Dual Effects of Lamb Shift in Quantum Thermodynamical Systems

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.

1. Introduction

In realistic scenarios, the physical system can hardly be completely isolated from its external environment. It is inevitable for an open system to exchange energy, information, or matter with its surroundings. The evolution of open systems is determined not only by the system’s intrinsic Hamiltonian but also by its coupling with environmental degrees of freedom [1]. There are a variety of nontrivial quantum phenomena, including decoherence and energy dissipation, in open systems. The master equation (ME) is one of the critical methods describing the dynamic evolution of open systems [1,2,3]. The most popular ME could count on the Gorini–Kossakowski–Lindblad–Sudarshan (GKLS) ME [4,5,6] based on the Born–Markov–Secular approximation, which has been extensively validated and applied in numerous studies [7,8,9,10,11,12]. Some other approaches, such as the coarse-graining method [13], the universal Lindblad equation [14], and the geometric–arithmetic master equation [15,16], are also used to derive the master equation without the secular approximation. In addition, some works especially focused on examining the validity of the ME [13,14,16,17,18,19]. Among these contributions, one can usually find the system’s energy level shift, known as the Lamb shift, caused by its interaction with its environment.

The Lamb shift typically manifests as an additional term as the energy level correction caused by the environment, which usually has a marginal contribution to the question of interest [1,2,3]. A numerical demonstration showed that the Lamb shift only weakly perturbs the system Hamiltonian [14]; Refs. [12,20] ignored the contribution of the Lamb shift by comparing the order of magnitude of the Lamb shift in the weak coupling limit; some studies have assumed that heat baths possess a huge bandwidth, resulting in an effective zero Lamb shift [21]. In this sense, the Lamb shift is safely neglected in many applications. For example, the ME, which neglects the Lamb shift, is used to analyze the interactions between light and matter, quantum interference of light, and quantum light sources, such as single-photon sources and lasers [22,23,24]. The decoherence in various quantum information processes [25,26,27,28] has been analyzed without considering the Lamb shift. The ME without the Lamb shift is used to study the energy exchange processes between the working substance and its heat reservoirs [29,30,31,32,33,34,35]. The Lamb shift commutes with the system’s Hamiltonian; hence, the steady state is independent of the Lamb shift, and so is the steady-state heat current [36]. Recently, quantum batteries have attracted increasing interest [37,38,39,40,41], where ME facilitates understanding the charging and discharging processes of quantum states, especially considering the environmental factors such as temperature and noise on battery performance, but one can find that the Lamb shift is not considered either.

However, recent studies have begun to reevaluate the significance of the Lamb shift, acknowledging its crucial role in accurately characterizing system–environment interactions. Numerical evidence has shown the substantial impact of the Lamb shift [10]. For instance, under specific parameter choices, the refined Lamb shift Hamiltonian yields more accurate results than models neglecting this contribution [42]. The discrepancies observed between Born–Markov methods and the stochastic Liouville equation with dissipation (SLED) at low temperatures have been attributed to the omission of the Lamb shift [43]. Moreover, a significant collective Lamb shift was experimentally demonstrated using two distant superconducting qubits [44]. In systems such as a giant artificial atom with multiple coupling points, the Lamb shift becomes a pronounced, frequency-dependent, and engineerable quantity that actively influences relaxation rates and enables tunable anharmonicity [45]. These energy shifts play an essential role in quantum thermodynamics. For example, in electric circuit systems [46,47], quantum refrigeration [48,49], quantum dots [50], and giant atoms [45,51], accurate accounting of the Lamb shift is critical for predicting heat currents and other thermodynamic quantities reliably [52].

In this paper, we find the significant influence of the Lamb shifts on the heat transport through two coupled atoms interacting with a heat bath, respectively. Here, we do not distinguish between the Lamb and Stark shifts in this paper for simplicity but rather collectively refer to the total of the environment-induced frequency shift of the system as the Lamb shift. We find that the heat current approaches an upper bound with the temperature difference increasing when the Lamb shift is not considered as usual [53,54,55,56]. In contrast, the heat current will monotonically increase with the temperature difference increasing if we consider the Lamb shift. This difference in heat currents persists when other forms of spectral densities are taken.

This paper is organized as follows. In Section 2, we give a brief description of our model and derive the master equations under the Born–Markov–Secular approximation. In Section 3, we calculate the Lamb shift. In Section 4, we compare the heat currents with and without the Lamb shift. We conclude with a summary in Section 5.

2. The Model

Let us consider two coupled two-level atoms (TLAs) interacting with a distinct thermal reservoir, respectively, as shown in Figure 1. The model has been widely studied in various cases [57,58,59,60]. The total Hamiltonian of the system and the reservoirs is

$$H=H_S+H_{B1}+H_{B2}+H_{SB1}+H_{SB2},$$

(1)

where

$$H_S={\displaystyle \frac{{\epsilon }_1}{2}}{\sigma }_1^z+{\displaystyle \frac{{\epsilon }_2}{2}}{\sigma }_2^z+g{\sigma }_1^x{\sigma }_2^x,$$

