Generation of Bright Squeezed Light from N Three-Level Atoms Pumped by a Coherent Light : Open Quantum System

The manuscript investigated the steady-state analysis of the squeezing and statistical properties of the light generated by N three-level atoms available in an open cavity pumped a coherent light and the cavity coupled to a two-mode vacuum reservoir. The results indicate that as the frequency increases, the local quadrature squeezing of the two-mode cavity light approaches the global quadrature squeezing. The effect of the spontaneous emission leads to an increase in the quadrature squeezing, but to a decrease in the mean photon number of the system. It is also found that, unlike the mean photon number and the variance of the photon number, the quadrature squeezing does not depend on the number of atoms. This implies that the quadrature squeezing of the two-mode cavity light is independent of the number of photons.

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Introduction
Squeezed states of light has played a crucial role in the development of quantum physics.Squeezing is one of the nonclassical features of light that have been extensively studied by several authors [1]- [8].In a squeezed state the quantum noise in one quadrature is below the vacuum-state level or the coherent-state level at the expense of enhanced fluctuations in the conjugate quadrature, with the product of the uncertainties in the two quadratures satisfying the uncertainty relation [1] [2] [4] [9].Because of the quantum noise reduction achievable below the vacuum level, squeezed light has potential applications in the detection of week signals and in low-noise communications [1] [3].Squeezed light can be generated by various quantum optical processes such as subharmonic generations [1]- [5] [10]- [12], four-wave mixing [13] [14], resonance fluorescence [6] [7], second harmonic generation [8] [15], and three-level laser under certain conditions [1] [3] [4] [9] [16]- [27].
Hence it proves useful to find some convenient means of generating a bright squeezed light.A three-level laser is a quantum optical device in which light is generated by three-level atoms in a cavity usually coupled to a vacuum reservoir via a single-port mirror.In one model of a three-level laser, three-level atoms initially prepared in a coherent superposition of the top and bottom levels are injected into a cavity and then removed from the cavity after they have decayed due to spontaneous emission [9] [16]- [21].In another model of a three-level laser, the top and bottom levels of the three-level atoms injected into a cavity are coupled by coherent light [22]- [27].It is found that a three-level laser in either model generates squeezed light under certain conditions.The superposition or the coupling of the top and bottom levels is responsible for the squeezed of the generated light.It appears to be quite difficult to prepare the atoms in a coherent superposition of the top and bottom levels before they are injected into the cavity.In addition, it should certainly be hard to find out that the atoms have decayed spontaneously before they are removed from the cavity.
In order to avoid the aforementioned problems, Fesseha [28] have considered that N two-level atoms available in a closed cavity are pumped to the top level by means of electron bombardment.He has shown that the light generated by this laser operating well above threshold is coherent and the light generated by the same laser operating below threshold is chaotic light.In addition, Fesseha [29] has studied the squeezing and the statistical properties of the light produced by a degenerate three-level laser with the atoms in a closed cavity and pumped by electron bombardment.He has shown that the maximum quadrature squeezing of the light generated by the laser, operating far below threshold, is 50% below the vacuum-state level.Alternatively, the three-level atoms available in a closed cavity and pumped by coherent light also generated squeezed light under certain conditions, with the maximum global quadrature squeezing is being 43% below the vacuum-state level [1].It appears to be practically more convenient to pump the atoms by coherent light than electron bombardment.
In this paper, we investigate the steady-state analysis of the squeezing and statistical properties of the light generated by a coherently pumped degenerate three-level laser with open cavity which is coupled to a two-mode vacuum reservoir via a single-port mirror.We carry out our calculation by putting the noise operators associated with the vacuum reservoir in normal order and by taking into consideration the interaction of the three-level atoms with the vacuum reservoir outside the cavity.

