Increase of coherent length in stimulated emission

[DaTario]

TL;DR Summary

Is there any theoretical prediction in QM regarding the increase of coherent length through a series of stimulated emissions as it seems to happen in the linear cavity of a ruby laser setup?

Hi All,

When I teach the basic structure of a laser setup, stimulated emission appears as a fundamental phenomenon. But in no reference I found a description that could account for the increase of coherent length of the EM laser field. According to my knowledge, one photon typically doesn't have even 1% of the coherence length of a laser beam. Sometimes I wonder if in each stimulated emission, the photon released by the atom stimulated by the passing resonant photon couples with it in a way that enlarges a bit the coherent length (as in situation B of figure bellow). But the most frequently found illustration for stimulated emission is similar to figure A.
My question is:
Is there any theoretical prediction in QM that aims at providing an elementary explanation of this increase of spatial coherence? Is there any reference that deals with this subject specifically?

Best Regards,
DaTario

 

[Cthugha]

You do not really need (much) QM for this. The temporal first order correlation function (its decay gives the coherence time) and the spectral power density of the light field are essentially Fourier transforms of each other. Accordingly, a narrower laser linewidth automatically means longer coherence times. So, all you need is the theoretical prediction for the linewidth of a laser. The theoretical lower limit it can reach due to QM is usually estimated by the Schawlow-Townes limit.

 

[DaTario]

Thank you, Cthugha.
I understand you are taking a semi-classical point of view, as you are dealing with EM fields that can be represented by sort of well behaved functions of time. My point here concerns the elementary quantum process of stimulated decay, as depicted in my attached figure in OP. Does the stimulated emission occur with an elementary increase of spatial coherence? (I know it is connected with time coherence as well)
My point is: A or B? and theoretical justification.

 

[Cthugha]

Well, stimulated emission is not really that quantum. In contrast to spontaneous emission, which really needs a lot of qm and in principle even qft for a correct description, stimulated emission and a classical driven antenna or dipole are not really different and a semiclassical approach works perfectly.

If you insist on translating it to a quantum picture: The excited state of an atom is stable in the absence of any external perturbation. So you need an external perturbation to put it into a superposition of the excited state and the ground state. Such a superposition state yields a time-varying probability amplitude for the spatial distribution of the electron and is akin to the quantum version of "accelerated charges radiate".
However, you do not have time-varying electron densities, but probability amplitudes. So these also do not get converted to photons, but to probability amplitudes for the a photon being added to the light field in the cavity. In the simple case of a single emitter, you would get an entangled state, which goes back and forth between the atom being in the excited state and the cavity containing n photons and the atom being in the ground state and the cavity containing n+1 photons. For spontaneous emission, the coherence time will be given by how long this superposition can be kept alive.

For stimulated emission, you instead usually have lots of emitters and lots of photons. So, if you want a mental picture, you rather have a collective superposition of many emitters and the cavity field. Still the basic picture of is similar. However, the state is obviously less prone to perturbartions to the atoms as dephasing of one atom out of 7 million atoms interacting with the light field is not as critical as dephasing of the only atom interacting with the light field.

Are you familiar with quantum optics in phase space?
If so, on a different note you can see the enhanced coherence time directly by plotting the Wigner function of the different states in phase space. Coherent states are essentially displaced vacuum states. with the displacement being equivalent to the coherent amplitude. Now as you increase the displacement, you can keep track of how well the phase of the state is defined. While it is completely undefined for the vacuum, the range of possible phases obviously becomes narrower, so the phase becomes defined better when increasing the displacement. As the displacement corresponds to the amplitude of the light field and is therefore directly related to the mean photon number of the light field, increasing the mean photon number of a coherent state automatically results in a better-defined phase and therefore enhanced coherence.

 

[DaTario]

Yes, I am somewhat acquainted with quantum optics. But I must confess that I have never related the displacement operator with stimulated emission. Are these two things fundamentally related? To me, the displacement operator represents an ideal chain of infinite processes of creation and anihilation of photons. Some representation with the help of Bloch sphere and the Jaynes-Cummings model still don't seem to help me understand the connection between stimulated emission and the gain of spatial coherence. I would ask you one more thing: you seem to have shown that if a coherent state grows in intensity we can infer that it is gaining spatial coherence (coherent length specifically). How is it so?

