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Journal of the Physical Society of Japan

Decoherence of the Laser Field in Energy Steady-State Operation

Mikio Fujii

Published Dec 15, 1999• 0 citations
Publisher

Abstract

The rate at which pure initial states satisfying the detailed balance condition on the photon number states deteriorate into mixtures in the decoherence process is computed for a laser field above threshold in the Scully-Lamb laser model. In steady-state laser operation, the field state suggested by Scully and Lamb is shown to be the least increase of the error area in the complex amplitude plane, thus becoming the most stable field state among the squeezed states characterized by the phase order parameter Δ.

DOI
10.1143/jpsj.68.3819

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The rate at which pure initial states satisfying the detailed balance condition on the photon number states deteriorate into mixtures in the decoherence process is computed for a laser field above threshold in the Scully-Lamb laser model. In steady-state laser operation, the field state suggested by Scully and Lamb is shown to be the least increase of the error area in the complex amplitude plane, thus becoming the most stable field state among the squeezed states characterized by the phase order parameter Δ.
The lifetimes of different pure states of the quantized field in a single-mode laser (Scully-Lamb model) are considered. For operation high above threshold, I have shown number states to have extremely short lifetimes: Long lifetimes require coherent superpositions of neighboring number states with a width of the order of the square root of the average photon number. Thus, the long-lived states approach classical states. I propose a definition of the lifetime, which is found to be maximized by states that resemble the conventional Glauber coherent states, although they are slightly narrower in photon-number distribution. This is in contrast to the steadystate photon-number distribution for an ensemble of identical lasers (as given by the diagonal steady-state density matrix), which is wider than a coherent state. Possible interpretations and implications of these results are discussed.
It has widely been accepted that the laser field is a coherent state with phase diffusion. For example, in quantum theory of the laser it is conventional that the initial field state is given as a coherent state. But strictly speaking, this initial condition is a hypothesis which should be examined through experiments in terms of the quantum fluctuations of the held. In this study a new steady-state solution for the laser field density matrix is suggested, whose initial photon state is a quadrature-phase squeezed state, and whose photon number statistics is just the same as the conventional ansatz.
A mathematical model was developed to obtain the population inversion of the atoms in laser field in a laser cavity by considering the electric field in the optical cavity and the atomic states of the medium to be quantized. The master equation of the density operator of the laser field was studied analytically and numerically. Using coherent states, the Fokker-Plank equation for the phase space density for the laser field was solved analytically for the time dependent and steady state situations. The laser field above the threshold can be represented by a randomly phased mixture of coherent states. As the pump parameter increases in the laser process, the phase density becomes narrower, tending toward a delta function, creating a coherence laser field.
A quantum-mechanical model for a multiphoton laser is proposed by an obvious generalization of the usual one-photon laser model. The authors consider a class of lasers which preserve detailed balance. This condition is ensured by assuming a multiphoton loss mechanism for the laser light of the same order as the multiphoton gain from the stimulated emission process. Solutions for the stationary density operator of the light field are obtained by a straightforward generalization of the techniques of Scully and Lamb. The multiphoton laser around threshold exhibits behaviour analogous to a second-order phase transition. A non-thermal distribution of multiphoton laser light is achieved above threshold.
The single-mode laser model as formulated by Haken and co-workers is considered. Within the approximation which takes into account only the interaction (at all orders in the coupling constant) between the mode and the single atoms, this model leads to a linear time-evolution equation for the Glauber quasiprobability distribution $P(\ensuremath{\beta},{\ensuremath{\beta}}^{*},t)$ of the field mode, containing derivatives of all orders in $\ensuremath{\beta}$ and ${\ensuremath{\beta}}^{*}$. This equation is solved exactly in the stationary situation. Thus the quantum effects due to the terms with derivatives of order higher than the second one are fully taken into account. It is shown that these terms have a shrinking effect on the stationary distribution above threshold, counteracting the broadening effect due to the second-derivative term. The photon and photoelectron distributions are given. Exact relations are deduced, which allow the calculation of all the moments of the steady-state distribution from the first one. This distribution is compared with the stationary solutions of Risken's equation, of P. Mandel's equation, and with the stationary distribution deduced by Weidlich, Risken, and Haken. The connection with the single-mode laser theory of Scully and Lamb is analyzed in the high-intensity region. The complete Scully-Lamb master equation is translated into the Glauber representation, yielding a linear equation for $P(\ensuremath{\beta},{\ensuremath{\beta}}^{*},t)$ containing derivatives of all orders in $\ensuremath{\beta}$ and ${\ensuremath{\beta}}^{*}$. Suitably connecting the parameters AAA, BBB, and CCC of the Scully-Lamb equation with the parameters of the model of Haken and co-workers, it is shown that there is perfect agreement between the strong-signal Scully-Lamb steady-state distribution and that of Weidlich, Risken, and Haken for all values of the pump parameter.
The rate at which pure initial states deteriorate into mixtures is computed for a harmonic oscillator interacting with an environment in thermal equilibrium. The decoherence process resulting from this interaction selects a set of states characterized by maximal stability (or minimal loss of predictive power) which can be quantified by the rate of increase in either linear or statistical entropy. In the weak coupling limit, coherent states are shown to produce the least entropy, thus becoming the natural counterparts of classical points in phase space.