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An analysis of lasing without inversion in a three level system in terms of vector representation
Optik 122 (2011) 95–98
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
An analysis of lasing without inversion in a three level system in terms of vector representation Farzana B. Hazarika a , S.N. Rai b , G.D. Baruah a,∗ a b
Department of Physics, Dibrugarh University, Dibrugarh 786004, India Department of Physics, Maharishi Markandeswar University, Mullarva, Ambala 133203, India
a r t i c l e
i n f o
Article history: Received 12 May 2009 Accepted 28 November 2009
a b s t r a c t In the present work we have worked out phasor diagrams with the help of the concept of moving vectors. We show that the area enclosed by the phasor diagrams behaves like lifetime of atomic states. We also make a comparative study of our model with the models suggested by other workers to explain lasing without inversion. © 2010 Elsevier GmbH. All rights reserved.
1. Introduction Quantum interference is a challenging principle of quantum theory. Essentially the concept states that elementary particles can not only be in more than one place at a given time (through superposition), but that an individual particle, such as photon can cross its own trajectory and interfere with the direction of its path. Debate over whether light is essentially particles or waves dates back over three hundred years. In the seventeenth century, Sir Isaac Newton (1642–1726) proclaimed that light consisted of particles. In the early part of seventeenth century, Christian Huygens (1629–1695) enunciated a convenient working principle to describe how the progress of the primary wavefront of a source of light is due to the generation of the secondary wavelets from every point of the primary wavefront. Thomas Young devised the double slit experiment to prove that it consisted of waves. Although the implications of Young’s experiment are difficult to accept, it has reliably yielded proof of quantum interference through repeated trials. According to Feynman et al. [1], each photon not only goes through both slits simultaneously but traverses every possible trajectory on the way to the target, not just in theory, but in fact. In order to see how this might possibly occur, experiments have focused on tracking the paths of individual photons. What happens in this case is that the measurement in some way disrupts the photons trajectories (in accordance with uncertainty principle), and somehow, the results of the experiment become what would be predicted by classical physics.
∗ Corresponding author. E-mail address: [email protected] (G.D. Baruah). 0030-4026/$ – see front matter © 2010 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2009.11.013
From what has been described above it is apparent that uncertainty principle is involved in explaining the phenomenon of interference and therefore the phenomenon is more appropriately termed as quantum interference whose research is being applied in a growing number of applications including quantum interference laser or LWI [2–17]. Under special conditions coherent atomic transitions can cancel absorption. The main idea of LWI is that absorption cancellation provides the possibilities to obtain light amplification even if the population of the upper level is less than the population of the lower level. Such a situation can be realized for instance, in a three level system, when two coherent atomic transitions destructive interfere and, hence, cancel absorption. To present the basic physics of LWI it is best to consider the theory of this effect in a three level -configuration and then to demonstrate how the concept of lasing without inversion can be realized experimentally. The configuration of -three levels atomic system is presented in Fig. 1. It is formed by upper levels |a and |c through interaction with electromagnetic fields E1 and E2 , respectively, in such way that only atomic transitions |a − |c and |a − |b are allowed. The physical reason for canceling absorption in this system is the uncertainty in atomic transitions |c − |a and |b − |a which results in destructive interference between them. This situation is similar to Young’s double slit problem, where interference is a consequence of uncertainty in determining through which of two slits the photon passed. In the present work we report a vector model to represent quantum interference laser in terms of phasor diagrams which have been worked out from the concept of transition probabilities and their time evolutions with suitable decay parameters being introduced. We also present a comparative study
96
F.B. Hazarika et al. / Optik 122 (2011) 95–98
Fig. 1. Three level atom in the -configuration interacting with two fields of frequencies WL1 and WL2 .
between different vector models representing three-level type maser. 2. Atom field interaction It may be noted that the atom decays to the lower level as a result of its interaction with vacuum. The theory of spontaneous emission of Weisskopf–Wigner [18] justified the inclusion of phenomenological decay rates a and b in the Schrödinger equation for the atomic probability amplitudes. The probability of stimulated absorption including damping is given by [19]
2 2 Ca (t)2 = 1 ℘E0 exp(−b t) sin(w − )t/2 4
h ¯
(w − )/2
(1)
3. Moving vectors The transition probability for arbitrary value of time may be plotted for different values of detuning (ω − ). Using Eq. (1) we have worked out the graph, which represents the time evolution of population in the upper state for different values of detuning at 90, 100, 110, 120 and 130 MHz. This is shown in Fig. 2. As shown in
Fig. 3. Decay versus area.
