6.1 Noise cancellation by means of cross-correlation

6.1.1 Introduction

In this section, we consider the interferometer configurations that use the idea of the cross-correlation of the shot and the radiation pressure noise sources discussed in Section 4.4. This cross-correlation allows the measurement and the back-action noise to partially cancel each other out and thus effectively reduce the sum quantum noise to below the SQL.

As we noted above, Eq. (378View Equation) tells us that this cross-correlation can be created by tuning either the homodyne angle Ο•LO, the squeezing angle πœƒ, or the detuning δ. In Section 4.4, the simplest particular case of the frequency-independent correlation created by means of measurement of linear combination of the phase and amplitude quadratures, that is, by using the homodyne angle Ο•LO ⁄= πβˆ•2, has been considered. We were able to obtain a narrow-band sensitivity gain at some given frequency that was similar to the one achievable by introducing a constant rigidity to the system, therefore such correlation was called effective rigidity.

However, the broadband gain requires a frequency-dependent correlation, as it was first demonstrated for optical interferometric position meters [148], and then for general position measurement case [82]. Later, this idea was developed in different contexts by several authors [81, 118, 159Jump To The Next Citation Point, 90Jump To The Next Citation Point, 70, 69, 149, 9]. In particular, in [90Jump To The Next Citation Point], a practical method of creation of the frequency-dependent correlation was proposed, based on the use of additional filter cavities, which were proposed to be placed either between the squeeze light source and the main interferometer, creating the frequency-dependent squeezing angle (called pre-filtering), or between the main interferometer and the homodyne detector, creating the effective frequency-dependent squeezing angle (post-filtering). As we show below, in principle, both pre- and post-filtering can be used together, providing some additional sensitivity gain.

It is necessary to note also an interesting method of noise cancellation proposed by Tsang and Caves recently [146]. The idea was to use matched squeezing; that is, to place an additional cavity inside the main interferometer and couple the light inside this additional cavity with the differential mode of the interferometer by means of an optical parametric amplifier (OPA). The squeezed light created by the OPA should compensate for the ponderomotive squeezing created by back-action at all frequencies and thus decrease the quantum noise below the SQL at a very broad frequency band. However, the thorough analysis of the optical losses influence, that as we show later, are ruinous for the subtle quantum correlations this scheme is based on, was not performed.

Coming back to the filter-cavities–based interferometer topologies, we limit ourselves here by the case of the resonance-tuned interferometer, δ = 0. This assumption simplifies all the equations considerably, and allows one to clearly separate the sensitivity gain provided by the quantum noise cancellation due to cross-correlation from the one provided by the optical rigidity, which will be considered in Section 6.3.

We also neglect optical losses inside the interferometer, assuming that γ2 = 0. In broadband interferometer configurations considered here, with typical values of 3 −1 γ ≳ 10 s, the influence of these losses is negligible compared to those of the photodetector inefficiency and the losses in the filter cavities. Indeed, taking into account the fact that with modern high-reflectivity mirrors, the losses per bounce do not exceed A ≲ 10− 4 arm, and the arms lengths of the large-scale GW detectors are equal to several kilometers, the values of −1 γ2 ≲ 1 s, and, correspondingly, −3 γ2βˆ•γ ≲ 10, are feasible. At the same time, the value of photodetector quantum inefficiency πœ–2d ≈ 1 − ηd ≈ 0.05 (factoring in the losses in the interferometer output optical elements as well) is considered quite optimistic. Note, however, that in narrow-band regimes considered in Section 6.3, the bandwidth γ can be much smaller and influence of γ 2 could be significant; therefore, we take these losses into account in Section 6.3.

Using these assumptions, the quantum noises power (double-sided) spectral densities (376View Equation), (377View Equation) and (378View Equation) can be rewritten in the following explicit form:

ℏ [ ] S𝒳𝒳 (Ω) = ------2--------2---- cosh2r + sinh 2rcos 2(πœƒ − Ο•LO) + πœ–2d , (386 ) 2M ٠𝒦 (Ω )sin Ο•LO ℏM Ω2 𝒦(Ω ) Sβ„±β„± (Ω) = -----2------(cosh2r + sinh 2rcos 2πœƒ), (387 ) S𝒳ℱ (Ω) = ----ℏ----[cosh2r cosΟ•LO + sinh 2r cos(2πœƒ − Ο•LO )], (388 ) 2 sin Ο•LO
where
2J γ 𝒦 (Ω) = --2--2----2- (389 ) Ω (γ + Ω )
is the convenient optomechanical coupling factor introduced in [90Jump To The Next Citation Point].

Eq. (381View Equation) and (385View Equation) for the sum quantum noise and its power (double-sided) spectral density in this case can also be simplified significantly:

∘ ------- [ ] ˆhn (Ω) = -2 --ℏ------1---HT [Ο•LO] (γ + iΩ)𝕂 (Ω)π•Šsqz[r,πœƒ]ˆavac(Ω) + (γ − iΩ)πœ–dˆn (Ω) , (390 ) L 2M J γsinΟ•LO 2ℏ [ ] Sh (Ω) = ----------------2---- HT [Ο•LO ]𝕂 (Ω )π•Šsqz[2r,πœƒ]𝕂 †(Ω )H [Ο•LO ] + πœ–2d , (391 ) M L2Ω2 𝒦 (Ω )sin Ο•LO

where

[ ] 1 0 𝕂 (Ω ) = − 𝒦 (Ω) 1 . (392 )

In Section 6.1.2 we consider the optimization of the spectral density (391View Equation), assuming that the arbitrary frequency dependence of the homodyne and/or squeezing angles can be implemented. As we see below, this case corresponds to the ideal lossless filter cavities. In Section 6.1.3, we consider two realistic schemes, taking into account the losses in the filter cavities.

