Photon occupation number

To calculate the rates of these transitions it is more convenient to express the radiation field in terms of the number of photons per mode of the radiation field, i.e. the photon occupation number n of each mode. The photon occupation number n is given by4... 

Experiments with Xe analogous to the previously described Xe experiment were carried out at both 90 K and 300 K. Specifically the time dependences of the populations in higher nd and ng states were measured subsequent to population of an nf state by a pulsed laser.19 The time dependences of the observed populations were fit to a model which yielded the radiative transfer rates from the initially excited state to the final states. Not surprisingly, as the temperature was reduced from 300 K to 90 K, not only was the overall radiative transfer reduced, but the transfer to the highest lying states was most sharply reduced, as expected from the dependence of the photon occupation number on frequency and temperature. For example the transfer from the Xe 25f to 26d,g states was reduced by a factor of 3 while the 25f-27d,g transfer was reduced by a factor closer to 5. 

Here q is the photon wavevector and e its polarization. The initial photon occupation number n and the matrix element for vacuum excitation (coupling element) determine the emission rate at the Doppler-shifted frequency 0). The imperative question is what is the appropriate coupling element to be used in Eq. (3) ... 

If, instead of (1.12), the total Hamiltonian for the molecule in the presence of light were H = Hq + the eigenstates of the system would become simple products of molecular and radiation field states T (r, t)>IXrad)> where the radiation field states Xrad> would depend on photon occupation numbers, energies and polarizations. Since no coupling of light with molecular states is implied in such a Hamiltonian, no transitions can occur between molecular states due to absorption or emission of light in this description. The simplest Hamiltonian that can account for spectroscopic transitions is therefore the one in Eq. 1.12. 

For an optical refrigerator, the photon occupation number n in the entropy flux rate (Eq. 16) has contributions from both the fluorescent photons of the refrigerator, rif v), and the ambient thermal photons, a(v). The latter is given... 

In spite of considering two-photon processes, we still find the energy levels (8.17) to be equidistant with respect to the photon occupation numbers n, -I- i. This suggests that it is permissible to neglect the effect of the electron-photon interaction on statistics and to occupy the electron states and the photon states according to Fermi statistics and to Bose statistics. We shall check this question in Section 8.4 by studying the creation and annihilation operators of the resulting quasi-electrons and quasi-photons. 

Let us now check whether the need for independent particles is more appropriately met by considering quasi-particles rather than interacting electrons and photons. We noted in Section 8.2 that the energy levels corresponding to the perturbed states (8.9) are equidistant with respect to the photon occupation numbers n,. This result holds in spite of the fact that the energy terms of order four arise from two-photon exchange interactions, suggesting that at least the quasi-photons represent a system of independent particles. 

We have to sum over all possible distributions of the quasi-electrons among the electron states i and over all quasi-photon occupation numbers n. Using the fact that the energy levels. ..) are... 

The renormalization of the quasi-photon energies entails a renormalization of the quasi-electron energies. The renormalized quasi-electron Hamiltonian results from the total Hamiltonian (8.31) by replacing the photon occupation numbers + i by coth hcoJ2kT). These are just the average occupation numbers given by the Bose distribution (16). 

The interaction (3.27) couples between eigenstates of Hq = Hm + At- Such states are direct products of eigenstates of Hu and of At and may be written as 1/, n ) where the index j (in our model j = 1,2) denotes the molecular state and is the set of photon occupation numbers. From (3.27) we see that Fmr is a sum of terms that couple between states of this kind that differ both in their molecular-state character and in the occupation number of one mode. 

If N2 and Nj represent the populations (or photon occupation numbers) of levels 2 and 1, respectively, under thermodynamic equilibrium, we can write the rate of change in the population of level 2 through spontaneous emission (decay for population which is why the - sign) as ]34]... 

Figure 5. From the point of view of correlations it corresponds to the formation of a correlation involving one photon k and two atomic states which has a finite lifetime (this is by definition the duration of the collision). From the point of view of occupation numbers it corresponds to a change of +1 for two atomic states and of +1 for a photon state.

