6.3 Optical rigidity

6.3.1 Introduction

We have seen in Section 4.3 that the harmonic oscillator, due to its strong response on near-resonance force, is characterized by the reduced values of the effective quantum noise and, therefore, by the SQL around the resonance frequency, see Eqs. (165View Equation, 172View Equation) and Figure 22View Image. However, practical implementation of this gain is limited by the following two shortcomings: (i) the stronger the sensitivity gain, the more narrow the frequency band in which it is achieved; see Eq. (171View Equation); (ii) in many cases, and, in particular, in a GW detection scenario with its low signal frequencies and heavy test masses separated by the kilometers-scale distances, ordinary solid-state springs cannot be used due to unacceptably high levels of mechanical loss and the associated thermal noise.

At the same time, in detuned Fabry–Pérot cavities, as well as in the detuned configurations of the Fabry–Pérot–Michelson interferometer, the radiation pressure force depends on the mirror displacement (see Eqs. (312View Equation)), which is equivalent to the additional rigidity, called the optical spring, inserted between the cavity mirrors. It does not introduce any additional thermal noise, except for the radiation pressure noise ˆFb.a., and, therefore, is free from the latter of the above mentioned shortcomings. Moreover, as we shall show below, spectral dependence of the optical rigidity K (Ω) alleviates, to some extent, the former shortcoming of the ‘ordinary’ rigidity and provides some limited sensitivity gain in a relatively broad band.

The electromagnetic rigidity was first discovered experimentally in radio-frequency systems [26]. Then its existence was predicted for the optical Fabry–Pérot cavities [25]. Much later it was shown that the excellent noise properties of the optical rigidity allows its use in quantum experiments with macroscopic mechanical objects [17Jump To The Next Citation Point, 23, 24]. The frequency dependence of the optical rigidity was explored in papers [32Jump To The Next Citation Point, 83Jump To The Next Citation Point, 33]. It was shown that depending on the interferometer tuning, either two resonances can exist in the system, mechanical and optical ones, or a single broader second-order resonance will exist.

In the last decade, the optical rigidity has been observed experimentally both in the table-top setup [48Jump To The Next Citation Point] and in the larger prototype interferometer [111].

6.3.2 The optical noise redefinition

In detuned interferometer configurations, where the optical rigidity arises, the phase shifts between the input and output fields, as well as between the input fields and the field, circulating inside the interferometer, depend in sophisticated way on the frequency Ω. Therefore, in order to draw full advantage from the squeezing, the squeezing angle of the input field should follow this frequency dependence, which is problematic from the implementation point of view. Due to this reason, considering the optical-rigidity–based regimes, we limit ourselves to the vacuum-input case only, setting π•Šsqz[r,πœƒ] = 𝕀 in Eq. (375View Equation).

In this case, it is convenient to redefine the input noise operators as follows:

√ -- √ --- √ --- γaˆnew = γ1ˆa + γ2ˆg, √ -- √ --- √ --- γ ˆgnew = ∘ γ1ˆg − ∘γ2ˆa, γ2- -γ- πœ–nˆnew = γ1 ˆg + γ1 πœ–dnˆ, (468 )
where
∘ ------ 1- γ1- πœ– = η − 1 and η = γ ηd (469 )
is the unified quantum efficiency, which accounts for optical losses both in the interferometer and in the homodyne detector.

Note that if the operators ˆa, ˆg, and ˆn describe mutually-uncorrelated vacuum noises, then the same is valid for the new ˆanew, ˆgnew, and nˆnew. Expressing Eqs. (371View Equation and 373View Equation) in terms of new noises (468View Equation) and renaming them, for brevity,

ˆanew → ˆa, ˆnnew → ˆn, (470 )
we obtain:
∘ ------- 𝒳ˆmeas (Ω) = --ℏ--------π’Ÿ-(Ω-)----HT (Ο•LO)[ℝ(Ω )ˆa(Ω ) + πœ–ˆn (Ω)], (471 ) 2M J γHT (Ο•LO )D (Ω) [ ]T β„±ˆ (Ω) = ∘2-ℏM--J-γ 1 𝕃 (Ω)ˆa (Ω), (472 ) b.a. 0
where ℝ(Ω ) is the lossless cavity reflection factor; see Eq. (550View Equation).

