Superradiant Many-Qubit Absorption Refrigerator

We show that the lower levels of a large-spin network with a collective antiferromagnetic interaction and collective couplings to three reservoirs may function as a quantum-absorption refrigerator. In appropriate regimes, the steady-state cooling current of this refrigerator scales quadratically with the size of the working medium, i.e., the number of spins. The same scaling is observed for the noise and the entropy production rate.

Figure 1

Pauli-type transition rates (arrow-headed solid and dashed lines) generated by the reservoirs, Eq. (2), connect the energy eigenstates (horizontal levels) of the maximum angular momentum and even parity sector of our system, Eq. (1), ($N$ is assumed odd). In suitable parameter regimes, the lowest three levels of this sector can be driven exclusively by the reservoirs (arrow-headed solid lines). For thermal reservoirs, all transitions prefer the downward direction. However, if this imbalance is for the hot reservoir significantly stronger than for the cold and work reservoirs, the system will on average traverse the cycle counterclockwise, absorbing energy from the cold reservoir (cooling functionality). Since near the system’s ground state all rates are boosted superradiantly, the net cooling current is then boosted as well.

Figure 2

Stationary current ${\overline{I}}_E^c$ and long-term fluctuations ${\overline{S}}_E^c$ (inset) as a function of the (odd) WF size $N$ (blue), see Eq. (1), together with the reduced three-level model (red). The solid lines correspond to the analytic formulas, Eqs. (G4) and (G5), for an infinite temperature work reservoir ${\beta }_w\to 0$. Parameters used are ${\epsilon }_c=2\mathrm{\Omega }{\epsilon }_h=6\mathrm{\Omega }$, ${\epsilon }_w=4\mathrm{\Omega }$, ${\delta }_c={\delta }_h=0.1\mathrm{\Omega }$, ${\delta }_w={10}^{-3}\mathrm{\Omega }$, ${\beta }_c=2{\mathrm{\Omega }}^{-1}$, ${\beta }_h=1{\mathrm{\Omega }}^{-1}$, ${\beta }_w={10}^{-3}{\mathrm{\Omega }}^{-1}$, ${\overline{\mathrm{\Gamma }}}_h={\overline{\mathrm{\Gamma }}}_w={\overline{\mathrm{\Gamma }}}_c=1\mathrm{\Omega }$.

Figure 3

Attainability of cooling as a function of the widths ${\delta }_{\nu }$ of the spectral densities of a type, Eq. (5). The color code shows the stationary current ${\overline{I}}_E^c$. The blue regions denote the setting where the machine functions as a refrigerator. (a) The cold reservoir spectral density width is fixed ${\delta }_c=0.1\mathrm{\Omega }$. (b) The work reservoir spectral density width is fixed ${\delta }_w={10}^{-3}\mathrm{\Omega }$. The size of the system forming the WF is $N=31$. Other parameters are the same as in Fig. 2.

Figure 4

Attainability of cooling (current above the dashed horizontal line) as a function of the temperature of the cold reservoir ${\beta }_c$ plotted for several values of the width of the work reservoir spectral density ${\delta }_w$. The vertical lines mark at which values of ${\beta }_c$ the current flips its sign for each curve. The vertical dotted line marks the maximal inverse temperature ${\beta }_c^{\max }=3{\beta }_h$ at which the reduced model can operate. The size of the system forming the WF is $N=31$. Other parameters are the same as in Fig. 2.

Figure 5

Relative coefficient of performance for cooling as a function of the inverse temperatures of the cold ${\beta }_c$ and the hot ${\beta }_h$ reservoirs. The blue region marks the positive cooling current ${\overline{I}}_E^c>0$ and the color intensity encodes the cooling performance $\kappa /\overline{\kappa }$, see Eqs. (F1) and (20). The red dots indicate the highest value of ${\overline{I}}_E^c$. Panels (a)–(d) correspond to different values of ${\delta }_{\nu }$. The red point in each panel marks the largest relative coefficient of performance. The dashed lines indicate ${\beta }_c={\beta }_h$ and ${\beta }_c=3{\beta }_h$, which is the temperature window in which the reduced model works as a cooler, see Appendix pp7. The thin green solid lines in all panels mark the equivalue lines of a constant cooling current ${\overline{I}}_E^c$ in the case of the reduced model. From left to right these values are ${\overline{I}}_E^c/({\overline{\mathrm{\Gamma }}}_c\mathrm{\Omega })=0.4,0.04,0.004$. The size of the system forming the WF is $N=31$. Other parameters are the same as in Fig. 2.

Figure 6

Mapping procedure for spectral densities. Since the original spectral density (black) does not decay fast enough, we consider the regularized version (red). The mapped spectral density (green) is of Lorentz-Drude type and rather structureless.

Figure 7

Thermalization time $t_{\mathrm{th}}$ as a function of the size of the WF given by the Hamiltonian (1). The black points are the numerical values computed using Eq. (C1) for integer $N$ and the red dashed line is $1/N^2$ fit. The initial state is a thermal state with ${\beta }_i=1{\mathrm{\Omega }}^{-1}$. The temperature of the heat bath is ${\beta }_f=4{\mathrm{\Omega }}^{-1}$.