Impurity reveals distinct operational phases in quantum thermodynamic cycles

We analyze the effect of impurity on the work output and efficiency of quantum Otto and quantum Carnot heat cycles, modeled as a single quantum particle in an infinite square well potential, which is the working substance. We solve this quantum mechanical system perturbatively up to first and second order in strength of the impurity for strong- and weak-coupling regimes, respectively. We derive the analytical expressions of work and efficiency for the strong-coupling regime to the first order in the strength parameter. The threshold value of the strength parameter in weak coupling is obtained up to which the numerical result agrees with the perturbative result for a repulsive and attractive impurity. To our surprise, an embedded impurity unlocks new operational phases in the system, such as a quantum heat engine, quantum refrigerator, and quantum cold pump. In addition, the efficiency of the quantum Otto heat engine is seen to reach Carnot efficiency for some parameter regimes. The cooling power and coefficient of performance of the quantum refrigerator and quantum cold pump are nontrivially affected by the impurity.

Figure 1

The potential $V\left(x\right)$ of ISW embedded with impurity at position $pL$ versus $x$. The length of the well is $L$ and strength $\lambda$ of the impurity.

Figure 2

(a) $kL$ vs position of the impurity $p$ (for strong-coupling case), plot for attractive impurity $f=0.02$ (in blue), and plot for repulsive impurity $f=-0.02$ (in red). (b) $kL$ vs $p$ (for weak-coupling case) plot for attractive impurity $f=1$ (in blue) and repulsive impurity $f=-1$ (in red). (c) $\kappa L$ vs $p$ for four different strength parameters $f=0.44$, 0.33, 0.2, and 0.1 (yellow, blue, red, and green). (d) $\frac{E}{|E_b|}$ vs $p$ at three different strength parameters $f=0.1,$ 0.04, and 0.02 (green, red, and blue).

Figure 3

Weak-coupling pertubative results. (a) $kL$ vs $p$ for the attractive $\delta$ function impurity with strength $f=1$. The first six energy levels are plotted both numerically (red solid) and perturbatively (black dashed). (b) $kL$ vs $p$ for the repulsive $\delta$ function with the strength parameter $f=-1$.

Figure 4

$\frac{E_g}{|E_b|}$ vs $p$ where black ($f>0$) and green solid lines ($f<0$) are the perturbative ground-state results for the attractive and repulsive $\delta$ function respectively which is plotted as a function of position $p$ of impurity. There are four colors green, red, blue, and black. Green ($f<0$) and black ($f>0$) are plotted analytically using perturbed ground-state expression, whereas red ($f<0$) and blue ($f>0$) are plotted numerically. The lowest, middle, and the topmost blue and red dots pair or black and green solid line pair are plotted at the strength parameter $\left|f\right|=0.1$, 0.5, and 1, respectively.

Figure 5

Temperature-entropy curve for QCC. $B\to A$ and $D\to C$ are adiabatic processes and $C\to B$ and $A\to D$ are isothermal processes.

Figure 6

Temperature-entropy curve for QOC. $S\left(B\right)$ and $S\left(D\right)$ are entropies at temperatures $T_h$ and $T_c$, respectively. $B\to A$ and $D\to C$ are adiabatic processes and $C\to B$ and $A\to D$ are isochoric processes.

Figure 7

Quantum Otto cycle operating between a hot bath at fixed temperature $T_h$ and a cold bath at fixed temperature $T_c$. It has two adiabatic ($B\to A$ and $D\to C$) and two isochoric ($C\to B$ and $A\to D$) strokes. Strength changes during the cycle between $f_h$ and $f_c$.

Figure 8

Density plots for work output in meV of QOC. (a) $T_h=25$ K, (b) $L=25$ nm, (c) $p=0.5$, (d) $p=0.15$.

Figure 9

Efficiency and coefficient of performance of QOC. (a) $T_h=25$ K, (b) $L=25$ nm, (c) $T_h=25$ K, (d) $L=25$ nm.

Figure 10

Quantum Otto cycle for varying length of ISW during the cycle. There are two adiabatic ($B\to A$ and $D\to C$) strokes and two isochoric ($C\to B$ and $A\to D$) strokes.

Figure 11

Density plots for work output of QOC. (a) Parameters: $L_h=100$ nm and $L_c=129$ nm, $T_h=2.49$ K (fixed for all cycles), after every cycle position and strength are changed. As the work is entirely negative, the system works as a refrigerator. (b) Work is expressed in meV. $L_h=100$ nm and $L_c=163$ nm, $T_h=5$ K. As the work is positive, the system works as a heat engine.

Figure 12

Density plots for work output of QCC. (a) Parameters: $L_h=100$ nm and $L_c=129$ nm, $T_h=2.49$ K (fixed for all cycles). As work is negative, system works as a refrigerator. (b) $L_h=100$ nm and $L_c=163$ nm, $T_h=5$ K. As work is positive, the system works as a heat engine.

Figure 13

Plot of efficiency versus position of the impurity in QOHE and QCHE, calculated with $L_h=100$ nm and $L_c=163$ nm, $f=5$ (fixed for all cycles), $T_h=5$ K(fixed for all cycles).

Figure 14

Quantum Otto cycle is shown working between $T_h$ and $T_c$. $B\to A$ and $D\to C$ are adiabatic strokes while $C\to B$ and $A\to D$ are isochoric strokes. Position of impurity varies between $p_h$ and $p_c$.

Figure 15

Density plots of work for QOC in meV. (a) Parameters: $p_h=0.1$ and $p_c=0.8$, $T_h=10$ K (fixed for all cycles), $20\mathrm{nm}<L<80\mathrm{nm}$ and the strength of impurity varies as $1<f<6$. (b) $p_h=0.1$, $p_c=0.8$, $L=40$ nm (fixed for all cycles). (c) $p_h=0.1$ and $p_c=0.8$, $f=5$ (fixed for all cycles). (d) Parameters: $p_h=0.1$ and $p_c$ = 0.8, $T_h=10$ K (fixed for all cycles), $20\mathrm{nm}<L<50\mathrm{nm}$ and the strength of impurity varies as $1<f<6$.

Figure 16

Density plots of work and efficiency of QOC. Parameters: (a) $p_h=0.2$ and $p_c=0.5$, $T_h=10$ K, $20\mathrm{nm}<L<80\mathrm{nm}$ and the strength of impurity $0.01<f<0.09$. (b) $p_h=0.2$, $p_c=0.5$, $T_h=10$ K. (c) $p_h=0.2$, $p_c=0.5$, $T_h=10$ K. (d) $p_h=0.2$, $p_c=0.8$, $f=0.03$.