Thermal Transistor Effect in Quantum Systems

We study a quantum system composed of three interacting qubits, each coupled to a different thermal reservoir. We show how to engineer it to build a quantum device analogous to an electronic bipolar transistor. We outline how the interaction among the qubits plays a crucial role for the appearance of the effect, also linking it to the characteristics of system-bath interactions that govern the decoherence and dissipation mechanisms of the system. By comparison with previous proposals, our model extends the regime of parameters where the transistor effect occurs, the effect being more robust with respect to small variations of the coupling parameters. Moreover, our model appears to be more realistic and directly connected in terms of potential implementations to feasible setups in the domain of quantum spin chains, such as molecular nanomagnets.

Figure 1

A system composed by three qubits interacting with each other according to the model in Eq. (2). Each of them is coupled to a different thermal environment as described in Eq. (3).

Figure 2

The three thermal currents defined in Eq. (7) (in the unit of $\kappa \hslash {\mathrm{\Delta }}^2$) exchanged by the environments with the system as a function of the control parameter $T_M$ (in the unit of $\hslash \mathrm{\Delta }/k_B$, as for $T_L$ and $T_R$) when the system is described by an $x$-axis Ising model (${\lambda }_x=1$ and ${\lambda }_y={\lambda }_z=0$). The solid red line refers to $J_L$, the dashed green line to $J_M$, and the dotted blue line to $J_R$. The parameters are ${\omega }_L=\mathrm{\Delta },{\omega }_M=0.1\mathrm{\Delta },{\omega }_R=0.2\mathrm{\Delta }$, ${\eta }_{LM}={\eta }_{MR}=\mathrm{\Delta },{\eta }_{LR}=0$, ${\kappa }_L={\kappa }_M=\kappa ,{\kappa }_R=100\kappa$, $T_L=10$, and $T_R=0.01$. The inset shows the currents in the presence of dephasing as in Eq. (10), with the same color code as in the main plot.

Figure 3

The amplification factors defined in Eq. (9) as a function of the control parameter $T_M$ (in the unit of $\hslash \mathrm{\Delta }/k_B$). In (a), the solid red line refers to ${\alpha }_L$ and the blue dotted line to ${\alpha }_R$. The inset shows $J_M$ (in the unit of $\kappa \hslash {\mathrm{\Delta }}^2$) versus $T_M$, including the local minimum responsible of the divergence of the amplification factors. The parameters have the same values as in Fig. 2. (b) ${\alpha }_L$ as a function of $T_M$ for different values of the coupling between the left qubit and the right qubit. The solid red line refers to the case ${\eta }_{LR}=\mathrm{\Delta }/10$, the dashed green line to ${\eta }_{LR}=\mathrm{\Delta }/3$, and the dotted blue line to ${\eta }_{LR}=\mathrm{\Delta }$.

Figure 4

The three thermal currents defined in Eq. (7) (in the unit of $\kappa \hslash {\mathrm{\Delta }}^2$) exchanged by the environments with the system as a function of the control parameter $T_M$ (in the unit of $\hslash \mathrm{\Delta }/k_B$) when the system is described by a $z$-axis Ising model (${\lambda }_x={\lambda }_y=0$ and ${\lambda }_z=1$). The solid red line refers to $J_L$, the dashed green line to $J_M$, and the dotted blue line to $J_R$. Empty markers refer to the currents observed in the presence of dephasing as in Eq. (10). Circles, squares, and triangles refer to $J_L$, $J_M$, and $J_R$, respectively. The parameters have the same values as in Fig. 2. The inset shows an enlargement including the working region of this setting.

Figure 5

Local temperatures of the three qubits defined in Eq. (11) as a function of the control parameter $T_M$ (in the unit of $\hslash \mathrm{\Delta }/k_B$). The plots in (a)–(c) refer, respectively, to the local temperature of the left qubit, the middle qubit, and the right qubit. The parameters have the same values as in Fig. 2.

Figure 6

Comparison between the global currents computed for the $x$-axis Ising model (red lines), the same ones as in Fig. 2, and the currents obtained with the local model (dotted blue lines) as a function of $T_M$ (in the unit of $\hslash \mathrm{\Delta }/k_B$). All the currents are the in unit of $\kappa \hslash {\mathrm{\Delta }}^2$. The plots in (a)–(c) refer, respectively, to the comparison of the current exchanged with the left thermostat, the middle thermostat, and the right thermostat.

Figure 7

The ratio $\zeta$ defined in Eq. (13) for the $x$-axis Ising model as a function of the mean value of the spurious couplings ${\overline{\lambda }}^{i,j}$. The dashed green line refers to ${\zeta }^z$ and the dotted blue line to ${\zeta }^y$, whereas the red solid line takes into account perturbations with ${\overline{\lambda }}^y={\overline{\lambda }}^z\ne 0$, and then it refers to ${\zeta }^{y,z}$. The inset shows the ratio $\zeta$ for the $z$-axis Ising model. The dashed green line refers to ${\zeta }^x$, coinciding with ${\zeta }^y$ (red diamonds). The dotted blue line refers to ${\zeta }^{x,y}$ and is obtained by our imposing ${\overline{\lambda }}^y={\overline{\lambda }}^z\ne 0$. The parameters of the unperturbed configuration are the same as in Fig. 2. The control temperature T_M is fixed equal to 0.650.

Figure 8

The three thermal currents defined in Eq. (7) (in the unit of $\kappa \hslash {\mathrm{\Delta }}^2$) exchanged by the environments with the system as a function of the control parameter $T_M$ (in the unit of $\hslash \mathrm{\Delta }/k_B$) in the case of the $x$-axis Ising model (${\lambda }_x=1$ and ${\lambda }_y={\lambda }_z=0$). The solid red line refers to $J_L$, the dashed green line to $J_M$, and the dotted blue line to $J_R$. The parameters have the same values as in Fig. 2, except for ${\eta }_{LM}$ and ${\eta }_{MR}$, whose values are reduced to ${\eta }_{LM}={\eta }_{MR}=\mathrm{\Delta }/100$. Empty markers refer to the local currents based on the local temperatures defined in Eq. (11). Circles, squares, and triangles refer to $J_L$, $J_M$, and $J_R$, respectively. The inset shows the local temperature of the left qubit as a function of the control parameter $T_M$ (both in the unit of $\hslash \mathrm{\Delta }/k_B$).