Thus, we have effectively reduced our lossy interferometer to the equivalent lossless one, but with less effective homodyne detector, described by the unified quantum efficiency η < ηd. Now we can write down explicit expressions for the interferometer quantum noises (376View Equation), (377View Equation) and (378View Equation), which can be calculated using Eqs. (552View Equation):

ℏ |π’Ÿ (Ω )|2 S𝒳 𝒳(Ω) = ---------2---2------2---2---- , 4M J γηΓ sin φ + Ω sin Ο•LO ℏM-J-γ(Γ 2-+-Ω2-) Sβ„± β„±(Ω) = |π’Ÿ(Ω )|2 , S𝒳 β„±(Ω) = ℏΓ-cos-φ-−-iΩ-cosΟ•LO-, (473 ) 2Γ sin φ − iΩ sin Ο•LO
where
∘ ------- Γ = γ2 + δ2, φ = Ο•LO − β, β = arctan δ-. (474 ) γ

6.3.3 Bad cavities approximation

We start our treatment of the optical rigidity with the “bad cavity” approximation, discussed in Section 6.1.2 for the resonance-tuned interferometer case. This approximation, in addition to its importance for the smaller-scale prototype interferometers, provides a bridge between our idealized harmonic oscillator consideration of Section 4.3.2 and the frequency-dependent rigidity case specific to the large-scale GW detectors, which will be considered below, in Section 6.3.4.

In the “bad cavity” approximation à ≫ Ω, the Eqs. (473View Equation) for the interferometer quantum noises, as well as the expression (374View Equation) for the optical rigidity can be significantly simplified:

2 S𝒳𝒳 = -----ℏΓ-------, Sβ„± β„± = ℏM--Jγ-, S 𝒳ℱ = ℏ-cotφ, (475 ) 4M Jγη sin2 φ Γ 2 2
and
M J δ K = ---2-. (476 ) Γ
Substituting these equations into the equation for the sum quantum noise (cf. Eq. (144View Equation)):
F 22 { 2 } S (Ω) = |K − M Ω |S 𝒳𝒳 + 2 Re [K − M Ω ]S𝒳 β„± + Sβ„± β„± = |Ke ff − M Ω2|2S𝒳 𝒳 + Seβ„±ffβ„±, (477 )
where
S S − |S |2 S Seβ„±ffβ„± = --𝒳𝒳--β„±β„±------𝒳ℱ-- and Keff = K − -𝒳ℱ-- (478 ) S𝒳 𝒳 S𝒳𝒳
stand for the effective back-action noise and effective optical rigidity, respectively, and dividing by SF SQL,f.m. defined by Eq. (161View Equation), we obtain the SQL beating factor (418View Equation):
2 1 [ 2 2 2 Γ 2 Jγ 2 ] ξ (Ω ) = --2 (Ω m − Ω ) --------2--+ --2(1 − ηcos φ) , (479 ) Ω 4Jγ ηsin φ Γ
where
2 Ke-ff -J- Ωm = M = Γ 2(δ − γη sin 2φ) (480 )
is the effective resonance frequency (which takes into account both real and virtual parts of the effective rigidity Ke ff). Following the reasoning of Section 4.3.2, it is easy to see that this spectral density allows for narrow-band sensitivity gain equal to
2Jγ ξ2(Ωm + ν) ≤ ξ2(Ωm ± Δ Ωβˆ•2 ) ≈ --2-2-(1 − ηcos2φ ) (481 ) Γ Ω m
within the bandwidth
∘ -------------------- ∘ ------------ Δ-Ω- -2J-γ- 2 2 2 --η-sin2-φ--- Ω = Γ 2Ω2 η sin φ(1 − η cos Ο•) = ξ (Ωm ± Δ Ωβˆ•2 ) 1 − η cos2φ . (482 ) m m
In the ideal lossless case (η = 1),
ΔΩ ----= ξ2(Ωm ± Δ Ω βˆ•2), (483 ) Ωm
in accord with Eq.(171View Equation). However, if η < 1, then the bandwidth, for a given ξ lessens gradually as the homodyne angle φ goes down. Therefore, the optimal case of the broadest bandwidth, for a given ξ, corresponds to φ = πβˆ•2, and, therefore, to SxF = 0 [see Eqs. (475View Equation)], that is, to the pure ‘real’ rigidity case with non-correlated radiation-pressure and shot noises. This result naturally follows from the above conclusion concerning the amenability of the quantum noise sources cross-correlation to the influence of optical loss.

