Juan Pablo Paz, world-record holder in number of talks given in the Paraty workshops, maintained his title with a very nice first talk at the workshop this year. His talk was titled “Thermodynamics of ion crystals”, and described how trapped ions can be used to study some delicate quantum thermodynamical properties.

It has been known for a while that anomalous results in heat transfer appear when we are dealing with low-dimensional quantum systems. One of the main motivations for this work is to prepare ion trap experiments that measure these effects directly, which hasn’t been done up to now. It is also important to understand heat transfer in ion traps, as heating is an important source of decoherence, and ion traps have been proven more and more useful as quantum simulators.

The effective dimensionality of the ion crystal can be changed by changing the aspect ratio of the Paul trap – Juan Pablo showed some cool animations illustrating this.

One quantity of interest is the local ion temperature, which can be defined using the mean value of the square of the momentum of ion i. As discussed in the question time, defining temperature this way may involve subtle conceptual issues, as it is important to take into account the way that the thermometer will couple to the system. The heat flow on ion i can be defined by calculating the mean force of ion j on i, then the power flowing into i, then doing a sum over j.

Given a few assumptions (ohmic reservoir, coupling of environment to single ions individually), they showed that the dynamics is Gaussian, and the system can be completely described by its characteristic function (the expectation value of the displacement operator). This enabled them to solve the problem exactly, imposing a hot reservoir in one end of the chain, and a cold one in the opposite end.

They could observe the validity of the 2nd Law directly in the exact result: heat flows from hot to cold. They could also prove a no-go theorem for linear quantum absorption refrigerators (QAR) – these have been discussed previously by Kosloff et al., but now we see the non-linearity used by Kosloff et al. is a necessary feature.

They also observe anomalous heat flow: in the Fourier Law, the thermal conductivity C now depends on the system size L. The deviation from an expected (classical) constant value is stronger for zig-zag 2D or 3D crystals, and depends also on disorder, which is obtained by considering fluctuations of the trapping potential, or “artificial decoherence” imposed by hand with laser pulses.

A second anomalous heat transport feature they observed was a non-linear profile in the local temperature between the hot and cold reservoirs. Their systems show an abrupt change in local temperature close to the reservoirs, with a plateau in between. Interestingly, disorder restores the linear, classical behaviour. This restoration is highly dependent on system dimensionality.

This work is especially timely given recent results by Plenio et al., discussing how to directly measure these thermodynamical quantitions in ion traps. Paz et al.’s technique is efficient and near-analytical, and should help in the analysis of these near-future experiments.