Model and Dynamics of Atomic and Cavity Mode Operators
Let us consider a system of N degenerate three-level atoms in cascade configuration are available in an open cavity and interacting with the two (degenerate) cavity modes.The top and bottom levels of the three-level atoms are coupled by coherent light.When a degenerate three-level atom in cascade configuration decays from the top level to the bottom levels via the middle level, two photons of the same frequency are emitted.For the sake of convenient, we denote the top, middle, and bottom levels of these atoms by k a , k b , and k c , respectively.We wish to represent the light emitted from the top level by 1 â and the light emitted from the middle by 2 â .In addition, we assume that the two cavity modes 1 a and 2 a to be at resonance with the two transitions The interaction of one of the three-level atoms with light modes 1 a and 2 a can be described at resonance by the Hamiltonian where and are lowering atomic operators, 1 â and 2 â are the annihilation operators for light modes 1 a and 2 a , and g is the coupling constant between the atom and the light mode 1 a or light mode 2 a .And the interaction of the three-level atom with the driving coherent light can be described at resonance by the Hamiltonian and Here  is the amplitude of the driving coherent light and g is coupling constant between the atom and coherent light.Thus upon combining Equations ( 1) and ( 4), the interaction of the three-level atom with the driving coherent light and cavity modes 1 a and 2 a is described at resonance by the Hamiltonian On the other hand, the degenerate three-level atoms available in an open cavity are coupled to a two-mode vacuum reservoir.The master equation for the three-level atom interacting with a two-mode vacuum reservoir has the form [1] where  is the spontaneous emission decay constant associated with the two modes 1 a and 2 a .Hence with the aid of Equation ( 7), the master equation describing the two-mode cavity light of a coherently pumped degenerate three-level atom would be We recall that the laser cavity is coupled to a two-mode vacuum reservoir via a single-port mirror.In addition, we carry out our analysis by putting the noise operators associated with the vacuum reservoir in normal order.Thus the noise operators will not have any effect on the dynamics of the cavity mode operators [1] [28].In view of this, we can drop the noise operators and write the quantum Langevin equation for the operators 1 â and 2 â as where k is the cavity damping constant.Then with the aid of Equation ( 7), we easily find The procedure of normal ordering the noise operators renders the vacuum reservoir to be a noiseless physical entity.We uphold the view point that the notion of a noiseless vacuum reservoir would turn out to be compatible with observation [30].Furthermore, making use of the master equation and the fact that where We see that Equations ( 14)-( 19) are nonlinear and coupled differential equations.Therefore, it is not possible to obtain the exact time-dependent solutions.We intend to overcome this problem by applying the large-time approximation [28].Then using this approximation scheme, we get from Equations ( 12) and ( 13) the approximately valid relations Upon substituting ( 23) and (24) into Equations ( 14)-( 19), we get where is the stimulated emission decay constant.We next sum Equations ( 25)-( 30) over the N three-level atoms.We then see that with the operators ˆa N , ˆb N , and ˆc N representing the number of atoms in the top, middle, and bottom levels.In addition, employing the completeness relation ˆ, we easily arrive at ˆˆˆ.
Furthermore, applying the definition given by (2) and setting for any k Following the same procedure, one can easily find Moreover, using the definition and taking into account Equations ( 47)-( 52), it can be readily established that Upon adding Equations ( 12) and ( 13), we have where In the presence of N three-level atoms, we can rewrite Equation (57) as in which  is a constant whose value remains to be fixed.The steady-state solution of Equation ( 57) is Taking into account of (60) and its adjoint, the commutation relation for the cavity mode operator is found to be and on summing over all atoms, we have stands for the commutator of   † ˆ, aa when the superposed light mode a is interacting with all the N three-level atoms.On the other hand, using the steady-state solution of Equation ( 59

Photon Statistics
Here we seek to obtain the global (local) mean photon number and the global (local) variance of the photon number for the two-mode cavity light beam at steady state.

The Global Mean Photon Number
We wish to calculate the mean photon number of the two-mode cavity light in the entire frequency interval.The steady-state solution of Equation ( 66) is given by On account of (67) together with (54), the mean photon number of the two-mode cavity light is expressible as . With the aid of ( 147) and ( 148), the mean photon number of the two-mode cavity light turns out to be We note that the global mean photon number takes for We observe from the plots in Figure 1 that the presence of spontaneous emission leads to a decrease in the global mean photon number of the two-mode cavity light beam.

Local Mean Photon Number
We seek to determine the mean photon number in a given frequency interval, employing the power spectrum for the two-mode cavity light.The power spectrum of a two-mode cavity light with central common frequency 0  is defined as On introducing (162) into Equation (71) and carrying out the integration, we readily get The mean photon number in the frequency interval between    and    is expressible as  d, where .We therefore observe that a large part of the total mean photon number is confined in a relatively small frequency interval.

The Global Variance of the Photon Number
The variance of the photon number for the two-mode cavity light is expressible as and using the fact that   ât is a Gaussian variable with zero mean, we arrive at Employing once more (67) and taking into account (55), we readily get so that in view of ( 146) and (69), there follows Now on account of Equations ( 68), (83), and (84), we readily find Equation (79) to be This can be put in the form In view of (69), we arrive at We immediately see from the plots in Figure 3 that the presence of spontaneous emission leads to a decrease in the global variance of the photon number of the two-mode cavity light beam.In addition, the global variance of the photon number of the two-mode cavity light increases with increasing  .
Finally, we note that the variance of the photon number takes for in which n is given by (70).This represents the normally ordered variance of the photon number for a chaotic state.

