Quantum optics

INTRODUCTION We are going to start by a sentence taken from Smoluchowski lecture of the famous physicist Nico Van Kampen in Warsaw sometime ago: "Now consider the laser… Then we obtain a stochastic differential equation of a quite different type than the naive Langevin equation We need a new Smoluchowski to solve such a problem" [1] It reflects three important aspects of the problem considered in my thesis At first, in quantum optics, the fluctuation theory which is an important part of statistical physics is widely used Secondly, we frequently then must deal with differential stochastic equations, which are very difficult to be solved They are analytically averaged only in some simple cases, whereas in most cases which are physically interesting, they are averaged by different approximation methods Finally, Poland has had a long tradition in this domain of physics The research works of Polish physicists are well-known worldwide Real laser is never perfectly monochromatic But in most considered cases, the monochromatic character of the laser is usually referred as its main advantage In practical applications, the laser is the most monochromatic among all the existing lights However, in currently performed experiments, one needs to research on the influence of spectral width of the laser to different phenomena Its spectral width changes (depending on the type of the laser) from about 10s-1 to about 109s-1 Several mechanisms lead to spectral line broadening There are different classifications of these mechanisms In the laser technique, the noises are classified into two forms: the "technical" noises, which include mechanical vibrations of the resonant cavity and other types of external noises, and "fundamental" noises due to quantum fluctuations Spectral broadenings are usually classified to be homogeneous or inhomogeneous ones Inhomogeneous broadening is a result of the inhomogeneous conditions in which the atoms of the active laser medium are placed An example for this type is the broadening caused by the motion of atoms in a gas laser As a result of the Doppler Effect, transitions between the same energy levels in different atoms create a radiation source with different frequencies On the other hand, broadening associated with the natural width of the energy levels of atoms, and broadening resulting from the collisions of atoms with each other and with the walls of the resonator are considered to be homogeneous broadenings This is the result of interactions of our system (atoms of the active medium plus the selected modes of the electromagnetic radiation) with other systems treated as the reservoirs (surroundings as a whole, thermal bath of other modes of radiation…) In the last time, a number of theoretical research works successfully applied an approach in which the laser field is treated as a stochastic process The spectral width is the resulting average of the phase or amplitude fluctuations of the field around the mean values (figure 01) Stochastic models of the laser light are described in part 1.1 of Chapter Imζ (t) ζ δϕ ϕ δζ ζ+δζ Reζ (t) Figure 01 The spectral width is the resulting average of the phase or amplitude fluctuations of the field around the mean values In general, most of the phenomena in quantum optics are of the type shown in figure 02 The laser is then treated as an external source to the atomic system Dynamic equations, which contain field parameters such as the phase, the amplitude or the intensity, become stochastic differential equations LASER ATOMIC SYSTEM Figure 02 A typical problem in quantum optics Averaging of solutions gives us the opportunity to reflect on the influence of the laser fluctuations on the atomic quantities of which we are interested This problem has been one of the central problems of quantum optics since the pioneering papers of Eberly and Agarwal [2] Character of the stochastic process which describes the laser field, depends on the laser model (see section 1.1 of Chapter 1) All current stochastic models of the laser have a common character: the laser is a classical electromagnetic field, which is a stationary Gaussian stochastic process with the finite correlation time Exact analytical averaging of stochastic equations with Gaussian noise, that has finite correlation time, is a difficult task Practically only the extreme case of white noise (i.e the correlation time is equal to zero) has been well researched Already in this simplest case modeling of laser by a stochastic process gives many interesting results In