Compared to classical optical coherence theory based on Maxwell's electromagnetic theory and Glauber's quantum optical coherence theory based on matrix mechanics formulation of quantum mechanics, quantum optical coherence theory based on Feynman's path integral formulation of quantum mechanics provides a novel tool to study optical coherence. It has the advantage of understanding the connection between mathematical calculations and physical interpretations better. Quantum optical coherence theory based on Feynman's path integral is introduced and reviewed in this paper. Based on the results of transient first-order interference of two independent light beams, it is predicted that the classical model for electric field of thermal light introduced by classical optical textbooks may not be accurate. The physics of two-photon bunching of thermal light and Hong-Ou-Mandel dip of entangled photon pairs is the same, which can be interpreted by constructive and destructive two-photon interference, respectively. Quantum optical coherence theory based on Feynman's path integral is helpful to understand the coherence properties of light, which may eventually lead us to the answer of the question: what is a photon?
Richard Feynman’s method of path integrals is based on the fundamental assumption that a system starting at a point 𝐴𝐴 and arriving at a point 𝐵𝐵 takes all possible paths from 𝐴𝐴 to 𝐵𝐵, with each path contributing its own (complex) probability amplitude. The sum of the amplitudes over all these paths then yields the overall probability amplitude that the system starting at 𝐴𝐴 would end up at 𝐵𝐵. We apply Feynman’s method to several optical systems of practical interest and discuss the nuances of the method as well as instances where the predicted outcomes agree or disagree with those of classical optical theory. Examples include the properties of beam-splitters, passage of single photons through Mach-Zehnder and Sagnac interferometers, electric and magnetic dipole scattering, reciprocity, time-reversal symmetry, the optical theorem, the Ewald-Oseen extinction theorem, far field diffraction, and the two-photon interference phenomenon known as the Hong-Ou-Mandel effect.
The nature of light, extending from the optical to the x-ray regime, is reviewed from a diffraction point of view by comparing field-based statistical optics and photon-based quantum optics approaches. The topic is introduced by comparing historical diffraction concepts based on wave interference, Dirac’s notion of photon self-interference, Feynman’s interference of space–time photon probability amplitudes, and Glauber’s formulation of coherence functions based on photon detection. The concepts are elucidated by a review of how the semiclassical combination of the disparate photon and wave concepts have been used to describe light creation, diffraction, and detection. The origin of the fundamental diffraction limit is then discussed in both wave and photon pictures. By use of Feynman’s concept of probability amplitudes associated with independent photons, we show that quantum electrodynamics, the complete theory of light, reduces in lowest order to the conventional wave formalism of diffraction. As an introduction to multi-photon effects, we then review fundamental one- and two-photon experiments and detection schemes, in particular the seminal Hanbury Brown–Twiss experiment. The formal discourse of the paper starts with a treatment of first-order coherence theory. In first order, the statistical optics and quantum optics formulations of coherence are shown to be equivalent. This is elucidated by a discussion of Zernike’s powerful theorem of partial coherence propagation, a cornerstone of statistical optics, followed by its quantum derivation based on the interference of single-photon probability amplitudes. The treatment is then extended to second-order coherence theory, where the equivalence of wave and particle descriptions is shown to break down. This is illustrated by considering two photons whose space–time probability amplitudes are correlated through nonlinear birth processes, resulting in entanglement or cloning. In both cases, the two-photon diffraction patterns are shown to exhibit resolution below the conventional diffraction limit, defined by the one-photon diffraction patterns. The origin of the reduction is shown to arise from the interference of two-photon probability amplitudes. By comparing first- and second-order diffraction, it is shown that the conventional first-order concept of partial coherence with its limits of chaoticity and first-order coherence has the second-order analogue of partial entanglement, with its limits corresponding to two entangled photons (“entangled biphotons”) and two cloned photons (“cloned biphotons”), the latter being second-order coherent. The concept of cloned biphotons is extended to the case of
n cloned photons, resulting in a
1/n reduction of the diffraction limit. In the limit of
nth-order collective state are shown to propagate on particle like trajectories, reproducing the 0th-order ray-optics picture. These results are discussed in terms of the linearity of quantum mechanics and Heisenberg’s space–momentum uncertainty principle. Finally, a general concept of coherence based on photon density is developed that in first order is equivalent to the conventional wave-based picture.
Two-photon interference of broadband chaotic light in a Michelson interferometer is theoretically and numerically studied with a two-photon-absorption detector when polarizations are taken into account. Under the frame of quantum optics, the theoretical analysis is based on two-photon interference and Feynman path integral theory. A two-photon coherence matrix is introduced to calculate the second-order interference pattern. Our research shows that polarization is another dimension, as well as time and space, to tune the second-order interference pattern in the two-photon interference process. Polarizations can act as switches to manipulate the interference process and may open the door to some new experimental schemes. The quantum optical vector model is also helpful to understand the physics of second-order interference when the light source is quantum.
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The second-order temporal interference of classical and nonclassical light at an asymmetrical beam splitter is discussed based on two-photon interference in Feynman's path integral theory. The visibility of the second-order interference pattern is determined by the properties of the superposed light beams, the ratio between the intensities of these two light beams, and the reflectivity of the asymmetrical beam splitter. Some requirements about the asymmetrical beam splitter have to be satisfied in order to ensure that the visibility of the second-order interference pattern of nonclassical light beams exceeds the classical limit. The visibility of the second-order interference pattern of photons emitted by two independent single-photon sources is independent of the ratio between the intensities. These conclusions are important for the researches and applications in quantum optics and quantum information when an asymmetrical beam splitter is employed.
The spatial second-order interference of two independent pseudothermal light beams in a Hong-Ou-Mandel interferometer is studied experimentally and theoretically. The similar cosine modulation in the second-order coherence function as the one with entangled-photon pairs in a Hong-Ou-Mandel interferometer is observed. Two-photon interference based on Feynman's path integral theory is employed to interpret the results. The experimental results and theoretical simulations agree with each other very well.