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March 31, 2005
Notifications should be working now!
Hi Class,
Now, I hope, you'll actually get a notification when I add an entry to the website.
Brian
Posted by Brian at 03:52 PM | Comments (0)
March 30, 2005
Note from Man Hong on Casimir effect
Hello Dr. DeMarco,
I intended to post a comment somewhere in the course blog, but I could find no way to do it... Anyway, there is a recent claim by Jaffe (MIT) saying that Casimir effect can be derived without reference to zero point energies.
Hopefully, this article would be interesting to people in this class.
Ref:
The Casimir Effect and the Quantum Vacuum, hep-th/0503158
Regards,
Man Hong
Posted by Brian at 10:58 AM | Comments (0)
Change in Homework 2
Hi Class,
I've made a slight change to problem 2 on Homework 2. Don't do the case for pi polarization. To do that problem correctly requires a lot of work -- much more than I want you to do.
Cheers,
Brian
Posted by Brian at 09:39 AM | Comments (0)
March 29, 2005
March 29 — class review;
Hi Class,
Casimir's wife's name was Josina Jonker; they were married in 1933.
Remember! No class on April 5, and the homework is due on April 7.
There is a pre-flight for Thursday.
We got embroiled in a discussion of g(2) at the end of class. Correlation functions are confusing, and I'll bring a concrete example to next class.
Today we talked about:
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Quantum States of light. We talked about Fock, or number, states. For these states, the expectation value of the electric field is 0, but the variance depends on n.
Coherent states. These states are produced by a laser (maybe) and classical currents (an antenna). Coherent states are a specific superposition of number states, where the probability distribution is Poisson-ian in the number state basis. Coherent states are also characterized by a single parameter that is a complex number. The expectation value of the field is finite, and shows classical-like behavior. That means that the electric field strength oscillates in time. What is not classical about coherent states is the extra quantum "fuzz" on the electric field, although the size of the fuzz as a fraction of the electric field strength shrinks as the expectation value of n grows.
We talked about the correlation function g(2) as something that distinguishes Fock from coherent states. More on this next time!
Cheers,
Brian
Posted by Brian at 04:16 PM | Comments (0)
March 27, 2005
Updated Homework #2
Hey Class,
I had to make a change to problem 2 on Homework number 2 (I forgot to normalize the cross-section by the incident intensity). A corrected version is on the website.
Brian
Posted by Brian at 08:12 AM | Comments (0)
March 26, 2005
Pre-flights ready for 3/29!
Hey Class,
The pre-flights for 3/29 are ready (and I scanned some more reading material).
Also: an important note: Class will be cancelled on Tuesday April 5.
See you Tuesday!
Cheers,
Brian
Posted by Brian at 02:04 PM | Comments (0)
March 22, 2005
Homework 2 update
Hey Class,
Peter has been beta testing Homework #2 for us. I've corrected one error that he found — the picture in problem 1 is not consistent with the Hamiltonian H0. I've corrected the online version. Stick with what I wrote down for H0 (see the updated picture), otherwise you have to define the detunings differently.
Also, wait to do problem 2 until we talk about spontaneous emission in class next week. We don't really have what most people mean by "scattering" yet. I'll basically give you the answer to this question in class.
Cheers,
Brian
Posted by Brian at 10:11 AM | Comments (0)
March 18, 2005
Note from Shizhong on the Casimir and Casimir-Polder effects
I checked the Casimir Effect. I should apologize for one misleading statement made in class. The first theoretical justification of the van der Waals force is made by F. London in 1930 using QM, who showed that there is a R^{-6} behaviour. In 1940, Overbeek did the experiment and he found that when distance between two particles is small, then we have R^{-6} dependence. However, if distance is large, then one has R^{-7} dependence. The R^{-7} dependence comes actually from the retarded potential. Casimir and Polder did the calculation taking into account the retarded potential and did find the R^{-7} potential.