Local Variance of the Photon Number
Here we wish to obtain the variance of the photon number in a given frequency interval, employing the spectrum of the photon number fluctuations for the superposition of light modes 1 a and 2 a .We denote the central common frequency of these modes by 0  .The spectrum of the photon number fluctuations for the superposed light modes can be expressed as where

 
and we have used the notation . With the aid of ( 90) and (91) and Equation (142), the photon number fluctuation can be expressed as Upon introducing (162)-(165) into Equation (92) and on carrying out the integration over  , the spectrum of the photon number fluctuations for the two-mode cavity light is found to be is given by (79).
Upon integrating both sides of (93) over  , one easily finds On the basis of Equation ( 94), we observe that  d R  represents the steady-state variance of the photon number for the two-mode cavity light in the interval between  and d   . We thus realize that the photonnumber variance in the interval between    and    can be written as in which 0      .Thus upon substituting (93) into Equation (95), we find so on carrying out the integration over  , applying the relation described by Equation (75), we readily get where   From the plots in Figure 4 that we easily find   0.5 0.7625 z  ,   We immediately observe that a large part of the total variance of the photon number is confined in a relatively small frequency interval.

Quadrature Squeezing
In this section we seek to calculate the quadrature squeezing of the two-mode cavity light in any frequency interval.

The Global Quadrature Squeezing
The squeezing properties of the two-mode cavity light are described by two quadrature operators defined as It can be readily established that Now upon replacing the atomic operators that appear in Equation ( 62) by their expectation values, the commutation relation for the two-mode light can write as † ˆ,, aa The variance of the quadrature operators is expressible as With the aid of ( 68), (83), and (104) together with (45), we obtain Finally, on account of ( 146) and ( 148), the global quadrature variance of the two-mode cavity light turns out at steady state to be It is then not difficult to observe that the two-mode cavity light beam is in a squeezed state and the squeezing occurs in the minus quadrature.
We next proceed to calculate the quadrature squeezing of the two-mode cavity light relative to the quadrature variance of the two-mode cavity vacuum state.We define the quadrature squeezing of the two-mode cavity light by Moreover, upon setting 0  in Equation (110), we see that which represents the quadrature variance of the two-mode cavity vacuum state.Hence on account of Equations ( 110) and (112), we arrive at We note that, unlike the mean photon number, the quadrature squeezing does not depend on the number of atoms.This implies that the quadrature squeezing of the two-mode cavity light is independent of the number of photons.We see from the plots in Figure 5 that the maximum global quadrature squeezing of the two-mode cavity light for 0   is 43.42% (and occurs at 0.1717  ) and for 0.1   is found 47.15% (and occurs at 0.2323  ).And for 0.2   , the maximum global quadrature squeezing is observed to be 50% below the vacuum-state level and this occurs when the three-level laser is operating at 0.303  .Moreover, upon setting 0   in Equation (113), we note that where c   .Equation ( 114) is indicates that the quadrature squeezing of the light produced by degenerate three-level laser with the N three-level atoms available inside a closed cavity pumped to the top level by electron bombardment which has been reported by Fesseha [1].

Local Quadrature Squeezing
Here we wish to obtain the quadrature squeezing of the two-mode cavity light in a given frequency interval.To this end, we first obtain the spectrum of the quadrature fluctuations of the superposition of light modes 1 a and 2 a .We define this spectrum for the two-mode cavity light by and 0  is the central frequency of the modes 1 a and 2 a .In view of Equation (142), we obtain Then on account of Equations ( 99), (100), (116), and (117), one can write Equation (118) as Upon substituting of ( 162)-(165) into Equation (119), we arrive at ˆˆˆˆˆˆˆ, e e .
k k a t a t a t a t a t a t a t a t kk This can be put in the form ˆ, e e k k a t a t a kk and ˆ, e e .
k k a t a t a kk Now introducing (122) into Equation (115) and on carrying out the integration over  , we find the spectrum of the minus quadrature fluctuations for a two-mode cavity light to be Upon integrating both sides of (123) over  , we get     .We thus realize that the variance of the minus quadrature in the interval between    and    is expressible as in which 0      .On introducing (123) into Equation ( 125) and on carrying out the integration over  , employing the relation described by Equation (75), we find where We the quadrature squeezing of the two-mode cavity light in the   frequency interval by Furthermore, upon setting 0  in Equation (126), we see that the local quadrature variance of a two-mode cavity vacuum state in the same frequency is found to be and   2 v a   is given by (112).Finally, on account of Equations ( 110), (112), and (129) along with (128), we readily get This shows that the local quadrature squeezing of the two-mode cavity light beams is not equal to that of the global quadrature squeezing.Moreover, we found from the plots in Figure 6 that the maximum local quadrature squeezing for 0   is 71.73% (and occurs at 0.06

