Quantum Optics: Fock State and Its Meaning for Light

[fisico30]

In quantum optics, I hear talking about a Fock state as a state with a fixed number of particles and indeterminate phase. What are they talking about? What type of light is that?

Does having indeterminate phase mean that we cannot predict the phase (therefore the instantaneous amplitude) of the field at a precise instant in time?
Thanks

 

[Redbelly98]

Yes, you can think of it as a state with a fixed number of particles, though I prefer the term "energy quanta". One example is the number of photons in an optical or microwave cavity. If that number is known precisely, the phase is completely uncertain ... just another example of the uncertainty principle at work.

 

[fisico30]

That makes sense, thank you. It would seem to me that we can control the intensity (therefore the number of quanta) exactly for any light field.
We always think of a laser as the most coherent source, which to me means that it has a precise phase...and enegy...

 

[Cthugha]

That makes sense, thank you. It would seem to me that we can control the intensity (therefore the number of quanta) exactly for any light field.
We always think of a laser as the most coherent source, which to me means that it has a precise phase...and enegy...

No, a coherent state has neither precise phase nor precise photon number. But it is a state of minimumm total uncertainty where the uncertainty is distributed equally between amplitude and phase.

In detail, the coherent state is thought to be the closest thing to a classical light field, which means that any measurement should not change the state of the field (as this is a feature of the classical formulation). However, this also means, that the destruction of a photon should leave the state unchanged, which is not possible with a state of precisely defined photon number.

 

[fisico30]

Thanks Cthugha for the clarification. I am independently studying the topic on the book by Saleh.
I guess i am struggling to move away from the classical picture of light fields.
To a question like "in an interval of 10 seconds, what is the number n of photons incident on the detector?" we can only answer with the probability based on Poisson distibution, for coherent light. We don't know the number with certainty, even if the intensity is constant.Surely, the more the mean optical power, the more photons arrive.

Why would the "registration of the photons be statistically independent". Why?
As you pointed out, the space and momentum uncertainties are both 1/2, for an uncertainty product equal to 1/4 (the minimum possible), for a coherent state.

a) i still do not get what this uncertainties are
b)they are defined as the uncertainties of the phasor components. But a phasor is just a mathematical too. What is real about these uncertainties. Is it because together, these two determine (only together) the amplitude and phase of the field?
But arent' phase and field amplitude always related? Can we have a phase fluctuation without an amplitude fluctuation? An example?
Also, your words"at the destruction of a photon should leave the state unchanged, which is not possible with a state of precisely defined photon number". Why can u elaborate on that, in simple terms?
thanks

 

[Cthugha]

I am independently studying the topic on the book by Saleh.

Great book. I do not know your exact level of understanding, but i will try to answer your questions as basic as possible.

a) i still do not get what this uncertainties are
b)they are defined as the uncertainties of the phasor components. But a phasor is just a mathematical too. What is real about these uncertainties. Is it because together, these two determine (only together) the amplitude and phase of the field?

There are several uncertainies, which can be applied to em-fields. The easiest one to understand is the uncertainty in terms of the field amplitude and phase. So the field amplitude uncertainty leads to uncertainty in the photon number and the phase uncertainty has some influence on the coherence properties of the light field. Honestly, I really can't explain this any better than just to say, that the amplitude and phase of an em-field do not have precise values, but a certain spread.

Why would the "registration of the photons be statistically independent".

Let's start with the classical point of view. Imagine some light field with constant intensity. Now you split the light beam and direct it towards two photodetectors, which take "single pictures" simultaneously. On every detector you will now see the same average number of counts. For simplicity assume a low count number of 1 detection every 10 pictures on a single photodetector. Now, you want to know, how often there will be a detection of a photon on both detectors simultaneously. In the case of statistical independence the probability of a joint detection will just be the product of the single detection probabilities. So you will expect 1 out of 100 pictures to show a joint detection (1 out of 10 on one detector times 1 out of 10 on the second detector). This is what one would expect to see from a classical point of view and this is, what one indeed sees, when coherent light is examined.

Also, your words"at the destruction of a photon should leave the state unchanged, which is not possible with a state of precisely defined photon number". Why can u elaborate on that, in simple terms?

Ok, there are several possibilities to interpret the result with the two photodetectors, I mentioned beforehand. The fact, that the detection of one photon does not change the possibility to detect another photon afterwards might seem trivial at first, but it isn't. The case I mentioned means, that the photon number fluctuations on both detectors are completely random and not correlated to each other.

Now consider the same setup, but a different light source: a single atom. A single atom will also produce a light field with constant average intensity, but it will only emit one photon at a time. So imagine you have again count rates of 1 detection every 10 pictures on both detectors. Taking into account, that the single atom can only emit one photon per cycle, you will now find, that there will be no joint detections at all. One photon can't be detected at both detectors simultaneously. So in this case the detection (and therefore destruction) of a photon changed the light field and the photon detections on both detectors are not statistically independent anymore. One could say, that the photon number fluctuations at both detectors are anticorrelated.

Now imagine an em-field with strongly fluctuating amplitude, but constant average intensity on a timescale, which is longer than the timescale of the fluctuations. Let me assume 1 count in 10 pictures per detector again. As the intensity/photon number is proportional to the square of the amplitude, you will again find, that the photon detections are not statistically independent, but you will have more joint detection than expected, because in the moments of high amplitude fluctuations, the photon number will be far above average and in the moments of low amplitude fluctuations, it will be far below. So in this case, the photon number fluctuations on both detectors will be correlated. Either the photon number is above average at both detectors simultaneously or it is below average at both detectors simultaneously. This is, what you would expect for thermal light like sunlight, the emission of a lightbulb and so on.

