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Canadian Journal of Physics

The two-photon laser beam as a breathing mode of the electromagnetic field

DAVID J. ROWE

Published Apr 1, 1978• 8 citations
Publisher

Abstract

It is predicted that the beam from a two-photon laser, if it can be made to operate in a coherent mode, will have properties quite different from the classical properties of the standard single-photon laser. Whereas the coherent beam of the single-photon laser is a dipole mode of the electromagnetic field, that of the two-photon laser is expected to be a monopole or quadrupole mode. It is predicted that the coherence properties of the two-photon laser will show up rather dramatically in non-linear processes.

DOI
10.1139/p78-057

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It is predicted that the beam from a two-photon laser, if it can be made to operate in a coherent mode, will have properties quite different from the classical properties of the standard single-photon laser. Whereas the coherent beam of the single-photon laser is a dipole mode of the electromagnetic field, that of the two-photon laser is expected to be a monopole or quadrupole mode. It is predicted that the coherence properties of the two-photon laser will show up rather dramatically in non-linear processes.
The statistical properties of the emission of a two-photon single-mode laser are theoretically analyzed in terms of quantum electrodynamics, using the kinetic equation for the electromagnetic field. It is found that in contrast to suggestions made by others, there is nothing unique in the properties of two-photon emission. Thus, the relative fluctuations of the numbers of photons in this radiation cannot be smaller than in coherent emission, and therefore two-photon emission in the form considered cannot be used to produce noiseless sources.
A quantum theory of a two-photon laser in a three-level atomic system of the cascade type is presented starting from an original microscopic Hamiltonian. The master equation of the laser operation is derived by using the method of Scully and Lamb [Phys. Rev. A 159, 208 (1967)] for a single-mode laser containing a two-level medium. The photon-number distribution of the laser is studied in detail. The curve of the distribution may have a two-peak structure. One of the peaks located at the origin comes from the single-photon process and reflects the thermal property of the laser, while another comes from the two-photon process and reflects the coherent property of the laser. We cannot find an exact criterion for the threshold of the two-photon laser. The two-photon laser can approximately be considered as a coherent state for certain values of A/C, which can even be much smaller than the threshold for the nonresonant single-mode laser in a two-level system, but it is much larger than the threshold for the resonant one.
Space-time description of pulsed laser radiation by means of coherent states is presented. The corresponding displacement operator contains space dependent annihilation and creation operators, not the "standard" operators corresponding to mode decomposition of the electromagnetic field. This allowed for a direct description of space and time dependence of the electromagnetic field of the pulse. The main characteristics of the pulse is its profile in the wave-vector space. It determines space-time structure of expectation values of the electromagnetic field carried by the pulse, and the distribution of photon modes as well. Determination of the latter required transition from the space-time, or field description to the photon mode approach. Expectation values of the electric field of a Gaussian pulse propagating in the z-direction and linearly polarized in the x-direction are presented graphically in the form of contour plots in the x-z plane. Apart of the dominant x-component electric field has a small longitudinal component in the z-direction. The dominant component of the magnetic field in the y-direction is acompannied by small components in z- and x-directions. The plane wave limit of the pulse can be implemented by shrinking transverse width of the profile to zero (transverse spatial extension goes to infinity), while keeping the photon density constant. One can then retrieve standard plane wave form of the pulse also in the dipole approximation.
The concept of the photon is introduced by discussion of the process of electromagnetic field quantization within a closed cavity or in an open optical system. The nature of a single-photon state is clarified by consideration of its behaviour at an optical beam splitter. The importance of linear superposition or entangled states in the distinctions between quantum-mechanical photon states and classical excitations of the electromagnetic field is emphasized. These concepts and the ideas of wave-particle duality are illustrated by discussions of the effects of single-photon inputs to Brown-Twiss and Mach-Zehnder interferometers. Both the theoretical predictions and the-confirming experimental observations are covered. The defining property of the single photon in terms of its ability to trigger one. and only one, photodetection event is discussed.
When a single quantum of electromagnetic field excitation is added to the same spatiotemporal mode of a coherent state, a new field state is generated that exhibits intermediate properties between those of the two parents. Such a single-photon-added coherent state is obtained by the action of the photon creation operator on a coherent state and can thus be regarded as the result of the most elementary excitation process of a classical light field. Here we present and describe in depth the experimental realization of such states and their complete analysis by means of a novel ultrafast, time-domain, quantum homodyne tomography technique clearly revealing their nonclassical character.
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.