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February 27, 2005
Final project topics appearing
Hey Class,
A few of you have noticed already that final project topics are appearing on the website. I will continue to update that list as ideas come to me. Your best bet will be to check that list and just to browse the home pages of different AMO groups.
Brian
Posted by Brian at 04:46 PM | Comments (0)
February 24, 2005
Class review — 2/24
Hi Class,
The pre-flights for Tuesday are all ready to go. Lecture notes and the VPython simulations from today will be posted sometime tomorrow. I'm also going to try to grade homework #1 this weekend.
Let me know if you have more trouble with the pdf files that I've posted.
Today we learned about:
Spin echo. This is a useful trick in a Ramsey experiment, which can take out the effect of a spread in Larmor frequencies (or detunings). Note that this trick ruins a Ramsey experiment for use as a clock.
Density matrices. We use density matrices to talk about real-life measurements of quantum systems. The density matric tells us everything we can know about an ensemble of quantum systems (which we always have to use to make measurements).
Time evolution of the density matrix. We can translate the Liouville equation into equations of motion for the matrix elements of the density matrix. Coherences drive changes in the population, and differences in the population drive changes in the coherences.
Alternative Bloch vector picture based on the density matrix. In this formalism, the x and y components of the Bloch vector are determined by the coherences, and the z component by the difference in the populations. A Bloch vector which has a length less than one represents a mixed state. Classical states all have a Bloch vector which lies on the z-axis.
The Liouville equation with decoherence added in. We're going to talk about this more next time, and look at some examples.
Cheers,
Brian
Posted by Brian at 04:29 PM | Comments (0)
February 22, 2005
Class review — 2/22
Hey Class,
The VPython files (and a link to the VPython website) are on the website class notes for today. Read the VPython website carefully — you need to install Python first, and then the visual module.
The lecture notes posted for 2/17 cover today's class. Pre-flights for Thursday are ready!
Today we talked about:
Visual simulations of the Bloch vector and Rabi oscillations. Play with them - you might learn something!
If you look at Rabi oscillations over an ensemble that has atoms with different Larmor frequencies or pulse areas, then you need to average what you see over that ensemble. Examples are finite transit times in a beam experiment, or the effects of an inhomogeneous field.
Ramsey experiments, which are confusing to just about everyone. You can play with the VPython files to get a feeling for how the lineshape comes about. You should be able to understand the lineshape for small detunings when the free precession time is long by using the Bloch vector picture. The key ideas:
Use a single rotating frame, at the drive frequency.
The first pi/2 pulse rotates the Bloch vector almost into the x-y plane.
During the free precession time, the Bloch vector precesses with an angular frequency equal to the detuning.
The second pi/2 pulse rotates the Bloch vector somewhere, with a projection onto the z-axis not equal to 1. One fact: the azimuthal angle in the plane before the second pulse is roughly equal to delta*t. Second fact: that angle is fixed with respect to the effective torque vector. If the effective torque vector is almost in the x-y plane, then the angle between the plane and the Bloch vector after the second pi/2 pulse is just delta*t. Then do some geometry!
Cheers,
Brian
Posted by Brian at 05:02 PM | Comments (0)
February 18, 2005
Updates to website
Hi Class,
The pre-flights for 2/22 are ready, the lecture notes from 2/17 are posted, and homework #1 solutions are up on the course site.
I need to think about possible topics for class projects more, so look for some info on that after the weekend.
Cheers,
Brian
Posted by Brian at 06:41 PM | Comments (0)
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 05:32 PM | Comments (0)
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:
All two-level systems are equivalent to a spin-1/2 particle in a magnetic field.
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.
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.
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).
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 10:08 PM | Comments (0)
February 11, 2005
Preflight for 2/15 ready
Hey Class,
The pre-flights for Tuesday, February 15 are ready. Check the class notes on the website for reading material. The lecture notes and Powerpoint slides from class on 2/10 are also on the website now.
--Brian
Posted by Brian at 03:39 PM | Comments (0)
February 10, 2005
Class review — 2/10
Hey Class,
Remember to check the blog for comments on the homework.
Pre-flights for Tuesday and the lecture notes from today will be ready by tomorrow (Friday) night.
Here's what we talked about today:
RF Paul traps. Classical motion of ions in a Paul trap is described by the Matheiu equations, which have exact solutions. An approximate solution can be put together using the "pseudopotential" approximation, where the slow, secular motion is separated from the fast micromotion at the RF frequency. Paul traps have regions of stable motion determined by the charge-to-mass ratio and the DC and RF voltages.
We started on 2-state systems! We'll use spin-1/2 language, where we always imagine a static magnetic field present along the z-direction.
Larmor precession. If a spin-1/2 particle is not prepared in an eigenstate, then the spin precesses around the magnetic field. Spin precession just means that áSxñ and áSyñ oscillate out of phase.
Mixing of states when we apply a static, transverse field. The new eigenstates rotate as the ratio of the transverse (B^) to z-field (B0) is varied. The energy of the two states also varies in a hyperbolic fashion as the ratio B^/B0changes.
Landau-Zener transitions. Note that the formula I showed today is only valid when B0 is changed with B^ fixed (that's the case that Zener considers in his paper). The beautiful two-state picture (with a transverse field) contains all of the Landau-Zener transition and Majorana physics in it.
See you next week!
Brian
Posted by Brian at 08:06 PM | Comments (0)
February 08, 2005
Class review — 2/8
Hi Class,
The pre-flights and reading material for Thursday are ready. I'll probably finish up magnetic and electric traps, and then move on to two-level systems.
Check the Homework #1 entry on the blog — new comments have already appeared!
