 Research article
 Open Access
Coherently driven degenerate threelevel laser with parametric amplifier
 Tewodros Y Darge^{1}Email author and
 Fesseha Kassahun^{1}
 Received: 1 January 2010
 Accepted: 24 March 2010
 Published: 24 March 2010
Abstract
We discuss the squeezing and statistical properties of the light produced by a coherently driven degenerate threelevel 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 normallyordered variance of the photon number.
PACS codes: 42.55.Ah, 42.50.Lc, 42.50.Ar
Keywords
 Cavity Mode
 Photon Number
 Quadrature Variance
 Bottom Level
 Pump Mode
1 Introduction
It has been established that a threelevel laser under certain conditions generates squeezed light [1–9]. In a cascade threelevel laser, threelevel atoms in a cascade configuration are injected into a cavity coupled to a vacuum reservoir via a singleport 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 threelevel 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 threelevel atom is called degenerate otherwise it is called nondegenerate.
Some authors have studied the squeezing and statistical properties of the light produced by a threelevel laser in which the crucial role is played by the superposition of the top and bottom levels [1–7]. Ansari [7] has predicted that such a laser can generate under certain conditions squeezed light. Furthermore, Lu and Zhu [2] have considered a nondegenerate threelevel laser with the atoms initially prepared in coherent superposition of the top and bottom levels. They have predicted a maximum of 50% interacavity twomode squeezing.
A threelevel 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 [9] have considered a degenerate threelevel 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 coherentstate level. Some authors [19, 20] have considered a threelevel laser whose cavity contains a parametric amplifier. Fesseha [19] has studied a threelevel 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 [20] have considered a degenerate threelevel 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 cnumber Langevin Equations
Here γ, considered to be the same for all the threelevels, 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 wellbehaved solution for ε > G. We then identify ε = G as the threshold condition.
Quadrature variance
Quadrature Variance  

η  β  G  ε 

0  0.1000  5.3135  5.3000  0.0731 
0.1000  0  5.4000  5.3000  0.0608 
This result indicates that a degenerate threelevel laser driven by a strong light behaves like a degenerate parametric oscillator [9].
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
6 Conclusions
In this paper we have seen the simplicity with which the squeezing and statistical properties of the light, generated by a coherently driven degenerate threelevel lasers whose cavity contains a parametric amplifier, could be analyzed with the aid of cnumber 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 coherentstate 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 coherentstate 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 normallyordered 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 normallyordered 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 superPoissonian. In addition, we have found that there is a finite probability to find odd number of photons inside the cavity.
Declarations
Acknowledgements
One of the authors, Tewodros, would like to thank the Abdus Salam ICTP for financial support.
Authors’ Affiliations
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