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

- Sintayehu Tesfa
^{1}Email author

**Received: **08 July 2008

**Accepted: **01 December 2008

**Published: **01 December 2008

## Abstract

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

## Keywords

## 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 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.

## Equation of evolution

*g*is the coupling constant taken to be the same for both transitions,

*â*is the annihilation operator for the cavity mode and Ω 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 |Φ

_{ A }(0)⟩ =

*C*

_{ a }(0)|

*a*⟩ +

*C*

_{ c }(0)|

*c*⟩, where

*C*

_{ a }(0) and

*C*

_{ c }(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

It is not difficult to notice that ${\rho}_{aa}^{(0)}$ and ${\rho}_{cc}^{(0)}$ are the probabilities for the atoms to be initially in the upper and lower energy levels, whereas ${\rho}_{ac}^{(0)}$ represents the initial atomic coherence.

*r*

_{ a }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

*γ*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

*κ*is the cavity damping constant, $A=\frac{2{r}_{a}{g}^{2}}{{\gamma}^{2}}$ is the linear gain coefficient,

It is worth noting that *N* is the mean photon number of the environment mode capable of coupling with the cavity radiation. As a result, *N* can be interpreted as the measure of the intensity of the biased noise fluctuations whereas $M=\sqrt{N(N+1)}$ as the degree of the correlation among the biased modes. Moreover, from the form of the master equation, one can realize that *C* corresponds to the gain but *D* to the lose of the cavity mode. On the other hand, *E* and *F* are related to the correlation of the generated radiation that indicates the existence of quantum features.

where *α*(*t*) is the c-number variable related to the cavity mode *â*(*t*) in the normal ordering, *μ* = $\frac{A}{B}$ (*D* - *C*) + *κ*, *β* = $\frac{A}{B}$ (*E* - *F*) + *κ* and *f*(*t*) is the corresponding stochastic noise force with the correlation properties:

*f*(

*t*)⟩ = 0,

Upon introducing,

*α*_{±} (*t*) = *α** (*t*) ± *α* (*t*), (13)

where

*λ*_{∓} = *μ* ∓ 2*β*

Therefore, formal integration of Eq. (14) leads to

*α*(*t*) = *a*_{+}(*t*)*α*(0) + *a*_{-}(*t*)*α* *(0) + *G*_{-}(*t*) + *G*_{+}(*t*)

It is not difficult to notice that a well behaved solution of Eq. (14) exists at steady state provided that *λ*_{∓} > 0. Hence *μ* = 2*β* is designated as a threshold condition. It perhaps worth mentioning that the squeezing as well as the statistical properties of the cavity radiation and output radiation are investigated with the aid of Eq. (16) when *μ* > 2*β*.

## Quadrature variances

where *F*_{
R
}(*t*) is the noise force associated with the environment modes and satisfies for broadband biased noise fluctuations the correlations:

⟨*F*_{
R
}(*t*)⟩ = 0,

⟨*F*_{
R
}(*t*) *F*_{
R
}(*t'*)⟩ = *κM δ*(*t* - *t'*)

It is possible to realize that the noise force related to the stochastic process (*f*(*t*)) represents the contribution of the vacuum fluctuations of the cavity as well as the environment modes and hence can be put in the form

*f*(*t*) = *F*_{
C
}(*t*) + *F*_{
R
}(*t*)

*F*

_{ C }(

*t*) is the noise force corresponding to the system in the cavity in the absence of the biased input. On the basis of the fact that the noise force of the environment

*F*

_{ R }(

*t*) does not correlate with

*F*

_{ C }(

*t*) and the system variables at the earlier times along with the fact that the contribution of the biased noise fluctuations should be taken from initial time

*t*= 0 to

*t*= ∞, one can verify at steady state that

*η*≤ 1. It is straight forward to verify when the three-level atoms are initially prepared to be in arbitrary atomic coherent superposition that ${\rho}_{cc}^{(0)}=\frac{1+\eta}{2}$ and ${\rho}_{ac}^{(0)}=\frac{\sqrt{1-{\eta}^{2}}}{2}$. Therefore, on the basis of Eqs. (4), (5), (6), (7), (8), (15) and (36), Eq. (35) finally takes at steady state the form

This reduces for *A* = 0 to $\Delta {a}_{\pm (out)}^{2}$ = 1 + 2*N* ± 2*M*, since there is only biased noise fluctuations in the cavity in this case.