Therefore, setting φ = π βˆ•2 in Eq. (479View Equation) and taking into account that

Ω2 Ω2 Jγ- = --qcos2 β, K--= Jδ-= -q-sin 2β, (484 ) Γ 2 2 M Γ 2 4
where Ω2q is the normalized optical power defined in Eq. (417View Equation), we obtain that
1 [ ( Ω2 )2 1 ] ξ2(Ω) = ---- -q-sin 2β − Ω2 ----------+ Ω2q cos2β . (485 ) 2Ω2 4 Ω2qη cos2β

Consider now the local minimization of this function at some given frequency Ω0, similar to one discussed in Section 6.1.2. Now, the optimization parameter is β, that is, the detuning δ of the interferometer. It is easy to show that the optimal β is given by the following equation:

2 Ω2 4Ω-0− --2--− --q(4η − 1)cos4β = 0. (486 ) Ω2q tanβ Ω20
This fifth-order equation for tan β cannot be solved in radicals. However, in the most interesting case of Ω0 β‰ͺ Ωq, the following asymptotic solution can easily be obtained:
π- 2Ω20 β ≈ 2 − Ω2 , (487 ) q
thus yielding
[( )2 2 ] 2 1- Ω2- -Ωq- 4Ω20 ξ (Ω ) ≈ 2 1 − Ω20 Ω20η + Ω2 . (488 ) q
View Image

Figure 43: Plots of the SQL beating factor (485View Equation) of the detuned interferometer, for different values of the normalized detuning: 0 ≤ β ≡ arctan(δβˆ•γ ) < π βˆ•2, and for unified quantum efficiency η = 0.95. Thick solid line: the common envelope of these plots. Dashed lines: the common envelopes (420View Equation) of the SQL beating factors for the virtual rigidity case, without squeezing, r = 0, and with 10 dB squeezing, 2r e = 10 (for comparison).

The function (485View Equation), with optimal values of β defined by the condition (486View Equation), is plotted in Figure 43View Image for several values of the normalized detuning. We assumed in these plots that the unified quantum efficiency is equal to η = 0.95. In the ideal lossless case η = 1, the corresponding curves do not differ noticeably from the plotted ones. It means that in the real rigidity case, contrary to the virtual one, the sensitivity is not affected significantly by optical loss. This conclusion can also be derived directly from Eqs. (485View Equation) and (488View Equation). It stems from the fact that quantum noise sources cross-correlation, amenable to the optical loss, has not been used here. Instead, the sensitivity gain is obtained by means of signal amplification using the resonance character of the effective harmonic oscillator response, provided by the optical rigidity.

The common envelope of these plots (that is, the optimal SQL-beating factor), defined implicitly by Eqs. (485View Equation) and (486View Equation), is also shown in Figure 43View Image. Note that at low frequencies, Ω β‰ͺ Ωq, it can be approximated as follows:

2 ξ2 (Ω) = 2Ω-- ≈ γ-, (489 ) env Ω2q δ
(actually, this approximation works well starting from ٠≲ 0.3Ωq). It follows from this equation that in order to obtain a sensitivity significantly better than the SQL level, the interferometer should be detuned far from the resonance, δ ≫ γ.

For comparison, we reproduce here the common envelopes of the plots of ξ2(Ω ) for the virtual rigidity case with η = 0.95; see Figure 36View Image (the dashed lines). It follows from Eqs. (489View Equation) and (420View Equation) that in absence of the optical loss, the sensitivity of the real rigidity case is inferior to that of the virtual rigidity one. However, even a very modest optical loss value changes the situation drastically. The noise cancellation (virtual rigidity) method proves to be advantageous only for rather moderate values of the SQL beating factor of ξ ≳ 0.5 in the absence of squeezing and ξ ≳ 0.3 with 10 dB squeezing. The conclusion is forced upon you that in order to dive really deep under the SQL, the use of real rather than virtual rigidity is inevitable.

Noteworthy, however, is the fact that optical rigidity has an inherent feature that can complicate its experimental implementation. It is dynamically unstable. Really, the expansion of the optical rigidity (374View Equation) into a Taylor series in Ω gives

M Jδ 2M J γδ K (Ω) ≈ ---2- + ----4--iΩ + ... (490 ) Γ Γ
The second term, proportional to iΩ, describes an optical friction, and the positive sign of this term (if δ > 0) means that this friction is negative.