And just to check: can you see the difference betweem situations A and B? Do you think that putting the question in this basis is meaninful and sound?

 

[Cthugha]

Yes, I am somewhat acquainted with quantum optics. But I must confess that I have never related the displacement operator with stimulated emission. Are these two things fundamentally related? To me, the displacement operator represents an ideal chain of infinite processes of creation and anihilation of photons. Some representation with the help of Bloch sphere and the Jaynes-Cummings model still don't seem to help me understand the connection between stimulated emission and the gain of spatial coherence. I would ask you one more thing: you seem to have shown that if a coherent state grows in intensity we can infer that it is gaining spatial coherence (coherent length specifically). How is it so?

And just to check: can you see the difference betweem situations A and B? Do you think that putting the question in this basis is meaninful and sound?

Well, the displacement operator e.g. determines the amplitude of a coherent state. And as you increase the mean photon number via stimulated emission, you just increase the coherent amplitude. It is just important to understand that the exponent of the displacement operator is a sum of annihilation and creation operators. So it is related to fields, not photon numbers (which are products of creation and annihilation operators). Actually, single photons are pretty irrelevant to stimulated emission in lasers. In lasers, you usually do not encounter any situation, where you have a well-defined photon number inside the cavity and you add exactly one to it via stimulated emission. Such pictures are pretty misleading. Stimulated emission is a phase-sensitive process and as such understood best in a field picture. The phase sensitive amplification essentially corresponds to increasing the coherent amplitude.

As you say that you are familiar with quantum optics in phase space: Below, the Wigner functions of two different coherent states are depicted (where the amplitude is not represented faithfully, but binary - all values above some threshold value are shown in one color and all values below threshold are just white). One has a small mean photon number and one has a larger mean photon number. As you can see, the figure also shows the phase uncertainty of the two states and the state of larger photon number clearly has a smaller phase uncertainty. This automatically results in enhanced coherence. Actually stimulated emission will result in "extending" this displacement vector and you the Wigner function gives you the relative weights for extending it along the corresponding direction. Obviously, due to this, the distribution will get narrower in phase space as the amplitude increases.

With respect to your A and B: I think this is misleading. It may be useful to discuss this picture for stimulated emission from a single isolated atom, but it is not the most useful picture for lasers.

 

[DaTario]

I am very grateful for you contribution, but let me ask you few more questions.
When you say that
"Actually, single photons are pretty irrelevant to stimulated emission in lasers."
it seems to me a bit like the sentence:
"Actually, stacking bricks are pretty irrelevant to the building of a house.",
in the sense that the model of stimulated emission seems to constitute a basic and elementary level to explain the appearance and the properties of the laser field.

Another question: the information content of a Wigner function like the one you have choosen to present is enough to allow us to infer the coherent length of this state?
My question in other words: does the Wigner function give a complete account of the whole spatial partern of the field state it represents?

Best regards

 

[Cthugha]

I am very grateful for you contribution, but let me ask you few more questions.
When you say that
"Actually, single photons are pretty irrelevant to stimulated emission in lasers."
it seems to me a bit like the sentence:
"Actually, stacking brics are pretty irrelevant to the building of a house.",
in the sense that the model of stimulated emission seems to constitute a basic and elementary level to explain the appearance and the properties of the laser field.

A single photon has a well-defined meaning. It is a highly non-classical state of a quantum field with an excitation number of exactly 1 and no photon number uncertainty. My emphasis is that for example having a coherent field of mean photon number 7 (and the typical photon number uncertainty of a Poissonian distribution), going to a mean photon number of 8 (also with some photon number uncertainty) via stimulated emission is a completely different thing than having a coherent field of mean photon number 7 and adding a single photon (without any photon number uncertainty) to it. The first state is pretty classical, while the second state will have non-classical features.

