4.3 Standard Quantum Limit

Recall the SQM in Section 4.1.2.

The SQM has non-correlated effective measurement and back-action noises that results in S š’³ā„±(Ω ) = 0. Apparently, under these conditions ˆ š’³ and ˆ ā„± turn into ˆxfl and ˆ F fl of Eqs. (112View Equation) and (114View Equation), respectively. Hence, we will use Sx(Ω ) instead of Sš’³š’³ (Ω) and SF (Ω ) instead of Sā„±ā„± (Ω) Then the uncertainty relation (148View Equation) transforms into:

ā„2 Sx(Ω )SF(Ω ) ≥ --. (149 ) 4

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 Ω. To derive this limit we assume noise sources xfl and Fb.a. to have minimal values allowed by quantum mechanics, i.e.

2 Sx(Ω )SF(Ω ) = ā„-. (150 ) 4
Then, using this condition, one can minimize SQM’s sum noise:
2 SF (Ω ) = --Sx(Ω-)- + SF (Ω) = --Sx(Ω-)- + ---ā„--- |χxx(Ω )|2 |χxx(Ω )|2 4Sx (Ω)

to yield:

SF (Ω) = ---ā„---- (151 ) SQL |χxx (Ω )|
, that is achieved when contributions of measurement noise and back-action noise to the sum noise are equal to each other, i.e., when
ā„- ----ā„----- Sx(Ω ) = 2|χxx(Ω )|, ⇐ ⇒ SF (Ω) = 2|χxx(Ω )|. (152 )

It is instructive to cite the forms of the SQL in other normalizations, i.e., for h-normalization and for x-normalization. The former is obtained from (151View Equation) via multiplication by 4āˆ•(M 2L2 Ω4):

4SF (Ω ) ShSQL(Ω ) = --SQL-----= -------4ā„--------. (153 ) M 2L2Ω4 M 2L2Ω4 |χxx (Ω )|
The latter is obtained fromEq. (151View Equation) using the obvious connection between force and displacement x (Ω ) = χxx(Ω )F(Ω ) :
SxSQL(Ω ) = |χxx (Ω)|2SFSQL(Ω ) = ā„|χxx (Ω)|. (154 )

These limits look fundamental. There are no parameters of the meter (only ā„ as a reminder of the uncertainty relation (116View Equation)), 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 brute-force methods of sensitivity improving cease working, and methods that allow one to blot out the back-action noise ˆ ā„± (t) 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.

4.3.1 Free mass SQL

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 (150View Equation) 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 fm in ā‰² 1Hz range. The sufficient degree of isolation is provided at frequencies much higher than fm, 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:

( )1 āˆ•4 Ω = --SF-- . (155 ) q M 2Sx
Using the uncertainty relation (150View Equation), the noise spectral densities Sx and SF can be expressed through Ωq as follows:
2 S = ---ā„---, S = ā„M--Ωq-. (156 ) x 2M Ω2q F 2
Therefore, the larger Ωq is, the smaller Sx is (the higher is the measurement precision), and the larger SF is (the stronger the meter back action is).

In the case of interferometers, Ω2 q is proportional to the circulating optical power. For example, for the toy optical meter considered above,

āˆ˜ --------- Ω = 8ωp-ā„0Š³2-, (157 ) q M c2
see Eqs. (123View Equation). Using this notation, and taking into account that for a free mass M,
χfx.mx.(Ω) = − --1--, (158 ) M Ω2
the sum quantum noise power (double-sided) spectral density can be written as follows:
ā„M Ω2 ( 4 ) SFf.m.(Ω ) = M 2Ω4Sx + SF = -----q- Ω--+ 1 . (159 ) 2 Ω4q
The SQL optimization (152View Equation) takes the following simple form in this case:
Ωq = Ω, (160 )
SFSQLf.m.(Ω ) = ā„M Ω2. (161 )

This consideration is illustrated by Figure 22View Image (left), where power (double-sided) spectral density (159View Equation) is plotted for three different values of Ωq. It is easy to see that these plots never dive under the SQL line (161View Equation), 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’.