 
) and for 0.1   is found 71.83% (and occurs at 0.06

 
).And for 0.2   , the maximum local quadrature squeezing is observed to be 71.88% ( and occurs at 0.06

 
).Furthermore, we note that the local quadrature squeezing approaches the global quadrature squeezing as  increases.

Conclusions
The steady-state analysis of the squeezing and statistical properties of the light produced by coherently pumped degenerate three-level laser with open cavity and coupled to a two-mode vacuum reservoir is presented.We carry out our analysis by putting the noise operators associated with the vacuum reservoir in normal order and by taking into consideration the interaction of the three-level atoms with the vacuum reservoir outside the cavity.We observe that a large part of the total mean photon number (variance of the photon number) is confined in a relatively small frequency interval.In addition, we find that the maximum global quadrature squeezing of the light produced by the system under consideration for 0

 
, the maximum global quadrature squeezing is observed to be 50% below the vacuum-state level and this occurs when the three-level laser is operating at 0.303  .Furthermore, results show that the presence of spontaneous emission leads to a decrease in the mean photon number and to an increase in the quadrature squeezing.
Moreover, we find that the maximum local quadrature squeezing for 0   is 71.73% (and occurs at 0.06

 
) and for 0.1   is 71.83% (and occurs at 0.06

 
).And for 0.2   , the maximum local quadrature squeezing is observed to be 71.88%(and occurs at 0.06

 
).In addition, we note from the plots in Figure 6 that as  increases, the local quadrature squeezing approaches the global quadrature squeezing.We observe that the light generated by this laser operating under the condition   is in a chaotic light.And we have also established that the local quadrature squeezing of the two-mode light is not equal to the global quadrature squeezing.
Furthermore, we point out that unlike the mean photon number and the variance of the photon number, the quadrature squeezing does not depend on the number of atoms.This implies that the quadrature squeezing of the two-mode cavity light is independent of the number of photons.

Appendix 1. Solutions of the Expectation Values of the Cavity (Atomic) Mode Operators
In order to determine the mean photon number and the variance of the photon number, and the quadrature squeezing of the two-mode cavity light in any frequency interval at steady state, we first need to calculate the solution of the equations of evolution of the expectation values of the atomic operators and cavity mode operators.To this end, the expectation values of the solution of Equation ( 66) is expressible as We notice that the solution of Equation ( 134) for  different from zero at steady state is In a similar manner, applying the large-time approximation scheme to Equation (32), we obtain † Finally, on account of (147) and (148) along with Euation (45), we find

Two-Time Correlation Functions
Here we seek to calculate the two-time correlation functions for the two-mode cavity light.To this end, we realize that the solution of Equation (66) can write as         .Thus upon substituting (72) into Equation (73), we find

Figure 1 .N
Figure 1.Plots of the global mean photon number [Equation (69)] versus  at steady state for 0.4 c   , 0.8 k  , 50 N  , and different values of  .

Figure 3 .N
Figure 3. Plots of the global variance of the photon number [Equation (87)] versus  at steady state for 0.4 c   , 0.8 k  , 50 N  , and different values of  .

Figure 5 .
Figure 5. Plots of the global quadrature squeezing [Equation (113)] versus  at steady state for 0.4 c   , 0.8 k  , and different values of  .
of Equation (124), we observe that  d S   is the steady-state variance of the minus quadrature in the interval between  and d  

Figure 6 .
Figure 6.Plot of the local quadrature squeezing [Equations (131)] versus  at steady state for We finally seek to determine the solution of the expectation values of the atomic operators at steady state.Moreover, the steady-state solution of Equations (34)-(36) yields ât is a Gaussian variable with zero mean.
Since the cavity mode operator and the noise operator of the atomic modes are not correlated, we see that On account of these results and on carrying out the integration of Equation (159) over   , we readily get