An increase in fluorescence caused by reducing a quenching interaction. Dye dilution can reduce Förster resonance energy transfer; the signal alone need not identify the process that diluted the dyes.

For purely dynamic quenching of a single emitting population, ${\Phi }_0/{\Phi }_Q={\tau }_0/{\tau }_Q=1+k_Q{\tau }_0[Q]$. The slope measures $k_Q{\tau }_0$, so determining a bimolecular quenching constant requires the unquenched lifetime or equivalent independent rate information.

Quenching by an excited-state encounter that introduces an additional decay rate $k_Q[Q]$. It reduces both fluorescence quantum yield and lifetime in the simple single-population model.

Loss of fluorescent population through nonemitting ground-state association. In a simple static-only case the surviving unbound emitters keep their lifetime, distinguishing it from dynamic quenching; mixtures and association-dependent spectra complicate that test.

The difference between electron and hole quasi-Fermi levels under nonequilibrium conditions. For ideal selective contacts their terminal separation sets $qV$; poor contacts can lose part of the internal splitting.

An admissible wavefunction chosen from a tractable family for the variational method. Its Hamiltonian expectation gives an upper bound on the ground-state energy; parameters can be adjusted to lower that bound.

Unrestricted minimization of a Rayleigh quotient seeks the ground state. To bound the $k$th excited energy from above, restrict normalized trial wavefunctions to the orthogonal complement of the exact lower-energy eigenspaces. In an exact orthonormal eigenbasis, all remaining expansion coefficients multiply energies at least $E_k$, so their weighted average is at least $E_k$. A known symmetry sector can supply some of these exclusions automatically. Orthogonality only to inaccurate approximate lower states does not automatically retain the same rigorous excited-state upper bound.

For a pure input target $|\psi ⟩$ and normalized output density operator ${\rho }_{\mathrm{out}}$, the displayed squared quantum fidelity measures how well a teleportation channel reproduces that target. A mean value additionally specifies the input ensemble and whether it is conditioned on success. Detection efficiency, conditional fidelity, heralding and overall success probability are distinct experimental quantities.

Dik Bouwmeester and colleagues demonstrated selected-outcome photon-polarization quantum teleportation. An ultraviolet pump traversed a nonlinear crystal twice: one passage supplied an entangled pair, the other the input photon and its trigger partner. The input and one resource photon interfered at a beam splitter; output-detector coincidences selected their antisymmetric Bell state. Narrow filters made their coherence time exceed the pump duration, preserving temporal indistinguishability.

The reported pump pulses were $200\mathrm{fs}$ long at $76\mathrm{MHz}$; $788\mathrm{nm}$ photons filtered to $4\mathrm{nm}$ bandwidth had a quoted $520\mathrm{fs}$ coherence time. A polarization analyzer measured the remote output. Trigger-conditioned fourfold coincidences excluded a double-pair background without subtraction. The selected Bell result had ideal probability $1/4$; this demonstration did not distinguish and actively correct all four outcomes.^[ref]

Labels an angular momentum eigenstate through the eigenvalue ${\hslash }^{2}j(j+1)$ of the squared angular momentum operator. The allowed projections are $m\hslash$, with $m=-j,\dots ,j$. Orbital angular momentum has integer $j$; spin can also have half-integer $j$.

The $p$-th percentile is a quantile corresponding to cumulative probability $p/100$.

Minimizing $({\omega }_0^2-{\omega }^2{)}^2+4{\beta }^2{\omega }^2$ gives a displacement maximum at $\sqrt{{\omega }_0^2-2{\beta }^2}$ only when $\beta <{\omega }_0/\sqrt{2}$. An underdamped oscillator with $1/2<Q\le 1/\sqrt{2}$ consequently has no nonzero-frequency displacement peak. The absorbed-power response instead peaks at ${\omega }_0$, with exact half-power bandwidth $2\beta$ for this model. Comparing bandwidth definitions requires naming the measured response.

A convex quadrilateral is a simple quadrilateral whose filled region is a convex set. In the nondegenerate case, each interior angle is strictly less than $\pi$, and its two diagonals of a polygon meet inside the region. A rectangle is a convex quadrilateral; a dart-shaped concave quadrilateral is not. If the diagonal lengths are $p,q$ and the angle between them is $\theta$, splitting about their intersection gives area $pq\sin \theta /2$. This is at most $pq/2$, with equality for perpendicular diagonals.

Mixing a mode with independent vacuum on a beam splitter of intensity transmission $\eta$ gives the displayed optical quadrature variance when vacuum variance is $1/4$. It approaches vacuum noise as transmission falls. Phase noise instead mixes squeezed and anti-squeezed axes, and cannot in general be represented by the same efficiency alone.

The formulas $a=m^2-n^2$, $b=2mn$, $c=m^2+n^2$ give a Pythagorean triple for integers $m>n>0$, because $(m^2-n^2{)}^2+(2mn{)}^2=(m^2+n^2{)}^2$. They give a primitive Pythagorean triple exactly when $m,n$ are coprime and have opposite integer parity. Every primitive Pythagorean triple arises this way after possibly exchanging the legs; scaling gives the nonprimitive triples.

A Pythagorean triple $(a,b,c)$ whose coordinates have greatest common divisor 1. Its two legs have opposite integer parity, and every Pythagorean triple is a unique positive integer multiple of a primitive Pythagorean triple up to the ordering of its legs.

The Berggren tree is a rooted tree of primitive Pythagorean triples, with three children at every node. Fix the convention $a$ odd, $b$ even and $c>0$. Starting from the column vector $(3,4,5{)}^{\mathsf{T}}$, multiply by any ofThe first children are $(5,12,13)$, $(21,20,29)$ and $(15,8,17)$. These unimodular matrices preserve the quadratic form $a^2+b^2-c^2$, integer parity and greatest common divisor. Every primitive Pythagorean triple occurs exactly once. F. J. M. Barning's report gives this matrix-generation theorem with the even leg first.^[ref]

An infinite descent argument explains completeness. Except at the root, the unique parent isIts hypotenuse is positive and smaller than $c$, since $c<a+b\le \sqrt{2}c$. The first two unmodified entries cannot both be negative; their signs select the inverse of $A$, $B$ or $C$. The odd entry cannot vanish; the even entry vanishes only at $(3,4,5)$. Repeated descent therefore reaches that root. Multiplying all three coordinates by a positive integer supplies every nonprimitive Pythagorean triple.

The temperature dependence of electric polarization at fixed strain.