Three qubits in less than three baths: Beyond two-body system-bath interactions in quantum refrigerators

We show that quantum absorption refrigerators, which have traditionally been studied as of three qubits, each of which is connected to a thermal reservoir, can also be constructed by using three qubits and two thermal baths, where two of the qubits, including the qubit to be locally cooled, are connected to a common bath. With a careful choice of the system, bath, and qubit-bath interaction parameters within the Born-Markov and rotating-wave approximations, one of the qubits attached to the common bath achieves a cooling in the steady state. We observe that the proposed refrigerator may also operate in a parameter regime where no or negligible steady-state cooling is achieved, but there is considerable transient cooling. The steady-state temperature can be lowered significantly by an increase in the strength of the few-body interaction terms existing due to the use of the common bath in the refrigerator setup. The proposed refrigerator built with three qubits and two baths is shown to provide steady-state cooling for both Markovian qubit-bath interactions between the qubits and canonical bosonic thermal reservoirs, and a simpler reset model for the qubit-bath interactions.

Figure 1

Schematic diagram representing the qubits-baths setup. The qubits 1 and 2 are in contact with the bath $B_1$ at temperature ${\tau }_1$, while the qubit 3 is connected to the bath $B_2$ at temperature ${\tau }_2$.

Figure 2

Dynamics of cold-qubit temperature in the weak- and strong-coupling operating regimes. We choose $g={10}^{-2}$ (“weak coupling”) in panels (a), (b), and (c), and $g=0.5$ (“strong coupling”) in panels (d), (e), and (f). The values of ${\kappa }_1$ differ in the different panels, with ${\kappa }_1={\kappa }_2=0$ for panels (a) and (d), ${\kappa }_1=1,{\kappa }_2=0$ for (b) and (e), ${\kappa }_1=2,{\kappa }_2=0$ for (c), and ${\kappa }_1=4,{\kappa }_2=0$ for (f). ${\kappa }_2$ is kept at zero throughout. The definitions of the types of dynamics denoted by S1–S3 are as given in Sec. 3b1. For all of the panels (a)–(f), the values of the qubit-bath interaction parameters are chosen to be (i) ${\delta }_1={10}^{-8},{\delta }_2={10}^{-4}$ for S1, (ii) ${\delta }_1={10}^{-4},{\delta }_2={10}^{-5}$ for S2, and (iii) ${\delta }_1={10}^{-4},{\delta }_2={10}^{-8}$ for S3. The horizontal black dashed lines in panels (a) and (d) correspond to the cold bath temperature ${\tau }_1$. For the other panels, the same coincide with the horizontal upper boundaries. All quantities plotted are dimensionless.

Figure 3

Behavior of steady-state temperature with three-body interaction. Variation of $T_1^s$ as a function of ${\kappa }_1$, with ${\kappa }_2=0$ in the (a) weak- ($g={10}^{-2}$) and (b) strong-coupling ($g=0.5$) regimes. The values of the qubit-bath interaction parameters are the same as in Fig. 2. All quantities plotted are dimensionless.

Figure 4

Variations of the steady-state temperature of the cold qubit as function of system parameters. Behaviors of $T_1^s$ with varying $\mathrm{\Delta }E$ are depicted in (a) (weak coupling: $g={10}^{-2}$) and (b) (strong coupling: $g=0.5$), while (c) exhibits the variation of $T_1^s$ with $g$. The values of the qubit-bath interaction parameters are ${\delta }_1={10}^{-4},{\delta }_2={10}^{-5}$. Note that the choice of the $({\kappa }_1,{\kappa }_2)$ pair dictates the relative strengths between the single-body term, and two-body, three-body, and four-body interactions. All quantities plotted are dimensionless.

Figure 5

Variation of steady-state temperature of the cold qubit with respect to difference of temperatures of the two baths. Variations of $T_1^s$ against $\mathrm{\Delta }\tau$ in the (a) weak- ($g={10}^{-2}$) and (b) strong-coupling ($g=0.5$) regimes are plotted here. The values of the qubit-bath interaction parameters are ${\delta }_1={10}^{-4},{\delta }_2={10}^{-5}$. All quantities plotted are dimensionless.

Figure 6

Response to four-body interaction term is insignificant. We analyze here the consequences of a four-body interaction term in the Hamiltonian representing the interaction between the bath $B_1$ and the qubits 1 and 2. For (a), we depict the temperature of cold qubit, $T_1\left(t\right)$, vs the scaled time. The operating regime of the machine is chosen to be the strong-coupling regime ($g=0.5$), and the values of the qubit-bath interaction parameters are ${\delta }_1={10}^{-4},{\delta }_2={10}^{-8}$. For (b) and (c) the operating regime is weak coupling $(g={10}^{-2})$ and strong coupling $(g=0.5)$, respectively, and the values of the qubit-bath interaction parameters are the same as in Fig. 2. Trends of $T_1^S$ with the increase of coupling of four-body interaction term, ${\kappa }_2$, are shown. All quantities plotted are dimensionless.

Figure 7

Local cooling in a three-qubit single-bath refrigerator. We set the values of the parameters as $\delta ={10}^{-4}$ and $E_1=1$ for all panels. In each panel, we plot for three values of $\mathrm{\Delta }E$, viz., $1,2,3$. Weak-coupling instances ($g={10}^{-2}$) are considered in panels (a) for ${\kappa }_1=1,{\kappa }_2=0$ and (b) for ${\kappa }_1=2,{\kappa }_2=0$, while strong-coupling ones ($g=0.5$) are in panels (c) for ${\kappa }_1=1,{\kappa }_2=0$ and (d) for ${\kappa }_1=4,{\kappa }_2=0$. The black dashed lines represent the bath temperature $\tau =1$. All quantities plotted are dimensionless.

Figure 8

Consequences of using reset model. (a) Dynamics of cold-qubit temperature in the weak-coupling ($g={10}^{-2}$) operating regime. The values of the qubit-bath interaction parameters are chosen to be (i) ${\delta }_1={10}^{-7.5},{\delta }_2={10}^{-3.5}$ for S1, (ii) ${\delta }_1={10}^{-3.5},{\delta }_2={10}^{-7.5}$ for S2, and (iii) ${\delta }_1={10}^{-3},{\delta }_2={10}^{-4}$ for S3. Variations of $T_1^s$ as functions of (b) $\mathrm{\Delta }E$ for $g={10}^{-2}$, (c) $g$, and (d) $\mathrm{\Delta }\tau$ for $g={10}^{-2}$ are presented. The reset probabilities are taken to be $p_1={10}^{-3.5}$ and $p_2={10}^{-2.5}$ for (b), (c), and (d). The black dashed line in panel (a) represent the cold bath temperature ${\tau }_1$. All quantities plotted are dimensionless.