*γ*and

*η*, where the degree of squeezing generally found to increase with the linear gain coefficient. In order to analyze the dependence of the degree of squeezing of the output radiation on the intensity of the environment modes, amplitude of the driving radiation and injected atomic coherence more closely, it is necessary to consider various cases of interest separately. To this effect, for Ω = 0

Despite the fact that a higher degree of squeezing is achievable for larger values of *A*, the linear gain coefficient is limited to smaller values so that the dependence of the degree of squeezing on the parameters under consideration is evident from the figures.

*η*. Even though the linear gain coefficient is limited to smaller values for convenience, squeezing of 71% occurs for

*N*= 0.4 and

*A*= 10 at

*η*= 0.25 whereas for

*N*= 0.1 squeezing of 53% occurs at

*η*= 0.22. Hence it is observed that the biased noise fluctuations enhance the degree of squeezing of the output radiation substantially. It is believed that the correlated emission initiated by the initially prepared atomic coherent superposition is accountable for the reduction of the fluctuations of the noise in one of the quadrature components below the classical limit in addition to the biased input, since three-level laser falls under phase-sensitive amplifier [7, 26].

*η*= 0), one gets

*γ*. It is found that a maximum squeezing of 71% occurs at Ω = 0.14

*γ*for

*A*= 10 and

*N*= 0.4. The degree of squeezing decreases with the amplitude of the driving radiation for larger values of Ω/

*γ*, but it increases with the intensity of the biased noise fluctuations throughout. Further manipulation reveals that squeezing of higher degree than 71% can be realizable in the system under consideration when the atoms are initially prepared in a possible maximum coherent superposition and externally pumped with radiation of relatively smaller amplitude. It is expected that the possibility for generating highly squeezed light by altering various parameters will make this system reliable and attractive source of squeezed light.

*A*= 0 that the degree of squeezing inside and outside the cavity is the same, since there is basically the same radiation in both regions. It is not difficult to see from Figs. 4 and 5 that the degree of squeezing outside the cavity can be greater than that inside the cavity for certain values of the initial preparation of the atomic coherent superposition. It is believed that this can be taken as one of the essential encouragements to utilize the generated squeezed radiation outside the cavity. It can also be deduced that when compared to the cavity radiation, the output radiation slowly varies with the initial preparation of the superposition and amplitude of the external radiation. Since the output radiation is the superposition of the radiation escaping through the coupler mirror and biased environment modes reflected from the same mirror, its degree of squeezing is undoubtedly influenced by the biased fluctuations more significantly. This must be the reason for getting nearly a uniform degree of squeezing for the output radiation close to the squeezing of the environment modes.

On the basis of the definition of the parameter *η* (Eq. (36)), it can be realized for *η* = 0 that ${\rho}_{aa}^{(0)}$ = ${\rho}_{cc}^{(0)}$ = ${\rho}_{ac}^{(0)}$ = 1/2, which corresponds to a maximum possible initial atomic coherence. But a case for which *η* = 1, ${\rho}_{aa}^{(0)}$ = ${\rho}_{ac}^{(0)}$ = 0 and ${\rho}_{cc}^{(0)}$ = 1, is related to the absence of injected atomic coherence at the beginning. One can readily see from Eq. (40) that there is no squeezing when the atoms are initially prepared with a maximum or minimum atomic coherence, if they are not pumped externally (Ω = 0) and there is no biased environment (*N* = 0). However, as shown in Figs. 2 and 4, the maximum squeezing occurs when the atoms are prepared with initial coherence close to the maximum possible value. It can also be observed from Figs. 3 and 4 that the external driving radiation initiates correlation between the states of the emitted radiation in the cascading process which leads to squeezing inside and outside the cavity for *η* = 0 even when *N* = 0. When there is no injected atomic coherence (*η* = 1) and *N* = 0, squeezing close to 50% is obtained for a very large amplitude of the external radiation (Ω ≫ *γ*). This result agrees with the recent claim that three-level laser in which the atoms are initially prepared in the bottom level and externally pumped by strong radiation resembles parametric oscillator [2, 14]. Moreover, comparison of the results shown in Figs. 2 and 3 reveals that a higher degree of squeezing can be obtained when the atoms are initially prepared with maximum atomic coherence and pumped externally with radiation of relatively smaller amplitude. Likewise, a maximum noise reduction when the atoms are injected into a resonant cavity with a maximum atomic coherence and pumped with external radiation for Λ three-level laser has been shown [3]. Though the external radiation induces the atomic coherence accountable for the squeezing, it is observed that pumping the atoms with a stronger radiation than required destroys the squeezing. For parameters under consideration the degree of squeezing is found to increase with the intensity of the biased noise fluctuations, however as recent calculation shows the degree of squeezing in the nondegenerate case can decrease with the intensity of the biased fluctuations [7] due to the phenomenon of electron shelving [27, 28].

