# Enhanced phase escape in the washboard potential

So I’m taking some tentative steps towards this open-notebook science / open-research movement, (at least whilst I remain in academia) so here are some examples of recent data I have been taking:

Disclaimer: If this post makes no sense at all, don’t worry. I’m writing a series of posts on ‘Introduction to Josephson Junctions’ which might be useful to read first. However they’re a lot harder to write than posts like this, and I haven’t quite got them up to standard yet.

Here’s a schematic of the washboard potential (the energy landscape of a current-biased Josephson Junction) under the influence of microwave irradiation. As the current is increased, the potential becomes more tilted and the height of the barrier defining the metastable minimum decreases.

The phase is a macroscopic quantum variable, so the wavefunction of the system comprises of discrete eigenstates with individual energies (well, almost… you can’t solve the S.E. exactly for this case as the state is not fully confined, but there are ‘likely’ energy states in the well). The phase of the junction can escape from the potential via thermal activation or quantum tunnelling through the barrier, after which it ‘rolls’ down the potential landscape like a ball on a washboard – hence the analogy. This is detected experimentally as a sudden increase in voltage across the junction (a constantly changing phase across a Josephson Junction corresponds to the appearance of a DC voltage).

Here is a plot of data showing the rate at which the phase escapes from the well as a function of bias current:

The escape is linear as a function of current when plotted on this (somewhat manipulated) scale. Which it should be. The red line is a fit to the thermal activation theory at the temperature of the measurement, 0.065K.

The escape process can also be shown as a histogram, where each escape event is binned according to the current at which it happened. The plot below is for a different junction and temperature, but the shape of the histogram is the same. The fitted line again denotes the expected value from thermal activation theory.

However occasionally you see deviations from this result. Here the escape rate data have a different structure:

This can also be visualised as a histogram, as explained in the standard case. However, here we see that the histogram is doubly-peaked:

I have denoted this process an ‘enhancement’ as the background slope of the escape rate appears to be higher than the fit from thermal activation theory (shown in red). Physically this corresponds to the phase escaping from a higher energy level in the potential well, such that you do not need to tilt the washboard as much to obtain a high probability of an escape event. The histogram gets narrower and the escape rate gets steeper accordingly.

One interesting point is that the enhancement seems to provide exactly a 50-50 level population, which suggests some kind of saturated equilibrium process (as opposed to, say a population inversion).

Also, in this case, the ‘splitting’ – which can be seen from the histogram data, is about 11nA in 8.58uA, or about 0.12%, which is quite small and shows the kind of resolution you can achieve in these experiments. Unfortunately this is difficult to convert to an energy level spacing, as you need to know the ideal critical current Ic0 and exactly where the energy levels are in the well to calculate this.

So what causes the enhancement? It could be a source of interference at a particular frequency, for example the nearby wireless networks. It could also be a thermal enhancement, if the energy level spacing in the washboard is well below kBT. But I’m not quite sure yet as to the exact origin of this behaviour.