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1 Available online at Archives of Physics Research, 2014, 5 (5):7-11 ( ISSN : CODEN (USA): APRRC7 Spontaneous absorption and Einstein s rate equation approximation R. Bora Bordoloi 1, R. Bordoloi 2, H. Konwar 3, J. Saikia 3 and G. D. Baruah 3 1 Department of Physics, Namrup College, Namrup, Assam 2 Women s College, Tinsukia, Assam, India 3 Centre for Laser and Optical Science, New Uchamati, Doomdooma, India ABSTRACT The nature of spontaneous absorption and its technical aspects are discussed. Einstein s derivation of Planck s black body energy distribution law has been examined with the inclusion of the process of spontaneous absorption. It has been shown that the inclusion of t he term of spontaneous absorption does not affect the distribution law. Self consistent model of the rate equation approximation justifies the inclusion of process of Spontaneous absorption. Key words: Spontaneous Absorption, Rate Equation Approximation. INTRODUCTION Spontaneous radiation, together with stimulated radiation and absorption is among the most fundamental processes in optics. They are primarily responsible for the interaction of radiation with matter. Light falling on matter is absorbed leaving the atoms in the excited state which spontaneously emits radiation with a spread of frequency inversely proportional to the decay time. The situation is modified when the atomic decay times are sufficiently long and the radiation is sufficiently intense. Then the radiation falling on the atom in the excited state, results in stimulated emission rather than stimulated absorption. The third process is needed to obtain Planck s Radiation law in Einstein s derivation [1]. It is stimulated emission which enables an atomic medium to amplify incident radiation and therefore is an essential part in light amplifiers and oscillators. Thus the word LASER has become the acronym for Light Amplification by Stimulated Emission of Radiation. The phenomenon of spontaneous emission is wellknown as it is responsible for most of the radiation around us and probability we owe our existence to it. Quantum theory of radiation [2,3] explains spontaneous emission as a consequence of induced zero point oscillation of electromagnetic field of empty space. This mechanism of spontaneous emission an isotropic perturbation always present, attributed in connection with the quantum theory to the all pervading zero point fluctuations of the electromagnetic field. The light excites the atom zero point fluctuation de-excites them, resulting in the re radiation of light. According to Ginzburg [4] the definition is equally applicable in the classical and quantum regions. He quotes from the book of Heitler [3], The interaction between the atom and the radiation field can cause these radiative transitions even if, in the initial state, no light quanta at all are present. Supposing the atom to be excited in the initial stage, then in the final state the number of light quanta will be increased from zero to some finite value. This process presents then a spontaneous emission of light. In the classical theory the situation is exactly the same. If the radiation field at a given moment is equal to zero (in this case it is obviously not necessary to worry about zero point oscillations of the electromagnetic field), but there is accelerated motion of a charge (charge in vacuum), then at a subsequent time radiation will appear, the energy of which will increase with time. This is spontaneous radiation. Spontaneous radiation appears because the state in which a mechanical sub system (an atom, a moving charge etc.) is at some level is moving in a specified manner but the radiation field is absent, is not a stationary Eigen state of the complete system (the mechanical sub system plus the electromagnetic field). 7

2 R. Bora Bordoloi et al Arch. Phy. Res., 2014, 5 (5):7-11 The zero point oscillations are a purely quantum effect, which is not present in the classical theory and which formally disappears as ћ 0. Zero point oscillations give rise to the effects like Lamb Shift [5], Casimir Effect [6], Laser Line width [7] etc. Stimulated emission has been observed as LASER and is now available in many forms over a wide spectral range from far U V to infra red. Similarly absorption (or, stimulated absorption) is ubiquitous and it is easily observed as dark lines in the absorption spectra. But what about Spontaneous absorption? Until now nobody has observed it and discussion on this topic is not to be found in any available literature. Since we are using the term of spontaneous absorption in Einstein s Rate Equation, it is worthwhile to justify it or otherwise, based on the work of Ginzburg [4] and Miloni [8]. Why there is a need to relate spontaneous radiation to zero point oscillations of the field? Spontaneous radiation is induced radiation of light quanta produced by the zero point oscillations. This is indicated by the review in 1935 by Weisskopf [9], who took an active part in the development of quantum theory [10]. In [9] he writes, From quantum theory there follows the existence of so called zero point oscillations; for example each oscillator in its lowest is not completely at rest but always is moving about its equilibrium position. Therefore electromagnetic oscillations also can never cease completely. Thus the quantum nature of the electromagnetic field has as its consequence zero point oscillations of the field strength in the lowest energy state, in