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 . Then its existence was predicted for the optical Fabry–Pérot cavities . 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  and in the larger prototype interferometer .
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: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,
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):
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:
Therefore, setting in Eq. (479) and taking into account that
Consider 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:
The 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:
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 corresponding characteristic instability time is equal to[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: 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):
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:mechanical resonance () and the optical resonance () of the interferometer . In order to clarify their origin, consider an asymptotic case of the weak optomechanical coupling, . In this case,
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:
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:second-order pole regime, described for the first time in , 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  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 . 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:
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
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
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:
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:
For a harmonic oscillator, using Eq. (170), we obtain
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
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:
In order to verify our narrow-band model, we optimized numerically the general normalized SNR for the broadband burst-type signals:
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,
Living Rev. Relativity 15, (2012), 5
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