Thermodynamics of precision in quantum nanomachines

Fluctuations strongly affect the dynamics and functionality of nanoscale thermal machines. Recent developments in stochastic thermodynamics have shown that fluctuations in many far-from-equilibrium systems are constrained by the rate of entropy production via so-called thermodynamic uncertainty relations. These relations imply that increasing the reliability or precision of an engine's power output comes at a greater thermodynamic cost. Here we study the thermodynamics of precision for small thermal machines in the quantum regime. In particular, we derive exact relations between the power, power fluctuations, and entropy production rate for several models of few-qubit engines (both autonomous and cyclic) that perform work on a quantized load. Depending on the context, we find that quantum coherence can either help or hinder where power fluctuations are concerned. We discuss design principles for reducing such fluctuations in quantum nanomachines and propose an autonomous three-qubit engine whose power output for a given entropy production is more reliable than would be allowed by any classical Markovian model.

Figure 1

Illustration of an autonomous two-qubit engine coupled to an infinite-dimensional load via a tripartite interaction. The system is maintained out of equilibrium by heat currents ${\dot{Q}}_1$ and ${\dot{Q}}_2$ exchanged with thermal reservoirs at temperatures $T_1$ and $T_2>T_1$. A portion of the energy flowing from the hot qubit (red) to the cold qubit (blue) is diverted to perform work on the load (gray).

Figure 2

The virtual qubit is the pair of states $\{{|0\rangle }_1{|1\rangle }_2,{|1\rangle }_1{|0\rangle }_2\}$ in the composite Hilbert space of the two engine qubits. Thermal baths drive transitions between the four engine eigenstates, generating population inversion in the virtual qubit, i.e., a negative temperature. Resonant coupling to the virtual qubit causes the load to thermalize with the negative virtual temperature, thus performing work.

Figure 3

Thermodynamics of precision for autonomous few-qubit engines. The product of relative power fluctuations and entropy production rate is plotted as a function of the coherent coupling $g$ relative to the average dissipation rate $p$. Results are shown for the two-qubit engine (2QE) with a reset (solid black line) or local Lindblad (dashed gray line) thermalization model, and for the three-qubit engine (3QE) model (dot-dashed blue line). The classical Markovian TUR bound is shown by the dotted black line. The parameters are ${\beta }_1{E}_1=3$ and ${\beta }_2{E}_2=1$.

Figure 4

Schematic of the autonomous three-qubit engine. The energy flow from the baths is intermediated by an additional qubit resonant with the load. This boosts quantum coherence associated with energy transport, allowing for reduced relative power fluctuations.

Figure 5

Schematic depiction of an Otto cycle where a qubit working medium is driven by a harmonic oscillator. The harmonic motion of the oscillator, stylized here as a coiled spring of extension $x$, modulates the level spacing of the qubit periodically. The qubit couples to hot and cold baths every half-period, thus changing the excited-state occupation probability (gray circles). The difference in heat absorbed from the hot and cold baths is converted into work done on the spring, driving oscillations of increasing amplitude.

Figure 6

The flywheel dynamics can be modeled as a discrete-time random walk in phase space, where each value of $\alpha$ represents a coherent state of the oscillator. The state of the qubit can be visualized as a spin pointing up or down, whose orientation determines the direction of the force on the oscillator (inset). If the spin flips its orientation after half the cycle, the effective driving force is resonant with the oscillatory motion, thus changing the displacement in phase space by $\pm 2d$. If the spin's orientation does not change, the effective force is constant and the displacement is unaltered after a full cycle.

Figure 7

Thermodynamics of precision for random walks. The TUR ratio after a large number of steps $N$ is plotted against the bias parameter $\chi$, which quantifies the entropy production per step. The result for the flywheel (blue dot-dashed line), corresponding to a discrete-time (DT) random walk over coherent states, is compared to a DT random walk over Fock states (gray dashed line) and a classical continuous-time (CT) random walk (solid black line). The parameters are chosen so that $p_0=0.6$.