Fig. 2, we may join all the five maxima in a group of detunings as an arrow and a number of these arrows can be drawn in this way. We observe that these vectors evolve in time. These vectors can be used to represent transition probabilities for the changes of population in the upper state. The zero transition probabilities at arbitrary time of 62 n (n = 1, 2, 3, . . .) indicate some type of beats in the process of interference. A phasor diagram has been constructed with the help of the moving vectors. The moving vectors can be represented by the equation dkn = ˝n × kn dt
(2)
where n = 1, 2, 3, . . ., ˝n is the driving field vector, kn is the amplitude vector. We now proceed to analyse the phasor diagrams as shown in Fig. 2 which are drawn with the help of the moving vectors. The vectors in the phasor diagram first close up as in the case of resonating cavity but uncoil again and expand. It is also observed that when decay is small the expansion is slow but for bigger decay the expansion is more. This is a general nature of all the phasor diagrams we
Fig. 2. Probability versus time graphs for decay = .001, .005, .01, .02, .03, .05 and their corresponding phasor diagrams.
F.B. Hazarika et al. / Optik 122 (2011) 95–98
have constructed with different decay constants and with the same set of detuning values. The phasor diagrams also indicate that as the decay increases the area decreases exponentially which may be illustrated by plotting decay versus area graph as shown in Fig. 3.As regards its physical significance it may be noted that the parameter is related to the lifetime as = 1/. This follows from the theory of spontaneous emission. In the present case the area behaves like lifetime. The result is new. 4. Density matrix in motion In this section we introduce the concept of the vector model of density matrix in motion [19]. This model has value not only in solving the equation of density matrix in motion but also in providing a physical picture of a particular system. The equation are equivalent to the Bloch equation [21] appearing in nuclear magnetic resonance. In this connection we may note that the phenomenon of the effect of an adiabatic rapid passage in inverting the population distribution of an assembly of nuclear spins has been recognized and predates the maser itself by approximately a decade. The technique is straightforward. In a system in which a radiation field far from resonance with a transition frequency in a material sample interacts with a sample, the field frequency is swept through the material resonance until it is again far off resonance but with the frequency deviation reversed. If the sweep rate has been adjusted to lie within relatively easily attainable limits in certain materials, it is found that the population of the material sample after the passage is inverted relative to its value prior to the passage, For laser media when decays a and b are small compared to in the Bloch model which is accurate and may be easier to use. We have in the geometrical model of density matrix a moving vector defined as [19] = − R +R ×B R
(3)
The R vector precesses clockwise about the effective field B with diminishing magnitude. It is possible to follow the R vectors for some nonzero values of detuning as in the case of moving vectors and observe the condition of interference. 5. Lasing without inversion and vector model of Ning Lu We now present an analysis of the salient features of the vector model given by Lu [22]. In this model the population trapping and the emission and absorption processes in -type lasers and masers can be visualized in terms of a vector model in the resonant case = 0 in which there is no mode pulling = ˝. The state vector of the coupled atom field system is written as: | =