the general case, we succeeded in calculating the average of stochastic equations only for problems in which the dynamic equations are either linear or non-linear with a special form [2] When the stochastic equation is highly nonlinear, the exact analytical average of the stochastic equation is unrealizable However, there are many methods of finding the approximate average One of them is the pregaussian noise method [3], in which it is a possibility of obtaining the exact analytical average even in the case of high nonlinearity of the problem Moreover, this noise very well approximates Gaussian noise Pregaussian noise is defined as the sum of a finite numbers of so-called telegraph noises In applications of quantum optics [4, 5] the pregaussian noise which consists of only a few telegraph noises, perfectly approximates Gaussian noise Thus pregaussian noise gives us the possibility to consider the influence of Gaussian noise when the other approximate methods fail Moreover, due to the fast convergence of this noise to Gaussian noise, we expect that this method gives more precise results in comparison with other approximation methods The scheme of my thesis looks as follows In the first Chapter, elementary concepts concerning the theory of stochastic processes are presented In particular we describe in detail the concept of pregaussian noise In the second Chapter we consider the phenomena in which the laser light is modeled by white noise Applications of the one-telegraph pregaussian noise in several optical phenomena and some extensions to the case of pregaussian noise with two telegraphs are shown in the third Chapter The last section contains our conclusions CHAPTER STOCHASTIC PROCESSES IN QUANTUM OPTICS 1.1 Stochastic models of the laser light 1.1.1 Single mode laser with fluctuations of the amplitude and the phase The laser theory is built within the framework of quantum theory on the interaction of electromagnetic field with an active material system One can start from a microscopic picture, which then leads to a stochastic model of the laser light All characteristic parameters of the stochastic processes describing the laser field can be defined by a complete microscopic theory By considering in more detail this theory, we can obtain a model, in which the radiation field emitted by the laser is described by the complex amplitude: ζ (t ) = (ζ + δζ (t ))e iϕ (t ) , (1.1) where ζ = const and ζ (t ) and ϕ (t ) are independent stochastic processes The theory that describes the homogeneous spectral width is presented in figure 1.1 [6] We assume that the considered system is described by a set of the operators {l} = {l1 , l , , l µ , }, for example for one mode of the radiation field {l} = {l , l + }, where l and l + are respectively annihilation and creation operators of photons For a two-level atom, that is, for a model of the atom in the active medium, we have {l} = {σ ,σ + ,σ z }, where σ and σ + are the combinations of Pauli matrices [7] We now introduce an additional set of the operators, which describe the reservoirs { m} = {m1 , m2 , , mµ , } For example, in the case of the thermal radiation, mµ can be annihilation or creation operators of the field quanta with energies ℏω µ If the interaction Hamiltonian contains only bilinear terms, then the Heisenberg equations which describe the evolution of the system, are linear with respect to the variables {l} , {m} Then the variables {m} can be eliminated As a result we obtain the equations which only contain the operators {l} and the initial values of the operators {m} : dli = f i ({l}) + Gi ({m(0)}) dt SYSTEMS (A) (1.2) RESERVOIRS (B) Pumping Atoms of the active medium σ, σ+, σz Vacuum fluctuations (spontaneous emission) Phonons or atoms colliding with each other Interaction ATOM + FIELD Radiation fields l, l+ Walls of the resonant cavity, vibration of the mirrors, thermal radiation Figure 1.1 The scheme which leads to the model given above The concrete values of the operators mi (0 ) cannot be known in advance However, from the properties of the given reservoir we can find their statistical properties and hence we can have statistical properties of forces Gi In general, these statistical properties cannot be analyzed in an exact manner Fortunately, the real correlation time of functions Gi (t ) is usually the smallest of all the characteristic times involved in the system, so we may assume that the two-time correlation function for the forces has the form: Gi (t )Gk (t ') = 2ℓ ik δ (t − t ') (1.3) Then the equation (1.2) is