which there are no light quanta in space. The zero point oscillations act on an electron in the same way as ordinary electricall oscillations do. They can change the eigen state of the electron, but only in a transition to a state with the lowest energy, since empty space can only take away energy, and not give it up. In this way spontaneous radiation arises as a consequence of the existence of these unique field strengths corresponding to zero point oscillations. Thus spontaneous radiation is induced radiation if light quanta produced by zero point oscillations of empty space. Research on the Lamb Shift led to the idea that this shift of an energy level and also the natural line width due to spontaneous emission could be attributed to the quantum mechanical zero point fluctuations of the electromagnetic field. In a paper published in 1948, Welton [11] wrote that spontaneous emission can be thought of as forced emission taking place under the action of the fluctuating field. From what has been described above it is understood that the zero point oscillation is concerned only in the case of a transition to a state with lowest energy, i.e. spontaneous emission. But in the case of a space which is not perfectly empty this statement needs to be qualified. Again in some works [12] it has been shown that in the ground state the influence of the vacuum field fluctuations is cancelled by the atomic dipole fluctuations, at least quantum mechanically, why there is no absorption (stimulated and spontaneous) from zero point field [12]. The idea of zero point fluctuations was not there in Einstein s derivation of Planck spectrum [1]. We may add here that in 1913 Einstein and Stern found that the black body spectrum of Planck could be derived by assuming that the dipole oscillation had a zero point energy ћω 0 [13]. There classical approach failed to account for the zero point energy which is ½ћω 0 for a field mode of frequency ω 0. With these backgrounds in mind we now make a bold assumption that the probability of Spontaneous Absorption is non zero and with this we proceed to derive Planck s radiation law using Einstein s Rate Equation Approximation [1]. We shall show that the inclusion of the Spontaneous Absorption term does not affect the distribution law derived by Einstein, which is also supported by Reservoir model [7] as we shall describe in section 4. In section 2, we briefly discuss our self consistent model [14] which justifies the inclusion of Spontaneous absorption. Fig. 1: Self consistent model of Einstein s rate equation approximation 2. Self Consistent model of Rate Equation Approximation. In this section we briefly illustrate our self consistency model of Einstein s Rate equation approximation [14]. It must be emphasized that almost alll the laws of nature exhibit the phenomenon of self consistency and in fact the phenomenon of self consistency is the essence of all physical laws. 8

3 An example is the geometrical model of the semi classical theory of laser [7]. Figure -1shows the geometrical model representing the self consistent model of Einstein s rate equation approximation. 3. Einstein s rate equation approximation and spontaneous absorption We shall now directly turn to the process of rate equation approximation. Before the creation of quantum mechanics, in the framework of the old quantum theory [15], the major problem was the description of the absorption and emission of light by an atom. In the absence of a good microscopic theory, the only approach was a detailed discussion involving the use of emission and absorption probabilities. This was done by Einstein [1, 17] who explicitly indicated an analogy with the process of radioactive decay. The following calculations are exactly the procedure adopted by Einstein but for the term spontaneous absorption. The experimentally observed spectral distribution of blackbody radiation is well fitted by the formula discovered by Planck [15]. ρ( ω, T ) = 3 hω 2 3 π c 1 hω exp( ) 1 κ T (1) where ω is the angular frequency, T is the absolute temperature, κ is the Boltzmann constant. Einstein revealed that this formula can be explained in terms of ensembles of atoms with discrete energy levels of various frequencies provided that (i) Spontaneous emission is included along with the process of stimulated emission and stimulated absorption (ii) The upper and lower levels are populated according to the Boltzmann distribution. In this section we follow this derivation but include information of spontaneous absorption, which was not used by Einstein. A careful examination of the four processes indicates that stimulated emission corresponds to stimulated absorption when the two levels are equally populated. Same is the case with spontaneous emission under similar circumstances. We consider appropriate expressions related to transitions in a system of atoms between states n and m with energies E n and E m (>E n ); and consider as in the work of Einstein [17] the radiation to be isotropic, unpolarized and to be propagating in vacuum. Generalization to the case of presence of a medium and with the inclusion of various polarizations of the normal modes is given by Ginzburg [18]. Let us assume that there are n 1 atoms in the lower state n and n 2 atoms in the upper state m and ρ (ω) is the radiation field density. We thus have = Rate of atoms going up due to spontaneous absorption = = Rate of atoms coming down to lower level n due to spontaneous emission= = Rate of atoms coming down due to stimulated