n (t) |n −i(ω +n˝)t
(4)
=a,b,c
where n (t) are the probability amplitudes. Using the Schrödinger equation i ˙ = H| the equation of motion for the probability amplitude is obtained as:
d dt
bn ian−1 cn
=
0 √ g1 n 0
√ −g1 n 0 √ −g2 n
0 √ g2 n 0
bn ian−1 cn
(5)
where atomic relaxation terms are neglected for simplicity. Each set of equations in (4) is put into the form of a vector model dSn = ˝n × Sn dt
(6)
97
where n = 1, 2, 3,. . . and the amplitude vector
Sn =
bn ian−1 cn
(7)
and the driving field vector √
−g2 n −sin √ ˝n = 0 =G n 0 √ g1 n cos
(8)
While the driving field vector ˝n is real, the amplitude Sn is complex. By separating Sn into real and imaginary parts Sn = 1n + 2n , the complex vector equation (5) can be decomposed into real vector equations, dln = ˝n × ln , dt where
1n =
Re bn −Im an−1 Re cn
l = 1, 2
(9)
,
2n =
Im bn Re an−1 Im cn
(10)
6. Discussion It is appropriate now to make a comparative study of the vector models as discussed in the earlier sections. The salient features in each case can be underlined as follows. In the vector model of density matrix as discussed by Feynman et al. [20], for some detuning (ω = / ), complete transition (upper to lower level) never occurs. Then the vector r precesses clockwise about the effective field B with frequency |B| = [(ω − )2 + ℘E0 /]2 . This model is presumably based on Rotating Wave Approximation. In the vector model proposed by Lu [22], when an atom is initially in the trapping state, none of the vectors rotates. One find population trapping but no absorption. The vector model proposed in our work involves a simplified treatment based on RWA. It may be noted that the vectors are obtained by joining the maxima at different detunings and these vectors evolve and also rotates with time. We call them as moving vectors. The rotations of these vectors are analogous to the rotation of vectors in other models. Specifically, we have drawn the phasor diagrams using our vectors and these phasor diagrams reflect one of the most important features of spontaneous emission, i.e. area enclosed by the vectors is inversely proportional to decay (). This is a manifestation of the fact that lifetime is inversely proportional to decay. References [1] R.P. Feynman, R.B. Leighton, M. Sands, The Feynman Lectures on Physics, vol. I, Addison-Wesley Publishing Co., Reading, Mass., 1965. [2] S.E. Harris, Function element using quantum interference effect, Phys. Rev. Lett. 62 (1989) 1033; A. Imamoglu, Interference of radiatively broadened resonance, Phys. Rev. Lett. A 40 (1989) 2835; S.E. Harris, J.J. Macklin, Lasers without inversion: single atom transient response, Phys. Rev. Lett. A 40 (1989) 4135. [3] M.O. Scully, S.Y. Zhu, A. Gavreilides, Degenerate quantum beat laser: lasing without inversion and inversion without lasing, Phys. Rev. Lett. 62 (1989) 2813. [4] N. Lu, J.A. Bergou, Quantum theory of a laser with injected atomic coherence: quantum noise quenching via nonlinear processes, Phys. Rev. A 40 (1989) 237. [5] A. Lyras, X. Tang, P. Lambropoulous, J. Zhang, Radiation amplification through autoionizing resonances without population inversion, Phys. Rev. A 40 (1989) 4131. [6] G.S. Agarwal, S. Ravi, J. Cooper, dc-field-coupled autoionising states for laser action without population inversion, Phys. Rev. A 41 (1990) 4721. [7] G.S. Agarwal, Dressed state lasers and masers, Phys. Rev. A 42 (1990) 686. [8] G.S. Agarwal, Theory of the laser operating due to gain on three photon mollow side band, Opt. Commun. 80 (1990) 37.
