I Atoms in a harmonic oscillator and number states

[jamie.j1989]

TL;DR Summary

I am confused about the relation between the number state with the annhilation and creation operatros and the number of atoms in the harmonic oscillator.

I am confused about the relation between the number state $|n\rangle$ with the annhilation and creation operators $a^\dagger$ and $a$ respectively, and the number of atoms in the harmonic oscillator. I'll try to express my current understanding, I thought the number states represent the quanta of enregy or the energy spacing of the oscillator, and the annihlation and creation operators represent the removal and imparting of a quantum of energy to the oscillator. When we take the expectation of the number operator $n=a^\dagger a$ with a state, say a coherent state, then we get the mean number state being occupied $\langle n\rangle$. This is now where my confusion lies, I see regularly in the literature (can provide if needed) $\langle n\rangle$ discussed as an average number of atoms, does it depend on the context? For example, if one atom exists in the oscillator, surely we can exite it to higher number states etc which is explained via the application of the creation operator on the initial number state, when does the differentiation between atom number and number state occur, when applying $a^\dagger$ and $a$?

 

[vanhees71]

The number operators count "phonons" not atoms. You deal with a single particle when solving a single-particle Schrödinger equation! The phonons of the harmonic oscillator are the most simple example for quasiparticles, which is also an utmost important concept in connection with many-particle systems, where often collective excitations (like sound waves in a solid, where the name "phonons" for the vibration modes of a harmonic oscillator comes from) behave approximately as particles in the description of QFT.

 

[jamie.j1989]

The number operators can also be used to count atoms, see here https://en.wikipedia.org/wiki/Fock_state, I've done a bit more reading and it does seem one needs to look at the context, for example in this paper https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.110.163604 they represent the total number of particles as $n=a^{\dagger}a+b^{\dagger}b$ where the operators $a^{\dagger},a$ and $b^{\dagger},b$ are bosonic mode annihilation and creation operators and act on different modes of a two input interferometer. However, now I'm confused at what a coherent state in this context actually means, if the number state $|n\rangle$ no longer represents the mode of the oscillator but the number of atoms. Does it just mean that if in some experiment with trapped atoms we measure the number of atoms in one mode with $n_a=a^{\dagger}a$, say through absorption imaging or some other means, then over many such measurements with identically prepared atoms we would measure a mean number corresponding to $\langle n_a\rangle$ with a variance in the measurements equal to the mean? In this case we could ascertain that the input to the mode $a$ was a coherent state i.e. $|\alpha_a\rangle$?

 

[vanhees71]

Of course, in QFT free fields are a collection of harmonic oscillators, and then you "count particles".

A coherent state is no Fock state, i.e., it is not a state with a determined number of particles. It can, e.g., describe a BEC. An example is the description of liquid helium in the superfluid phase, where one way to describe the superfluid part as a coherent state. Another example are coherent states of Cooper pairs in superconductors. For a quite good first review about coherent states, see

https://en.wikipedia.org/wiki/Coherent_state

 

[PeterDonis]

The number operators can also be used to count atoms

In scattering experiments, yes. But you're not talking about a scattering experiment. You're talking about a harmonic oscillator that's made of some fixed number of atoms, like a crystal. The thing that is oscillating in this case is certainly not the number of atoms, and there is no useful QFT description in terms of an atom number operator that can have creation and annihilation operators applied to it.

The oscillations in this case are, as @vanhees71 said, phonons--various modes of vibration of the object being described (e.g., a crystal), none of which change how many atoms it has. A Fock state in this case is a state with a definite number of phonons, and a coherent state is a state which does not have a definite number of phonons, but is an eigenstate of the phonon annihilation operator.

 

[vanhees71]

Indeed, in the case of a solid (idealized as a perfect crystal lattice) you have of course a set of normal modes which are quantized. In the first approximation you indeed get something like a QFT made with these normal modes with creation and annihilation operators for excitation and deexcitation of these modes. What you then get is something like Debye's famous model for the specific heat. The "particles" you describe here are, however, not atoms or electrons but these collective excitations. Since the math looks very similar as when you describe particles within QFT one calls these excitations quasiparticles. In this example the quasiparticles are called phonons, because it's nothing else than quantized sound waves in the solid. The coherent states with sufficient intensity describes quantum-mechanically macroscopic lattice vibrations. Another important state is the thermal state (which is a mixed rather than a pure state) describing the solid in thermal equilibrium. From this you can get models for the specific heat, and that's one of the first applications of quantum theory to thermal-equilibrium many-body (quasiparticle) states and solid-state physics (Einstein, Debye).