Short time cycles of purely quantum refrigerators

Four stroke Otto refrigerator cycles with no classical analog are studied. Extremely short cycle times with respect to the internal timescale of the working medium characterize these refrigerators. Therefore, these cycles are termed sudden. The sudden cycles are characterized by the stable limit cycle, which is the invariant of the global cycle propagator. During their operation the states of the working medium possess significant coherence which is not erased in the equilibration segments due to the very short time allocated. This characteristic is reflected in a difference between the energy entropy and the Von Neumann entropy of the working medium. A classification scheme for sudden refrigerators is developed allowing simple approximations for the cooling power and coefficient of performance.

Figure 1

Top ($\alpha$): An example of an extreme sudden cycle with analytical scheduling, in the ($\Omega ,S_E$) plane (curved shape) and ($\Omega ,S_{VN}$) plane (shrunken rectangle shape), together with the isotherms corresponding to the cold/hot bath temperatures, $T_c/T_h$. The cooling power is $Q_c/\tau =1.2\times {10}^{-6}$. The cycle parameters are $T_c=14$, $T_h=15$, $J=2$, ${\omega }_c=0.1$, and ${\omega }_h=6$. The time allocations are ${\tau }_c,{\tau }_{ch},{\tau }_h,{\tau }_{hc}=0.9$, 0.00035, 0.00025, and 0.00035; ${\kappa }_h^{\downarrow }=0.36$; and ${\kappa }_c^{\downarrow }=0.328$. Fig. 1: The corresponding cycle trajectory in the $H,L,C$ space.

Figure 2

A typical sudden cycle of the refrigerator with additional time allocation on the segments. Top ($\alpha$): In the ($\Omega ,{\mathcal{S}}_E$) and ($\Omega ,{\mathcal{S}}_{\mathrm{VN}}$) plane (rectangle), together with the isotherms corresponding to the cold/hot bath temperatures, $T_c/T_h$. Note ${\mathcal{S}}_E>{\mathcal{S}}_{\mathrm{VN}}$. The cycle parameters are $T_c=2.175$, $T_h=2.9$, $J=2.5$, ${\omega }_c=2.5$, and ${\omega }_h=10$. The time allocations are ${\tau }_c,{\tau }_{ch},{\tau }_h,{\tau }_{hc}=0.2,0.21,0.44,0.21$ and ${\kappa }_{h,c}=0.36,0.328$. Fig. 2: The corresponding cycle trajectory in the $H,L,C$ space. The longer time allocation on the hot isomagnetic segment and the larger frequency ${\Omega }_h$ allows us to complete approximately 3/4 of a period.

Figure 3

The coherence $\tilde{\mathcal{C}}$, Eq. (9), as a function of $\Omega$ for cycles with different time allocations on the adiabats. The cycle parameters correspond to the data of Fig. 1 except for the different ${\tau }_{hc}={\tau }_{ch}$. Notice the decrease in coherence when the time allocation on the adiabats increases.

Figure 4

A sudden cycle with additional time allocated on the isomagnetic segments. The cycle parameters are $J=2.5$, $T_h=2.9$, $T_c=2.175$, ${\omega }_h=10$, ${\omega }_c=2.5$, $k_h\downarrow =3.35$, $k_c\downarrow =0.328$ ${\tau }_h=0.442$, ${\tau }_{hc}=0.00744$, ${\tau }_c=0.527$, and ${\tau }_{ch}=0.00824$. Top ($\alpha$): In the $(\Omega ,{\mathcal{S}}_E)$ and $(\Omega ,{\mathcal{S}}_{\mathrm{VN}})$ planes. Fig. 3: The corresponding trajectory in the $H,L,C$ space. Linear scheduling of $\omega \left(t\right)$.

Figure 5

Top ($\alpha$): Cycle trajectories in the $H,L,C$ space which complete approximately one revolution on the hot isomagnetic segment. ${\omega }_h{\tau }_h=6.252$. The data for the three cycles are $J=2.5$, $T_h=10$, $T_c=9$, ${\omega }_c=2.5$, $k_h\downarrow =0.36$, $k_c\downarrow =0.328$, ${\tau }_{hc}={\tau }_{ch}=0.065625$, and ${\tau }_c=0.008375$. From large to small cycles: ${\omega }_h=156.3$, 78.15, and 39.075 and therefore ${\tau }_h=0.04$, 0.08, and 0.16. Bottom ($\beta$): The same cycles in the $(\Omega ,{\mathcal{S}}_E)$ and $(\Omega ,{\mathcal{S}}_{\mathrm{VN}})$, planes.