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F.B. Hazarika et al. / Optik 122 (2011) 95–98
[9] A. Lezama, Y. Zhu, M. Kanskar, T.W. Mossberg, Radiative emission of driven two level atoms into the modes of an enclosing optical cavity, Phys. Rev. A 41 (1990) 1576. [10] M. Lewenstein, Y. Zhu, T.W. Mossberg, Two photon gain and lasing in strongly driven two level atoms, Phys. Rev. Lett. 64 (1990) 3131. [11] S.Y. Zhu, E.E. Fill, Non linear theory of lasers without inversion with two upper lasing levels, Phys. Rev. A 42 (1990) 5684. [12] J.A. Bergou, P. Bogar, Quantum theory of a noninversion laser with injected atomic coherence, Phys. Rev. A 43 (1991) 4889. [13] A. Imamoglu, J.E. Field, S.E. Harris, Lasers without inversion: a closed lifetime broadened system, Phys. Rev. Lett. 66 (1991) 1154. [14] G.S. Agarwal, Origin of gain in systems without inversion in bare or dressed states, Phys. Rev. A 44 (1991) R28. [15] N. Lu, P.R. Berman, Lasing without inversion in dressed state lasers, Phys. Rev. Lett. A 44 (1991) 5965. [16] M.O. Scully, K. Wodkiewicz, M.S. Zubairy, J. Bergou, N. Lu, J. Meyer ter Vehn, Two photon correlated-spontaneous emission laser: quantum noise quenching and squeezing, Phys. Rev. Lett. 60 (1988) 1832; N. Lu, F.X. Zhao, J. Bergou, Nonlinear theory of two photon correlated spontaneous emission laser: a coherently pumped two-level-two photon laser, Phys. Rev. A 39 (1989) 5189;
[17]
[18] [19] [20]
[21] [22]
N. Lu, S.Y. Zhu, Quantum theory of two photon correlated-spontaneousemission lasers: exact atom field interaction Hamiltonian approach, Phys. Rev. A 40 (1989) 5735. B.R. Mollow, Stimulated emission and absorption near resonance for driven systems, Phys. Rev. A 5 (1972) 2217; F.Y. Wu, S. Ezekiel, M. Ducley, B.R. Mollow, Observation of amplification in a strongly driven two level system at optical frequencies, Phys. Rev. Lett. 38 (1977) 1077. V.F. Weisskopf, E. Wigner, Berechnung der naturlichen Linienbrette auf Grund der Diracschen Lichttheorte (in German), Z. Phys. 63 (1930) 54. M. Sargent III, M.O. Scully, W.E. Lamb Jr., Laser Physics, Addison-Wesley Publishing Company, Reading, Massachusetts, 1974. R.P. Feynman, F.L. Vernon, R.W. Hellwarth, Geometrical representation of the Schrodinger equation for solving laser problems, J. Appl. Phys. 28 (1957) 49. F. Bloch, Nuclear induction, Phys. Rev. 70 (1946) 460. N. Lu, -type lasers and masers: lasing without inversion and a vector model, Phys. Rev. A 45 (7) (1992).
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
An analysis of lasing without inversion in a three level system in terms of vector representation Farzana B. Hazarika a , S.N. Rai b , G.D. Baruah a,∗ a b
Department of Physics, Dibrugarh University, Dibrugarh 786004, India Department of Physics, Maharishi Markandeswar University, Mullarva, Ambala 133203, India
a r t i c l e
i n f o
Article history: Received 12 May 2009 Accepted 28 November 2009
a b s t r a c t In the present work we have worked out phasor diagrams with the help of the concept of moving vectors. We show that the area enclosed by the phasor diagrams behaves like lifetime of atomic states. We also make a comparative study of our model with the models suggested by other workers to explain lasing without inversion. © 2010 Elsevier GmbH. All rights reserved.
1. Introduction Quantum interference is a challenging principle of quantum theory. Essentially the concept states that elementary particles can not only be in more than one place at a given time (through superposition), but that an individual particle, such as photon can cross its own trajectory and interfere with the direction of its path. Debate over whether light is essentially particles or waves dates back over three hundred years. In the seventeenth century, Sir Isaac Newton (1642–1726) proclaimed that light consisted of particles. In the early part of seventeenth century, Christian Huygens (1629–1695) enunciated a convenient working principle to describe how the progress of the primary wavefront of a source of light is due to the generation of the secondary wavelets from every point of the primary wavefront. Thomas Young devised the double slit experiment to prove that it consisted of waves. Although the implications of Young’s experiment are difficult to accept, it has reliably yielded proof of quantum interference through repeated trials. According to Feynman et al. [1], each photon not only goes through both slits simultaneously but traverses every possible trajectory on the way to the target, not just in theory, but in fact. In order to see how this might possibly occur, experiments have focused on tracking the paths of individual photons. What happens in this case is that the measurement in some way disrupts the photons trajectories (in accordance with uncertainty principle), and somehow, the results of the experiment become what would be predicted by classical physics.