The Casimir physics in this problem is that the R-dependence of the potential is _INDEPENDENT_ of the atom shell structure. Since it is tempting to argue that the van der Waals force comes from the fluctuation of electric dipoles in each of the atom, it is hard to understand why the shell structure of atom does not matter so much. It is at this time, when Casimir met Bohr(1947), Bohr suggested that this kind of universal R-dependence may have something to do with vacuum energy. Casimir did the calculation and did find the result R^{-7}. The more elabrate calculation can be found in the following paper Spruch L et al PRA _18_ 845(1978).
With best wishes,
Shizhong
Posted by Brian at 09:35 AM | Comments (0)
Class Review — 3/17
Hey Class,
Lecture notes on stimulated Raman transitions / lambda-systems are up on the website now. Pre-flights for next class will go up some time next week.
Have a great Spring Break!
Yesterday we talked about:
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Quantization of the EM field. The procedure to turn the EM field into something qauntum mechanical is to first quantize the field and then second quantize the field. We use the Coulomb, or transverse, gauge for working with quantum EM fields (and in QED).
First quantization is purely classical, and just means writing down the fields as expansions in an othornormal basis set of functions. In class, we did this by considering the fields in free space and choosing a box as our quantization volume. We're going to care about the vector potential A when we deal with light interacting with an atom, so we write down the wave equation for A and solve it; E and B are just derivatives of A.
Then we write down the total energy in the field, and that's our classical Hamiltonian. Cleverly, we can invent real valued coordinates Q and P and show that this Hamiltonian is just a simple harmonic oscillator Hamiltonian in Q and P. The fields are made quantum mechanical (this is second quantization) by the usual procedure — the coordinates are made operators and we introduce raising and lowering operators. The Hamiltonian is then diagonal in the mode occupation basis, or photon number basis.
At this point, the vacuum energy shows up. This is real, and hard to understand. The vacuum energy has physical consequences, such as in the Casimir and Casimir-Polder effects.
Cheers,
Brian
Posted by Brian at 09:23 AM | Comments (0)
March 15, 2005
3/15 — Class Review
Hey Class,
Pre-flights for next class are up!
Today we talked about:
----------------------
Stimulated Raman transitions. We can drive Rabi oscillations between two ground states in a three-state system by applying light at two different frequencies. Adiabatic elimination is an approximation that lets us ignore the excited states. More or less, the beat frequency from the two beams drives the transition between ground states.
Recoil. When atoms undergo stimulated Raman transitions, they recoil. This can be worked out from the semi-classical theory, but a photon picture becomes more useful. Recoil makes stimulated Raman transitions really useful for making atom interferometers.
Cheers,
Brian
Posted by Brian at 03:12 PM | Comments (0)
March 13, 2005
Homework 2 ready
Hey class,
Homework #2 is ready and on the website. It's due April 5!
Brian
Posted by Brian at 03:05 PM | Comments (0)
March 11, 2005
Class review — 3/10
Hey Class,
There is no pre-flight for Tuesday — we are playing pre-flight catchup. Lecture notes will be posted sometime today or tomorrow.
Remember! You need to tell me something about your project by March 15!
In the summary below (check the blog), there is a comment on Matt White's comment on the so-called "dipole approximation".
Yesterday in class we talked about:
The AMO secret handshake — what AMO physicists really mean by polarization. We talk about light having three polarizations: right-circular, left-circular, and pi.
To calculate how light really couples ground and excited electronic states, we have to calculate matrix elements of r. There is a big long formula for working out the angular-momentum part of that matrix element, but most AMO people use handy charts. The reduced matrix element for r must be measured, and is on the order of ea0.
So-called "lin perp lin" lattices work because when light is far detuned from a resonance (but not too far compared with the fine structure), the potential experienced by atoms is polarization-dependent (it's also state-dependent).
We can see clean, direct Rabi oscillations between ground and excited states (without spontaneous emission) if we use a "forbidden" transition. We saw an example in 40Ca+.
