Number and phase uncertainties of theq-analog quantized field
Charles A. Nelson, Mitchell H. Fields · Physical Review A · 1995
The q-analog coherent states \ensuremath{\Vert}z${\mathrm{〉}}_{\mathit{q}}$ are used to identify some of the canonical physical properties of the single-mode q-analog quantized radiation field in the \ensuremath{\Vert}z${\mathrm{〉}}_{\mathit{q}}$ ``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{ e}1, the fractional uncertainties do still approach zero for the usual number operator, N, and the N-Hamiltonian ${\mathit{H}}_{\mathit{N}}$\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${\mathrm{〉}}_{\mathit{q}}$ 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${\mathrm{〉}}_{\mathit{q}}$ classical limit. The \ensuremath{\Vert}z${\mathrm{〉}}_{\mathit{q}}$ 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.