The corresponding characteristic instability time is equal to

Γ 4 τinst = -----. (491 ) 2Jγδ
In principle, this instability can be damped by some feedback control system as analyzed in [32Jump To The Next Citation Point, 43]. However, it can be done without significantly affecting the system dynamics, only if the instability is slow in the timescale of the mechanical oscillations frequency:
Γ 2 J Ωm τinst = ------≈ -------≫ 1. (492 ) 2γΩm 2Ω3m ξ2
Taking into account that in real-life experiments the normalized optical power J is limited for technological reasons, the only way to get a more stable configuration is to decrease ξ, that is, to improve the sensitivity by means of increasing the detuning. Another way to vanquish the instability is to create a stable optical spring by employing the second pumping carrier light with opposite detuning as proposed in [48, 129]. The parameters of the second carrier should be chosen so that the total optical rigidity must have both positive real and imaginary parts in Eq. (490View Equation):
( ) ( ) M-J1δ1- M-J2δ2- 2M--J1γ1δ1 2M--J2γ2δ2 Ksum(Ω ) = K1 (Ω) + K2 (Ω ) ≈ Γ 21 + Γ 22 + iΩ Γ14 + Γ 42 + ... (493 )
that can always be achieved by a proper choice of the parameters J1,2, γ1,2 and δ1,2 (δ1δ2 < 0).

6.3.4 General case

Frequency-dependent rigidity.
In the large-scale laser GW detectors with kilometer-scale arm cavities, the interferometer bandwidth can easily be made comparable or smaller than the GW signal frequency Ω. In this case, frequency dependences of the quantum noise spectral densities (376View Equation), (377View Equation) and (378View Equation) and of the optical rigidity (374View Equation) influence the shape of the sum quantum noise and, therefore, the detector sensitivity.

Most quantum noise spectral density is affected by the effective mechanical dynamics of the probe bodies, established by the frequency-dependent optical rigidity (374View Equation). Consider the characteristic equation for this system:

− Ω2[(γ − iΩ )2 + δ2] + Jδ = 0. (494 )
In the asymptotic case of γ = 0, the roots of this equation are equal to
∘ ---------------- ∘ ---------------- 2 ∘ -4------ 2 ∘ -4------ Ω (m0)= δ-− δ- − J δ, Ω (o0) = δ-+ δ- − J δ, (495 ) 2 4 2 4
(hereafter we omit the roots with negative-valued real parts). The corresponding maxima of the effective mechanical susceptibility:
1 χeffxx(Ω) = -------------2 (496 ) K (Ω ) − M Ω
are, respectively, called the mechanical resonance (Ωm) and the optical resonance (Ωo) of the interferometer [32]. In order to clarify their origin, consider an asymptotic case of the weak optomechanical coupling, 3 J β‰ͺ δ. In this case,
∘ ------ J K (0) Ωm ≈ --3 = -----, Ωo ≈ δ. (497 ) δ M
It is easy to see that Ωm originates from the ordinary resonance of the mechanical oscillator consisting of the test mass M and the optical spring K [compare with Eq. (476View Equation)]. At the same time, Ωo, in this approximation, does not depend on the optomechanical coupling and, therefore, has a pure optical origin – namely, sloshing of the optical energy between the carrier power and the differential optical mode of the interferometer, detuned from the carrier frequency by δ.

In the realistic general case of γ ⁄= 0, the characteristic equation roots are complex. For small values of γ, keeping only linear in γ terms, they can be approximated as follows:

[ (0) ] [ (0) ] Ω = Ω(0) 1 + ∘-iγ-Ωm---- , Ω = Ω (0) 1 − ∘--iγΩ-o---- . (498 ) m m δ4 − 4J γ o o δ4 − 4J γ
Note that the signs of the imaginary parts correspond to a positive dumping for the optical resonance, and to a negative one (that is, to instability) for the mechanical resonance (compare with Eq. (490View Equation)).
View Image

Figure 44: Roots of the characteristic equation (494View Equation) as functions of the optical power, for γ βˆ•δ = 0.03. Solid lines: numerical solution. Dashed lines: approximate solution, see Eqs. (498View Equation)

In Figure 44View Image, the numerically-calculated roots of Eq. (494View Equation) are plotted as a function of the normalized optical power Jβˆ•δ3, together with the analytical approximate solution (498View Equation), for the particular case of γ βˆ•δ = 0.03. These plots demonstrate the peculiar feature of the parametric optomechanical interaction, namely, the decrease of the separation between the eigenfrequencies of the system as the optomechanical coupling strength goes up. This behavior is opposite to that of the ordinary coupled linear oscillators, where the separation between the eigenfrequencies increases as the coupling strength grows (the well-known avoided crossing feature).

