract (“Spontaneous Emission Probabilities at Radio Frequencies”, Phys. Rev. 69, 681, 1946). The enhancement is the Purcell factor.

F = 3Q(λ/n)³ / (4π²V)

F: Purcell factor, emission rate in the cavity over the free-space rate; Q: quality factor of the cavity; λ: free-space wavelength of the transition (m); n: refractive index of the cavity medium; V: mode volume of the cavity (m³)

Barnetta

Spin an unmagnetised iron rod and it becomes magnetised. Samuel Barnett showed this in 1915 (Phys. Rev. 6, 239): rotation acts on the electrons' angular momentum like a magnetic field, so their spins tend to line up. It is the reverse of the Einstein–de Haas effect, and gave g ≈ 2 for iron.

M ∝ Ω

M: magnetisation (A/m); Ω: angular speed of rotation (rad/s)