The q-analog coherent states \ensuremath{\Vert}z
〉q are used to identify some of the canonical physical properties of the single-mode q-analog quantized radiation field in the \ensuremath{\Vert}z
〉q ``classical limit'' where \ensuremath{\Vert}z\ensuremath{\Vert} is large. In this quantum-optics-like limit, the fractional uncertainties of most physical quantities (momentum, position, amplitude, phase) which characterize the quantum field are shown to be O(1), and only vanish as O(1/\ensuremath{\Vert}z\ensuremath{\Vert}) when q=1. In contrast to this more-quantum-like behavior for q\ensuremath{\ne}1, the fractional uncertainties do still approach zero for the usual number operator, N, and the N-Hamiltonian
HN\ensuremath{\equiv}\ensuremath{\Elzxh}\ensuremath{\omega}(N+1/2) which describes a free q-boson gas. An empirical signature for q-boson counting statistics is that (\ensuremath{\Delta}N
)2/〈N〉\ensuremath{\rightarrow}0 as \ensuremath{\Vert}z\ensuremath{\Vert}\ensuremath{\rightarrow}\ensuremath{\infty}. Properties of the q-analog generalizations of the phase operators of Susskind and Glogower (SG) and of the phase operator \ensuremath{\varphi}${\mathrm{^}}_{\mathit{q}}$ of Pegg and Barnett are investigated. In contrast to the manifest q deformed properties of SG operators for moderate \ensuremath{\Vert}z${\mathrm{\ensuremath{\Vert}}}^{2}$, the ``Hermitian'' phase operator \ensuremath{\varphi}${\mathrm{^}}_{\mathit{q}}$ still exhibits almost normal classical behavior in the \ensuremath{\Vert}z
〉q basis. In particular, the conventional (approximate) number-phase uncertainty relation \ensuremath{\Delta}N\ensuremath{\Delta}\ensuremath{\varphi}${\mathrm{^}}_{\mathit{q}}$\ensuremath{\ge}1/2 and approximate commutation relation [N,\ensuremath{\varphi}${\mathrm{^}}_{\mathit{q}}$]=i are found to follow for the single-mode q-analog quantized field. So N and \ensuremath{\varphi}${\mathrm{^}}_{\mathit{q}}$ are almost canonically conjugate operators in the \ensuremath{\Vert}z
〉q classical limit. The \ensuremath{\Vert}z
〉q coherent states minimize this uncertainty relation for moderate \ensuremath{\Vert}z${\mathrm{\ensuremath{\Vert}}}^{2}$. q-analog generalizations of the P, Q, and W phase-space representations are treated in the Appendix.
The q-analogue coherent states {vert_bar}z >{sub q} are used to identify physical signatures for the presence of a q-analogue quantized radiation field in the {vert_bar} >{sub q} classical limit where {vert_bar}z{vert_bar} is large. In this quantum-optics-like limit, the fractional uncertainties of most physical quantities (momentum, position, amplitude, phase) which characterize the quantum field are O(1). They only vanish as O(1/{vert_bar}z{vert_bar}) when q = 1. However, for the number operator, N, and the N-Hamiltonian for a free q-boson gas, H{sub N} = {Dirac_h}{omega}(N + 1/2), the fractional uncertainties do still approach zero. A signature for q-boson counting statistics is that ({Delta}N){sup 2}/<N> {yields} 0 as {vert_bar}z{vert_bar} {yields} {infinity}. Except for its O(1) fractional uncertainty, the q-generalization of the Hermitian phase operator of Pegg and Barnett, {phi}{sub q}, still exhibits normal classical behavior. The standard number-phase uncertainty-relation, {Delta}N {Delta}{phi}{sub q} = 1/2, and the approximate commutation relation, [N,{phi}{sub q}] = i, still hold for the single-mode q-analogue quantized field. So, N and {phi}{sub q} are almost canonically conjugate operators in the {vert_bar}z >{sub q} classical limit. The {vert_bar}z >{sub q} CS`s minimize this uncertainty relation for moderate {vert_bar}z{vert_bar}{sup 2}.
For a quantized mode of the radiation field, the operator whose classical analog is the ordinary phase factor of the mode amplitudes has been shown to be nonunitary. A mathematically rigorous formulation of the phase P is given on the basis of the canonical factorization theorem. All other phase quantities are defined in terms of P and P†. Many of the seemingly complex features of phase operators are found to be simple direct consequences of the general mathematical theory. It is easily seen that P is a partial isometry but not a unitary operator. In contrast to the amplitude operator, it is found that P is not a spectral operator and the set of phase eigenstates is not complete. Mathematically precise operator relations, including the rigorous statement of commutation rules, are developed. For each of the phase operators, a complete spectral analysis is given, with the nature of the spectrum, spectral decomposition (if any), and eigenstates shown explicitly. The values of the various phase operators are compared between phase states and coherent states.
The formalism that we have previously developed for the phase difference between two quantized electromagnetic fields, which is intimately connected with the measurement process, is explored further, both theoretically and experimentally. We calculate the higher moments of the measured cosine and sine operators for certain two-mode Fock states \ensuremath{\Vert}
n1,
n2〉, and show how the measurement itself exerts a bias on the outcome. We find that the corresponding phase difference becomes uniform over the interval 0 to 2\ensuremath{\pi} only in the limit
n1,
n2\ensuremath{\rightarrow}\ensuremath{\infty}. On the other hand, the phase difference associated with the product of a coherent state \ensuremath{\Vert}v〉 with a Fock state can be random for large \ensuremath{\Vert}v\ensuremath{\Vert}, because of the availability of an infinite number of photons. Several of our theoretical predictions are compared with predictions based on the Susskind-Glogower and the Pegg-Barnett operators, and they are also tested by experiment. We find that the experimental results confirm our theory in every case, and this includes tests of the higher moments of the measured cosine operator.
Following the development of the quantum theory of radiation and with the advent of the laser, the states of the field that most nearly describe a classical electromagnetic field were widely studied. In order to realize such ‘classical’ states, we will consider the field generated by a classical monochromatic current, and find that the quantum state thus generated has many interesting properties and deserves to be called a coherent state . An important consequence of the quantization of the radiation field is the associated uncertainty relation for the conjugate field variables. It therefore appears reasonable to propose that the wave function which corresponds most closely to the classical field must have minimum uncertainty for all times subject to the appropriate simple harmonic potential. In this chapter we show that a displaced simple harmonic oscillator ground state wave function satisfies this property and the wave packet oscillates sinusoidally in the oscillator potential without changing shape as shown in Fig. 2.1. This coherent wave packet always has minimum uncertainty, and resembles the classical field as nearly as quantum mechanics permits. The corresponding state vector is the coherent state |α〉, which is the eigenstate of the positive frequency part of the electric field operator, or, equivalently, the eigenstate of the destruction operator of the field. Classically an electromagnetic field consists of waves with welldefined amplitude and phase. Such is not the case when we treat the field quantum mechanically.