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Journal of Modern Optics

Field Fluctuations and Cross-correlation in the Two-photon Laser

S. Swain

Published Jan 1, 1988• 9 citations
Publisher

Abstract

We consider the theory of the two-photon laser using the Scully–Lamb theory developed for ordinary single-photon lasers, concentrating on the calculation of the linewidths and the cross-correlation coefficient. In a previous publication we have shown that the quantum-mechanical noise in the modes of a two-mode laser may be highly correlated. Here we show that the same situation appertains to the two-photon laser, that is the two modes are strongly positively cross-correlated when the laser is operating well above threshold. The two-photon laser has the advantage of tunability over the two-mode or quantum-beat laser. The point of using modes whose noise properties are strongly correlated is that in an appropriate two-photon experiment the noises may effectively ‘cancel out’ leaving a very low noise signal. We present a brief development of the two-photon laser theory and show the connection between our work and that of other approaches.

DOI
10.1080/09500348814550151

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We consider the theory of the two-photon laser using the Scully–Lamb theory developed for ordinary single-photon lasers, concentrating on the calculation of the linewidths and the cross-correlation coefficient. In a previous publication we have shown that the quantum-mechanical noise in the modes of a two-mode laser may be highly correlated. Here we show that the same situation appertains to the two-photon laser, that is the two modes are strongly positively cross-correlated when the laser is operating well above threshold. The two-photon laser has the advantage of tunability over the two-mode or quantum-beat laser. The point of using modes whose noise properties are strongly correlated is that in an appropriate two-photon experiment the noises may effectively ‘cancel out’ leaving a very low noise signal. We present a brief development of the two-photon laser theory and show the connection between our work and that of other approaches.
Thomas et al. (1981, 1982) proposed the generation of cross correlation between two laser fields interacting with a three-level system as a means of reducing noise and subsequently exploited this property in the observation of very narrow Ramsey fringes. Cross correlation has been discussed theoretically by Dalton and Knight (1982, 1983) and shown to have interesting effects in population trapping. For such effects to be important, the cross correlation coefficient must be as large as possible. The authors discuss the degree of correlation between the two modes of a two-mode laser using the approach of Scully and Lamb (1967, 1968) and show it can be large. They also evaluate the linewidths of the two laser modes. They find that if the laser parameters for the two modes are equal, the two-mode linewidth is one half the value of the linewidth of the corresponding single-mode laser, well above threshold.
Using the quantum theory of a non-degenerate parametric amplifier, we discuss the phase correlations between signal and idler modes well above threshold. Particular attention is paid to the influence which fluctuation of the driving laser has on these correlations. We point out that the signal and idler mode phase correlation may be tuned by control of the driving laser field intensity and linewidth. For coherent driving, the mode phases are negatively cross-correlated. We stress the qualitative importance of these mode correlations for a class of two-photon resonant excitation experiments in atoms. In particular, the non-degenerate parametric oscillator represents a concrete model of a source with negatively cross-correlated phase fluctuations.
A quantum theory of a two-photon laser in a three-level atomic system of the cascade type is presented starting from an original microscopic Hamiltonian. The master equation of the laser operation is derived by using the method of Scully and Lamb [Phys. Rev. A 159, 208 (1967)] for a single-mode laser containing a two-level medium. The photon-number distribution of the laser is studied in detail. The curve of the distribution may have a two-peak structure. One of the peaks located at the origin comes from the single-photon process and reflects the thermal property of the laser, while another comes from the two-photon process and reflects the coherent property of the laser. We cannot find an exact criterion for the threshold of the two-photon laser. The two-photon laser can approximately be considered as a coherent state for certain values of A/C, which can even be much smaller than the threshold for the nonresonant single-mode laser in a two-level system, but it is much larger than the threshold for the resonant one.
The quantum-statistical properties of the optical field of a two-mode laser with two coupled transitions have been studied by means of a generalization of the Scully and Lamb theory. The photon number distribution and the mode intensity distribution are obtained in the steady state for a system of homogeneously broadened atoms in resonance with the laser field. It is shown that the mode-coupling constant $\ensuremath{\xi}$ is unity and that near threshold the results of earlier treatments are recovered. Furthermore, it is shown that certain limit measures for the relative intensity fluctuations predicted by the semiclassical Fokker-Planck treatments based on the third-order theory are valid even in the limit of arbitrarily high gain levels.
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.
We develop a nonlinear theory of a two-photon correlated-spontaneous-emission laser (CEL) by using an effective interaction Hamiltonian for a two-level system coupled by a two-photon transition. Assuming that the active atoms are prepared initially in a coherent superposition of two atomic levels involved in the two-photon transition, we derive a master equation for the field-density operator by using our quantum theory for coherently pumped lasers. The steady-state properties of the two-photon CEL are studied by converting the field master equation into a Fokker-Planck equation for the antinormal-ordering Q representation of the field-density operator. Because of the injected atomic coherence, the drift and diffusion coefficients become phase sensitive. This leads to laser phase locking and an extra two-photon CEL gain. The laser field can build up from a vacuum in the no-population-inversion region, in contrast to an ordinary two-photon laser for which triggering is needed. We find an approximate steady-state solution of the Q representation for the laser field, which consists of two identical peaks of elliptical type. We calculate the phase variance and, for any given mean photon number, obtain the minimum variance in the phase quadrature as a function of the initial atomic variables. Squeezing of the quantum noise in the phase quadrature is found and it exhibits the following features: (1) it is possible only when the laser intensity is smaller than a certain value; (2) it becomes most significant for small mean photon number, which is achievable in the no-population-inversion region; and (3) a maximum of 50% squeezing can be asymptotically approached in the small laser intensity limit. As a by-product we also study the ordinary two-photon laser and find, e.g., photon-number variance and laser linewidth.