For a system operator c c c
H I = i ℏ ∫ d ω κ 2 π ( b ω † c − c † b ω ) . H_I=i\hbar\int d\omega\,\sqrt{\frac{\kappa}{2\pi}}
\left(b_\omega^\dagger c-c^\dagger b_\omega\right). H I = i ℏ ∫ d ω 2 π κ ( b ω † c − c † b ω ) . The notes warn to keep sign conventions consistent. With the convention above, an arbitrary system operator a a a
a ˙ = − i ℏ [ a , H s ( t ) ] − [ a , c † ] ( κ 2 c + κ b i n ) + ( κ 2 c † + κ b i n † ) [ a , c ] . \dot a
=-\frac{i}{\hbar}[a,H_s(t)]
-[a,c^\dagger]\left(\frac{\kappa}{2}c+\sqrt{\kappa}\,b_{\rm in}\right)
+\left(\frac{\kappa}{2}c^\dagger+\sqrt{\kappa}\,b_{\rm in}^\dagger\right)[a,c]. a ˙ = − ℏ i [ a , H s ( t )] − [ a , c † ] ( 2 κ c + κ b in ) + ( 2 κ c † + κ b in † ) [ a , c ] . For c = a c=a c = a
a ˙ = − i ℏ [ a , H s ( t ) ] − κ 2 a − κ b i n . \dot a=-\frac{i}{\hbar}[a,H_s(t)]-\frac{\kappa}{2}a-\sqrt{\kappa}\,b_{\rm in}. a ˙ = − ℏ i [ a , H s ( t )] − 2 κ a − κ b in . The input-output relation is
b o u t − b i n = κ c ( t ) . b_{\rm out}-b_{\rm in}=\sqrt{\kappa}\,c(t). b out − b in = κ c ( t ) . For a bosonic mode,
[ a , a † ] = 1. [a,a^\dagger]=1. [ a , a † ] = 1. Dimensionless quadratures:
X = a + a † 2 , P = a − a † 2 i , [ X , P ] = i 2 . X=\frac{a+a^\dagger}{2},\qquad
P=\frac{a-a^\dagger}{2i},\qquad
[X,P]=\frac{i}{2}. X = 2 a + a † , P = 2 i a − a † , [ X , P ] = 2 i . A thermal state is diagonal in the Fock basis:
ρ = N exp ( − β ℏ ω a † a ) , \rho=\mathcal N\exp(-\beta\hbar\omega a^\dagger a), ρ = N exp ( − β ℏ ω a † a ) , where N \mathcal N N ρ \rho ρ n ˉ \bar n n ˉ
⟨ a n ⟩ = ⟨ ( a † ) n ⟩ = 0 , ⟨ a † a ⟩ = n ˉ , ⟨ a a † ⟩ = n ˉ + 1. \langle a^n\rangle=\langle(a^\dagger)^n\rangle=0,\qquad
\langle a^\dagger a\rangle=\bar n,\qquad
\langle aa^\dagger\rangle=\bar n+1. ⟨ a n ⟩ = ⟨( a † ) n ⟩ = 0 , ⟨ a † a ⟩ = n ˉ , ⟨ a a † ⟩ = n ˉ + 1. The quadrature variances are
σ X 2 = ⟨ X 2 ⟩ = 1 4 ⟨ { a , a † } ⟩ = 1 4 ( 2 n ˉ + 1 ) , σ P 2 = σ X 2 . \sigma_X^2=\langle X^2\rangle
=\frac14\langle\{a,a^\dagger\}\rangle
=\frac14(2\bar n+1),
\qquad
\sigma_P^2=\sigma_X^2. σ X 2 = ⟨ X 2 ⟩ = 4 1 ⟨{ a , a † }⟩ = 4 1 ( 2 n ˉ + 1 ) , σ P 2 = σ X 2 . Using the physical convention
X ′ = ℏ 2 ( a + a † ) , P ′ = ℏ 2 a − a † i , [ X ′ , P ′ ] = i ℏ , X'=\sqrt{\frac{\hbar}{2}}(a+a^\dagger),\qquad
P'=\sqrt{\frac{\hbar}{2}}\frac{a-a^\dagger}{i},
\qquad [X',P']=i\hbar, X ′ = 2 ℏ ( a + a † ) , P ′ = 2 ℏ i a − a † , [ X ′ , P ′ ] = i ℏ , the variances become
σ X ′ 2 = σ P ′ 2 = ℏ 2 ( 2 n ˉ + 1 ) . \sigma_{X'}^2=\sigma_{P'}^2
=\frac{\hbar}{2}(2\bar n+1). σ X ′ 2 = σ P ′ 2 = 2 ℏ ( 2 n ˉ + 1 ) . The vacuum value is therefore ℏ / 2 \hbar/2 ℏ/2
For thermal input-output fields in the white-noise limit,
[ a i n ( t ) , a i n † ( t ′ ) ] = δ ( t − t ′ ) , [a_{\rm in}(t),a_{\rm in}^\dagger(t')]=\delta(t-t'), [ a in ( t ) , a in † ( t ′ )] = δ ( t − t ′ ) , ⟨ a i n † ( t ) a i n ( t ′ ) ⟩ = n ˉ i n δ ( t − t ′ ) , \langle a_{\rm in}^\dagger(t)a_{\rm in}(t')\rangle
=\bar n_{\rm in}\delta(t-t'), ⟨ a in † ( t ) a in ( t ′ )⟩ = n ˉ in δ ( t − t ′ ) , and the opposite ordering is
⟨ a i n ( t ) a i n † ( t ′ ) ⟩ = ( n ˉ i n + 1 ) δ ( t − t ′ ) . \langle a_{\rm in}(t)a_{\rm in}^\dagger(t')\rangle
=(\bar n_{\rm in}+1)\delta(t-t'). ⟨ a in ( t ) a in † ( t ′ )⟩ = ( n ˉ in + 1 ) δ ( t − t ′ ) . Reference
Gardiner and Collett, Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation .
Phys. Rev. A 31, 3761 (1985) .
Gardiner, Parkins, and Collett, Input and output in damped quantum systems. II. Methods in non-white-noise situations and application to inhibition of atomic phase decays .
J. Opt. Soc. Am. B 4, 1683 (1987) .