(2)

with ${\epsilon }_1\ge {\epsilon }_2$, ${\sigma }_i^z$ and ${\sigma }_i^x$ denoting the Pauli matrices, and g denoting the coupling strength of two qubits. For simplicity, we adopt natural units by setting the reduced Planck constant $\hslash =1$ and the Boltzmann constant $k_B=1$. $H_{B1}$ and $H_{B2}$ are the Hamiltonians of the two thermal reservoirs, which are characterized by a collection of independent harmonic oscillators, where

$$H_{Bj}=\sum _n{\omega }_{n,j}{b}_{j,n}^{†}{b}_{j,n}$$

(3)

with the summation running over all discrete modes of the jth thermal reservoir. $H_{SB1}$ and $H_{SB2}$ are the interaction Hamiltonians between the system of interest and the reservoirs, where

$$\begin{array}{c}\hfill H_{S{B}_j}={\sigma }_j^x\sum _n{g}_{j,n}\left(b_{j,n}+b_{j,n}^{†}\right)={\sigma }_j^x{B}_j^x\end{array}$$

(4)

with $B_j^x={\displaystyle \sum _n}{g}_{j,n}\left(b_{j,n}+b_{j,n}^{†}\right)$.

Figure 1. 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 ${\epsilon }_1$ and ${\epsilon }_2$.

To obtain the dynamics of the system, we would like to derive the master equation. Following the standard process [1,61], we would like to first give the eigensystems of $H_S$ as $H_S\left|s_i\right.⟩=s_i\left|s_i\right.⟩$, where the eigenvectors read

$$\begin{array}{c}\left|s_1\right.⟩=\cos {\displaystyle \frac{\phi }{2}}\left|0,0\right.⟩-\sin {\displaystyle \frac{\phi }{2}}\left|1,1\right.⟩,\left|s_2\right.⟩=\sin {\displaystyle \frac{\phi }{2}}\left|0,0\right.⟩+\cos {\displaystyle \frac{\phi }{2}}\left|1,1\right.⟩,\hfill \end{array}$$

(5)

$$\begin{array}{c}\left|s_3\right.⟩=\cos {\displaystyle \frac{\theta }{2}}\left|1,0\right.⟩+\sin {\displaystyle \frac{\theta }{2}}\left|0,1\right.⟩,\left|s_4\right.⟩=-\sin {\displaystyle \frac{\theta }{2}}\left|1,0\right.⟩+\cos {\displaystyle \frac{\theta }{2}}\left|0,1\right.⟩,\hfill \end{array}$$

(6)

and the eigenvalues are $s_1=-\beta ,s_2=\beta ,s_3=\alpha ,s_4=-\alpha$ with

$$\alpha =\sqrt{{\displaystyle \frac{{\left({\epsilon }_1-{\epsilon }_2\right)}^2}{4}}+g^2},\beta =\sqrt{{\displaystyle \frac{{\left({\epsilon }_1+{\epsilon }_2\right)}^2}{4}}+g^2},\tan \phi ={\displaystyle \frac{2g}{{\epsilon }_1+{\epsilon }_2}},\tan \theta ={\displaystyle \frac{2g}{{\epsilon }_1-{\epsilon }_2}}.$$

(7)

Thus, we can derive the eigenoperators $V_{j\mu }$ with $[H_S,V_{j\mu }]=-{\omega }_{j\mu }{V}_{j\mu }$ [1,61] as

$$\begin{array}{c}{V}_{1,1}=\sin {\varphi }_{+}\left(\left|s_3\right.⟩\left.⟨s_2\right|-\left|s_1\right.⟩\left.⟨s_4\right|\right),V_{1,2}=\cos {\varphi }_{+}\left(\left|s_1\right.⟩\left.⟨s_3\right|+\left|s_4\right.⟩\left.⟨s_2\right|\right),\hfill \\ V_{2,1}=\cos {\varphi }_{-}\left(\left|s_3\right.⟩\left.⟨s_2\right|+\left|s_1\right.⟩\left.⟨s_4\right|\right),V_{2,2}=\sin {\varphi }_{-}\left(\left|s_1\right.⟩\left.⟨s_3\right|-\left|s_4\right.⟩\left.⟨s_2\right|\right),\hfill \end{array}$$

with

$${\varphi }_{+}={\displaystyle \frac{\theta +\phi }{2}},{\varphi }_{-}={\displaystyle \frac{\theta -\phi }{2}},$$

(8)

and the corresponding eigenfrequencies ${\omega }_{j\mu }$ as

$${\omega }_{11}={\omega }_{21}=\beta -\alpha ,{\omega }_{12}={\omega }_{22}=\beta +\alpha .$$

Later we will use ${\omega }_{\mu }$ instead of ${\omega }_{j\mu }$ since ${\omega }_{j\mu }$ is independent of j in our model. Based on the eigenoperators, one can rewrite ${\sigma }_j^x={\displaystyle \sum _{\mu }}\left(V_{j\mu }+V_{j\mu }^{†}\right)$. Accordingly, the interaction Hamiltonian can also be rewritten as

$$\begin{array}{c}\hfill H_{S{B}_j}\equiv B_j^x\sum _{\mu }\left(V_{j\mu }+V_{j\mu }^{†}\right).\end{array}$$

(9)

With the previous preliminary knowledge, one can directly get the master equation, subject to the Born–Markov–Secular approximation, as

$${\displaystyle \frac{d\rho }{dt}}=-i[H_S+H_{LS},\rho ]+{\mathcal{L}}_1\left(\rho \right)+{\mathcal{L}}_2\left(\rho \right),$$

