Consecutive measurements of photon number and quantum phase

We introduce the conditional probability to consider consecutive measurements of photon number and quantum phase of a single mode. Let P be the conditional probability to measure the phase α with precision ∆α, given a previous measurement of k photons with precision ∆k. Two upper bounds of the probability P are derived. For arbitrary given precisions ∆k and ∆α, these bounds refer to the variation of k, α, and the state vector ψ in Hilbert space. The first (weaker) bound is given by the inequality P ≤ ξ, with ξ = ∆α (∆k+1) 2π . It is nontrivial for measurements with ξ < 1. As our main result the least upper bound of P is determined. We obtain an analytical representation of this bound in the asymptotic limit ∆k → ∞ and ∆α → 0 such that ξ > 0 is fixed. Finally, we present a rigorous prove that the well-known Heisenberg limit in precision phase measurement can never be attained with measurement probabilities greater than 1/π. The classical picture for the evolution of a single-mode electromagnetic field is simply determined by an amplitude (specifying the strength of the field) and a phase (specifying the zeros of the field). In quantum theory, the field strength may be specified exactly in terms of photon number N . On the other hand, the concept of electromagnetic phase as an observable quantity is a long-standing problem of quantum optics and it has been the question whether there exists a phase observable that is canonically conjugate to the number observable for a single-mode field. The quantum mechanical description of phase was first considered by London [1] and Dirac [2]. An obvious way of defining an operator for the phase is by polar decomposition of the photon annihilation operator â = e √ N̂ . The phase operator φ̂ defined in this way is equivalent to that considered by Dirac [2], who obtained the commutator [φ̂, N̂ ] = i by employing the correspondence between commutators and classical Poisson bracket. Formally, this would imply the uncertainty relation