similar to the Langevin equation for the theory of Brownian motion [8] The method described above is applied by Haken [9] in laser theory with homogeneous broadenings Further approximation methods (as the method of adiabatic elimination of atomic variables etc.) lead to the equation for the complex laser field (1.1) The linearization around the stationary solution of this equation provides to following equations of the Langevin type, which are independent from each other: ϕɺɺ + θϕɺ = G(t ) , (1.4a) δζɺ (t ) + ςδζ (t ) = ℑ(t ) , (1.4b) where G (t ) and ℑ(t ) are uncorrelated white noises which are Gaussian processes with the following properties: G (t ) = , G (t )G (t ') = 2lδ (t − t ') , (1.5a) ℑ(t ) = , ℑ(t )ℑ(t ') = 2mδ (t − t ') (1.5b) It follows from (1.4) that the amplitude and the derivative of the phase are socalled Orstein-Uhlenbeck processes [8] They are Gaussian processes with constant average value (usually equal to zero) x(t ) = const and correlation function: x(t )x(t ') = σ 02 exp(− θ t − t ' ) (1.6) Using the Gaussian property of x(t ) , we can show from (1.6) that the OrsteinUhlenbeck process is a Markov process [10] Fluctuations of the amplitude are usually much smaller than the fluctuations of the phase Therefore amplitude fluctuations are practically neglected in the description of one-mode laser Such model is called the phase-diffusion model [10] Formalism presented above leads to the homogeneous broadening of the laser When the inhomogeneous broadenings are taken into account, we should average the final results over the statistical distribution of the corresponding parameter which is involved in dynamic equations and related to the inhomogeneity of the active medium For example in the description of the gas laser, this parameter is the atomic velocity with Maxwell distribution 1.1.2 The laser model with pump fluctuation In seventies of last century it was commonly thought that laser theory, which was developed simultaneously at three places: Lamb School [9], Bell Telephone Laboratories [11] in the United States and Haken Group [6] in Stuttgart, Germany, describes very well the coherent properties of the laser light However in 1981 Mandel and his coworkers [12, 13] showed that in an experiment with the single-mode dye laser, fluctuation phenomena observed have quite different characteristics than that predicted in the standard model [6, 11] Kaminishi et al [12] have already applied Haken theory to describe these experimental results Haken theory presented schematically in the previous subsection leads to an equation for the complex amplitude of the electric field: ( ) ζɺ (t ) = ς (t ) − G1 ζ ζ + ξ (t ) , (1.7) where ς is the pump parameter, G1 (always positive) is the saturated parameter of the active medium which causes the stabilization of the laser action above the threshold, and ξ (t ) is the Gaussian white noise describing vacuum fluctuations or spontaneous emissions which are usually omitted (figure 1.1) However, equation (1.7) is not suitable enough to explain the experimental results [12] Nevertheless, it was for the first time suggested that pump fluctuations (see figure 1.1) could play an important role in single-mode dye laser For realizing this idea, Graham et al [14] assumed that the pump parameter ς is the white noise and ξ (t ) in equation (1.7) was omitted From this assumption the multiplicative process [15] has been considered instead of the additive process Then this equation can be solved exactly by means of an analytic method Experimental results obtained in [12] were explained well by this procedure As shown in [13], the above theory could still not describe well enough some other unpublished experimental results They proposed that the white noise should be replaced by a coloured noise because the relaxation time of the pump noise was not small enough in comparison with other characteristic times of the dye laser system However then the equation: ( ) ζɺ (t ) = ς (t ) − G1 ζ ζ , (1.8) could not be solved analytically The theory of coloured noises is well developed [16], but only in the special cases it gives us exact analytic solutions Dixit and Sahni [17] simulated numerically the colored noise and obtained the results which were in accordance with the experimental results of Short et al 1.1.3 Multimode laser and chaotic light We consider another mechanism leading to the broadening of laser light It occurs in multimode lasers Complex amplitude of the field emitted by the laser has the form [18]: M ζ (t ) = ∑ ζ k e −i (ω t +ϕ ) , k k =1 k (1.9) where M is the number of modes, ω k are their frequencies with respect to the average frequency of ωL , ζ k are constant amplitudes and ϕ k are mutually independent stochastic phases We accept the natural assumption that these phases are uniformly distributed in the interval [0,2π ] It follows from the properties of phases ϕ k that ζ (t ) = 0, (1.10a) M ζ * (t )ζ (t ') = ∑ ζ k2 e −iω (t −t ' ) , (1.10b) ζ (t )ζ (t ') = ζ * (t )ζ * (t ') = (1.10c) k k =1 One can calculate so-called characteristic functional of the process ζ (t ) [2, 18]: [−i ζ (t )v (t )dt − i ∫ ζ ϕ M [v(t ), v* (t )] = e ∫ * (t )v * (t )dt ] = Π J (2 vk ) , M k =1 (1.11) where vk = ∫ e −iω t v(t )dt and J is the zero-order Bessel function [19] We assume k that the number of modes M tends to infinity, whereas their amplitudes decrease to zero, such that I (t ) = ζ (t ) M = ∑ ζ k2 (1.12) k =1 is constant Using the fact that [20] J 0' ( z ) z ≅ , z→ J (z ) (1.13) when ζ k → we obtain { } ϕ ∞ [v(t ), v * (t )] = exp − ∫∫ v(t ) ζ * (t )ζ (t ') v * (t ')dtdt ' 10 (1.14) W (E ) E a0 Figure 2.7 Photoelectron spectrum for the bare energies of discrete atomic states E1 = 0.5 , E2 = 7.5 ; laser frequency EL = 0.05 ; autoionizing widths γ = γ = 0.5 ; and the coherent part b0 = 0.5 Finite value of the asymmetry parameter: When asymmetry parameters q1 and q2 are finite, for the degenerate case (E21 = 0) , the photoelectron spectra are presented in the figures 2.8 and 2.9 which vary considerably when compared with the spectra described in the figures 2.4 and 2.5 The photoelectron spectrum, when the chaotic part is missing (a = ) , and for the big values of asymmetry parameters are shown in figure 2.8 The photoelectron spectrum is symmetric and has only one peak for the case of weak field A sharp wedge occurs in the photoelectron spectrum when the amplitude of the field increases This photoelectron spectrum is identical to that found by Leoński et al [45] However, when the chaotic part is not absent (a ≠ ) , the spectrum is presented in figure 2.9, the sharp wedge in the spectrum vanishes- - 43 - only one, symmetric peak remains in the spectrum but its intensity is smaller than in the case of a = W (E ) b0 E Figure 2.8 Photoelectron spectrum in the case of the degenerate case in with E1 = E2 = 0.5 ; the chaotic part a = 0.0 ; laser frequency EL = 0.5 ; autoionizing widths γ = γ = 0.5 , and the asymmetry parameters q1 = 90, q = 100 W (E ) b0 E Figure 2.9 Photoelectron spectrum in the case of the degenerate case in with E1 = E2 = 0.5 ; the chaotic part a = 0.5 ; laser frequency EL = 0.5 ; autoionizing widths γ = γ = 0.5 , and the asymmetry parameters q1 = 90, q = 100 - 44 - Figures 2.10 and 2.11 present the spectra for the nondegenerate case Figure 2.10 describes the spectrum when the chaotic part is absent and for the smaller values of b0 , the spectrum displays two peaks However, with an growth in b0 , the left peak diminishes quickly, whereas the right peak increases As the left peak vanishes, the right peak starts to split into two peaks Figure 2.11 describes the spectrum when the chaotic component is present, the two-peak structure of the spectrum becomes smooth with an growth in b0 This process is quicker than that discussed in the case when the chaotic part was missing According to the theory of Fano [44], The appearance of Fano zeros in the photoelectron spectra manifest some quantum interferences which can appear in the system due to the fact that the system can attain the continuum states via autoionizing levels and by direct transitions If the external electromagnetic field fluctuates then these interferences are weakened Therefore we can predict that Fano zeros vanish in the photoelectron spectra of our model We can easily see that the Fano zeros really not exist in all presented figures W (E ) b0 E Figure 2.10 Photoelectron spectrum for the bare energies of discrete atomic states E1 = 2.0 , E2 = 5.0 ; laser frequency EL = 2.0 ; autoionizing widths γ = 0.1 , γ = 0.9 ; the asymmetry parameters q1 = 2.0, q = 10.0 ; and the chaotic part a0 = 0.0 - 45 - W (E ) b0 E Figure 2.11 Photoelectron spectrum for the bare energies of discrete atomic states E1 = 2.0 , E2 = 5.0 ; laser frequency EL = 2.0 ; autoionizing widths γ = 0.1 , γ = 0.9 ; the asymmetry parameters q1 = 2.0, q = 10.0 ; and the chaotic part a = 0.5 In summary, we considered in this section a laser-induced autoionizing model, which was introduced earlier in [45], when two discrete states embedded in one continuum replaced one autoionizing state In this model, we