(10)

where $\rho$ is the density matrix of the TLAs, $H_{LS}$ is the energy correction, i.e., the Lamb shift, and ${\mathcal{L}}_j\left(\rho \right)$ are the dissipators given by

$${\mathcal{L}}_j\left(\rho \right)=\sum _{\mu =1}^2\left[{\Gamma }_j\left({\omega }_{\mu }\right)\left(V_{j\mu }\rho V_{j\mu }^{†}-{\displaystyle \frac{1}{2}}\left\{\rho ,V_{j\mu }^{†}{V}_{j\mu }\right\}\right)+{\Gamma }_j\left(-{\omega }_{\mu }\right)\left(V_{j\mu }^{†}\rho V_{j\mu }-{\displaystyle \frac{1}{2}}\left\{\rho ,V_{j\mu }{V}_{j\mu }^{†}\right\}\right)\right].$$

(11)

Here ${\Gamma }_j\left(\omega \right)={\int }_{-\infty }^{\infty }ds{e}^{i\omega s}⟨B_j^x\left(s\right)B_j^x⟩$ is the Fourier transform of the reservoir correlation function $⟨B_j^x\left(s\right)B_j^x⟩$, so one can have

$$\begin{array}{c}{\Gamma }_j\left({\omega }_{\mu }\right)=2J_j\left({\omega }_{\mu }\right)[{\overline{n}}_j\left({\omega }_{\mu }\right)+1],\end{array}$$

(12)

$$\begin{array}{c}{\Gamma }_j(-{\omega }_{\mu })=2J_j\left({\omega }_{\mu }\right){\overline{n}}_j\left({\omega }_{\mu }\right),\end{array}$$

(13)

where ${\overline{n}}_j\left(\omega \right)={(\exp \left({\beta }_j\omega \right)-1)}^{-1}$ are the average photon number and

$$J_j\left(\omega \right)=\pi \sum _n{\left|g_{j,n}\right|}^2\delta \left(\omega -{\omega }_n\right)$$

(14)

are the spectral densities of the heat reservoirs. In practical calculations, we typically replace the discrete sum over infinitely many delta-function-like modes with a continuous spectral density function. For instance, in this work, we consider an Ohmic-type thermal reservoir with a high-frequency cutoff ${\omega }_D$, whose spectral density takes the conventional form [1,62]:

$$J_j\left(\omega \right)={\displaystyle \frac{{\gamma }_j\omega }{1+{\left(\omega /{\omega }_D\right)}^2}}.$$

(15)

This Drude cutoff provides a physically meaningful extension of the standard Ohmic model by incorporating a frequency-dependent damping term, thereby offering a more accurate representation of the underlying physical processes. This regularization scheme modifies the spectral density through a Lorentzian damping factor, which naturally introduces a smooth high-frequency cutoff. The resulting spectral density remains finite across all frequencies, resolving the unphysical divergence that occurs in the simple Ohmic case. This regularization is essential for constructing realistic models of quantum dissipation, as it properly accounts for the finite response times of physical environments while maintaining the characteristic linear frequency dependence at low energies. The cutoff frequency parameter simultaneously determines both the high-frequency roll-off and the characteristic timescale of environmental correlations.

The Lamb shift $H_{LS}$ in Equation (10) reads

$$H_{LS}=\sum _{j\mu }\left[S_j\left({\omega }_{\mu }\right)V_{j\mu }^{†}{V}_{j\mu }+S_j\left(-{\omega }_{\mu }\right)V_{j\mu }{V}_{j\mu }^{†}\right],$$

(16)

where

$$\begin{array}{c}{S}_j\left({\omega }_{\mu }\right)={\displaystyle \frac{1}{\pi }}P.V.\underset{0}{\overset{\infty }{\int }}{J}_j\left(\omega \right)\left({\displaystyle \frac{{\overline{n}}_j\left(\omega \right)+1}{{\omega }_{\mu }-\omega }}+{\displaystyle \frac{{\overline{n}}_j\left(\omega \right)}{{\omega }_{\mu }+\omega }}\right)d\omega ,\end{array}$$

(17)

$$\begin{array}{c}{S}_j\left(-{\omega }_{\mu }\right)=-{\displaystyle \frac{1}{\pi }}P.V.\underset{0}{\overset{\infty }{\int }}{J}_j\left(\omega \right)\left({\displaystyle \frac{{\overline{n}}_j\left(\omega \right)}{{\omega }_{\mu }-\omega }}+{\displaystyle \frac{{\overline{n}}_j\left(\omega \right)+1}{{\omega }_{\mu }+\omega }}\right)d\omega ,\end{array}$$

(18)

and $P.V.$ denotes the Cauchy principal value of the integral. The more detailed derivation is provided in Appendix A. Within our parameter range, we establish the following hierarchy of parameters:

$${\gamma }_j^{-1}\sim {\omega }_D/g\gg {\omega }_{\mu }/g\sim 1\gg {\gamma }_j.$$

(19)

Noting that the eigenfrequencies satisfy ${\omega }_2-{\omega }_1=2\alpha \ge g\gg {\gamma }_j$. This establishes a crucial relationship between two characteristic timescales. The system’s dynamical timescale ${\tau }_S={|{\omega }_1-{\omega }_2|}^{-1}$ is much shorter than the reservoir correlation time ${\tau }_R={\gamma }_j^{-1}$, which ensures the validity of the secular approximation. Therefore, the global master equation approach is rigorously justified in this parameter regime, providing a consistent description of the open quantum system dynamics while properly accounting for the system’s coherent evolution and dissipative processes.

