The SQM has noncorrelated effective measurement and backaction noises that results in . Apparently, under these conditions and turn into and of Eqs. (112) and (114), respectively. Hence, we will use instead of and instead of Then the uncertainty relation (148) transforms into:
The SQL is the name for an ultimate lower bound of a sum noise spectral density the SQM can, in principle, have at any given frequency . To derive this limit we assume noise sources and to have minimal values allowed by quantum mechanics, i.e.
Then, using this condition, one can minimize SQM’s sum noise:

to yield:
, that is achieved when contributions of measurement noise and backaction noise to the sum noise are equal to each other, i.e., whenIt is instructive to cite the forms of the SQL in other normalizations, i.e., for normalization and for normalization. The former is obtained from (151) via multiplication by :
The latter is obtained fromEq. (151) using the obvious connection between force and displacement :These limits look fundamental. There are no parameters of the meter (only as a reminder of the uncertainty relation (116)), and only the probe’s dynamics is in there. Nevertheless, this is not the case and, actually, this limit can be beaten by more sophisticated, but still linear, position meters. At the same time, the SQL represents an important landmark beyond which the ordinary bruteforce methods of sensitivity improving cease working, and methods that allow one to blot out the backaction noise from the meter output signal have to be used instead. Due to this reason, the SQL, and especially the SQL for the simplest test object – free mass – is usually considered as a borderline between the classical and the quantum domains.
In the rest of this section, we consider in more detail the SQLs for a free mass and for a harmonic oscillator. We also assume the minimal quantum noise requirement (150) to hold.
The free mass is not only the simplest model for the probe’s dynamics, but also the most important class of test objects for GW detection. Test masses of GW detectors must be isolated as much as possible from the noisy environment. To this end, the design of GW interferometers implies suspension of the test masses on thin fibers. The real suspensions are rather sophisticated and comprise several stages slung one over another, with mechanical eigenfrequencies in range. The sufficient degree of isolation is provided at frequencies much higher than , where the dynamics of test masses can be approximated with good precision by that of a free mass.
Let us introduce the following convenient measure of measurement strength (precision) in the first place:
Using the uncertainty relation (150), the noise spectral densities and can be expressed through as follows: Therefore, the larger is, the smaller is (the higher is the measurement precision), and the larger is (the stronger the meter back action is).In the case of interferometers, is proportional to the circulating optical power. For example, for the toy optical meter considered above,
see Eqs. (123). Using this notation, and taking into account that for a free mass , the sum quantum noise power (doublesided) spectral density can be written as follows: The SQL optimization (152) takes the following simple form in this case: giving:This consideration is illustrated by Figure 22 (left), where power (doublesided) spectral density (159) is plotted for three different values of . It is easy to see that these plots never dive under the SQL line (161), which embodies a common envelope for them. Due to this reason, the sensitivities area above this line is typically considered as the ‘classical domain’, and below it – as the ‘quantum domain’.
The simplest way to overcome the limit (161), which does not require any quantum tricks with the meter, is to use a harmonic oscillator as a test object, instead of the free mass. It is easy to see from Eq.(151) that the more responsive the test object is at some given frequency (that is, the bigger ) is, the smaller its force SQL at this frequency is. In the harmonic oscillator case,
with standing for the oscillator mechanical eigenfrequency, and the sum quantum noise power (doublesided) spectral density equal to Due to a strong response of the harmonic oscillator near resonance, the first (measurement noise) term in Eq. (159) goes to zero in the vicinity of . Therefore, reducing the value of , that is, using weaker measurement, it is possible to increase the sensitivity in a narrow band around . At the same time, the smaller is, the more narrow the bandwidth is where this sensitivity is achieved, as can be seen from the plots drawn in Figure 22 (right).Consider, in particular, the following minimax optimization of the narrowband sensitivity. Let be the detuning from the resonance frequency. Suppose also
In this case, the sum noise power (doublesided) spectral density (163) can be approximated as follows: Then require the maximum of in a given frequency range be as small as possible.It is evident that this frequency range has to be centered around the resonance frequency , with the maximums at its edges, . The sum noise power (doublesided) spectral density at these points is equal to
The minimum of this expression is provided by Substitution of this value back into Eq. (165) gives the following optimized power (doublesided) spectral density: with Therefore, the harmonic oscillator can provide a narrowband sensitivity gain, compared to the free mass SQL (161), which reads and can be further written accounting for the above optimization as: Of course, the oscillator SQL, equal to cannot be beaten is this way, and the question of whether the sensitivity (171) is the ‘true’ beating of the SQL or not, is the question to answer (and the subject of many discussions).
Above, we have discussed, in brief, different normalizations of the sum noise spectral density and derived the general expressions for the SQL in these normalizations (cf. Eqs. (153) and (154)). Let us consider how these expressions look for the free mass and harmonic oscillator and how the sensitivity curves transform when changing to different normalizations.

In the case of a free mass with the above expression transforms as:

and that results in the following power (doublesided) spectral density formula:
and results in the following formula for free mass SQL in normalization: The plots of these spectral densities at different values of are given in the left panel of Figure 23.As for the harmonic oscillator, similar formulas can be obtained taking into account that . Thus, one has:

that results in the following power (doublesided) spectral density formula:
and results in the following formula for free mass SQL in normalization:The corresponding plots are drawn in the right panel of Figure 23. Despite a quite different look, in essence, these spectral densities are the same force spectral densities as those drawn in Figure 22, yet tilted rightwards by virtue of factor . In particular, they are characterized by the same minimum at the resonance frequency, created by the strong response of the harmonic oscillator on a nearresonance force, as the corresponding forcenormalized spectral densities (163, 172).
Spectral density of and the corresponding SQL are equal to
In the free mass case, the formulas are the same as in normalization except for the multiplication by :
with SQL equal to:In the harmonic oscillator case, these equations have the following form:
The corresponding plot of the harmonic oscillator power (doublesided) spectral density in normalization is given in Figure 24.
Note that the curves display a sharp upsurge of noise around the resonance frequencies. However, the resonance growth of the displacement due to signal force have a long start over this seeming noise outburst, as we have shown already, leads to the substantial sensitivity gain for a nearresonance force. This sensitivity increase is clearly visible in the force and equivalent displacement normalization, see Figures 22 and 23, but completely masked in Figure 24.
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Living Rev. Relativity 15, (2012), 5
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