Mark Mitchison
Trinity College Dublin
ZOOM LINK TO JOIN IN: http://s.ic.fo/QTD_Transport191020
Monday Oct 19, 2020 / 20:00-20:30 CEST
Thermodynamics of precision in quantum nano-machines
Fluctuations strongly affect the dynamics and functionality of nanoscale thermal machines. Recent developments in stochastic thermodynamics have shown that fluctuations in many far-from-equilibrium systems are constrained by the rate of entropy production via so-called thermodynamic uncertainty relations. These relations imply that increasing the reliability or precision of an engine’s power output comes at a greater thermodynamic cost. Here we study the thermodynamics of precision for small thermal machines in the quantum regime. In particular, we derive exact relations between the power, power fluctuations, and entropy production rate for several models of few-qubit engines (both autonomous and cyclic) that perform work on a quantised load. Depending on the context, we find that quantum coherence can either help or hinder where power fluctuations are concerned. We discuss design principles for reducing such fluctuations in quantum nano-machines, and propose an autonomous three-qubit engine whose power output for a given entropy production is more reliable than would be allowed by any classical Markovian model.
Hi Mark! Interesting talk! I am starting to be interested in TUR since I “see” them emerging in several contexts and I liked your simple formulation: SNR*(Entropy Prod. rate) \geq 2. I also noticed that all TURs you mentioned later in your presentation are lower bounded by 2. Is it only due to a proper choice of normalisation or is it really a universal lower bound? If so, is there some physical meaning/interpretations associated with this 2?
Hi Camille, thanks! I am not sure if I have much intuition for why the bound takes the precise value of 2. This value isn’t just a trivial normalisation, but it is also not completely general. Even in classical systems there exist other TUR bounds that hold under different assumptions (e.g. see the recent review https://www.nature.com/articles/s41567-019-0702-6). In our study, we found other examples where the bound is less than 2, but I didn’t have time to show them explicitly in the talk. For example, we derived a TUR bound for the three-qubit engine of approximately 1.2 (see https://arxiv.org/abs/2009.11303 for details). I think that in general that the TUR ratio of quantum systems can be less than 2 when coherences come into play. For example, Guarnieri et al. (https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.1.033021) derived a general TUR for non-equilibrium steady states in quantum systems that are not too strongly driven. Here the bound is exactly 1, so fluctuations can be even smaller than in our examples.
Alright, thank you very much! Interesting!
Hi Mark, thanks for the nice talk. I have a question: Since you define work as the average energy in the load, it is not exactly the same as the work defined in terms of energy exchange with a classical field. So I find interesting that you still found a TUR using this definition. Could you elaborate a bit on that? In particular, if one makes the connection with the classical characterization one knows that in general the energy stored in the load cannot be completely extracted by an external driving.
Hi Ivan, thank you! Yes, you raise an important point. We define the work very simply as the energy of the load. Note that we don’t consider only the mean energy: fluctuations in the load’s energy are treated as part of the fluctuating work output. (If you like, I guess you could think of our “stochastic work” as being defined by a two-point measurement on the load, but we were only interested in the first two “cumulants”.)
As you say, not all of the load’s energy is unitarily extractable. A more refined analysis could consider the extractable work, e.g. as quantified by the ergotropy or the non-equilibrium free energy. To be honest with you, I wanted to look at this, but we decided that we had already enough material and so left it for a future study 🙂 If, for example, one quantifies work output by the ergotropy, one certainly finds a kind of performance limitation connected to the TUR. However, it is not really a TUR per se because the ergotropy itself is not a fluctuating quantity (as usually defined). I did a *very* brief analysis of this idea in my review (https://arxiv.org/abs/1902.02672), but I only looked at a simple random-walk model there. I would definitely be interested to see how it plays out in a more thorough analysis. (The introduction of the above review also explains as best as I can why I prefer to think about work in terms of genuine dynamical systems instead of classical fields, whenever possible!)
I should also highlight this nice paper by García-Pintos et al. (https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.125.040601) which has a similar spirit but focusses on non-equilibrium free energy.
Thanks a lot for the elaborate the reply Mark!
Hi Mark,
I have a couple of questions.
The TUR is valid in a steady-state, right? Does the model (two baths and the infinite ladder) reach a steady-state?
You consider a very big $g$ in one of the plots, but the model (resetting) is a “local master equation”, which are thermodynamically consistent for small $g$ I think… so any comment?
Thanks!!
Hi Felipe, thanks for the questions! The standard TUR is valid in stationary state, although there are non-stationary extensions (e.g. see Timpanaro et al. https://arxiv.org/abs/1904.07574). In our case, the system does reach a steady state in the sense that the currents (and qubit energies) are time-independent. However, since the load’s energy is increasingly continuously in time, d\rho/dt is not zero, where \rho is the state of the qubits and the load.
About the value of g: the plot showed values where g and p (p=thermalisation rate) are on the same order as each other. Both can still be “small”. For the validity of the local master equation one typically assumes that g and p are small in comparison to the local subsystem energies. More importantly, one should assume that g and p are small compared to the temperatures (or that the temperature is effectively zero), see e.g. Hofer et al. (https://arxiv.org/abs/1707.09211). Both of these conditions can be easily satisfied for the parameters we consider. Indeed, the energies and temperatures individually are arbitrary in the plots, we specify only their ratios.
Actually, though, the reset thermalisation model is a kind of collision model. Therefore, one can also think about it in terms of the work done by refreshing the qubits during each collision, as you know very well 🙂 (see https://arxiv.org/abs/1509.04223 and https://arxiv.org/abs/1808.10450 for any others who may be reading this). In this picture, g can be arbitrarily large without “violating thermodynamic laws” as long as one accounts for this work done. In this case, the work done in the quasi-stationary state is actually zero because [H_0, H_int] = 0. This is similar to your results for the quantum absorption refrigerator (https://arxiv.org/abs/1710.00245), which has practically the same Hamiltonian.
Gracias!!