Another question: the information content of a Wigner function like the one you have choosen to present is enough to allow us to infer the coherent length of this state?
My question in other words: does the Wigner function give a complete account of the whole spatial partern of the field state it represents?

Well, you still would need to know the mode (spectral width) of the light field and whether the state is stationary and will show diffusion in phase space over time. Just for the record: For pretty much every real laser, the coherence time is actually given by mechanical or similar instabilities.

 

[DaTario]

I guess I understand your example of N=7 to N=8. But it seems we are stcuck in the middle of an impass: at one side we have an idealized Fock state added to the field while at the other side we have an idealized exponential contribution (which contains an infinite number of instantaneous processes - sum of powers...). What would be the best way to represent the elementary gain of the laser?

Now I see that through the addition of a Fock state we simply add radiation without any phase reference. So we must try a way that envolves, for instance, coherent superpositions.

Regarding the second question, do you confirm that Wigner function gives us information regarding an ensemble of experiments on the field done at a given position in space and at a certain time after the preparation of the sate, and has no information associated with the duration of the pulse or its spatial coherence?

 

[Cthugha]

I guess I understand your example of N=7 to N=8. But it seems we are stcuck in the middle of an impass: at one side we have an idealized Fock state added to the field while at the other side we have an idealized exponential contribution (which contains an infinite number of instantaneous processes - sum of powers...). What would be the best way to represent the elementary gain of the laser?

The displacement operator contains an infinite number of infinitesimally small translations of the field. That is a reasonable way to represent it. As a mental picture, I would always consider the atom as a driven dipole. QM really has pretty much nothing to add here.

Regarding the second question, do you confirm that Wigner function gives us information regarding an ensemble of experiments on the field done at a given position in space and at a certain time after the preparation of the sate, and has no information associated with the duration of the pulse or its spatial coherence?

Well, the Wigner function is only defined with respect to some mode of interest. In experiments this would be given by a local oscillator acting as a reference. The properties of this reference define the mode of interest, which includes temporal duration and spatial shape of the mode. The same mode of a light field may result in very different Wigner functions with respect to different reference beams. You of course need to know what this mode is to be able to interpret the Wigner function.

 

[DaTario]

One problem that I see in this discussion is related to the photon concept. I have never been told about it but soon I in the physics course it appeared to me that when dealing with cavity quantum eletrodynamics two definitions of photon show up. The first one is related to the atom and is what the atom emits in terms of EM field due to downward transitions of its electron. The second is related to the modes of the cavity - photon as a quantum of occupation of energy in some mode. Assuming the cavity is resonant with the electronic transition, the atomic photon, once generated in the correct direction, must give rise to a cavity photon, but just after a time of the order of 2L/c. It seems reasonable to say that these photonic structures are not identical.
And I see myself trying to understand stimulated emission in this context. So it seems relevant to clearify to what kind of photon I am talking about. My primary goal was to understand this process in terms of the atomic photons. It seems that your reasoning is based on cavity photons and I understand its practical value, its pragmatism. But do you think it is possible to understand stimulated emission of an isolated atom? In my opinion this phenomenon has to include some sort of mechanism that increases spatial coherence (as in situation B in the figure above).

 

[vanhees71]

Photons are single-particle-Fock states of the quantized (free) electromagnetic field. That's true in both "free-space QED" as in "cavity QED". If the transition energy of the atom in the cavity is not a resonance frequency of the cavity, it cannot be emitted within the cavity, because it simply doesn't exist to begin with!

 

[DaTario]

I guess it is not so clear cut, as you posed. Ok, we have modifications, due to detuning, of the time it takes for the atomic system to decay. Complete supression of emission is somewhat a rare effect, isn't it? A "not so resonant" atomic photon can be part of the bunch of photons that lives inside a cavity. Interference effects will limit its amplitude but it does not mean that the photon is forbiden to enter the cavity. The greater the detuning the smaller the probability of this photon entering the cavity.

I remember that my point is what model of stimulated emission can we produce, within the framework of cavity QED, so as to account for the gain in spatial coherence relatively to the same coherence property observed in single photons.

