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. (165, 172) and Figure 22. 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. (171); (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. (312)), 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 , and, therefore, is free from the latter of the above mentioned shortcomings. Moreover, as we shall show below, spectral dependence of the optical rigidity 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 [17, 23, 24]. The frequency dependence of the optical rigidity was explored in papers [32, 83, 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 [48] and in the larger prototype interferometer [111].

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 . 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 in Eq. (375).

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

where is the unified quantum efficiency, which accounts for optical losses both in the interferometer and in the homodyne detector.Note that if the operators , , and describe mutually-uncorrelated vacuum noises, then the same is valid for the new , , and . Expressing Eqs. (371 and 373) in terms of new noises (468) and renaming them, for brevity,

we obtain: where is the lossless cavity reflection factor; see Eq. (550).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 . Now we can write down explicit expressions for the interferometer quantum noises (376), (377) and (378), which can be calculated using Eqs. (552):

where

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 , the Eqs. (473) for the interferometer quantum noises, as well as the expression (374) for the optical rigidity can be significantly simplified:

and Substituting these equations into the equation for the sum quantum noise (cf. Eq. (144)): where stand for the effective back-action noise and effective optical rigidity, respectively, and dividing by defined by Eq. (161), we obtain the SQL beating factor (418): where is the effective resonance frequency (which takes into account both real and virtual parts of the effective rigidity ). 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 within the bandwidth In the ideal lossless case (), in accord with Eq.(171). However, if , then the bandwidth, for a given lessens gradually as the homodyne angle goes down. Therefore, the optimal case of the broadest bandwidth, for a given , corresponds to , and, therefore, to [see Eqs. (475)], 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 in Eq. (479) and taking into account that

where is the normalized optical power defined in Eq. (417), we obtain thatConsider now the local minimization of this function at some given frequency , similar to one discussed in Section 6.1.2. Now, the optimization parameter is , that is, the detuning of the interferometer. It is easy to show that the optimal is given by the following equation:

This fifth-order equation for cannot be solved in radicals. However, in the most interesting case of , the following asymptotic solution can easily be obtained: thus yieldingThe function (485), with optimal values of defined by the condition (486), is plotted in Figure 43 for several values of the normalized detuning. We assumed in these plots that the unified quantum efficiency is equal to . In the ideal lossless case , 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. (485) and (488). 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. (485) and (486), is also shown in Figure 43. Note that at low frequencies, , it can be approximated as follows:

(actually, this approximation works well starting from ). 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, .For comparison, we reproduce here the common envelopes of the plots of for the virtual rigidity case with ; see Figure 36 (the dashed lines). It follows from Eqs. (489) and (420) 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 in the absence of squeezing and 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 (374) into a Taylor series in gives

The second term, proportional to , describes an optical friction, and the positive sign of this term (if ) means that this friction is negative.The corresponding characteristic instability time is equal to

In principle, this instability can be damped by some feedback control system as analyzed in [32, 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: Taking into account that in real-life experiments the normalized optical power is limited for technological reasons, the only way to get a more stable configuration is to decrease , 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. (490): that can always be achieved by a proper choice of the parameters , and ().

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

In the asymptotic case of , the roots of this equation are equal to (hereafter we omit the roots with negative-valued real parts). The corresponding maxima of the effective mechanical susceptibility: are, respectively, called the mechanical resonance () and the optical resonance () of the interferometer [32]. In order to clarify their origin, consider an asymptotic case of the weak optomechanical coupling, . In this case, It is easy to see that originates from the ordinary resonance of the mechanical oscillator consisting of the test mass and the optical spring [compare with Eq. (476)]. At the same time, , 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 .In the realistic general case of , the characteristic equation roots are complex. For small values of , keeping only linear in terms, they can be approximated as follows:

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. (490)).In Figure 44, the numerically-calculated roots of Eq. (494) are plotted as a function of the normalized optical power , together with the analytical approximate solution (498), for the particular case of . 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:

then, in the asymptotic case of , the eigenfrequencies become equal to each other: If , then some separation retains, but it gets smaller than , 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 , 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 , and their depth (inversely proportional to their width) hinges on the bandwidth .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.

Two opposite examples are drawn in Figure 45. 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 , , and , which translates to , and the planned technical noise). The optimization performed in [93] gave the quantum noise spectral density, labeled as ‘Broadband’ in Figure 45. 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 , 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 45. Here, despite one order of magnitude less optical power used (), 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.

