News from the Front, XVIII: de Sitter Black Holes and Continuous Heat Engines

Another title for this could be “Making sense of de Sitter black hole thermodynamics”, I suppose. What I’m going to tell you about is either a direct correspondence or a series of remarkable inspiring coincidences. Either way, I think you will come away agreeing that there is certainly something interesting afoot.

It is an idea I’d been tossing around in my head from time to time over years, but somehow did not put it all together, and then something else I was working on years later, that was seemingly irrelevant, helped me complete the puzzle, resulting in my new paper, which (you guessed it) I’m excited about.

It all began when I was thinking about heat engines, for black holes in anti-de Sitter, which you may recall me talking about in posts here, here, and here, for example. Those are reciprocating heat engines, taking the system through a cycle that -through various stages- takes in heat, does work, and exhausts some heat, then repeats and repeats. And repeats.

I’ve told you the story about my realisation that there’s this whole literature on quantum heat engines that I’d not known about, that I did not even know of a thing called a quantum heat engine, and my wondering whether my black hole heat engines could have a regime where they could be considered quantum heat engines, maybe enabling them to be useful tools in that arena…(resulting in the paper I described here)… and my delight in combining 18th Century physics with 21st Century physics in this interesting way.

All that began back in 2017. One thing I kept coming back to that really struck me as lovely is what can be regarded as the prototype quantum heat engine. It was recognized as such as far back as 1959!! It is a continuous heat engine, meaning that it does its heat intake and work and heat output all at the same time, as a continuous flow. It is, in fact a familiar system – the three-level maser! (a basic laser also uses the key elements).

A maser can be described as taking in energy as heat from an external source, and giving out energy in the form of heat and work. The work is the desired maser action, at a particular frequency. Natural masers are detectable all over nature, in fact, most notably in astrophysics! Here’s a page about that. (And the Hubble Jupiter photo at the top is from there.) Typically, you have a medium with molecules that have a finite set of energy levels, and those levels are pumped with energy in such a way that there is then an instability to emission of energy in a particular band – the maser action.

Scovil and Shultz-du Bois (in another of those amazing Bell Labs papers from that time) showed that even though the basic maser medium relies on discrete quantum energy levels, key aspects of the system can be cast into classic thermodynamic language, and the condition for the masing action to take place is that the efficiency of the system is equal to or below the classic Carnot efficiency!

The key elements are shown in the figure (from my paper), with three levels with populations n1, n2, and n3. The presence of three levels makes it natural to have three characteristic energy differences, and via Boltzmann you can associate those differences to population ratios in three different ways, associating a temperature to each. For example, n3/n1 = exp(-w_H/T_H) < 1. There's thermal energy in from the input heat bath, encouraging the quanta to jump up to level 3. They can relax by jumping down to level 2, giving up some thermal energy to heat bath at temperature T_C. States can also jump down from level 2 to level 1, emitting some energy at that frequency. Under normal circumstances, n2 < n1, and that also is just thermal energy flow. BUT, if you are in a situation where you have population inversion, that last jump’s output of energy is not heat flow, but work. Population inversion is when there are more quanta in state 2 than in state 1, i.e., n2/n1 > 1. You can’t write this as a thermal state a la Boltzmann any more. It’s really a non-equilibrium situation.

A bit of simple algebra (try it!) shows that population inversion is achieved precisely when the efficiency (energy of the 2-1 channel divided by energy of the 1-3 channel) is less than or equal to 1 – T_C/T_H, the classic Carnot efficiency of a heat engine! This is the awesome Scovil & Shultz-Du Bois observation – a maser is a quantum heat engine.

“So what has all this ancient 1959 physics got to do with us 21st Century sophisticates?”, I hear you ask. Excellent question. I think I’ll put that all into a second part of this post, as I have to dash off to do some other things now. But the key elements are that a long puzzle about black holes in de Sitter spacetimes (spacetimes with positive cosmological constant – our own universe might be such a spacetime) has been that there are two natural temperatures when you come to do black hole thermodynamics. There’s the temperature associated to the black hole’s horizon, but there’s also the temperature associated to the cosmological horizon. And the whole system is unstable because there ought to be energy flow due to the two (different) temperatures, and so the whole spacetime solution must evolve. People working in that area of physics have long been puzzled about how to make sense of the thermodynamics, how to put all these elements together.

But I’ve just shown you a simple system that naturally incorporates two temperatures and an instability in a way that makes a lot of sense. Two additional elements are needed in de Sitter: a highest frequency (otherwise you can’t pump to population inversion) and a channel for doing work.

All those elements are present for a generic de Sitter black hole, as I’ll explain. (The key, highest frequency, aspect is a new feature that helped me put it all together, and connects to my Schottky peak discovery of a couple months back.) Could it be that there’s a natural combination of these things that’s simply the operation of a continuous heat engine?

The answer is YES! I’ll describe that all in a follow up post. (Or you can just read the paper.)

-cvj