Theory - Time of Flight Measurements

A time of flight (TOF) measurement is literally what it sounds like - it is the measurement of time it takes to “fly” through a certain distance. These measurement appear in various experimental contexts in physics. In our AMO experiment, we use TOF measurements to figure out the temperature of our cold atom cloud.

Once the atoms are prepared in our trap, we know that they will be at a very low temperature. However, we don’t really have any practical thermometer to measure the temperature we’ve achieved! In order to do this, we can use some thermal and statistical physics. The temperature of an atomic cloud can be inferred from the velocity of the atoms. Thus, if we release the atoms from the trap, the cold atom cloud will start expanding. We can release the atoms from the trap and let it expand for some amount of time. Then, we measure the size of the cloud. This is our TOF measurement, and the time it takes to expand a certain distance gives us the initial velocity of our cloud, giving us a measurement of the temperature. Mathematically, we use the Maxwell-Boltzmann velocity distribution to find our velocity.

Let’s now go over, practically, how a TOF measurement works. Once the atoms are trapped, we then turn off the MOT coils to let the atomic cloud expand. Using a CCD camera for imagining, we take a measurement after a given expansion time has passed. We can use these fluorescence measurements to determine the Maxwell-Boltzmann distribution and obtain the mean initial velocity of the atomic cloud.

The particle number density under expansion is given by

$$n(\vec{r}, t) = \frac{e^{r^2 / (2\sigma_f ^2)}}{(2\pi \sigma_f^2)^{3/2}} ,$$

Where the important values are $t$, the expansion time, and $\sigma_f$ the cloud width after the time of expansion. This quantity can be calculated as

$$\sigma_f^2 = \sigma_i^2 + k_BTt^2/m,$$

Where $\sigma_i$ is the initial width, $m$ is the atomic mass, and $T$ is the temperature. Intuitively, we can reason as so: for heavier atoms, the expansion is slower, and the expansion is proportional to both the temperature and the time. We should expect that low temperature clouds expand very slowly. Notice that the function of $n$ is effectively a Gaussian distribution. When imaging, we can fit a Gaussian to the cloud to determine the standard deviation, or the width $\sigma_f$ after some time. Once we determine this value, we can use the measurement, the initial width of the cloud, and the mass of the atoms to determine the temperature.

References

• Wikipedia Link on TOF measurements in physics

• TOF Measurements for Mg Atoms (ETH Zurich)