suppose that the laser light is separated into two parts: coherent and white noise Subsequently, we established and solved exactly a system of coupled stochastic integro-differential equations and thus received the results describing the dynamics of autoionization for the double Fano model The results displayed here contain certain additional terms, in contrast to those described in [46], which make our results more exact In particular, accurate analytical expressions for the photoelectron spectrum was obtained and we compared these results with those described in [45, 46] The photoelectron spectra for the finite and infinite values - 46 - of the asymmetry parameters were considered Moreover, we displayed photoelectron spectra for the cases of degenerate and nondegenerate autoionizing levels Our results for both strong and weak fluctuation regimes are discussed the detailed As in [59], the amplitudes of the real laser used in experiments always fluctuate, so we believe that our model described here is more realistic than that given in [45] In the last section of this Chapter we discuss a other quantum interference phenomenon, namely EIT in Λ-like systems which contain a continuum coupled to one or two autoionizing states Furthermore the strictly deterministic control laser field will be replaced by that described by white noise - 47 - CONCLUSIONS When the laser fluctuations are taken into account in real experiments, we use the stochastic processes to model the laser light Then the dynamical equations involved in considered problems become stochastic differential equations Averaging of solutions gives us the opportunity to reflect on the influence of the laser fluctuations on the atomic quantities of which we are interested This problem has been one of the central problems of quantum optics since the pioneering papers of Eberly and Agarwal [2] Almost all stochastic models of the laser have a common character: the laser is a classical electromagnetic field, which is a stationary Gaussian stochastic process with the finite correlation time Exact analytical averaging of stochastic equations with Gaussian noise, that has finite correlation time, is a difficult task, sometimes a “mission impossible” Practically only the extreme case of white noise (i.e the correlation time is equal to zero) has been well researched Already in this simplest case modeling of laser by a stochastic process gives many interesting results In this thesis, we obtained some new and interesting results in the optical phenomena in which the laser light is modeled by white noise [59, 60, 61, 87] Then we can average exactly the equations involved in the problems These phenomena concerned two classes of the quantum interference intensively studied in the last three decades, namely the autoionization processes and electromagnetically induced transparency At first we considered an external field-driven double Fano model, in which we assumed that the external electromagnetic field was separated into coherent component and white noise Thus we solved a set of coupled stochastic integrodifferential equations involving the Fano model with two discrete levels and obtained an exact expression to discuss the photoelectron spectrum and compared it with the outcomes of previous papers As a result, when the white - 48 - noise of the external electromagnetic field is present, Fano zeros disappear in the photoelectron spectra and the peaks of photoelectron spectra were smaller in comparison with the case when the white noise is absent Secondly, electric susceptibility of a laser-dressed atomic medium was considered for a model Λ-like system which includes two lower states and a structured continuum This structured continuum involves one or two autoionizing states embedded in the flat continuum (dressed continuum) It was also assumed that the strong control field of EIT could be separated into two components, which are coherent part and white noise By solving a set of coupled stochastic integro-differential equations in the stationary regime, we obtained the exact formulae determining the electric susceptibility for EIT The dispersion and absorption spectra of the medium susceptibility were derived The results shown that when the white noise is present, slope of the dispersion curves and the depth of the transparency window decrease, the zero point shifts to the right, the position and width of the transparency window change dramatically in comparison with the case when the white noise of the strong control field is absent In