6.1.2 Frequency-dependent homodyne and/or squeezing angles

Classical optimization.
As a reference point, consider first the simplest case of frequency independent homodyne and squeezing angles. We choose the specific values of these parameters following the classical optimization, which minimizes the shot noise (386View Equation) without taking into account the back action. Because, the shot noise dominates at high frequencies, therefore, this optimization gives a smooth broadband shape of the sum noise spectral density.

It is evident that this minimum is provided by

π Ο•LO = 2-, πœƒ = 0. (393 )
In this case, the sum quantum noise power (double-sided) spectral density is equal to
[ −2r 2 ] Sh(Ω ) = ---2ℏ--- e----+-πœ–d + 𝒦 (Ω)e2r . (394 ) M L2 Ω2 𝒦 (Ω)
It is easy to note the similarity of this spectral density with the ones of the toy position meter considered above, see Eq. (173View Equation). The only significant differences introduced here are the optical losses and the decrease of the optomechanical coupling at high frequencies due to the finite bandwidth γ of the interferometer. If πœ–d = 0 and γ ≫ Ω, then Eqs. (173View Equation) and (394View Equation) become identical, with the evident correspondence
ℐ ℐ0Π³2 ↔ -c. (395 ) γτ
In particular, the spectral density (394View Equation) can never be smaller than the free mass SQL Sh (Ω ) SQLf.m. (see (174View Equation)). Indeed, it can be minimized at any given frequency Ω by setting
−r∘ -−2r----2 𝒦 (Ω ) = e e + πœ–d, (396 )
and in this case,
Sh (Ω ) ∘ --------- ξ2(Ω) ≡ -h---------= 1 + πœ–2de2r ≥ 1. (397 ) SSQLf.m.(Ω )

The spectral density (394View Equation) was first calculated in the pioneering work of [38], where the existence of two kinds of quantum noise in optical interferometric devices, namely the measurement (shot) noise and the back action (radiation pressure) noise, were identified for the first time, and it was shown that the injection of squeezed light with πœƒ = 0 into the interferometer dark port is equivalent to the increase of the optical pumping power. However, it should be noted that in the presence of optical losses this equivalence holds unless squeezing is not too strong, e− r > πœ–d.

View Image

Figure 34: Examples of the sum quantum noise spectral densities of the classically-optimized (Ο•LO = πβˆ•2, πœƒ = 0) resonance-tuned interferometer. ‘Ordinary’: J = JaLIGO, no squeezing. ‘Increased power’: J = 10JaLIGO, no squeezing. ‘Squeezed’: J = JaLIGO, 10 dB squeezing. For all plots, γ = 2π × 500 s−1 and ηd = 0.95.

The noise spectral density curves for the resonance-tuned interferometer are drawn in Figure 34View Image. The default parameters for this and all subsequent similar plots are chosen to be close to those planned for the Advanced LIGO interferometer: the value of J = JaLIGO ≡ (2π × 100 )3 s−3 corresponds to the circulating power of ℐ = 840 kW arm, L = 4 km, and M = 40 kg; the interferometer bandwidth −1 γ = 2π × 500 s is close to the one providing the best sensitivity for Advanced LIGO in the presence of technical noise [93Jump To The Next Citation Point]; 10 dB squeezing (2r e = 10), which corresponds to the best squeezing available at the moment (2011) in the low-frequency band [102, 150, 151]); ηd = 0.95 can be considered a reasonably optimistic estimate for the real interferometer quantum efficiency.

Noteworthy is the proximity of the plots for the interferometer with 10 dB input squeezing and the one with 10-fold increased optical power. The noticeable gap at higher frequencies is due to optical loss.

Frequency dependent squeezing angle.
Now suppose that the homodyne angle can be frequency dependent, and calculate the corresponding minimum of the sum noise spectral density (391View Equation). The first term in square brackets in this equation can be rewritten as:

V Tβ„™ [πœƒ]π•Šsqz[2r,0]β„™†[πœƒ]V = e2r(Vc cosπœƒ + Vs sin πœƒ)2 + e− 2r(− Vcsin πœƒ + Vs cosπœƒ)2, (398 )
where
[ ] Vc † V ≡ V = 𝕂 (Ω )H [Ο•LO]. (399 ) s
It is evident that the minimum of (398View Equation) is provided by
Vc tan πœƒ = − V--= − cot Ο•LO + 𝒦 (Ω) (400 ) s
and is equal to V TV e−2r. Therefore,
h ---------2ℏ----------[ T † −2r 2] S (Ω ) = M L2Ω2𝒦 (Ω )sin2 Ο• H [Ο•LO ]𝕂(Ω )𝕂 (Ω )H [Ο•LO]e + πœ–d { [ LO ] 2 } = ---2ℏ--- ------1-------− 2 cotΟ• + 𝒦 (Ω ) e−2r + ------πœ–d------ . (401 ) M L2Ω2 𝒦 (Ω) sin2 Ο•LO LO 𝒦 (Ω) sin2 Ο•LO
Thus, we obtaine a well-known result [90Jump To The Next Citation Point] that, using an optimal squeezing angle, the quantum noise spectral density can be reduced by the squeezing factor e− 2r in comparison with the vacuum input case. Note, however, that the noise contribution due to optical losses remains unchanged. Concerning the homodyne angle Ο•LO, we use again the classical optimization, setting
π- Ο•LO = 2. (402 )
In this case, the sum noise power (double-sided) spectral density and the optimal squeezing angle are equal to
[ −2r 2 ] Sh(Ω ) = --2ℏ---- e----+-πœ–d+ 𝒦(Ω )e−2r (403 ) M L2 Ω2 𝒦 (Ω)
and
tan πœƒ = 𝒦 (Ω). (404 )

