3. Rate-Equation Model¶
PDF pages 97–106
3.1 Quantization¶
Chapter 3 Rate-Equation Model Before using a proper quantum model of the atom, we will use a simple model of the atom that includes discrete energy levels. However, we will not include any coherence effects, so the resulting rate equations constitute a sort of ‘‘semi-quantum’’ model of the atom. At the same time, we must treat the field with a discrete model, invoking the idea of photons, so that energy exchange between an atom and the field occurs only in multiples of ¯h\omega. Even with the language of photons, we will stick to a strictly semiclassical treatment, not really treating the atoms or the field quantum mechanically. With this rudimentary model and some simple arguments, we can derive a number of important results without the full apparatus of quantum mechanics. 3.1 Quantization To elaborate, we will start with the observation that the energy in an electromagnetic field is quantized. This means that a monochromatic field of frequency \omega (typically restricted to some ‘‘quantization volume’’ such as an optical cavity) has possible energies given by E = n + 1 ¯h\omega, (3.1) where n is a nonnegative integer, representing the number of photons in the field. This may be familiar as the energy-level structure of the quantum harmonic oscillator. The photon number is always defined with respect to a particular mode (fixing the direction, polarization, and frequency characteristics). The energies for atoms and molecules are also quantized, although the exact energy-level structure depends on the specific atom or molecule. If we denote the quantized energies by En, then the differences in energy levels correspond to frequencies via ∆Emn := Em −En = ¯h\omegamn. (3.2) The idea is that atoms with an energy difference ∆Emn prefer to interact with resonant fields of frequency \omegamn. In this case, the energy of a single photon matches the atomic energy difference, and energy is conserved. There are different types of transitions, generally corresponding to different types of radiation. Electronic transitions in atoms are the most energetic of the type we will consider, and they correspond to visible optical frequencies. Vibrational transitions in a molecule correspond to different amplitudes and types of motion internal to the molecule, and generally correspond to radiation in the infrared. Rotational transitions in molecules have yet lower energy, and they correspond to microwave radiation (which enables the maser, the microwave predecessor to the laser).
3.3 Einstein Rate Equations¶
Chapter 3. Rate-Equation Model 3.2 Fundamental Light–Atom Interactions There are three fundamental interactions between light and atoms. In all cases we will consider only a two- level atom with ground-state energy E1 and excited-state energy E2. We will also assume resonant light,
- Absorption (stimulated). In the absorption process, a photon is destroyed and the atom is promoted to the excited state. More generally, if there are n photons to start with in some resonant mode, then there are n −1 photons after the absorption process. E1 E2 photon E1 E2 (no photon) 2. Stimulated Emission. This process involves the atom initially being in the excited state, in the presence of n photons in some resonant mode. After the stimulated-emission event, the atom is demoted to the ground state and the field is left with n + 1 photons. In some sense, this process is the opposite of stimulated absorption, although absorption ending with 0 photons is possible while stimulated emission beginning with 0 photons is not. E1 E2 n photons E1 E2 no+o1 photons 3. Spontaneous Emission. This process is much like stimulated emission, but when the atom is de- moted, a photon is created in some mode that is initially unpopulated. Thus, a photon can go into a wide range of possible modes by spontaneous emission. It is possible to view spontaneous emission as stimulated emission due to quantum vacuum fluctuations in addition to classical radiation reaction. E1 E2 (no photons) E1 E2 1 photon Generally, we can associate stimulated absorption and emission with a single mode that is already populated, or singled out by some other means, such as an optical cavity. We can associate spontaneous emission additionally with all other modes. 3.3 Einstein Rate Equations Now let’s consider an ensemble of two-level atoms interacting with light. Let N1,2 denote the number density of atoms with energy E1,2. Then the Einstein rate equation for the excited state is1 dN2 dt
(3.3) (Einstein rate equation) Here, \rho(\omega) is the energy density of the electromagnetic field (the energy density in the frequency interval \omega to \omega + d\omega). The first term corresponds to spontaneous emission, and we can see that it reduces the excited-state population, even in the absence of any field. The second and third terms are proportional to 1A. Einstein, ‘‘Zur Quantentheorie der Strahlung,’’ Physikalische Zeitschrift 18, 121 (1917), translation ‘‘On the Quantum Theory of Radiation’’ by Alfred Engel appears in The Collected Papers of Albert Einstein, Volume 6, The Berlin Years: Writings, 1914-1917 (Princeton University Press, 1997) p. 200 (ISBN: 0691017344).
