Classical analog of quantum phase

Daniela Dragoman, Mircea Dragoman

Jian Bo Fu, Wei Fang, Yongzheng Ye

In this paper, we introduce an optical analogy to quantum Fourier tanformation based on a pseudorandom phase ensemble. The optical analogy also brings about exponential speedup over classical Fourier tanformation. Using the analogy, we demonstrate three classcial fields to realize Fourier transform similar to three quantum particles.

Nicola Cufaro Petroni

Charles Jaffé, Sheldon Kanfer, Paul W. Brumer

We introduce an analog, in classical phase space, of the fundamental building block of time dependence in quantum mechanics, i.e., the superposition state. Applications to two-level systems and quantum beats illustrate this conceptually pleasing phase-space picture of the phase interference between energy eigenstates which comprises pure-state quantum dynamics.

J H Hannay

Charles A. Nelson, Mitchell H. Fields

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{\ne}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.