As a result, if the optomechanical coupling reaches the critical value:

3 J = δ-, (499 ) 4
then, in the asymptotic case of γ = 0, the eigenfrequencies become equal to each other:
(0) (0) -δ-- Ω m = Ω o = Ω0 ≡ √2-. (500 )
If γ > 0, then some separation retains, but it gets smaller than 2 γ, which means that the corresponding resonance curves effectively merge, forming a single, broader resonance. This second-order pole regime, described for the first time in [83], promises some significant advantages for high-precision mechanical measurements, and we shall consider it in more detail below.

If J < Jcrit, then two resonances yield two more or less separated minima of the sum quantum noise spectral density, whose location on the frequency axis mostly depends on the detuning δ, and their depth (inversely proportional to their width) hinges on the bandwidth γ.The choice of the preferable configuration depends on the criterion of the optimization, and also on the level of the technical (non-quantum) noise in the interferometer.

View Image

Figure 45: Examples of the sum noise power (double-sided) spectral densities of the detuned interferometer. ‘Broadband’: double optimization of the Advanced LIGO interferometer for NS-NS inspiraling and burst sources in presence of the classical noises [93Jump To The Next Citation Point] (J = JaLIGO ≡ (2π × 100)3 s−3, Γ = 3100 s−1, β = 0.80, Ο•LO = π βˆ•2 − 0.44). ‘High-frequency’: low-power configuration suitable for detection of the GW signals from the millisecond pulsars, similar to one planned for GEO HF [169Jump To The Next Citation Point] [J = 0.1JaLIGO, −1 Γ = 2π × 1000 s, β = πβˆ•2 − 0.01, Ο•LO = 0]. ‘Second-order pole’: the regime close to the second-order pole one, which provides a maximum of the SNR for the GW burst sources given that technical noise is smaller than the SQL [Stech = 0.1SSQL, J = JaLIGO, Γ = 1050 s− 1, β = πβˆ•2 − 0.040, Ο•LO = 0.91]. In all cases, ηd = 0.95 and the losses part of the bandwidth − 1 γ2 = 1.875 s (which corresponds to the losses − 4 Aarm = 10 per bounce in the 4 km long arms).

Two opposite examples are drawn in Figure 45View Image. The first one features the sensitivity of a broadband configuration, which provides the best SNR for the GW radiation from the inspiraling neutron-star–neutron-star binary and, at the same time, for broadband radiation from the GW burst sources, for the parameters planned for the Advanced LIGO interferometer (in particular, the circulating optical power ℐarm = 840 kW, L = 4 km, and M = 40 kg, which translates to 3 −3 J = JaLIGO ≡ (2π × 100 ) s, and the planned technical noise). The optimization performed in [93] gave the quantum noise spectral density, labeled as ‘Broadband’ in Figure 45View Image. It is easy to notice two (yet not discernible) minima on this plot, which correspond to the mechanical and the optical resonances.

Another example is the configuration suitable for detection of the narrow-band GW radiation from millisecond pulsars. Apparently, one of two resonances should coincide with the signal frequency in this case. It is well to bear in mind that in order to create an optical spring with mechanical resonance in a kHz region in contemporary and planned GW detectors, an enormous amount of optical power might be required. This is why the optical resonance, whose frequency depends mostly on the detuning δ, should be used for this purpose. This is, actually, the idea behind the GEO HF project [169]. The example of this regime is represented by the curve labeled as ‘High-frequency’ in Figure 45View Image. Here, despite one order of magnitude less optical power used (J = 0.1JaLIGO), several times better sensitivity at frequency 1 kHz, than in the ‘Broadband’ regime, can be obtained. Note that the mechanical resonance in this case corresponds to 10 Hz only and therefore is virtually useless.

The second-order pole regime.
In order to clarify the main properties of the second-order pole regime, start with the asymptotic case of γ → 0. In this case, the optical rigidity and the mechanical susceptibility (496View Equation) read

-M--Jδ-- K (Ω) = δ2 − Ω2 , (501 ) 1 1 δ2 − Ω2 χexffx,dbl(Ω) = --------------= ------------------. (502 ) K (Ω ) − M Ω2 M Jδ − δ2Ω2 + Ω4
If condition (499View Equation) is satisfied, then in the close vicinity of the frequency Ω 0 (see Eq. (500View Equation)):
|Ω − Ω0 | β‰ͺ Ω0, (503 )
the susceptibility can be approximated as follows:
2 χeff,dbl(Ω) ≈ ------Ω0----- . (504 ) xx M (Ω20 − Ω2 )2
Note that this susceptibility is proportional to the square of the susceptibility of the ordinary oscillator,
osc -----1------ χ xx(Ω) = M (Ω20 − Ω2 ), (505 )
and has the second-order pole at the frequency Ω0 (thus the name of this regime).

