Does quantum optics explain beat notes from classical optics.

Discussion in 'Physics & Math' started by al onestone, Jan 21, 2013.

  1. al onestone Registered Senior Member

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    IN classical optics you can add two independent sources of light at a detector to produce an optical beat note with frequency equal to the difference frequency.

    Assume source 1 has state ψ(ω1) and source two has state ψ(ω2) where ω1 and ω2 are the frequencies/phases of sources 1 and 2.

    The addition of the two sources at a detector will give an optical beat with frequency equal to the difference of the two sources, ½(ω1-ω2). This can be exploited to gain coherence/interference between independent sources of light as in the link below (from the 60's)

    http://www.haverford.edu/physics/lo...e - Pfleegor Mandel/PfleegorMandelp1084_1.pdf

    Has this interference collection method vanished from history? Has anyone used this since? Is there an explaination of it that makes it meaningless in a modern context. Mandel explains the theory in the paper with modern quantum optical explaination, so did this method just become outdated?
     
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  3. eram Sciengineer Valued Senior Member

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    Since it has been given a quantum mechanical interpretation, the old method has not become outdated, but incorporated into the current method.
     
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  5. al onestone Registered Senior Member

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    I noticed in Ballentine's text there is an explaination of "quantum beats" which is the same thing as the classical beat frequency. The effect is a temporal coherence phenomenon.

    The effect of two-independent-source interference collection is a method that combines the temporal coherence phenomenon of the quantum beat with a spatial interference/coherence effect.
     
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  7. Q-reeus Banned Valued Senior Member

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    Have tried elsewhere on several occasions to get sensible feedback re the following setup which to my mind calls into serious question the notion that photons only self-interfere:

    Take the classical situation of a high Q cavity resonator fed via a probe maintained at a constant rms excitation current. It's well known the cavity fields E, B, then rise linearly with time until such point that the rms internal fields are an appreciable fraction of their equilibrium value, tapering off exponentially to zero growth rate eventually. That appreciable departure from linear growth could take many thousands or even millions of cycles depending on the particular Q value. Within the essentially linear regime, field energies rise quadratically as a function of time (W ~ E^2, therefore t^2), not linearly, since there is constructive interference occurring. No surprise classically, but how is this to be squared with the notion each photon being pumped into the cavity is only self-interfering? It should go without saying the classical superposition principle is assumed to hold - independent of the internal field values, rms rate of photon injection into the cavity is constant given the constant rms probe feed current.

    Problem as I see it is that according to QM, there is no photon mutual interference (at least not in the sense that supposedly a laser beam say of a given rms intensity has a photon rms density directly proportional to that intensity). Hence since the cavity rms photon injection rate is constant, QM surely predicts a linear rise of field energy: W ~ t. But we know from all experience with e.g. microwave cavity resonators, we actually have W ~ t^2 in accordance with above description. Can someone explain how all this is reconciled? Note there is no suggestion of any energy conservation problem - to maintain a constant probe rms current requires a continually ramping feed voltage and there is a proper power balance at all times. It's the distribution of energy density that seems problematic - not from a classical perspective, but according to this QM thing about 'non-interference' between photons. Hope this is considered on-topic and not a thread hijack.

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    Last edited: Jan 31, 2013
  8. al onestone Registered Senior Member

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    If you have an optical cavity with a high Q factor, no appreciable loss, and you continuously pump it with photons, then the energy density does increase linearly. I don't know whee your getting this idea that

    What could it possibly have to do with interference? Interference does not affect the energy density.

    I'm not an expert on this type of physics

    but I would question this t^2 dependence. There simply is no way to increase the energy non-linearly when you pump the cavity linearly.
     
  9. Q-reeus Banned Valued Senior Member

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    Not true. Cavity at resonance presents a case of complete constructive interference. There are various modes and ways of viewing things, but consider a cavity comprised of a small length of waveguide in fundamental mode, shorted at far end and with some obstacle providing the coupling aperture between so-formed cavity and rest of otherwise identical line. High Q is synonymous with a small cavity coupling factor k (where k= |transmitted field amplitude/incident field amplitude|).

