Energy/particle number uncertainty in a laser beam

[Sonderval]

If I understand things correctly, the coherent state of a laser beam implies that the photon quantum field is a superposition of states with different particle numbers.
This also implies that a laser beam is not in an energy eigenstate.

To get energy conservation, I assume that this corresponds to the laser medium that creates the laser beam to exist in an overlap of different states with different numbers of atoms excited, so there is entanglement between the state in the laser beam and the state in the laser medium. If, for example, I start with 100 atoms in the excited state (and I measured that very precisely so that there is no initial uncertainty of energy), the part of the laser wave function that contains one photon would be entangled with the state of the laser medium where 99 atoms are excited (one photon emitted) and so on.
Is this a correct picture?

If so, what would happen to the laser beam if I later on measure the energy content of the laser medium (for example using a very precise weighing apparatus)? Would the state of the laser beam collapse to a state of well-defined photons?

And, if so, to spin the idea further, would this mean that it is impossible to create a laser beam if I were to weigh the laser medium continuously?

Not that any of this seems impossible, but it does seem weird to me and is not something that I ever read anywhere. So, is it all rubbish? If so, why?

 

[mfb]

If I understand things correctly, the coherent state of a laser beam implies that the photon quantum field is a superposition of states with different particle numbers.

Why? What is wrong with exactly n photons in a beam, with coherent phases?

 

[Sonderval]

How can I have an exact photon number and coherence?
For any exact number state of a quantum field $|n\cdot k\rangle$ (n particles of momentum k), the expectation value of the field is zero, is it not?
According to Wikipedia (and lots of other sources), the photon state in a laser beam is
$|\alpha\rangle =e^{-{|\alpha|^2\over2}}\sum_{n=0}^{\infty}{ \alpha^n\over\sqrt{n!}}|n\rangle =e^{-{|\alpha|^2\over2}}e^{\alpha\hat a^\dagger}|0\rangle$
i.e., it is a superposition of all possible quantum numbers.
From analogy to the harmonic oscillator, that is what one should expect because a photon field of definite value corresponds to a harmonic oscillator state of definite position.

 

[Cthugha]

If I understand things correctly, the coherent state of a laser beam implies that the photon quantum field is a superposition of states with different particle numbers.
This also implies that a laser beam is not in an energy eigenstate.

Correct. Coherent states are eigenstates of the photon annihilation operator.

To get energy conservation, I assume that this corresponds to the laser medium that creates the laser beam to exist in an overlap of different states with different numbers of atoms excited, so there is entanglement between the state in the laser beam and the state in the laser medium. If, for example, I start with 100 atoms in the excited state (and I measured that very precisely so that there is no initial uncertainty of energy), the part of the laser wave function that contains one photon would be entangled with the state of the laser medium where 99 atoms are excited (one photon emitted) and so on.
Is this a correct picture?

It is not exactly that easy because the dynamics of the gain medium and the cavity photon number vary on slightly different timescales. You need to consider all three or four levels for typical lasers operating with full inversion. As a photon is emitted from some atom into the cavity, the atom is indeed in its ground state for a short time, but due to the optical pumping process into some high level, it will quickly change back to the excited state again. The photon is emitted into the cavity and will typically perform something between a few hundred and a few million round trips before it is randomly transmitted through one of the cavity mirrors. For large gas lasers or similar designs, the photon round trip time is typically quite large compared to the timescale on which the gain medium dynamics take place, so that there is not really entanglement between the emitted photon and the emitting atom for a long time. If there was entanglement, one would see antibunching of the emission as the emission of a photon would simultaneously mean that the number of possible emitters reduces for a while. Some results on microcavity lasers may imply that things like that happen on short timescales (few picoseconds), but the results may also be interpreted in a slightly different manner.

Basically you get a Poissonian photon number distribution because it is the distribution describing statistically independent random events. The number of atoms that will emit a photon during some certain time interval is indeed distributed in a manner that each emission event is independent of the other ones at a given mean cavity photon number.

If so, what would happen to the laser beam if I later on measure the energy content of the laser medium (for example using a very precise weighing apparatus)? Would the state of the laser beam collapse to a state of well-defined photons?

The problem is that it is quite complicated to measure the photon number (or energy in the field) reliably without manipulating it. You can do so using quantum non-demolition measurements (although these are not really measurements in the intuitive sense and usually only give averaged results) and may then watch the gradual collapse of the photon number distribution to some well defined photon number. This has been done in:
"Progressive field-state collapse and quantum non-demolition photon counting" by C. Guerlin et al., Nature 448, 889-893 (2007).
If you do not have a subscription, the paper is also available at arXiv:http://arxiv.org/abs/0707.3880.

 

[Sonderval]

@Cthuga
Thanks for the answer and the link, that looks like a cool paper. (I'll read it tonight.)

You are of course right, the interaction with the laser medium makes things a bit complicated.

The problem is that it is quite complicated to measure the photon number (or energy in the field) reliably without manipulating it

That was why I had the idea of looking at the cavity - of course after I switched off whatever was responsible for the inversion.

To circumvent the problem with the interaction of photons and cavity, I could try something like this:
1. Pump as many atoms as possible into excited states.
2. Switch off pumping device
3. Put mirrors in operation (so no lasing before this step), simultaneously measure energy of cavity
4. Get a single laser pulse out of the system
5. Enclose system in perfect mirrors, so no more energy can escape
6. Measure energy content of the cavity again.

O.k., that's surely not possible to do in practice, but as a thought experiment I should still have some entanglement between the photons in the laser beam and the atoms in the cavity (otherwise, without any entanglement, I could violate energy conservation, at least for a single measurement, if not on average). So at step 6, the number of photons should become

 

[Cthugha]

I am not quite sure it is easy to achieve a good single shot laser pulse on a timescale short compared to the dynamics of the gain medium, but let me just assume it is possible somehow.

You might be able to achieve something roughly similar to your idea by means of cavity optomechanics. The guys in that field use cavities where one or two of the mirrors are actually nano-cantilevers which act as mechanical oscillators. So you may actually link the photon field to mechanical oscillations. However, that field is not exactly my cup of tea, but if you have a look at current literature, you may find some proposal going into a similar direction to what you have in mind.

 

[Sonderval]

Thanks for the hint - I've read a bit about those, but as far as I know, nobody ever did something like this (probably because it would be a very delicate experiment to achieve some result that is theoretically obvious).