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CRC Press eBooks

Interferometry via the Dirac Notation

F.J. Duarte

Published Oct 29, 2013• 0 citations
Publisher

Abstract

The genesis of quantum optics can be found in the Dirac discussion on interference as disclosed in The Principles of Quantum Mechanics . This is a masterful and prophetic discussion that begins by considering a roughly monochromatic light source. The discussion continues by considering a beam of light consisting of a large number of photons. Besides Dirac, useful references on interferometry include Steel, Meaburn and Born and Wolf. The Hanbury Brown–Twiss effect originates in interferometric measurements performed by an “intensity interferometer” used for astronomical observations. A diagram of the stellar intensity interferometer used to determine the diameter of stars is depicted. Two-beam interferometers are optical devices that divide and then recombine a light beam. It is on recombination of the beams that interference occurs. The most well-known two-beam interferometers are the Sagnac interferometer, the Mach–Zehnder interferometer, and the Michelson interferometer.

DOI
10.1201/b16055-10

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The genesis of quantum optics can be found in the Dirac discussion on interference as disclosed in The Principles of Quantum Mechanics . This is a masterful and prophetic discussion that begins by considering a roughly monochromatic light source. The discussion continues by considering a beam of light consisting of a large number of photons. Besides Dirac, useful references on interferometry include Steel, Meaburn and Born and Wolf. The Hanbury Brown–Twiss effect originates in interferometric measurements performed by an “intensity interferometer” used for astronomical observations. A diagram of the stellar intensity interferometer used to determine the diameter of stars is depicted. Two-beam interferometers are optical devices that divide and then recombine a light beam. It is on recombination of the beams that interference occurs. The most well-known two-beam interferometers are the Sagnac interferometer, the Mach–Zehnder interferometer, and the Michelson interferometer.
Optical interferometry was at the heart of the revolution which ushered in the new era of twentieth century physics. For example, the Michelson interferometer was used to show that there is no detectable motion relative to the ‘ether’; a key experiment in support of special relativity. It is a wonderful tribute to Michelson that the same interferometer concept is central to the gravity-wave detectors which promise to provide new insights into general relativity and astrophysics in the twenty-first century. Similar tales can be told about the Sagnac and Mach–Zehnder interferometers as discussed in this chapter. We further note that the intensity correlation stellar interferometer of Hanbury- Brown and Twiss was a driving force in ushering in the modern era of quantum optics. We are thus motivated to develop the theory of field (amplitude) and photon (intensity) correlation interferometry. In doing so we will find that the subject provides us with an exquisite probe of the micro and macrocosmos, i.e., quantum mechanics and general relativity. With these thoughts in mind we here develop a framework to study the quantum statistical correlations of light. We will motivate the quantum correlation functions of the field operators from the standpoint of photodetection theory. Many experimentally observed quantities, such as photoelectron statistics and the spectral distribution of the field, can be related to the appropriate field correlation functions. These correlation functions are essential in the description of Young's double-slit experiment and the notion of the power spectrum of light.
The photon in the interferometer This chapter introduces many of the ideas of quantum theory by exploring three specific “case studies” of quantum systems. Each is an example of a qubit , a generic name for the simplest type of quantum system. The concepts we develop will be incorporated into a rigorous mathematical framework in the next chapter. Our business here is to provide some intuition about why that mathematical framework is reasonable and appropriate for dealing with the quantum facts of life. Interferometers In Section 1.2 we discussed the two-slit interference experiment with a single photon. In that experiment, the partial waves of probability amplitude were spread throughout the entire region of space beyond the two slits. It is much easier to analyze the situation in an interferometer , an optical apparatus in which the light is restricted to a finite number of discrete beams . The beams may be guided from one point to another, split apart or recombined as needed, and when two beams are recombined into one, the result may show interference effects. At the end of the interferometer, one or more sensors can measure the intensity of various beams. (A beam is just a possible path for the light, so there is nothing paradoxical in talking about a beam of zero intensity.) Figure 2.1 shows the layout of a Mach–Zehnder interferometer , which is an example of this kind of apparatus.
Quantum optics was born in the first years of the 20th century. The first papers leading to further ideas of quanta of light were presented by Max Planck and Albert Einstein. Their pioneering papers laid the foundations for the theory of quantum optics. However, one should remember that for the birth of the formal theory of quantisation of light, the development of quantum mechanics was necessary. Interestingly, the word ’photon’ appeared for the first time in a paper by a Gilbert Lewis, a chemist, in 1926 and was later used by P.A.M. Dirac, a physicist, the next year. The modern sense of the term quantum optics was established in 1956 when Hanbury Brown and Twiss built the intensity interferometr and performed their famous experiment in which correlations between two light beams were measured. Although the results of their experiment could be explained on the basis of the classical theory of light, with some quantumcomponent to the process of photodetection, their result is commonly accepted as the beginning of the modern quantum optics era. The next milestone in the development of quantum optics was the invention of laser in 1960. The physical properties of the light generated by lasers were considerably different from those characterising the light generated by classical thermal sources. This fact was a great impulse leading to further development of quantum optics’ ideas. The first of them were presented in the fundamental papers written by Glauber, describing new states of light quantum coherent states.
Two-photon interferometry started to play an important role in quantum optics since 1980’s. In that time, Dirac was criticized to be mistaken because he stated in his book, The Principles of Quantum Mechanics , that “photon only interferes with itself. Interference between two different photons never occurs”. Two-photon interference was considered as the interference between two photons. In fact, neither classical theory nor quantum theory suggested the interference between different photons. Classical theory views optical interference the result of coherent superposition between electromagnetic waves. The two-photon interference picture provided reasonable interpretation to that observation too: under the condition of complete overlapping, when the two biphoton amplitudes superpose in phase the joint-detection of the two distant photodetectors reaches its maximum counting rate as the sign of “correlation”. When the optical path difference is increased from zero to unbalanced values, a coincidence curve of “dip” is observed.
Hanbury-Brown and Twiss (HBT) effect is the foundation for stellar intensity interferometry. However, it is a phase insensitive two-photon interference effect. Here we extend the HBT interferometer by mixing intensity-matched reference fields with the input fields before intensity correlation measurement. With the freely available coherent state serving as the reference field, we experimentally demonstrate the phase sensitive two-photon interference effect when the input fields are thermal fields in either continuous wave or non-stationary pulsed wave and measure the complete complex second-order coherence function of the input fields without bringing them together from separate locations. Moreover, we discuss how to improve the signal level by using the more realistic continuous wave broadband anti-bunched light fields as the reference field. Our investigations pave the way for developing new technology of remote sensing and interferometric imaging with applications in long baseline high-resolution astronomy.