    Initially an incident wave almost entirely reflects at obstacle but there is a small transmitted wave of relative amplitude k. At resonance, transmitted wave reflects from far end and returns to aperture end with just the right phase that upon reflection again from aperture end, it is completely in-phase with the next incident wave transmitted wave crest. Thus has amplitude almost exactly equal to 2k. And so on it goes - 3k, 4k, 5k, ... Eventually internal field becomes sufficiently large that 'bleed' from within essentially matches incident amplitude, occurring roughly when internal cavity field relative amplitude is 1/k. Linear range of cavity amplitude buildup can be over millions of cycles if Q is in the millions (some optical cavities can have Q's in the billions). That process is interference all the way. Amplitude, not intensity, thus builds up linearly with time initially, and only gradually tapers off to an asymptotic constant value. A lengthy derivation can be found e.g. here:
    http://www.jpier.org/PIER/pier78/15.07090605.Wen.pdf
    see sections 5.2, 5.3, fig's. 6, 10.
    Depends what you mean by 'linearly'. Classic EM is clear on this - constant amplitude feed implies initially linear wrt time cavity field amplitude buildup, hence parametric intensity buildup. How under such a situation do you manage to get net cavity photon number to be parametric wrt time, as required if photons only self-interfere? Doesn't make sense. There is clearly a constant rate of photon injection into cavity via incident wave feed through obstacle. The picture then is of a linear rate of net cavity photon number growth initially, tapering off to zero net rate eventually.

    As per #4, a detailed balance will show energy is always conserved - IF classical EM is followed. Try getting such a balance on the basis of self-interfering-only photons, while sticking to established transmission and reflection coefficients. An impossibility methinks. But don't take my word for it - see if you can get such a balance. My approach though is to just concentrate on the known fact of linear growth in cavity field amplitude, consistent with classical EM, but totally at odds with this legacy of Dirac you quote in your #17 here:
    http://www.sciforums.com/showthread.php?106574-Laser-intersection&p=3069164&viewfull=1#post3069164
    "These were the papers that prooved Dirac's famous quote "Each photon only interferes with itself.""
    Sure looks like we have a serious issue here. And no-one has so far ventured an answer, here or elsewhere I have tried.
     
    Last edited: May 11, 2013
  10. al onestone Registered Senior Member

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    I would look at the frequency model of the light that you're examining in the cavity. When you say the energy density changes non-linearly, this is only possible if the interference is selecting out specific frequency components as in a Fabry-Perot cavity.
     
  11. Q-reeus Banned Valued Senior Member

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    Huh? It is naturally assumed the source of incident wave is monochromatic. And that everything is linear. And that cavity is high Q, which implies acting as an excellent frequency filter anyway. As for Fabry-Perot cavity, that is simply a particular configuration of cavity that is particularly high Q, but otherwise is no different in basic principle to any other. So I cannot see the logic in what you say above. Have you read the relevant parts in the article I linked to? Do you dispute that, as shown there, at resonance it is the field amplitudes, not intensities, that rises linearly with time - given a fixed rms amplitude feed as modeled there? A basic characteristic of any classical high Q oscillator driven by a constant amplitude excitation at resonant frequency. General character is always that oscillator amplitude goes like ctsin(ωt), with ct the envelope part and c a system time constant.

    And it should be clear a fixed amplitude feed condition implies a constant net rate of photon injection into cavity. As per #4, this further implies the feed voltage is continually adjusted upward so as to counter opposing cavity field (this is based on a stub probe as feed element into cavity). In #6 scenario, with reactive obstacle in place rather than a feed probe, and constant amplitude incident field, effect is much the same. With however the effective net 'probe voltage' gradually diminishing owing to increasing amplitude of transmission back from cavity field towards incident wave source. If as assumed the cavity itself is lossless, what happens over time is that the cavity seen as a terminating load changes from being effectively a short initially, to an increasingly lossy load in between as cavity field steadily rises, to finally an open circuit when cavity field has maxed out to steady-state value of ~ E√Q, where E is the rms amplitude of source incident wave.

    You are probably stuck on the notion that if the source field amplitude is constant, the power draw is constant and thus cavity energy density must rise linearly to suit. Not correct. Almost all power reflects back to source initially and finally, but in between there is destructive interference between reflected incident wave and cavity transmitted wave - power draw rises quadratically in keeping with quadratically rising cavity energy - in the approx. linear ramp cavity field regime. Conservation of energy is guaranteed at all stages by the unitary character of scattering matrix for a lossless reactive obstacle.

    There is an evident unresolved clash between the driven classical oscillator at resonance - initially linearly ramping cavity rms field, thus quadratically ramping intensity = cavity energy density, and this notion of photons only self-interfering, which if true would have the field amplitude initially ramping as square root of time. In keeping with the linearly ramping photon density. Well classical EM is shouting out here that evidently photons happily mutually interfere. And I imagine the same type of enigma would apply to an acoustic situation, where presumably phonons supposedly only self-interfere. Quite a headache.
     
  12. Fednis48 Registered Senior Member

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    After thinking about this for a while, I think the photons in your thought experiment actually do interfere with each other in some sense. But there's a catch: the interference is a second-order effect, mediated by the cavity, so it is still correct so say that two photons will never directly interfere with each other.