Matt White made a comment about electric quadrupole transitions being possible in our semi-classical picture if we expand the operator:
to higher order. I think that he is basically right. But, expanding that operator to higher order (or just not expanding it at all, which gives you recoil) is basically admitting to photons. We'll see that this operator is intimately connected to the idea of a photon.
Cheers,
Brian
Posted by Brian at 10:14 AM | Comments (0)
March 09, 2005
Pre-flight back online
Hey everyone,
Ellen mentioned to me that the pre-flight claimed it was already closed. I fixed that problem, and updated one question. Let me know if you have any trouble!
Brian
Posted by Brian at 03:07 PM | Comments (0)
March 08, 2005
Class review — 3/8
Hey Class,
The pre-flight for Thursday is ready. Remember that you need to tell me something about your final project soon!
Today we talked about some new things:
Dipole traps. A focused laser beam can trap atoms! The trap depth and trapping frequencies are functions of the light intensity and detuning. A far-detuned trap can confine all Zeeman levels in a hyperfine ground state manifold, unlike a magnetic trap. Traps can be red-detuned (focused beam) or blue-detuned (focused beam with a dark spot in the middle).
Optical lattices. One type of optical lattice is formed from a standing wave of light. Lattices can be 1, 2, or 3 dimensional.
Cheers,
Brian
Posted by Brian at 06:23 PM | Comments (0)
March 03, 2005
Class review — 3/3
Hey Class,
Boy am I tired. Expect many updates to the website tomorrow, like the last couple set of lecture notes and the pre-flights for Tuesday.
Today an issue came up about the energies of the "dressed states" of an atom interacting with light. I worked out the details very carefully after class. I was right about which state is the ground state. Remember — we're working in the interaction picture, and to get the total energy we have to add the energy from H0. This type of solution is fundamentally different that what we did with the two-level problem before. I will explain the solution very carefully on Tuesday. You can check out the lecture notes on this topic once they're up tomorrow.
Today we learned about:
T1 processes are not the same as decay. They go along with an interaction with a bath. T1 processes cannot occur without T2 processes at the same time.
A model: classical atom interacting with classical light. When the electron gets moving, it radiates. This is equivalent to a damping force, so the whole problem looks like a damped, forced harmonic oscillator. The emitted radiation has a Lorentzian lineshape. Our ability to drive the atom has the same Lorentzian lineshape.
The physics of a quantum atom interacting with classical light. This just looks like a two-level system! We looked at the physics in the limit of far detuning, where the ground state shift is
For "red-detuned" light, the atom is attracted to regions of high intensity.
Cheers,
Brian
Posted by Brian at 08:15 PM | Comments (0)
March 01, 2005
Class review — 3/1/2005
Hey Class,
Note that there is no pre-flight for Thursday. I want to get the pre-flights and the classes synchronized. Also, there was a problem with the example that I gave regarding Rabi oscillations in the presence of a T1-type process. I will clear that up next class. And, the lecture notes from today need some cleaning up; they will probably not be posted before Thursday.
Today we talked about:
We reviewed the equations of motion for the density matrix and Bloch vector from last class.
We considered a T2-type process in a Ramsey experiment. With a fluctuating magnetic field present during the free-evolution time, decoherence results from an ensemble measurement. Note that decoherence is showing up with only pure Hamiltonian evolution. With T2-type decoherence present, the coherences decay exponentially in time.
We started talking about light, and all of the ways that light can interact with an atom. Only two of these processes (plain Rabi oscillations and stimulted Raman transitions) can be explained semi-classically (quantum atom but classical field). You can't get spontaneous emission without quantizing the electro-magnetic field.
We laid out a cheesy (mmm...cheese) classical model where light is an oscillating electric field driving an electron that is bound harmonically to the nucleas. The electron will only be driven strongly when the electric field oscillates at the resonance frequency, and when the electron moves it will radiate!
Cheers,
Brian
Posted by Brian at 04:08 PM | Comments (0)