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

while that of the second-order pole object grows quadratically: It follows from Eq. (151) that due to this strong response, the second-order pole test object has a reduced value of the SQL around 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

If the frequency is close to : then and (compare with Eq. (165)), where the frequency is defined by Eq. (155).The same minimax optimization as performed in Section 4.3.2 for the harmonic oscillator case gives that the optimal value of is equal to

and in this case, Comparison of Eqs. (513) and (171) shows that for a given SQL-beating bandwidth , the second-order pole system can provide a much stronger sensitivity gain (i.e., much smaller value of ), than the harmonic oscillator, or, alternatively, much broader bandwidth for a given value of . It is noteworthy that the factor (513) can be made smaller than the normalized oscillator SQL (see Eq. (171)), which means beating not only the free mass SQL, but also the harmonic oscillator one.This consideration is illustrated by the left panel of Figure 46, where the factors for the harmonic oscillator (170) and of the second-order pole system (511) are plotted for the same value of the normalized back-action noise spectral density , as well as the normalized oscillator SQL (171).

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

Suppose that the interferometer parameters satisfy approximately the second-order pole conditions. Namely, introduce a new parameter defined by the following equation: where the frequency is defined by Eq. (500), and assume that Keeping only the first non-vanishing terms in , , and in Eq. (514), we obtain that where It follows from Eq. (517) that the parameter is equal to the separation between the two poles of the susceptibility .It is evident that the spectral density (517) represents a direct generalization of Eq. (510) in two aspects. First, it factors in optical losses in the interferometer. Second, it includes the case of . We show below that a small yet non-zero value of allows one to further increase the sensitivity.

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 (453) can be approximated as follows:

It is convenient to normalize both the signal force and the noise spectral density by the corresponding SQL values, which gives: where is the dimensionless integral sensitivity measure, which we shall use here, andFor a harmonic oscillator, using Eq. (170), we obtain

This result is natural, since the depth of the sum quantum-noise spectral-density minimum (which makes the dominating part of the integral (521)) in the harmonic oscillator case is inversely proportional to its width , see Eq. (171). As a result, the integral does not depend on how small the minimal value of is.The situation is different for the second-order pole-test object. Here, the minimal value of is proportional to (see Eq. (513)) and, therefore, it is possible to expect that the SNR will be proportional to

Indeed, after substitution of Eq. (511) into (521), we obtain: 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 . Suppose also that this spectral density does not vary much within our frequency band of interest . Then the factor can be approximated as follows:

where Concerning quantum noise, we consider the regime close (but not necessarily exactly equal) to that yielding the second-order pole, that is, we suppose . 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. (517), one gets In the Appendix A.3, we calculate integral Eq. (526) and optimize it over and . The optimization gives the best sensitivity, for a given value of , is provided by In this case, The pure second-order pole regime (), with the same optimal value of , provides slightly worse sensitivity: The optimized function (528) is shown in Figure 46 (left) for the particular case of . In Figure 46 (right), the optimal SNR (530) is plotted as a function of the normalized technical noise .In order to verify our narrow-band model, we optimized numerically the general normalized SNR for the broadband burst-type signals:

where is the sum quantum noise of the interferometer defined by Eq. (514). The only assumption we have made here is that the technical noise power (double-sided) spectral density does not depend on frequency, which is reasonable, since only the narrow frequency region around contributes noticeably to the integral (532). The result is shown in Figure 46 (right) for two particular cases: the ideal (no loss) case with , and the realistic case of the interferometer with , and (which corresponds to the loss factor of per bounce in the long arms; see Eq. (322)). The typical optimized quantum noise spectral density (for the particular case of ) is plotted in Figure 45.It is easy to see that the approximations (528) work very well, even if and, therefore, the assumptions (516) 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 . It follows from Eqs. (518) and (529) that this case corresponds to the proportionally reduced bandwidth of the interferometer,

(for a typical value of ). Therefore, the loss-induced part of the total bandwidth , which has no noticeable effect on the unified quantum efficiency [see Eq. (469)] for the ‘normal’ broadband values of , degrades it in this narrow-band case. However, it has to be emphasized that the degradation of , for the reasonable values of , is only about a few percent, and even for the quite unrealistic case of , does not exceed 25%.

Living Rev. Relativity 15, (2012), 5
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