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February 17, 2005
Class review — 2/17
Hi Class,
A couple of notes:
Two questions came up regarding the homework. One is why I believe that that I can take a Hamiltonian like:
use the projection theorem:
and rewrite the Hamiltonian like:
Another question came up regarding the operator:
and why I claim that this operator behaves like a rank-1 irreducible tensor for any matrix element.
I have put together cogent explanations of these two points, and I will discuss them at the beginning of class on Tuesday.
The lineshape that I drew for an n-pi pulse is wrong (I went back and looked at some data!) — I mixed up the Rabi and Ramsey envelopes in class. I will clear this up on Tuesday. I have a hard time visualizing how that particular problem works for finite detuning using a Bloch vector picture. So, I'm working on a 3-D Bloch vector simulation that we can play with on Tuesday. I'm using VPython, which is a great tool that I'll show you.
Homework solutions, the pre-flight for Tuesday, and possible topics for the final project will all be posted sometime tomorrow.
Tuesday we'll cement our understanding of quantum two-levels systems and Bloch vectors, talk about Ramsey / clock experiments, and how density matrices and decoherence work in two-level systems.
Today we talked about:
A real two-level quantum system: Hypefine states in an atom. There are polarization selection rules that tell you which states can be connected by an oscillating magnetic field. State detection can be done using resonance fluorescence, which we'll talk about soon. For that type of state detection, one state looks bright and the other dark.
The quantum two-level problem in the presence of an oscillating drive can be solved using a rotating frame, which amounts to defining new coefficients for wavefunctions. See the lecture notes for the gory details, and for an alternative, more modern solution. Using the rotating frame solution, we find new eigenstates which have an energy that depends on the Rabi frequency and detuning.
Rabi's formula for Rabi oscillations can be derived by writing one of the bare eigenstates (the eigenstates in the absence of the coupling) in terms of the new eigenstates. Then, you just time evolve the state using exponential factors, and calculate the probability to be found in the other bare state. Rabi oscillations occur at a rate that depends on the Rabi frequency and detuning. The lineshape associated with Rabi oscillations is surprisingly complicated.
Cheers,
Brian
Posted by Brian at February 17, 2005 05:32 PM
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