    I think the key is to recognize that the laser photons are not the same as the cavity photons. At its heart, a photon is an excitation of an oscillator mode. The laser modes are different from the cavity modes, so they can be treated as two separate, coupled collections of photons. The Hamiltonian for the laser shining on the cavity will contain terms that look like:
    \(\begin{equation}a_{cavity}a^\dag_{laser}+a^\dag_{cavity}a_{laser}\end{equation}\)
    In words, photons in the laser mode are being annihilated to create photons in the cavity mode, and vice-versa. Depending on whether the incoming laser photons are in- or out-of-phase with the photons already in the cavity, this could certainly lead to interference.
     
  13. Q-reeus Banned Valued Senior Member

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    Not seeing an answer in that. As I understand it, the QED picture of photon annihilation and creation is supposed to be via correspondence principle completely consistent in the classical regime with the classical picture of wave scattering and absorption in which superposition of wave amplitudes hold (assuming linear media). Otherwise all of electrical & communications engineering would be a shambles - but superposition of amplitudes works.

    Consider just the case of an electrically short dipole antenna, radiating into free space. There is some fixed rms current in the dipole, corresponding to a fixed rms power output, hence a fixed rms photon generation rate. Now suppose a second identical antenna is placed very close to the first one - lateral separation distance d << λ, and both are forced to oscillate with identical rms currents as for single dipole. One expects photon output rate for each dipole should be a function only of it's own current hence independent of the other's output. Hence net photon generation rate should be double that of single dipole. Hence net power radiated simply doubles.

    Wrong! Power rises as essentially the square of net rms current, hence goes up 2^2 = 4-fold (Input impedance doubles for each dipole thus power is conserved). Something confirmed by many observations of antenna array behavior. Here there is no possibility of having different modes as explanation. To reconcile quadratic relation between net current amplitude and output power, one has to assume photon generation in each dipole is somehow a function of both current and ambient, externally sourced field, which makes no sense to me. The source is oscillating charge - hence current, not current plus externally sourced field. This becomes a reductio ad absurdum situation - subdivide any current source into smaller and smaller elements, and net photon generation rate can be made arbitrarily small thereby.
    In general case of arbitrary spacing of antenna dipoles, it's clear that overall radiation pattern is very much governed by interference between radiators. Classically just what is predicted and found, but how can this square with non-interference between photons? Doesn't that imply simply summing of radiator intensities as say for incoherent radiators (e.g. frosted light-bulbs) - hence zero interference?!

    And the same kind of difficulty arises in may other situations - single-mode waveguide junctions etc. At best, mutual interference is required to make any sense of scattering relations. Something seems very basically amiss here. Hard if not impossible imo to justify photon concept at all in these classical EM scenarios. How does QED explain such cases?
     
    Last edited: May 14, 2013
  14. Fednis48 Registered Senior Member

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    I take issue with the following statement: "One expects photon output rate for each dipole should be a function only of it's own current hence independent of the other's output." Just like in the cavity example, energy is jumping from one oscillator into another; in this case, motional excitations of the electrons in the antennae are turning into excitations of the free-space electromagnetic field. Just like with the cavity problem, we could write this as pairs of creation/annihilation operators, except that this time we're coupling motional oscillator modes to bosonic modes rather than coupling two bosonic modes to each other. The result of this interaction depends on the state of both the motional oscillator and the electromagnetic oscillator, so in general the rate at which one antenna creates photons in the electromagnetic mode will depend on how many photons the other antenna has already put there.

    I'm pretty sure that for any example of interference where different photons seem to interfere with one another, there will be a step in which two distinguishable oscillators (spatially separated lasers, different antennae, etc) are coupled to a common output mode, allowing them to interact via that coupling.

    After thinking about this, I realized that that does sound an awful lot like separate photons interfering, so I asked one of my more E&M-minded colleagues about it. He pointed out that if we're considering continuous lasers or steadily-driven antennae, the natural basis is the "coherent state" basis, which puts everything in a messy superposition of photon number states. At some level, it's probably still correct to say that photons only interfere with themselves, but that doesn't give much physical insight when the photons can't even be meaningfully counted. In an experiment involving definite-photon-number states, the "only with itself" principle is more useful and meaningful.
     
  15. Q-reeus Banned Valued Senior Member

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    Thanks for that input. Interesting. This is the first response that sort of makes some sense to me. So what appears to be interference between photons is explained by mutual coupling - to the vacuum in the case of e.g. antenna currents. The difficulty is in trying to make sense of it physically. Given the 1/r^2 Coulomb field of an electron, I would naively expect any excitation of vacuum modes to be strongly confined to region closest to electron itself, hence minimally influenced by distant field sources. Evidently not so and it all just comes out of the formal math. The notorious problem of not being able to visualize QM/QED.
     
  16. Fednis48 Registered Senior Member

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    ^This.

    Also, glad I could help! It was a good question; I also learned a bit thinking it through and asking my colleague for clarification.
     

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