Limit on attainable laser intensities implied by any small violation of Bose statistics

Abstract

Effects of a small violation of Bose statistics are investigated by treating the photon as a q boson, i.e., the commutation relation of creation and annihilation operators is replaced by ${\mathit{aa}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$-${\mathit{qa}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$a=1 with -1\ensuremath{\le}q\ensuremath{\le}1, so that q=1 is a Bose particle and q=-1 is a Fermi particle. A theory of q-boson emission is modeled on the usual theory of photon emission, and the physical consequences of a small deviation of q from unity are determined. The fact that a and ${\mathit{a}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$ are bounded operators for all q1 implies that stimulated emission ceases when the intensity is sufficiently large in any simple model. The specific relationship between q and the maximum attainable intensity depends on the model, although rather insensitively. Two simple models are examined in detail. In both we find that stimulated emission saturates when there are approximately 4/(1-q${)}^{2}$ quanta per mode. This implies a maximum attainable intensity for 1-eV lasers of \ensuremath{\approxeq}3/(1-q${)}^{2}$ MW/${\mathrm{cm}}^{2}$ so that a deviation of q from unity as large as ${10}^{\mathrm{\ensuremath{-}}6}$ is quite consistent with the existence of current high-intensity lasers. One of the two models constructed, which appears very natural in the mathematical formalism of the generalized commutator algebra, is found to give a q extension of the exclusion principle. Specifically it predicts that the growth of the occupation number in time will drop from quadratic to logarithmic when the occupation number reaches a value of \ensuremath{\approxeq}[2/(1-q)${]}^{2}$. Peculiarities in the large-t behavior near q=0 and -1 are examined briefly.