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# Precision Measurement of Gravity with Cold Atoms in an Optical Lattice and Comparison with a Classical Gravimeter

## The Paper

The .pdf file can be found here.

An external link to the paper is here.

The DOI is 10.1103/PhysRevLett.106.038501

Monday's PowerPoint Presentation can be found here.

## Discussion

I will be checking this page quite frequently up until JC starts Monday at 3:35, so please feel free to post questions!

### Extra resources

• A short synopsis of this paper can be found here.
• A more detailed exposition can be found here
• Also see this page, which contains a description and great diagram for a nearly identical setup. They trap Yb atoms instead, so their laser frequencies are different (“green” MOT instead of “red”), but the energy diagram and principle of operation is exactly the same.

### Overview

They measured $g$ in three different ways. In the order presented in the paper:

1. By wiggling the magnitude of an optical lattice and finding a resonant frequency where the trapped atoms spread apart.
2. Using a laser interferometer where one arm is free to fall
3. By watching oscillations of atoms trapped in an optical lattice.

These three methods are compared. #2 is the most accurate, and the easiest to understand. #3 is less accurate, but easier to understand than #1, so I'll start with this one.

Their conclusion is that the precision and accuracy of method #1 (as verified by method #2) is closing in on some of the best available measurements, and might someday enable experimental tests of whatever gravitational contortions the theorists can dream up.

### Method 3: Bloch oscillations

• Step 1: Load Sr atoms into a magneto-optical trap (MOT). (More on this later.)
• Step 2: Create a standing wave by reflecting a laser beam back on itself. Due to the Stark effect (E-fields!), the neutral Rb atoms gather at the points of lowest laser intensity. If you were to look at the atoms with the CCD camera at this point, you would only see a fuzzy blob! The resolution of the camera is not high enough to distinguish individual “lattice sites”.
• Step 3: The timing of all this is critical. Once the atoms are loaded into the optical lattice, they wait for a specific time duration and then suddenly turn off the lattice. As we watch with the CCD camera, we'll see the cloud of atoms begin to separate depending on the momentum distribution they had before the lattice was switched off.
• Step 4: Plot several pretty pictures as a function of $t$. VOILA! The momentum distribution is periodic with time! But how can this be?

#### Bloch oscillations explained

Okay, so here is my brief (and likely overly simplistic) introduction to Bloch oscillations.

• First off, if you have a periodic potential (such as $V\left(x\right)=V\left(x+a\right)$), this will give rise to a periodic dispersion relation (such as $E\left(p\right)=E\left(p+h/a\right)$). If you believe me and could care less about the details, great. If not, go read up on Bloch's theorem.
• Suppose we apply a constant force $F$, so that we have $p\left(t\right)=Ft$. Because of the oscillatory dispersion relation, this means that the energy will oscillate in time! i.e. $E\left(p\right)=E\left(Ft\right)$.
• Now consider the (group) velocity, given simply by $v_g=dE/dp=\dot{E}/F$. So the group velocity is also oscillatory! This means that in real space, the particle will oscillate back and forth.
• We don't observe this in normal crystal lattices because of the short scattering time, i.e. friction due to impurities, phonons, etc. However, an optical lattice provides a near-ideal system and Bloch oscillations can be observed for thousands of cycles!
• It's easy to calculate the frequency. Given a lattice periodicity $a$, you'll have to take my word for it that the dispersion relation has a period of $h/a$, where $h$ is the Planck constant. In one period then, we have $Ft=h/a$, giving a Bloch frequency of $\nu_B=1/t=Fa/h$. Now, the periodicity in our case is just half the wavelength of laser beam used for the lattice ($a=\lambda_L/2$) and the force is just gravity ($F=mg$). This yields the expression given in the paper: $\nu_B=mg\lambda_L/2h$. Tada!

So, to summarize, a constant force applied to particles in a nearly ideal lattice will cause them to oscillate in time. By measuring the frequency of this oscillation, we can determine the magnitude of the force. They use this to measure the local gravitational constant to 6 ppm. It's not as accurate as the other methods due to the systematic errors that enter in.

### Method 1: Amplitude oscillations

This method is not quite as straight-forward. By modulating the amplitude (think “shaking”)of the optical lattice at a multiple of the Bloch frequency, it causes atoms to tunnel to neighboring lattice sites, and the “fuzzy blob” that we can see with the CCD grows. By sweeping frequency while watching the size of the blob, they are able to determine the Bloch frequency very accurately.

### Extra terms

If you don't know what a MOT is, spend 5 - 10 mins and read the wikiP article. It explains it better than I can here. When the PRL paper refers to a “blue” MOT and “red” MOT, they literally mean the color of light used to trap the atoms. The blue MOT uses more energetic light and can be thought of as a “rough” trap. The red MOT is less energetic, meaning the atoms receive less of a kick when they absorb a photon. This allows them to cool the Sr atoms even further. ($0.6 \mu K$!)

Another term that may be unfamiliar is a “Zeeman Slower”. This is simply a “cone” of coils meant to produce a weak magnetic field at one end, and a strong field at the other. When Sr atoms come out of the oven, they have a more or less well defined velocity. But once the atoms receive a “kick” from the incident laser beam, they will no longer be in resonance. By applying a more intense magnetic field as the atoms slow down, the resulting Zeeman shift “retunes” the resonant frequency so that most of the atoms can still be slowed by the same frequency of laser light.

### Questions

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groups/journal_club/2011spring/week4.txt · Last modified: 2011/02/13 21:56 by geppert