This ‘double-resonance’ feature creates a stronger response to the external forces with spectra concentrated near the frequency Ω0, than in the ordinary harmonic oscillator case. Consider, for example, the resonance force F0sinΩ0t. The response of the ordinary harmonic oscillator with eigenfrequency Ω0 on this force increases linearly with time:

xosc(t) = -F0t--sinΩ0t, (506 ) 2M Ω0
while that of the second-order pole object grows quadratically:
( ) x (t) = − -F0- t2cosΩ t − sin-Ω0t- . (507 ) dbl 8M 0 Ω0
It follows from Eq. (151View Equation) that due to this strong response, the second-order pole test object has a reduced value of the SQL around Ω0 by contrast to the harmonic oscillator.

Consider the quantum noise of the system, consisting of this test object and the SQM (that is, the Heisenberg’s-uncertainty-relation–limited quantum meter with frequency-independent and non-correlated measurement and back-action noises; see Section 4.1.1), which monitors its position. Below we show that the real-life long-arm interferometer, under some assumptions, can be approximated by this model.

The sum quantum noise power (double-sided) spectral density of this system is equal to

h 4 eff,dbl 2 S (Ω ) = M-2L2-Ω4(|χxx (Ω)| S𝒳𝒳 + Sβ„±β„± ). (508 )
If the frequency Ω is close to Ω0:
Ω = Ω0 + ν, |ν| β‰ͺ Ω0, (509 )
then
( ) h --2ℏ---- 16ν4- 2 S (Ω0 + ν) ≈ M L2Ω4 Ω2 + Ω q , (510 ) 0 q
and
Sh (Ω + ν) 1 ( 16ν4 ) ξ2(Ω0 + ν) ≡ -h-----0--------≈ ---2 ---2-+ Ω2q , (511 ) SSQLf.m.(Ω0 + ν) 2Ω 0 Ω q
(compare with Eq. (165View Equation)), where the frequency Ω q is defined by Eq. (155View Equation).

The same minimax optimization as performed in Section 4.3.2 for the harmonic oscillator case gives that the optimal value of Ωq is equal to

Ωq = ΔΩ, (512 )
and in this case,
|| ( Δ Ω )2 ξ2(Ω0 + ν)| ≤ ξ2(Ω0 ± Δ Ω βˆ•2) = ---- . (513 ) |ν|≤ΔΩβˆ•2 Ω0
Comparison of Eqs. (513View Equation) and (171View Equation) shows that for a given SQL-beating bandwidth Δ Ω, the second-order pole system can provide a much stronger sensitivity gain (i.e., much smaller value of ξ2), than the harmonic oscillator, or, alternatively, much broader bandwidth Δ Ω for a given value of 2 ξ. It is noteworthy that the factor (513View Equation) can be made smaller than the normalized oscillator SQL 2|ν|βˆ•Ω0 (see Eq. (171View Equation)), which means beating not only the free mass SQL, but also the harmonic oscillator one.
View Image

Figure 46: Left panel: the SQL beating factors ξ2 for Ωq βˆ•Ω0 = 0.1. Thick solid: the second-order pole system (511View Equation); dots: the two-pole system with optimal separation between the poles (528View Equation), (529View Equation); dashes: the harmonic oscillator (170View Equation); thin solid – SQL of the harmonic oscillator (171View Equation). Right: the normalized SNR (526View Equation). Solid line: analytical optimization, Eq. (530View Equation); pluses: numerical optimization of the spectral density (514View Equation) in the lossless case (η = 1); diamonds: the same for the interferometer with J = (2π × 100)3 s−3, ηd = 0.95 and the losses part of the bandwidth γ2 = 1.875 s−1 (which corresponds to the losses Aarm = 10− 4 per bounce in the 4 km long arms).

This consideration is illustrated by the left panel of Figure 46View Image, where the factors ξ2 for the harmonic oscillator (170View Equation) and of the second-order pole system (511View Equation) are plotted for the same value of the normalized back-action noise spectral density (Ωq βˆ•Ω0)2 = 0.01, as well as the normalized oscillator SQL (171View Equation).

