How a particle tells time

One of the first things you learn about quantum mechanics is that particles have a wavelength, and thus a frequency. If the particle is in rest, this frequency is the Compton frequency and proportional to the particles’ rest mass. It appears in the wavefunction of the particle at rest as a phase. This means basically the particle oscillates, even if it doesn’t move, with a frequency directly linked to its mass.

The precision of atomic clocks in use today relies on the precise measurement of transition frequencies between energy levels in atoms which serve as reference for an oscillator. But via the Compton wavelength, the mass of a (stable) particle is also a reference for an oscillator. Can one therefore use a single particle to measure the
passing of time?

This is the question Holger Müller and his collaborators from the University of Berkeley have addressed in a neat experiment that was published in the recent issue of Science:

A Clock Directly Linking Time to a Particle's Mass
Shau-Yu Lan, Pei-Chen Kuan, Brian Estey, Damon English, Justin M. Brown, Michael A. Hohensee, Holger Müller
Science, DOI: 10.1126/science.1230767

As you can tell from the title of the article, the answer is Yes, one can use a single particle to measure time! They have done it, with the particle in question a Cesium atom, and call it a “Compton clock.” The main difficulty is that the oscillation frequency is very high, far beyond what is measurable today. To make it indirectly measureable, they had to cleverly combine two main ingredients, an atomic interferometer and a frequency comb.

The atomic interferometer works as follows. The atom is hit by two laser pulses, one pulse with frequency a little higher than the laser’s direct output frequency, and one with a frequency a little lower. This splits the wavefunction of the atom. A couple more precisely timed laser pulses are then used to let the wavefunction converge again. It interfers with itself and the interference pattern can be measured in repeating this process.

The relevant aspect of the atom interferometry here is that the phase accumulated by each part of the wave-function depends on the output frequency of the laser, the difference in frequency between the two pulses (tiny in comparison to the output frequency), as well as on the path taken. The path-dependent phase itself depends on the mass of the atom because the two parts of the wavefunction are not in rest with each other. So then the experimentalist can turn a knob and change the difference between the frequencies of the two pulses until the interference pattern vanishes. If the interference pattern vanishes, one then has a fixed relation between the mass of the particle, the output frequency of the laser, and the difference between the pulse frequencies.

So far, so good. If one now knows the frequency of the laser, one can measure the particle’s mass by looking at the frequency split of the pulses needed to get the interference to vanish. Alas, this is not what one wants for the purpose of a clock, which should not rely on an additional, external, measurement.

This is where the frequency comb comes in. In 2005, frequency combs brought a Nobel Prize to John Hall and Theodor Hänsch. Before the invention of the frequency comb, it was not possible to accurately determine absolute frequencies in the optical range. Relative frequencies, yes, but not absolute ones. They’re just too fast to be counted by any electronic means. Frequency combs address this issue by relating very high optical frequencies to considerably lower frequencies, which then can be counted. This is done by pulsing a low frequency signal . If one takes the Fourier transformation of such a pulsed signal, one obtains (ideally) a series of peaks – the frequency comb – whose positions are exactly known (these are the higher harmonics of the low frequency signal). If one knows the pulse pattern of the laser comb one can then substitute the measurement of a very high frequency with that of a considerably lower frequency. Ingenious!

And more ingenuity. Mueller and his collaborators use a frequency comb to self-reference the (tiny) difference in the laser pulses with the output frequency of the laser. The relation between both is then known and given by the pulse pattern of the frequency comb. This way, one gets rid of one parameter and has a direct relation between a measurable frequency and the mass of the particle: It’s a clock!

For what the precision of this clock is concerned however, it is orders of magnitude below today’s state-of-the-art atomic clocks. So unless there are truly dramatic improvements to atom interferometry, nobody is going to use the Compton clock in practice any time soon.

But this clock works both ways. It doesn’t only relate a mass to time (oscillation frequency), but also the other way round. Thus, one can use the Compton clock to measure mass if one has a time reference. With the "Avogadro Project", an enourmously precisely manufactured silicon crystals containing an accurately known number of atoms, one can scale up a single atom to a large number and macroscopic masses. This way the Compton clock might one day be used to define a standard of mass.