The sum quantum noise power (double-sided) spectral density (403View Equation) is plotted in Figure 35View Image for the ideal lossless case and for η = 0.95 d (dotted line). In both cases, the optical power and the squeezing factor are equal to J = JaLIGO and 2r e = 10, respectively.

View Image

Figure 35: Examples of the sum quantum noise power (double-sided) spectral densities of the resonance-tuned interferometers with frequency-dependent squeezing and/or homodyne angles. Left: no optical losses, right: with optical losses, ηd = 0.95. ‘Ordinary’: no squeezing, Ο•LO = π βˆ•2. ‘Squeezed’: 10 dB squeezing, πœƒ = 0, Ο• = π βˆ•2 LO (these two plots are provided for comparison). Dots [pre-filtering, Eq. (403View Equation)]: 10 dB squeezing, Ο•LO = π βˆ•2, frequency-dependent squeezing angle. Dashes [post-filtering, Eq. (408View Equation)]: 10 dB squeezing, πœƒ = 0, frequency-dependent homodyne angle. Dash-dots [pre- and post-filtering, Eq. (410View Equation)]: 10 dB squeezing, frequency-dependent squeeze and homodyne angles. For all plots, J = J aLIGO and γ = 2π × 500 s− 1.

Frequency dependent homodyne angle.
Suppose now that the squeezing angle corresponds to the classical optimization:

πœƒ = 0 (405 )
and minimize the resulting sum noise spectral density:
[ 2 ] Sh(Ω ) = ---2ℏ--- cosh-2r-+-sinh-2r-cos2Ο•LO--+-πœ–d− 2e2r cotΟ• + 𝒦 (Ω )e2r (406 ) M L2 Ω2 𝒦 (Ω )sin2 Ο•LO LO
with respect to Ο•LO. The minimum is provided by the following dependence
𝒦 (Ω) cotΟ•LO = -----2-−2r, (407 ) 1 + πœ–de
and is equal to
[ ] h ---2ℏ--- e−2r +-πœ–2d- ---πœ–2d----- S (Ω) = M L2Ω2 𝒦 (Ω ) + 1 + πœ–2e− 2r𝒦 (Ω ) . (408 ) d
The sum quantum noise spectral density (408View Equation) is plotted in Figure 35View Image for the ideal lossless case and for ηd = 0.95 (dashed lines).

Compare this spectral density with the one for the frequency-dependent squeezing angle (pre-filtering) case, see Eq. (403View Equation). The shot noise components in both cases are exactly equal to each other. Concerning the residual back-action noise, in the pre-filtering case it is limited by the available squeezing, while in the post-filtering case – by the optical losses. In the latter case, were there no optical losses, the back-action noise could be removed completely, as shown in Figure 35View Image (left). For the parameters of the noise curves presented in Figure 35View Image (right), the post-filtering still has some advantage of about 40% in the back-action noise amplitude √ -- S.

Note that the required frequency dependences (404View Equation) and (407View Equation) in both cases are similar to each other (and become exactly equal to each other in the lossless case πœ– = 0 d). Therefore, similar setups can be used in both cases in order to create the necessary frequency dependences with about the same implementation cost. From this simple consideration, it is possible to conclude that pre-filtering is preferable if good squeezing is available, and the optical losses are relatively large, and vice versa. In particular, post-filtering can be used even without squeezing, r = 0.

Frequency dependent homodyne and squeezing angles.
And, finally, consider the most sophisticated configuration: double-filtering with both the homodyne angle Ο•LO and the squeezing angle πœƒ being frequency dependent.

Concerning the squeezing angle, we can reuse Eqs. (400View Equation) and (401View Equation). The minimum of the spectral density (401View Equation) in Ο•LO corresponds to

𝒦 (Ω) cot Ο•LO = ----2-2r, (409 ) 1 + πœ–de
and is equal to
[ ] h ---2ℏ--- e−-2r +-πœ–2d ---πœ–2d---- S (Ω ) = M L2 Ω2 𝒦 (Ω) + 1 + πœ–2e2r𝒦 (Ω ) . (410 ) d
It also follows from Eqs.(400View Equation) and (409View Equation) that the optimal squeezing angle in this case is given by
2 tan πœƒ = ---πœ–d----𝒦(Ω ). (411 ) e−2r + πœ–2d

It is easy to see that in the ideal lossless case the double-filtering configuration reduces to a post-filtering one. Really, if πœ–d = 0, the spectral density (410View Equation) becomes exactly equal to that for the post-filtering case (408View Equation), and the frequency dependent squeezing angle (411View Equation) degenerates into a constant value (405View Equation). However, if πœ–d > 0, then the additional pre-filtering allows one to decrease more the residual back-action term. For example, if e2r = 10 and ηd = 0.95 then the gain in the back-action noise amplitude √ -- S is equal to about 25%.