∗ Corresponding author. E-mail address: [email protected] (G.D. Baruah). 0030-4026/$ – see front matter © 2010 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2009.11.013
From what has been described above it is apparent that uncertainty principle is involved in explaining the phenomenon of interference and therefore the phenomenon is more appropriately termed as quantum interference whose research is being applied in a growing number of applications including quantum interference laser or LWI [2–17]. Under special conditions coherent atomic transitions can cancel absorption. The main idea of LWI is that absorption cancellation provides the possibilities to obtain light amplification even if the population of the upper level is less than the population of the lower level. Such a situation can be realized for instance, in a three level system, when two coherent atomic transitions destructive interfere and, hence, cancel absorption. To present the basic physics of LWI it is best to consider the theory of this effect in a three level -configuration and then to demonstrate how the concept of lasing without inversion can be realized experimentally. The configuration of -three levels atomic system is presented in Fig. 1. It is formed by upper levels |a and |c through interaction with electromagnetic fields E1 and E2 , respectively, in such way that only atomic transitions |a − |c and |a − |b are allowed. The physical reason for canceling absorption in this system is the uncertainty in atomic transitions |c − |a and |b − |a which results in destructive interference between them. This situation is similar to Young’s double slit problem, where interference is a consequence of uncertainty in determining through which of two slits the photon passed. In the present work we report a vector model to represent quantum interference laser in terms of phasor diagrams which have been worked out from the concept of transition probabilities and their time evolutions with suitable decay parameters being introduced. We also present a comparative study
96
F.B. Hazarika et al. / Optik 122 (2011) 95–98
Fig. 1. Three level atom in the -configuration interacting with two fields of frequencies WL1 and WL2 .
between different vector models representing three-level type maser. 2. Atom field interaction It may be noted that the atom decays to the lower level as a result of its interaction with vacuum. The theory of spontaneous emission of Weisskopf–Wigner [18] justified the inclusion of phenomenological decay rates a and b in the Schrödinger equation for the atomic probability amplitudes. The probability of stimulated absorption including damping is given by [19]
2 2 Ca (t)2 = 1 ℘E0 exp(−b t) sin(w − )t/2 4
h ¯
(w − )/2
(1)
3. Moving vectors The transition probability for arbitrary value of time may be plotted for different values of detuning (ω − ). Using Eq. (1) we have worked out the graph, which represents the time evolution of population in the upper state for different values of detuning at 90, 100, 110, 120 and 130 MHz. This is shown in Fig. 2. As shown in
Fig. 3. Decay versus area.
Fig. 2, we may join all the five maxima in a group of detunings as an arrow and a number of these arrows can be drawn in this way. We observe that these vectors evolve in time. These vectors can be used to represent transition probabilities for the changes of population in the upper state. The zero transition probabilities at arbitrary time of 62 n (n = 1, 2, 3, . . .) indicate some type of beats in the process of interference. A phasor diagram has been constructed with the help of the moving vectors. The moving vectors can be represented by the equation dkn = ˝n × kn dt
(2)
where n = 1, 2, 3, . . ., ˝n is the driving field vector, kn is the amplitude vector. We now proceed to analyse the phasor diagrams as shown in Fig. 2 which are drawn with the help of the moving vectors. The vectors in the phasor diagram first close up as in the case of resonating cavity but uncoil again and expand. It is also observed that when decay is small the expansion is slow but for bigger decay the expansion is more. This is a general nature of all the phasor diagrams we
Fig. 2. Probability versus time graphs for decay = .001, .005, .01, .02, .03, .05 and their corresponding phasor diagrams.
