Effective shot noise and radiation-pressure noise

2.3 Features of noise spectral density in SR interferometers

2.3.2 Effective shot noise and radiation-pressure noise

black holes and/or neutron stars is given by µS

N

¶2

= 4 Z _+∞

0

|h(f)|²

Sh(f) df . (2.37)

Using the Newtonian quadrupole approximation, for which the waveform’s Fourier transform is

|h(f)|²∝f^−7/3, and introducing in the above integral a lower cutoff due to seismic noise at Ωs= 0.1γ (fs'10 Hz), we get for the parameters used in Fig. 2.2:

(S/N)1

(S/N)conv '1.83, (S/N)2

(S/N)conv '1.98, (2.38)

where (S/N)1, (S/N)2 and (S/N)conv use for the noise spectral density either that of the first quadratureb^C₁ or the second quadratureb^C₂ or the conventional interferometer, respectively. A more thorough analysis of signal-to-noise ratio for inspiraling binaries inevitably requires the specification of the readout scheme. We assume a homodyne readout scheme here, while the heterodyne scheme will be discussed in Chapter 4.

commutation relations (Eq. (2.19) in Chapter 3)

[F(Ω),F^†(Ω⁰)] = 0 = [Z(Ω),Z^†(Ω⁰)], [Z(Ω),F^†(Ω⁰)] =−2πi~δ(Ω−Ω⁰). (2.41) We shall refer to Z andF as the effectiveshot noise andeffective radiation-pressure force, respec- tively, because we will show in Chapter 3 that for a SR interferometer the real back-action force acting on the test masses is not proportional to the effective radiation-pressure noise, but instead is a combination of the two effective observablesZandF. When the shot noise and radiation-pressure noise are correlated, the real back-action force does not commute with itself at different times, ¹² which makes the analysis in terms of real quantities more complicated than in terms of the effective ones. We prefer to discuss our results in terms of real quantities separately (Chapter 3), in a more formal context which uses the description of a GW interferometer as a linear quantum-measurement device [4].

The noise spectral density, written in terms of the effective operators Z andF, reads [4]

Sh= 1 L²

©S_ZZ+ 2R^xx<[S_FZ] +R²xxS_FFª

, (2.42)

where the (one-sided) cross spectral density of two operators is expressible, by analogy with Eq. (2.31), as

1

2S_AB(Ω)2π δ(Ω−Ω⁰) =1

2hA(Ω)B^†(Ω⁰) +B^†(Ω⁰)A(Ω)i. (2.43) In Eq. (2.42) the terms containing S_ZZ, S_FF and <[S_FZ] should be identified as effective shot noise, back-action noise and a term proportional to the effective correlation between the two noises, respectively [4]. Relying on the commutators (2.41) between the effective field operators one can derive (see Ref. [4] and Chapter 3) the following uncertainty relation for the (one-sided) spectral densities and cross correlations ofZ andF:

S_ZZS_FF −S_ZFS_FZ ≥~². (2.44)

Equation (2.44) does not, in general, impose a lower bound on the noise spectral density Eq. (2.42).

However, in a very important type of measurement it does, namely, for interferometers with uncor- related shot noise and back-action noise, e.g., LIGO-I/TAMA/Virgo. In this caseS_ZF = 0 =S_FZ [7] and inserting the vanishing correlations into Eqs. (2.43), (2.44), one easily finds that the noise

12We will show in Chapter 3 that as a consequence of this identification the antisymmetric mode of motion of the four mirrors acquires an optical-mechanical rigidity and a SR interferometer responds to GW signal like an optical spring. This phenomenon was already observed in optical bar detectors by Braginsky’s group [9].

spectral density has a lower bound which is given by the standard quantum limit, i.e.,

Sh(Ω)≥S_h^SQL(Ω)≡ 2|Rxx(Ω)|~

L² = 8~

mΩ²L² =h²_SQL(Ω). (2.45) From this it follows that to beat the SQL one must create correlations between shot noise and back-action noise.

