Virtual qubits, virtual temperatures, and the foundations of thermodynamics

We argue that thermal machines can be understood from the perspective of “virtual qubits” at “virtual temperatures”: The relevant way to view the two heat baths which drive a thermal machine is as a composite system. Virtual qubits are two-level subsystems of this composite, and their virtual temperatures can take on any value, positive or negative. Thermal machines act upon an external system by placing it in thermal contact with a well-selected range of virtual qubits and temperatures. We demonstrate these claims by studying the smallest thermal machines. We show further that this perspective provides a powerful way to view thermodynamics, by analyzing a number of phenomena. This includes approaching Carnot efficiency (where we find that all machines do so essentially by becoming equivalent to the smallest thermal machines), entropy production in irreversible machines, and a way to view work in terms of negative temperature and population inversion. Moreover we introduce the idea of “genuine” thermal machines and are led to considering the concept of “strength” of work.

Figure 1

Schematic diagram of bath virtual qubits. (a) Two baths, one at temperature $T_1$ and one at temperature $T_2$. In each bath we identify two arbitrary energy eigenstates, ${|g\rangle }_{{}_{B_i}}$ and ${|e\rangle }_{{}_{B_i}}$, with spacing ${\mathcal{E}}_i$. (b) These two baths can be viewed as a single composite bath. Two energy eigenstates of this composite system are ${|G\rangle }_{{}_{B_v}}\equiv {|e\rangle }_{{}_{B_1}}{|g\rangle }_{{}_{B_2}}$ and ${|E\rangle }_{{}_{B_v}}\equiv {|g\rangle }_{{}_{B_1}}{|e\rangle }_{{}_{B_2}}$. The “virtual temperature” ${\mathcal{T}}_v$ of this “virtual qubit” can take on any value—positive or negative—depending on ${\mathcal{E}}_1$ and ${\mathcal{E}}_2$.

Figure 2

Schematic diagram of the smallest thermal machines. (a) The smallest machine comprises two qubits, one with energy spacing $E_1$ in thermal contact (wavy line) with a bath at temperature $T_1$ and the other with energy spacing $E_2$ in thermal contact with a bath at temperature $T_2$. (b) The ultimately more relevant way to view this system is that the “machine virtual qubit” with spacing $E_v=E_2-E_1$, is in thermal contact with virtual qubits in the composite bath at the virtual temperature $T_v$. The local energy-level spacings $E_1$ and $E_2$ allow the machine virtual qubit to “filter” out a single virtual temperature from the composite bath.

Figure 3

Schematic diagram of the smallest machine interacting with an isolated external qubit. An isolated external system—here a single qubit—is placed into thermal contact with the machine virtual qubit, which has matching energy-level spacing. This interaction is depicted by the wavy line. The net effect is that the external system is placed in thermal contact with virtual qubits in the composite bath at temperature $T_v$, mediated by the machine virtual qubit.

Figure 4

Schematic diagram of the smallest machine interacting with an open external qubit. If the external system also interacts with its own environment at temperature $T_3$ then it will behave as any system in contact with two thermal baths at two differing temperatures.

Figure 5

Schematic diagram of the smallest heat engine. (a) The weight, an equispaced system, unbounded from above and below, interacts with the machine qubits via the interaction depicted by vertical arrows. This interaction induces transitions between the degenerate energy eigenstates ${|0\rangle }_{{}_1}{|1\rangle }_{{}_2}{|n\rangle }_{{}_w}\leftrightarrow {|1\rangle }_{{}_1}{|0\rangle }_{{}_2}{|n+1\rangle }_{{}_w}$. The energies $E_1$ and $E_2$ are chosen such that the “forward” transition in which the weight is lifted is biased in favor of the “backward” transition in which the weight is lowered. As such, the weight is lifted on average, and hence this system produces work. (b) From the viewpoint of the virtual qubit, the biasing condition says that the virtual qubit must have a population inversion or, in other words, must be at a negative temperature.

Figure 6

Schematic diagram of idealized thermal machines, showing the flow of heat and entropy for (a) a refrigerator, (b) a heat pump, and (c) a heat engine.

Figure 7

Graph of virtual temperature against bath temperature. We hold fixed the local energy-level spacings $E_1$ and $E_2$, as well as the bath temperature $T_1$, and allow the bath temperature $T_2$ to vary. When $T_2<T_1$, the virtual temperature $T_v$ becomes smaller than either environmental temperature and the machine is a refrigerator. For $T_1<T_2<\frac{E_2}{E_1}{T}_1$, $T_v$ becomes larger than $T_1$ and $T_2$ and hence the machine is a heat pump. Finally, for $\frac{E_2}{E_1}{T}_1<T_2$, $T_v$ becomes negative and the machine functions as a heat engine.

Figure 8

Schematic diagram of thet smallest machine interacting with an isolated external system. An isolated external system—here an $N$ level equispaced system—is placed into thermal contact (wavy lines) with the machine virtual qubit, which has matching energy-level spacing. The net effect is that the external system is placed in thermal contact with virtual qubits in the composite bath at temperature $T_v$, mediated by the machine virtual qubit, and reaches a Boltzmannian at temperature $T_v$. This holds independent of whether $T_v$ is positive or negative, in which case the state is an “inverted Boltzmannian.”