The Smallest Thermal Machines

Are there limits to how small one can make a thermodynamic machine? At first sight, this question appears to contradict itself, as thermodynamics usually begins with a kind of macroscopic picture. However, as Nicolas Brunner at the University of Bristol, UK, and colleagues describe in a paper appearing in Physical Review E, this constraint can be circumvented, if one focuses on a special kind of architecture [1]. As a starting point, we may think of a nanoscopic object like a single molecule or even a two-level system (qubit) being in contact with a large heat bath. This contact will ultimately lead to a local thermal equilibrium between the object and the bath. (Temperature controls the relative occupancy of the energy states on the qubit.) We may also imagine a situation where a nanoscopic object interconnects two baths with different temperatures, thus enabling a heat current [2]. Taking this one step further, we can build up the ingredients of a fundamental thermodynamic machine by connecting three separate baths with a nanoscopic node—a black box that is designed to exchange heat between the baths in a particular way [Fig. 1(a)]. By definition, baths can only interchange heat and we’ll assume the box cannot store energy, so heat currents to or from the three heat baths have to add up to zero. Clearly, this could not happen if the box was just another heat bath. Let’s imagine this black box has three separate “sockets,” each represented by a two-level qubit that can have an energy E(1), E(2), or E(3), respectively. Let’s also assume there is an interaction between the three qubits that allows them to transmit energy to each other, but that these transitions are constrained by E(2) − E(1) = E(3). Put differently, socket 3 can’t be excited unless socket 2 transfers its energy to socket 1. This machinery, which is a consequence of the socket qubits being correlated with one another on the most elementary level, is all we need to produce an energy switch, which can, for example, act as a heat pump. Brunner et al. propose a convenient tool [1] for designing these correlations by introducing the concept of a “virtual qubit” with a “virtual temperature” [Fig. 1(b)]. Within each qubit we arbitrarily select a pair of states, which thus specifies an effective spin or qubit. The quantum system is thus reduced to three qubits with fixed occupation probabilities. Grouping any two qubits makes an effective four-level system. Out of the possible transitions within the resulting enlarged system there is but one two-particle transition, which would involve the transfer of energy between the two subsystems considered. This new qubit is the “virtual qubit.” The transfer energy will be nonzero, if the energies associated with the original two qubits are different, as the authors assume. The probability that the virtual qubit is in a particular state is fixed by its virtual temperature. As for any two-level system, the probability that the higher energy state is occupied relative to the probability that the lower energy state is occupied is exp(−∆E/T )). In the present case, this temperature is neither a local temperature of a real subsystem, nor does it relate to the four-level composite subsystem as a whole. In this sense it is virtual, and it can be positive or negative. So far, this is only a partial description, nothing is going to happen, no currents. But now Brunner et al. ask what happens if this virtual qubit, with its virtual temperature, is resonantly coupled to a third qubit, which is in contact with its own bath T3. They assume the coupling is weak and can be modeled by a specifically designed interaction Hamiltonian. (The assumption of weak coupling makes sure that each level is still predominantly localized in one of the subsystems.) Effectively, this brings us back to the picture of two baths interconnected by an effective node: we expect a flow of heat, the direction of which is dictated by thermalization. It seems as though nothing is new.