F.B. Hazarika et al. / Optik 122 (2011) 95–98
have constructed with different decay constants and with the same set of detuning values. The phasor diagrams also indicate that as the decay increases the area decreases exponentially which may be illustrated by plotting decay versus area graph as shown in Fig. 3.As regards its physical significance it may be noted that the parameter is related to the lifetime as = 1/. This follows from the theory of spontaneous emission. In the present case the area behaves like lifetime. The result is new. 4. Density matrix in motion In this section we introduce the concept of the vector model of density matrix in motion [19]. This model has value not only in solving the equation of density matrix in motion but also in providing a physical picture of a particular system. The equation are equivalent to the Bloch equation [21] appearing in nuclear magnetic resonance. In this connection we may note that the phenomenon of the effect of an adiabatic rapid passage in inverting the population distribution of an assembly of nuclear spins has been recognized and predates the maser itself by approximately a decade. The technique is straightforward. In a system in which a radiation field far from resonance with a transition frequency in a material sample interacts with a sample, the field frequency is swept through the material resonance until it is again far off resonance but with the frequency deviation reversed. If the sweep rate has been adjusted to lie within relatively easily attainable limits in certain materials, it is found that the population of the material sample after the passage is inverted relative to its value prior to the passage, For laser media when decays a and b are small compared to in the Bloch model which is accurate and may be easier to use. We have in the geometrical model of density matrix a moving vector defined as [19] = − R +R ×B R
(3)
The R vector precesses clockwise about the effective field B with diminishing magnitude. It is possible to follow the R vectors for some nonzero values of detuning as in the case of moving vectors and observe the condition of interference. 5. Lasing without inversion and vector model of Ning Lu We now present an analysis of the salient features of the vector model given by Lu [22]. In this model the population trapping and the emission and absorption processes in -type lasers and masers can be visualized in terms of a vector model in the resonant case = 0 in which there is no mode pulling = ˝. The state vector of the coupled atom field system is written as: | =
n (t) |n −i(ω +n˝)t
(4)
=a,b,c
where n (t) are the probability amplitudes. Using the Schrödinger equation i ˙ = H| the equation of motion for the probability amplitude is obtained as:
d dt
bn ian−1 cn
=
0 √ g1 n 0
√ −g1 n 0 √ −g2 n
0 √ g2 n 0
bn ian−1 cn
(5)
where atomic relaxation terms are neglected for simplicity. Each set of equations in (4) is put into the form of a vector model dSn = ˝n × Sn dt
(6)
97
where n = 1, 2, 3,. . . and the amplitude vector
Sn =
bn ian−1 cn
(7)
and the driving field vector √
−g2 n −sin √ ˝n = 0 =G n 0 √ g1 n cos
(8)
While the driving field vector ˝n is real, the amplitude Sn is complex. By separating Sn into real and imaginary parts Sn = 1n + 2n , the complex vector equation (5) can be decomposed into real vector equations, dln = ˝n × ln , dt where
1n =
Re bn −Im an−1 Re cn
l = 1, 2
(9)
,
2n =
Im bn Re an−1 Im cn
(10)
6. Discussion It is appropriate now to make a comparative study of the vector models as discussed in the earlier sections. The salient features in each case can be underlined as follows. In the vector model of density matrix as discussed by Feynman et al. [20], for some detuning (ω = / ), complete transition (upper to lower level) never occurs. Then the vector r precesses clockwise about the effective field B with frequency |B| = [(ω − )2 + ℘E0 /]2 . This model is presumably based on Rotating Wave Approximation. In the vector model proposed by Lu [22], when an atom is initially in the trapping state, none of the vectors rotates. One find population trapping but no absorption. The vector model proposed in our work involves a simplified treatment based on RWA. It may be noted that the vectors are obtained by joining the maxima at different detunings and these vectors evolve and also rotates with time. We call them as moving vectors. The rotations of these vectors are analogous to the rotation of vectors in other models. Specifically, we have drawn the phasor diagrams using our vectors and these phasor diagrams reflect one of the most important features of spontaneous emission, i.e. area enclosed by the vectors is inversely proportional to decay (). This is a manifestation of the fact that lifetime is inversely proportional to decay. References [1] R.P. Feynman, R.B. Leighton, M. Sands, The Feynman Lectures on Physics, vol. I, Addison-Wesley Publishing Co., Reading, Mass., 1965. [2] S.E. Harris, Function element using quantum interference effect, Phys. Rev. Lett. 62 (1989) 1033; A. Imamoglu, Interference of radiatively broadened resonance, Phys. Rev. Lett. A 40 (1989) 2835; S.E. Harris, J.J. Macklin, Lasers without inversion: single atom transient response, Phys. Rev. Lett. A 40 (1989) 4135. [3] M.O. Scully, S.Y. Zhu, A. Gavreilides, Degenerate quantum beat laser: lasing without inversion and inversion without lasing, Phys. Rev. Lett. 62 (1989) 2813. [4] N. Lu, J.A. Bergou, Quantum theory of a laser with injected atomic coherence: quantum noise quenching via nonlinear processes, Phys. Rev. A 40 (1989) 237. [5] A. Lyras, X. Tang, P. Lambropoulous, J. Zhang, Radiation amplification through autoionizing resonances without population inversion, Phys. Rev. A 40 (1989) 4131. [6] G.S. Agarwal, S. Ravi, J. Cooper, dc-field-coupled autoionising states for laser action without population inversion, Phys. Rev. A 41 (1990) 4721. [7] G.S. Agarwal, Dressed state lasers and masers, Phys. Rev. A 42 (1990) 686. [8] G.S. Agarwal, Theory of the laser operating due to gain on three photon mollow side band, Opt. Commun. 80 (1990) 37.
