Performance of autonomous quantum thermal machines: Hilbert space dimension as a thermodynamical resource

Multilevel autonomous quantum thermal machines are discussed. In particular, we explore the relationship between the size of the machine (captured by Hilbert space dimension) and the performance of the machine. Using the concepts of virtual qubits and virtual temperatures, we show that higher dimensional machines can outperform smaller ones. For instance, by considering refrigerators with more levels, lower temperatures can be achieved, as well as higher power. We discuss the optimal design for refrigerators of a given dimension. As a consequence we obtain a statement of the third law in terms of Hilbert space dimension: Reaching absolute zero temperature requires infinite dimension. These results demonstrate that Hilbert space dimension should be considered a thermodynamic resource.

Figure 1

The smallest possible fridge comprising three energy levels. Throughout this paper, couplings to ${\beta }_{\mathrm{c}}$ will be denoted by (blue) downward arrows, couplings to ${\beta }_{\mathrm{h}}$ by (red) upward arrows, and the virtual qubit by an (orange) arrow in the direction consistent with the machine (upward for the fridge, downward for the engine).

Figure 2

Sketch of machines discussed in the present work. We consider several generalizations of the simplest qutrit machine (top left). We first discuss single-cycle machines (top right), which can then be extended to multicycle machines (bottom right). Second, we study concatenated qutrit machines (bottom left).

Figure 3

Performance of machines as a function of dimension. The accessible virtual qubit, characterized by the bias $Z_{\mathrm{v}}$ and the norm $N_{\mathrm{v}}$, is shown for single-cycle machines (green dots), multicycle machine (blue dots), and concatenated qutrit machines (red dots). As a comparison we also show the machines discussed in Ref. [31] (purple dots). The dimension of the machine (i.e., the number of levels) is indicated next to each point, for all machines except the qutrit; there, the number $k$ of concatenated machines is given (hence the dimension is exponentially larger, $3^k$).

Figure 4

Sketch of the optimal single-cycle refrigerator, for an even number of levels $n$.

Figure 5

(a) Starting from the qutrit fridge, and adding a fourth level $|4\rangle$, the norm of the virtual qubit can be increased to $N_{\mathrm{v}}=1$, while maintaining the same bias $Z_{\mathrm{v}}$. This four-level fridge thus outperforms the qutrit fridge. (b) The four-level fridge viewed as a tensor product of the virtual qubit, now becoming a real qubit since $N_{\mathrm{v}}=1$, and a simpler thermal cycle. Note the coupling to the hot bath is now nonlocal, between the levels $|0\rangle \otimes {|e\rangle }_{\mathrm{v}}$ and $|1\rangle \otimes {|g\rangle }_{\mathrm{v}}$.

Figure 6

(a) Starting from a 5-level fridge, and adding 3 levels (dashed lines), the norm of the virtual qubit can be boosted to $N_{\mathrm{v}}=1$ while maintaining the same bias $Z_{\mathrm{v}}$. (b) The resulting 8-level fridge can be viewed as a tensor product of a 4-level cycle and the virtual qubit, which is now a real one since $N_{\mathrm{v}}=1$.

Figure 7

By concatenating two qutrit machines, one obtains a better fridge, outperforming the simple qutrit fridge. Specifically, the new machine now consists of a qutrit fridge (left), which is boosted via the use of a qutrit heat engine (right). The role of this heat engine is to create an effectively hotter temperature (hotter than $T_{\mathrm{h}}$) in order to fuel the fridge.

Figure 8

Concatenating many qutrit machines to form an engine.

Figure 9

Relationship between the steady-state virtual temperature and the length of the cycle. We consider various equilibration time scales, ${\tau }_{\mathrm{s}}=1$ (green, diamond), ${\tau }_{\mathrm{s}}=10$ (orange, square) and ${\tau }_{\mathrm{s}}=100$ (blue, dot). All other parameters are kept fixed: time scale of all thermal couplings of the cycle ${\tau }_{\beta }=1$, bath temperatures ${\beta }_{\mathrm{h}}=0.05$, ${\beta }_{\mathrm{c}}=0.2$, and energies $E_{\max }=2$, and $E_{\mathrm{v}}=1$ (as in Fig. 3).

Figure 10

Length of the optimal cycle vs equilibration time scale ${\tau }_{\mathrm{s}}$. Other parameters are the same as in Fig. 9.

Figure 11

Different methods of amplifying the virtual qubit of an arbitrary cycle. (a) Amplification that maintains the energy and bias of the virtual qubit. (b) Amplification that modifies (possibly amplifies) the bias of the virtual qubit. (c) Amplification that flips the bias of the virtual qubit.

Figure 12

Engine formed out of the concatenation of many qutrit machines.