Basically, my question is whether there are methods to monitor mass position with an interaction Hamiltonian followed by a measurement of the detector wave function, as in an Arthurs and Kelly scheme. As far as I know, the measurement of the particle (or mirrors in the Gravitational Wave Detectors) are governed by simple reflection and detection of photons, and there isn't an interaction Hamiltonian. Or is that not true?

Edit: Per request, giving an example of a one detector Arthurs Kelly measurement scheme:

A free mass (whose position x is to be measured) $\psi(x)$ in the basis $\{x,p\}$ and a detector $\phi(Q)$ in the basis $\{Q,P\}$ are made to interact for time $\tau$ with a pulse approximation (individual Hamiltonian evolutions are taken to be negligible during the short time $\tau$) according to an interaction Hamiltonian

$H_{int} = K\hat{x}\hat{P} $

with the coupling constant $K = \frac{1}{\tau}$

Because of the Hamiltonian, in time $\tau$, the joint system goes from $\Psi(x,Q,t=0) = \psi(x)\phi(Q)$ to $\Psi(x,Q,t=\tau) = \psi(x)\phi(Q-x)$, at which point, $<\hat{Q}>$ equals the position $x_0$ of the particle $\psi$ at the start of the interaction.

Arthurs and Kelly outlined this type of scheme in their paper On the simultaneous measurement of a pair of conjugate observables in 1965, but they did not give an experimental realization of the Hamiltonian. Such Hamiltonians from then have been utilized in many mass position measuring schemes, but it is not clear to me if there is any experimental realization of such schemes.


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