Quantum State of the Fermionic Carriers in a Transport Channel Connecting Particle Reservoirs

Round 1

Reviewer 1 Report

This is a very good work. The authors address a problem of current importance and obtain new significant results. I recommend publication.

Author Response

We thank the Referee for the positive evaluation of our work.

Reviewer 2 Report

Decoherence and thermalization are not the same phenomenon. Mesoscopic systems are a good place to observe the difference, e.g. in the Aharonov-Bohm effect in mesoscopic rings, where quantum interference persists despite multiple scattering on impurities at finite (small) temperature. In this work, the authors replace a true thermalized particle reservoir (which would be characterized only by a temperature and chemical potential) with a mesoscopic quantum system with a Wigner-Dyson energy level distribution (deterministic but chaotic). The spirit is much like replacing a real sequence of coin tosses with a pseudo-random number generator. The interest is to observe decoherence effects in a single-chain transmission channel attached to the model reservoir, as a function of time. In the second half of the paper, the authors derive a corresponding master equation and use it to consider two reservoirs connected by the transmission channel.

My first objection is that the role of physical temperature is unclear. If E in Eq. 4 is an eigenenergy of the system, then the temperature in the same equation is not an independent variable. (If it were, the internal energy U would appear in the equation instead of E.) In this microcanonical situation, one must compare the ensuing temperature fluctuations with the internal dynamics of the system to see that the reservoir is really in the deterministic (albeit chaotic) regime which the authors treat microscopically.

In a microcanonical setting, both the temperature and the entropy can behave contrary to usual (canonical) intuition, because homogeneity and concavity are not guaranteed, see e.g. D.H.E. Gross, https://doi.org/10.1063/1.1901658 . One cannot assume without discussion that the authors' common-sense application of the von Neumann entropy avoids this pitfall.

The second objection concerns adding a second reservoir at the level of the master equation, which is macroscopic by construction. In other words, the standard model of thermal contact is assumed for the compound distribution of the two reservoirs. The authors prudently note that their construction is only applicable at high temperature, but the reader is left guessing what that would be in the present case (1, 10, 100 K?). Again, it is not clear that the authors' modelling corresponds to two real mesoscopic reservoirs connected by a wire of atomic dimensions, because the alternative possibility has not been excluded, that they would decohere by thermalization before having time to realize that their spectrum happens to be chaotic.

In this context, the statement on line 92, "for infinite reservoir temperature we have complete equilibration," is somewhat worrying. Surely one is entitled to complete equilibration at any temperature, of course with the usual temperature fluctuations associated with a microcanonical setup?

I have no objection in principle to the microscopic modelling of particle reservoirs, as long as the context and scope of such modelling is made clear to the reader. With electronics approaching atomic dimensions, one must be careful to differentiate realistic models predicting non-thermal physical regimes available to real devices - despite their being in contact with physical, truly random, thermostats - from artificial models constructed to make theoretical points. "Which points?" is also a good question, because to my eyes the conclusion is significantly more circumspect than the abstract, so it is not really clear where exactly is progress being made.

When the authors provide sufficient context to make these issues clear to the general reader for the particular case of their model, the paper can be published.

Author Response

We agree with the Referee that decoherence and thermalization are not the same phenomenon and, in fact, do not make such a statement in the manuscript. However, we disagree that our reservoir is not a true particle reservoir, as well as we disagree that coin toss is a truly random generator. Indeed, the coin dynamics obeys the deterministic equation of motion and we can use this system as a random generator only because its dynamics is chaotic and, hence, exponentially sensitive to the initial condition. It is a commonly accepted viewpoint that one finds a rigorous foundation of the statistical approach in classical physics only by appealing to classical chaos.

Analogously, the quantum statistical mechanics can be derived by using the theory of quantum chaos. This is a rather different way to obtain the known equations (like, for example, the Fermi-Dirac or Bose-Einstein distributions) which does not involve the notion of a thermostat and canonical or micro-canonical distributions. This approach is especially suitable for studying finite systems and we would like to thank the Referee for drawing our attention to relevant studies of finite systems which use the "traditional" approach.

We accept the Referee criticism concerning the modelling of the second reservoir by means of the Lindblad relaxation operators. Of course, it would be nice to model it microscopically but not macroscopically. Unfortunately, for the considered chain length L=6 such simulation are not feasible. (Yet one can address the equilibration between two reservoirs through a point contact L=1. These studies are in progress.)

