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. (378) tells us that this cross-correlation can be created by tuning either the homodyne angle , the squeezing angle , or the detuning . 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 , 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, 159, 90, 70, 69, 149, 9]. In particular, in [90], 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, . 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 . In broadband interferometer configurations considered here, with typical values of , 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 , and the arms lengths of the large-scale GW detectors are equal to several kilometers, the values of , and, correspondingly, , are feasible. At the same time, the value of photodetector quantum inefficiency (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 can be much smaller and influence of 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 (376), (377) and (378) can be rewritten in the following explicit form:

where is the convenient optomechanical coupling factor introduced in [90].Eq. (381) and (385) for the sum quantum noise and its power (double-sided) spectral density in this case can also be simplified significantly:

where

In Section 6.1.2 we consider the optimization of the spectral density (391), 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.

It is evident that this minimum is provided by

In this case, the sum quantum noise power (double-sided) spectral density is equal to It is easy to note the similarity of this spectral density with the ones of the toy position meter considered above, see Eq. (173). 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 of the interferometer. If and , then Eqs. (173) and (394) become identical, with the evident correspondence In particular, the spectral density (394) can never be smaller than the free mass SQL (see (174)). Indeed, it can be minimized at any given frequency by setting and in this case,The spectral density (394) 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 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, .

The noise spectral density curves for the resonance-tuned interferometer are drawn in Figure 34. 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 corresponds to the circulating power of , , and ; the interferometer bandwidth is close to the one providing the best sensitivity for Advanced LIGO in the presence of technical noise [93]; 10 dB squeezing (), which corresponds to the best squeezing available at the moment (2011) in the low-frequency band [102, 150, 151]); 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.

The sum quantum noise power (double-sided) spectral density (403) is plotted in Figure 35 for the ideal lossless case and for (dotted line). In both cases, the optical power and the squeezing factor are equal to and , respectively.

Compare this spectral density with the one for the frequency-dependent squeezing angle (pre-filtering) case, see Eq. (403). 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 35 (left). For the parameters of the noise curves presented in Figure 35 (right), the post-filtering still has some advantage of about 40% in the back-action noise amplitude .

Note that the required frequency dependences (404) and (407) in both cases are similar to each other (and become exactly equal to each other in the lossless case ). 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, .

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

and is equal to It also follows from Eqs.(400) and (409) that the optimal squeezing angle in this case is given byIt is easy to see that in the ideal lossless case the double-filtering configuration reduces to a post-filtering one. Really, if , the spectral density (410) becomes exactly equal to that for the post-filtering case (408), and the frequency dependent squeezing angle (411) degenerates into a constant value (405). However, if , then the additional pre-filtering allows one to decrease more the residual back-action term. For example, if and then the gain in the back-action noise amplitude is equal to about 25%.

We have plotted the sum quantum noise spectral density (410) in Figure 34, 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 (410), there exists an optimal value of the coupling factor (that is, the optical power) which provides the minimum to the sum quantum noise spectral density at any given frequency :

The minimum is equal to This limitation is severe. The reasonably optimistic value of quantum efficiency that we use for our estimates corresponds to . It means that without squeezing () one is only able to beat the SQL in amplitude by The gain can be improved using squeezing and, if then, in principle, arbitrarily high sensitivity can be reached. But depends on only as , and for the 10 dB squeezing, only a modest value of can be obtained.In our particular case, the fact that the additional noise associated with the photodetector quantum inefficiency 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. (410) 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, . Therefore, this method calls for long filter cavities. In particular, in the original paper [90], 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.

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 in those devices is typically much larger than the mechanical frequencies . If one takes, e.g., the power transmissivity value of for the ITMs and length of arms equal to , then , 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 can be approximated as:

where Note that in this approximation, the noise spectral densities (386), (387) and (388) turn out to be frequency independent.In Figure 36, the SQL beating factor

is plotted for the sum quantum noise spectral density with the following values of homodyne and squeezing angles and factoring in the “bad cavity” condition (416). The four panes correspond to the following four combinations: (upper left) no losses () and no squeezing (); (lower left) no losses () and 10 dB squeezing (); (upper right) with losses () and no squeezing (); (lower right) with losses () and 10 dB squeezing(). In each pane, the family of plots is shown that corresponds to different values of the ratio , ranging from 0.1 to .The minima of these plots form the common envelope, given by Eqs. (410) and (416):

which is also plotted in Figure 36. It is easy to see that in the ideal case of , there is no limitation on the SQL beating factor, provided a sufficiently small ratio of : However, if , then function (420) has a minimum in at [compare with Eq. (412)], equal to (413).