3. Lamb Shift

Now, let us focus on the Lamb shift. Define

$$\begin{array}{c}{\Delta }_{j\mu }={\displaystyle \frac{2{\omega }_{\mu }}{\pi }}P.V.{\int }_0^{\infty }{\displaystyle \frac{J_j\left(\omega \right){\overline{n}}_j\left(\omega \right)}{{\omega }_{\mu }^2-{\omega }^2}}d\omega ,\end{array}$$

(20)

$$\begin{array}{c}{\Delta }_{j\mu }^{+}={\displaystyle \frac{1}{\pi }}{\int }_0^{\infty }{\displaystyle \frac{J_j\left(\omega \right)}{{\omega }_{\mu }+\omega }}d\omega ,\hfill \end{array}$$

(21)

$$\begin{array}{c}{\Delta }_{j\mu }^{-}={\displaystyle \frac{1}{\pi }}P.V.{\int }_0^{\infty }{\displaystyle \frac{J_j\left(\omega \right)}{{\omega }_{\mu }-\omega }}d\omega ,\hfill \end{array}$$

(22)

then the Lamb shift can be rewritten as

$$H_{LS}=\sum _{n=1}^4{\Delta }_n\left|s_n\right.⟩\left.⟨s_n\right|,$$

(23)

where

$$\begin{array}{cc}\hfill {\Delta }_1& =-\left({\Delta }_{1,1}+{\Delta }_{1,1}^{+}\right){\sin }^2{\varphi }_{+}-\left({\Delta }_{2,2}+{\Delta }_{2,2}^{+}\right){\sin }^2{\varphi }_{-}\hfill \\ \hfill & -\left({\Delta }_{1,2}+{\Delta }_{1,2}^{+}\right){\cos }^2{\varphi }_{+}-\left({\Delta }_{2,1}+{\Delta }_{2,1}^{+}\right){\cos }^2{\varphi }_{-},\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill {\Delta }_2& =\left({\Delta }_{1,1}+{\Delta }_{1,1}^{-}\right){\sin }^2{\varphi }_{+}+\left({\Delta }_{2,2}+{\Delta }_{2,2}^{-}\right){\sin }^2{\varphi }_{-}\hfill \\ \hfill & +\left({\Delta }_{1,2}+{\Delta }_{1,2}^{-}\right){\cos }^2{\varphi }_{+}+\left({\Delta }_{2,1}+{\Delta }_{2,1}^{-}\right){\cos }^2{\varphi }_{-},\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill {\Delta }_3& =-\left({\Delta }_{1,1}+{\Delta }_{1,1}^{+}\right){\sin }^2{\varphi }_{+}+\left({\Delta }_{2,2}+{\Delta }_{2,2}^{-}\right){\sin }^2{\varphi }_{-}\hfill \\ \hfill & +\left({\Delta }_{1,2}+{\Delta }_{1,2}^{-}\right){\cos }^2{\varphi }_{+}-\left({\Delta }_{2,1}+{\Delta }_{2,1}^{+}\right){\cos }^2{\varphi }_{-},\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill {\Delta }_4& =\left({\Delta }_{1,1}+{\Delta }_{1,1}^{-}\right){\sin }^2{\varphi }_{+}-\left({\Delta }_{2,2}+{\Delta }_{2,2}^{+}\right){\sin }^2{\varphi }_{-}\hfill \\ \hfill & -\left({\Delta }_{1,2}+{\Delta }_{1,2}^{+}\right){\cos }^2{\varphi }_{+}+\left({\Delta }_{2,1}+{\Delta }_{2,1}^{-}\right){\cos }^2{\varphi }_{-}.\hfill \end{array}$$

(27)

It is easy to deduce that the Lamb shift commutes with the system’s Hamiltonian $H_S$. It acts as a modification to the energy levels of the system. This implies that the Lamb shift does not alter the fundamental structure of the Hamiltonian but rather fine-tunes the energy levels within the system, showcasing its role as a subtle yet pivotal adjustment factor in open quantum system dynamics. From the equations above, we can obtain the increments of the transition frequencies from $\left|s_2\right.⟩\to \left|s_3\right.⟩$ and $\left|s_2\right.⟩\to \left|s_4\right.⟩$ as

$$\begin{array}{c}\hfill {\delta }_1={\Delta }_3-{\Delta }_2=\left(2{\Delta }_{1,1}+{\Delta }_{1,1}^{\prime }\right){\sin }^2{\varphi }_{+}+\left(2{\Delta }_{2,1}+{\Delta }_{2,1}^{\prime }\right){\cos }^2{\varphi }_{-},\end{array}$$

(28)

$$\begin{array}{c}\hfill {\delta }_2={\Delta }_4-{\Delta }_2=\left(2{\Delta }_{2,2}+{\Delta }_{2,2}^{\prime }\right){\sin }^2{\varphi }_{-}+\left(2{\Delta }_{1,2}+{\Delta }_{1,2}^{\prime }\right){\cos }^2{\varphi }_{+},\end{array}$$

(29)

where

$${\Delta }_{j\mu }^{\prime }={\Delta }_{j\mu }^{+}+{\Delta }_{j\mu }^{-}={\displaystyle \frac{2{\omega }_{\mu }}{\pi }}P.V.{\int }_0^{\infty }{\displaystyle \frac{J_j\left(\omega \right)}{{\omega }_{\mu }^2-{\omega }^2}}d\omega .$$

(30)