98
F.B. Hazarika et al. / Optik 122 (2011) 95–98
[9] A. Lezama, Y. Zhu, M. Kanskar, T.W. Mossberg, Radiative emission of driven two level atoms into the modes of an enclosing optical cavity, Phys. Rev. A 41 (1990) 1576. [10] M. Lewenstein, Y. Zhu, T.W. Mossberg, Two photon gain and lasing in strongly driven two level atoms, Phys. Rev. Lett. 64 (1990) 3131. [11] S.Y. Zhu, E.E. Fill, Non linear theory of lasers without inversion with two upper lasing levels, Phys. Rev. A 42 (1990) 5684. [12] J.A. Bergou, P. Bogar, Quantum theory of a noninversion laser with injected atomic coherence, Phys. Rev. A 43 (1991) 4889. [13] A. Imamoglu, J.E. Field, S.E. Harris, Lasers without inversion: a closed lifetime broadened system, Phys. Rev. Lett. 66 (1991) 1154. [14] G.S. Agarwal, Origin of gain in systems without inversion in bare or dressed states, Phys. Rev. A 44 (1991) R28. [15] N. Lu, P.R. Berman, Lasing without inversion in dressed state lasers, Phys. Rev. Lett. A 44 (1991) 5965. [16] M.O. Scully, K. Wodkiewicz, M.S. Zubairy, J. Bergou, N. Lu, J. Meyer ter Vehn, Two photon correlated-spontaneous emission laser: quantum noise quenching and squeezing, Phys. Rev. Lett. 60 (1988) 1832; N. Lu, F.X. Zhao, J. Bergou, Nonlinear theory of two photon correlated spontaneous emission laser: a coherently pumped two-level-two photon laser, Phys. Rev. A 39 (1989) 5189;
[17]
[18] [19] [20]
[21] [22]
N. Lu, S.Y. Zhu, Quantum theory of two photon correlated-spontaneousemission lasers: exact atom field interaction Hamiltonian approach, Phys. Rev. A 40 (1989) 5735. B.R. Mollow, Stimulated emission and absorption near resonance for driven systems, Phys. Rev. A 5 (1972) 2217; F.Y. Wu, S. Ezekiel, M. Ducley, B.R. Mollow, Observation of amplification in a strongly driven two level system at optical frequencies, Phys. Rev. Lett. 38 (1977) 1077. V.F. Weisskopf, E. Wigner, Berechnung der naturlichen Linienbrette auf Grund der Diracschen Lichttheorte (in German), Z. Phys. 63 (1930) 54. M. Sargent III, M.O. Scully, W.E. Lamb Jr., Laser Physics, Addison-Wesley Publishing Company, Reading, Massachusetts, 1974. R.P. Feynman, F.L. Vernon, R.W. Hellwarth, Geometrical representation of the Schrodinger equation for solving laser problems, J. Appl. Phys. 28 (1957) 49. F. Bloch, Nuclear induction, Phys. Rev. 70 (1946) 460. N. Lu, -type lasers and masers: lasing without inversion and a vector model, Phys. Rev. A 45 (7) (1992).