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}.
The 'classical limit' of the q-analog quantized radiation field is studied paralleling conventional quantum optics analyses. The q-generalizations of the phase operator of Susskind and Glogower and that of Pegg and Barnett are constructed. Both generalizations and their associated number-phase uncertainty relations are manifestly q-independent in the n greater than g number basis. However, in the q-coherent state z greater than q basis, the variance of the generic electric field, (delta(E))(sup 2) is found to be increased by a factor lambda(z) where lambda(z) greater than 1 if q not equal to 1. At large amplitudes, the amplitude itself would be quantized if the available resolution of unity for the q-analog coherent states is accepted in the formulation. These consequences are remarkable versus the conventional q = 1 limit.
A family of q -coherent states is constructed allowing us to obtain a new quantized version of the harmonic oscillator. These q -states are normalized and form an overcomplete set resolving the unity with respect to the appropriate Jackson measure if 0 < q < 1. We only consider here those values of q such that q −1 is a Pisot number. In this case the q -deformed integers ([ n ] q ) form Fibonacci-like sequences of integers. We study the main physical characteristics of the corresponding quantum oscillator: localization in the configuration and in the phase spaces, probability distributions and related statistical features and semi-classical phase space trajectories whose periodicity is related with the fact that q is an algebraic number.
Semi-coherent states (semi-CSs) were previously introduced and their nonclassical properties studied. In this paper, we first propose a general formalism for the construction of ‘nonlinear semi-CSs’ as an extension of semi-CSs and then their explicit form in the number state basis is generally deduced. Next, as some physical appearances of the formalism, after choosing two well-known nonlinearity functions, nonclassical features and quantum statistical properties of the associated states are studied by evaluating quadrature squeezing, higher order squeezing and the Mandel parameter, as well as their phase properties and number and phase squeezing of the introduced states. Finally, the results are compared with the properties of semi-CSs.
At the end of the preceding chapter, we showed that the photon number states | n 〉 have a uniform phase distribution over the range 0 to 2π. Essentially, then, there is no well-defined phase for these states and, as we have already shown, the expectation value of the field operator for a number state vanishes. It is frequently suggested (see, for example, Sakurai) that the classical limit of the quantized field is the limit in which the number of photons becomes very large such that the number operator becomes a continuous variable. However, this cannot be the whole story since the mean field 〈 n | Ê x | n 〉 = 0 no matter how large the value of n . We know that at a fixed point in space a classical field oscillates sinusoidally in time. Clearly this does not happen for the expectation value of the field operator for a number state. In this chapter we present a set of states, the coherent states, which do give rise to a sensible classical limit; and, in fact, these states are the “most classical” quantum states of a harmonic oscillator, as we shall see. Eigenstates of the annihilation operator and minimum uncertainty states In order to have a non-zero expectation value of the electric field operator or, equivalently, of the annihilation and creation operators, we are required to have a superposition of number states differing only by ±1.