3.4 Relations Between the Einstein Coefficients¶
3.4 Relations Between the Einstein Coefficients \rho(\omega), and correspond to stimulated emission and absorption, respectively, as we can see from their overall signs. By convention, the constant A21 is called the Einstein A coefficient, while B21 and B12 are called the Einstein B coefficients. At this point we are simply postulating that the three processes contribute to the atomic evolution in this form, with the rate coefficients yet to be determined. The Einstein A coefficient here represents the rate at which energy is lost from the atom. We can thus identify A21 = \gamma, where \gamma is the damping rate from the Lorentz atom. The connection of the Lorentz atom with the B coefficients is less clear, in part because the classical model gets this wrong (hence the necessity of patching the classical solution with the oscillator strength). We will defer this comparison until we derive the cross section for the two-level atom. To be consistent, N1 +N2 must add up to some constant, assuming that we really have two-level atoms and that the atoms stay in place (something that even works fairly well for gas lasers as long as we modify
dN1 dt
(3.4) (steady-state solution) for the ground-state population N1. We can gain some valuable insight by looking at the equilibrium behavior of the rate equations. Steady
N2 N1 =
(3.5) If the energy levels are not degenerate, it turns out that B12 = B21, as we will see shortly. That is, stimulated emission and absorption are exactly symmetric from the rate-equation point of view. Then we can rewrite the steady-state solution as N2 N1 = A21
. (3.6) We can see from this that N2 < N1 in steady state. This result has an important result for using atoms as a gain medium for a laser: there is no steady-state population inversion in a two-level system, and hence there is no net gain of light transmitted through a medium composed of two-level atoms. This is because on average, absorption (attenuation) occurs more often than stimulated emission (amplification). In the limit of large intensity, \rho(\omega) −\rightarrow \infty, the populations equalize. This points to an important effect that is missed by the Lorentz model: atomic saturation. For small excitation, N2/N1 is proportional to \rho(\omega), but as the excitation increases, the slope of N2/N1 decreases, dropping to zero for large intensities. We will treat this point more carefully after establishing some more results regarding the rate coefficients. 3.4 Relations Between the Einstein Coefficients Now we briefly outline Einstein’s derivation of the relation between the A and B coefficients. If the energy levels are degenerate, we can define the degeneracy factors g1,2 as the number of ways of having energy E1,2. For example g1,2 = 2J1,2 + 1 for atomic angular-momentum states. Then the steady-state population ratio from Eq. (3.4) can be written also via Boltzmann statistics as N2 N1 = g2 g1
(3.7) Solving for \rho(\omega),
B21 B12g1 B21g2
. (3.8)
3.5 Line Shape and Spectral Distributions¶
Chapter 3. Rate-Equation Model This is equivalent to the Planck blackbody distribution2
\lambda3
(3.9) if we make the identifications g2B21 = g1B12 (3.10) (relation between B coefficients) and A21 B21 = 4h \lambda3 . (3.11) (relation between A and B coefficients) Recall that \lambda here is the wavelength within the atomic medium. Remarkably, this simple thermodynamic argument reproduces the full quantum result that we will derive later. Essentially, this is because the Planck distribution is valid for a particular set of quantum (thermal) states (at some level, a proper summation over field modes is buried in the Planck distribution, as evidenced by the correct frequency dependence of \omega3). This is sufficient to establish the relationship between the coefficients, since they are independent of the quantum state. 3.5 Line Shape and Spectral Distributions So far, we’ve considered only monochromatic light and two-level atoms with sharply defined energy levels. Now it’s time to improve our model of the two-level atom and its interaction with light. We will first introduce a line-shape function s(\omega) to model the fact that the energy levels have some width. The line shape is defined such that s(\omega) d\omega is the probability that a spontaneously emitted photon will have frequency between \omega and \omega + d\omega. We can also interpret this as the relative probability of stimulated emission or absorption of a photon with frequency between \omega and \omega + d\omega. Since s(\omega) represents a probability density, it is appropriately normalized: Z \infty
(3.12) Note that as in our discussion of coherence in Chapter 2, we are using a ‘‘one-sided spectrum’’ that ranges only over positive frequencies. In terms of the ‘‘two-sided spectrum’’ s\pm(\omega) with both positive and negative
When we apply the line shape and sum over all frequencies, the rate equation becomes dN2 dt = −A21N2 −B21N2 Z \infty
Z \infty
(3.13) Qualitatively, we can picture the line shape function as a relatively sharply peaked distribution centered around the resonant optical frequency \omega0. dw/2p ~ 10oo -10 o Hz wº/2p ~ 10oo -10 o Hz w s(w) Often, s(\omega) turns out to be a Lorentzian, a Gaussian, or a convolution of the two (a Voigt profile). The line-shape function models transition width due to spontaneous emission, collisions, Doppler shifts in gas lasers, and local crystal structure effects on the dopant atoms. Note that in the absence of radiation, 2P. W. Milonni and M.-L. Shih, ‘‘Zero-point energy in early quantum theory,’’ American Journal of Physics 59, 684 (1991) (doi: 10.1119/1.16772).