Remark. It should be clear that this transition to an occupation number description is a purely algebraic step. In this respect it is similar to what in quantum mechanics is denoted by the misleading term second quantization . The only difference is that here we deliberately eliminate the information about the identity of the molecules , whereas in quantum mechanical applications (e.g., to photons or... 

In its most general physical use, occupation number is an integer denoting the number of particles that can occupy a well-defined physical state. For fermions it is 0 or 1, and for bosons it is any integer. This is because only zero or one fermion(s), such as an electron, can be in the state defined by a specified set of quantum numbers, while a boson, such as a photon, is not so constrained (the Pauli exclusion principle applies to fermions, but not to bosons). In chemistry the occupation number of an orbital is, in general, the number of electrons in it. In MO theory this can be fractional. 

At this stage, it is convenient to assume a very low density of excited dipoles. In other words, we assume that the exciting external source is sufficiently weak so that at each instant the probability of finding a given dipole in an excited state is very small compared to 1. In this condition, the system satisfies the linear-response approximation. Since the elementary excitations are very dilute (i.e., the occupation numbers are very small), all statistics are equivalent. For the convenience of further calculations (e.g. interaction with photons), the operators B B are assumed to obey Bose statistics21,22 ... 

The natural line width of the spectral line is a significant result of the dissipative quantum process which accompanies the spontaneous emission of an atom. We will treat this emission process in a dissipative two-state model. We consider the two states of the atom as the zeroth and the first occupation number state of a linearly damped oscillator. In this case, the spontaneous emission of a photon is the consequence of the transition from the first occupations number state to the equilibrium state of the damped oscillator. In this model, the spectrum density of the emitted photon follows from Equation (92)... 

Suppose that initially there was a single excited mode labeled with an index n. Because of the linearity of the process, one may assume that the mean number of photons in this mode was v = 1. Then the mean occupation number of the m-th mode at x > 0 equals... 

Here V denotes the quantization volume, and e 1 is the unit polarization vector for the radiation mode characterized by wavevector k, polarization A and circular frequency co = c k where it appears, an overbar denotes complex conjugation. The polarization vector is considered a complex quantity specifically to admit the possibility of circular or elliptical polarizations. Associated with each mode (k, A) are a Hermitian conjugate pair of photon annihilation and creation operators, and k / , respectively, which operate eigenstates of //raci with m(k, A) photons (m being the mode occupation number) as follows... 

If a photon of frequency v 12 collides with an ion where an electron occupies level 1, that electron may be excited into level 2, with a probability B12- This transition probability (or oscillator strength) is effectively the bound-bound absorption coefficient for a single line, abb(zq.2) = B12 When multiplied by the occupation number of each level 1, and summed over transitions between all levels in all ions, the total monochromatic bound-bound opacity is obtained... 

The basic expression for the quantization of the electromagnetic field is the expansion Eq(54). In the quantized theory the numbers Ck,, C x become operators of the creation C x and the annihilation Ck,x of photons. These operators are acting on the state vector < ) that is defined in the Fock space (occupation number space). The C xt Ck, operators satisfy the commutation relations ... 

Only the valence Compton profiles are needed for the reconstruction of the momentum density and the occupation number density. So one has to subtract an appropriate core Compton profile. Furthermore the contribution of the multiple scattered photons to the measured spectra has to be taken into account (for example by a Monte Carlo simulation [6]). Additionally one has to take heed of the fact that the efficiency of the spectrometer is energy dependent, so the data must be corrected for energy dependent effects which are the absorption in the sample and in the air along the beam path, the vertical acceptance of the spectrometer and the reflectivity of the analyzing crystal. The relativistic derivation of the relationship between the Compton cross section and the Compton profile leads to a further correction factor [7j. Finally a background subtraction and a normalization of the valence profiles to the number of valence... 

The main difference between the quantum theoretical susceptibilities dss( i,j) according to Eqs. (7.68X (7.77) and the respective classical quantities (5.24), (5.25) is the dependence of the former on temperature via the occupation numbers / . /t The retarded dispersion energy between the particles considered, in accordance with the underlying electron-photon-exchange interaction, depends both on Bose statistics and on Fermi statistics. 

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