To evaluate the effect of the Lamb shift, the critical task is to determine the values of ${\Delta }_{j\mu }$ and ${\Delta }_{j\mu }^{\prime }$. Fortunately, these values can be determined using the definition of the Cauchy principal value integral

$$P.V.\underset{0}{\overset{\infty }{\int }}{\displaystyle \frac{d\omega }{{\omega }_{\mu }-\omega }}=\underset{\eta \to 0}{\lim }\left({\int }_0^{{\omega }_{\mu }-\eta }{\displaystyle \frac{d\omega }{{\omega }_{\mu }-\omega }}+{\int }_{{\omega }_{\mu }+\eta }^{\infty }{\displaystyle \frac{d\omega }{{\omega }_{\mu }-\omega }}\right).$$

(31)

Applying the residue theorem, we have

$$\begin{array}{c}{\Delta }_{j\mu }={\displaystyle \frac{J_j\left({\omega }_{\mu }\right)}{\pi }}\left(\ln {\displaystyle \frac{{\omega }_D}{{\omega }_{\mu }}}+{\displaystyle \frac{\pi }{{\beta }_j{\omega }_D}}+R_{j\mu }\right),\end{array}$$

(32)

$$\begin{array}{c}{\Delta }_{j\mu }^{\prime }=-{\displaystyle \frac{2J_j\left({\omega }_{\mu }\right)}{\pi }}\ln {\displaystyle \frac{{\omega }_D}{{\omega }_{\mu }}},\end{array}$$

(33)

where

$$R_{j\mu }={\displaystyle \frac{2\pi }{{\beta }_j}}\sum _{k=1}^{\infty }{\displaystyle \frac{{\omega }_{\mu }^2-{\omega }_D{\omega }_k}{\left({\omega }_{\mu }^2+{\omega }_k^2\right)\left({\omega }_D+{\omega }_k\right)}}$$

(34)

is a series with respect to ${\omega }_k$, and ${\omega }_k=2k\pi /{\beta }_j$ is the Matsubara frequency. Using the Euler–Maclaurin formula, we can obtain an estimation as

$$R_{j\mu }=\ln {\displaystyle \frac{\sqrt{4{\pi }^2+{\omega }_{\mu }^2{\beta }_j^2}}{2\pi +{\omega }_D{\beta }_j}}+O\left({\beta }_j\right).$$

(35)

The error of this estimation is clearly illustrated in Figure 2, and we can observe a constant increasing ${\delta }_{\mu }$ as the temperature increases from the inset. A more detailed discussion on this is provided in Appendix B. Note that the negative initial value of ${\delta }_{\mu }$ suggests the Lamb shift’s effect on heat current from reducing to enhancing contributions with increasing temperature difference.

Figure 2. $R_{2,1}$ and its estimation vs. the relative temperature difference $\Delta T/{\omega }_D$. Here we set $T_1=1, T_2=1+\Delta T, {\gamma }_1={\gamma }_2=0.01, {\omega }_D=50, {\epsilon }_1=3, {\epsilon }_2=2, k=0.5$. The green line represents the estimation of $R_{2,1}$, and 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.

4. Heat Currents

We have obtained all the analytic expressions of the Lamb shift and dissipators of the master Equation (10); hence, we can get the system’s dynamics. Here, we are interested in the steady-state behavior of the system. We have solved the density matrix of the system and found that only the diagonal entries do not vanish so that we can express the steady-state reduced density matrix as ${\rho }^S=diag[{\rho }_{11}^S,{\rho }_{22}^S,{\rho }_{33}^S,{\rho }_{44}^S],$ where [54]

$${\rho }_{11}^S={\displaystyle \frac{X^{-}{Y}^{-}}{XY}}, {\rho }_{22}^S={\displaystyle \frac{X^{+}{Y}^{+}}{XY}}, {\rho }_{33}^S={\displaystyle \frac{X^{-}{Y}^{+}}{XY}}, {\rho }_{44}^S={\displaystyle \frac{X^{+}{Y}^{-}}{XY}},$$

with

$$\begin{array}{cc}\hfill & X^{+}=J_1\left({\omega }_1\right){\overline{n}}_1\left({\omega }_1\right){\sin }^2{\varphi }_{+}+J_2\left({\omega }_1\right){\overline{n}}_2\left({\omega }_1\right){\cos }^2{\varphi }_{-},\hfill \\ \hfill & Y^{+}=J_1\left({\omega }_2\right){\overline{n}}_1\left({\omega }_2\right){\cos }^2{\varphi }_{+}+J_2\left({\omega }_2\right){\overline{n}}_2\left({\omega }_2\right){\sin }^2{\varphi }_{-},\hfill \\ \hfill & X^{-}=J_1\left({\omega }_1\right)\left({\overline{n}}_1\left({\omega }_1\right)+1\right){\sin }^2{\varphi }_{+}+J_2\left({\omega }_1\right)\left({\overline{n}}_2\left({\omega }_1\right)+1\right){\cos }^2{\varphi }_{-},\hfill \\ & Y^{-}=J_1\left({\omega }_2\right)\left({\overline{n}}_1\left({\omega }_2\right)+1\right){\cos }^2{\varphi }_{+}+J_2\left({\omega }_2\right)\left({\overline{n}}_2\left({\omega }_2\right)+1\right){\sin }^2{\varphi }_{-},\hfill \\ \hfill & X=X^{+}+X^{-},Y=Y^{+}+Y^{-}.\hfill \end{array}$$

One can find that the above steady-state density matrix ${\rho }^S$ is the same as the one without considering the Lamb shift. This is consistent with the usual understanding that the Lamb shift does not affect the system’s eigenstates but only the eigenvalues.

