# In what sense are shot noise and photon pressure canonically-conjugate variables in the LIGO interferometer?

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This week I saw a seminar by Kip Thorne and he mentioned that in the LIGO interferometer, the photon shot noise is actually canonically conjugate to the noise induced by photon pressure acting on the mirrors; he gave a good explanation during the talk but I can no longer remember the details. However, instead of digging that up and keeping some half-hearted answer to myself, I thought it'd be better to have this settled in more detail and in a more public venue.

So: in what sense are photon pressure and shot noise related in the LIGO interferometer?

This post imported from StackExchange Physics at 2017-07-05 16:23 (UTC), posted by SE-user Emilio Pisanty

edited Jul 5, 2017
I imagine photon pressure $P$ looks like $P \propto \langle \hat{n} \rangle$. In that case maybe he was referring to the shot "noise" coming from the distribution of $n$ in a coherent state.

This post imported from StackExchange Physics at 2017-07-05 16:23 (UTC), posted by SE-user DanielSank
@DanielSank Possibly, yes (and yes, I know that shot noise is just a convenient classical fiction, but it maps uniquely into the uncertainty in photon number of a coherent state). Thorne showed what looked like a full formal treatment, down to a canonical commutation relationship between quadratures identified as those two observables, and that's what I would like answers to reconstruct.

This post imported from StackExchange Physics at 2017-07-05 16:23 (UTC), posted by SE-user Emilio Pisanty
Ah. I see what you want. Upvote because I want it too.

This post imported from StackExchange Physics at 2017-07-05 16:23 (UTC), posted by SE-user DanielSank

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In the following, I will try to answer the question via two ways. First, handwavingly and probably more intuitive, in the particle picture - here, one can see that "shot noise" and "radiation pressure (shot) noise" are two sides of the same coin. Then, I will try to sketch the same in the language of quadrature operators (again not rigorously, I'm an experimentalist). I believe this is what you (and Kip Thorne) had in mind. I will in the following relate to a linear cavity with a moving endmirror, the simplest interferometer with radiation pressure interaction, but the same applies to more complicated configurations. (And I beg to excuse missing constants, wrong signs etc.)

While measuring a laser beam's power (i.e. counting photons), the poisson distribution of photons causes inherent fluctuations in the measurement. The accuracy of measuring the phase shift due to some length change in a cavity scales linearly with power inside the cavity (i.e. linearly with the number of photons $N$). The fluctuations of photons scale with the square root of photon number. The overall uncertainty thus can be decreased proportionally to $1/\sqrt N$ with more photons/higher power. This corresponds to a stronger measurement. At the same time, the photons transfer momentum onto the moving endmirror. This force (radiation pressure) is fluctuating and scales with the squareroot of $N$. The deflection of the moving mirror due to that is linear to first order, so the radiation pressure shot noise messes up my measurement $\propto\sqrt{N}$. I disturb my system the more, the stronger I want to measure it. (The perfect balance between more accuracy/stronger measurement and less disturbance at a given Fourier frequency is called "Standard Quantum Limit".)

Now, more formally, the quadrature picture. Look at the standard radiation pressure interaction (= optomechanical) Hamiltonian in the interaction picture [1]: $$H\propto g_0a^\dagger ax_m,$$ where $a$ is the cavity mode, $g_0$ the coupling strength and $x_m$ the (dimension-less) mirror position ($p_m$ will be its momentum). The photon number couples to the mirror position. Linearising this via $a \rightarrow \alpha + a$ for a strong intracavity field with amplitude $\alpha$ leads to $$H\propto g(a+a^\dagger)x_m = g x_a x_m,$$ where $g = g_0 \alpha$ and $x_a$ the amplitude quadrature such that $a\propto x_a+ip_a$ and $p_a$ the phase quadrature with commutation relation $[x_a,p_a]=i$, such that $[a, a^\dagger]=1$. From this, we can derive the Heisenberg equations of motion, which are (including decay and input noise, i.e. coupling to a bath): \begin{align} \dot x_a &=-\kappa_a x_a-\sqrt{\kappa_a}x_a^\text{in},\\ \dot p_a &=-\kappa_a p_a-gx_m-\sqrt{\kappa_a}p_a^\text{in},\\ \dot x_m &=\omega_m p_m,\\ \dot p_m &=-\omega_m x_m - gx_a-\gamma p_m + \sqrt{\gamma}F. \end{align} Here, $\kappa_a$ and $\gamma$ are the linewidths of the cavity and the endmirror, respectively, and $F$ is some, maybe thermal, force, acting on the mirror and causing it to move.

What we measure in the end is the cavity output field. The output quadratures in frequency space are (solving the EOMs and using input-output relations, for the latter see [2]) \begin{align} x_\text{out} &= -\kappa_a \chi_a x_\text{in},\\ p_\text{out} &= e^{i\phi} p_\text{in} - \sqrt{\kappa_a}g\chi_a x_m^0 + g^2\chi_a^2\chi_m x_\text{in}, \end{align} where we used several abbreviations: some phase $e^{i\phi}=\frac{i\omega-\kappa_a/2}{i\omega+\kappa_a/2}$, the cavity transfer function $\chi_a=(i\omega+\kappa_a/2)^{-1}$, the mechanical transfer function $\chi_m=\omega_m(\omega^2-\omega_m^2-i\omega\gamma)^{-1}$ and the (under some force) evolving position $x_m^0=\sqrt{\gamma}\chi_m F$ we want to measure.

From the above, one sees that possible length changes of the cavity we like to detect are encoded in the phase quadrature of the cavity and thus can be found in the output phase quadrature $p_\text{out}$. This can be measured either with a homodyne setup or with a standard photon-counter-type detector, which essentially measures $a_\text{out}^\dagger a_\text{out}\propto x_\text{out}^2+p_\text{out}^2$. For now, we choose to measure the phase quadrature directly, and infer the mirror position to be \begin{align} \hat x_m &= -\frac{p_\text{out}}{\sqrt{\gamma}\chi_mF}\\ &= x_m^0 - \frac{e^{i\phi}}{\sqrt{\kappa_a}g\chi_a}p_\text{in} - \sqrt{\kappa_a}g\chi_a\chi_m x_\text{in}. \end{align} Note that instead of measuring $x_m^0$ we measure something different. There are two additional contributions to our estimation, the first one stemming from input phase fluctuations, scaling with $1/g\propto 1/\sqrt{N}$ (as did the shot noise in the particle picture above), the second one scaling with $g\propto\sqrt{N}$ (as did the radiation pressure noise). Note also that we have vacuum fluctuations of two quadratures coming in which cannot be made zero at the same time. This becomes clearer looking at the position spectral density which is (assuming uncorrelated quadratures) $$S_{xx}=\frac{(\Delta p_\text{in})^2}{\kappa_a\chi_a^2g^2} + \kappa_a\chi_a^2g^2(\Delta x_\text{in})^2$$ Here, you see the canonically conjugate pair you were looking for. You could squeeze one quadrature and antisqueeze the other, but that has the same effect as using more or less power. Also, it can be optimised only for one measurement frequency at a time, which is not of too much use for a broadband detector like LIGO. The two quadratures can be made to correlate which leads towards Quantum Non-Demolition measurements, but that is an entirely different topic.

[1] Law, Phys. Rev. A 51 (1995), 2537.

[2] Gardiner and Zoller, Quantum Noise, Heidelberg (2004).

This post imported from StackExchange Physics at 2017-07-05 16:23 (UTC), posted by SE-user dodi
answered Jun 29, 2017 by (20 points)

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