View Image

Figure 22: Sum quantum noise power (double-sided) spectral densities of the Simple Quantum Meter for different values of measurement strength Ωq (red ) < Ωq (green) < Ωq(blue). Thin black line: SQL. Left: free mass. Right: harmonic oscillator

4.3.2 Harmonic oscillator SQL

The simplest way to overcome the limit (161View Equation), 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.(151View Equation) that the more responsive the test object is at some given frequency Ω (that is, the bigger χxx (Ω )) is, the smaller its force SQL at this frequency is. In the harmonic oscillator case,

1 χosxcx(Ω) = ----2-----2-, (162 ) M (Ω0 − Ω )
with Ω0 standing for the oscillator mechanical eigenfrequency, and the sum quantum noise power (double-sided) spectral density equal to
2 [ 2 2 2 ] SF (Ω) = M 2(Ω2 − Ω2)2S + S = ā„M--Ωq- (Ω-0 −-Ω-)-+ 1 . (163 ) osc 0 x F 2 Ω4q
Due to a strong response of the harmonic oscillator near resonance, the first (measurement noise) term in Eq. (159View Equation) goes to zero in the vicinity of Ω0. Therefore, reducing the value of Ωq, that is, using weaker measurement, it is possible to increase the sensitivity in a narrow band around Ω0. At the same time, the smaller Ωq is, the more narrow the bandwidth is where this sensitivity is achieved, as can be seen from the plots drawn in Figure 22View Image (right).

Consider, in particular, the following minimax optimization of the narrow-band sensitivity. Let ν = Ω − Ω 0 be the detuning from the resonance frequency. Suppose also

|ν| ā‰Ŗ Ω0. (164 )
In this case, the sum noise power (double-sided) spectral density (163View Equation) can be approximated as follows:
2 ( 2 2 ) SF (Ω + ν) = ā„M--Ωq- 4Ω0ν--+ 1 . (165 ) osc 0 2 Ω4q
Then require the maximum of F S osc in a given frequency range ΔΩ be as small as possible.

It is evident that this frequency range has to be centered around the resonance frequency Ω0, with the maximums at its edges, ν = ± Δ Ωāˆ•2. The sum noise power (double-sided) spectral density at these points is equal to

ā„M Ω2q( Ω20Δ Ω2 ) SFosc(Ω0 ± Δ Ωāˆ•2 ) = ------- ----4-- + 1 . (166 ) 2 Ωq
The minimum of this expression is provided by
āˆ˜ ------- Ω = Ω Δ Ω. (167 ) q 0
Substitution of this value back into Eq. (165View Equation) gives the following optimized power (double-sided) spectral density:
( ) F ā„M-Ω0-- -4ν2 Sosc(Ω0 + ν ) = 2 Δ Ω + Δ Ω , (168 )
SF (Ω ± Δ Ωāˆ•2 ) = ā„M Ω Δ Ω. (169 ) osc 0 0
Therefore, the harmonic oscillator can provide a narrow-band sensitivity gain, compared to the free mass SQL (161View Equation), which reads
F ( 2 2) 2 Sosc(Ω0-+-ν-) 1- 4ν-- Ω-q ξosc = SFSQLf.m.(Ω0 ) ā‰ƒ 2 Ω2q + Ω20 , (170 )
and can be further written accounting for the above optimization as:
| SFosc(Ω0 + ν)| SFosc(Ω0 ± ΔΩ āˆ•2) Δ Ω --F---------|| ≤ ----F------------= ---. (171 ) S SQLf.m.(Ω0) |ν|≤Δ Ωāˆ•2 SSQLf.m.(Ω0 ) Ω0
Of course, the oscillator SQL, equal to
SF = ā„M |Ω2 − Ω2| ≈ 2ā„M Ω |ν| (172 ) SQLosc 0 0
cannot be beaten is this way, and the question of whether the sensitivity (171View Equation) is the ‘true’ beating of the SQL or not, is the question to answer (and the subject of many discussions).

4.3.3 Sensitivity in different normalizations. Free mass and harmonic oscillator

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. (153View Equation) and (154View Equation)). 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.

