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Physical Review A

Interference between a fluorescent photon and a classical field: An example of nonclassical interference

C. K. Hong, Z. Y. Ou, L. Mandeļ

Published Apr 1, 1988• 11 citations
Publisher

Abstract

Fluorescent photons emitted from an atom will interfere with a classical field only if the atom is not in a pure excited state. This is a strictly quantum-mechanical condition, which can be tested, in principle, by allowing the light emitted from an atom undergoing Rabi oscillations to interfere with the coherent pumping field. The theory of this process is discussed.

DOI
10.1103/physreva.37.3006

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Physical Review A•Jun 1, 1993•9 citations
Fluorescent photons emitted from an atom will interfere with a classical field only if the atom is not in a pure excited state. This is a strictly quantum-mechanical condition, which can be tested, in principle, by allowing the light emitted from an atom undergoing Rabi oscillations to interfere with the coherent pumping field. The theory of this process is discussed.
It has been predicted that when fluorescent photons emitted from an atom in the process of resonance fluorescence interfere with a portion of the coherent driving field, the visibility of the resulting interference exhibits nonclassical features and is zero at the moment when the atom is fully excited and when the fluorescent intensity is greatest.1 We show theoretically that a similar conclusion applies to interference between the fluorescent photons and the elastically scattered part of the driving field, which can be separated out with narrow band filters in practice. This involves the calculation of certain fourth order, three-time correlation functions that have not previously been studied. A possible application of the theory to an interference experiment will be discussed.
A comprehensive review is given of investigations, both theoretical and experimental, that demonstrate interference to take place even when the fields involved are produced by independent sources. The physical situation is rather different for coherent and incoherent fields. In the first case, where the sources are lasers, a conventional interference pattern can be observed which is adequately described by the classical theory. In the present paper, special attention is paid to the question of whether this interference persists when the two laser beams become strongly attenuated. In the second case, the sources are individual atoms excited by a pumping mechanism, that emit spontaneously and, hence, independently from each other. In those circumstances, no interference pattern can show up. However, it becomes evident from both the classical and the quantum-mechanical theory that interference effects can still be established by observing intensity correlations rather than the intensity itself. This point is discussed in greater detail. The pioneering experiments of Forrester, Gudmundson and Johnson, and Brown and Twiss are reviewed in this context. Especially interesting from the theoretical point of view is the case of two emitting atoms, since then the classical and the quantum-mechanical description differ significantly, the quantum theory predicting the intensity correlations to be distinctly stronger than those following from classical considerations. This specific quantum-mechanical feature is shown to be intimately connected with the corpuscular aspect of light.
We demonstrate the existence of nonclassical correlations in the radiation of two atoms, that are coherently driven by a continuous laser source. The photon-photon correlations of the fluorescence light show a spatial interference pattern not present in a classical treatment. A feature of this phenomenon is that bunched and antibunched light is emitted in different spatial directions. The calculations are performed analytically. It is pointed out that the correlations are induced by state reduction due to the measurement process when the detection of the photons does not distinguish between the atoms. It is interesting to note that the phenomena show up even without any interatomic interaction.
Two-photon excitation of a three-level atom in a ladder configuration $(\stackrel{\ensuremath{\rightarrow}}{1}\stackrel{\ensuremath{\rightarrow}}{2}3)$ by simultaneous illumination with fields in squeezed vacuum and coherent states results in quantum interference for the excitation process. The particular configuration considered here is one for which the signal and idler output fields of a subthreshold nondegenerate optical parametric oscillator are in resonance with the two-stepwise dipole atomic transitions $(\stackrel{\ensuremath{\rightarrow}}{1}2,\stackrel{\ensuremath{\rightarrow}}{2}3),$ while a ``reference oscillator'' field is in two-photon resonance with the quadrupole transition $(\stackrel{\ensuremath{\rightarrow}}{1}3).$ In an extension of the work of Ficek and Drummond [Phys. Rev. A 43, 6247 (1991)], a theoretical formulation based on the full quantum master equation for the problem is presented. The combined effects of quantum interference and the nonclassical character of the squeezed state are investigated, and offer the potential for a new detection strategy for quantum fluctuations of the electromagnetic field with ultrahigh frequencies (10's--100's THz). Based on the theory developed, we analyze quantum interference in excitation in several special cases relevant to experimental realizations, including the effects of a small focusing angle of the squeezing onto the atoms, and unusual population inversions. Special emphasis is given to identifying intrinsically quantum optical field effects versus classical field effects. Procedures that could distinguish between the two (i.e., classical and nonclassical) are suggested.
We investigate quantum interference and classical-interference effects when three-level atoms in the ``$\ensuremath{\Lambda}$'' or ``V'' configuration interact with both a cavity-field mode and an external driving field mode. We describe electromagnetically induced transparency ($\ensuremath{\Lambda}$ configuration), as well as a ``field-induced transparency'' (V configuration), and we go beyond the weak cavity-field studies of previous papers by utilizing a Heisenberg-picture density matrix approach. We find that under certain circumstances the cavity field evolves to be equal in magnitude to, but $180\ifmmode^\circ\else\textdegree\fi{}$ out of phase with, the external pump field when the pump-field frequency equals the cavity frequency, a surprising effect found in two-level atoms in a cavity excited by an external pump field that persists in three-level atoms. The better the cavity, the quicker this buildup occurs. When the cavity field reaches this ``out-of-phase'' condition, the resonance fluorescence from the atom in the cavity goes to zero. This is a purely classical-interference effect between the two out-of-phase fields, with the resonance fluorescence going to zero at the same time as the two excited state populations go to zero. This is quite different from the quantum interference that occurs under the right circumstances when the state populations are coherently driven into a linear combination that is decoupled from any applied field and population is trapped in the excited states (``coherent population trapping''). When the cavity frequency is tuned to the frequency at which quantum interference would occur, the two interference effects compete with each other. The cavity field begins to evolve toward the $180\ifmmode^\circ\else\textdegree\fi{}$ out-of-phase condition, thereby approaching a classical destructive interference condition with the pump field. Then at some point before 100% classical interference occurs, depending on the cavity damping rate and atom/cavity coupling strength, the atom builds up its coherence between state populations to produce the quantum interference effect. At that point, the atom becomes transparent and resonance fluorescence disappears, including fluorescence into the cavity, thereby causing the cavity field to die away.
Some coherence properties of the light emitted from a two-level atom which is excited by a monochromatic coherent field are investigated. It is shown that the elastically scattered light is coherent, and that this compoment of the emitted field can be separated from the total field by narrow-band filtering at the driving frequency. These results are then applied to the problem of interference between the fluorescent and the elastically scattered field. An interference experiment is analyzed in which the detection of one emitted photon signifies the beginning of a Rabi cycle and the detection of a subsequent photon fixes the instant in the Rabi cycle when the interference is detected. It is shown that the visibility of the interference is near zero at the time when the fluorescence is maximum.