If we define ψ ( t ) = 1 √ $_{σ}$√$_{2}$$_{π}$ e t 2 4 σ 2 $_{, such that 2}$√$_{2 ln 2}$$_{σ}$ is the pulse width at FWHM, we get The resulting visibility is If we rewrite this with the dimensionless ratio Δ = τ /σ and if we define a probability distribution for this ratio we can find the average visibility We also perform the same calculations using FWHMs instead of variances. We use the variables where ψ ( t ) is a gaussian wavepacket and ψ ( ω ) its Fourier transform. The output state will be If photons come in from either side separated by a time delay τ , the corresponding state and its projection in time will be For τ much larger than the wavepacket length the last two terms are completely independent; for τ = 0 they cancel each other and the photons bunch. The amplitude corresponding to photons exiting the beamsplitter from different outputs | Ψ$_{do}$ 〉 is The final result takes the same form in new units: such that the total probability of photons not bunching is ρ (Δ) = 1 $_{σ$_{Δ}$}$√$_{2}$$_{π}$ $_{e}$− Δ 2 2 σ 2 Δ , (21) V = ∫ ρ (Δ) V (Δ) d Δ (22) 1 = √ 1+ 1 $_{2}$σ 2 Δ . (23) $_{Δ}$′ = $_{2}$√$_{2 ln 2}$$_{τ}$ σ (24) r$_{j}$ = σ$_{Δ}$ $_{2}$√$_{2 ln 2}$ . (25) V = 1 √ 1+ 1 $_{2}$r 2 j . (26) P = ∫ ∫ [ ψ ( t$_{a}$ ) ψ ( t$_{b}$ − τ ) − ψ ( t$_{a}$ − τ ) ψ ( t$_{b}$ )] $^{2}$dt$_{a}$ dt$_{b}$. (16) 〈 t$_{a}$ |〈 t$_{b}$ || Ψ$_{do}$ 〉 = ψ ( t$_{a}$ ) ψ ( t$_{b}$ − τ ) − ψ ( t$_{a}$ − τ ) ψ ( t$_{b}$ ) , (15) | Ψ$_{i}$ 〉 = $_{a}$† $_{0}$b † τ | 0 〉 (5) = ∫ ψ ( ω$_{a}$ ) ψ ( ω$_{b}$ ) e iω$_{b}$τ (6) ·| ω$_{a}$ 〉| ω$_{b}$ 〉 dω$_{a}$dω$_{b}$ (7) s.t. 〈 t$_{a}$ |〈 t$_{b}$ || Ψ$_{i}$ 〉 = ψ ( t$_{a}$ ) ψ ( t$_{b}$ − τ ) (8) V (Δ) = e − Δ 2 4 , (20) V ( τ ) = P $^{(}$∞$^{)}$− P ( τ ) P $^{(}$∞) (18) = $_{e}$− τ 2 4 σ 2 . (19) P ( τ ) = 2 · (1 − $_{e}$− τ 2 4 σ 2 ) . (17) a † → 1 √$_{2}$ ( ic † + d † ) (3) b † → 1 √$_{2}$ ( c † + id † ) . (4) 7 ·| ω$_{a}$ 〉| ω$_{b}$ 〉 dω$_{a}$dω$_{b}$ ] . (14) + ∫ ψ ( ω$_{a}$ ) ψ ( ω$_{b}$ )( e iω$_{a}$τ − e iω$_{b}$$^{τ}$) (13) + i ψ $^{2}$( ω$_{b}$ ) e ω$_{b}$ dω$_{b}$ 0 (12) = 1 $_{2}$[ i ∫ ψ $^{2}$( ω$_{a}$ ) e iω$_{a}$$^{τ}$| ω$_{a}$ 〉 dω$_{a}$ | 0 〉 (11) iω$_{b}$$^{τ}$| | Ψ$_{o}$ 〉 = 1 $_{2}$[ $_{ic}$† $_{0}$c † τ + $_{id}$† $_{0}$d † τ (9) + $_{c}$† $_{0}$d † τ − c † $_{$_{τ}$d}$† $_{0}$] | 0 〉 (10)