"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.)
Do they define it that way or do you assume that? A single mode is just that: something with a unimodal distribution. You do not have to express that in a monochromatic plane wave basis. Single mode thermal light can be tens of nanometers broad spectrally and is nevertheless single mode, if it follows just a single mode probability distribution. "All fields possessing first-order coherence, we show, may be regarded as ones in which only a single mode is excited; this mode need not, however, be a monochromatic one." (U. M. Titulaer and R. J. Glauber, "Density Operators for Coherent Fields", Phys. Rev. 145, 1041–1050 (1966)).
Thank you for the great reference. It gave me a new perspective on the problem, but my underlying concern with the derivation of the natural line width remains. Let's consider a new set of orthogonal modes (as described by Titulaer and Glauber), which can also be used to define a coherent state, a state that keeps its "shape" as it propagates, etc. These new modes can be a superposition of multiple modes from the previous basis. That means the new coherent state is not monochromatic in terms of the original modes. As Titulaer and Glauber did, we can use L to enumerate the new modes, where k still enumerates the original modes. I can thus imagine that the derivation of the natural line width could be done within context of a new single mode L. If the final result of the derivation is still within the same mode (as was part of my concern), one could transform the resultant state back into the original basis where one would find a broadened (non-monochromatic) spectrum. Perhaps that sounds reasonable on the surface, but it would be deceptive. If the chosen L basis in which the derivation is performed is already a combination of multiple spectral modes (k basis), then such a chosen basis would already impose non-monochromaticity on the field--it's spectrum in our desired basis would of course have some spectral shape and width to begin with. Further manipulations within a single mode L would still result in the same combination of the multiple spectral modes (k's). There can be no new spectral information obtained in this way. So back to the original concern, that the derivation in question shows no mechanism of creating new modes from the original mode, no matter the basis.
They define it that why mathematically. In a full mathematical description of the entire field, it is the sum over all k-vectors (including direction and magnitude) and over the polarization (which is defined by a set of mutually orthogonal vectors which are also orthogonal to a particular k-vector). One mode is therefore specified by a single k-vector and polarization. In the case of the line-width, which represents a spread of energies and/or frequencies, one can perhaps effectively ignore the directionality and polarization of the field. But it's not possible to describe a wave with multiple frequency components with a single frequency. (By frequency, I also mean wave number, k, related by [itex]\nu[/itex]=kc.) Strictly speaking, for the discussion of the line width where we ignore the direction of the radiation, it doesn't need to be a plane wave. It could be a combination of other basis waves. However, to represent a "single mode" --a single frequency-- in the strictest sense, the set of basis waves (whether planar or speherical, etc.) must be of the same frequency. In that case, it simplifies the problem then to just consider a single direction, or in other words a plane wave. Your use of "single mode" here must be describing a practical, real world field generated by an ensemble of real atoms. A "single mode" distribution in this sense will indeed have a certain width spectrally because of how the field is generated. That's exactly why they call it the "natural line width", because it can supposedly be described by the interaction of the atomic systems with the vacuum modes. But the point of the quantum optics derivation attempts to explain the origin of the line width. It can't be explained just by calling a narrow spectral peak "single mode". When I read "single mode probability distribution", I interpret that to be a delta function, defined exactly at one mode. A sharply peaked spectrum (i.e. probability distribution) with some width is only "single mode" in that it might not be possible to get narrower because of the "natural line width". We just come back to the question of "where does the natural line width come from".
I still do not get why you want to identify a single mode with a single frequency. It is typically not in quantum optics and it rarely makes sense to do that. Personally, I think it is even very unfortunate that the single frequency basis is often called "mode" because it leads to erroneous implications. Why a delta function? As correctly stated by Glauber, all you need is first-order coherence. Or equivalently all photons of interest need to be indistinguishable. There is absolutely no need for monochrmaticity to arrive at a single mode. Well, all that really matters is that the photons are indistinguishable despite the large spectral width. Well, you change the weights of the initial spectral "modes" and can obviously also gain finite contributions from spectral "modes" which had vanishing contribution initially. I do not really see the problem here. Unfortunately I do not have the Scully/Zubairy here. Do you have the exact chapter or page? Maybe I see the exact problem when reading the text again. Correct. This is why it is kind of odd to really call these "modes". I know a lot of people do that, but I think doing so is unfortunate at best.