particular, the characteristic parameter of white noise is an important parameter to play the controlling role the propagation group velocity of light in the medium We trust that the our models, which the laser light is modeled by white noise, are more realistic than models that described in the case when the white noise is absent because the amplitudes of the laser light used in experimental always contain some fluctuation component Other class of exactly soluble models is related to the pregaussian noise method [3], in which it is a possibility of obtaining the exact analytical average even in the case of high nonlinearity of the problem Moreover, this noise very well approximates Gaussian noise Pregaussian noise is defined as the sum of a finite numbers of so-called telegraph noises In applications of quantum optics [4, - 49 - 5] the pregaussian noise which consists of only a few telegraph noises, perfectly approximates Gaussian noise Thus pregaussian noise gives us the possibility to consider the influence of Gaussian noise when the other approximate methods fail Moreover, due to the fast convergence of this noise to Gaussian noise, we expect that this method gives more precise results in comparison with other approximation methods Even in the case of one telegraph noise we obtained some interesting results [104, 115, 117], namely we discussed the influence of collision fluctuations on optical phenomena The fluctuations are taken into account by a simple shift of the deterministic detuning, involved in a set of optical Bloch equations by collision frequency noise that is modeled by a random telegraph noise The exact solution of stochastic Bloch equations was obtained This solution was applied to consider in detail the Mollow spectrum of RF for the case of an arbitrary detuning of the laser frequency with the velocity of the emitter which remains constant or obey Maxwell-Boltzman distribution When the collision fluctuations are present, the maximum values of the peaks are lower in comparison with the case when the collision fluctuations are absent but the spectrum remains symmetric The asymmetric of the spectrum is only caused by the detuning Finally, we discussed the model of a nonlinear system with random telegraph noises [117, 131] Then the exact differential equations for the stationary probability distribution of fluctuation in this nonlinear system under the influence of both one and two telegraphs noises are obtained In its application, the stationary probability distribution of fluctuation was calculated analytically and the existence of noise induced phase transitions in a Raman Ring Laser was showed Specially, we showed explicitly the so-called noise reduction in Raman Ring Laser This is seemly interesting phenomenon and only appear in the nonlinear system, namely when the values of the noise parameters in the nonlinear systems increase, the steady probability distribution shrinks It has been shown that the Stokes output of Raman Ring Laser tends to the stabilisation - 50 - under influence of the broad-band telegraph pump The construction of the injected telegraph pump signal is much easer in comparison with the case of Gaussian signal, so this phenomenon could be realised experimentally in a much easer manner than for the case of Gaussian pump In the future, we will consider in detail the RF spectrum for the case of an arbitrary detuning of the laser frequency in which our formalism must be extended to the case of the theory without the assumption of the RWA Next, we will show explicitly noise reduction in the nonlinear system for the case of twotelegraph pregaussian pump We predict that noise reduction process in Raman Ring Laser will be faster in comparison with the case of one-telegraph noise - 51 - BIBLIOGRAPHY [1] Cao Long Van, Stochastic Models of Isolated Collisions: Applications to Optical Phenomena, LAMP Conference’89, Trieste, Italy, 1989 [2] J H Eberly, Phys Rev Lett 37, 1383 (1976) G H Agarwal, Phys Rev Lett 37, 1387 (1976) [3] K Wódkiewicz, B W Shore and J H Eberly, Phys Rev A 30, 2390 (1984) J H Eberly, K Wódkiewicz and B W Shore, Phys Rev A 30, 2381 (1984) [4] V Cao Long and K Wódkiewicz, J Phys B 19, 1925 (1986) V Cao Long and S Janeczko, Z Phys B 62, 531 (1986) [5] V Cao Long, Z Phys B 65, 535 (1987) [6] M Sargent, M O Scully and W E Lamb, Laser Physics, Addison Wesley, Reading 1974 [7] L Allen, J H Eberly and K Rzążewski, Rezonans Optyczny, Państwowe Wydawnictwo Naukowe, Warszawa 1981 [8] G E Uhlenbeck and L S Ornstein, Phys Rev 36, 823 (1930) [9] H