Abstract
We use the P-distribution to show that the familiar values 1, 2 and 3 of the normalized second order correlation function at equal times corresponding to a coherent state, a thermal state and a highly squeezed vacuum are a consequence of the number of dimensions these states take up in quantum phase space. Whereas the thermal state exhibits rotational symmetry and thus extends over two dimensions, the squeezed vacuum factorizes into two independent one-dimensional phase space variables, and in the limit of large squeezing is therefore a one-dimensional object. The coherent state is a point in the phase space of the P-distribution and thus has zero dimensions. The fact that for photon number states the P-distribution is even narrower than that of the zero-dimensional coherent state suggests the notion of ‘negative’ dimensions.
Highlights
A virtual photon borrowed from the vacuum cannot create a photoelectron
We use the P-distribution to show that the familiar values 1, 2 and 3 of the normalized second order correlation function at equal times g(2)(0) corresponding to a coherent state, a thermal state and a highly squeezed vacuum are a consequence of the number of dimensions these states take up in quantum phase space
In the present article we show using the familiar expressions [5] for g(2)(0) corresponding to a thermal state, a coherent state, a squeezed vacuum and photon number states that the P-distribution reflects the number of dimensions the underlying quantum state takes up in phase space
Summary
A virtual photon borrowed from the vacuum cannot create a photoelectron For this reason a photodetector measures [1] normally ordered products of annihilation and creation operators of the electromagnetic field [2]. Even for an elementary state, such as a squeezed state the corresponding P-distribution involves [5] an infinite number of derivatives of delta functions Despite this complication a careful analysis of the emerging integrals. The most intriguing case with respect to g(2)(0) as a measure of dimensionality is the class of photon number states They have the same rotational symmetry as a thermal state but involve a finite number of derivatives of delta functions. In this sense the corresponding P-distributions are narrower than coherent states.
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