To get the effect of the Lamb shift, we begin to study the heat current, which is defined as [1,63]

$${\mathcal{J}}_j=Tr\left((H_S+H_{LS}){\mathcal{L}}_j\left(\rho \right)\right).$$

(36)

One can easily check that $[H_S+H_{LS},{\rho }^S]=0$; hence, from the master Equation (10), we have

$${\mathcal{L}}_1\left({\rho }^S\right)+{\mathcal{L}}_2\left({\rho }^S\right)={\displaystyle \frac{d{\rho }^S}{dt}}=0,$$

(37)

which further implies that the two heat currents satisfy the conservation relation ${\mathcal{J}}_1=-{\mathcal{J}}_2$. From Equation (36), we can give the explicit form of the heat current as

$${\mathcal{J}}_1^{\delta }=\sum _{\mu =1}^2{A}_{\mu }\left({\overline{n}}_1\left({\omega }_{\mu }\right)-{\overline{n}}_2\left({\omega }_{\mu }\right)\right)\left({\omega }_{\mu }+{\delta }_{\mu }\right)$$

(38)

with

$$\begin{array}{c}{A}_1=2{\sin }^2{\varphi }_{+}{\cos }^2{\varphi }_{-}{\displaystyle \frac{J_1\left({\omega }_1\right)J_2\left({\omega }_1\right)}{X}},\hfill \end{array}$$

(39)

$$\begin{array}{c}{A}_2=2{\sin }^2{\varphi }_{-}{\cos }^2{\varphi }_{+}{\displaystyle \frac{J_1\left({\omega }_2\right)J_2\left({\omega }_2\right)}{Y}},\hfill \end{array}$$

(40)

and ${\delta }_i$ is from Equations (28) and (29). We want to emphasize that $\delta$ in the heat current ${\mathcal{J}}_1^{\delta }$ is the signature of the Lamb shift; $\delta ={\delta }_i=0$ corresponding to ${\mathcal{J}}_1^0$ means that the Lamb shift is not considered. Thus, one can easily obtain the difference in heat currents with and without considering the Lamb shift as

$$\Delta {\mathcal{J}}_1^{\delta }=\sum _{\mu =1}^2{A}_i\left(n_1\left({\omega }_{\mu }\right)-n_2\left({\omega }_{\mu }\right)\right){\delta }_{\mu }.$$

(41)

Notice that ${\mathcal{J}}_1^{\delta }$ is always negative when $T_2>T_1$ from Equation (A34), which is consistent with the second law of thermodynamics. There is no issue of the direction of heat current in this model because we are using the global approach for the master equation [64,65,66]. Next, we are only concerned with the magnitude of the heat currents, so the remainder of the discussion focuses on the absolute value of the heat currents.

From Equation (41), it can be seen that when ${\delta }_{\mu }$ is negative, the Lamb shift has a suppressing effect on the heat current, as illustrated by Figure 3. Note that when $\Delta T>{\omega }_D/2$, ${\delta }_1$ is already positive, but since the reduction effect of the terms with ${\delta }_2$ is greater, $\Delta {\mathcal{J}}_1^{\delta }$ is still negative at this point. In fact, when the signs of ${\delta }_{\mu }$ are not the same, the size of $\Delta {\mathcal{J}}_1^{\delta }$ will depend on the competition between terms with ${\delta }_1$ and ${\delta }_2$.

Figure 3. The heat currents ${\mathcal{J}}_1^{\delta }$ vs. the relative temperature difference $\Delta T/{\omega }_D$. Here we set ${\epsilon }_1=3, {\epsilon }_2=2, T_1=0.1, T_2=0.1+\Delta T, {\gamma }_1={\gamma }_2=0.02, {\omega }_D=100, k=0.5$. 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.

From Equations (28), (29), (32) and (33), our analysis reveals that the value of omega exerts a substantial influence on delta. This can be attributed to the direct influence of the cutoff frequency on the Lamb shift, suggesting that the impact of the Lamb shift on the heat current is amplified with increasing cutoff frequency. Indeed, if the cutoff frequency tends to infinity, the Lamb shift will diverge.

Our analysis reveals that $\Delta {\mathcal{J}}_1^{\delta }$ exhibits a consistently developing linear dependence on large temperature differences, independent of the specific value of ${\omega }_D$, as illustrated by Figure 4. Namely, when $T_1$ as the lower temperature is fixed, the heat current ${\mathcal{J}}_1^0$ will approach an upper bound, but ${\mathcal{J}}_1^{\delta }\to \infty$ with the temperature increment $\Delta T\to \infty$. To show this, let us take the derivative of ${\mathcal{J}}_1^0$ with respect to $\Delta T$; then we have

$$\begin{array}{c}\hfill {\displaystyle \frac{d{\mathcal{J}}_1^0}{d\Delta T}}=2K_1{\displaystyle \frac{J_1\left({\omega }_1\right){\sin }^2{\varphi }_{+}+J_2\left({\omega }_1\right){\cos }^2{\varphi }_{-}}{X^2}}{\omega }_1+2K_2{\displaystyle \frac{J_2\left({\omega }_2\right){\sin }^2{\varphi }_{-}+J_1\left({\omega }_2\right){\cos }^2{\varphi }_{+}}{Y^2}}{\omega }_2,\end{array}$$