Today we talked about:
Earnshaw's and Wing's theorem. The upshot is that we cannot trap charged particles using static electric fields, and we can only trap atoms with negative magnetic moments in magnetic traps. Wing's theorem is also a cool example of what you can do with vector calculus.
Quadrupole traps. You can trap atoms using quadrupole traps, but the hole of death will limit the density. The hole of death comes from non-adiabatic Marjorana transitions.
The adiabatic theorem. This is deep thing, man. It tells us that any transformation of a Hamiltonian is adiabatic if it is slow enough compared with "internal" timescales. There is also a specific statement of the adiabatic theorem for quantum mechanical systems, which relates the fractional rate of change of the Hamiltonian to the characteristic energy gap in the system.
Majorana transitions. These happen for a magnetic dipole in a field which changes direction, and cause trouble near the middle of a quadrupole trap. Near the center of a quadrupole trap, the Larmor frequency is small, and non-adiabatic spin-flip transitions become more likely.
TOP traps. This is a quadrupole trap with an additional rotating bias field. The hole death gets moved away from the atoms in this kind of trap.
Ioffe-Pritchard traps. These are a complicated mess of coils, which confine atoms using magnetic forces. They use static fields, and nominally have a bias field with a fixed direction.
Whew! See you Thursday!
Brian
Posted by Brian at 03:46 PM | Comments (0)
February 04, 2005
Homework #1
Due 2/17!
You can use this entry as a discussion forum for homework #1 questions.
Notes:
Problem 3: Find a pair of "magic states" at finite (non-zero) field, and not in the high-field limit. One Zeeman level should be in the F=2 manifold, and the other from the F=1 manifold.
There was a missing
in the parameter "x" for the lecture notes on the Breit-Rabi formula. Updated lecture notes and Powerpoint slides are on the website now.You may want to do problem 2 before problem 1.
Problem 1(a): The text should read "Think about what the projection theorem tells you about <S> and <J>."
Problem 1(b): Think about why I said to evaluate the matrix element for one pair of mF states. What does the Wigner-Eckart theorem tell you about the matrix elements of I·S? Are we working with a scalar or a vector operator?
Note that the real question for Problem 1 is to find b.
In problem 1, the atom cannot actually be He, and the electron configuration cannot be (1s)(2s). In order to get P1/2 and P3/2 states, the total S=1/2. This is not possible with two electrons! So, imagine that this is some other multi-electron atom with an odd number of electrons.
Posted by Brian at 11:24 AM | Comments (1)
February 03, 2005
Class Review — 2/3
The pre-flights for Tuesday should be up tomorrow; I'll send around another email. The readings are already set, so see the class notes on the website for 2/8. Slogging through the material by Ghosh is not for the faint-of-heart. Brian King's thesis is a better place to start to learn about modern rf traps.
Review from today:
Today we learned all about the fun Zeeman effect, which is just how anything with a magnetic moment interacts with a static magnetic field.
The real story that we care about is for alkali atoms in magnetic fields. Some good things to keep in mind:
In the low field limit, the Zeeman effect is treated as a pertubation to the hyperfine Hamiltonian. The energy shift is m0gFmFB. Note that gF has a different sign for states with different F! That means that sometimes the m>0 states increase in energy with increasing magnetic fields, and sometimes the m<0 states increase in energy with increasing magnetic fields. This happens because the hyperfine interaction mixes together states with different magnetic moments (which is always dominated by the electron magnetic moment).
Always remember how the stretched states shift with field. These states have a fixed magnetic moment, which is, to a good approximation, the electron magnetic moment.
At high fields, the states group by their mJ quantum numbers. In this limit, the hyperfine interaction is treated as a pertubation to the Zeeman Hamiltonian. The hyperfine interaction breaks the degeneracy between states with the same mJ.
At intermediate fields, we have to solve the problem exactly. For the ground state this is not too hard — see the homework! The magnetic moment of the hyperfine states depends on field, which is because the makeup of the states in the |mI,mS> basis is changing!
We also started talking about traps today. We learned that nature foils our ability to trap both neutral and charged atoms. But, experimentalists are clever, and we trap all kinds of atoms anyway! We'll learn more next class.
Cheers,
Brian
Posted by Brian at 02:30 PM | Comments (0)
February 01, 2005
Class Review — 2/1
Hi Class,
I was talking trash at the end of class today. In fact, the hyperfine ground state splitting is small compared with room temperature, so the two states will be equally occupied. We see this any time we do spectroscopy in a room temperature vapor cell! Most AMO experiments begin at temperatures below 1 mK, and then only the lower ground state is occupied. The lifetime of the upper ground state is very long, and we will estimate it when we talk about light interacting with atoms.
Here is a review of class today:
Forget trying to calculate the electronic structure of real atoms. Screening is a serious effect in alkali atoms, for example. We use a semi-empirical formulation (quantum defect theory) for estimating quantities, and look up transition wavelengths in databases when we really care. The NIST database and the data from Dan Steck and Mike Gehm are valuable resources.
Hyperfine structure comes from the interaction of the electron with the nuclear electric and magnetic moments. The hyperfine interactions are diagonal in the F=I+J basis, where F is the total angular momentum. So, we construct states as |F,mF>, which are superpositions of states in the |mI,mS> basis. The hyperfine splittings (ground and excited states) can be calculated using some formulas and the A and B constants, which are measured.
Isotope shifts are important, since optical transitions can be different for different isotopes of the same atom. A big contributor to isotope shifts is the finite volume of the nucleus. I forgot to mention that there are other effects, such as effects due to the internal structure of the nucleus (non-uniform charge and current distributions).
These reviews will soon appear on a blog that I am starting for the class. You will be able to leave comments on the lectures and use it as a discussion board. More information to follow!
Brian
Posted by Brian at 02:30 PM | Comments (0)