Now return to the quantum noise of a real interferometer. With account of the noises redefinition (468View Equation), Eq. (385View Equation) for the sum quantum noise power (double-sided) spectral density takes the following form:

{ [ ]2 Sh (Ω) = ---ℏ---------------1---------- γ2 − δ2 + Ω2 + -J-(δ − γ sin 2Ο•LO ) M L2J γΩ2 sin2Ο•LO + Γ 2 sin2 φ Ω2 ( J )2 || J δ||2} +4γ2 δ − ---sin2 Ο•LO + πœ–2||π’Ÿ (Ω) − ---|| . (514 ) Ω2 Ω2
Suppose that the interferometer parameters satisfy approximately the second-order pole conditions. Namely, introduce a new parameter Λ defined by the following equation:
J(δ − γ sin 2Ο•LO) = Ω40 − Ω20Λ2, (515 )
where the frequency Ω0 is defined by Eq. (500View Equation), and assume that
2 2 2 ν ∼ Λ ∼ Ω0 γ β‰ͺ Ω 0. (516 )
Keeping only the first non-vanishing terms in ν2, Λ2, and γ in Eq. (514View Equation), we obtain that
( ) 2ℏ 1 { [( Ω γ )2 ] } Sh(Ω0 + ν ) ≈ ------4- --- (4ν2 − Λ2)2 + πœ–2 4 ν2 − Λ2 + √-0-sin2 Ο•LO + 4γ2Ω20 + Ω2q(,517 ) M L2 Ω0 Ω2q 2
where
2 √ -- 2 Ωq = 2γ Ω0(1 + cos Ο•LO ). (518 )
It follows from Eq. (517View Equation) that the parameter Λ is equal to the separation between the two poles of the susceptibility eff,dbl χxx.

It is evident that the spectral density (517View Equation) represents a direct generalization of Eq. (510View Equation) in two aspects. First, it factors in optical losses in the interferometer. Second, it includes the case of Λ β„= 0. We show below that a small yet non-zero value of Λ allows one to further increase the sensitivity.

Optimization of the signal-to-noise ratio.
The peculiar feature of the second-order pole regime is that, while being, in essence, narrow-band, it can provide an arbitrarily-high SNR for the broadband signals, limited only by the level of the additional noise of non-quantum (technical) origin. At the same time, in the ordinary harmonic oscillator case, the SNR is fundamentally limited.

In both the harmonic oscillator and the second-order pole test object cases, the quantum noise spectral density has a deep and narrow minimum, which makes the major part of the SNR integral. If the bandwidth of the signal force exceeds the width of this minimum (which is typically the case in GW experiments, save to the narrow-band signals from pulsars), then the SNR integral (453View Equation) can be approximated as follows:

2 ∫ ∞ ρ2 = |hs(Ω0-)|- ----dν-----. (519 ) π −∞ Sh(Ω0 + ν )
It is convenient to normalize both the signal force and the noise spectral density by the corresponding SQL values, which gives:
2 M-L2-Ω30- 2 2 ρ = 4ℏ |hs(Ω0 )|σ , (520 )
where
∫ ∞ σ2 = -1-- ----dν----- (521 ) πΩ0 −∞ ξ2(Ω0 + ν )
is the dimensionless integral sensitivity measure, which we shall use here, and
M L2Ω2 ξ2(Ω0 + ν) = -------0Sh(Ω0 + ν). (522 ) 4ℏ

For a harmonic oscillator, using Eq. (170View Equation), we obtain

σ2 = 1. (523 )
This result is natural, since the depth of the sum quantum-noise spectral-density minimum (which makes the dominating part of the integral (521View Equation)) in the harmonic oscillator case is inversely proportional to its width ΔΩ, see Eq. (171View Equation). As a result, the integral does not depend on how small the minimal value of 2 ξ is.

The situation is different for the second-order pole-test object. Here, the minimal value of ξ2 is proportional to (Δ Ω)−2 (see Eq. (513View Equation)) and, therefore, it is possible to expect that the SNR will be proportional to

σ2 ∝ ---1---× ΔΩ ∝ -1-- ∝ 1-. (524 ) (Δ Ω )2 Δ Ω ξ
Indeed, after substitution of Eq. (511View Equation) into (521View Equation), we obtain:
σ2 = √1--Ω0-= ---1---. (525 ) 2 Ωq 2ξ(Ω0)
Therefore, decreasing the width of the dip in the sum quantum noise spectral density and increasing its depth, it is possible, in principle, to obtain an arbitrarily high value of the SNR.

