Minimal universal quantum heat machine

In traditional thermodynamics the Carnot cycle yields the ideal performance bound of heat engines and refrigerators. We propose and analyze a minimal model of a heat machine that can play a similar role in quantum regimes. The minimal model consists of a single two-level system with periodically modulated energy splitting that is permanently, weakly, coupled to two spectrally separated heat baths at different temperatures. The equation of motion allows us to compute the stationary power and heat currents in the machine consistent with the second law of thermodynamics. This dual-purpose machine can act as either an engine or a refrigerator (heat pump) depending on the modulation rate. In both modes of operation, the maximal Carnot efficiency is reached at zero power. We study the conditions for finite-time optimal performance for several variants of the model. Possible realizations of the model are discussed.

Figure 1

Inset: Illustration of the QHM by double-well qubits (with periodically modulated tunneling barrier) embedded between cold and hot baths. (a) Sinusoidal-modulation effects [Eqs. (18) and (19)]. Harmonic (Floquet) peaks of the response superimposed on rising cold and hot bath spectra $G^j\left(\omega \right)=A^j{\omega }^3$(top) and on the spectra of the same baths recalculated in the presence of different filter modes, transforming these baths’ spectra into skewed Lorentzians [according to Eqs. (40) and (41)]. The filtered spectra obey condition A1 (left) or A2 (right). The parameters (in arbitrary units) used to calculate the spectra are $A^C=1,A^H=1/10$ and, for the graph on the left, ${\gamma }_f^H=22$, ${\gamma }_f^C=1$, ${\omega }_f^H=13,$ and ${\omega }_f^C=1$ and, for the graph on the right, ${\gamma }_f^H=1$, ${\gamma }_f^C=2$, ${\omega }_f^H=13,$ and ${\omega }_f^C=20$. (b) Same, under condition B (phase-flip modulation), for rising hot-bath spectra and cold-bath spectrum with cutoff.

Figure 2

(a) Currents and power as function of the machine modulation for an ideal heat machine. For $\Delta <{\Delta }_{cr}$ it operates as a heat engine and for $\Delta >{\Delta }_{cr}$ as a refrigerator. (b) Efficiency for the same machine. The Carnot bound is reached at $\Delta ={\Delta }_{cr}$. Inset: Periodic modulation of three-level impurities embedded between hot and cold baths. The modulating field is detuned from the $|e\rangle -|u\rangle$ transition by ${\Delta }_u$ and exerts periodic ac Stark shift on $|e\rangle$.