4.4 Quantum noise

4.4.1 Photoelectron shot noise

For gravitational-wave signals to be detected, the output of the interferometer must be held at one of a number of possible points on an interference fringe. An obvious point to choose is halfway up a fringe since the change in photon number produced by a given differential change in arm length is greatest at this point (in practice this is not at all a sensible option and interferometers are operated at, or near, a dark fringe – see Sections 5.1 and 5.4). The interferometer may be stabilised to this point by sensing any changes in intensity at the interferometer output with a photodiode and feeding the resulting signal back, with suitable phase, to a transducer capable of changing the position of one of the interferometer mirrors. Information about changes in the length of the interferometer arms can then be obtained by monitoring the signal fed back to the transducer.

As mentioned earlier, it is very important that the system used for sensing the optical fringe movement on the output of the interferometer can resolve strains in space of 2 × 10–23 Hz–1/2 or lower, or differences in the lengths of the two arms of less than −19 1∕2 10 m ∕Hz, a minute displacement compared to the wavelength of light ≃ 10–6 m. A limitation to the sensitivity of the optical readout scheme is set by shot noise in the detected photocurrent. From consideration of the number of photoelectrons (assumed to obey Poisson statistics) measured in a time τ it can be shown [181] that the detectable strain sensitivity depends on the level of laser power P of wavelength λ used to illuminate the interferometer of arm length L, and on the time τ, such that:

[ ]1∕2 1- -λhc--- detectable strain in time τ = L 2π2P τ , (7 )
[ ]1∕2 detectable strain (Hz )−1∕2 = 1- -λhc- , (8 ) L π2P
where c is the velocity of light, h is Planck’s constant and we assume that the photodetectors have a quantum efficiency ≃ 1. Thus, achievement of the required strain sensitivity level requires a laser, operating at a wavelength of 10–6 m, to provide 6 × 106 power at the input to a simple Michelson interferometer. This is a formidable requirement; however, there are a number of techniques which allow a large reduction in this power requirement and these will be discussed in Section 5.

4.4.2 Radiation pressure noise

As the effective laser power in the arms is increased, another phenomenon becomes increasingly important arising from the effect on the test masses of fluctuations in the radiation pressure. One interpretation on the origin of this radiation pressure noise may be attributed to the statistical uncertainty in how the beamsplitter divides up the photons of laser light [133Jump To The Next Citation Point]. Each photon is scattered independently and therefore produces an anti-correlated binomial distribution in the number of photons, N, in each arm, resulting in a √ --- ∝ N fluctuating force from the radiation pressure. This is more formally described as arising from the vacuum (zero-point) fluctuations in the amplitude component of the electromagnetic field. This additional light entering through the dark-port side of the beamsplitter, when being of suitable phase, will increase the intensity of laser light in one arm, while decreasing the intensity in the other arm, again resulting in anti-correlated variations in light intensity in each arm [109Jump To The Next Citation Point, 110Jump To The Next Citation Point]. The laser light is essentially in a noiseless “coherent state” [154] as it splits at the beamsplitter and fluctuations arise entirely from the addition of these vacuum fluctuations entering the unused port of the beamsplitter. Using this understanding of the coherent state of the laser, shot noise arises from the uncertainty in the phase component (quadrature) of the interferometer’s laser field and is observed in the quantum fluctuations in the number of detected photons at the interferometer output. Radiation pressure noise arises from uncertainty in the amplitude component (quadrature) of the interferometer’s laser field. Both result in an uncertainty in measured mirror positions.

For the case of a simple Michelson, shown in Figure 3View Image, the power spectral density of the fluctuating motion of each test mass m resulting from fluctuation in the radiation pressure at angular frequency ω is given by [133Jump To The Next Citation Point],

( ) 2 -4P-h--- δx (ω ) = m2ω4c λ , (9 )
where h is Planck’s constant, c is the speed of light and λ is the wavelength of the laser light. In the case of an interferometer with Fabry–Pérot cavities, where the typical number of reflections is 50, displacement noise δx due to radiation pressure fluctuations scales linearly with the number of reflections, such that,
( ) δx2 (ω ) = 502 × -4P-h--- . (10 ) m2 ω4cλ
Radiation pressure may be a significant limitation at low frequency and is expected to be the dominant noise source in Advanced LIGO between around 10 and 50 Hz [166Jump To The Next Citation Point]. Of course the effects of the radiation pressure fluctuations can be reduced by increasing the mass of the mirrors, or by decreasing the laser power at the expense of degrading sensitivity at higher frequencies.

4.4.3 The standard quantum limit

Since the effect of photoelectron shot noise decreases when increasing the laser power as the radiation pressure noise increases, a fundamental limit to displacement sensitivity is set. For a particular frequency of operation, there will be an optimum laser power within the interferometer, which minimises the effect of these two sources of optical noise, which are assumed to be uncorrelated. This sensitivity limit is known as the Standard Quantum Limit (SQL) and corresponds to the Heisenberg Uncertainty Principle, in its position and momentum formulation; see [133Jump To The Next Citation Point, 109, 110Jump To The Next Citation Point, 221Jump To The Next Citation Point].

Firstly, it is possible to reach the SQL at a tuned range of frequencies, when dominated by either radiation-pressure noise or shot noise, by altering the noise distribution in the two quadratures of the vacuum field. This effect can be achieved “by squeezing the vacuum field”. There are a number of proposed designs for achieving this in future interferometric detectors, such as a “squeezed-input interferometer” [110, 300], a “variational-output interferometer” [307] or a “squeezed-variational interferometer” using a combination of both techniques. This technique may be of importance in allowing an interferometer to reach the SQL at a particular frequency, for example, when using lower levels of laser power and otherwise being dominated by shot noise. Experiments are under way to incorporate squeezed-state injection as part of the upgrades to current gravitational-wave detectors, and where a squeezing injection bench has already been installed in the GEO600 gravitational-wave detector, which expects to be able to achieve an up-to-6 dB reduction in shot noise using the current interferometer configuration [301]. Similar experiments are also under way to demonstrate variational readout, where ponderomotive squeezing arises from the naturally-occurring correlation of radiation-pressure noise to shot noise upon reflection of light from a mirror [117, 271]

Secondly, if correlations exist between the radiation-pressure noise and the shot-noise displacement limits, then it is possible to bypass the SQL, at least in principle [221]. There are at least two ways by which such correlations may be introduced into an interferometer. One scheme is where an optical cavity is constructed, where there is a strong optical spring effect, coupling the optical field to the mechanical system. This is already the case for the GEO600 detector, where the addition of a signal recycling cavity creates such correlation, where signal recycling is described in Section 5.2. Other schemes of optical springs have been studied, such as optical bars and optical levers [103, 101]. Another method is to use suitable filtering at optical frequencies of the output signal, by means of long Fabry–Pérot cavities, which effectively introduces correlation [201, 116].

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