3.3 How to calculate spectral densities of quantum noise in linear optical measurement?

In this section, we give a brief introduction to calculation of the power spectral densities of quantum noise one usually encounters in linear optical measurement. In optomechanical sensors, as we have discussed earlier, the outgoing light carries the information about the measured quantity (e.g., the displacement due to GW tidal forces) in its phase and (sometimes) amplitude quadratures. The general transformation from the input light characterized by a vector of quadrature amplitudes ˆa (Ω) = {ˆac(Ω ),ˆas(Ω )}T to the readout quantity of a meter is linear and can be written in spectral form as:
ˆY (Ω) = 𝒴𝒴𝒴 †(Ω)ˆa(Ω ) + G(Ω ) = 𝒴∗c(Ω )ˆac(Ω) + 𝒴s∗(Ω )ˆas(Ω) + G (Ω), (85 )
where G (Ω ) is the spectrum of the measured quantity, 𝒴c,s(Ω ) are some complex-valued functions of Ω that characterize how the light is transformed by the device. Quantum noise is represented by the terms of the above expression not dependent on the measured quantity G, i.e.,
( ) ( ) NˆY (Ω ) = 𝒴𝒴𝒴†(Ω )aˆ(Ω) = 𝒴∗(Ω ) 𝒴 ∗(Ω) ⋅ ˆac(Ω) . (86 ) c s ˆas(Ω )

The measure of quantum noise is the power spectral density SY (Ω) that is defined by the following expression:

† † 2πSY (Ω )δ(Ω − Ω ′) = ⟨ψ|NˆY (Ω ) ∘ ˆN Y(Ω′)|ψ ⟩ = ⟨ ˆNY (Ω) ∘NˆY (Ω′)⟩. (87 )
Here |ψ ⟩ is the quantum state of the light wave.

In our review, we will encounter two types of quantum states that we have described above, i.e., vacuum |vac⟩ and squeezed vacuum |sqz0(r,ϕ)⟩ states. Let us show how to calculate the power (double-sided) spectral density of a generic quantity ˆY (Ω) in a vacuum state. To do so, one should substitute Eq. (86View Equation) into Eq. (87View Equation) and obtain that:

SvaYc(Ω ) = 𝒴𝒴𝒴 †(Ω)⟨ˆa (Ω) ∘ ˆa †(Ω )⟩vac𝒴𝒴𝒴 (Ω) = 𝒴𝒴𝒴 †(Ω)𝕊vac(Ω)𝒴𝒴𝒴 (Ω) 𝒴𝒴𝒴†(Ω )𝒴𝒴𝒴(Ω ) |𝒴 (Ω )|2 + |𝒴 (Ω)|2 = -----------= --c----------s-----, (88 ) 2 2
where we used the definition of the power spectral density matrix of light in a vacuum state (65View Equation)7. Similarly, one can calculate the spectral density of quantum noise if the light is in a squeezed state |sqz (r,ϕ)⟩ 0, utilizing the definition of the squeezed state density matrix given in Eq. (83View Equation):
sqz † † † 1 † S Y (Ω) = 𝒴𝒴𝒴 (Ω)⟨ˆa (Ω ) ∘ ˆa (Ω )⟩sqz𝒴𝒴𝒴 (Ω ) = 𝒴𝒴𝒴 (Ω )𝕊sqz(Ω )𝒴𝒴𝒴 (Ω ) = -𝒴𝒴𝒴 (Ω)ℙ &#x0 2 2 2 = |𝒴c-(Ω-)|-(cosh2r + sinh2r cos2 ϕ) + |𝒴s-(Ω)|-(cosh2r − sinh 2rcos 2ϕ) 2 2 − Re [𝒴c(Ω )𝒴 ∗s(Ω )]sinh 2rsin2 ϕ. (89 )

It might also be necessary to calculate also cross-correlation spectral density SY Z(Ω) of ˆY(Ω ) with some other quantity ˆZ(Ω ) with quantum noise defined as:

ˆ † ∗ ∗ NZ (Ω) = 𝒵𝒵𝒵 (Ω )ˆa (Ω) = 𝒵 c(Ω)ˆac(Ω ) + 𝒵 s(Ω )ˆas(Ω ).