We have plotted the sum quantum noise spectral density (410View Equation) in Figure 34View Image, right (dash-dots). This plot demonstrates the best sensitivity gain of about 3 in signal amplitude, which can be provided employing squeezing and filter cavities at the contemporary technological level.

Due to the presence of the residual back-action term in the spectral density (410View Equation), there exists an optimal value of the coupling factor 𝒦 (Ω ) (that is, the optical power) which provides the minimum to the sum quantum noise spectral density at any given frequency Ω:

1 𝒦 (Ω) = ---r + πœ–der, (412 ) πœ–de
The minimum is equal to
h ---4ℏ--- −r Smin = M L2Ω2 πœ–de . (413 )
This limitation is severe. The reasonably optimistic value of quantum efficiency ηd = 0.95 that we use for our estimates corresponds to πœ–d ≈ 0.23. It means that without squeezing (r = 0) one is only able to beat the SQL in amplitude by
∘ --h--- ξ ≡ Smin-= √ πœ–- ≈ 0.5. (414 ) min ShSQL d
The gain can be improved using squeezing and, if r → ∞ then, in principle, arbitrarily high sensitivity can be reached. But ξ depends on r only as e−rβˆ•2, and for the 10 dB squeezing, only a modest value of
ξ ≈ 0.27 (415 ) min
can be obtained.

In our particular case, the fact that the additional noise associated with the photodetector quantum inefficiency πœ– > 0 d does not correlate with the quantum fluctuations of the light in the interferometer gives rise to this limit. This effect is universal for any kind of optical loss in the system, impairing the cross-correlation of the measurement and back-action noises and thus limiting the performance of the quantum measurement schemes, which rely on this cross-correlation.

Noteworthy is that Eq. (410View Equation) does not take into account optical losses in the filter cavities. As we shall see below, the sensitivity degradation thereby depends on the ratio of the light absorption per bounce to the filter cavities length, Afβˆ•Lf. Therefore, this method calls for long filter cavities. In particular, in the original paper [90Jump To The Next Citation Point], filter cavities with the same length as the main interferometer arm cavities (4 km), placed side by side with them in the same vacuum tubes, were proposed. For such long and expensive filter cavities, the influence of their losses indeed can be small. However, as we show below, in Section 6.1.3, for the more practical short (up to tens of meters) filter cavities, optical losses thereof could be the main limiting factor in terms of sensitivity.

Virtual rigidity for prototype interferometers.
The optimization performed above can be viewed also in a different way, namely, as the minimization of the sum quantum noise spectral density of an ordinary interferometer with frequency-independent homodyne and squeezing angles, yet at some given frequency Ω0. In Section 4.4, this kind of optimization was considered for a simple lossless system. It was shown capable of the narrow-band gain in sensitivity, similar to the one provided by the harmonic oscillator (thus the term ‘virtual rigidity’).

This narrow-band gain could be more interesting not for the full-scale GW detectors (where broadband optimization of the sensitivity is required in most cases) but for smaller devices like the 10-m Hannover prototype interferometer [7], designed for the development of the measurement methods with sub-SQL sensitivity. Due to shorter arm length, the bandwidth γ in those devices is typically much larger than the mechanical frequencies Ω. If one takes, e.g., the power transmissivity value of T ≳ 10 −2 for the ITMs and length of arms equal to L ∼ 10 m, then 5 −1 γ ≳ 10 s, which is above the typical working frequencies band of such devices. In the literature, this particular case is usually referred to as a bad cavity approximation.

In this case, the coupling factor 𝒦 (Ω ) can be approximated as:

Ω2q 𝒦 (Ω) ≈ Ω2-, (416 )
where
2J Ω2q = ---. (417 ) γ
Note that in this approximation, the noise spectral densities (386View Equation), (387View Equation) and (388View Equation) turn out to be frequency independent.
View Image

Figure 36: Plots of the locally-optimized SQL beating factor ξ(Ω ) (418View Equation) of the interferometer with cross-correlated noises for the “bad cavity” case Ω0 β‰ͺ γ, for several different values of the optimization frequency Ω0 within the range √ --- 0.1 × Ωq ≤ Ω0 ≤ 10 × Ωq. Thick solid lines: the common envelopes of these plots; see Eq. (420View Equation). Left column: ηd = 1; right column: ηd = 0.95. Top row: no squeezing, r = 0; bottom row: 10 dB squeezing, e2r = 10.

In Figure 36View Image, the SQL beating factor

∘ ------------ ---Sh(Ω-)-- ξ (Ω ) = Sh (Ω) (418 ) SQLf.m.
is plotted for the sum quantum noise spectral density Sh (Ω) with the following values of homodyne and squeezing angles
𝒦 (Ω0 ) πœ–2d cotΟ•LO = -----2-2r, tan πœƒ = -−-2r----2𝒦 (Ω0), (419 ) 1 + πœ–de e + πœ–d
and factoring in the “bad cavity” condition (416View Equation). The four panes correspond to the following four combinations: (upper left) no losses (η = 1 d) and no squeezing (r = 0); (lower left) no losses (η = 1 d) and 10 dB squeezing (2r e = 10); (upper right) with losses (ηd = 0.95) and no squeezing (r = 0); (lower right) with losses (ηd = 0.95) and 10 dB squeezing(2r e = 10). In each pane, the family of plots is shown that corresponds to different values of the ratio Ω0 βˆ•Ωq, ranging from 0.1 to √ --- 10.

