In ordinary notation of quantum optics we describe coherent state as a state with infinite photon distribution. If we try to analyze it on infinite time, we will obtain an event with infinite photon number (infinite energy). I believe it's "non physical". How can we interpret infinite fock number? Is it some kind of divergence in quantum theory, or mathematical approximation only?
Indeed, if one observes a source of fixed intensity for infinite amount of time, a photon count must be infinite. Practically it means that the longer you observe the source, the more energy you accumulate in direct proportionality.
Pavel,
your question is a good one that many have ruminated over. you have hit upon the difference between a commensurable number (having a common basis, measure, or standard of comparison) and an incommensurable magnitude. perhaps you can determine how this works into the Fock number. Here is a quote from my manuscript that may help you figure it out.
It would be good to study the Dedekind cut.
"When applied to certain integers the square root operation was also found to be responsible for the introduction of the set of irrationals.. The direct and indirect relevance debate between the rational and the irrational sets concerns the difference between a commensurable number and an incommensurable magnitude. This difference also caused much debate over physicality vs. mathematical construct. This set was not rigorously understood for years until formally defined by the Dedekind cut" (PDF) Quantum wave function realism, time, and the imaginary unit i. Available from: https://www.researchgate.net/publication/322386956_Quantum_wave_function_realism_time_and_the_imaginary_unit_i [accessed Mar 22 2020].
you may also find this manuscript enlightening or interesting.
I have to add some refinements. I know that infinite mean energy exists on infinite time. My question is about different thing.
I have source of coherent quantum state. When I make N (in natural numbers) measurements on finite time T, I collect selective photon distribution which describe quantum state realized on time NT. Mean photon number depends from T. If N tends to infinity, I have a a very small but non-zero probability to collect in time T infinite number of photons. How can it be if the energy of the field is finite?
Pasha, this is not a correct limit for a Poisson process. Typically, when you assume N->\infty, you assume that T->0 so that NT=\lambda is finite. But, formally, the field with infinite number of photons and non-zero frequency indeed has infinite energy. When Planck first figured it out more than a hundred years ago, he was amazed.
Dear Pavel Gostev, I should add to all interesting answers in this thread that in the limit of large α, the statistics of a coherent state is equivalent to that of a classical stable wave, I elaborate:
The coherent state by definition is an eigenstate of the annihilation operator in the Heisenberg picture, and the formal solution of the eigenvalue equation is the vacuum state displaced to a location α in phase space, that is, the unitary displacement operator D(α) operate on the vacuum. So as a conclusion I guess that a state with infinite photon distribution is a classical wave.
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