Single-Atom Heat Machines Enabled by Energy Quantization

Quantization of energy is a quintessential characteristic of quantum systems. Here we analyze its effects on the operation of Otto cycle heat machines and show that energy quantization alone may alter and increase machine performance in terms of output work, efficiency, and even operation mode. We show that this difference in performance occurs in machines with inhomogeneous energy level scaling, while quantum machines with homogeneous level scaling behave like classical machines. Our results demonstrate that quantum thermodynamics enables the realization of classically inconceivable Otto machines, such as those with an incompressible working substance. We propose to measure these effects experimentally using a laser-cooled trapped ion as a microscopic heat machine.

Figure 1

(a) The adiabatic introduction of an infinite $\delta$ function barrier does not change the energy of a classical ideal gas (left), precluding classical work extraction, but shifts the energy of the quantum ground state (right) due to the nonzero amplitude of its wave function at the barrier position, enabling quantum work extraction. (b) A quantum Otto cycle heat engine using the infinite well potential and the $\delta$ barrier. The cycle is composed of two adiabatic strokes (connecting states $A$ and $B$, and $C$ and $D$) where the potential is adiabatically deformed and the ion does not interact with any thermal bath, and two “isochoric” or “isoparametric” strokes [31] (connecting states $B$ and $C$, and $D$ and $A$) where the potential is kept constant while the ion equilibrates with the cold and the hot bath, respectively see the SM [35] Sec. VII. Work exchange results from the energy shift of the ground state, while the excited state remains unshifted (for $L_c=L_h$): work extraction $W_{CD}$ takes place after thermalization with the cold bath and therefore at high ground-state population, whereas work injection $W_{AB}$ takes place after thermalization with the hot bath and therefore at lower ground-state population. The difference between the ground-state populations results in net work extraction $|W_{CD}|>|W_{AB}|$ at constant volume [36].

Figure 2

(a) Efficiency normalized to Carnot efficiency for a classical (see SM [35] Sec. III) and quantum (see SM [35] Sec. I) Otto engine with a barrier of the form $g\delta (x)$ [see Fig. 1]. The plane is divided into three areas depending on the classical operation mode [see Eq. (1) and the discussion below it]. The temperature ratio is $(T_h/T_c)=12$, $\gamma =3$, $g$ is given in units of the critical value $g_{\mathrm{cri}}=(2{\hslash }^2/m{L}_c)$, and units are assumed such that $(2{\hslash }^2/m{L}_h)=1$. (b) Classical (dashed) and quantum (continuous) normalized efficiency as function of $T_c$, for $r=1.4$ (thick) and $r=1$ (thin), and fixed ratio $(T_h/T_c)$ for an ideal gas. The temperature dependence is a signature of the quantum machine. The number of populated levels depends on the temperature.

Figure 3

(a) The proposed experimental potential [see Eq. (4)]. (b) The calculated work (top) and normalized efficiency (bottom) for the classical (left) and quantum limit (right). Here ${\kappa }_h=1$ and ${\omega }_h=1\mathrm{MHz}$ have been chosen. The white areas correspond to the heater. The $X$ indicates the parameters used for Fig. 3, ${\kappa }_c=1.7$ and ${\omega }_c={\omega }_h$, where the classical heater operates as a heat engine in the quantum limit. (c) The calculated work and efficiency. The classical limit is obtained at $(1/\xi )\to 0$, where the work becomes constant and positive (work injection). In contrast, at the quantum limit, $(1/\xi )=1$, work is extracted. Cycle parameters: $T_h/T_c=41.6$, ${\overline{n}}_c=0.033$.