What is the concept of a pure state in the electromagnetic field?

In summary, Ballentine's Quantum Mechanics book discusses pure states and non-pure (mixed) states. He explains that polarized monochromatic light from a laser can be considered a pure state of the electromagnetic field, while unpolarized monochromatic radiation and black body radiation are examples of nonpure states. However, there is some discussion about the definition of "pure state" in this context, as well as the possibility of pure thermal states. Overall, the invention of the laser was groundbreaking as it provided a source of coherent light and also allowed for the study of quantum effects such as entanglement. Additionally, there is ongoing research on thermalization without ensemble or time averaging.
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pellman
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In Ballentine's Quantum Mechanics book, as part of a discussion of pure states vs non-pure (mixed) states, he says

Polarized monochromatic light produced by a laser can approximate a pure
state of the electromagnetic field. Unpolarized monochromatic radiation and
black body radiation are examples of nonpure states of the electromagnetic
field.
I think that he is referring to the classical EM field and using "state" in a more broad sense than quantum states, though I could be wrong about that.

So in what sense is unpolarized light not a pure state? I would think that, just as superpositions of pure state quantum wave functions give another pure state, that superpositions of the EM field would also give "pure states". That is, any solution of Maxwell's equations would be considered a pure state.

But really I am muddled on this concept.
 
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I think that restricting the view of lasers to QM makes life very hard. If you regard 'ordinary light' as coming from a large number of individual atoms, each of with produced its photon at a random time then this is similar to a large number of radio transmitters with their antennae, all with slightly different frequencies and with the antennae in random directions. That is your impure state and it's down to bandwidth (line width?) When the radiation is stimulated, magically all the outputs from all the atoms become so close in phase that it's as if they'd all been produced from one continuous energy transition and the whole beam of light is regarded as being in a pure state. As for the polarisation, the antennae are all lined up to be parallel and produce a single polarisation plane. (Or circular / elliptical , I guess)
 
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Thanks
 
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Quantum mechanically laser light is (nearly) a coherent state and as such a pure state, while "thermal" light is in a mixed state, which is in the case of ideal black-body radiation is described by the thermal-equilibrium Statistical Operator
$$\hat{R}=\frac{1}{Z} \exp(-\beta \hat{H}),$$
where ##\beta=1/k_{\text{B}} T## with ##T## the temperature, and the partition sum
$$Z=\mathrm{Tr} \exp(-\beta \hat{H}).$$
The invention of the laser was so ground breaking for science and applications alike, because

(a) after all one had a source of (nearly perfectly) coherent light. Indeed the coherent state with a high intensity can be very accurately described also as a coherent classical electromagnetic wave. The most recent breakthrough in this direction is the possibility to have such light sources also in the X-ray regime, where the coherent light can be used for high-precision measurements with applications reaching from materials science (like condensed-matter physics) to biology.

but also because

(b) it provided a possibility to create sources of single photons and entangled two- (and even few-) photon Fock states, which brought many "gedanken experiments" a la Einstein, Podolsky, and Rosen to the realm of real experiment with the possibility to check the astonishing quantum effects related with entanglement, among other things the disprove of the possibility of a local deterministic hidden-variable theory (Bell's inequality) to an overwhelming degree of significance. Also this branch, which is the real "quantum optics", i.e., optics that cannot be understood within classical Maxwell theory (and it is less trivial to find observables realizable in the real world that prove the quantum nature of electromagnetic waves than even some modern quantum textbooks suggest, but that's another story).
 
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vanhees71 said:
Quantum mechanically laser light is (nearly) a coherent state and as such a pure state, while "thermal" light is in a mixed state, which is in the case of ideal black-body radiation is described by the thermal-equilibrium Statistical Operator
$$\hat{R}=\frac{1}{Z} \exp(-\beta \hat{H}),$$
where ##\beta=1/k_{\text{B}} T## with ##T## the temperature, and the partition sum
$$Z=\mathrm{Tr} \exp(-\beta \hat{H}).$$

But some pure states have thermal properties, so couldn't thermal light also be described by a pure state?

Here are some examples of pure thermal states
http://arxiv.org/abs/1302.3138
http://arxiv.org/abs/1309.0851
http://www2.yukawa.kyoto-u.ac.jp/~new-noneq2015.ws/Presentation/Sugiura.pdf
 
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That sounds interesting, but it seems also to be a bit misleading to call these states "pure states", because there's an random process underlying, defining stochastic pure states rather than what you call a "pure state" in the usual sense of standard quantum theory. I've not yet understood, what's the point of this construction (despite that it seems to be a pretty interesting mathematical idea to do Monte-Carlo Simulations of equilibrium many-body systems).
 
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vanhees71 said:
That sounds interesting, but it seems also to be a bit misleading to call these states "pure states", because there's an random process underlying, defining stochastic pure states rather than what you call a "pure state" in the usual sense of standard quantum theory. I've not yet understood, what's the point of this construction (despite that it seems to be a pretty interesting mathematical idea to do Monte-Carlo Simulations of equilibrium many-body systems).

The point of the construction is similar to Boltzmann kinetic theory: how do we reconcile irreversibility in thermodynamics with the reversible laws of mechanics?

So can a pure quantum system thermalize without averaging? The existence of pure thermal states suggests that it is possible in many cases.

To add to the above references (which did not mention the approach to equilibrium), here is another one which talks about thermalization without ensemble or time averaging (strong thermalization): http://arxiv.org/abs/1007.3957 (http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.106.050405).
 
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FAQ: What is the concept of a pure state in the electromagnetic field?

1. What is a pure state of the EM field?

A pure state of the EM field refers to a state in which the electromagnetic field is in a single, definite quantum state. This means that the field has a well-defined energy, momentum, and polarization at a particular point in space and time.

2. How is a pure state of the EM field different from a mixed state?

A pure state of the EM field is in a single, definite quantum state, while a mixed state is a combination of multiple quantum states. In a mixed state, the field's properties are not well-defined and can be described by a probability distribution.

3. What are the properties of a pure state of the EM field?

A pure state of the EM field has well-defined energy, momentum, and polarization at a specific point in space and time. It also follows the laws of quantum mechanics, such as superposition and uncertainty principle.

4. How is a pure state of the EM field described mathematically?

A pure state of the EM field can be described mathematically using the wave function, which is a mathematical representation of the field's properties. The wave function can be used to calculate the probability of finding the field in a particular state.

5. Can a pure state of the EM field exist in nature?

Yes, a pure state of the EM field can exist in nature. However, it is difficult to create and observe in a laboratory setting due to the field's interaction with its environment. Most electromagnetic fields in nature are in mixed states, as they are constantly interacting with different particles and fields.

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