3.5.2 Nearly Monochromatic Light¶
3.5 Line Shape and Spectral Distributions the rate equation is dN2/dt = −A21N2, which has an exponentially damping solution. As we discussed before, the Fourier transform of an exponential is a Lorentzian, so the line shape for spontaneous emission (the ‘‘natural line shape’’) is Lorentzian, with a half-width at half maximum of A21. Collisions are often modeled by a spontaneous-emission-like term, and thus also lead to Lorentzian line shapes. Doppler shifts lead to Gaussian line shapes because the Maxwell–Boltzmann velocity distribution is Gaussian. If multiple, independent broadening effects contribute, their combined effect can be modeled by the convolution of the individual line shapes. Now we will consider two limiting cases for the light spectrum. Both are important in understanding laser operation, but the second is the more useful case for comparing to coherent quantum light–atom interactions. 3.5.1 Broadband Light Light is broadband (relative to the transition) if \rho(\omega) is much broader than s(\omega). Then we can evaluate the integral in Eq. (3.13) by noting that \rho(\omega) varies slowly over the width of s(\omega), so that we can pull it out of the integral: Z \infty
Z \infty
(3.14) Thus, we recover the previous rate equations, corresponding to Eq. (3.3), with sharp energy levels. 3.5.2 Nearly Monochromatic Light For nearly monochromatic light, the field spectrum is narrow, so s(\omega) is much broader than \rho(\omega). Thus, we can evaluate the integral with the same slowly varying approximation as for the broadband case: Z \infty
Z \infty
(3.15) The integral on the right-hand side is the total field energy density, summed over all frequencies. Let’s denote this simply by \rho. Then the rate equation becomes dN2 dt
(3.16) where we have written s(\omega) in place of s(\omegafield). The total energy density is related to the total intensity I
dN2 dt
¯h\omega N2 −g2 g1 N1 . (rate equation, monochromatic light) (3.17) Here, we have defined the absorption cross-section
\lambda2 4 s(\omega). (3.18) (cross section) The cross section has the dimensions of area, and is defined such that \sigma(\omega)I is the power absorbed by a single atom when irradiated by intensity I (in the weak-excitation limit). Note that for a Lorentzian line shape s(\omega),
∆\omega
(3.19) the resonant cross section \sigma(\omega0) is given by
∆\omega \lambda2 2\pi . (3.20)
3.6 Absorption Coefficient and Saturation¶
Chapter 3. Rate-Equation Model For homogeneous broadening, ∆\omega is the natural line width given by ∆\omega = A21, so that the natural cross section is
2\pi . (3.21) (natural, on-resonance cross section) This answer is consistent with a fully quantum-mechanical calculation, and this is the same cross-section that we used before [Eq. (1.35)] to derive the form of the oscillator strength. This relation also establishes the relation of the Einstein B coefficient to the classical model of the atom. Note that this answer assumes an average over all possible atomic orientations, since the blackbody distribution of Eq. (3.9) assumes isotropic radiation. For atomic dipole moments aligned with the field polarization, the resonant cross section is
2\pi , (3.22) since the coupling that would normally be ‘‘distributed’’ among three orthogonal directions is concentrated into one. 3.6 Absorption Coefficient and Saturation Let us assume that nearly monochromatic light of frequency \omega passes through a vapor of two-level atoms. Evidently, from the rate equation (3.17), the rate per unit volume at which atoms are being promoted to the excited state by the pumping field is
¯h\omega N2 −g2 g1 N1 . (3.23) We define the absorption coefficient by dI
(3.24) (absorption coefficient definition) Then we multiply the expression (3.23) by the photon energy ¯h\omega to obtain the rate of energy absorption per unit volume, or equivalently the rate of intensity absorption per unit length, which matches the right-hand side of Eq. (3.24). Thus, we find
N2 −g2 g1 N1 . (3.25) We can get the population difference here from the steady state of the rate equation (3.17). This gives N2 N1 = g2 g1 \sigmaI ¯h\omegaA21 1 + \sigmaI ¯h\omegaA21 . (3.26) Noting that N1 + N2 = N, which implies N2 −g2 g1 N1 N = g1 g2 N2 N1 −1 g1 g2 N2 N1 + g1 g2 = − g2/g1 1 + 1 + g2 g1 \sigmaI ¯h\omegaA21 . (3.27) Putting this result into Eq. (3.25), we find for the absorption coefficient
g2 g1
1 + 1 + g2 g1
¯h\omegaA21 . (3.28)
