 is fluctuation symmetry and coherence in quantum heat transport with the unified quantum master equation. Hi, yeah, just give me one moment. All right, can everyone see that? I can see it, yes. Thanks. Okay, thanks for the introduction. Yeah, I have cut down on the number of E phrases in the title here, but essentially what I'm going to be presenting is sort of a grab bag of different insights that come out of using quantum master equations that take into account coherences, particularly in the context of heat transport. So heat exchange with one or more thermal reservoirs. This is mostly covered in a paper that just came out in PRE by myself and my PhD supervisor, Javier Segal, who you will be hearing next, actually. Yeah, so I'll quickly kind of try and motivate why it's interesting to think about coherence in open quantum systems, although it seems like a lot of the talks in this session have been talking about similar topics anyway. And then I'll talk about how we get quantum master equations of Linnbad form that can take into account coherences and go a little bit into the properties of these master equations with respect to thermodynamics and statistics, and also kind of looking at different master equations comparing when each is useful. So I'll be focusing a lot kind of or at least coming back to this V model. As an open quantum system, that's a couple to thermal reservoirs where there's some ground state, and then there are two excited states that are nearly degenerate. They're separated by this splitting delta, which is taken to be very small in comparison to the energy separation between the ground state and the excited states. And the ground state is dissipatively coupled to either of the two excited states via coupling to thermal reservoirs. Yeah, so you'll get heat transport in the situation where there are multiple reservoirs at different temperatures. In general, like while the conventional kind of understanding is that coupling to the environment kills off coherence, we find that for systems like this that have eigenstates close in energy, coherences between nearly degenerate states become relevant. So for instance, in non-equilibrium steady states, even in the in the long time limit, we see that this row two, three, this off diagonal element of the density operator will be generally nonzero. And also in transient regime, even in not in non-equilibrium situations, but just kind of during this process of thermalization, there are coherences that are long lived and they die off actually on a timescale that's related to this splitting. It's proportional actually to the splitting to the power of minus two. So the approach that we are going to take towards studying these kinds of systems is using quantum master equations. So again, you have some quantum system like a few levels, a couple to an environment, which is described by its own Hamiltonian. And then some interaction Hamiltonian that couples the system in bath. Generally, the way that quantum master equations are derived from microscopic principles is to take the evolution of the system plus bath to be just unitary, but then trace over the environment. So say like we don't want to track the environmental degrees of freedom and just get an equation of motion for the system, the reduced density operator of the system. And that involves making the born Markov approximations. So essentially related to weak coupling and the balance being memoryless. And what you end up with when you do that is the Redfield equation. So basically, there's a part that looks like unitary evolution plus this whole kind of dissipative part, which actually describes the interaction between the baths and the system. But the Redfield equation on its own, it's often kind of avoided because it doesn't have all the properties that we would hope for a master equation to have. Namely, it doesn't preserve the positivity of the density operator. So you can end up with things like negative probabilities, etc. And the way to avoid that is to ultimately get a master equation of Linn-Blad form. It's kind of the most general form to preserve the positivity and all the properties of the density operator that we like. So a very common approach towards doing that is to look at the Redfield equation and notice that there's this phase factor here, which will oscillate anytime that these two frequencies, which are associated with energy splitings in the system, are not the same. So what you can do is you can say, okay, once I integrate up, all these oscillating terms won't end up mattering. So we'll only keep terms where these two frequencies are exactly the same. And that's not the secular approximation. That's what gets you the fully secular on a master equation. So that equation has a lot of good properties. It is, it preserves positivity of the density matrix for sure. It also satisfies fluctuation symmetry. So it's thermodynamically consistent. But it basically just looks like kinetic rate equations for population to jump between the energy eigenstates. So in other words, it basically ignores coherence in the energy eigenbasis. And that's not always going to be a good move. So if you consider the possibility that there might be some nearly degenerate states in the system, then you can have terms where these two frequencies here are pretty close. And so the timescale for this oscillation is actually pretty slow. And it might start to rival the timescale for the dissipative dynamics set by this rate gamma. So it might not be a good move to just ignore these terms. So instead, we keep them and we, we use the quantum master equation that is of lindblad form, but also does can account for coherence. So that equation has actually been derived kind of independently a number of different times in the last few years. In one of