So the classical hypothesis, that the measurement of the intensity/photon number at one moment does not change the photon detection probability for later moments is not always true. However the situations, in which this classical hypothesis is right, can not have a definite photon number. Imagine a state with exactly 5 photons. Now one is detected and thereby destroyed. It is obvious, that the probability for detecting a photon for a state with 4 photons will be different from the probability for detecting a photon for a state with 5 photons. So to cancel this effect, the photon number noise needs to be on the order of the average photon number, which is a feature of the Poisson distribution.

 

[fisico30]

Great answers. I will metabolize it all and get back to you to prove my uderstanding.
I like the Saleh book very much. Readable, detailed... Do you have any other book suggestion that works as good ?
Thanks for sharing your knowledge.

 

[Cthugha]

"Optical coherence and quantum optics" by Mandel and Wolf is another great book.

Also "One Hundred Years of Light Quanta", the nobel lecture given by Roy Glauber, gives a very interesting first overview of the topic.

 

[fisico30]

Hello Cthugha,
got all the references you mentioned.thanks.
you explain things so well that I must ask/bug you with another bugging question: A laser beam, coming out of a laser source operated in single mode, shows a spectrum that has large peak frequency(the longitudinal mode freq) and a bandwidth around 1 millionth of that peak frequency.
This is the spectrum that we would get both mathematically (doing a Fourier transform) and experimentally(with a spectrum analyzer).
But these frequencies are the frequencies of infinite, eternal sinusoidal waves that only exist in theory. Are these frequencies, in the frequency band, just fictitious, artificial then? Or do they, somehow, really reflect the way atoms in the material oscillate, like small diapasons?
But those atom emitters must start maybe change (due to perturbations) and end the way they emit their signal. I can envision those small emitters emit Gaussian or Lorentian type, short or long finite time and space pulses, that then all overlap to give the total field. What is really happening?
Thanks again for any help!

 

[Cthugha]

[...]
A laser beam, coming out of a laser source operated in single mode, shows a spectrum that has large peak frequency(the longitudinal mode freq) and a bandwidth around 1 millionth of that peak frequency.
This is the spectrum that we would get both mathematically (doing a Fourier transform) and experimentally(with a spectrum analyzer).
But these frequencies are the frequencies of infinite, eternal sinusoidal waves that only exist in theory. Are these frequencies, in the frequency band, just fictitious, artificial then? Or do they, somehow, really reflect the way atoms in the material oscillate, like small diapasons?
But those atom emitters must start maybe change (due to perturbations) and end the way they emit their signal. I can envision those small emitters emit Gaussian or Lorentian type, short or long finite time and space pulses, that then all overlap to give the total field. What is really happening?

Well, this is a very broad topic, so excuse me, if I write a little bit concerning a lot of topics, but cover none of these in depth. I do not know, what exactly you are interested in and what your level of knowledge is, so this is only a very general first information.

Ok, lasers are a bit more complicated, so let me begin with optical transitions of single atoms. Ideally speaking the transition from an excitred state towards the ground state should be sharp and happen at a well defined energy. However, you are right that Fourier transform theory tells us, that this is not strictly possible. Taking this to a "more quantum" point of view, this corresponds to the time energy uncertainty. So one finds correspondingly that the minimal linewidth of an atomic transition is determined by the lifetime of the excited state leading to a Lorentzian shape of the linewidth. A short lifetime means a large bandwidth. As to the question, which properties of the atom these frequencies reflect, I can just tell you, that the simplest model for the wavefunction in question develops from the initial state towards the final state with a superposition in between. This summer there was a talk at our department by Friedrich Herrmann, who works in the field of didactics of physics. He presented a simple model, which already shows some of the basic properties of a radiating atom. You can find some of his pictures here: http://www.hydrogenlab.de/
The pictures are available in English. There are also videos, but those are only in German under the points "atomare Übergänge in 3D" and "atomare Übergänge in 2D".

Now a laser is a bit more complicated as there is no free spontaneous decay of the excited state anymore, but stimulated emission, in which a photon causes the excited atom to emit another photon, which is indistinguishable from the first one, so there is a fixed phase relationship between photons and not just some random superposition of independent emitters. Here the linewidth is determined by several broadening factors. First of all a pulse of short duration (for example a fs pulse) will of course have a broader linewidth than a cw laser due to the time energy uncertainty mentioned before. Then there are inhomogenous broadening mechanisms like the doppler shift due to velocity distributions of the atoms in a gas laser, which leads to a Gaussian broadening of the linewidth.
Of course even the design of the laser cavity itself is important here. As you might imagine, the coherence properties are related to the frequency spectrum of the emitted light. One might imagine that light with a small bandwidth should be more coherent than light with a huge bandwidth. This is true. Mathematically speaking, the first order correlation function is the Fourier transform of the frequency spectrum. So a small bandwidth causes a broad first order correlation and therefore also a long coherence time because the coherence time is defined by the decay of the first order correlation function. So one can imagine, that also the average time a photon stays inside the cavity is related to the coherence time and bandwidth because the longer a photon is inside a cavity the more stimulated emission can occur in principle. This is accounted for by the so called quality factor of the laser cavity. It is defined as the central frequency of the cavity divided by the the frequency bandwidth (so small bandwith means large quality), but it is also proportional to the number of oscillation periods of the field, after which the stored energy inside the cavity has decreased to a value of 1/e of the initial value (so large quality also means that a photon stays rather long inside the cavity on average).