(42)

where

$$\begin{array}{cc}\hfill K_1& ={\displaystyle \frac{d{\overline{n}}_2\left({\omega }_1\right)}{d\Delta T}}{\sin }^2{\varphi }_{+}{\cos }^2{\varphi }_{-}{J}_1\left({\omega }_1\right)J_2\left({\omega }_1\right)\left(2{\overline{n}}_1\left({\omega }_1\right)+1\right),\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill K_2& ={\displaystyle \frac{d{\overline{n}}_2\left({\omega }_2\right)}{d\Delta T}}{\sin }^2{\varphi }_{-}{\cos }^2{\varphi }_{+}{J}_1\left({\omega }_2\right)J_2\left({\omega }_2\right)\left(2{\overline{n}}_1\left({\omega }_2\right)+1\right).\hfill \end{array}$$

(44)

Both $K_1$ and $K_2$ are two constant positive quantities. Thus, one can easily obtain that ${\mathcal{J}}_1^0$ is a monotonically increasing function of $\Delta T$. However, simple calculations can show that

$$\begin{array}{cc}\hfill \underset{\Delta T\to \infty }{\lim }{A}_1|{\overline{n}}_1\left({\omega }_1\right)-{\overline{n}}_2\left({\omega }_1\right)|& =J_1\left({\omega }_1\right){\sin }^2{\varphi }_{+},\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill \underset{\Delta T\to \infty }{\lim }{A}_2|{\overline{n}}_1\left({\omega }_2\right)-{\overline{n}}_2\left({\omega }_2\right)|& =J_1\left({\omega }_2\right){\cos }^2{\varphi }_{+},\hfill \end{array}$$

(46)

and from Equation (38) we have

$$\underset{\Delta T\to \infty }{\lim }{\mathcal{J}}_1^0=J_1\left({\omega }_1\right){\omega }_1{\sin }^2{\varphi }_{+}+J_1\left({\omega }_2\right){\omega }_2{\cos }^2{\varphi }_{+},$$

(47)

which remains constant since the system’s parameters are fixed. Thus, Equation (47) serves as the supremum of ${\mathcal{J}}_1^0$. Namely, without considering the Lamb shift, the heat current has an upper bound with the temperature difference tending to infinity.

Figure 4. The heat current difference $\Delta {\mathcal{J}}_1^{\delta }$ vs. the temperature difference $\Delta T$. Here we set ${\epsilon }_1=3$, ${\epsilon }_2=2, T_1=1, T_2=1+\Delta T, {\gamma }_1={\gamma }_2=0.02, k=0.5$. 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.

Now, let us turn to the case of the Lamb shift. Based on Equations (28), (29), (32) and (33), one finds that ${\delta }_{\mu }$ can be rewritten as

$${\delta }_{\mu }=P_{\mu }+Q_{\mu }\Delta T+{\displaystyle \frac{1}{\pi }}{Q}_{\mu }{\omega }_D{R}_{2,\mu },$$

(48)

where

$$\begin{array}{cc}\hfill & P_1={\displaystyle \frac{2J_1\left({\omega }_1\right)}{\pi }}\left({\displaystyle \frac{\pi }{{\beta }_1{\omega }_D}}+R_{1,1}\right){\sin }^2{\varphi }_{+}+{\displaystyle \frac{2J_2\left({\omega }_1\right)}{{\beta }_1{\omega }_D}}{\cos }^2{\varphi }_{-},\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill & P_2={\displaystyle \frac{2J_1\left({\omega }_2\right)}{\pi }}\left({\displaystyle \frac{\pi }{{\beta }_1{\omega }_D}}+R_{1,2}\right){\cos }^2{\varphi }_{+}+{\displaystyle \frac{2J_2\left({\omega }_2\right)}{{\beta }_1{\omega }_D}}{\sin }^2{\varphi }_{-},\hfill \end{array}$$

(50)

and

$$Q_1={\displaystyle \frac{2J_2\left({\omega }_1\right)}{{\omega }_D}}{\cos }^2{\varphi }_{-},Q_2={\displaystyle \frac{2J_2\left({\omega }_2\right)}{{\omega }_D}}{\sin }^2{\varphi }_{-}.$$

(51)

Substituting Equations (45), (46) and (48) into Equation (41), one can obtain

$$\begin{array}{c}\hfill \underset{\Delta T\to \infty }{\lim }{\displaystyle \frac{\Delta {\mathcal{J}}_1^{\delta }}{\Delta T}}=J_1\left({\omega }_1\right)Q_1\left(1+{\displaystyle \frac{{\omega }_D}{\pi }}\underset{\Delta T\to \infty }{\lim }{\displaystyle \frac{R_{2,1}}{\Delta T}}\right){\sin }^2{\varphi }_{+}\\ \hfill +J_1\left({\omega }_2\right)Q_2\left(1+{\displaystyle \frac{{\omega }_D}{\pi }}\underset{\Delta T\to \infty }{\lim }{\displaystyle \frac{R_{2,2}}{\Delta T}}\right){\cos }^2{\varphi }_{+}.\end{array}$$

(52)

Considering Equations (34) and (35), we have

$$\underset{\Delta T\to \infty }{\lim }{R}_{2,\mu }/\Delta T=0.$$

(53)

Substituting Equation (53) into Equation (52), we will arrive at

$$\underset{\Delta T\to \infty }{\lim }{\displaystyle \frac{\Delta {\mathcal{J}}_1^{\delta }}{\Delta T}}=J_1\left({\omega }_1\right)Q_1{\sin }^2{\varphi }_{+}+J_1\left({\omega }_2\right)Q_2{\cos }^2{\varphi }_{+}.$$

