Theory Of Radiation And Absorption Of Deformed Field Quanta

We calculate the intensity of spontaneous radiation of a system of non-linear quantum field, where the non-linearity is due to deformations of the Poison brackets of the generalized coordinates and momenta.


Introduction
Starting from the seminal works by [1] and [2], quantum systems with deformed Poison brackets have attracted much attention in various fields of theoretical physics. Of a special interest is the so called space with the "minimal length," which means that the deformation of the Poison bracket is quadratic in momentum for both coordinates and momenta. This has first been introduces in Refs. [3,4]. Such a deformation leads to a non-zero minimal root-mean-square value of the coordinate. There is an increasing number of papers on the subject, and the literature review would require a separate article.
The deformation with the minimal length can in principle be extended to an arbitrary space of generalized coordinates and momenta which are natural for the description of various models. In particular, one can study the electromagnetic field represented as a set of oscillators with the deformed commutator relations for coordinates and momenta, which generates the minimal length. Obviously, in this case we have the deformation of the field itself rather then the deformation of the real space. This model of the electromagnetic field has been considered in Refs. [5,6]. The Casimir effect in such a deformed field has been studied in Ref. [7], where the Casimir energy was calculated for the one-, two, and three-dimensional cases. It has been shown that the deformation suppresses the interaction between the confining boundaries.
In the present work, we study the interaction of the deformed electromagnetic field with atomic systems in the coordinate undeformed space. We shall not discuss 1 This article has been written for the special issue of the Condensed Matter Physics dedicated to the famous Ukrainian theoretical physicist Professor Igor Stasyuk. Professor Stasyuk exemplifies a highly erudite, passionate physicist who never leaves any un-clarified aspects in physical and mathematical mechanisms of models he proposes for the explanation of physical phenomena. I greatly appreciate scientific (and beyond scientific) discussions during the many years of our acquaintance which started when we were students. These numerous discussions have always been for me warm intellectual "festivals", and will always stay as such in my heart.
the properties of the field equations (which are non-linear) or problems related to their Lorentz and gauge invariance which will be considered in a separate paper, we shall rather propose a model of interaction of such fields with atomic systems. We shall also calculate the intensity of spontaneous radiation as a function of the deformation parameter.

Basic equations
We study the interaction of an atomic system with the electromagnetic field, which Hamiltonian after the decomposition into a set of oscillators is given by: where k is the wave-vector, α denotes the polarization, ω k = ck is the frequency (where c is the speed of light in the vacuum).Q k,α ,P k,α are the generalized coordinate and momenta operators with the deformed Poison brackets: where β ≥ 0 is the deformation parameter, and all remaining commutators are equal zero. As it is well-known, such commutation relations lead to the "minimal length" in the coordinate space, Q 2 k,α =h √ β [3,4]. Because in this case we do not deform the ordinary "physical" space (in which our field lives) but rather the commutation relations of the dynamical variables (fields), we have the deformation of the field itself rather then the space-deformation.
Let us introduce new operatorsq k,α andp k,α as follows: It is easy to show that they are canonically conjugate operators: The Hamiltonian (2.1) becomeŝ . (2.5) The equation of motion of such a field have been studied in Ref. [5,6], where the Hamiltonian was presented as an expansion in ordinary annihilation and creation operators. The field equations are non-linear but can be treated perturbatively.
As it is usual for oscillatory systems, we present the magnetic vector potential A as follows In terms of the new, canonically conjugate operators (2.3) we have: As usual, the operator of the interaction of the field with an atom is: wherep is the momentum operator (of the electron in an atom), e = −|e| is the electron charge and m the mass. Therefore, we propose the following model of the interaction of the deformed field with an atomic system: we assume that all the expressions of the ordinary theory of the electromagnetic field are valid, except that there are deformed commutation relations for the operatorsQ k,α andP k,α . Such a non-linear field is a simple model which allows us to study the effects of the deformation of the Poison brackets on the properties of physical systems.

Spontaneous radiation
Let us assume that an atom is in the excited state |2 (with the energy E 2 ) while the field is in the ground state | . . . , 0, . . . . As a result of the interaction of the atom with the field, the atom jumps to a level with the lower energy E 1 and radiates a light quantum with the energyhω = E 2 − E 1 . The field, therefore, jumps to the state with one phonon, i.e., | . . . , 0, N k,α = 1, 0 . . . . The transition probability rate of a system "field-plus-atom" to jump from the initial state |i = |2 | . . . , 0, . . . to the final state |f = |1 | . . . , 0, N k,α , 0, . . . is given by: where the energy of the field quantum, according to Eq. (3.23), is We take into account the first term of the interaction operator (2.8), which is linear in the vector potential.
The intensity of spontaneous radiation I k,α is defined in a usual way as the amount of energy with the given polarization α radiated by an atom in unit time on the resonance frequency ω = (E 2 − E 1 )/h per space angle, i.e., (4.28) By inserting Eqs. (4.24)-(4.27) into Eq. (4.28) we obtain, after integration: Taking into account Eq. (4.25), we finally get, after simple transformations, and the dimensionless deformation parameter is In the limiting case of no deformation (β = 0) we have g(0) = 1 (see Eq. (4.29)), and therefore one gets for the intensity I k,α = e 2 ω 2 2πm 2 c 3 |p 12 | 2 , p 12 = 1|e ikr (e k,αp )|2 , (4.32) which is the well-known expression for the intensity of radiation for the non-deformed field.
In the opposite limit of large deformation parameter (ω ≫ 1), the term ∼ sin(kr) vanishes (see Eq. (4.29)), and hence the quadrupole radiation vanishes too. The reason is that the sin(kr) term is the main contribution to the quadrupole radiation in the long wavelength limit. Finally we note that the function g(ω) in the case of considerable deformations (ᾱ = 1) has the following asymptotic form: (4.33)

Intensity of the dipole radiation
Within the dipole-transition approximation, the wave-vector k → 0 (cos(kr) → 1 and sin(kr) → 0), hence p c 12 = (e k,α p 12 ) and p s 12 = 0 (see Eq. (4.26)). We therefore get: (5.35) We therefor conclude that with increasing of the deormation parameter the intensity decreases as β −1/2 . The intensity of the dipole interaction as a function of the dimensionless deformation parameterω is shown in Fig. 1

Conclusion
We have considered a theory of spontaneous radiation taking into account the creation of one photon. Because the field is non-linear, the transitions between any two states are allowed, not only between the two neighboring, as in the case of the ordinary oscillator. In order to study such processes in the case of more then one photon, but in the linear approximation (in the vector potential A in the interaction operator (2.8)) and neglecting all the higher order terms in an expression for the quantum transition rates, we take the following matrix elements instead of (4.27): It is obviously that these integrals are not equal zero only at even n + n ′ .
In general case the integrals in Eq. (6.36) cannot be expressed in terms of elementary functions. For some particular values of n and n ′ , however, the integrals can be easily evaluated, as for instance for n = 1 and n ′ = 0 (see Eq. (4.27)). Evidently, many-photon processes are absent for β = 0 while their role increases with increasing the deformation parameter β.