The noise spectral density in h-normalization can be obtained using Eq. (12View Equation). Where the SQM is concerned, the sum noise in h-normalization reads

[ ] h 2 ˆxfl(Ω) hsum (Ω) ≡ š’©ˆ (Ω ) = − -----2- ------- + ˆFfl(Ω ) . M L Ω χxx (Ω )

In the case of a free mass with χxx(Ω ) = − 1 āˆ•(M Ω2) the above expression transforms as:

2ˆx (Ω ) 2Fˆ hfs.mu.m(Ω ) = --fl----− ----fl-- L M L Ω2

and that results in the following power (double-sided) spectral density formula:

[ ] ( 4) Sh (Ω ) = -4- Sx + -SF--- = ---2ā„--- 1 + Ω-q (173 ) f.m. L2 M 2Ω4 M L2Ω2q Ω4
and results in the following formula for free mass SQL in h-normalization:
4 ā„ ShSQLf.m.(Ω ) = ----2-2-. (174 ) M L Ω
The plots of these spectral densities at different values of Ωq are given in the left panel of Figure 23View Image.
View Image

Figure 23: Sum quantum noise power (double-sided) spectral densities of the Simple Quantum Meter in the h-normalization for different values of measurement strength: Ωq (red ) < Ωq(green) < Ωq (blue ). Thin black line: SQL. Left: free mass. Right: harmonic oscillator.

As for the harmonic oscillator, similar formulas can be obtained taking into account that χosxcx(Ω) = 1āˆ• (M (Ω20 − Ω2 )). Thus, one has:

2(Ω2 − Ω2 )ˆxfl(Ω) 2 ˆFfl hossucm(Ω ) = − ---0-----2-------− ------2 L Ω M LΩ

that results in the following power (double-sided) spectral density formula:

[( 2)2 ] [ 4( 2 2 2) ] Sh (Ω ) = -4- 1 − Ω-0 S + --SF-- = --2-ā„--- Ω-q 1 + (Ω-0 −-Ω-)- (175 ) osc L2 Ω2 x M 2Ω4 M L2 Ω2q Ω4 Ω4q
and results in the following formula for free mass SQL in h-normalization:
h 4ā„|Ω20 − Ω2| S SQLosc(Ω) = ------2-4--. (176 ) M L Ω

The corresponding plots are drawn in the right panel of Figure 23View Image. Despite a quite different look, in essence, these spectral densities are the same force spectral densities as those drawn in Figure 22View Image, yet tilted rightwards by virtue of factor 1āˆ•Ω4. In particular, they are characterized by the same minimum at the resonance frequency, created by the strong response of the harmonic oscillator on a near-resonance force, as the corresponding force-normalized spectral densities (163View Equation, 172View Equation).

Another normalization that is worth considering is the actual probe displacement, or x-normalization. In this normalization, the sum noise spectrum is obtained by multiplying noise term ˆF š’© (Ω) in Eq. (139View Equation) by the probe’s susceptibility

ˆ ˆxsum(Ω ) = ˆxfl(Ω) + χxx(Ω )Ffl(Ω ). (177 )
It looks rather natural at a first glance; however, as we have shown below, it is less heuristic than the force normalization and could even be misleading. Nevertheless, for completeness, we consider this normalization here.

Spectral density of ˆxsum (Ω) and the corresponding SQL are equal to

Sx (Ω ) = S (Ω ) + |χ (Ω )|2S (Ω), (178 ) x xx F SxSQL (Ω ) = ā„|χxx(Ω )|. (179 )

In the free mass case, the formulas are the same as in h-normalization except for the multiplication by 4āˆ•L2:

[ ] ( ) x SF ā„ Ω4q Sf.m.(Ω) = Sx + M-2Ω4- = 2M--Ω2- 1 + Ω2- (180 ) q
with SQL equal to:
x --ā„-- SSQLf.m.(Ω ) = M Ω2. (181 )

In the harmonic oscillator case, these equations have the following form:

S ā„ [Ω4 ( (Ω2 − Ω2 )2)] Sxosc(Ω) = Sx + ------2F------= ------- --q 1 + --0-------- , (182 ) M 2(Ω 0 − Ω2 )2 2M Ω2q Ω4 Ω4q x ā„ SSQLosc(Ω ) = ----2----2-. (183 ) m |Ω0 − Ω |

The corresponding plot of the harmonic oscillator power (double-sided) spectral density in x-normalization is given in Figure 24View Image.

View Image

Figure 24: Sum quantum noise power (double-sided) spectral densities of Simple Quantum Meter and harmonic oscillator in displacement normalization for different values of measurement strength: Ωq (red ) < Ωq(green) < Ωq (blue ). Thin black line: SQL.

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 G have a long start over this seeming noise outburst, as we have shown already, leads to the substantial sensitivity gain for a near-resonance force. This sensitivity increase is clearly visible in the force and equivalent displacement normalization, see Figures 22View Image and 23View Image, but completely masked in Figure 24View Image.

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