In the amended version of the manuscript we made few changes to stress that:

a) We use "an alternative" approach to the statistical mechanics where the reservoir temperature is determined by its energy. This approach allows us to consider atomic reservoirs where one typically has 10^4-10^5 atoms in a trap and these atoms are isolated from the environment (i.e., no contact with a thermostat).

b) The infinite reservoir temperature corresponds to the situation where all trap modes are equally (up to fluctuations due to finite system size) occupied. Correspondingly, the low-temperature limit corresponds to the case where the mode occupations are non-uniform, showing a pronounced step at the Fermi energy.

c) We use the information entropy to characterize the quantum state of the carriers, which is not the same as the thermodynamic entropy.

d) The main results (main progress) are the first-principle analysis of the quantum state of the carriers in the transport channel and justification of the master equation (15) in the case of infinite reservoir temperature.

We hope that these changes make the paper more accessible for the general reader (which was the main objection of the Referee).

Reviewer 3 Report

This is an interesting paper which presents in full detail the dynamics of fermions in a conducting lead, connected to one or two reservoirs. Its main virtue lies precisely in the figures, which permit to see these dynamics as they evolve in time. From a theoretical point of view, it adopts Von Neumann entropy as a means of gauging decoherence. This is a standard method, although not fully satisfactory. The authors themselves note its limitations in sect. 3.2, that lead them to consider partial SPDMs.

Moreover, it is shown in this paper that the true dynamics well agrees with the results of a master equation approach. Since the numerical solution of the former is clearly limited to a small number of particles, states, and lattice sites, it would be rather interesting if the authors, in a subsequent publication, could present a detailed examination of the errors involved in the master equation approximation, on their scaling laws, and on the range of parameters achievable in numerical analysis and, of course, in real laboratory experiments. I repeat that this is a suggestion for a future publication; in the present paper, though, one could add one figure superimposing two curves from fig. 2 with two corresponding curves from fig. 4, so that one could appreciate the degree of approximation provided by the master equation approach.  I have also a minor remark concerning these figures. The subscript n appearing in the vertical axes (P_n, S_n) does not agree with caption and text, where the letter i is used. Also, one would like to see in the caption the correspondence between colors and values of i. I also have an untutored question: what about i=0, i.e. the probability that no particle is in the channel? I guess that one can obtain it by subtracting all remaining probabilities from the value one. Is this correct? Can one see also this curve?

Finally, what is the color coding in Fig. 1? The usual color code bar can be added to the side.

Author Response

We would like to thank the Referee for positive evaluation of our work, careful reading of the paper, and posed questions for future studies. Following the Referee recommendations/remarks we added the colour bar in Fig.1, extended the caption to Fig.2, and corrected the axis labelling in Fig.4.

Round 2

Reviewer 2 Report

The paper is sufficiently clarified now for the readers to reach their own opinion of the work presented, which was the purpose of the review. Thus I have no further objections to publication. In particular, it is clear that the authors' modelling is what I originally called artificial. I have no objection to that as a conscious choice, but I was surprised to realize from the authors' response that they seem to believe it actually possible to have a finite closed autonomous system, e.g. an electronic device which would not be subject to random perturbations from the substrate.

The physical reality which I referred to in my review is that the only autonomous system not subject to the "usual" thermalization is the universe as a whole (see e.g. the introduction in the Landau-Lifschitz book on thermodynamics). Any real closed system with a boundary is thermalized by truly random perturbations across the boundary, which cannot be removed even in principle, let alone by a discussion whether they can be called coin tosses.

It is true that thermodynamics can also be based on variants of the ergodic hypothesis, but such constructions do not reflect physical reality, unlike the ensemble construction, which does. As long as the authors persist in ergodic modelling, they miss on potentially the most interesting aspect of their work, which is the possibility of constructing real mesoscopic devices which decohere autonomously despite operating at a finite temperature, as in the example of the Aharonov-Bohm effect. In particular biological systems are a fascinating source of templates for nanoelectronics operating even at room temperature, see e.g. https://doi.org/10.1021/acs.chemrev.5b00298 . I hope the authors can be motivated to consider these wider issues eventually.

Comments for author File: Comments.txt

Author Response

The second report of the Reviewer does not request/require any changes in the manuscript. It is just an exchange of personal viewpoints on fundamental questions in physics and we are very thankful to the Referee for extremely clear formulation of their viewpoint which we will certainly keep in mind.