emission = () = Rate of atoms going up due to stimulated absorption = () Here we have considered the probability of spontaneous absorption to be different from that of stimulated absorption, but afterward it will be taken as equal. At this stage we would like to emphasize the fact that in laser spontaneous emission always appears as a noise and until now nobody has been able to separate this noise from stimulated emission. Similar considerations may be applicable in the case of spontaneous emission where stimulated absorption appears as noise. The word stimulated absorption has been used in place of absorption simply because of the fact that when a radiation of photon density () of right frequency is incident on the lower level n the atoms in that level receive the radiation, absorb it and get stimulated to a higher level. This is just the mirror image of Stimulated emission which occurs when the radiation is sufficiently strong and falls in the excited state with lot of population in it. We have the basic assumptions n = n exp( hω κt ) 2 1 h ω = E E m B = B = B = B nm nm mn n W + W = W + W (2) W mn mn nm nm = W (3) 9

4 The last term in the RHS of equ (2) is new. We thus have n A + B n = n B + n B 2 mn mn 2 1 nm 1 nm { } n B n B = n A + n B 1 nm 2 mn 2 mn 1 nm n 1 n1 B B nm mn = A + B mn nm n n 2 2 exp( ω κt ) B B = A + exp( hω κt ) B { h } nm mn mn nm Amn exp( hω κt ) Bnm = + exp( hω κt ) B B exp( hω κt ) B B nm mn nm mn A 1 exp( hω κt ) = + B exp( hω κt ) 1 exp( hω κt ) 1 (4) Consider the second term in RHS of eqn (4). If the quantity exp ( h ωκt) is replaced by AB the term becomes equivalent to the first term which is the formula for the energy density of the black body radiation. There is no overall change in the distribution function. Let us now consider the light field damping by atomic beam reservoir [7] to supplement eqn (4) and obtain additional information. 4. Light field damping by atomic beam reservoir Let us consider the reservoir concept in simple terms which may be illustrated by coupling a simple harmonic oscillator, a time dependent electric field to a beam of two level atoms, some of which are initially excited. This is shown in Fig. 2. Fig. 2. Diagram shows passage of a beam of two level atoms initially either in upper state or lower state. Atoms initially in the upper state pass through the cavity at the rate per second and atoms initially in the lower state pass through the cavity at the rate. The atoms interact with the field bringing it to an equilibrium temperature T In Fig. 2 the atomic energy distribution is characterized by a temperature T given by Boltzmann distribution. 1 3 = 567( ħ:;) (5) where < = and < > are number of atoms per second in the upper and lower levels which pass through a laser cavity. The effect of the beam on the simple harmonic oscillator is to bring it to the equilibrium temperature T. The derivation of this fact can be found in reference 7. Using eqs (5) and (4) we have () =? + 1 ABC(ħDEF)G ABC(ħDEF)G (6) As may be seen from Eqn (6) that both the terms in the L.H.S. are representative of Planck s distribution law provided 1 3 =? = ħi D J K I L J Here A is the transition probability due to spontaneous emission and is the number of atoms per second in the lower levels which pass through a laser cavity. Thus the reservoir theory justifies the inequality as represented by eqn (7). 10

5 CONCLUSION From what has been discussed above it is appropriate to draw few conclusions. We have observed that in the steady state temperature T, inclusion of spontaneous absorption term in the rate equation approximation, results in the same distribution law as derived by Einstein, but with additional information. With increasing frequency ω the ratio 1 3 increases. This means that at extremely short wavelength the spontaneous emission noise becomes more important and this fact affects the gain in a laser. In this case the ratio 1 3 behaves like the ratio AB, as it should. The idea of relating spontaneous radiation to zero point oscillation is discussed in its historical aspects. There is a scope to consider the process of spontaneous absorption. Acknowledgement One of the authors (R. B. Bordoloi) is thankful to the University Grant Commission for award of a teacher fellowship (Grant No.F4-59TF2012NERO13823, 08, Oct, 2013.) REFERENCES [1] A. Einstein, Z Phys., 18, 121 (1917). [2] V.F.Weisskopf and E.Wigner, Z. Phys. 63, 54(1930) [3] W.Heitler, Quantum Theory of Radiation, Oxford (1936) [4] V.N.Ginzburg. Sov. Phys. Usp 1983, 26(8), 713. [5] W.E.Lamb Jr. and R.C. Rutherford, Phys. Rev. 1947, 72, 241 [6] H.B.G. Casimir,Proc. K.Ned.Wet. 1948, 51,793, Physica 1953, 19,846. [7] M. Sargent III, M.O.Scully and W.E.Lamb Jr. Laser Phys. 1974, p292. [8] P.W.Miloni, Am. J. Physics 1984, 52(4), 340. [9] V.F.Weisskopf, Naturwissenschaften, 1935, 27, 631. [10] V.F.Weisskopf, Phys. Today, 1981, No 11, 69. [11] T.A.Welton, Phys. Rev., 1948, 74, [12] V.M.Fain and Ya I. Khanin, Quantum Electronics, 1969, (MIT, Cambridge, MA), Vol 1, Sec 28 and 29. [13] P.W. Miloni, Am. J. Phys. 1981, 49, 177. [14] R. B. Bordoloi, and G. D. Baruah et al., Indian J. Science, 2012, 1(1), 58. [15] De ter Haar, The Old Quantum Theory, 1967, Pergamon Press, NY. [16] M.Planck, Ver. Deut. Phys. Ges. 1900, 2, 237; Ann. Physik. 1901, 4, 553. [17] A. Einstein, Sobranie nauchnykh truvov (Collected Scientific Works) Moscow, Nauka, 1966,Vol 3, pp 386. [18] V.L.Ginzburg, Teoreticheskaya Fizika, 1981 (Theoretical Physics and Astrophysics) Nauka, Moscow. 11