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Quantum and Semiclassical Optics Journal of the European Optical Society Part B

The N-atom laser below saturation: a density matrix approach without large-N scaling

Michael Fleischhauer, S Sandhager

Published Jun 1, 1995• 1 citation
Publisher

Abstract

The authors present a method of deriving a reduced density operator equation for a quantized field interacting with an arbitrary number of atoms that does not neglect atom-atom correlations. As a particular example they analyse a three-level laser operating well below saturation and solve the density matrix equations in steady state. For weak pumping the system is equivalent to the Scully-Lamb model (1966) and the corresponding equations are recovered. Using the saturation photon number as a (large) scaling parameter one can derive a Fokker-Planck equation for the Glauber-P-distribution. The stationary solution of the reduced density matrix equation is compared with the solution of the Fokker-Planck equation and-for the case of a single-atom laser-with a numerical solution of the total density matrix equation.

DOI
10.1088/1355-5111/7/3/012

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The authors present a method of deriving a reduced density operator equation for a quantized field interacting with an arbitrary number of atoms that does not neglect atom-atom correlations. As a particular example they analyse a three-level laser operating well below saturation and solve the density matrix equations in steady state. For weak pumping the system is equivalent to the Scully-Lamb model (1966) and the corresponding equations are recovered. Using the saturation photon number as a (large) scaling parameter one can derive a Fokker-Planck equation for the Glauber-P-distribution. The stationary solution of the reduced density matrix equation is compared with the solution of the Fokker-Planck equation and-for the case of a single-atom laser-with a numerical solution of the total density matrix equation.
A quantum theory of two-photon correlated-spontaneous-emission lasers (CEL's) is developed, starting from the exact atom-field interaction Hamiltonian for cascade three-level atoms interacting with a single-mode radiation field. We consider the situation where the active atoms are prepared initially in a coherent superposition of three atomic levels and derive a master equation for the field-density operator by using a quantum theory for coherently pumped lasers. The master equation is transformed into a Fokker-Planck equation for the antinormal-ordering Q function. The drift coefficients of the Fokker-Planck equation enable us to study the steady-state operation of the two-photon CEL's analytically. We have studied both resonant two-photon CEL for which there is no threshold, and off-resonant two-photon CEL for which there exists a threshold. In both cases the initial atomic coherences provide phase locking, and squeezing in the phase quadrature of the field is found. The off-resonant two-photon CEL can build up from a vacuum when its linear gain is larger than the cavity loss (even without population inversion). Maximum squeezing is found in the no-population-inversion region with the laser intensities far below saturation in both cases, which are more than 90% for the resonant two-photon CEL and nearly 50% for the off-resonant one. Approximate steady-state Q functions are obtained for the resonant two-photon CEL and, in certain circumstances, for the off-resonant one.
A quantum theory of a two-level single-mode laser with injected atomic coherence is developed by generalizing the Scully-Lamb laser theory to a form appropriate for the analysis of a coherently pumped laser. We assume that the active atoms are prepared initially in a coherent superposition of the upper and lower levels, and we derive the master equation for the field density operator by treating the interaction of the laser field with many active atoms simultaneously. It is shown that the photon-number distribution can be exactly Poissonian. The laser operation is analyzed in terms of the Fokker-Planck equation for the laser field. Both the intensity and phase diffusion coefficients are phase sensitive and, for stable laser operation, become much smaller than those of an ordinary laser. Consequently, the injected atomic coherence reduces both the photon-number noise and phase noise simultaneously. The intensity diffusion coefficient can vanish exactly, and at the same time the phase diffusion coefficient can become very small. This leads to spontaneous-emission noise quenching in the photon-number distribution, and the laser field can become very close to a coherent state. A scheme to generate the proper form of the initial atomic coherence necessary for the quantum noise quenching is proposed and analyzed.
The quantum theory of two-level atom single-mode laser system of Sargent, Scully and Lamb is extended to two-mode laser in a three-level atomic system with common lower level, and master equation is obtained for the system with all the three levels being pumped. There appear four terms in the master equation that are absent in the single-mode case and those interpreted as 2-photon processes play an especial role in the present case. The master equation is represented by a probability flow diagram in two-dimensions with the photon numbers n1 and n2 of the two-mode as variables. The probability flow diagram can be extended to infinity, and each arrow of it represents a term of the right hand side of master equation. By taking sum over one of the variables n1 or n2, the master equation is reduced to two equations, each of them can be represented by an one-dimensional probability flow diagram. The equation of motion under steady state is obtained by considering the correspondence between the macroscopic balance and the microscopic balance.A parameter H is introduced to reduce the equation of motion and to facilitate the mathematical procedure and a formal solution is thus obtained. Using the equation of motion and its formal solution various operation characteristics are discussed. Among them are: the threshold condition for each mode; the condition of only one mode operation; the variation of photon statistics between single-mode operation and two-mode operation and the variation of photon statistics with lower level being pumped and not pumped.
The transient behavior of a laser is discussed using the quantum theory as did Scully and Lamb. The formal solution of the density-matrix equation is expressed in terms of exponentially decaying eigenmodes. Some of the lower decay constants are obtained numerically. The equations for the moments of the density matrix are then derived and solved by a truncation method. The equations of motion are integrated numerically for the case where the average number of photons in a laser cavity has the realistically large value 1.3 \ifmmode\times\else\texttimes\fi{} 105{10}^{5}105. An alternative Fokker-Planck-equation approach is discussed.
We derive the equation of motion for the quantum-mechanical radiation density matrix of a gas laser to lowest order in the dimensionless coupling constant. Our derivation is fully quantum mechanical and we can calculate the coherence properties from the radiation density matrix. Our model consists of NNN two-level systems interacting with radiation in a cavity in the presence of dissipation, pumping, and collisions. The method we use is a generalization of the Bogoliubov derivation of the kinetic equation for a small parameter. Our derivation holds for any physically realizable pump power, and near threshold reduces to Lamb's near-threshold theory. With the equation of motion for the radiation density matrix, we obtain solutions both for when the average field is nonzero and for when it is zero. The steady-state electromagnetic density is the same in both cases except for a small spontaneous-emission term. We show that the reason gas lasers do not satisfy rate equations is the existence of zeroth-order correlations between the internal atomic variables and atomic center-of-mass variables. It is these same zeroth-order correlations which are responsible for the Lamb dip. Our derivation includes collisions and reduces the calculations of their effect to quadrature.
Starting from the master equation for density operator of the atoms and the field mode and applying the operator method of adiabatic elimination of the atomio variables developed by Haake and Lewenstein, we derived the Fokker-Planck Equation for Wigner function of the lightfield and its steady-state solution. With a Gaussian approximation to the solution, analytical and numerical results on the photon statistics for various pumping strengths are obtained. Comparisons with previous works are made.