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

where and are the two-photon quadrature amplitude vectors of the input and output beams, stands for noise fields entering the cavity due to optical losses (which are assumed to be in a vacuum state), where is the power transmittance of the input/output mirror, is the factor of power loss per bounce, is the filter cavity length, is its half-bandwidth, and is its detuning.In order to demonstrate how the filter cavity works, consider the particular case of the lossless cavity. In this case,

and the reflection matrix describes field amplitude rotations with the frequency-dependent rotation angle: where The phase factor 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 and squeezing factor and thus can be described by the following two-photon quadrature vector

where the quadratures vector describes the vacuum state. After reflecting off the filter cavity, this light will be described with the following expression [see Eq. (75)], where also describes the light field in a vacuum state. Thus, the pre-filtering indeed rotates the squeezing angle by a frequency-dependent angle .In a similar manner, we can consider the post-filtering schemes. Consider a homodyne detection scheme with losses, described by Eq. (272). Suppose that prior to detection, the light described by the quadrature vector , reflects from the filter cavity. In this case, the photocurrent (in Fourier representation) is proportional to

where 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 .It is easy to see that the necessary frequency dependencies of the homodyne and squeezing angles (404) or (407) (with the second-order polynomials in in the r.h.s. denominators) cannot be implemented by the rotation angle (432) (with its first order in 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 . However, when

which is actually the case for the planned second and third generation GW detectors, the factor can be well approximated by Eq. (416) 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.

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

which yields the following expression for a power (double-sided) spectral densityIn the ideal lossless filter cavity case, taking into account Eq. (431), this spectral density can be simplified as follows:

[compare with Eq. (391)]. In this case, the necessary frequency dependence of the squeezing angle (404) can be implemented by the following filter cavity parameters: whereAlong similar lines, the post-filtering scheme drawn in the right panel of Figure 37 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 , 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 . Assuming that the squeezing angle is defined by Eq. (405) and again using Eqs. (390) and (423), we obtain that the sum quantum noise and its power (double-sided) spectral density are equal to

and In the ideal lossless filter cavity case, factoring in Eq. (431), this spectral density takes a form similar to (391), but with the frequency-dependent homodyne angle: where The necessary frequency dependence (407) of this effective homodyne angle can be implemented by the following parameters of the filter cavity: 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 (441) and (447) into Eqs. (440) and (445), respectively, taking Condition (437) into account, results in spectral densities for the ideal frequency dependent squeezing and homodyne angle, see Eqs. (403) and (408).

In the general case of lossy filter cavities, the conditions (404) and (407) 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:

This inequality can be rewritten as the following condition for the filter cavity specific losses: In particular, for our standard parameters used for numerical estimates (, , ), we obtain , and there should be (the r.h.s. corresponds, in particular, to a 50 m filter cavity with the losses per bounce ).Another more crude limitation can be obtained from the condition that should be small compared to the filter cavity bandwidth :

Apparently, were it not the case, the filter cavity would just cease to work properly. For the same numerical values of and as above, we obtain: (for example, a very short 2.5 m filter cavity with or 25 m cavity with ).

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 [131, 124]. In this case, the SNR is equal to (see [58])

where is a factor that does not depend on the interferometer parameters, and 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, .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 and , providing this gain, we choose to normalize the SNR by the value corresponding to the ordinary interferometer (without the filter cavities):

with power (double-sided) spectral density [see Eq. (394)].We optimized numerically the ratio , with filter cavity half-bandwidth and detuning as the optimization parameters, for the values of the specific loss factor ranging from (e.g., very long 10 km filter cavity with ) to (e.g., 10 m filter cavity with ). Concerning the main interferometer parameters, we used the same values as in all our previous examples, namely, , , and .

The results of the optimization are shown in Figure 38. In the left pane, the optimal values of the filter cavity parameters and 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 and are virtually the same as , while the specific loss factor satisfies the condition (448), and starts to deviate sensibly from only when approaches the limit (451). 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).

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 (phase quadrature) at high frequencies to (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 in the denominator of Eq. (445)]. In the pre-filtering case there is no such effect, for the value of , 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 39 for several typical values of the specific loss factor, and for the same values of the rest of the parameters, as in Figure 38. For comparison, the spectral densities for the ideal frequency-dependent squeezing angle Eqs. (403) and homodyne angle (408) are also shown. These plots clearly demonstrate that providing sufficiently-low optical losses (say, ), 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.

Living Rev. Relativity 15, (2012), 5
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