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Physical Review A

Quantum Electrodynamics of Intense Photon Beams

Iwo Białynicki‐Birula, Zofia Białynicka-Birula

Published Dec 1, 1973• 29 citations
Publisher

Abstract

A new formula for induced transition amplitudes in intense photon beams is derived which relates the photon-number description to the coherent-field description. According to this formula transition amplitudes describing processes in which $m$ photons are added to the intense one-mode beam or subtracted from that beam are equal to the Fourier coefficients of the corresponding amplitude in the external field, expanded with respect to the phase of the classical field. The same formula is used to study the expectation values of dynamical variables in $n$-photon states leading to the conclusion that for large $n$ such states are very much like classical statistical states with a given amplitude and evenly distributed phase.

DOI
10.1103/physreva.8.3146

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A new formula for induced transition amplitudes in intense photon beams is derived which relates the photon-number description to the coherent-field description. According to this formula transition amplitudes describing processes in which mmm photons are added to the intense one-mode beam or subtracted from that beam are equal to the Fourier coefficients of the corresponding amplitude in the external field, expanded with respect to the phase of the classical field. The same formula is used to study the expectation values of dynamical variables in nnn-photon states leading to the conclusion that for large nnn such states are very much like classical statistical states with a given amplitude and evenly distributed phase.
Photons and electrons One of the great technical achievements of the second half of the 20th century was the invention of the laser. With its help extremely intense light could be produced. Furthermore, it even became posssible to excite a single resonator mode so that both the frequency and the propagation direction of the emitted radiation field are extraordinarily sharp. The mere existence of such high-intensity single-mode fields – the real problem that was ingeniously solved was to generate them – is no matter of surprise. We know from classical electromagnetic theory that the amplitude of a monochromatic electromagnetic field can take on arbitrarily large values. This is a direct consequence of the linearity of Maxwell's equations. What does this mean for quantized radiation fields? Certainly, the mentioned property of classical fields must also be retained in the quantum mechanical description. Since the electromagnetic energy is quantized in ‘packets’ of magnitude h ν (photons), the number of photons in a given mode must be unlimited, in principle. All this is very well known. However, it becomes noteworthy when we compare the behaviour of photons with that of electrons. Amazingly, it turns out that an electronic quantum state can be occupied by one electron, at maximum. (Strictly speaking, in the description of the electronic state the spin variable must be included.) This drastic difference in the collective behaviour of photons and electrons could, in fact, not be expected from what we know from single photons and electrons. Both of them have particle-like as well as wave-like properties. However, when many of them are present, they exhibit quite different a demeanour.
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By imposing the condition of non-vanishing expectation values for the amplitude and phase of field operators, pure quantum states are identified composed of two consecutive number states. These pure states also deliver noise-free radiation modes restricting the “half-photon noise” to the expectation value of the lowest level of dynamic and coherent number states. As a result, instantaneous phase-sensitive amplification of photons is easily controlled and direct evaluation of time or distance - varying wavefront distributions of photons and phases can be carried out for sub-Poissonian distributions of photons without the need for quasi-probabilities.
In a concise, but accessible for the first acquaintance form the procedure for the quantization of linear oscillator is set out. By analogy with this procedure the procedure of quantization (second quantization) of classical Maxwell’s electrodynamics is set up. The physical sense of the wave functions arguments of transverse electromagnetic field and its Fourier transformation is set up. One pay attention as for quantum coherent (almost a classic) states of the electromagnetic field and for photonics Fock states. Attention is drawn to the fact of absence the power of the universal content of such concepts as field amplitude, phase and number of particles (photons), which are used by experimenter’s to describe the states of a quantized field. The semi quantitative description the interaction processes of a quantum electromagnetic field with substance is set up. Specified situations are shown in which the discrepancy between the predictions of classical and quantum electrodynamics is noticeable at the macroscopic level.
A general approximation method is developed for the treatment of bound quantum systems in the presence of an intense external electromagnetic field of long wavelength. The method is based upon a unitary transformation which approximately removes the electromagnetic field from the problem. The accuracy of the technique increases with the number of photons involved in any given process; and in the case of large photon number, the results have a simple, concise form resembling first-order perturbation theory. It is possible to handle intense-field problems with small numbers of photons also, but then the formalism takes on a complexity analogous to the corresponding perturbation expressions for the same number of photons. The theory is extended to the case where two electromagnetic field modes are present, one or both of which may be intense. Bound-bound transitions in a hydrogenlike atom are calculated as an illustration of the application of the method. One of the explicit nonlinear effects to appear is an intensity-dependent deviation of the transition probability behavior from In{I}^{n}In for nnn-photon transitions, where III is the field intensity. It is demonstrated that intensity effects are much more probable with bound electrons than with free electrons.
The Hill method is used in an analysis of the Schrodinger equation for a quatum system subjected to a time-periodic nonmonochromatic field. An algebraic equation is obtained and its degree in respect of exp(iET), where E is the quasienergy and T is the period of the external field, is equal to the number of levels. The coefficients of this equation are expanded in powers of the intensity parameter into a series with a factorial convergence even when one of the field frequencies is in exact resonance with one of the transitions. The Fourier components of the density matrix are expressed in terms of quasienergy considered as a function of the parameters of the problem and this gives approximate expressions outside the perturbation theory framework. Various results are obtained for a two-level system and, in particular, a formula is derived for the description of the natural lifetime of a quasienergy state.