Dipole, harmonic oscillator, and the coherent state

[Karthiksrao]

Dear all,

I am aware that a weakly driven dipole can be modeled as a damped driven simple harmonic oscillator.

If I have to model the dipole as being driven by a classical monochromatic electromagnetic wave, would the corresponding simple harmonic oscillator then be in coherent state ?

In other words would the SHO be having a poisson distribution of photons in its energy levels (just like, if the SHO was in equilibrium with a thermal reservoir it would be populated according to Bose-Einstein distribution corresponding to the reservoir temperature T) ?

Thanks!

 

[Cryo]

What do you mean by dipole? A dipole antenna?

Assuming what you mean is that you have some small metallic or dielectric rod, ignore loss, you can treat it as harmonic resonator, and equivalently, as quantum harmonic oscillator. Incident electromagnetic waves would couple to this resonator through dipole interaction. Then you will get a driven harmonic oscillator. If the driver is a simple sine-wave you will get coherent state in your oscillator. Whilst this treatment will lead you to creation/annihilation operators that have bosonic commutation relations, I would not call the excitations of your oscillator photons.

All of this is for zero temperature. If you want finite temperature. There will be more things to worry about.

 

[Karthiksrao]

Thanks Cryo for your reply. I want to isolate just the interaction of the harmonic resonator with the sine wave. So we can assume its at zero temperature.

I was indeed looking at the excitations as photons. Its interesting that you say these are not photons which are populating the higher energy states. Now that I think of it, I guess you are right because at zero temperature you do expect the photons to be in the ground state.

However, since the coherent state is a superposition of Fock states, I was picturing the the quantum harmonic oscillator in a coherent state to have its higher energy levels to also be populated - since the higher number Fock states do have definite number of photons. I'm clearly missing something elementary. What is wrong with my picturization ?

Thanks

 

[Cryo]

IMHO, 'photon' is a term reserved for excitations of quantum electromagnetic field, usually some sort of propagating field. When you talk about your dipole (antenna) I am picturing a metallic or dielectric bar that houses some sort of mode. Sure, with enough paper you can try to express this mode in terms of photons, but it will not enlighten you, and it will confuse other people that are not used to your terminology. In any case, I don't think it is too important.7

Coherent state can be represented as superposition of Fock states, because the latter provides a complete and orthogonal basis set. Indeed, if your oscillator is driven by a sine-wave you will get a state that has non-zero amplitude for Fock-states with multiple quanta.

What I think you are missing is maths. Words are imprecise and subjective. Can you treat the problem of a driven quantum harmonic oscillator? Can you find the solution for the Hamiltonian:

$\hat{H}=\hbar\omega\left(\hat{a}^{\dagger}\hat{a}+\frac{1}{2}\right)+\alpha\exp\left(-\gamma\left| t\right|\right)\sin\left(\omega t\right)\left(\hat{a}^\dagger+\hat{a}\right)$

i.e. given initial state ($t\to -\infty$) is $|0\rangle$, what is the final state ($t\to \infty$). Where $\omega$ is frequency, $\gamma\to 0$ is a vanishingly small quantity with units of frequency, $t$ is time, $\hat{a}$ is the anihilation operator for you resonator, and $\alpha$ is a real-valued constant with units of energy.

Once you can do this everything becomes clear.

I would suggest E. Merzbacher "Quantum Mechanics", Chapter 10, but many other texts do it too.