Effect of biased noise fluctuations on the output radiation of coherent beat laser

Effect of biased noise fluctuations on the degree of squeezing as well as the intensity of a radiation generated by a one-photon coherent beat laser is presented. It turns out that the radiation exhibits squeezing inside and outside the cavity under certain conditions. The degree of squeezing is enhanced by the biased noise input significantly in both regions. Despite the presence of the biased environment modes outside the cavity, the degree of squeezing outside the cavity can be greater than or equal to or even less than the cavity radiation depending on the initial preparation of the atomic superposition and amplitude of the external driving radiation. But the intensity of the radiation is found to be lesser outside the cavity regardless of these parameters. PACS Codes: 42.50.Dv, 42.50.Ar, 42.50.Gy, 32.80.Bx Introduction In recent years, interaction of three-level atoms with radiation has attracted a great deal of interest in relation to the strong correlation induced particularly during the cascading transitions [1-13]. It is common knowledge by now that the atomic coherence in such a system is accountable for the squeezing of the emitted radiation. The atomic coherence can be induced in a three-level cascade scheme by coupling the upper energy level |a and lower energy level |c , between which a direct transition is dipole forbidden, with external radiation [1-6] or by preparing the atom initially in coherent superposition of these two levels [7-12] or using these mechanisms at the same time [13]. In addition to these options, Xu and Hu [14] have considered the two-step cascade coherent excitation. For the sake of convenience, the amplification of light when spontaneously emitted photons in the cascade transitions are correlated by the atomic coherence resulting from the initial preparation of the superposition and external driving mechanism can be taken as coherent beat laser (CBL). The initially prepared atoms are assumed to be injected into the cavity at constant rate and removed after they spontaneously decay to energy levels that are not Published: 1 December 2008 PMC Physics B 2008, 1:17 doi:10.1186/1754-0429-1-17 Received: 8 July 2008 Accepted: 1 December 2008 This article is available from: http://www.physmathcentral.com/1754-0429/1/17 © 2008 Tesfa This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/ licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Page 1 of 18 (page number not for citation purposes) PMC Physics B 2008, 1:17 http://www.physmathcentral.com/1754-0429/1/17 involved in the lasing process. It is a well known fact that in three-level cascading process, basically, two photons are generated. If the two photons have identical frequency, the three-level atom is referred to as a degenerate three-level atom. In actual experimental setting, the cavity is unavoidably coupled to the fluctuations in the external environment modes. As a result, the quantum properties in such a system is limited by the leakage through the mirror and inevitable amplification of the quantum fluctuations in the cavity. In other words, the squeezing of the cavity radiation in particular is degraded since the vacuum field has fundamentally no definite phase. In this accord, various authors have studied similar scheme coupled to vacuum reservoir when the atomic coherence is induced by external driving radiation and when the atoms are initially prepared to be in the upper energy level [1], lower energy level [2] and arbitrary coherent superposition of the two levels. It has been argued that the three-level laser in these cases resemble the corresponding parametric oscillator in the strong driving limit. However, replacing the ordinary vacuum reservoir by squeezed vacuum can enhance the squeezing of the cavity radiation [15,16]. Based on this, the idea of coupling the cavity radiation of phase-sensitive amplifier to biased noise fluctuations of broadband environment modes has been explained recently [7]. Since the squeezing in the phase-sensitive laser corresponds to unequal gain and unequal noise in the quadrature phases, coupling the cavity to biased noise fluctuations is expected to lead to enhancement of the degree of squeezing as long as the environment modes are biased in the right quadrature. In the present day state of the art technology, biased noise fluctuations can be generated, for instance, by optical feedback loop [17]. In light of this, the effects of the biased noise fluctuations (squashed light and twin beams) on the radiative properties of a three-level cascade atom have been discussed by Wiseman and co-worker [18-20] earlier. Though previous works deal mainly with the degree of squeezing and statistical properties of the cavity radiation, it is the