3.6 Absorption Coefficient and Saturation Note that in the case of equal degeneracies, g1 = g2, we have
¯h\omegaA21 . (3.29)
For small intensities, this expression is equivalent to the classical expression (1.33), which had the constant value \sigma(\omega)N. For large intensities, the absorption coefficient falls to zero. This is the effect of saturation or optical bleaching of the medium. On resonance, the absorption coefficient becomes
\sigma0N 1 + 2 \sigma0I ¯h\omega0A21 , (3.30)
2\sigma0 , (3.31) (saturation intensity) so that we can write the resonant absorption coefficient as
\sigma0N 1 + I Isat . (3.32) The saturation intensity gives the intensity scale over which saturation sets in. Specifically, we see that the absorption coefficient drops to half the small-signal value when I = Isat. Again, this is one important feature of the light–matter interaction that the classical model misses: the harmonic oscillator can be excited to arbitrarily high amplitudes, but in a quantum-mechanical atom, the best excitation is when the maximum number of atoms are pumped into the excited state.
3.7 Exercises¶
Chapter 3. Rate-Equation Model 3.7 Exercises Problem 3.1 We said that a medium of two-level atoms is no good as a laser gain medium, since the ground state ends up with more population than the excited state, and so absorption wins out over stimulated emission (i.e., loss wins out over amplification). The simplest change that we can make to achieve a population inversion, where the excited state is more populated than the ground state, is to add a third level. The level scheme is shown here. pump R13 A21o(slow) A32o(fast) laser transition } The new level (with highest energy) decays quickly, while the laser (2 −\rightarrow 1) transition decays slowly.
laser),
¯h\omega , (3.33) where \sigma(\omega) is the absorption cross section for the 1 −\rightarrow 3 transition, and \omega is the frequency of the pumping field. Then we can write the rate equations for the three-level atom as dN3 dt
dN2 dt = A32N3 −A21N2 dN1 dt
(3.34) where of course one of the equations is redundant since N1 +N2 +N3 must be a constant of the motion. The key to why this scheme gives a population inversion on the laser transition (1 −\rightarrow 2) is that atoms will be promoted to level 3 after absorbing a pump photon, and they will quickly decay to the metastable level 2 before stimulated emission by the pump field returns them to level 1. In this way the pump depletes level 1 and populates level 2 without trying to return the atoms back to level 1. (a) Got all that? Outstanding. Now find the steady-state solution in the limit where A32 is by far the fastest time scale in the problem. Under what conditions does a population inversion occur on the laser transition? What should you do to get the best possible inversion? You can do this directly, but here is the fancy-schmancy way. Note that level 3 decays quickly, and the decay term is like a damping term for N3. Thus, we can assume N3 is always in quasiequilibrium with respect to N1 and N2, which evolve comparatively slowly. So we can take dN3/dt \approx 0 to obtain an approximate (‘‘adiabatic’’) expression for N3. Now use this to adiabatically eliminate the N3’s in the other two rate equations. By now you should have a set of effective rate equations for a two-level atom. Finding the steady state of these new equations is a piece of cake. (b) Now that you’re comfortable with the three-level atom, let’s turn it upside down and consider the inverted three-level laser scheme shown here.
3.7 Exercises pump R02 A10o A21o laser transition } Argue qualitatively that in the best possible cases (optimal pumping and decay rates), for the same pumping rate, laser cross section, and atomic number density, the small-signal gain coefficient for the usual scheme is twice the small-signal gain coefficient for the inverted scheme shown here. Recall that
\sigma(\omega)[Ne −Ng], where ‘‘e’’ and ‘‘g’’ refer to the excited and ground levels of the laser transition, respectively. Note that you could go solving a new set of rate equations in steady state, but if you do some thinking, you will realize that you don’t really need to. Just use the analysis of the two-level atom to reason out what the steady-state populations would be in the optimal cases. (c) Give a qualitative argument for why the saturation intensity for the inverted three-level scheme will be twice that of the saturation intensity for the usual three-level scheme. Assume the laser cross sections are the same in both cases.