Energy in microwave cavity quantized?

[Spinnor]

Consider the microwave cavity used in a particle accelerator to accelerate particle bunches. Before a bunch of particles enters the microwave cavity can we, if only in principle, quantum mechanically describe the state of the microwave cavity as a proper sum of microwave photon number states, summing over all possible momentum?

If yes, after the particle bunch has passed through the cavity and gained energy at the expense of the energy in the cavity can we say that the new state of the microwave cavity has fewer microwave quanta?

Can we say that each particle in the accelerated bunch of particles absorbs many microwave quanta while in the cavity?

Thanks!

 

[tech99]

Consider the microwave cavity used in a particle accelerator to accelerate particle bunches. Before a bunch of particles enters the microwave cavity can we, if only in principle, quantum mechanically describe the state of the microwave cavity as a proper sum of microwave photon number states, summing over all possible momentum?

If yes, after the particle bunch has passed through the cavity and gained energy at the expense of the energy in the cavity can we say that the new state of the microwave cavity has fewer microwave quanta?

Can we say that each particle in the accelerated bunch of particles absorbs many microwave quanta while in the cavity?

Thanks!

It seems to me that the particles we are talking about are, for example, electrons, and not photons. So we are dealing with electricity. Even the cavity resonator action is described by the motion of the electrons in its walls. And electricity is not quantised.

 

[Spinnor]

So classically we can determine the electromagnetic modes of a microwave cavity resonator. These classical modes can then be quantized? The modes can then have 0, 1, 2, 3, ... quanta in each mode? Even though engineers probably use classical electrodynamics to design these cavities they can still be treated quantum mechanically?

Thanks!

 

[Swamp Thing]

Even though engineers probably use classical electrodynamics to design these cavities they can still be treated quantum mechanically?Thanks!

A microwave field corresponds more or less to what is called a coherent state, where you have a pretty large number of photons that satisfy certain conditions that define such a state. A laser beam is an example of a coherent state.

When you have enough photons in the coherent state, Maxwell's equations are good enough approximations to work with.

How many photons are enough to say that they can be modeled classically? A coherent state with a smallish number of photons has a Poisson statistical behavior that amounts to a significant noise source. But if this noise is less than the thermal noise at the working temperature, one can ignore photon noise and model the whole thing as a classical electromagnetic wave.

At frequencies up to perhaps a few hundred gigahertz, this is usually the case -- unless we are thinking of really, really low temperatures and using SQUIDS etc. for our measurements.

 

[PeterDonis]

a coherent state, where you have a pretty large number of photons

A coherent state is not an eigenstate of the photon number operator, so it does not have a definite number of photons. So this description is not really correct as you state it. You could say something like "a pretty large expectation value for photon number", which would be better. Is that what you mean?

 

[Fred Wright]

For an interesting example of the quantum properties of resonant cavities Google:"A Schrödinger cat living in two boxes".

 

[Swamp Thing]

You could say something like "a pretty large expectation value for photon number", which would be better. Is that what you mean?

Yes, that's it. Thanks.

 

[Spinnor]

When you have enough photons in the coherent state, Maxwell's equations are good enough approximations to work with.

That is my understanding. But my question was in principle that this microwave cavity that can be considered classically can also be treated quantum mechanically, and must be able to be treated this way as we are told all classical physics is quantum at its heart? If that is true then we can think of the accelerated particles as absorbing some number of real microwave photons thus gaining energy at the expense of energy in the cavity?

Thanks!

 

[f95toli]

That is my understanding. But my question was in principle that this microwave cavity that can be considered classically can also be treated quantum mechanically, and must be able to be treated this way as we are told all classical physics is quantum at its heart?

Sure, this is done all the time. Microwave cavities are key building blocks in most quantum circuits. There is even a whole sub-field of QM called cavity-QED (quantum electrodynamics) where -traditionally- large (several cm in size) 3D high-Q.resonators are made to exchange energy with e.g. Rydberg atoms (the analogous field for planar resonators/qubits is called circuit-QED).

Have a look at e.g. the motivation for awarding Serge Haroche a few years ago.