Using the definition of cross-spectral density SYZ(Ω ) similar to (87View Equation):

2πSY Z (Ω)δ(Ω − Ω ′) = ⟨ψ | ˆNY (Ω) ∘Nˆ†Z(Ω ′)|ψ⟩ = ⟨NˆY (Ω ) ∘ Nˆ†Z(Ω′)⟩, (90 )
one can get the following expressions for spectral densities in both cases of the vacuum state:
SvYacZ(Ω) = 𝒴𝒴𝒴 †(Ω)⟨ˆa (Ω ) ∘ ˆa †(Ω )⟩vac𝒵𝒵𝒵 (Ω) = 𝒴𝒴𝒴 †(Ω)𝕊vac(Ω)𝒵𝒵𝒵 (Ω) 𝒴𝒴𝒴†(Ω )𝒵𝒵𝒵 (Ω ) 𝒴 ∗(Ω)𝒵 (Ω ) + 𝒴∗(Ω )𝒵 (Ω ) = ------------= --c-----c-------s-----s----, (91 ) 2 2
and the squeezed state:
sqz † † † SYZ (Ω) = 𝒴𝒴𝒴 (Ω )⟨ˆa(Ω ) ∘ ˆa (Ω )⟩sqz𝒵𝒵𝒵(Ω ) = 𝒴𝒴𝒴 (Ω )𝕊sqz(Ω )𝒵𝒵𝒵(Ω ) 1- † = 2 𝒴𝒴𝒴 (Ω )ℙ[− ϕ ]𝕊sqz[2r, 0]ℙ [ϕ ]𝒵𝒵𝒵 (Ω ) 𝒴 ∗(Ω)𝒵 (Ω) 𝒴 ∗(Ω)𝒵 (Ω) = --c-----c---(cosh 2r + sinh 2r cos2ϕ ) +--s-----s---(cosh 2r − sinh 2r cos2ϕ ) 2 2 𝒴-∗s(Ω)𝒵c-(Ω-) +-𝒴∗c(Ω-)𝒵s(Ω-) − 2 sinh2r sin2ϕ. (92 )

Note that since the observables Yˆ(t) and Zˆ(t) that one calculates spectral densities for are Hermitian, it is compulsory, as is well known, for any operator to represent a physical quantity, then the following relation holds for their spectral coefficients 𝒴c,s(Ω ) and 𝒵c,s(Ω):

∗ ∗ 𝒴 c,s(Ω) = 𝒴c,s(− Ω ), and 𝒵 c,s(Ω ) = 𝒵c,s(− Ω). (93 )
This leads to an interesting observation that the coefficients 𝒴c,s(Ω) and 𝒵c,s(Ω ) should be real-valued functions of variable s = iΩ.

Now we can make further generalizations and consider multiple light and vacuum fields comprising the quantity of interest:

N ˆ ˆ ∑ † Y (Ω) → NY (Ω ) = 𝒴𝒴𝒴i(Ω )aˆi (Ω ), (94 ) i=1
where ˆai(Ω) stand for quadrature amplitude vectors of N independent electromagnetic fields, and 𝒴𝒴𝒴 †(Ω) i are the corresponding complex-valued coefficient functions indicating how these fields are transmitted to the output. In reality, the readout observable of a GW detector is always a combination of the input light field and vacuum fields that mix into the output optical train as a result of optical loss of various origin. This statement can be exemplified by a single lossy mirror I/O-relations given by Eq. (34View Equation) of Section 2.2.4.

Thus, to calculate the spectral density for such an observable, one needs to know the initial state of all light fields under consideration. Since we assume aˆi(Ω ) independent from each other, the initial state will simply be a direct product of the initial states for each of the fields:

⊗N |Ψ âŸ© = |ψi⟩, i=1

and the formula for the corresponding power (double-sided) spectral density reads:

∑N ∑N ∑N SY (Ω ) = 𝒴𝒴𝒴†(Ω )⟨ψi|ˆai(Ω ) ∘ ˆa†(Ω)|ψi⟩𝒴𝒴𝒴i(Ω ) = 𝒴𝒴𝒴 †(Ω)𝕊i(Ω )𝒴𝒴𝒴i(Ω) = S i=1 i i i=1 i i=1 i
with 𝕊i(Ω ) standing for the i-th input field spectral density matrix. Hence, the total spectral density is just a sum of spectral densities of each of the fields. The cross-spectral density for two observables Yˆ(Ω ) and ˆZ (Ω) can be built by analogy and we leave this task to the reader.

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