Photon operators

The vacuum expectation value of an odd number of (Heisenberg) photon operators vanishes. 

However, because of the correlated motion of the electrons, many-electron processes will also occur. (Looking at the many-particle effects in this way, the photon operator is a single-particle operator and electron-electron interactions have to be incorporated explicitly into the wavefunction. It is, however, also possible to describe the combined action of the electrons as an induced field which adds to the external field of the photoprocess, i.e., the transition operator becomes modified. Generally, the influence of the electron-electron interaction can be represented by modifying the wavefunction or the operator or by modifying both the wavefunction and the operator [DLe55, CWe87].) Of all the possible processes, only the important two-electron processes restricted to electron emission will be considered here. In many cases they can be divided into two different classes (see Fig. 1.3) t... 

Renaming the electron numbers shows that the first and fourth terms in the matrix element M i(M1,ms) and also the second and third terms are identical, and they can be combined. As a next step one calculates the action of the photon operator on the single-particle wavefunctions. Omitting for simplicity the wavefunction... 

The single-electron matrix elements for the passive and the active electrons contain different projections of the electron spin. Since neither the photon operator nor the unity operator (in the overlap matrix element) acts on the spin, the quantum numbers M, and ms are fixed by the corresponding spin of the formerly bound ls-electron. This yields... 

It is also possible to derive this result by incorporating the condition (2.5) below from the beginning by using the factorization of a two-electron function into a symmetrical spatial function and an antisymmetrical spin function, see equ. (1.16).) The expression in the braces indicates that the two electrons in the final state have opposite spins, i.e., the photoprocess reaches a singlet final state. This can be easily understood, because in LS-coupling spin-orbit effects are absent, and the photon operator does not act on the spin. Therefore, the selection rule... 

This result reflects the fact that the photon operator, as a one-particle operator, interacts with the active ls-electron only, ejecting it into a wave characterized asymptotically by Ka or, alternatively, Kb, while the passive ls-electron leads to... 

Finally, it should be noted that treating one of the otherwise equivalent electrons in equ. (7.46) individually is frequently used for calculating matrix elements with a one-electron operator (e.g., the photon operator) acting on equivalent electrons. Similarly, if two-electron operators play a role, like in the Coulomb interaction between electrons, then it is convenient to separate two electrons from the equivalent electrons. This is done using the coefficients of fractional grandparentage (for more details see [Cow81]). 

The dipole matrix element on the right-hand side of equ. (8.19b) is called the length form of the matrix element, because the vector r acts as the photon operator (see the discussion of equ. (1.28a) in which the name dipole approximation is also explained). Equ. (8.16) can then be replaced by... 

The Is and 2s orbitals which are affected by neither the photoionization nor the Auger process are omitted for simplicity.) If these wavefunctions are constructed from single-electron orbitals of a common basis set (the frozen atomic structure approximation), the photon operator as a one-particle operator allows a change of only one orbital. Hence, the photon operator induces the change 2p to r in these matrix elements ... 

In the dipole approximation which happens to be accurate enough for our purposes, an effective two-photon operator Q (E) acting on the states a), 6) and c) is defined as... 

Here, at is the Dirac a-matrix, A(r, t) is the vector potential of the field (divA = 0), and Ho is the Hamiltonian of an isolated atom whose energy in the initial state of the reaction is denoted by Ef. H0 i) = E i). Equation (5) defines the effective two-photon operator Q(2 ui,u> ) which has the dimension of L3 and is a straightforward relativistic generalization of its nonrelativistic counterpart that has been first introduced in [28] to describe the process of two-photon absorption. Generally, the matrix elements of u>,u> ) can be expressed explicitly as... 

Some conclusions about the role of quantum noise in the long-time behavior of the solutions for the SHG process can be drawn by closer inspection of the operator equations of motion for the number of photon operators and their approximate solutions for the expectation values [38,48], From the equations of motion (56) it is easy to derive the equations for the number of photons operators Na = b+a and Nb = b1 b in the form... 

A way to overcome the difficulties in the definition of the Hermitian phase operator has been proposed by Pegg and Barnett [40,45]. Their method is based on a contraction of the infinite-dimensional Hilbert-Fock space of photon states Within this method, the quantum phase variable is determined first in a finite 5-dimensional subspace of //, where the polar decomposition is allowed. The formal limit, v oc is taken only after the averages of the operators, describing the physical quantities, have been calculated. Let us stress that any restriction of dimension of the Hilbert-Fock space of photons is equivalent to an effective violation of the algebraic properties of the photon operators and therefore can lead to an inadequate picture of quantum fluctuations [46]. 

The canonical quantization of the field has introduced by Dirac [1] (see also Refs. 2-4,10,11,14,15,26,27) is provided by the substitution of the photon operators, forming a representation of the Weyl-Heisenberg algebra, into the... 

We now turn to the construction of the dual representation of the photon operators, providing the field counterpart of the SU(2) phase representation of the atomic variables. It is easily seen that the atomic exponential of the SU(2) phase operator (41) takes [in the representation of dual states (46)] the following diagonal form... 

The dual representation of the photon operators (67) reflects the transmission of phase information from the atomic transition to the radiation field via the integral of motion (70). This statement can be illustrated with the aid of the Jaynes-Cummings model (34). Employing the atomic phase states (46), we can introduce the dual representation of the atomic operators (35) as follows ... 

We now note that the operators (69) and (84) introducing the radiation phase are defined in terms of bilinear forms in the photon operators. At first glance, such a definition runs counter to the original idea by Dirac to determine the Hermitian quantum phase via linear forms in the photon operators [1] (see also Refs. 38, 42, and 44). Leaving aside Dirac s problem of existence of a Hermitian quantum phase variable of a harmonic oscillator, we should emphasize that the use of bilinear forms seems to be quite reasonable from the physical point of view. It can be argued in the following way ... 

The phase information is transmitted from the quantum source (atom) to photons via the conservation laws. In fact, only three physical quantities are conserved in the process of radiation energy, linear momentum, and angular momentum [26]. All of them are represented by the bilinear forms in the photon operators. 

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