Haken, Z Phys 190, 327 (1966) [10] J L Doob, Annals of Math 43, 351 (1942) [11] W H Louisell, Quantum Statistical Properties of Radiation, John Wiley and Sons, New York 1973 [12] K Kaminishi, R Roy, R Short and L Mandel, Phys Rev A 24, 370 (1981) [13] R Short, L Mandel and R Roy, Phys Rev Lett 49, 647 (1982) [14] R Graham, M Höhnerbach and A Shenzle, Phys Rev Lett 48, 1396 (1982) [15] R F Fox, J Math Phys 13, 1196 (1972) [16] M Kuś, Doctoral Thesis, Warsaw University 1983 [17] S N Dixit and P Sahni, Phys Rev Lett 50, 1273 (1982) [18] P Zoller, G Alber and R Salvador, Phys Rev A 24, 298 (1981) [19] M Abramowitz and I A Stegun, Handbook of Mathematical Functions, Natl Bur Stand Washington 1964 [20] E Janke, F Emde and F Lösch, Specjalnyje funkcji, Nauka, Moskwa 1977 - 52 - BIBLIOGRAPHY [21] A T Georges and P Lambropoulos, Phys Rev A 20, 991 (1979) [22] P Zoller, Phys Rev A 20, 1019 (1979) [23] A T Georges, Phys Rev A 21, 2034 (1980) [24] L Arnold, Stochastic Differential Equations: theory and applications, Wiley-Interscience, New York 1974 [25] J L Doob, Stochastic Processes, John Wiley and Sons, New York 1990 [26] N G Van Kampen, Phys Rep C 24, 173 (1976) [27] R P Feynman and A R Hibbs, Quantum Mechanics and Path Integrals, Mc Graw-Hill, New York 1964 [28] B Simon, Functional Integrals and Quantum Physics, Acad Press, New York 1979 [29] R F Fox, Phys Rep 48, 179 (1978) [30] I Białynicki – Birula, Postępy Fizyki XVII, 359 (1966) [31] W I Kliatskin, Stochasticzeskjie urawnienija i wołny w słuczajnonieodnorodnych sriedach, Nauka, Moskwa 1980 [32] M C Wang and G E Uhlenbeck, Rev Mod Phys 17, 323 (1945) [33] P Zoller and F Ehlotzky, J Phys B 10, 3023 (1977) [34] R Kubo, J Math Phys 4, 174 (1963) [35] S N Dixit, P Zoller and P Lambropoulos, Phys Rev A 21, 1289 (1980) [36] K Wódkiewicz, Z Phys B 42, 95 (1981) [37] K Wódkiewicz, J Math Phys 20, 45 (1971) [38] N G Van Kampen, Stochastic Processes in Physics and Chemistry, NorthHolland Publishing Company 2007 [39] J M Sancho and M San-Miguel, Z Phys B 36, 357 (1980) [40] K Wódkiewicz, B W Shore and J H Eberly, J Opt Soc Am B 1, 398 (1984) [41] K Rzążewski and J Eberly, Phys Rev Lett 47, 408 (1981) [42] K Rzążewski and J Eberly, Phys Rev A 27, 2026 (1983) [43] P Lambropoulos and P Zoller, Phys Rev A 24, 370 (1981) [44] U Fano, Phys Rev 124, 1866 (1961) [45] W Leoński, R Tanaś and S Kielich, J Opt Soc Am B 4, 72 (1987) - 53 - BIBLIOGRAPHY [46] Cao Long Van and M Trippenbach, Z Phys B 63, 267 (1986) [47] J Peřina Jr, A Lukš, W Leoński and V Peřinová, Phys Rev A 83, 053416 (2011) J Peřina Jr, A Lukš, W Leoński and V Peřinová, Phys Rev A 83, 053430 (2011) [48] W Leoński, R Tanaś and S Kielich, J Phys B 21, 2835 (1988) [49] W Leoński and V Buzek, J Mod Opt 37, 1923 (1990) [50] W Leoński, J Opt Soc Am B 10, 244 (1993) [51] A Raczyński, M Rzepecka, J Zaremba and S Zielińska-Kaniasty Opt Commun 266, 552 (2006) [52] M Lewenstein and K Rzążewski, Phys Rev A 61, 022105 (2000) [53] B Luk'yanchuk, N I Zheludev, S A Maier, N J Halas, P Nordlander, H Giessen and C T Chong, Nature materials 9, 707 (2010) [54] A E Miroshnichenko, S Flach and Y S Kivshar, Rev of Mod Phys 82, 2257 (2010) [55] S Feneuille, S Liberman, J Pinard and A Taleb, Phys Rev Lett 42, 1404 (1979) [56] Yu I Heller, V F Lukinykh, A K Popov and V V Slabko, Phys Lett A 82, 4, (1981) [57] P E Coleman and P L Knight, J Phys B 15, 235 (1982) [58] K Wódkiewicz, Phys Rev A 19, 1686 (1979) [59] K Doan Quoc, V Cao Long and W Leoński, Phys Scr T 147, 014008 (2012) [60] V Cao Long, W Leoński, K Doan Quoc and L Chu Van, Proc Natl Conf Theor Phys 35, 42 (2010) [61] K Doan Quoc, V Cao Long and W Leoński, Phys Scr 86, 045301 (2012) [62] S E Harris, J E Field and A Imamoğlu, Phys Rev Lett 64, 1107 (1990) [63] M Fleischhauer, A Imamoğlu and J P Marangos, Rev Mod Phys 77, 634 (2005) [64] W Gawlik, Optical nonlinearity and atomic coherences w: Modern Nonlinear Optics, Part 3, eds: M Evans and S Kielich Advances in Chemical Physics Series, LXXXV, 734 (1994) [65] J P Marangos, J Mod Optics, 45, 471 (1998) [66] A Imamoğlu and S E Harris, Opt Lett 14, 1344 (1989) - 54 - BIBLIOGRAPHY [67] K J Boller, A Imamoğlu and S E Harris, Phys Rev Lett 66, 2593 (1991) [68] J-J Su and I A Yu, Chin J Phys 41 (2003) [69] S Jin, Y Li and M Xiao, Opt Commun 119, 90 (1995) [70] S Stenholm, Foundations of Laser Spectroscopy, Wiley, New York 1984 [71] M Yan, E G Rickey and Y Zhu, J Opt Soc Am B 18, 1057 (2001) [72] J Gea-Banacloche, Y-q Li, S-z Jin and M Xiao, Phys Rev A 51, 576 (1995) [73] D J Fulton, S Shepherd, R R Moseley, B