Thanks again Cthugha for your helpful comments. Part of my intention is to weed out my own incorrect assumption, so I really appreciate your responses. I identify a single mode with a single frequency because of how I defined a "mode" previously. As I already wrote a couple times, ignoring the direction and polarization, consider the simple relationship in vacuum [itex]\nu = kc[/itex], with c the speed of light, [itex]\nu[/itex] the frequency, k defines the mode. I think I defined very clearly what I understand a mode to be. Since you feel it so unfortunate that I and others have this notion, would you please give your precise definition of a "mode"? I realize that the dispersion relation of the wave may change so that [itex]\nu = kc[/itex] doesn't hold in all systems. I also see now from Glauber that a new mode basis can be defined where a mode is no longer specified by a single k vector. I already conceded in my reply that they don't have to be related one-to-one, but I think choosing to stay in the basis where they are directly related is reasonable. Also, the book I'm following often uses the frequency in defining the field and operators and implies the relationship between mode and frequency that I use, especially when describing classical and semi-classical solutions. I take that back since it's not clear what we agree is a "mode"... and what sort of probability distribution you mean. This is getting to the heart of the problem. By specifying a particular mode, one is therefore specifying "weights of the initial spectral 'modes'". But your "obvious" assertion is exactly the opposite of what I'm trying to explain... namely, that the derivations I'm concerned with indicate in no way that the weights of the component spectral modes are changing... no initially vanishing contribution is evolving into a finite contribution. They are defined for a single mode only, so the weights are "set" and don't change. Specifically, I'm looking at section 11.4 (page 341) of "Quantum Optics". If you do read it, be aware that 11.4.1 uses a classical approach, which is apparent in the form of the field in Eq. (11.4.1). It might be a bit deceiving, since the notation might look like it's a quantum mechanical treatment, but the authors even say it's classical in the beginning of the next section. The "fully quantum mechanical" arguments are very briefly outlined in section 11.4.2. The "single-mode-ness" of the equations and field representations are not stated explicitly, but they are implied by the notation (carried over from the previous chapters which are predominantly for a single mode), and Zubairy explained that the line width is described by a single mode of the cavity and emission from the atoms.
Well, let me put it this way: I feel that there are many books or courses focusing entirely on the spectral decomposition and referring to these as "the" modes instead of some possible choice. This heavy focus is unfortunate. So let me try to give several (hopefully) identical explanations of what I consider an optical mode from several different viewpoints: 1) Any assembly of indistinguishable photons. If one cannot distinguish these photons in principle, they need to be considered as a single mode. Unfortunately the concept of indistinguishability in quantum optics is also full of pit traps. This directly relates to point 2, so I continue the discussion there. 2) Any field possessing first order coherence. This is close to Glauber's way of explaining it. As all photons inside a coherence volume are inherently indistinguishable, this is basically just a different aspect of point 1. First-order coherence implies a fixed phase relationship between fields at different times and positions. This does not require monochromaticity. However, the spectral distribution limits the timescale over which a fixed phase relationship can be kept (first order temporal coherence is just the Fourier transform of the power spectral density). As a visualization, consider the intensity distribution caused by the interference of two beams. This will have three components. One proportional to the square of the first field, one proportional to the square of the second field and one corresponding to cross terms containing both fields. These are obviously interference terms. They also exist if the two beams are not of the same frequency. This keeps true, if you go away from the classical intensity down to the single photon level. Even for single photon detection events there may be interference terms due to two fields of different frequency. If a photon is detected due to these cross-terms, one cannot in principle say that it came from which of the fields it originated - although they have different frequencies. So I consider anything which shows genuine interference (or equivalently first order coherence) as one mode. 3) Anything with a unimodal photon number distribution. Any light field which has a photon number distribution given by just one distribution (Poissonian or Bose-Einstein, it does not really matter). This is also a consequence of indistinguishability. Identical photons can be described by a single mode distribution. In this respect, spectrally broad light fields can form a single mode. The obvious example is spectrally broad thermal light. In some sense another example are short laser pulses. Short laser pulses (say 100 fs) typically have a spectrum which is about 10 nm broad. Ultrashort laser pulses may be even broader. Despite this spectral width, the pulses show nice interference patterns. The community working on laser optics, however, typically also uses the convention of calling the spectral basis "modes" and therefore calls fs lasers mode-locked. In a nutshell: I also consider superpositions of the monochromatic "modes" as single modes if they are superposed with a well defined phase relationship. Thanks for the info. That book was not the one I learned quantum optics from back in the days, but I will try to see whether there is a copy available here. Without looking at the approach and the equations they use, I am afraid my comments will be somewhat handwaving. edit: I do not know whether the explanation given in the Scully/Zubairy is good or handwaving. For a rather detailed analysis (for a lossless cavity, however, so strong coupling is discussed in detail) see Phys. Rev. Lett. 51, 550–553 (1983) (Theory of Spontaneous-Emission Line Shape in an Ideal Cavity) by J. J. Sanchez-Mondragon et al. and some of the articles which are citing this one and cited articles therein. edit: Do they by chance give the relative magnitude of light-matter coupling and cavity decay rate in the book. Or put differently, do they assume weak or strong coupling? Do they discuss a regime where the Purcell effect becomes important or is it just a very lossy cavity? It is quite uncommon to introduce a cavity and only consider simple spontaneous emission without considering complicated stuff like strong coupling and such stuff. Another good overview of what can happen when introducing a cavity is given in one of Jon Keeling's lecture notes:http://www.st-andrews.ac.uk/~jmjk/keeling/teaching/quantum-optics.pdf, starting from chapter 3.
