Coherently driven degenerate three-level laser with parametric amplifier
© Darge and Kassahun 2010
Received: 1 January 2010
Accepted: 24 March 2010
Published: 24 March 2010
We discuss the squeezing and statistical properties of the light produced by a coherently driven degenerate three-level laser with a parametric amplifier. We consider the case in which the atoms injected into the cavity are prepared in a coherent superposition of the top and bottom levels and with these levels coupled by the pump mode emerging from the parametric amplifier. It so happens that the presence of the parametric amplifier increases the squeezing and the mean photon number significantly. Furthermore, it is found that the maximum interacavity squeezing is 93% in the presence of the coupling and when the superposition has no contribution (η = 0). On the other hand, the maximum interacavity squeezing turns out to be 94% in the absence of the coupling. This squeezing is due to the parametric amplifier and the superposition. In addition, our calculation shows that one effect of coupling the top and bottom levels is to decrease the mean and the normally-ordered variance of the photon number.
PACS codes: 42.55.Ah, 42.50.Lc, 42.50.Ar
It has been established that a three-level laser under certain conditions generates squeezed light [1–9]. In a cascade three-level laser, three-level atoms in a cascade configuration are injected into a cavity coupled to a vacuum reservoir via a single-port mirror. The injected atoms may initially be prepared in a coherent superposition of the top and bottom levels and/or these levels may be coupled by strong coherent light after they are injected into the cavity. The superposition or the coupling of the top and bottom levels is responsible for the interesting nonclassical features of the generated light. When a three-level atom in a cascade configuration makes a transition from the top to the bottom level via the intermediate level, two photons are generated. If the two photons have the same frequency, the three-level atom is called degenerate otherwise it is called nondegenerate.
Some authors have studied the squeezing and statistical properties of the light produced by a three-level laser in which the crucial role is played by the superposition of the top and bottom levels [1–7]. Ansari  has predicted that such a laser can generate under certain conditions squeezed light. Furthermore, Lu and Zhu  have considered a nondegenerate three-level laser with the atoms initially prepared in coherent superposition of the top and bottom levels. They have predicted a maximum of 50% interacavity two-mode squeezing.
A three-level laser in which the top and bottom levels of the atoms injected into the cavity are coupled by a strong light has also been studied by different authors [7–9]. Ansari et al  have considered a degenerate three-level laser, with the atoms initially in the upper level and with the top and bottom levels of the atoms coupled by coherent light. They have shown that this system behaves like a parametric oscillator for sufficiently strong coherent light. They have also predicted that such a system can generate squeezed light over large range of the amplitude of the coherent light.
Furthermore, it has been predicted theoretically [10–16] and subsequently confirmed experimentally [17, 18] that a parametric oscillator produces light with a maximum interacavity squeezing of 50% below the coherent-state level. Some authors [19, 20] have considered a three-level laser whose cavity contains a parametric amplifier. Fesseha  has studied a three-level laser whose cavity contains a degenerate parametric amplifier, and with the injected atoms prepared initially in coherent superposition of the top and bottom levels. He has shown that the effect of the parametric amplifier is to increase the interacavity squeezing by a maximum of 50%. He has also pointed out that since the presence of the parametric amplifier also leads to a significant increase in the mean photon number, the system can produce a bright and highly squeezed light. Moreover, Alebachew and Fesseha  have considered a degenerate three-level laser whose cavity contains a parametric amplifier, with the top and bottom levels of the injected atoms coupled by the pump mode emerging from the parametric amplifier. They have studied this system for the specific case in which the number of atoms initially in the top and bottom levels are equal. They have found that this system generates under certain conditions a highly squeezed light. The squeezing in this case is exclusively due to the parametric amplifier and the coupling of the top and bottom levels.
2 c-number Langevin Equations
Here γ, considered to be the same for all the three-levels, is the atomic decay constant and A is the linear gain coefficient.
3 Quadrature Squeezing of the Cavity Mode
We observe that the equation of evolution of α-(t), described by (31), does not have a well-behaved solution for ε > G. We then identify ε = G as the threshold condition.
This result indicates that a degenerate three-level laser driven by a strong light behaves like a degenerate parametric oscillator .
It is apparent that ε is the only parameter representing the parametric amplifier. And inspection of Eq. (53) shows that one effect of the parametric amplifier is to decrease the value of the variance of the minus quadrature.
4 Quadrature Squeezing of the Output Mode
5 Photon Statistics of the Cavity Mode
5.1 Mean and variance of the photon number
5.2 Photon number distribution
In this paper we have seen the simplicity with which the squeezing and statistical properties of the light, generated by a coherently driven degenerate three-level lasers whose cavity contains a parametric amplifier, could be analyzed with the aid of c-number Langevin equations. Applying the solutions of these equations, we have calculated the quadrature variance for the cavity and output modes. Our results show that the presence of the parametric amplifier increases the squeezing of the light generated by the system under consideration, while the driving light has no effect on the squeezing. Furthermore, it so happens that for small values of the amplitude of the pump mode, the coupling of the top and bottom levels enhances the degree of the intracavity squeezing significantly for η = 0. Otherwise, it leads to a decrease in the intercavity squeezing.
It so turns out that for η = 0, A = 100, and κ = 0.8 the maximum interacavity squeezing is 93% below the coherent-state level (occurs at β = 0.1). This squeezing is exclusively due to the parametric amplifier and the coupling of the top and bottom levels. Furthermore, for η = 0.1 and the above values of A and κ the maximum interacavity squeezing is found to be 94% below the coherent-state level (occurs at β = 0). This squeezing is due to the parametric amplifier and the superposition of the top and bottom levels. In addition, we have shown that the cavity mode squeezing is greater than the output mode squeezing by 19%. On the other hand, we have determined via the Q function the mean and the normally-ordered variance of the photon number and the photon number distribution for the cavity mode. From the results we have found, we note that the driving coherent light and the parametric amplifier increase the mean of the photon number significantly. We have seen that one effect of the coupling of the top and bottom levels is to decrease the mean and normally-ordered variance of the photon number. This could be due to stimulated emission induced by the pump mode. The photons emitted this way do not contribute to the mean photon number of the cavity mode. Furthermore, we have also observed that the photon number statistics is super-Poissonian. In addition, we have found that there is a finite probability to find odd number of photons inside the cavity.
One of the authors, Tewodros, would like to thank the Abdus Salam ICTP for financial support.
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