The minima of these plots form the common envelope, given by Eqs. (410View Equation) and (416View Equation):

1 [ Ω2 πœ–2 Ω2q] ξ2min(Ω0 ) = -- (e−2r + πœ–2d)-02 + ----d2-2r--2 , (420 ) 2 Ω q 1 + πœ–de Ω 0
which is also plotted in Figure 36View Image. It is easy to see that in the ideal case of πœ–d = 0, there is no limitation on the SQL beating factor, provided a sufficiently small ratio of Ω0 βˆ•Ωq:
2 e−-2r Ω20- ξmin(Ω) = 2 Ω2. (421 ) q
However, if πœ–d > 0, then function (420View Equation) has a minimum in Ωq at
∘ ----------- Ω = Ω -1--+ πœ– er, (422 ) q 0 πœ–der d
[compare with Eq. (412View Equation)], equal to (413View Equation).

6.1.3 Filter cavities in GW interferometers

Input/output relations for the filter cavity.
In essence, a filter cavity is an ordinary Fabry–Pérot cavity with one partly transparent input/output mirror. The technical problem of how to spatially separate the input and output beam can be solved in different ways. In the original paper [90Jump To The Next Citation Point] the triangular cavities were considered. However, in this case, an additional mirror in each cavity is required, which adds to the optical loss per bounce. Another option is an ordinary linear cavity with additional optical circulator, which can be implemented, for example, by means of the polarization beamsplitter and Faraday rotator (note that while the typical polarization optics elements have much higher losses than the modern high-quality mirrors, the mirrors losses appear in the final expressions inflated by the filter cavity finesse).

In both cases, the filter cavity can be described by the input/output relation, which can be easily obtained from Eqs. (290View Equation) and (291View Equation) by setting Ec,s = 0 (there is no classical pumping in the filter cavity and, therefore, there is no displacement sensitivity) and by some changes in the notations:

ˆo(Ω ) = ℝf(Ω )ˆi(Ω ) + 𝕋f (Ω )qˆ(Ω), (423 )
where ˆi and ˆo are the two-photon quadrature amplitude vectors of the input and output beams, ˆq stands for noise fields entering the cavity due to optical losses (which are assumed to be in a vacuum state),
[ ] 1 γ2f1 − γ2f2 − δ2f + Ω2 + 2iΩ γf2 − 2γf1δf ℝf (Ω) = ------- 2γ δ γ2 − γ2 − δ2+ Ω2 + 2iΩ γ , (424 ) π’Ÿf(Ω-)---[ f1 f ] f1 f2 f f2 2√ γf1γf2 γf − iΩ − δf 𝕋f (Ω) = ---------- δ γ − iΩ , (425 ) π’Ÿf (Ω) f f π’Ÿf (Ω) = (γf − iΩ )2 + δ2, (426 ) f γf1 = cTf- (427 ) 4Lf cAf γf2 = ---, (428 ) 4Lf
where Tf is the power transmittance of the input/output mirror, Af is the factor of power loss per bounce, L f is the filter cavity length,
γf = γf1 + γf2 (429 )
is its half-bandwidth, and δf is its detuning.

In order to demonstrate how the filter cavity works, consider the particular case of the lossless cavity. In this case,

ˆo(Ω) = ℝf (Ω)ˆi(Ω ), (430 )
and the reflection matrix describes field amplitude rotations with the frequency-dependent rotation angle:
ℝ (Ω ) = β„™[πœƒ (Ω )]eiβf(Ω ), (431 ) f f
where
πœƒ (Ω ) = arctan ---2γf-δf---, (432 ) f γ2f − δ2f + Ω2 eiβf(Ω) = |π’Ÿf-(Ω-)|. (433 ) π’Ÿf (Ω )
The phase factor iβf(Ω) e is irrelevant, for it does not appear in the final equations for the spectral densities.

Let us now analyze the influence of the filter cavities on the interferometer sensitivity in post- and pre-filtering variational schemes. Start with the latter one. Suppose that the light, entering the interferometer from the signal port is in the squeezed state with fixed squeezing angle πœƒ and squeezing factor r and thus can be described by the following two-photon quadrature vector

aˆ = π•Š [r,πœƒ]ˆavac, (434 ) sqz
where the quadratures vector ˆavac describes the vacuum state. After reflecting off the filter cavity, this light will be described with the following expression
[ ] vac iβf(Ω) [ ] vac′ oˆ(Ω) = β„™ πœƒf (Ω ) π•Šsqz[r,πœƒ]ˆa e = π•Šsqz r,πœƒ + πœƒf(Ω) ˆa , (435 )
[see Eq. (75View Equation)], where [ ] ˆavac′ = β„™ πœƒf(Ω ) ˆavaceiβf(Ω) also describes the light field in a vacuum state. Thus, the pre-filtering indeed rotates the squeezing angle by a frequency-dependent angle πœƒf(Ω ).