Figure 6

Top ($\alpha$): Equation (26) is presented when changing the values of ${\tau }_c$, for two different ${\tau }_h$ values, as denoted on the figure: ${\tau }_h=0.00075$, red line; ${\tau }_h=0.00025$, green line. Bottom ($\beta$): The functions $\exp \left({\Gamma }_c{\tau }_c\right)$ and $\cos \left({\Omega }_c{\tau }_c\right)$ are shown as functions of ${\tau }_c$. Their crossing points compare well to the exact points (top panel). The other parameters are $J=2$, ${\tau }_{ch}={\tau }_{hc}=0.00035$, ${\omega }_c=0.1$, ${\omega }_h=6$, $T_h=15$, and $T_c=14$, as in Fig. 1.

Figure 7

Top ($\alpha$): ${\mathcal{S}}_E$ and ${\mathcal{S}}_{\mathrm{VN}}$ as a function of $\Omega$ cycles for three different sets of times, ${\tau }_h$, ${\tau }_{ch}={\tau }_{hc}$, ${\tau }_c=$ are: 0.000405, 0.4194, 0.029538, green (a), 0.00045, 0.466, 0.03282, orange (b), and 0.00036, 0.3728, 0.026256, non refrigeration cycles, and 0.000225, 0.233, 0.01641, magenta (c) cycles. Bottom ($\beta$): Continuation of the Top by lowering the cycle time proportionally. ${\tau }_h$, ${\tau }_{ch}={\tau }_{hc}$, and ${\tau }_c=$ are magenta (c) as on the Top, 0.0001125, 0.1165, 0.008205, green (d), 0.00005625, 0.05825, 0.0041025 maroon (e). The parameters for all the cycles are: $J=1.25$, $T_h=4$, $T_c=3.6$, ${\omega }_h=11$, ${\omega }_c=6.5$, ${\kappa }_h^{\downarrow }=0.36$, ${\kappa }_c^{\downarrow }=0.0656$ see Eq. (29).

Figure 8

The trajectories in the space $H,L,C$ of the cycles of Fig. 7 where the cycles are indicated by letters corresponding to the cycles of Fig. 7.

Figure 9

$Q_c$ as a function of ${\tau }_{\mathrm{adi}}$ with some internal scaling as denoted in the figure in order to achieve the exact results by the approximation of Eq. (29), which approximate the cycles of Fig. 7, with the same parameters. The fineness of the sign change is obvious.

Figure 10

A map of parameter regions of time allocation leading to refrigeration regions (blue circles) and nonrefrigeration regions (red squares). The cycle of Fig. 1 is located in the blue island in the lower left corner. The general parameters are $J=2$, $T_c=14$, $T_h=15$, ${\omega }_c=0.1$, ${\omega }_h=6$, ${\kappa }_h^{\downarrow }=0.36$, and ${\kappa }_c^{\downarrow }=0.328$. The parameters used fit Case 1: Eq. (26).

Figure 11

The $1/\left(\mathrm{COP}\right)$ as a function of inverse cooling power, $1/{\mathcal{P}}_c$. The data correspond to the conditions of approximation $Q_c^{\mathrm{appr}2}$, Eq. (27). The data for both figures are $J=2.5$, $T_h=10$, $T_c=9$, ${\omega }_c=2.5$, ${\tau }_c=0.008375$, ${\tau }_{ch}={\tau }_{hc}=0.065625$, and also the condition ${\omega }_h{\tau }_h=6.252$. In addition, ${\omega }_h=625.2$ is the largest ${\omega }_h$ value for both plots. The differences between the plots are the $\kappa$ values. For the orange circle plot, ${\kappa }_c^{\downarrow }=0.328$ and ${\kappa }_h^{\downarrow }=0.36$, where three points correspond to the three cycles of Fig. 5, whose ${\tau }_h$ values are denoted in the figure. The largest ${\tau }_h$ is $0.32$. For the red diamond plot, ${\kappa }_h^{\downarrow }=3.6$ and ${\kappa }_c^{\downarrow }=3.28$. For this figure the largest ${\tau }_h$ is $0.5157905$.

Figure 12

$\ln \left({\mathcal{P}}_c\right)$ as a function of $J/T_c$ for the three classes of sudden refrigerators. The cycle parameters for type 1 correspond to Fig. 1 where $T_c/T_h$ was kept constant and $T_c$ was varied. The cycle parameters of type 2 correspond to Fig. 2 where $T_c/T_h$ was kept constant and $T_c$ was varied. The cycle parameters of type 3b correspond to Fig. 7 where $T_c/T_h$ was kept constant and $T_c$ was varied.