 

[Cthugha]

I do not see what an "atomic photon" is supposed to be. The standard definition is indeed the one as a Fock state of the quantized field. In really old days the individual monochromatic modes that arise when doing a Fourier decomposition of the light field have been called photons as well, but using this term in this way is discouraged and outdated.

The reason why complete suppression of emission is rare is the same reason, why huge Purcell factors are rarely encountered: it is pretty impossible to create cavities that are that perfect in all three dimensions. Usually, one needs quite low mode volumes to achieve this and then reality gets in the way. People tried to do that e.g. for quantum dots in micropillars and achieved quite nice Purcell factors, but etching the pillars usually introduces some surface roughness, which decreases the effect.

With respect to stimulated emission by a single atom: this is a complicated topic and it depends on a lot of parameters. What are the photon statistics of your incoming light field? What is its spectrum? What is the pulse length?

Several papers gave theoretical descriptions of stimulated emission for single atoms, e.g. Phys. Rev. Lett. 108, 143602 (2012) (https://arxiv.org/abs/1204.4668) or NJP 14, 083029 (2012) (https://arxiv.org/abs/1208.5732) or OSA continuum 1, 772 (2018) (https://arxiv.org/abs/1806.04862). Reference 24 in the last paper also shows some experiments with respect to this question, which show that the spectrum of the two-photon state may even be broadened beyond the natural linewidth.

The standard increase in coherence seen for lasers is in a nutshell really just a result of the fact that coherent states of larger amplitude have a better-defined phase than coherent states of lower amplitude. This works well for large numbers, but for single atoms the entanglement effects due to the effective non-linearity of the atom (which just means that it cannot absorb two photons) are much more prominent.

 

[vanhees71]

Of course, I talked about an idealized completely "lossless" cavity. As @Cthugha rightfully states that's not achievable in reality.

 

[DaTario]

Thank you both for the contributions, and specially to @Cthugha for the references.
When an atom emits a photon by electronic decay and in free space, what comes out in terms of EM field is what I was referring to as atomic photon (simple propagating wave).
This process may work differently if the excited atom is inside a large resonant cavity with no photons (vaccum), but it seems to me that the EM field this atom emits initially has important differences in relation to the usual stationary mode of the cavity where we will see the population N = 1 long after the decay.

 

[Zarqon]

I haven't quite read all of the answers so far, but I'll just try to answer your original question.

The increase in coherence length of a laser has nothing to do with the atoms at all. The coherence properties of a laser comes from the mirrors used in the laser cavity (of course some times you have to pick mirrors that match your medium in order to get maintain the right level of population inversion). This is exactly the same mechanism as in a conventional Fabry-Perot cavity without a lasing medium in it.

In short, the finesse of a cavity is given only by it's mirror's reflectivity, R, (in absence of other loss mechanisms), and the finesse is also "free spectral range"/linewidth, where the free spectral range is given by the length, L, of the cavity. Thus, R and L together determines the linewidth of the cavity, which is more or less 1/coherence time. So from that you see that the coherence time is not decided by atomic parameters at all (in the simple case), but only the cavity ones.

 

[DaTario]

But, Zarqon, what would be, according to your argument, the use of the gain medium (atomic part) for the laser?
From the rate equations approach, for instance, we see that field and atomic equations are coupled, so it seems that atomic variables play an essential role in principle.

 

[Zarqon]

Well, the gain medium provides the inverted population, ready to give away photons. But the coherence/phase of the photons that are created through stimulated emission is set by the seed photons that first decayed (randomly) and then travalled around the cavity. The act of going around the cavity and surviving means that those photons had the right phase/frequency to build up constructive interference, given the cavity constraints.

The higher the reflectivity of the mirrors are, the more round trips the photons will make, and thus the more precise their frequency/phase have to be in order to still be constructive with each other even after so many round trips.

 

[DaTario]

You are saying that a lossy cavity can spoil the coherence and it has somewhat the taste of the fluctuation–dissipation theorem (FDT), which connects dissipation with fluctuations (phase noise, in this case). But it also seems that you are underestimating the role played by the stimulated emission. Note its presence in the name of laser, :)