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Number-phase Uncertainty Relations

M. Beck, D.T. Smithey and M.G. Raymer

The uncertainty principle lies at the heart of quantum mechanics. Any pair of noncommuting variables satisfy a form of uncertainty relationship, and this sets bounds on the measurement precision achievable for these quantities. In quantum optics, one such pair of variables are the photon number and the phase of an optical field. They satisfy an uncertainty relation ΔnΔφ≥(1/2)|‹|Ψ [n, φ] |Ψ›| where |Ψ› represents the state of the field. One important point to note is that the lower bound of this relation depends on the state. Thus, one needs a way of determining the values of both the uncertainty product and the expectation value of the commutator [n, φ].