In a similar manner, we can consider the post-filtering schemes. Consider a homodyne detection scheme with losses, described by Eq. (272View Equation). Suppose that prior to detection, the light described by the quadrature vector ˆb, reflects from the filter cavity. In this case, the photocurrent (in Fourier representation) is proportional to

T [ [ ] iβ (Ω) ] i− (Ω ) ∝ H [Ο•LO] β„™ πœƒf(Ω ) ˆb(Ω)e f + πœ–dˆn(Ω ) = HT [Ο• − πœƒ (Ω )][ˆb(Ω ) + πœ– ˆn ′(Ω )]eiβf(Ω ), (436 ) LO f d
where [ ] ˆn′ = β„™ − πœƒf(Ω )nˆe −iβf(Ω) again describes some new vacuum field. This formula demonstrates that post-filtering is equivalent to the introduction of a frequency-dependent shift of the homodyne angle by − πœƒf(Ω ).

It is easy to see that the necessary frequency dependencies of the homodyne and squeezing angles (404View Equation) or (407View Equation) (with the second-order polynomials in Ω2 in the r.h.s. denominators) cannot be implemented by the rotation angle (432View Equation) (with its first order in 2 Ω polynomial in the r.h.s. denominator). As was shown in the paper [90], two filter cavities are required in both these cases. In the double pre- and post-filtering case, the total number of the filter cavities increases to four. Later it was also shown that, in principle, arbitrary frequency dependence of the homodyne and/or squeezing angle can be implemented, providing a sufficient number of filter cavities [35].

However, in most cases, a more simple setup consisting of a single filter cavity might suffice. Really, the goal of the filter cavities is to compensate the back-action noise, which contributes significantly in the sum quantum noise only at low frequencies ----- ٠≲ Ω = ∘ 2Jβˆ•γ q. However, when

γ > J1βˆ•3, (437 )
which is actually the case for the planned second and third generation GW detectors, the factor 𝒦 (Ω ) can be well approximated by Eq. (416View Equation) in the low-frequency region. In such a case the single filter cavity can provide the necessary frequency dependence. Moreover, the second filter cavity could actually degrade sensitivity due to the additional optical losses it superinduces to the system.

Following this reasoning, we consider below two schemes, each based on a single filter cavity that realize pre-filtering and post-filtering, respectively.

Single-filter cavity-based schemes.
The schemes under consideration are shown in Figure 37View Image. In the pre-filtering scheme drawn in the left panel of Figure 37View Image, a squeezed light source emits frequency-independent squeezed vacuum towards the filter cavity, where it gets reflected, gaining a frequency-dependent phase shift πœƒf(Ω ), and then enters the dark port of the main interferometer. The light going out of the dark port is detected by the homodyne detector with fixed homodyne angle Ο•LO in the usual way.

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Figure 37: Schemes of interferometer with the single filter cavity. Left: In the pre-filtering scheme, squeezed vacuum from the squeezor is injected into the signal port of the interferometer after the reflection from the filter cavity; right: in the post-filtering scheme, a squeezed vacuum first passes through the interferometer and, coming out, gets reflected from the filter cavity. In both cases the readout is performed by an ordinary homodyne detector with frequency independent homodyne angle Ο•LO.

Following the prescriptions of Section 6.1.2, we suppose the homodyne angle defined by Eq. (402View Equation). The optimal squeezing angle should then be equal to zero at higher frequencies, see (404View Equation). Taking into account that the phase shift introduced by the filter cavity goes to zero at high frequencies, we obtain that the squeezing angle πœƒ of the input squeezed vacuum must be zero. Combining Eqs. (390View Equation) and (423View Equation) taking these assumptions into account, we obtain the following equation for the sum quantum noise of the pre-filtering scheme:

∘ ------- 2 ℏ { [ ] } ˆhsum(Ω ) = − -- ------HT [πβˆ•2] (γ + iΩ)𝕂 (Ω) ℝf (Ω)π•Šsqz[r,0]ˆavac(Ω ) + 𝕋f (Ω)ˆq(Ω ) + (γ − iΩ )πœ–dˆn(Ω &#x00 L 2M J γ
which yields the following expression for a power (double-sided) spectral density
h -----2ℏ------{ T [ † † ] † 2} S (Ω ) = M L2 Ω2𝒦 (Ω ) H [π βˆ•2]𝕂(Ω ) ℝf(Ω )π•Šsqz[2r,0]ℝf(Ω ) + 𝕋f(Ω )𝕋 f(Ω )𝕂 (Ω)H [π βˆ•2] + πœ–d(4.39 &#x0

In the ideal lossless filter cavity case, taking into account Eq. (431View Equation), this spectral density can be simplified as follows:

2ℏ [ ( ) ] Sh(Ω ) = ----2-2------HT [πβˆ•2]𝕂 (Ω )π•Š 2r,πœƒf(Ω ) 𝕂 †(Ω )H [πβˆ•2] + πœ–2d (440 ) M L ٠𝒦 (Ω)
[compare with Eq. (391View Equation)]. In this case, the necessary frequency dependence of the squeezing angle (404View Equation) can be implemented by the following filter cavity parameters:
γf = δf = γf0, (441 )
where
∘ ---- γf0 = Jβˆ•γ. (442 )

