Small quantum absorption refrigerator in the transient regime: Time scales, enhanced cooling, and entanglement

A small quantum absorption refrigerator, consisting of three qubits, is discussed in the transient regime. We discuss time scales for coherent dynamics, damping, and approach to the steady state, and we study cooling and entanglement. We observe that cooling can be enhanced in the transient regime, in the sense that lower temperatures can be achieved compared to the steady-state regime. This is a consequence of coherent dynamics but can occur even when this dynamics is strongly damped by the dissipative thermal environment, and we note that precise control over couplings or timing is not needed to achieve enhanced cooling. We also show that the amount of entanglement present in the refrigerator can be much larger in the transient regime compared to the steady state. These results are of relevance to future implementations of quantum thermal machines.

Figure 1

Three-qubit quantum thermal machine. Three two-level quantum systems (qubits) with energy gaps $E_C,E_R$, and $E_H$ are coupled to separate thermal reservoirs at temperatures $T_C,T_R$, and $T_H$ with coupling rates $p_C,p_R$, and $p_H$. The qubits interact collectively as described in the text, with interaction strength $g$. When operating as a fridge, the machines cools qubit $C$, i.e., brings it to a temperature below $T_C$.

Figure 2

Plots vs. time of [(a), (b)] the distance to the steady state $D(\rho \left(t\right),{\rho }_{\infty })$, [(c), (d)] the cold qubit temperature $T_c\left(t\right)$, [(e), (f)] bipartite entanglement $W_{R|CH}\left(t\right)$, and [(g), (h)] genuine tripartite entanglement $W_{CRH}\left(t\right)$. We use two sets of parameters given by $E_C=1,E_H=100,T_C=T_R=1,T_H=100,p_C=p_H={10}^{-5},p_R={10}^{-3}$, and $g={10}^{-2}$ (blue curves), $g={10}^{-4}$ (red curves), and the system is initially in a thermal state with each qubit equilibrated to its bath. In (c) and (d) the minimal cold qubit temperature attainable by unitary dynamics alone is indicated (dashed horizontal), and in (d) also the period of the unitary dynamics (dashed vertical). In (e)–(h) the maximal entanglement extractable from the initial state by the unitary dynamics alone is indicated (dashed).

Figure 3

(a) Damping rate of coherent dynamics vs. interaction strength for $E_C=1,E_H=100,T_C=1,T_R=1,T_H=100,p_C={10}^{-4},p_R={10}^{-3},p_H={10}^{-4}$. The limiting values $p_C+p_R+p_C$ and $3(p_C+p_R+p_C)/4$ for small and large $g$, respectively, are indicated (dashed). (b) Asymptotic decay rate vs. interaction strength for $E_C=1,E_H=100,T_C=1,T_R=1,T_H=100,p_C={10}^{-4},p_R={10}^{-3},p_H={10}^{-5}$ (blue) and $p_C=p_H=2\times {10}^{-5}$ (red). The limiting values $min\{p_C,p_R,p_H\}=p_H$ and $\sim p_H+p_C/4$ for the case of all $p_i$ different are indicated (dashed).

Figure 4

(a) Cold qubit temperature vs. interaction strength. The lowest cold qubit temperature attained within a fixed time of $t=500$ is shown (solid) as well as the steady-state value (dashed), for $E_C=1,E_H=100,T_C=T_R=1,T_H=100,p_C=p_H={10}^{-5}$, and $p_R={10}^{-3}$. (c) Cold qubit temperature vs. time corresponding to the bath couplings of (a) and $g=5\times {10}^{-3}$. The lowest temperature attainable from the initial state by coherent dynamics is indicated (dotted) as well as the steady-state value (dashed). (b) and (d) are the same as (a) and (c), with $p_C={10}^{-4}$. Interestingly, the transient temperature of the cold qubit is significantly lower compared to the steady-state value.

Figure 5

A minimum cold qubit temperature is reached in finite time for $E_C=1,E_H=100,T_C=T_R=1,T_H=100,p_C={10}^{-5},p_R={10}^{-3},p_H={10}^{-5},g={10}^{-4}$. The steady-state temperature is indicated (dashed) and half a period $\pi /2g$ of coherent evolution without dissipation (dotted).