AN EXPERIMENT TO MEASURE OPTOMECHANICAL SQUEEZING AT THE 40M PROTOTYPE INTERFEROMETER
4.1 Conceptual overview of "quantum noise" in an interferometer
C h a p t e r 4
AN EXPERIMENT TO MEASURE OPTOMECHANICAL
photons during the process of reflection. For the dual-recycled optical topology adopted by the LIGO interferometers, radiation pressure noise is the dominant con- tributor at low frequencies, while shot noise dominates at high frequencies, although the specific parameters of the interferometer such as the laser power circulating in the Fabry-Pérot arm cavities and the type of readout (i.e. DC readout / RF readout / BHD) must be known to quantify these statements. The insight of [53,54] was that a proper analysis of these two noise sources must take into account any correlations that exist between them.
A convenient technique to analyze the quantum noise of a particular optical con- figuration is to use the Input-Output (I/O) relations. "Inputs" refer to optical fields that enter open ports, such as the symmetric port where the PSL beam pumps the interferometer, while "Outputs" refer to fields that are read out at various open ports, such as the field exiting the AS port. Being a MIMO device, each output field can in general have contributions from multiple inputs. These can be derived using the Adjacency matrix approach described in Appendix C.1, but using the two-photon representation of the fields described in [61]. For LIGO-like optical topologies, the pertinent I/O relation is that for the output field at the AS port (which is where the gravitational wave signal is expected). In the two-photon formalism notation [62,63], this will take the general form
"
b1 b2
#
|{z}
output field
=
"
C11 C12 C21 C22
#
| {z }
Optomechanics, C
"
a1 a2
#
|{z}
input vacuum
noise
+
"
D1 D2
#
|{z}
optical gain,
D
h hSQL
|{z}
DARM signal
, (4.1)
where h ≡ δLL is the differential arm strain and hSQL ≡ q
8~
mΩ2L2 is the Standard Quantum Limit (SQL) for an interferometer with test masses of mass m and arm cavity length of L at the signal frequencyΩ. The generalized version of Eq. (4.1) taking into account optical losses entering the interferometer at various ports, ig- nored in the above, can be found as Eqs. 5.6-5.12 in [54]. Thefrequency-dependent matrix coefficientsCi j(Ω)andDi(Ω)are functions of the interferometer parameters, such as injected laser power, arm cavity bandwidth etc. Eq. (4.1) may be interpreted as follows: the output field at the AS port is the sum of the vacuum noise entering the AS port (which may be squeezed or unsqueezed), modified by the optomechanical properties of the interferometer, and the DARM signal scaled by the optomechanical gain of the interferometer.
It is conventional manipulate such I/O relations in units where unsqueezed vacuum has unit variance, and is represented as
"
a1 a2
#
=
"
1 1
#
. In general, we will measure some linear combination of the quadrature fields b1 and b2, with bζ ≡ b1sinζ + b2cosζ, withζbeing the "homodyne angle", discussed in greater detail in Chapter5.
Therefore, if|bζ| < 1, the variance of the measured field is below that of the vacuum, and is, therefore, squeezed. The dimensions of the fields in these normalized units is pnphotons/Hz. However, LIGO noise budgets are usually plotted as "signal-referred"
- i.e. the transfer function of the interferometer, in units of
√
W/h or
√
W/m where his the differential arm strain, is used to convert a measured noise to the equivalent DARM displacement (or strain) which would have produced the same output field (it is convenient to represent electric fields in units of
√
Wrather than S.I. units). To convertbζ from units ofp
nphotons/Hz to
√
W/h, we multiply the former by a factor of
√
2~ω0, whereω0is the laser frequency - the detailed derivation may be found in Appendix A of [64].
In order to use the interferometer as a source of squeezed vacuum1, the goal is to determine the optical configuration such that |bζ| < 1 for some ζ, over some appreciable range of frequencies Ω. Note that the squeezing is generated by the elementsC12orC21, which convert phase quadrature fluctuations into the amplitude quadrature or vice-versa (depending on the configuration of the interferometer). For us to be able to measure this generated squeezing, all other noise sources, such as differential arm motion due to Brownian noise of the dielectric coatings which contribute to bζ via the matrix elements D1 and D2, must be low enough that the noise variance of bζ remains lower than that of unsqueezed vacuum. For the 40m, at f = Ω/2π < 100 Hz, the displacement noises are typically so large that they far exceed the quantum noise (which is the input vacuum transformed by the matrixC).
The systematic optimization study to determine the configuration that gives the best chance of measuring optomechanical squeezing is described in detail in Chapter 3
1A reasonable question is why we would want to generate squeezed vacuum in this way. At the time of writing, squeezed vacuum generation using non-linear crystals is a mature technology, and both LIGO observatories have been operating with≈2 dB of squeezed vacuum injected into their AS ports for the entire O3 observing run [65]. However, using the optomechanical interaction may offer some advantages - for example, the losses due to imperfect polishing of the non-linear crystals far exceed that of super-polished mirrors, and it is known that losses limit the amount of squeezing that can be realized. Therefore, an ultra-low-loss optical cavity may allow even stronger squeezed fields to be generated. One reason non-linear crystals are the preferred technology is because the optomechanical interaction is inherently weak, and other classical noise sources would swamp any generated squeezing. However, recent technological advances have made it possible to mitigate many of these, making the idea worth pursuing.
of [60].
It is worth emphasizing that the goal of this particular experiment isnotto maximize the signal-to-noise ratio of the DARM signal (which is the primary objective at terrestrial observatories, since better SNR would presumably lead to more precise astrophysical measurements and probes of deviations from General Relativity). The quadrature which should be selected for that purpose is different from what is selected for the optomechanical squeezing experiment. We seek the quadrature at which the ratio of noise variance inbζ to unsqueezed vacuum is minimized. Other ways of validating the quantum-mechanical nature of the interferometer, encoded in the matrixC, include injecting well-characterizedsqueezedvacuum into the AS port, measure the output field, and inferring theoptomechanicalsqueezing operation by mapping the relationship between the two - this was the approach adopted in [47].