Quantum refrigerators and biological cooling

In a colloquium last week at PI, Sandu Popescu talked about the smallest possible refrigerators. He was motivated to explore this question by considering how biological systems manage to regulate temperatures despite their exposure to various elements of their external environment. The thinking was some form of biological cooling process might be involved.  This entry is an attempt to explain his most fascinating ideas on small cooling engines.

The logical place to start, of course, is to define what a refrigerator is. If you can still recall what you learned in basic thermodynamics, then you'll know that when you have two objects in contact, one hot and one cold, then heat will naturally flow from the hot object to the cold object until the two settle at the same temperature.

Maybe you'll remember something like the diagram above. It shows the essential details of how a refrigerator works: you have two baths or reservoirs at different temperatures. To make heat flow in the opposite direction, you will need an external source of work to pump out the heat. As long as you have that, it's possible to force heat to flow in reverse. 

By the way, in physics, work is what you do when you apply a force F on a box and it moves a distance d in that direction. In that case, W = Fd. In thermodynamics, work is anything that provides free energy, and by 'free' we mean it can do something useful, say, to push a box.

In a regular fridge, the source of external work is actually the liquid that flows through the pipes shown below. This cooling fluid or refrigerant is usually pure ammonia. The refrigerant runs through a cycle as depicted--in cold gas form, it enters a compressor that produces a high-pressure hot liquid. After it passes through a spiral it encounters an expansion valve which is mostly just a hole that allows the refrigerant to suddenly go from high pressure to much lower pressure. The quick change in pressure causes the fluid to expand and quickly evaporates. 

In this process of evaporation, the gas cools down its surroundings by taking heat away from it. By making the fluid run through another spiral, you can create a cooling area from which most of the heat will be collected. And to ensure that you don't run out of fluid to evaporate you just make the cold gas go back into the compressor and the cycle simply repeats.

If you try to compare the usual refrigerator with the thermodynamic diagram, then you can see that the cooling area spiral and compressor roughly correspond to the cold bath while the high-pressure fluid spiral and expansion valve represent the hot bath.

Now, we may ask, how small can we make these baths be? It turns out that you can have something as small as two atoms to do a similar cooling process, as depicted in the picture above. What we have here are three systems with two energy levels each. In the jargon of physics, you would call them three qubits, short for a quantum bit. It doesn't matter what the qubits really are so it might be convenient to think of them as three atoms and consider any two energy levels for its atomic vibrations, represented by dots in the picture. The lower horizontal bars indicate a lower or ground energy level or  while the higher bars indicate a higher or excited energy level.

Why atomic vibrations? In normal life-size objects, we know that an object is hotter because its molecules vibrate more. Thus it is consistent to think of the horizontal bars in the diagram as vibrational energy levels.

After that, you need the three atoms interact with their neighbor: atom 1 interacts with atom 2, and atom 2 interacts with 1 and 3. Then place atoms 1 and 2 at some temperature T_A and atom 3 at a temperature T_B that is bigger than T_A. The crucial property of the atoms is shown in the box below the diagram: the energy difference between the two levels in atom 2 must be equal to the total of the energy difference between the two levels in atoms 1 and 3. 

If everything is in place, then the indicated process happens with a significant probability: the atom 1 energy goes down, the atom 2 energy goes up and the atom 3 energy goes down. This means is that the vibrational energy of atom 1 is reduced by the presence of atoms 2 and 3. This is the cooling process at its core. In this example, atoms 2 and 3 represent the quantum refrigerator.

You may be wondering what's quantum about it. That has to do with the mathematical details of how to guarantee that this cooling process occurs for our three atoms. (In physics-speak, there is some interaction Hamiltonian for three qubits and a Lindbland master equation describing the evolution of each qubit, and you're looking for a particular steady-state solution of that equation.)

Maybe that's all good and nice but what has this got to do with biological systems? The initial hope was to develop a simple model for describing biological refrigeration but it seems finding a situation where the three-qubit model applies is highly unlikely. But there is at least one speculation for where a relatively simple biological cooling process may be discovered: catalysis by enzymes.

Catalysis is the speeding up of a chemical reaction by way of some extra substance called a catalyst. In living organisms, many biochemical processes occur at astounding rates because of protein-based catalysts called enzymes. The diagram above illustrates a simple type of biological catalysis: you have an enzyme with two primary docking sites. On the left, there is a site for adenosine triphosphate or ATP, shown as a gray blob.  On the right, there is an active site for the reaction molecules or reactants, shown here as light and dark violet ovals. It is the chemical reaction involving these two molecules that the enzyme speeds up.    

ATP is the major source of energy for cellular reactions,  released when its any of its phosphate links break (for example, ATP -> ADP + P). In this case, ATP is our external source of work. When an ATP molecule attaches onto the enzyme, the active site changes its shape as indicated, in the process trapping molecules that react with each other at an enhanced rate. 

How does an enzyme increase reaction rates? Popescu believes that it basically acts as a refrigerator for the reactants when they reside in the active site. In this case, you can think of the enzyme as an analogue of atoms 2 and 3 in the three-atom refrigerator, and the reactants are the objects to be cooled. By lowering their vibrational energies, reactants, it becomes much easier for the reactants to chemically combine given that they are already confined near each other.

Is that really what happens? Actually we don't know--it's the sort of thing you need experiments to figure out exactly what's going on. Unfortunately, Popescu is having trouble getting people to do the needed experiments. You see, biologists want to explore it only if it really happens but we won't know if it does until we look (roughly what he said). For now, it remains an interesting speculation. But what it tells us is despite the complexity of biological systems, it's still possible to find seemingly complicated biochemical processes that can be described in a relatively simple way using quantum physics, and perhaps even involve quantum effects.  


References:

N. Linden, S. Popescu, and P. Skrzypczyk, "How small can thermal machines be? The smallest possible refrigerator", Physical Review Letters 105, 130401 (2010).

H. J. Briegel, S. Popescu, "Intra-molecular refrigeration in enzymes", arXiv:0912.2365v1 [quant-ph] (2009).