Phase-space interference in extensive and nonextensive quantum heat engines

Quantum interference is at the heart of what sets the quantum and classical worlds apart. We demonstrate that quantum interference effects involving a many-body working medium is responsible for genuinely nonclassical features in the performance of a quantum heat engine. The features with which quantum interference manifests itself in the work output of the engine depends strongly on the extensive nature of the working medium. While identifying the class of work substances that optimize the performance of the engine, our results shed light on the optimal size of such media of quantum workers to maximize the work output and efficiency of quantum energy machines.

Figure 1

Quantum Otto engine cycle in the eigenenergy ($E_k$) and the occupation probability $p_k$ space of an arbitrary energy level $k$ of the spin system. In the quantum adiabatic stages, $D\to A$ and $B\to C$, the system is disconnected from the heat baths and model parameters ${\gamma }_x,{\gamma }_h$ changes between ${\gamma }_x^L,{\gamma }_y^L$ and ${\gamma }_x^H,{\gamma }_y^H$. In the isochoric stages, $A\to B$ and $C\to D$, the system is coupled to a hot bath at temperature $T_H$ and a cold bath at temperature $T_L$, respectively, so that $p_k$ changes between $p_k^L$ and $p_k^H$.

Figure 2

Interference of a squeezed vacuum state $|r\rangle$ with Fock number states $|k\rangle$ in the oscillator phase space $X,Y$, where $X$ and $Y$ denote the displacement and momentum, respectively. The ellipse and the concentric circular bands represent the squeezed state and the Fock states, respectively. The ellipse cuts two overlap regions, with area $A=A(k,r)$ on the annulus associated with $|k\rangle$. The dashed lines indicate two possible paths connecting the overlap regions. The shaded area between the dashed lines is denoted by $\varphi =k\pi$. The interference can be envisioned as a Young interferometer where the overlap regions correspond to the two slits, or a path interference of the trajectories between the overlap regions, or superposition of two waves originating from the overlap regions and propagating in opposite directions. Transition probability between the states is determined by $P\left(k\right)=4A{cos}^2(\varphi /2)$.

Figure 3

Interference of a spin state $|n\rangle$ with respect to the quantization axis $x$ and a spin state $|m\rangle$ with respect to the quantization axis $y$. Spin states are represented by circular bands about their corresponding quantization axes. Their overlaps are shown by the symmetric regions (1) and (2), each with area $a_{nm}^S$. Among four possible paths from (1) to (2), two paths are shown with solid black curves. The surface area between the indicated paths on the Bloch sphere of radius $R_S$ is shown by $A_{nm}^S$. The transition probability between the states is determined by $P(n,m;S)=4(a_{nm}^S/2\pi R_S){cos}^2{\varphi }_{nm}^S$, with ${\varphi }_{nm}^S$ being an interference angle determined by $A_{nm}^S$ and $R_S$.

Figure 4

Dependence of (a) the internal energy $U_B$ at point $B$ and (b) work output of a quantum Otto cycle in Fig. 1 with a working substance of interacting spins on the number of spins $N=2S$ with total spin $S$. The interaction is described by the LMG model. The upper red curve in (a) and connected red markers (b) are for the perturbative calculation, while the lower blue curve in (a) and the blue marking points in (b) are for the exact diagonalization method. We use dimensionless parameters scaled by ${\gamma }_x^L$ and take ${\gamma }_x^H=1.01$, ${\gamma }_x^L=1$, ${\gamma }_y^H=0.01$, and ${\gamma }_y^L=0.02$. The temperatures of the heat baths are $T_H=0.4$ and $T_L=0.1$.

Figure 5

Dependence of (a) the internal energy per particle $U_B/N$ at point $B$, (b) work output $W$, and efficiency $\eta$ of a quantum Otto cycle in Fig. 1 with a working substance of interacting spins on the number of spins $N=2S$ with total spin $S$. The interaction is described by the LMG model. The lower red curves in (a) and (b), and the middle red curve in (c) are for the extensive description of the LMG model, while the upper blue curve in (a) and the blue marking points in (b) and (c) are for the nonextensive description. We use dimensionless parameters scaled by ${\gamma }_x^L$ and take the spin-spin interaction parameters as ${\gamma }_x^H=1.01$, ${\gamma }_x^L=1$, ${\gamma }_y^H=0.01$, and ${\gamma }_y^L=0.02$. Temperatures of the heat baths are taken to be $T_H=0.4$ and $T_L=0.1$.

Figure 6

Population change $P_n^B-P_n^D$ of unperturbed levels $n$ for (a) the total spin $S=8$ and (b) $S=17/2$. Other $S$ values exhibit similar behavior. We use dimensionless parameters scaled by ${\gamma }_x^L$ and take the spin-spin interaction parameters as ${\gamma }_x^H=1.01$, ${\gamma }_x^L=1$, ${\gamma }_y^H=0.01$, and ${\gamma }_y^L=0.02$. Temperatures of the heat baths are taken to be $T_H=0.4$ and $T_L=0.1$.

Figure 7

Comparison of semiclassical (red squares) and exact evaluation (blue circles) of transition probabilities $P(n,m)$ between $x$-quantized spin states $|n\rangle$ and $y$-quantized spin states $|m\rangle$ for (a) $n=0$ and total spin $S=21$ and (b) $n=1/2$ and $S=21/2$. Transition probability differences (c) $P(3/2,S)-P(1/2,S)$ and (d) $P(1,S)-P(0,S)$ show better agreement between the semiclassical and exact evaluations.

Figure 8

(a) Dependence of interference angle $\mathrm{\Phi }(n,m=S;S)$ (scaled with $\pi /2$) on the total spin $S$. Spin state for the quantization axis $x$ ($y$) is denoted by $|n\rangle$ ($|m=S\rangle$). Plot markers of empty orange diamond, empty red triangle, filled green diamond, and filled blue triangle are for $n=0$, $n=1/2$, $n=1$, and $n=3/2$, respectively. (b) Top view of the Bloch sphere. Areas of overlap between the $|n\rangle$ and $|m=S\rangle$ decreases with $n$. The shaded area $A_{3/2,S}^S$ is the smallest.

Figure 9

Dependence of (a) interference cosine (squared) ${cos}^2\mathrm{\Phi }(n,m=S;S)$ and (b) $P(n,m=S;S)$ on the total spin $S$. Plot markers of empty orange diamond, empty red triangle, filled green diamond, and filled blue triangle are for $n=0$, $n=1/2$, $n=1$, and $n=3/2$, respectively.