A treatment of photon counting in the Michelson ...



A treatment of photon counting in the Michelson interferometer using a quantized radiation field but not many specifics is given in Loudin’s section 6.9.[1] This includes a reference to the potential use of such devices in the detection of gravitational waves.[2]

[pic]

Figure 1 Based on Loudin’s 6.10 Schematic Michelson interferometer

The calculation involves various expectation values of photon creation and destruction operators defined by (where I use + to indicate ()

[pic] Eqn 1

The photon-count distribution for count time T at the detector is

[pic] Eqn 2

where m is the number of photons detected. The operator N imposes normal ordering ,all d( operators to the left of the d operators, on the electric field operators that follow it. The operator [pic] is a density operator equal to |(>> 30 cm, the electromagnetic waves at the minimum will on average fail to cancel by (n where n is the total number of photons hitting the system. Thus in this limit equation 4 for coherent light becomes

[pic] Eqn 7

and the total uncertainty in the measurement for coherent light becomes

[pic]

In section 4.11 Physical properties of the single-mode coherent states, Loudin derives the Poisson distribution as the probability of finding n photons in a state with mean number |(|2. A much more physical and intuitive “derivation” of this same result is found in Infrared Detectors and Systems[3] though I fear it suffers from the photons as billiard balls approximation.

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[1]R. Loudin, The Quantum Theory of Light second edition, Clarendon Press, Oxford (1983) pp 242-247.

[2]K. S. Thorne , Rev. Mod. Phys. 52, 285 (1980)

[3]E.L. Dereniak and G.D. Boreman, Infrared Detectors and Systems, Wiley (1996) pp 158-161

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