Along similar lines, the post-filtering scheme drawn in the right panel of Figure 37View Image can be considered. Here, the squeezed-vacuum produced by the squeezor first passes through the interferometer and then, coming out, gets reflected from the filter cavity, gaining a frequency-dependent phase shift, which is equivalent to introducing a frequency dependence into the homodyne angle, and then goes to the fixed angle homodyne detector. Taking into account that this equivalent homodyne angle at high frequencies has to be πβˆ•2, and that the phase shift introduced by the filter cavity goes to zero at high frequencies, we obtain that the real homodyne angle must also be πβˆ•2. Assuming that the squeezing angle is defined by Eq. (405View Equation) and again using Eqs. (390View Equation) and (423View Equation), we obtain that the sum quantum noise and its power (double-sided) spectral density are equal to

∘ ------- 2 ℏ 1 ˆhsum (Ω) = − -- ----------------------(0)- L 2M Jγ HT [π βˆ•2]ℝf(Ω ) 1 ×HT [π βˆ•2]{(γ + iΩ)ℝ (Ω)𝕂 (Ω )π•Š [r,0]aˆvac(Ω ) + (γ − iΩ )[𝕋 (Ω )qˆ(Ω) + πœ– ˆn(Ω )]} (443 & f sqz f d
and
h -----2ℏ----------------1--------- S (Ω) = M L2Ω2 𝒦(Ω )||HT [π βˆ•2]ℝ (Ω )(0)||2 { [ f 1 ] } × HT [π βˆ•2] ℝf(Ω )𝕂(Ω )π•Šsqz[2r,0]𝕂 †(Ω )ℝ†f(Ω ) + 𝕋f(Ω )𝕋 †f(Ω )H [π βˆ•2] + πœ–2d . (444 )
In the ideal lossless filter cavity case, factoring in Eq. (431View Equation), this spectral density takes a form similar to (391View Equation), but with the frequency-dependent homodyne angle:
2ℏ [ [ ] [ ] ] Sh (Ω ) = ----2--2--------2------- HT Ο•LO (Ω ) 𝕂 (Ω )π•Šsqz[2r,0]𝕂 †(Ω )HT Ο•LO(Ω ) + πœ–2d , (445 ) M L ٠𝒦 (Ω )sin Ο•LO (Ω)
where
Ο•LO(Ω ) = πβˆ•2 − πœƒf(Ω ). (446 )
The necessary frequency dependence (407View Equation) of this effective homodyne angle can be implemented by the following parameters of the filter cavity:
γ = δ = ∘---γf0-----. (447 ) f f 1 + πœ–2de−2r
Note that for reasonable values of loss and squeezing factors, these parameters differ only by a few percents from the ones for the pre-filtering.

It is easy to show that substitution of the conditions (441View Equation) and (447View Equation) into Eqs. (440View Equation) and (445View Equation), respectively, taking Condition (437View Equation) into account, results in spectral densities for the ideal frequency dependent squeezing and homodyne angle, see Eqs. (403View Equation) and (408View Equation).

In the general case of lossy filter cavities, the conditions (404View Equation) and (407View Equation) cannot be satisfied exactly by a single filter cavity at all frequencies. Therefore, the optimal filter cavity parameters should be determined using some integral sensitivity criterion, which will be considered at the end of this section.

However, it would be a reasonable assumption that the above consideration holds with good precision, if losses in the filter cavity are low compared to other optical losses in the system:

γf2-≈ γf2-β‰ͺ πœ–2. (448 ) γf γf0 d
This inequality can be rewritten as the following condition for the filter cavity specific losses:
Af 4γf0 2 L--≲ -c--πœ–d. (449 ) f
In particular, for our standard parameters used for numerical estimates (J = JaLIGO, γ = 2π × 500 s− 1, ηd = 0.95), we obtain γf0 ≈ 280 s−1, and there should be
Af- −7 −1 L ≲ 2 × 10 m , (450 ) f
(the r.h.s. corresponds, in particular, to a 50 m filter cavity with the losses per bounce Af = 10− 5).

Another more crude limitation can be obtained from the condition that γf2 should be small compared to the filter cavity bandwidth γ f:

Af 4γf0 ---≲ ----. (451 ) Lf c
Apparently, were it not the case, the filter cavity would just cease to work properly. For the same numerical values of J and γ as above, we obtain:
Af- −6 −1 L ≲ 4 × 10 m , (452 ) f
(for example, a very short 2.5 m filter cavity with Af = 10−5 or 25 m cavity with Af = 10 −4).

Numerical optimization of filter cavities.
In the experiments devoted to detection of small forces, and, in particular, in the GW detection experiments, the main integral sensitivity measure is the probability to detect some calibrated signal. This probability, in turn, depends on the matched filtered SNR defined as

∫ ∞ |h (Ω )|2 dΩ ρ2 = --s----- --- (453 ) −∞ Sh (Ω) 2π
with h (Ω ) s the spectrum of this calibrated signal.

In the low and medium frequency range, where back-action noise dominates, and wherein our interest is focused, the most probable source of signal is the gravitational radiation of the inspiraling binary systems of compact objects such as neutron stars and/or black holes [131Jump To The Next Citation Point, 124]. In this case, the SNR is equal to (see [58])

∫ 2πfmax Ω −7βˆ•3 dΩ ρ2 = k0 -h-------, (454 ) 0 S (Ω )2π
where k0 is a factor that does not depend on the interferometer parameters, and fmax is the cut-off frequency that depends on the binary system components’ masses. In particular, for neutron stars with masses equal to 1.4 solar mass, fmax ≈ 1.5 kHz.

