The Phase Operator

Abstract

More particularly, the sets {|n〉 : n = 0, 1, · · · s} and {|θm〉 : m = 0, 1, · · · s} are two mutually unbiased bases for s as |〈n|θm〉|2 = 1/(s + 1) for n, m = 0, 1, · · · , s. The exponential phase operator is defined in terms of the phase state basis as ei φˆθ = s∑ m=0 eiθm |θm〉〈θm|, (2.4) or equivalently, in terms of the energy eigenstates as |n〉〈n + 1| + ei(s+1)θ0 |s〉〈0|. (2.5) The subscript θ serves as a reminder that the values of phase variable θm lie in the range from θ0 to θ0 + 2π . This construction ensures that the phase states are the eigenstates of ei φˆθ , i.e., ei φˆθ |θm〉 = eiθm |θm〉. (2.6) The exponential phase operator is unitary, because n=0 |n〉〈n| = 1ˆ s (2.7) where 1ˆ s is the identity operator on s . The Hermitian phase operator representing the phase angle itself was introduced in (Pegg and Barnett 1988b Paper 2.2) and (Barnett and Pegg 1989 Paper 2.3) as simply φˆθ = s∑ m=0 θm|θm〉〈θm|. (2.8) The operator on the left hand sides of Eqs. (2.4) and (2.5) is the exponential of i φˆθ .Moreover, the operator corresponding to an arbitrary function of the phase angle f (θ) was introduced in Barnett and Pegg (1992 Paper 3.3) and also discussed in Pegg and Barnett (1997a Paper 2.6) as f (φˆθ ) = s∑ m=0 f (θm)|θm〉〈θm|. (2.9) It follows that the Hermitian conjugate of ei φˆθ is the operator (ei φˆθ )† = s∑ m=0 e−iθm |θm〉〈θm| = e−i φˆθ . (2.10) This construction on the finite dimensional spaces elegantly sidesteps the problems faced with the infinite dimensional Hilbert space discussed in the previous chapter. Its simplicity is particularly appealing. The question remains, however, of how to deal with states such as | f 〉 = ∑∞n=0 cn|n〉 which reside in the infinite dimensional Hilbert space. For each value of s, there exists a phase operator defined on s ; this leads naturally to sequences of operators, such as e.g. φˆθ |s=S, φˆθ |s=S+1, φˆθ |s=S+2 · · · which are phase angle operators defined on the sequence of spaces S, S+1, S+2, · · · for arbitrarily large integer S. The space s contains a state | f 〉s which approximates | f 〉 to arbitrary precision for a sufficiently large value of s, such that the modulus of the overlap 〈 f | f 〉s is as close to unity as desired. Thus the issue of including states such as | f 〉 entails dealing with such sequences as s tends to infinity.