Experimental Demonstration of Quantum Effects in the Operation of Microscopic Heat Engines

The ability of the internal states of a working fluid to be in a coherent superposition is one of the basic properties of a quantum heat engine. It was recently predicted that in the regime of small engine action, this ability can enable a quantum heat engine to produce more power than any equivalent classical heat engine. It was also predicted that in the same regime, the presence of such internal coherence causes different types of quantum heat engines to become thermodynamically equivalent. Here, we use an ensemble of nitrogen vacancy centers in diamond for implementing two types of quantum heat engines, and experimentally observe both effects.

Figure 1

Quantum heat engine schematics. (a) Three basic heat engine types for a three-level system. Thermal coupling to a cold (hot) bath is represented by a blue (red) arrow. The unitary operation induced during the work stroke is represented by the letter $U$. The four-stroke engine (right) is equivalent to the Otto engine when $U$ induces a full population swap. (b) Left: The relevant levels of the ${\mathrm{NV}}^{-}$ center, and the optically induced incoherent couplings between them. Solid arrows represent excitation, and dashed (dotted) arrows represent spin preserving (nonpreserving) decay. The width of the arrow represents the transition rate. Right: The effective three-level ${\mathrm{NV}}^{-}$ heat-engine. The circles represent the steady-state populations. The wavy black arrows represent microwave driving and stimulated emission, extracting work.

Figure 2

Quantum thermodynamic signatures. (a) Beating the stochastic bound: Measured power output of the two-stroke engine (dots) vs the action per cycle, the theoretically predicted power (red line), and the stochastic bound (blue line) calculated for the measured Rabi frequency used in the experiment. Error bars (shaded regions) mark the experimental (theoretical) $1\sigma$ uncertainty. For the lowest action, the measured power is $2.4\sigma$ above the stochastic bound. (b) Work output per cycle of the two-stroke engine vs thermal stroke duration in units of dephasing time $T_2^{*}=75\mathrm{ns}$. Red dots (line) present the measured data (theoretical prediction). The blue line is the stochastic bound. The work-stroke length and Rabi frequency are fixed, while the optical excitation rate is adjusted to keep the thermal action constant. The insets on the left (right) depict cycles with a short (long) thermal stroke. The measured work output decreases to below the bound.

Figure 3

Quantum heat machine equivalence of two-stroke and continuous engine types. The dots present the power output of the two-stroke engine, running at a duty cycle of $1/3$, measured for different values of the action per cycle (varied by changing the cycle time). Each data set is for a different peak Rabi frequency, as indicated in the legend. The same thermal coupling rate, ${\gamma }_{\text{th}}=1.76\mathrm{MHz}$, is applied for all data sets. The error bars present the measurement uncertainty. The shaded regions represent the measured output powers of the continuous engine and their uncertainties for Rabi frequencies and thermal coupling rates that are $1/3$ and $2/3$, respectively, of those applied in the corresponding two-stroke engines, such that the mean values are the same. The theory predictions for the continuous (two-stroke) engine are given by the dashed (solid) lines. For small actions, the convergence in performance of the two engine types is clearly seen.