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February 15, 2005

Class Review — 2/15

Hi Class,

Remember — homework #1 is due Thursday. Check the blog for corrections and hints! I picked these problems because I think that you will learn a lot from doing them, so investing some time is probably worth your while.

There's no pre-flight for Thursday. We're going to work on the quantum two-level problem, and fun stuff like Ramsey's method of separated oscillatory fields. I'll also introduce density matrices and decoherence. You can review relevant material from Cohen-Tannoudji's and Eberly's book if you like.

Today we talked about:

  1. All two-level systems are equivalent to a spin-1/2 particle in a magnetic field.

  2. Classical magnetic resonance. This gets expectation values for the quantum system correct, but cannot get the fluctuations right. Using the rotating frame, one can see that the effect of an oscillating transverse field is to create a static effective torque. The direction of the effective torque depends on the detuning and the magnitude of the oscillating field. The spin vector precesses around the effective torque (call it a magnetic field, if you will) in the rotating frame.

  3. The Bloch, or pseudo-spin vector, can be rotated to point in any direction by using a combination of pulses with different phases and durations.

  4. The pulse-area theorem. The rotation angle associated with the action of a time-dependent oscillating field only depends on the pulse area (time-integral of the magnitude of the oscillating field for a resonant drive).

  5. Adiabatic rapid passage (ARP). It can be tough to flip a spin using a pi-pulse if you don't have good control or knowledge over the Larmor frequency and transverse field magnitude. ARP is another way to flip the spin — you just start with the drive far detuned, and sweep across resonance. In the rotating frame, the pseudo-spin adiabatically follows the effective magnetic field.

Cheers,

Brian

Posted by Brian at February 15, 2005 10:08 PM

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