## Mean photon number

It can be inferred from Eq. (45) that radiation can be generated when initially all atoms are prepared to be in the lower energy level and if there is no external driving radiation contrary to previous reports, since the atoms can absorb the biased input to be excited to the upper energy levels from which emission of the required radiation is possible. It is not difficult to notice that, as a result of the external pumping, it is possible to produce a strong radiation from the laser even when the atoms are initially prepared to be in the lower energy level. This demonstrates that the mechanism of lasing without population inversion is readily evident in this scheme.

*N*= 0. One gets the strongest radiation when all atoms are initially prepared to be in the upper energy level as expected. In addition, for

*η*= 0 Eq. (45) takes the form

*γ*. Though there are biased noise environment modes outside the cavity, as can readily be seen from Fig. 7, the mean photon number of the radiation in the cavity is much greater than the corresponding output radiation in many cases. This is actually related to the fact that in the good cavity limit (

*κ*small) much of the cavity radiation stays in the cavity. It is also clearly shown in Fig. 6 that the mean photon number increases with the intensity of the biased noise fluctuations. Even though it has been shown in previous work that the mean photon number increases with the linear gain coefficient, the values of

*A*are limited to smaller values so that the dependence of the mean photon number on the intensity of the biased noise fluctuations and amplitude of the driving radiation is evident from the figures. It is possible in principle to generate a strong radiation from the system under consideration by varying the rate at which the atoms are injected into the cavity, intensity of the biased environment modes, amplitude of the driving radiation and the way in which the atoms are initially prepared. Further manipulation reveals that there is a possibility for the intensity of the radiation to be equal inside and outside the cavity for some values of the parameters under consideration.

## Conclusion

Detailed analysis of the degree of squeezing and intensity of the output radiation generated by a degenerate coherent beat laser coupled to broadband biased environment modes via one of the coupler mirrors is presented. It is found that the output as well as the cavity radiation exhibits squeezing under certain conditions pertaining to the initial preparation of the atomic coherent superposition, strength of the driving radiation and intensity of the biased fluctuations. Though the external driving radiation induces the atomic coherence accountable for the squeezing, pumping the atoms externally with a strong radiation results considerable reduction in the degree of squeezing and intensity of the radiation in both regions. Hence I cannot see the practical advantage of pumping the atoms with a strong radiation in this respect. That is why this study is confined mainly to weaker driving regime. Nonetheless intense radiation with a substantial degree of squeezing can be generated by the driving mechanism specially when the atoms are initially prepared with equal probability to be in the lower and upper energy levels, where there is no squeezing in the absence of the driving radiation and when *N* = 0. It is also shown that it is possible to get squeezed radiation when the atoms are initially prepared to be in the lower energy level, where there is no radiation at all in the absence of driving process and *N* = 0. It can be realized that driving mechanism can be considered as an option for generating squeezed radiation when it is technically difficult to prepare the atoms initially in arbitrary coherent superposition. Due to the additional atomic coherence induced by the driving process, it is reasonable to expect the enhancement of the degree of squeezing when the atoms initially prepared with arbitrary atomic coherent superposition are externally pumped; the fact that cannot be confirmed in this study for all cases.

On the other hand, coupling the cavity to broadband biased environment modes is found to improve the degree of squeezing and intensity of the output as well as the cavity radiation. Contrary to this, the degree of squeezing in the nondegenerate three-level cascade laser does not always increase with the intensity of the biased noise fluctuations due to the phase competition in the correlation resulting from two different modes [7]. Comparison of the squeezing inside and outside the cavity reveals that the squeezing of the output radiation can be greater than or equal to or less than the cavity radiation depending on the parameters under consideration. Since the effect of the biased noise fluctuations would be prominent when there is small number of emitted photons, the degree of squeezing of the output radiation would be larger than the cavity radiation when relatively fewer atoms are initially prepared to be in the upper energy level. It is also found that though the mean photon number of the output radiation can be close to the cavity radiation, the intensity of the radiation in the cavity is much better in many instances. Hence despite the envisaged practical challenges to utilize the system under consideration, it is believed that the possibility of getting a strong squeezed light outside the cavity is not something that one brushes off without trying.

## Declarations

## Authors’ Affiliations

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