Induced transitions, Dipole Approx.

[logic smogic]

In a review question, we are asked to consider a particle of mass m and charge q in a 1-D harmonic oscillator potential V(x). Light is shined on the harm. osc. with E-field $$E=E_{o}cos((\omega)t-kx)$$, where $$k=\omega/c$$.

(Fine, so far. It seems like a Rabi frequency problem, similar to an Ammonia maser, which we studied. I'm assuming we can consider this charge in a harm. osc. as being analagous to an electron in a H-atom?

I'm not exactly sure how to cross over the math for it though... Also, it's clear that $$V(x)=\frac{1}{2}m(\omega_{ho})^2x^2$$ is the potential of the charge, I think. Back to the question...)

The light frequency $$\omega$$ is chosen to be resonant with the transition from the ground state to the first excited state of the harm. osc. Find an expression for the Rabi frequency for light induced transitions between the ground state and first excited state assuming the dipole approximation is valid.

--
Okay, when I started on this question, I reviewed the Ammonia maser.

For that, the Rabi frequency $$\Omega$$ is given by $$\Omega=\eta/\hbar$$, where $$\eta=Ed_{o}$$.
When the frequency is "resonant", it is
$$\omega=\omega_{o}=\frac{2A}{\hbar}$$,
where A is the Bohr radius.

I'm not really sure if I'm thinking about this correctly. Can I cross these results over from the (very different) H-atom? Or in the context of the ground-to-first state transition, is it allowed?

Also, what is the "dipole approximation"? I know that it has to do with assuming that the wavelength of the radiation causing the transition is large compared to the size of the system making the transition, but how does that play into this problem quantitatively?

Is there an expression/equation I can use to ammend the assumptions above? There's nothing in my text on it, and I couldn't find anything more than qualitative on the web.

Thanks much for any pointers!

 

[vanesch]

In a review question, we are asked to consider a particle of mass m and charge q in a 1-D harmonic oscillator potential V(x). Light is shined on the harm. osc. with E-field $$E=E_{o}cos((\omega)t-kx)$$, where $$k=\omega/c$$.

(Fine, so far. It seems like a Rabi frequency problem, similar to an Ammonia maser, which we studied. I'm assuming we can consider this charge in a harm. osc. as being analagous to an electron in a H-atom?

I'm not exactly sure how to cross over the math for it though... Also, it's clear that $$V(x)=\frac{1}{2}m(\omega_{ho})^2x^2$$ is the potential of the charge, I think. Back to the question...)

The light frequency $$\omega$$ is chosen to be resonant with the transition from the ground state to the first excited state of the harm. osc. Find an expression for the Rabi frequency for light induced transitions between the ground state and first excited state assuming the dipole approximation is valid.

--
Okay, when I started on this question, I reviewed the Ammonia maser.

For that, the Rabi frequency $$\Omega$$ is given by $$\Omega=\eta/\hbar$$, where $$\eta=Ed_{o}$$.
When the frequency is "resonant", it is
$$\omega=\omega_{o}=\frac{2A}{\hbar}$$,
where A is the Bohr radius.

I'm not really sure if I'm thinking about this correctly. Can I cross these results over from the (very different) H-atom? Or in the context of the ground-to-first state transition, is it allowed?

Also, what is the "dipole approximation"? I know that it has to do with assuming that the wavelength of the radiation causing the transition is large compared to the size of the system making the transition, but how does that play into this problem quantitatively?

Is there an expression/equation I can use to ammend the assumptions above? There's nothing in my text on it, and I couldn't find anything more than qualitative on the web.

Thanks much for any pointers!

I have to say I'm a bit rusty on Rabi oscillations, and I don't recall anymore any precise result. That's why I refrained from answering, but given that nobody has answered yet, I'll give it a try.
Rabi oscillations (if I recall correctly) are the time-dependent solution of a 2-state system with energies E0 and E1, with corresponding stationary states psi_0 and psi_1. The Hamiltonian is hence given by $$H_0 = |\psi_0> E0 <\psi_0| + |\psi_1>E1<\psi_1|$$ over a 2-dimensional Hilbert space, on which one introduces a small perturbation which is harmonic:
$$\Delta H = \lambda \exp(i \omega t)$$ where $$\omega \sim \frac{E1-E0}{\hbar}$$ times a unity matrix.

The dipole approximation means that we do not have to consider the spatial variation of the electric field in the EM wave (in other words, that we can consider that the E-field where it matters, is only dependent on time and not on position).

As such, if we can neglect the other states of the harmonic oscillator but the ground state and the first excited state, and if we can use the dipole approximation, then the problem becomes equivalent to a Rabi problem as pointed out above.

Now, the only thing I remember about the Rabi problem is that it is exactly soluble, but I don't remember the solution.

 

[logic smogic]

Ok, thanks. That certainly helps a bit.