Of course, it is possible only if there are no other noise sources in the interferometer except for the quantum noise. Consider, though, a more realistic situation. Let there be an additional (technical) noise in the system with the spectral density Stech(Ω). Suppose also that this spectral density does not vary much within our frequency band of interest ΔΩ. Then the factor 2 σ can be approximated as follows:

1 ∫ ∞ dν σ2 = ---- -2------------2---, (526 ) πΩ0 −∞ ξ (Ω0 + ν ) + ξtech
where
S (Ω ) ξ2tech = ---tech---0--. (527 ) SSQLf.m.(Ω0)
Concerning quantum noise, we consider the regime close (but not necessarily exactly equal) to that yielding the second-order pole, that is, we suppose 0 ≤ Λ β‰ͺ Ω 0. In order to simplify our calculations, we neglect the contribution from optical loss into the sum spectral density (we show below that it does not affect the final sensitivity much). Thus, as follows from Eq. (517View Equation), one gets
1 [(4ν2 − Λ2 )2 ] ξ2(Ω0 + ν) = ---2 ------2-----+ Ω2q . (528 ) 2Ω 0 Ω q
In the Appendix A.3, we calculate integral Eq. (526View Equation) and optimize it over Λ and Ωq. The optimization gives the best sensitivity, for a given value of 2 ξtech, is provided by
Λ = Ωq = Ω0 ξtech. (529 )
In this case,
1 σ2 = -√-------. (530 ) 2 2ξtech
The pure second-order pole regime (Λ = 0), with the same optimal value of Ω q, provides slightly worse sensitivity:
σ2 = ∘---√1-----. (531 ) 6 3 ξtech
The optimized function (528View Equation) is shown in Figure 46View Image (left) for the particular case of Λ = Ωq = 0.1Ω0. In Figure 46View Image (right), the optimal SNR (530View Equation) is plotted as a function of the normalized technical noise ξ2tech.

In order to verify our narrow-band model, we optimized numerically the general normalized SNR for the broadband burst-type signals:

2ℏ ∫ ∞ dΩ βˆ•Ω σ2burst = -----2--2 --h-------h--, (532 ) πM L Ω 0 −∞ S (Ω) + Stech
where Sh is the sum quantum noise of the interferometer defined by Eq. (514View Equation). The only assumption we have made here is that the technical noise power (double-sided) spectral density
2ℏ Shtech = ----2-2-ξ2tech (533 ) M L Ω0
does not depend on frequency, which is reasonable, since only the narrow frequency region around Ω 0 contributes noticeably to the integral (532View Equation). The result is shown in Figure 46View Image (right) for two particular cases: the ideal (no loss) case with η = 1, and the realistic case of the interferometer with 3 −3 J = (2π × 100 ) s, ηd = 0.95 and −1 γ2 = 1.875 s (which corresponds to the loss factor of Aarm = 10− 4 per bounce in the 4 km long arms; see Eq. (322View Equation)). The typical optimized quantum noise spectral density (for the particular case of ξ2 = 0.1 tech) is plotted in Figure 45View Image.

It is easy to see that the approximations (528View Equation) work very well, even if ξtech ∼ 1 and, therefore, the assumptions (516View Equation) cease to be valid. One can conclude, looking at these plots, that optical losses do not significantly affect the sensitivity of the interferometer, working in the second-order pole regime. The reason behind it is apparent. In the optical rigidity based systems, the origin of the sensitivity gain is simply the resonance increase of the probe object dynamical response to the signal force, which is, evidently, immune to the optical loss.

The only noticeable discrepancy between the analytical model and the numerical calculations for the lossless case, on the one hand, and the numerical calculations for the lossy case, on the other hand, appears only for very small values of ξ2 ∼ 0.01 tech. It follows from Eqs. (518View Equation) and (529View Equation) that this case corresponds to the proportionally reduced bandwidth of the interferometer,

γ ∼ Ω0 ξ2 ∼ 10 s−1 (534 ) tech
(for a typical value of 3 − 1 Ω0 ∼ 10 s). Therefore, the loss-induced part of the total bandwidth γ2, which has no noticeable effect on the unified quantum efficiency η [see Eq. (469View Equation)] for the ‘normal’ broadband values of γ ∼ 103 s−1, degrades it in this narrow-band case. However, it has to be emphasized that the degradation of σ2, for the reasonable values of ξ2 tech, is only about a few percent, and even for the quite unrealistic case of 2 ξtech = 0.01, does not exceed ∼ 25%.


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