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Journal of Mathematical Physics

Minimum Uncertainty Product, Number-Phase Uncertainty Product, and Coherent States

R. Jackiw

Published Mar 1, 1968• 205 citations
Publisher

Abstract

The number-phase uncertainty products proposed by Carruthers and Nieto are studied to determine whether they are minimized by coherent states. It is found that coherent states do not minimize these products. States that do minimize some of the uncertainty products are constructed. Variational techniques for the study of arbitrary uncertainty products are developed.

DOI
10.1063/1.1664585

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The minimization of the uncertainty relation $2\ensuremath{\Delta}x\ensuremath{\Delta}p\ensuremath{\ge}\ensuremath{\hbar}$ characterizes the coherent states. The spincoherent states of Radcliffe minimize the uncertainty relation $2\ensuremath{\Delta}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{J}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{u}})\ensuremath{\Delta}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{J}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}})\ensuremath{\ge}|〈\stackrel{\ensuremath{\rightarrow}}{\mathrm{J}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{w}}〉|$ (where $\stackrel{\ensuremath{\rightarrow}}{\mathrm{u}}$, $\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}$, $\stackrel{\ensuremath{\rightarrow}}{\mathrm{w}}$ form an arbitrary orthonormal basis), but this property does not characterize them. Here, we analyze that situation and suggest an alternative to restore the parallelism.

Physical Review A•Aug 1, 1978•18 citations
The number-phase uncertainty products proposed by Carruthers and Nieto are studied to determine whether they are minimized by coherent states. It is found that coherent states do not minimize these products. States that do minimize some of the uncertainty products are constructed. Variational techniques for the study of arbitrary uncertainty products are developed.
Nieto and Simmons have defined and studied coherent states for arbitrary potentials V(q)V(q)V(q). We show how to extend their results to certain time-dependent potentials V(q,t)V(q,t)V(q,t). For each V(q)V(q)V(q) there is a V(q,t)V(q,t)V(q,t), for which we can construct time-dependent coherent states.
The number-phase uncertainty product for displaced number states of the harmonic oscillator is examined within the framework of three different approaches: (i) the Carruthers-Nieto number-phase uncertainty relation in terms of the Susskind-Glogower sine and cosine phase operators, (ii) a similar relation to this calculated with the Pegg-Barnett unitary phase operator, and (iii) the number-phase uncertainty relation arising from the Pegg-Barnett Hermitian phase operator. The corresponding number-phase uncertainty product is calculated extending from average photon numbers 30 down to m for the first few classes of displaced number states (m=0,1,...,5). It is found that, for a displaced number state with a reasonable average number of excited quanta, all three rival phase formalisms yield similar number-phase uncertainty products, tending, for increasingly large magnitude of the displacement parameter, to the constant value m+1/2. On the other hand, for a small number of excited quanta it is found that, according to the first two formalisms, the number-phase uncertainty product for a given class of displaced number states tends to the maximum value \ensuremath{\surd}2m+1 /(\ensuremath{\surd}m+1 - \ensuremath{\surd}m ), while the third phase formalism predicts an entirely different behavior; with decreasing magnitude of displacement parameter, after passing through a maximum, the number-phase uncertainty product falls off eventually to zero making, in particular, the search for minimum number-phase uncertainty states futile. It is argued in favor of this last result, and the possibility of experimental verification, in the realm of quantum optics, is briefly considered.
The minimization of the uncertainty relation $2\ensuremath{\Delta}x\ensuremath{\Delta}p\ensuremath{\ge}\ensuremath{\hbar}$ characterizes the coherent states. The spincoherent states of Radcliffe minimize the uncertainty relation $2\ensuremath{\Delta}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{J}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{u}})\ensuremath{\Delta}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{J}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}})\ensuremath{\ge}|〈\stackrel{\ensuremath{\rightarrow}}{\mathrm{J}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{w}}〉|$ (where $\stackrel{\ensuremath{\rightarrow}}{\mathrm{u}}$, $\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}$, $\stackrel{\ensuremath{\rightarrow}}{\mathrm{w}}$ form an arbitrary orthonormal basis), but this property does not characterize them. Here, we analyze that situation and suggest an alternative to restore the parallelism.