A.2 Proof of Eq. (376), (377) and (378)

In the lossless case [see Eq. (380View Equation)], the noises (double-sided) power spectral densities (376View Equation), (377View Equation) and (378View Equation) are equal to
---ℏ--- † Sx (Ω) = 4M Jγ ψ 𝕊sqz[2r,0]ψ, † SF (Ω) = ℏM Jγϕ 𝕊sqz[2r,0]ϕ, ℏ- † SxF (Ω) = 2 ψ 𝕊sqz[2r,0]ϕ, (547 )
were
[ ] ψc -ℙ(𝜃)ℝ-†H-- ψ = ψs = (0 1)𝕃 †H , (548 ) [ ] [ ] ϕ = ϕc = ℙ(𝜃)𝕃 † 1 , (549 ) ϕs 0
and
ℝ (Ω) = 2γ 𝕃(Ω ) − 1. (550 )
Therefore,
2 S S − |S |2 = ℏ--[ψ †ð•Š [2r,0]ψ ⋅ ϕ †ð•Š [2r,0]ϕ − |ψ †ð•Š [2r,0]ϕ |2] x F xF 4 sqz sqz sqz ℏ2 ( 2) ℏ2 ( † † † 2) = --- |ψcϕs − ψsϕc| = --- ψ ψ ⋅ ϕ ϕ − |ψ ϕ | . (551 ) 4 4
Note that the squeezing factor r has gone.

Taking into account that

ℝ ℝ† = 𝕀, ℝ𝕃 † = 𝕃 (552 )
we obtain that
[ ] [ ]T [ ] HT 𝕃 1 † ----1----- † 1 † 1 † -----[-0]- ψ ψ = || [1 ]||2, ϕ ϕ = 0 𝕃𝕃 0 , ψ ϕ = 0 (553 ) ||H 𝕃 || HT 𝕃 1 0
and
[ ]T [ ] || [ ]||2 1 𝕃 𝕃† 1 − |HT 𝕃 1 | 2 ℏ2--0---------0----|------0-|-- SxSF − |SxF| = 4 || [ ]||2 . (554 ) |HT 𝕃 0 | | 1 |
It is easy to show by direct calculation that
| [ ]|2 | [ ]|2 [ ]T [ ] || T 1 || || T 0 || 1 † 1 |H 𝕃 0 | + |H 𝕃 1 | = 0 𝕃𝕃 0 , (555 )
which proves Eq. (379View Equation).
  Go to previous page Go up Go to next page