D Sinclair and M H Dunn, Phys Rev A 52, 2302 (1995) [74] J R Boon, E Zekou, D McGloin and M H Dunn, Phys Rev A 59, 4675 (1999) [75] H X Chen, A V Durrant, J P Marangos and J A Vaccaro, Phys Rev A 58, 1545 (1998) [76] A Rostami and G Rostami, Opt Commun 240, 133 (2004) [77] M D Lukin, Rev Mod Phys 75, 457 (2003) [78] S E Harris, J E Filed and A Kasapi, Phys Rev A 46, 29 (1992) [79] M Fleischhauer and M D Lukin, Phys Rev A 65, 022314 (2002) [80] D Dziczek, B Ziętek and S Chwirot, Acta Phys Pol A 106, 13 (2004) [81] N S Ginsberg, S R Garner and L V Hau, Nature 445, 623 (2007) [82] K Kowalski, V Cao Long, K Dinh Xuan, M Głódź, B Nguyen Huy and J Szonert, Comput Methods Sci Technol 16, 131 (2010) [83] P L Knight, M A Lauder and B J Dalton, Phys Rep 190, (1990) [84] Marlan O Scully and M Suhail Zubairy, Quantum optic, Cambridge University Press 1997 [85] A Raczyński, J Zaremba, Phys Rep 235, (1993) [86] T Bui Dinh, W Leoński, V Cao Long and J Peřina Jr, Electromagnetically induced transparency with a double Fano continuum, to be published; Bui Dinh Thuan, Doctoral Thesis, University of Zielona Góra, 2013 [87] K Doan Quoc, V Cao Long and W Leoński, Electromagnetically induced transparency for Λ-like systems with a double Fano continuum and broadband coupling laser, to be published - 55 - BIBLIOGRAPHY [88] A Corney, Atomic and Laser Spectroscopy, Oxford 1977 [89] W Heitler, The Quantum Theory of Radiation, Oxford Univ Press London 1954 [90] C Cohen-Tannoudji, Les Houches 1, (1975) [91] K Wódkiewicz and J H Eberly, Ann Phys N Y 101, 574 (1976) [92] B R Mollow, Phys Rev 188, 1969 (1969) [93] F Schuda, C R Stroud and Jr M Hercher, J Phys B 7, L198 (1974) [94] H Walther, Les Houches 1, 125 (1975) [95] H J Carmichael and D F Walls, J Phys B 8, 177 (1975) [96] H J Kimble and L Mandel, Phys Rev A 13, 2123 (1976) [97] B R Mollow and Miller, Ann Phys N Y 52, 464 (1969) [98] M E Smithers and H S Freedhoff, J Phys B 8, L209 (1975) [99] C R Stroud Jr., Phys Rev A 63, 1044 (1971) [100] L Mandel and E Wolf, Coherence and Quantum Optics, Plenum Press, New York 1972 [101] C Cohen-Tannoudji and S Reynaud, J Phys B 10, 345 (1977) [102] V Cao Long, Acta Physica Polonica A 57, 613 (1980) [103] Cao Long Van, Doctoral Thesis, Warsaw University 1979 [104] K Doan Quoc, V Cao Long and W Leoński, Mollow spectrum influenced by collisions: a stochastic model, to be published [105] F Bayrakceken, Spectrochimica Acta A 60, 923 (2004) [106] A Nick Vamirakas, Yong Zhao, Chao-Yang Lu and Mete Atature, Nature Physics 5, 198 (2009) [107] Zhang Xue-Hua and Hu Xiang-Ming, Chin Phys B 20, 114205 (2011) [108] S Gerber et al., Phys Rev Lett 102, 183601 (2009) [109] Kazuyuki Nakayama, Yutaka Yoshikawa, Hisatoshi Matsumoto, Yoshio Torii, and Takahiro Kuga, Optics Express 18, 6604 (2010) [110] K Wódkiewicz, Phys Rev A 19, 1686 (1979) [111] J H Eberly and K Wódkiewicz, J Opt Soc Am 67, 1252 (1977) [112] B R Mollow, Phys Rev A 15, 1023 (1977) [113] J L Carlsten et al., Phys Rev A 15, 1029 (1977) [114] R H Liu, W H Tan, Chin Phys Lett 16, 23 (1999) - 56 - BIBLIOGRAPHY [115] K Doan Quoc, V Cao Long and W Leoński, Modeling of the influence of buffer gas' collisions on Mollow spectra, submitted to EJP [116] A G Leonov, A A Panteleev, A N Starostin and D I Chekhov, Zh Eksp Teor Fiz 58, 959 (1993) [117] V Cao Long and K Doan Quoc, Opt and Quantum Electron 43, 137 (2012) [118] K Wódkiewicz, Acta Physica Polonica A 63, 191 (1983) [119] V E Shapiro and V M Loginov, Physica A 91, 563 (1978) [120] Kazuo Kitahara, Werner Horsthemke, René Lefever and Yukata Inaba, Prog Theor Phys 64, (1980) [121] M Lewenstein and K Rzążewski, Optics Comm 63, 174 (1987) [122] Bui Bang Doan and Cao Long Van, Internal Report IC/91/234, International Center of Theoretical Physics, Trieste, Italy 1991 [123] J.M Sancho and M San Miguel, Prog Theor Phys 69, 1085 (1983) [124] E.A Novikov, Soviet phys JETP 20, 1290 (1965) [125] P A Roos, L S Mei and J L Carsten, Applied Optics 42, 5517 (2003) [126] Z Luo, X Yuan, W Ye, C Zeng and J Ji, Optics Express 18, 19407 (2010) [127] A K Zamzuri, U S Ismail, M H Al-Mansoori, M S Islam and M A Mahdi, Laser Phys Lett 8, 62 (2011) [128] A Teubel, K Rzążewski, J Opt Soc Am B 6, 550 (1989) [129] P L Jones, U Gaubatz, U Hefter, K Bergmann, and B Wellegehausen, Appl Phys Lett 42, (1983); U Hefter, J Eichert, and K Bergmann, Opt Commun 52, 330 (1985) [130] X M Hu and J S Peng, J Phys B 31, L563 (1998) [131] K Doan Quoc, V Cao Long and W Leoński, Noise reduction in Raman Ring Laser by two-telegraph pregaussian pump, to be published - 57 -