Photonic Heat Rectification in a System of Coupled Qubits

We theoretically investigate a quantum heat diode based on two interacting flux qubits coupled to two heat baths. Rectification of heat currents is achieved by asymmetrically coupling the qubits to the reservoirs, which are modeled as dissipative $RLC$ resonators. We find that the coherent interaction between the qubits can be exploited to enhance the rectification factor, which otherwise would be constrained by the temperatures and couplings of the baths. Remarkably high values of the rectification ratio, up to $\mathcal{R}\sim 3.5$, can be obtained for realistic system parameters, with an enhancement up to approximately $230\mathrm{\%}$ compared to the noninteracting case. The system features the possibility of manipulating both the rectification amplitude and direction, allowing for an enhancement or suppression of the heat flow to a chosen bath. For the parameter regime in which rectification is maximized, we find a significant increase of the rectification above a critical interaction value, which corresponds to the onset of a nonvanishing entanglement in the system. Finally, we discuss the dependence of the rectification factor on the bath temperatures and couplings.

Figure 1

Left: a graphic illustration of the qubit heat diode. Two interacting flux qubits (green) are mutually coupled to each other via the inductance $M_{12}$ and to two thermal reservoirs with coupling factors $g_{L,R}$. The two reservoirs reside at temperatures $T_L$ and $T_R$ (red and blue). Right: the circuit diagram corresponding to the investigated system.

Figure 2

(a) The normalized power $\hslash P_R^{\pm }/{ϵ}_0^2$ transmitted to the right bath as a function of the dimensionless applied external flux $q$ for different values of the interaction $\tilde{\gamma }=\gamma /{ϵ}_0$. The continuous and dashed lines indicate, respectively, the direct- and reverse-thermal-bias configurations with $k_B{T}_L={ϵ}_0/5$ and $k_B{T}_R={ϵ}_0/20$. (b) The rectification ratio $\mathcal{R}=|P_R^{+}/P_R^{-}|$ extracted from the curves in (a). The gray dashed line indicates the absence of rectification. (c) The full dependence of $\mathcal{R}$ as a function of $q$ and $\tilde{\gamma }$. The black dashed line over the white area corresponds to points of absence of rectification ($\mathcal{R}=1$). The red and blue areas correspond, respectively, to regions of direct- and reverse-rectification direction. (d) The maximal rectification ${\chi }_{max}$ as a function of $\tilde{\gamma }$ for different $T_L$ at fixed $k_B{T}_R={ϵ}_0/20$ is shown as an highlighted curve. The solid and dashed lines correspond to the quantities $\ln {\mathcal{R}}_{max}$ and $-\ln {\mathcal{R}}_{min}$. The turning points at ${\tilde{\gamma }}_c$, associated with a switch of the rectification direction, are shown as white circles. The red curve corresponds to the temperature bias shown in (c). (e) The entanglement $\mathcal{E}$ corresponding to the same parameter values as in (d). In all plots, ${ϵ}_0=1$, $\mathrm{\Delta }=0.1$, $\hslash {\omega }_{LC}=10{ϵ}_0$, $Q_L=Q_R=10$, $R_L=R_R=1\mathrm{\Omega }$, $g_L=0.75$, and $g_R=0.25$ are assumed.

Figure 3

(a) The dependence of the rectification ratio $\mathcal{R}$ on the left and right bath temperatures $T_{L,R}$ for $g_L=0.75$ and $g_R=0.25$. The blue cut corresponds to the axis $T_L=T_R$, while the red one corresponds to fixed left-bath temperature $k_B{T}_L={ϵ}_0/5$. (b) The two cuts shown in (a) display the variation of $\mathcal{R}$ as a function of the right-bath temperature $T_R$. The green circle corresponds to the value $k_B{T}_R={ϵ}_0/20$ employed in Fig. 2. The gray dashed lines correspond to lower values of interaction $\tilde{\gamma }=\{0,0.1\}$ at $q=0$. (c) The dependence of $\mathcal{R}$ on the qubit-bath coupling strengths $g_{L,R}$. The blue line corresponds to the symmetric case of identical coupling $g_L=g_R$, while the red line corresponds to the condition of fixed $g_L=0.75$. (d) The two cuts displayed in (c), showing the dependence of $\mathcal{R}$ on $g_R$. The green circle corresponds to the condition $g_R=0.25$ presented in Fig. 2. The gray dashed lines correspond to lower interaction values $\tilde{\gamma }=\{0,0.1\}$ at $q=0$. In all plots, ${ϵ}_0=1$, $\mathrm{\Delta }=0.1$, $\hslash {\omega }_{LC}=10{ϵ}_0$, $Q_L=Q_R=10$, and $R_L=R_R=1\mathrm{\Omega }$ are employed.