output radiation that is accessible to the experimenter and can readily be utilized. In this respect, it appears that there is renewed interest in comparing the squeezing and intensity of the radiation inside the cavity with the outside radiation using the input-output relation introduced by Gardiner and Collett [21]. It is found that the squeezing of the output radiation for the degenerate parametric oscillator coupled to broadband squeezed vacuum reservoir is less than the cavity radiation contrary to earlier expectations [22]. In addition, it has been shown recently that the output radiation can have a better degree of squeezing than the cavity radiation for degenerate correlated emission laser coupled to broadband squeezed vacuum reservoir (Tesfa S, unpublished data). Despite these efforts, experimental realization of the squeezed state indicates that the band of the squeezed light is typically in the order of the atomic line width [23] which automatically defies the broadband approximation. It, hence, seems imperative studying the squeezing of the radiation outside when the cavity is coupled to experimentally realizable broadband environment modes. That is why the output of the degenerate CBL whose cavity is coupled to broadband biased noise fluctuations is considered. Moreover, this work is confined to the regime of lasing without population inversion due to the limitation imposed by the uncerPage 2 of 18 (page number not for citation purposes) PMC Physics B 2008, 1:17 http://www.physmathcentral.com/1754-0429/1/17 tainty condition. The quadrature variances and mean photon number of the output radiation are calculated in the linear and adiabatic approximation schemes in the good cavity limit. The situation in which the atoms are initially prepared with equal probability to be in the upper and lower energy levels is taken as a particular case. An equal emphasis is given to the comparison of the squeezing and intensity of the output and cavity radiations. For clarity the schematic representation of the quantum system under consideration is given in Fig 1. Equation of evolution Interaction of externally pumped cascade three-level atom with the cavity radiation can be described in the rotating-wave approximation and interaction picture by the Hamiltonian of the form where g is the coupling constant taken to be the same for both transitions, â is the annihilation operator for the cavity mode and  is a real-positive constant proportional to the amplitude of the driving radiation. The initial state of a three-level atom is taken to be |A (0) = Ca (0)|a + Cc (0)|c , where Ca (0) and Cc (0) are the probability amplitudes for the atom to be in the upper and lower energy levels. This consideration corresponds to the case when the three-level atom is initially prepared to be in arbitrary coherent superposition of the upper and lower energy levels, in which the initial density operator for the atom would be Schematic representation of a coherently pumped degenerate three-level atom in a cascade configuration Figure 1 Schematic representation of a coherently pumped degenerate three-level atom in a cascade configuration. The transitions from |a  |b and from |b  |c at frequency a are taken to be resonant with the cavity, whereas the transition from |a  |c is dipole forbidden. However, the transition from |c  |a is induced by pumping the atoms externally with a resonant radiation of frequency 2a. Moreover, biased noise fluctuations enter the cavity via one of the coupler mirrors. ˆ [ˆ(| | | |) (| | | |)ˆ ] [| | | † H ig a a b b c b a c b a i c a AR = 〉〈 + 〉〈 + 〉〈 + 〉〈 + 〉〈 − Ω 2 a c 〉〈 |], (1) ˆ ( ) | | | | | | | | ( ) ( ) ( ) ( )      A aa ac ca cc a a a c c a c c 0 0 0 0 0 = 〉〈 + 〉〈 + 〉〈 + 〉〈 . (2) Biased noise fluctuations rate Ω |a> ωa |b> ωa |c> Constant Page 3 of 18 (page number not for citation purposes) PMC Physics B 2008, 1:17 http://www.physmathcentral.com/1754-0429/1/17 It is not difficult to notice that and are the probabilities for the atoms to be initially in the upper and lower energy levels, whereas represents the initial atomic coherence. It is assumed that the atoms are injected into the cavity at a constant rate ra and removed after sometime, which is long enough for the atoms to decay spontaneously to levels that do not contribute to the lasing process. The atomic spontaneous decay rate  is taken to be the same for the involved levels. Applying the linear and adiabatic approximation schemes in the good cavity limit [8], the time evolution of the density operator for the cavity mode of degenerate CBL coupled to broadband biased noise fluctuations via a single port-mirror is found using the standard method [24] along with the recently discussed idea [18,25] to be where  is the cavity damping constant, is the linear gain coefficient, aa ( ) 0  cc ( ) 0