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We write

$$(\Delta z)^2 = (\Delta z)_{PC}^2 + (\Delta z)_{RP}^2 + (\Delta z)_{SQL}^2$$

to indicate the contributions to the total error from photon counting error, radiation pressure error and the Standard Quantum Limit error to the total error of an Michelson interferometer. And the SQL term is written as

$(\Delta z)_{SQL} = \sqrt{\frac{\hbar\tau}{m}}$

where m is the mass of the mirrors.

But the Interferometer uses two mirrors of masses m. So shouldn't the SQL itself be a sum of errors like

$(\Delta z)_{SQL,1}^2 + (\Delta z)_{SQL,2}^2 = \frac{2\hbar\tau}{m}$?

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  • $\begingroup$ I tried to figure out but didn't manage. A new issue that come out: the light on mirrors is incident at 45 deg (changes its path of 90 deg). Thus a back-and-forth uncertainty in mirror position of $\delta$ will bring an uncertainty in light path of $\sqrt{2}\delta$. Another 2 factor! $\endgroup$
    – patta
    Commented Apr 12, 2019 at 7:56
  • $\begingroup$ The only reason I can imagine for the disappearance of your 2 is entanglement of the two mirrors, as their relative position is measure by a single in-phase photon $\endgroup$
    – patta
    Commented Apr 12, 2019 at 8:08
  • $\begingroup$ I was speaking of the Michelson Interferometer, where the light is incident at 90 degrees, and not Mach-Zehdner Interferometer. Not sure if this is relevant though, since the two Interferometers are supposed to be mathematically equivalent. $\endgroup$
    – lAPPYc
    Commented Apr 12, 2019 at 10:45

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