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Optics Communications

Photon statistics in a single-mode two-photon laser: Exact versus effective interaction hamiltonian

Ning Lu

Published Jun 1, 1990• 4 citations
Publisher

Abstract

We study the operation and the photon statistics of a single-mode two-photon laser, using both the exact and the effective interaction hamiltonians. We obtain simple analytical expressions of the mean photon number and of the photon-number variance from both hamiltonians. The exact (effective) hamiltonian predicts that the two-photon laser can (cannot) build up from a vacuum without triggering, and the normalized photon-number variance is 118 (32) far above “threshold”. The domain of validity of the effective hamiltonian for the photon-number variance is found to be much smaller than that for the mean photon number.

DOI
10.1016/0030-4018(90)90435-v

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We study the operation and the photon statistics of a single-mode two-photon laser, using both the exact and the effective interaction hamiltonians. We obtain simple analytical expressions of the mean photon number and of the photon-number variance from both hamiltonians. The exact (effective) hamiltonian predicts that the two-photon laser can (cannot) build up from a vacuum without triggering, and the normalized photon-number variance is 118 (32) far above “threshold”. The domain of validity of the effective hamiltonian for the photon-number variance is found to be much smaller than that for the mean photon number.
A quantum theory of a two-photon laser is developed by making use of the antinormal-ordering Q function. Starting from the microscopic atom-field interaction Hamiltonian for cascade three-level atoms and including saturable absorbers, we first present a master equation for the field-density operator, and then transform the master equation into a Fokker-Planck equation for the Q function. The Q-function approach enables us to study the two-photon laser analytically, obtaining simple expressions for nonlinear gain, mean photon number, frequency pulling, natural linewidth, and photon-number variance in a unified method. We find that the field in a two-photon laser will build up from a vacuum without triggering if its linear gain is larger than the cavity loss. Also hysteresis can occur in the two-photon laser even without triggering. With an overall two-photon resonance, the normalized photon-number variance approaches a common value 11/8 well above ``threshold,'' independent of the one-photon detuning. We compare these results with previous results obtained from an effective interaction Hamiltonian. We find that the domain of validity of the effective Hamiltonian for the photon-number variance is much smaller than that for the mean photon number. Well above ``threshold'' the effective Hamiltonian overestimates both the natural linewidth and the photon-number variance.
In this dissertation, we study the quantum theory of two-photon micromaser and laser. To begin with, we study the photon statistics of the two-photon micromaser by considering the interaction of a quantized single-mode field with an effective two-level atom through intermediate states. We present an exact solution for the photon distribution function and study the behaviour of mean photon number and photon number fluctuations. A periodic "collapse" and "revival" type behaviour is predicted. We find that the photon number fluctuations exhibits a periodic sub-Poissonian behaviour with respect to the interaction time. We then discuss the photon statistics of two-photon laser and derive an exact expression for photon distribution function. We find that photon distribution function for two-photon laser is wider than the Poisson distribution function. We also calculate the natural line-width of a two-level two-photon laser by using a Fokker-Planck approach. It is shown that, apart from a constant, the gain contribution to the laser linewidth is equal to the two-photon gain parameter and is independent of the mean number of photons in the field. Finally, we study the two-photon transitions in multi-level atomic system. In this case first we discuss the photon statistics of three-level two-photon micromaser and laser. Then we consider the two-photon transition in four-level atomic system by considering dye laser as an example.
We have studied the statistical and squeezing properties of the cavity light generated by a two-level laser. This optical system contains N two-level atoms available in a cavity coupled to a single-mode vacuum reservoir. They are pumped to the top level from the bottom level by means of the electron bombardment. Applying the steady-state solutions of the equations of evolution of the expectation values of the atomic operators and the quantum Langevin equation, we obtained the global and local photon statistics of the single-mode light beam. We have found that, for the two-level laser operating well above the threshold, the uncertainties in the plus and minus quadratures are equal and satisfy the minimum uncertainty relation. In view of this, we have identified the light generated by the laser operating well above threshold to be coherent. On the other hand, the light generated by the laser operating at threshold is found to be chaotic. From the obtained results, we have also observed that a large part of the local mean photon number, the local photon number variance, and the local quadrature variance are confined in a relatively narrow frequency interval.
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.
Starting from the microscopic Hamiltonian of the three-level model and making use of the slowly varying amplitude approximation and the two-photon rotating wave approximation, we derive an effective Hamiltonian for the two-level two photon model under conditions where the two-photon process dominates the other radiartive processes, so that an exact relation between the three-level model and the two-level two-photon model is established. The effective Hamiltonian includes a term come from the optical Stark effect which was neglected in the conventional treatment. The coefficient of this term may have the same order of magnitude as the effective coupling constant. Based on this effective Hamiltonian, the standard method of Haken's laser theory is applied to study the threshold, the mean photon number, the self-induced frequency shift, the photon statistics and the fluctuation in steady states. Results are compared with previous ones which neglect the Strk effect.
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.