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arXiv:hep-th/9605052

Effective Field Theory for Coherent Optical Pulse Propagation

Q‐Han Park, Hyung‐Joon Shin

Published May 8, 1996• 1 citation
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Abstract

Hidden nonabelian symmetries in nonlinear interactions of radiation with matter are clarified. In terms of a nonabelian potential variable, we construct an effective field theory of self-induced transparency, a phenomenon of lossless coherent pulse propagation, in association with Hermitian symmetric spaces $G/H$. Various new properties of self-induced transparency, e.g. soliton numbers, effective potential energy, gauge symmetry and discrete symmetries, modified pulse area, conserved $U(1)$-charge etc. are addressed and elaborated in the nondegenerate two-level case where $G/H = SU(2)/U(1)$. Using the $U(1)$-charge conservation, a new type of analysis on pulse stability is given which agrees with earlier numerical results.

DOI
10.48550/arxiv.hep-th/9605052

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Physical Review A•Nov 1, 1977•4 citations
Hidden nonabelian symmetries in nonlinear interactions of radiation with matter are clarified. In terms of a nonabelian potential variable, we construct an effective field theory of self-induced transparency, a phenomenon of lossless coherent pulse propagation, in association with Hermitian symmetric spaces G/HG/HG/H. Various new properties of self-induced transparency, e.g. soliton numbers, effective potential energy, gauge symmetry and discrete symmetries, modified pulse area, conserved U(1)U(1)U(1)-charge etc. are addressed and elaborated in the nondegenerate two-level case where G/H=SU(2)/U(1)G/H = SU(2)/U(1)G/H=SU(2)/U(1). Using the U(1)U(1)U(1)-charge conservation, a new type of analysis on pulse stability is given which agrees with earlier numerical results.
We introduce a notion of ``matrix potential'' to nonlinear optical systems. In terms of a matrix potential ggg, we present a gauge-field-theoretic formulation of the Maxwell-Bloch equation that provides a semiclassical description of the propagation of optical pulses through resonant multilevel media. We show that the Bloch part of the equation can be solved identically through ggg and the remaining Maxwell equation becomes a second-order differential equation with a reduced set of variables due to the gauge invariance of the system. Our formulation clarifies the (non-Abelian) symmetry structure of the Maxwell-Bloch equations for various multilevel media in association with symmetric spaces G/HG/HG/H. In particular, we associate the nondegenerate two-level system for self-induced transparency with G/H=SU(2)/U(1)G/H=\mathrm{SU}(2)/\mathrm{U}(1)G/H=SU(2)/U(1) and three-level $\ensuremath{\Lambda}$ or VVV systems with G/H=SU(3)/U(2)G/H=\mathrm{SU}(3)/\mathrm{U}(2)G/H=SU(3)/U(2). We give a detailed analysis for the two-level case in the matrix potential formalism, and address various properties of the system including soliton numbers, effective potential energy, gauge and discrete symmetries, modified pulse area, conserved topological, and nontopological charges. The nontopological charge measures the amount of self-detuning of each pulse. Its conservation law leads to a different type of pulse stability analysis that explains earlier numerical results.
Above a critical power threshold for a given pulse width, a short pulse of coherent traveling-wave optical radiation is observed to propagate with anomalously low energy loss while at resonance with a two-quantum-level system of absorbers. The line shape of the resonant system is determined by inhomogeneous broadening, and the pulse width is short compared to dissipative relaxation times. A new mechanism of self-induced transparency, which accounts for the low energy loss, is analyzed in the ideal limit of a plane wave which excites a resonant medium with no damping present. The stable condition of transparency results after the traversal of the pulse through a few classical absorption lengths into the medium. This condition exists when the initial pulse has evolved into a symmetric hyperbolic-secant pulse function of time and distance, and has the area characteristic of a "$2\ensuremath{\pi}$ pulse." Ideal transparency then persists when coherent induced absorption of pulse energy during the first half of the pulse is followed by coherent induced emission of the same amount of energy back into the beam direction during the second half of the pulse. The effects of dissipative relaxation times upon pulse energy, pulse area, and pulse delay time are analyzed to first order in the ratio of short pulse width to long damping time. The analysis shows that the $2\ensuremath{\pi}$ pulse condition can be maintained if losses caused by damping are compensated by beam focusing. In an amplifying inhomogeneously broadened medium an analytic "$\ensuremath{\pi}$ pulse area" solution is presented in the limit of a sharp leading edge of the pulse. The dynamics of self-induced transparency are studied for the particular effects of Doppler velocities upon a resonant gas. The analysis of transparency for random orientations of dipole moments associated with degenerate rotational states yields modified forms of self-induced transparency behavior, which indicates a finite pulse energy loss as a function of distance in some cases. The effect of self-induced transparency on the photon echo is considered. Experimental observations of self-induced transparency have been made in a ruby sample at resonance with a pulsed ruby-laser beam. Single and multiple $2\ensuremath{\pi}$ pulse outputs have been observed, and pulse areas measured in the range of $2\ensuremath{\pi}$. The experimental results are compared with the predictions of the ideal-plane-wave theory. Deviations from the ideal-plane-wave theory are discussed. An analysis is made of the effect of a transverse mode of the propagating beam upon the transparency properties of the pulse.
We report on computer-generated solutions to the coupled matter-field equations which provide the first evidence for distortionless optical pulse propagation in an absorbing medium of three-level atoms. The associated phenomenon represents the three-level equivalent of self-induced transparency in two-level atoms. Results have application to the propagation of short, coherent optical pulses in a medium in which the atoms have two distinct transition frequencies in near resonance with the pulse carrier frequency. We relate in particular to the case of pulse propagation in a magnetically perturbed sodium vapor where the pulse carrier frequency is tuned near the frequencies of the well-known sodium DDD lines. Here we obtain particular solutions corresponding to three and five peak pulses with "areas" of approximately $6\ensuremath{\pi}$ and $10\ensuremath{\pi}$, respectively. The solutions are strongly supported by a number of analytical results. In particular, simple algebraic relations are found which give the populations of the levels in terms of the field variables.