Before investigating those correlations in a SR inteferometer, we shall first show how such cor- relations can be built up statically in a conventional (LIGO-I/TAMA/Virgo) interferometer by implementing frequency-independent homodyne detection at some angle ζ [8, 7]. By identifying in the interferometer output (2.34) the terms independent of m as effective shot noise and those inversely proportional tomas effective back-action noise, we get the effective field operatorsZζ^conv

andFζ^conv:

Zζ^conv(Ω) = e^iβL hSQL

√2K (a2+a1 tanζ), Fζ^conv(Ω) = ~e^iβ√ 2K

L hSQL a1. (2.46) [We remind the readers that hSQL∝1/√m and that K ∝1/m.] Evaluating the spectral densities of those operators using Eqs. (2.43) and (2.32), we obtain the following expressions for the spectral densities and their static correlations:

S_Z^conv_ζZζ(Ω) = L²h²_SQL

2K (1 + tan²ζ), S_F^conv_ζFζ(Ω) = 2K~²

L²h²_SQL, S_Z^conv_ζFζ(Ω) =~tanζ=S_F^conv_ζZζ(Ω). (2.47) By inserting these in Eq. (2.42) and optimizing the coupling constant K, we see that the SQL can be beaten for any 0< ζ < π/2, i.e., whenever there are nonvanishing correlations. See Refs. [7, 8]

for further details.

Let us now derive the correlations between shot noise and back action noise in SR interferometers.

Because in this case the correlations are built up dynamically by the SR mirror and are present in all quadratures, as an example, we limit ourselves to the two quadraturesb^C₁ andb^C₂. Identifying in Eqs. (2.29), (2.30) the effective shot and back-action noise terms due to their m dependences, we obtain the effective field operatorsZ¹,Z²,F¹andF²

Z¹(Ω) =−e^iβL hSQL

√2K

£a1(−2ρcos 2β+ (1 +ρ²) cos 2φ) +a2(−1 +ρ²) sin 2φ¤ cscφ

τ(1 +e^2iβρ) ,

Z²(Ω) =−e^iβL hSQL

√2K

£a1(1−ρ²) sin 2φ+a2(−2ρcos 2β+ (1 +ρ²) cos 2φ)¤ secφ

τ(−1 +e^2iβρ) ,(2.48)

and

F¹(Ω) = ~e^iβ√ 2K L hSQL

£a1(1 +ρ²) cosφ+a2(−1 +ρ²) sinφ¤ τ(1 +e^2iβρ) , F²(Ω) = ~e^iβ√

2K L hSQL

£a1(−1 +ρ²) cosφ+a2(1 +ρ²) sinφ¤

τ(−1 +e^2iβρ) . (2.49)

Evaluating the spectral densities of the above operators through Eqs. (2.43) and (2.32) we obtain the following expressions:

S_F1F1(Ω) = ~²2K L²h²_SQL

1 +ρ⁴+ 2ρ²cosφ (1−ρ²) (1 +ρ²+ 2ρcos 2β), S_F2F2(Ω) = ~²2K

L²h²_SQL

1 +ρ⁴−2ρ²cosφ

(1−ρ²) (1 +ρ²−2ρcos 2β), (2.50) and

S_Z1Z1(Ω) = L²h²_SQL 2K

£4(−1 +ρ²)² cos²φ+ (−2ρcos 2β+ (1 +ρ²) cos 2φ)² csc²φ¤ (1−ρ²) (1 +ρ²+ 2ρcos 2β) , S_Z2Z2(Ω) = L²h²_SQL

2K

£4(−1 +ρ²)² sin²φ+ (−2ρcos 2β+ (1 +ρ²) cos 2φ)² sec²φ¤ (1−ρ²) (1 +ρ²−2ρcos 2β) .(2.51) Finally, for the correlations between the shot noise and back-action noise we get

S_F1Z1(Ω) =S_Z1F1(Ω) =−~£

(−1 +ρ²)²−2ρ(1 +ρ²) cos 2β+ 4ρ² cos 2φ¤ cotφ (1−ρ²) (1 +ρ²+ 2ρcos 2β) , S_F2Z2(Ω) =S_Z2F2(Ω) = ~£

(−1 +ρ²)²+ 2ρ(1 +ρ²) cos 2β−4ρ² cos 2φ¤ tanφ

(1−ρ²) (1 +ρ²−2ρcos 2β) . (2.52) These correlations depend on the sideband angular frequency Ω and are generically different from zero. However, when φ = 0 and φ = π/2 the correlations are zero. We shall analyze these two extreme configurations in the following section.

2.3.3 Two special cases: extreme signal-recycling and resonant-sideband-