Dual effects of Lamb Shift in Quantum Thermodynamical Systems

TL;DR

This work shows that environment-induced energy level shifts, i.e. the Lamb shift, can decisively influence quantum heat transport. By analyzing two coupled two-level systems each connected to its own bath, the authors derive a global master equation including the Lamb shift term $H_{LS}$ and compute the steady-state heat current. They find a dual behavior: the Lamb shift suppresses heat flow at small temperature differences, while at large gradients it drives a linear growth in the current, potentially exceeding the zero-Lamb shift bound and even diverging as the temperature difference grows without bound. The results highlight the importance of accounting for $H_{LS}$ in quantum thermodynamics and suggest new avenues for controlling heat flow in quantum devices.

Abstract

The Lamb shift as an additional energy correction induced by environments usually has a marginal contribution and hence is neglected. We demonstrate that the Lamb shift, which modifies the energy levels, can influence the heat current to varying extents. We focus on the steady-state heat current through two coupled two-level atoms, respectively, in contact with a heat reservoir at a certain temperature. We find that the Lamb shift suppresses the steady-state heat current at small temperature gradients, while at large gradients, the heat current is restricted by an upper bound without the Lamb shift but diverges when it is included. These results not only demonstrate the Lamb shift's critical role in quantum heat transport but also advance our understanding of its impact in quantum thermodynamics.

Figures (5)

• Figure S1: The schematic illustration of our model, where the dashed line represents weak coupling, and the solid line represents strong coupling. The temperatures of two heat reservoirs are $T_{1}$ and $T_{2}$, the energy separation of two qubits are $\varepsilon _{1}$ and $\varepsilon _{2}$.
• Figure S2: ${R_{2,1}}$ and its estimation vs. the relative temperature difference $\Delta T/\omega _{D}$. Here we set $T_{1}=1,T_{2}=1+\Delta T,\gamma _{1}=\gamma _{2}=0.01,\omega _{D}=50,\varepsilon _{1}=3,\varepsilon _{2}=2,k=0.5.$ The green line represents the estimation of ${R_{2,1}}$, the dashed black line represents the exact value of ${R_{2,1}}$. For the inset, the red line represents $\delta _{1}$ and the blue line represents $\delta _{2}$ vs the relative temperature difference $\Delta T/\omega _{D}$ in the same regime.
• Figure S3: The heat currents ${\mathcal{J}_{1}^{\delta }}$ vs the relative temperature difference $\Delta T/\omega _{D}$. Here we set $\varepsilon _{1}=3,\varepsilon _{2}=2,T_{1}=0.1,T_{2}=0.1+\Delta T,\gamma _{1}=\gamma _{2}=0.02,\omega _{D}=100,k=0.5.$ This figure illustrates the suppression of heat flow induced by the Lamb shift. For the inset, the red line represents $\delta _{1}$ and the blue line represents $\delta _{2}$ vs the relative temperature difference $\Delta T/\omega _{D}$ in the same regime.
• Figure S4: The heat current difference $\Delta \mathcal{J}_{1}^{\delta }$ vs the temperature difference $\Delta T$. Here we set $\varepsilon _{1}=3,\varepsilon _{2}=2,T_{1}=1,T_{2}=1+\Delta T,\gamma _{1}=\gamma _{2}=0.02,k=0.5.$ For the insets, $\omega_D=10,20,50,100$, respectively. It can be seen that as $\omega_D$ increases, the temperature difference corresponding to the point where $\Delta \mathcal{J}_{1}^{\delta }$ changes from negative to positive becomes larger.
• Figure S5: The heat currents ${\mathcal{J}_{1}^{\delta }}$ vs the relative temperature difference $\Delta T/\omega _{D}$. Here we set $T_{1}=1,T_{2}=1+\Delta T,\gamma _{1}=\gamma _{2}=0.01,\omega _{D}=50,\varepsilon _{1}=3,\varepsilon _{2}=2.5,k=0.5.$ For the blue line, $J^{(1)}(\omega )=\frac{\gamma \omega }{1+(\omega /\omega _{D})^{2}};$ for the red line, $J^{(2)}(\omega )=\gamma \omega$ when $\omega <\omega _{D}$, $J^{(2)}(\omega )=0$ when $\omega \geq \omega _{D};$ for the black line, $J^{(3)}(\omega )=\gamma \omega exp(-\omega ^{2}/\omega _{D}^{2}).$ The dashed lines represent the heat currents $\mathcal{J}_{1}^{0}$ of the same regime, but the Lamb shift is not considered. For the inset, the solid lines represent $\delta _{1}$, and the dotted lines represent $\delta _{2}$ vs the relative temperature difference $\Delta T/\omega _{D}$ in the same regime; The correspondence of colors remains consistent with the previous text.