As I already said, I appreciate the discussion and I've learned something here, but I'm not hopeful that I can reconcile our differences in definitions at the same time I'm trying to understand the minutia of the book I am already learning from. The concepts you outlined in defining a "mode" are not foreign to me, but rather than defining a "mode", they are simply properties of the field that can exist for various distributions of different modes. Frankly, your various definitions of modes seem rather abstract. Can all of those points be described with self-consistent mathematical forms? Perhaps that's not a universal requirement, but the definition I gave of a mode has a very simple, straightforward mathematical form. The other various properties, like coherence, etc. do not define a mode, but rather are the result of operations on the field which can be defined as a superposition/distribution of modes. I'll stop here since I fear that we'll just go in circles trying to define the terms. From my circle, Zubairy and Scully are considered leaders in the field. I suppose they know their stuff, but from other reviews I've read, their book is actually not the most clear text to learn from. I will study further the references you gave. So, I now have more to think about, but just more that I don't understand.
All those points are a consequence of first-order coherence (or indistinguishablility, if you like), which has a solid mathematical foundation. It is in some sense the generalization of monochromaticity (photons from a monochromatic field are necessarily indistinguishable, but indistinguishable photons are not necessarily monochromatic. It is an attempt to have a formalism which actually allows to solve some things. Monochromatic fields necessarily have infinite duration, which makes calculations of time-resolved stuff more or less impossible. Keeping indistinguishability over a finite amount of time only is what leads to a first-order coherence description. But right - this indeed seems rather abstract unless one really runs into these problems. Yes, they are. To be honest from your description I thought they placed more emphasis on the influence of the cavity. So some of these references might be a bit over the edge. I did not manage to get a grip on a physical copy of that book, but I could browse the e-book for a few minutes. Scully/Zubairy are kind of trying to avoid the problem. They begin with the correct hint that an exact treatment would use equation 4.3.14. The sections around this equation (about double slit experiments and such stuff) already show the problem. They intend to use quasimonochromatic light, but also say that the equation in question relates first-order coherence and the power spectral density via a Fourier transform. This will obviously be quite trivial for monochromatic light. It is right as the definitions for the field operators they use are the general sum over all modes (equations 1.1.30 and 1.1.31), but the restriction to monochromatic light is somewhat pathological. Getting back to chapter 11.4.1, the classical approach is pretty cheap. They say their field is proportional to [tex]exp(i(\theta(t)-\nu_0 t))[/tex], where the theta is just a random variable with Gaussian distribution. This is not different from assuming that the frequency itself is a random variable with Gaussian distribution, so essentially they already put the line width in. They just derive a microscopic formalism to back it up, but basically they just make the replacement [tex]\nu(t)=\nu_0+\theta'(t)[/tex] here and perform ensemble averaging. This is where it gets a bit fishy. One can still claim that one treats a monochromatic field and treats a diffusing phase as a small perturbation which allows to keep the monochromatic mode picture without really having a monochromatic mode. It is kind of intuitive as it allows to keep a "this is the mode and that is the effect causing broadening"-picture, but it is not really thorough. Still it saves a lot of time and pages in the book. Strictly speaking a true monochromatic mode must have non-diffusing phase for all eternity, however treating this as a perturbation instead of going to a treatment involving several modes is way less painful.