Creating number states (states with a definite number of photons) in a cavity is not trivial and most oft the time the field is -as hare already been mentioned- best thought of as being coherent and you need to be at temperatures kBt<<H*F (~300 mK or so for a typical cavity).
That said; it is often useful in CQED type measurements to measure the energy in the cavity in terms of the average photon number <n>; you sometimes get some interesting physics when <n><1 even if the state is "only" coherent.

 

[Spinnor]

Creating number states (states .with definite number of photons) in a cavity is not trivial

Please clarify my understanding, any state of the electromagnetic field is the proper sum, distribution, of photon number states, with a proper sum over momentum?

Thanks! Work calls, damn.

 

[tech99]

Consider the microwave cavity used in a particle accelerator to accelerate particle bunches. Before a bunch of particles enters the microwave cavity can we, if only in principle, quantum mechanically describe the state of the microwave cavity as a proper sum of microwave photon number states, summing over all possible momentum?

If yes, after the particle bunch has passed through the cavity and gained energy at the expense of the energy in the cavity can we say that the new state of the microwave cavity has fewer microwave quanta?

Can we say that each particle in the accelerated bunch of particles absorbs many microwave quanta while in the cavity?

Thanks!

May I comment from an engineering perspective - sorry I should not be on this Quantum Physics section!
It seems to me that we are not dealing with radiation in the situation you describe, but the motion of charges. A cavity is an energy store, and the fields inside it may be said to be linked to the motion of the electrons in its walls. It resembles an LC circuit, where the energy flows alternately between L and C, and in a perfect case there are zero losses and no aggregate energy flow. The E and H fields in the cavity are in antiphase, unlike an EM wave in which they are in-phase, and they occur in different positions. So I would suggest that there is no radiation in the cavity, just separate E and H fields, and I cannot therefore see how photons are going to be detected.
Further than this, if the Q of the cavity is infinitely high, the bandwidth will be zero, so how can the energy within it be modulated with quantum noise and consist of photons?
I feel that the acceleration of the particles you describe is similar to an electric motor, not connected to radiation.
Now I admit that if a cavity is connected to an antenna, or shall we say a slot is cut in it, then we will see radiation. The slot radiates because the electrons in the surrounding conductor are accelerated by the fields in the cavity, and not by allowing the interior "radiation" to leak out. The slot will radiate 50% of its photons back into the cavity. And the loss of the energy radiated away by the slot will create a loss resistance, the radiation resistance, which will lower the Q of the cavity to allow the interior waves to be modulated with quantum noise, now showing the presence of photons.
Once again apologies for what I do realize must be a naive view.

 

[f95toli]

May I comment from an engineering perspective - sorry I should not be on this Quantum Physics section!
It seems to me that we are not dealing with radiation in the situation you describe, but the motion of charges. A cavity is an energy store, and the fields inside it may be said to be linked to the motion of the electrons in its walls. It resembles an LC circuit, where the energy flows alternately between L and C, and in a perfect case there are zero losses and no aggregate energy flow. The E and H fields in the cavity are in antiphase, unlike an EM wave in which they are in-phase, and they occur in different positions. So I would suggest that there is no radiation in the cavity, just separate E and H fields, and I cannot therefore see how photons are going to be detected.

That does not matter. The E and H fields in the cavity in these experiments will be quantized meaning they can be described by using the concept of photons; you do not need "radiation" a such. A mode in a 3D cavity if just a standing wave and if you want you can sort-of think of this as a photon bouncing between the two walls, note also that the probability of coupling to a photon has its maximum at the anti-nodes of the mode (which, btw, does NOT say anything about the photon having a "wavefunction").
There is no fundamental difference between this and say an optical Fabry-Perot cavity (two mirrors); the physics is exactly the same, it is only the wavelength that is different.

More generally you can quantize the equations for just about any microwave component you can think of; you can have single microwave photons trapped in normal planar LC circuits made up of an inductor and a capacitor if the Q value is high enough (this is what I use in my experiments); and it is also possible to quantize e.g. the equations for say the propagation along a coplanar waveguide.
There is PLENTY of material available online, just google Cavity-QED or circuit-QED.