Since our goal here is not the maximal value of the SNR itself, but rather the relative sensitivity gain offered by the filter cavity, and the corresponding optimal parameters γf1 and δf, providing this gain, we choose to normalize the SNR by the value corresponding to the ordinary interferometer (without the filter cavities):

∫ 2πfmax Ω −7βˆ•3 dΩ ρ20 = k0 -h-------, (455 ) 0 S0(Ω )2π
with power (double-sided) spectral density
[ ] h ---2ℏ--- 1 +-πœ–2d S0 (Ω ) = M L2Ω2 𝒦 (Ω) + 𝒦 (Ω ) (456 )
[see Eq. (394View Equation)].

We optimized numerically the ratio ρ2βˆ•ρ20, with filter cavity half-bandwidth γf1 and detuning δf as the optimization parameters, for the values of the specific loss factor Afβˆ•Lf ranging from 10−9 (e.g., very long 10 km filter cavity with A = 10 −5 f) to 10−5 (e.g., 10 m filter cavity with − 4 Af = 10). Concerning the main interferometer parameters, we used the same values as in all our previous examples, namely, J = JaLIGO, −1 γ = 2 π × 500 s, and ηd = 0.95.

The results of the optimization are shown in Figure 38View Image. In the left pane, the optimal values of the filter cavity parameters γ f1 and δ f are plotted, and in the right one the corresponding optimized values of the SNR are. It follows from these plots that the optimal values of γf1 and δf are virtually the same as γf0, while the specific loss factor Afβˆ•Lf satisfies the condition (448View Equation), and starts to deviate sensibly from γf0 only when Af βˆ•Lf approaches the limit (451View Equation). Actually, for such high values of specific losses, the filter cavities only degrade the sensitivity, and the optimization algorithm effectively turns them off, switching to the ordinary frequency-independent squeezing regime (see the right-most part of the right pane).

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Figure 38: Left: Numerically-optimized filter-cavity parameters for a single cavity based pre- and post-filtering schemes: half-bandwidth γf1 (solid lines) and detuning δf (dashed lines), normalized by γf0 [see Eq. (442View Equation)], as functions of the filter cavity specific losses Af βˆ•Lf. Right: the corresponding optimal SNRs, normalized by the SNR for the ordinary interferometer [see Eq. (455View Equation)]. Dashed lines: the normalized SNRs for the ideal frequency-dependent squeeze and homodyne angle cases, see Eqs. (403View Equation) and (408View Equation). ‘Ordinary squeezing’: frequency-independent 10 dB squeezing with πœƒ = 0. In all cases, J = JaLIGO, γ = 2π × 500 s−1, and ηd = 0.95.

It also follows from these plots that post-filtering provides slightly better sensitivity, if the optical losses in the filter cavity are low, while the pre-filtering has some advantage in the high-losses scenario. This difference can be explained in the following way [87]. The post-filtration effectively rotates the homodyne angle from Ο•LO = πβˆ•2 (phase quadrature) at high frequencies to Ο•LO → 0 (amplitude quadrature) at low frequencies, in order to measure the back-action noise, which dominates the low frequencies. As a result, the optomechanical transfer function reduces at low frequencies, emphasizing all noises introduced after the interferometer [see the factor sin2Ο•LO (Ω) in the denominator of Eq. (445View Equation)]. In the pre-filtering case there is no such effect, for the value of Ο•LO = πβˆ•2, corresponding to the maximum of the optomechanical transfer function, holds for all frequencies (the squeezing angle got rotated instead).

The optimized sum quantum noise power (double-sided) spectral densities are plotted in Figure 39View Image for several typical values of the specific loss factor, and for the same values of the rest of the parameters, as in Figure 38View Image. For comparison, the spectral densities for the ideal frequency-dependent squeezing angle Eqs. (403View Equation) and homodyne angle (408View Equation) are also shown. These plots clearly demonstrate that providing sufficiently-low optical losses (say, −8 Af βˆ•Lf ≲ 10), the single filter cavity based schemes can provide virtually the same result as the abstract ones with the ideal frequency dependence for squeezing or homodyne angles.

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Figure 39: Examples of the sum quantum noise power (double-sided) spectral densities of the resonance-tuned interferometers with the single filter cavity based pre- and post-filtering. Left: pre-filtering, see Figure 37View Image (left); dashes – 10 dB squeezing, Ο•LO = π βˆ•2, ideal frequency-dependent squeezing angle (404View Equation); thin solid – 10 dB squeezing, Ο• = πβˆ•2 LO, numerically-optimized lossy pre-filtering cavity with −9 − 7 −6.5 − 6 Afβˆ•Lf = 10 , 10 10 and 10. Right: post-filtering, see Figure 37View Image (right); dashes: 10 dB squeezing, πœƒ = 0, ideal frequency-dependent homodyne angle (407View Equation); thin solid – 10 dB squeezing, πœƒ = 0, numerically optimized lossy post-filtering cavity with A βˆ•L = 10−9, 10−8 and 10− 7 f f. In both panes (for the comparison): ‘Ordinary’ – no squeezing, Ο•LO = πβˆ•2; ‘Squeezed’: 10 dB squeezing, πœƒ = 0, Ο•LO = πβˆ•2

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