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Message ID: 15688     Entry time: Tue Nov 24 16:51:29 2020
 Author: gautam Type: Update Category: PonderSqueeze Subject: Ponderomotive squeezing in aLIGO

Summary:

On the call last week, I claimed that there isn't much hope of directly measuring Ponderomotive Squeezing in aLIGO without some significant configurational changes. Here, I attempt to quantify this statement a bit, and explicitly state what I mean by "significant configurational changes".

Optomechanical coupling:

The I/O relations will generally look something like:

$\begin{bmatrix} b_1\\ b_2 \end{bmatrix} = \begin{bmatrix} C_{11} & C_{12}\\ C_{21} & C_{22} \end{bmatrix} \begin{bmatrix} a_{1}\\ a_2 \end{bmatrix} + \begin{bmatrix} D_1\\ D_2 \end{bmatrix} \frac{h}{h_{\mathrm{SQL}}}$.

The. magnitudes of the matrix elements C_12 and C_21 (i.e. phase to amplitude and amplitude to phase coupling coefficients) will encode the strength of the Ponderomotive squeezing.

For the inital study, let's assume DC readout (since there isn't a homodyne readout yet even in Advanced LIGO). This amounts to setting $\zeta = \phi$ in the I/O relations, where the former angle is the "homodyne phase" and the latter is the "SRC detuning". For DC readout, the LO quadrature is fixed relative to the signal - for example, in the usual RSE operation, $\zeta = \phi = \frac{\pi}{2}$. So the quadrature we will read out will be purely $b_1$ (or nearly so, for small detunings around RSE operation). The displacement noises will couple in via the $D_1$ matrix element. Attachment #1 and Attachment #2 show the off-diagonal elements of the "C" matrix for detunings of the SRC near RSE and SR operation respectively. You can see that the optomechanical coupling decays pretty rapidly above ~40 Hz.

SRC detuning:

In this particular case, there is no benefit to detuning the SRC, because we are assuming the homodyne angle is fixed, which is not an unreasonable assumption as the quadrature of the LO light is fixed relative to the signal in DC readout (not sure what the residual fluctuation in this quantity is). But presumably it is at the mrad level, so the pollution due to the orthogonal anti-squeezed quadrture can be ignored for a first pass I think. I also assume ~10 degrees of detuning is possible with the Finesse ~15 SRC, as the linewidth is ~12 degrees.

Noise budget:

To see how this would look in an actual measurement, I took the data from Lee's ponderomotive squeezing paper, as an estimate for the classical noises, and plotted the quantum noise models for a few representative SRC detunings near RSE operation - see Attachment #3. The curves labelled for various phis are the quantum noise models for those SRC detunings, assuming DC readout. I fudged the power into the IFO to make my modelled quantum noise curve at RSE line up with the high frequency part of the "Measured DARM" curve. To measure Ponderomotive Squeezing unambiguously, we need the quantum noise curve to "dip" as is seen around 40 Hz for an SRC tuning of 80 degrees, and that to be the dominant noise source. Evidently, this is not the case.

The case for balanced homodyne readout:

I haven't analyzed it in detail yet - but it may be possible that if we can access other quadratures, we might benefit from rotating away from the DARM quadrature - the strength of the optomechanical coupling would decrease, as demonstrated in Attachments #1 and #2, but the coupling of classical noise would be reduced as well, so we may be able to win overall. I'll briefly investigate whether a robust measurement can be made at the site once the BHD is implemented.

 Attachment 1: QN_heatmap_RSE.pdf  170 kB
 Attachment 2: QN_heatmap_SR.pdf  172 kB
 Attachment 3: noiseBudget.pdf  168 kB
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