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keyboarder
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"Quantum Optics," by Scully and Zubairy, attempts to explain the natural line width of an emitting atomic ensemble (in a resonant cavity with leakage) using random spontaneous emission events of the atoms (which in turn is due to interactions with the vacuum field). The basic idea and corresponding mathematics describe a single resonant frequency with random phase shifts, where the phase shifts are due to the randomness of the spontaneous emission events. The net effect of the random phase shifts result in the line width--the spread of the spectrum from a "pure", single mode emission. The resonant cavity is important because it is supposed to constrain the field to a single mode/frequency, but due to the leakage at one side, the interaction with a reservoir of vacuum modes leads to the broadening.
The given mathematics start with a single-mode description (e.g. P-representation), which time evolution is described by some phase-dispersion equation, but ultimately ends in a description of the same mode. By definition, a particular mode is exactly that: A single, particular frequency in a particular direction with a particular polarization. Although the distribution of numbers states (or, likewise, coherent states) within that mode may change in time, the entire calculation and the result are still in the same mode--of the exact same single frequency.
The whole point of the discussion in the book was to justify and explain the natural line width in a pure quantum mechanical picture. Although discrete eigenstates are defined, no mechanism is outlined which transforms one mode into another. How can mathematical manipulation of the same mode really explain the evolution of the line width, which must include many modes?
For a pure coherent state, one can show mathematically that the classic expectation value of the field results in radiation of the same frequency as the mode of the coherent state (and its component number states). It's therefore easy to assume that the mode of the quantum mechanical states always translate directly to the same mode (frequency) of the classical field. But is this always true? Although the derivation I speak of is all done within a single mode of the field states, can the observed classical frequency of the field be different? If the answer is "no, the modes should be the same", then I think the quantum explanation fails to explain the line width. If "yes", then what is the real meaning of a "mode" and what is the relationship between the quantum mode and the classical mode/frequency?
(Classically, the explanation using a phase-dispersion equation seems reasonable. In that case, the generation of different modes, aside from the central resonant mode, is not constrained by discrete eigenstates of the field, so that the field can be described by a time and spatial phase dependence which leads to the broadened modes. Also, the Fourier-transform of a classical signal of a "single frequency" with random phase shifts results in a broadening of the spectrum--the many random discontinuities destroy the pure monochromaticity of the overall spectrum.)
The given mathematics start with a single-mode description (e.g. P-representation), which time evolution is described by some phase-dispersion equation, but ultimately ends in a description of the same mode. By definition, a particular mode is exactly that: A single, particular frequency in a particular direction with a particular polarization. Although the distribution of numbers states (or, likewise, coherent states) within that mode may change in time, the entire calculation and the result are still in the same mode--of the exact same single frequency.
The whole point of the discussion in the book was to justify and explain the natural line width in a pure quantum mechanical picture. Although discrete eigenstates are defined, no mechanism is outlined which transforms one mode into another. How can mathematical manipulation of the same mode really explain the evolution of the line width, which must include many modes?
For a pure coherent state, one can show mathematically that the classic expectation value of the field results in radiation of the same frequency as the mode of the coherent state (and its component number states). It's therefore easy to assume that the mode of the quantum mechanical states always translate directly to the same mode (frequency) of the classical field. But is this always true? Although the derivation I speak of is all done within a single mode of the field states, can the observed classical frequency of the field be different? If the answer is "no, the modes should be the same", then I think the quantum explanation fails to explain the line width. If "yes", then what is the real meaning of a "mode" and what is the relationship between the quantum mode and the classical mode/frequency?
(Classically, the explanation using a phase-dispersion equation seems reasonable. In that case, the generation of different modes, aside from the central resonant mode, is not constrained by discrete eigenstates of the field, so that the field can be described by a time and spatial phase dependence which leads to the broadened modes. Also, the Fourier-transform of a classical signal of a "single frequency" with random phase shifts results in a broadening of the spectrum--the many random discontinuities destroy the pure monochromaticity of the overall spectrum.)