News from the Front, XVI: Toward Quantum Heat Engines

(The following post is a bit more technical than usual. But non-experts may still find parts helpful.)

A couple of years ago I stumbled on an entire field that I had not encountered before: the study of Quantum Heat Engines. This sounds like an odd juxtaposition of terms since, as I say in the intro to my recent paper:

The thermodynamics of heat engines, refrigerators, and heat pumps is often thought to be firmly the domain of large classical systems, or put more carefully, systems that have a very large number of degrees of freedom such that thermal effects dominate over quantum effects. Nevertheless, there is a thriving field devoted to the study—both experimental and theoretical—of the thermodynamics of machines that use small quantum systems as the working substance.

It is a fascinating field, with a lot of activity going on that connects to fields like quantum information, device physics, open quantum systems, condensed matter, etc.

Anyway, I stumbled on it because, as you may know, I’ve been thinking (in my 21st-meets-18th century way) about heat engines a lot over the last five years since I showed how to make them from (quantum) black holes, when embedded in extended gravitational thermodynamics. I’ve written it all down in blog posts before, so go look if interested (here and here).

In particular, it was when working on a project I wrote about here that I stumbled on quantum heat engines, and got thinking about their power and efficiency. While working on that project, I had a very happy thought: Could I show that holographic heat engines (the kind I make using black holes) -at least a class of them- are actually, in some regime, quantum heat engines? That would be potentially super-useful and, of course, super-fun.

The blunt headline statement is that they are, obviously, because every stage of the engine can be mapped (via holography – these are AdS black holes) to an operation on some dual quantum field theory. There’s quantum in the name right there, so we’re done. No? No. That’s like saying that the steam engines back in the 17th Century are quantum heat engines because we now know that the underlying description of water/steam is, at the molecular/atomic level, a quantum one. We don’t say that because it’s a system that has firmly gone over into the classical thermodynamic regime by virtue of having a large number of degrees of freedom in play: quantum correlations have long been lost in the noise of thermal fluctuations.

The same is true in holography, broadly speaking. The underlying QFT is at large N (in order to make the gravity nice to work with, as Maldacena¹ showed us). Some positive power of N counts the number of degrees of freedom. These are thermalised, and the thermodynamics is not touching the quantum aspects. This is all nicely described by AdS black holes, as taught to us by Witten². Engineering it all into a sequence of operations on the black holes that looks like a heat engine still has us at large N. So no chance of connecting to a quantum engine. Right?

Maybe yes, but maybe not. Not if you look a bit more carefully. Not if you find a way to select out a small subsystem from this large N system and have that be, in a sense, the heat engine’s component. That’s what I’ve found how to do! And that’s what’s in the paper that I wrote recently (You can find it here. It also has lots of references to the literature that I have not included here.).

The key observation was actually in a different paper I wrote a few weeks ago. I hit upon it because I was trying to make sense of the heat engine issue and realised it has more general applications and so wrote it up as a separate paper. Here’s the gist:

People have been playing with the extended black hole thermodynamics (not the regular, traditional sort that has no pressure and volume) for some time now, and have written lots of interesting papers, many of them arguing that when you apply it to various black hole solutions, it is strikingly similar to thermodynamics of familiar systems. I believed it too, until I looked at it the right way, and realised that they are hugely different: They barely have any (traditional) degrees of freedom at all! In thermodynamics, a good measure of the number of (traditional) degrees of freedom in play is the behaviour the specific heat at constant volume, C_V, as a function of temperature T. It tells you where the system stores its energy in excitations, as you raise the temperature. Einstein’s classic 1906 paper linking observed specific heats to an underlying quantum mechanical model – basically inventing the field of modern solid state physics – is all about this. I noticed that nobody had studied C_V(T) before for the various black holes they’d been studying, save to note that for the simplest black holes, C_V=0. But I realised that if C_V=0, that means those extended thermodynamics systems have no (traditional) degrees of freedom at all. Put differently, in the extended thermo, the degrees of freedom should be devided up into those that can be excited without changing the volume, and those that can’t. The latter are counted by a power of N, and the former count to zero for many black hole examples (like schwarzschild, reissner-nordstrom, etc). This is really different from ordinary matter.

Moreover, when I examined examples that have C_V non zero, the function C_V(T) displays a peak, then vanishes at high T. (This was the motivation for the title of the paper I wrote a few weeks back pointing this out: “Specific Heats and Schottky Peaks…”) Such peaks are well known in, e.g., condensed matter physics, and represent a finite window of possible energy excitations – an earmark of a ”small” subset of the large N system. The examples are either (non-diagonally) charged black holes, or rotating black holes, and the charge or angular momentum becomes a parameter that can be used to tune just how big or small that window of available energies is. This tunabilty is the key.

Anyway, the point here is this: What if you could make an engine that uses that small subsector in the crucial thermodynamics of the engine? If the heat flows in and out, through coupling to the external heat baths – the thermal processes that control the efficiency of the engine – involved only that subsector, it would give us the chance to achieve the goal of making contact with a quantum heat engine regime. Those thermalisation steps would need to happen at constant volume, so that we talk to only those states. Then we isolate the other processes from thermal contact with external systems, i.e., they are adiabatic steps.

What we’ve just made here is an Otto cycle (look at the first picture of this post), and it is one of the most keenly discussed cycles in the quantum heat engine literature. (It’s also the thermodynamic cycle that underlies a lot of classical combustion engines, such as your car engine.) My papers construct a lot of Otto cycles from black holes (these are actually the first such cycles made from black holes) and analyse their properties. I’ll let you read the details, but the really nice thing is that in the limit where one tunes the subsector to be very small, the efficiency formula for these black hole engines is exactly of the classic form that it is shown that quantum (also, classical) heat engines satisfy.

So, what’s this good for? Well, let me say that at the outset that a key parameter that characterises the system, gamma, the adiabatic exponent, is very large in these examples. Ordinary materials, and the simple quantum systems people play with in the lab, currently, have gamma of order a few. So this construction is unlikely to give us models of *exactly* the kind people are building in labs. (Although exploring different black hole examples may yield smaller gamma than I got with these.) BUT. Getting gamma right isn’t really the point. The point is to find a laboratory – a source of solvable models – where one can explore key issues of pertinence to quantum heat engines. Issues that of keen interest theoretically and experimentally. These include things like how quantum effects make themselves manifest in issues like operating at finite power, and how efficiency might be enhanced if the working system has special features (like a critical point), and so forth. There might be universal aspects of these kinds of engines that can be captured and characterised in this setting.

My point here is that I’ve shown that this setting can well be such a context for exploration. Black holes supply very tractable model thermodynamic systems where the entire equation of state can be explored, and where every step of the engine can be interrogated. Some of the quantum aspects may well be accessible along the key steps of the engine by using the holographic dual description. The constant volume process needs to be better understood in the holographic language, but I am sure that it is possible to derive an effective model of that “small” subsector in that context. As is well known, there are powerful geometrical tools that can be brought to bear here that can capture some useful quantum physics.

Anyway, this is all very exciting to me (and I hope to you). This is potentially a new vein of very useful physics that connects black holes and gravity to physics issues of much wider concern. Who knows where it might lead?

I hope you enjoy reading the paper! (Here.)

-cvj

^1,2If you’re not familiar with the basics of the AdS/CFT correspondence, that connects certain theories of gravity (they have negative cosmological constant) to certain theories without gravity (gauge field theories), there are many reviews available. Here’s a technical one. Here’s a less technical one.