those derivations, it's referred to as the unified quantum master equations. I've got them all panel listed here, which is the name that I'm using here. So what you do is you, you still throw away the quickly oscillating terms, but you keep the slower ones. And you achieve lindblad form instead by just ignoring the difference between close frequencies when it comes to evaluating the rates. So this QME doesn't completely ignore coherence. It actually describes some coupling between coherences of nearly degenerate states and populations. Yeah, not going to go into very much detail about this, but you can, as with any quantum master equation, you can carry out full counting statistics and get a generating function that in our case for heat transport kind of contains all of the properties of the distribution over quantities of heat exchange with the bath. And for the unified master equation, it can be shown to satisfy fluctuation symmetry, which is equivalent to having that fluctuation theorems for energy exchange that guarantee things like the second law being satisfied at the level of averages, no heat flow from cold to hot or any weird properties like that. So that's exemplified here, just also numerically, how we show both the real and imaginary parts of this generating function at chi and at the shifted value coincide. And the nice thing about that is that while the Redfield equation in general does not satisfy that, in the case where you have nearly degenerate states, so this delta, this splitting between the two excited states in the context of that B model, when it's small, the unified equation actually approximates the Redfield equation much better than the fully secular master equation does. And again, you can look at that in the context of, yes, the coherences themselves, the fully secular equation, as long as there are no strict degeneracies, will assume that coherences just vanish, whereas here you can see the unified equation gives predictions much closer to Redfield. So in general, this equation could be very useful for situations with nearly degenerate states. It's arguably much more accurate than the fully secular equation in these contexts, but it also has these nice properties like Lindblad form, the dynamic consistency, etc. One other thing that I did want to just quickly mention, actually, is that when we do the same full counting statistics using full Redfield, so without that approximation, while it's true, we do see violations of the fluctuation symmetry, as kind of seen here with these deviations. For one thing, it does seem to always be satisfied when, for the real part, excuse me, and also for the imaginary part for small values of the counting field, chi. So in other words, the generating function is thermodynamically consistent, you know, it's good, at least up to second ordering, chi. And so if you're familiar kind of with full counting statistics, up to the second derivatives of this function shouldn't necessarily manifestly demonstrate any violations of the second law of thermodynamics, right? So the mean and the various, everything should be thermodynamically consistent with respect to those. So five minutes, sure, I'll be done in a minute. Kind of the last thing we looked at here was just these transport relations. So here, the Green-Cubo relation, essentially like a version of the fluctuation dissipation relation, and also this next transport relation about concerning higher order derivatives of the statistics of the heat current. For all three master equations we considered, including Redfield, we see that these are satisfied. So these are related with fluctuation symmetry. And because they only involve these first two cumulants, even the Redfield equation, which generally violates fluctuation symmetry, it's still actually, we've still found that they satisfies these relations in linear response. So this I'll just kind of quickly mention, the unified QME has also been looked at in the transient regime. So not considering non-equilibrium steady states, but just looking at this thermalization process. And as I mentioned in one of the first slides, there's this lifetime for these long-lived coherences that depends on the splitting delta. And so it's been found that the unified equation, despite ignoring that splitting, in calculating the rates, it still does actually capture it and it gives correct predictions for this lifetime. So with that, I'll wrap up. Yeah, basically just kind of wanted to talk about the way that this unified quantum master equation or this quantum master equation that's taken a few different names can be useful for situations where you have nearly degenerate states in systems that are public to the environment. Thanks. Thank you very much for that nice talk. I guess we have some time for questions. The floor is open. Jin Wang. Yeah, I'm not sure if you know of our work in 2014. We look at the same problem of three-level systems coupled with the two vast and show that a non-equilibrium steady state with the Redfield equations will have steady state coherence, residual coherence, while if you're coupled with one vast, equilibrium vast, you're going to have zero coherence. That work was published in 2014 on JCC. Yeah, so we kind of came into this with that understanding that you see coherences in non-equilibrium steady states, but we kind of wanted to give this master equation more of like a close look as to specifically kind of what properties it has and how we actually get those coherences when we do this microscopic derivation. Yeah, so I can send you the reference. Great, thank you. Thanks for that. Any other questions? Okay, well, thank you for that very interesting talk, and now we come to the last talk of the session, because as I had announced for those of you who may have not been there, the