Further than this, if the Q of the cavity is infinitely high, the bandwidth will be zero, so how can the energy within it be modulated with quantum noise and consist of photons?

The Q is not infinitely high. In most experiments you want the loaded Q value to be somewhere between 10^4-10^9; it really depends on the experiment and what you want to do.

 

[tech99]

That does not matter. The E and H fields in the cavity in these experiments will be quantized meaning they can be described by using the concept of photons; you do not need "radiation" a such. A mode in a 3D cavity if just a standing wave and if you want you can sort-of think of this as a photon bouncing between the two walls, note also that the probability of coupling to a photon has its maximum at the anti-nodes of the mode (which, btw, does NOT say anything about the photon having a "wavefunction").
There is no fundamental difference between this and say an optical Fabry-Perot cavity (two mirrors); the physics is exactly the same, it is only the wavelength that is different.

More generally you can quantize the equations for just about any microwave component you can think of; you can have single microwave photons trapped in normal planar LC circuits made up of an inductor and a capacitor if the Q value is high enough (this is what I use in my experiments); and it is also possible to quantize e.g. the equations for say the propagation along a coplanar waveguide.
There is PLENTY of material available online, just google Cavity-QED or circuit-QED.
The Q is not infinitely high. In most experiments you want the loaded Q value to be somewhere between 10^4-10^9; it really depends on the experiment and what you want to do.

Thank you for your answer. I know you are right, I just want to explore the issue. I presume you can observe quantum noise in an LC circuit or high Q cavity, showing the presence of photons. I presume the higher number of trapped photons causes a smoothing effect.
I also find the idea of a capacitor or inductor adding noise to a microwave current to be difficult and I wonder if you can observe this noise?
I am puzzled that, if the fields of an electron are quantised, how can the motion of an electron be not quantised, as the two are intimately linked?

 

[Spinnor]

I am puzzled that, if the fields of an electron are quantised, how can the motion of an the electron be not quantised, as the two are intimately linked?

With a typical electron gun for an old style tv, electrons can be accelerated to any energy. A free electron can have any kinetic energy. But the osscilating electrons and electric and magnetic fields inside a microwave cavity seem to form some kind of mass and spring system the energy of which is quantized? The mass/spring system must be one maifistation of the electromagnetic field interacting with charge?

 

[rootone]

My kitchen microwave broke down tonight,
I'm getting used to those thing only last for two years, and replacements not very expensive.
Odd though, it still makes the right noises and the machine gets on hot on top, but the food is not heated at all.
Any explanation for that would be interesting.

 

[Baluncore]

Odd though, it still makes the right noises and the machine gets on hot on top, but the food is not heated at all.
Any explanation for that would be interesting.

The HT fuse in line with the HT stick rectifier has blown because the HV capacitor failed. (75% probable).

 

[tech99]

With a typical electron gun for an old style tv, electrons can be accelerated to any energy. A free electron can have any kinetic energy. But the osscilating electrons and electric and magnetic fields inside a microwave cavity seem to form some kind of mass and spring system the energy of which is quantized? The mass/spring system must be one maifistation of the electromagnetic field interacting with charge?

Thank you. But is there any evidence for the motion of electrons in a purely reactive circuit at high frequencies becoming quantised? Do you observe quantum noise in your cavities and LC circuits?

 

[Spinnor]

Even the cavity resonator action is described by the motion of the electrons in its walls.

Here is a thought experiment that I think shows that it is more then just the motion of the electrons. Consider two metal cubes of side L and 2L which are our microwave cavities. The larger cube has 4 times the surface area but 8 times the volume. Let the lowest mode of the smaller cube be excited in both the small and large cube. Let both cubes have the same maximum electric field. We are told the number of likely microwave quanta is proportional to electric field energy density times the volume. It seems that the total current in both cubes will go as the surface area of each cube but the number of quanta will go as the volume of each cube. If so the number of quanta is not proportional to the number of electrons moving around. Electrons are involved but so are quanta?

This idea seemed clearer this morning, where do I go wrong?

Thanks!