(54)

This indicates that the $\Delta {\mathcal{J}}_1^{\delta }$ increases linearly with the temperature difference in the regime $\Delta T\to \infty$. That is, the heat current with the Lamb shift in the regime $\Delta T\to \infty$, the sum of ${\mathcal{J}}_1^0$ and $\Delta {\mathcal{J}}_1^{\delta }$, can exceed the upper bound Equation (47) due to the linearly increasing $\Delta {\mathcal{J}}_1^{\delta }$. Such a result is also explicitly illustrated in Figure 5. It can be seen that at slight temperature differences, the dashed line is below the dotted line, but the difference is not significant. With the temperature increasing, all the dashed lines are tightly below the corresponding orange solid line. In contrast, the dotted lines increase linearly and exceed the supremum corresponding to the orange solid line. In addition, it can be observed that when ${\epsilon }_1+{\epsilon }_2$ is a constant, the smaller the $\left|{\epsilon }_1-{\epsilon }_2\right|$ is, the smaller the temperature difference is at which the heat current with the Lamb shift surpasses the supremum of the heat current corresponding to the orange solid line.

Figure 5. The heat currents ${\mathcal{J}}_1^{\delta }$ vs. the relative temperature difference $\Delta T/{\omega }_D$. Here we set $T_1=1, T_2=1+\Delta T, {\gamma }_1={\gamma }_2=0.01, {\omega }_D=50, k=0.5$. For the blue dotted line, ${\epsilon }_1=3, {\epsilon }_2=2$; for the red dotted line, ${\epsilon }_1=2.75, {\epsilon }_2=2.25$; for the green dotted line, ${\epsilon }_1=2.5, {\epsilon }_2=2.5$. The dashed black lines represent the heat currents ${\mathcal{J}}_1^0$ of the same regime, but the Lamb shift is not taken into account, and the orange solid lines represent the supremum of ${\mathcal{J}}_1^0$.

We have chosen $J^{\left(1\right)}\left(\omega \right)={\displaystyle \frac{\gamma \omega }{1+{(\omega /{\omega }_D)}^2}}$ as the spectral density of the reservoirs in the previous study. We can also choose a simple, discontinuous cutoff as

$$\begin{array}{cc}& J^{\left(2\right)}\left(\omega \right)=\gamma \omega ,\omega <{\omega }_D,\hfill \\ & J^{\left(2\right)}\left(\omega \right)=0,\omega \ge {\omega }_D;\hfill \end{array}$$

(55)

or a cutoff for exponential decay as

$$J^{\left(3\right)}\left(\omega \right)=\gamma \omega exp(-{\omega }^2/{\omega }_D^2).$$

(56)

Notice that all three represent the Ohmic-type heat reservoir. Furthermore, since ${\omega }_{\mu }\ll {\omega }_D,$ $J^{\left(1\right)}\left({\omega }_{\mu }\right)-J^{\left(2\right)}\left({\omega }_{\mu }\right)$ and $J^{\left(3\right)}\left({\omega }_{\mu }\right)-J^{\left(2\right)}\left({\omega }_{\mu }\right)$ are equivalent infinitesimals of the same order $O({\omega }_{\mu }^2/{\omega }_D^2)$. Therefore, the results obtained without the Lamb shift should be very similar, as indicated by the dashed lines in Figure 6. We can also see that, considering the Lamb shift, the influence of $J^{\left(1\right)}$ on the heat current lies between $J^{\left(2\right)}$ and $J^{\left(3\right)}$, and the different Lamb shifts corresponding to these three spectral densities all lead to a linear increase in ${\delta }_{\mu }$ with the increasing temperature difference, eventually resulting in a linear increase in the heat current.

Figure 6. The heat currents ${\mathcal{J}}_1^{\delta }$ vs. the relative temperature difference $\Delta T/{\omega }_D$. Here we set $T_1=1, T_2=1+\Delta T, {\gamma }_1={\gamma }_2=0.01, {\omega }_D=50, {\epsilon }_1=3, {\epsilon }_2=2.5, k=0.5$. For the blue line, $J^{\left(1\right)}\left(\omega \right)={\displaystyle \frac{\gamma \omega }{1+{(\omega /{\omega }_D)}^2}};$ for the red line, $J^{\left(2\right)}\left(\omega \right)=\gamma \omega$ when $\omega <{\omega }_D$, $J^{\left(2\right)}\left(\omega \right)=0$ when $\omega \ge {\omega }_D;$ for the black line, $J^{\left(3\right)}\left(\omega \right)=\gamma \omega exp(-{\omega }^2/{\omega }_D^2).$ 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.

5. Conclusions

We investigate the influence of the Lamb shift on heat transport in a two-qubit system coupled to thermal reservoirs at different temperatures. Our results demonstrate a dual role of the Lamb shift in regulating heat current:

• At small temperature differences, the Lamb shift suppresses the steady-state heat current;
• In contrast, for large temperature differences, the system exhibits distinct behavior—while the heat current saturates to an upper bound when neglecting the Lamb shift, its inclusion leads to a divergent heat current as the temperature gradient approaches infinity.

Our findings yield profound insights into the fundamental mechanisms of quantum heat transport, revealing how quantum coherence and system-reservoir interactions collectively govern heat current at the quantum level. It should be noted that the observed phenomenon tends to be less pronounced under small temperature differences. Future work could extend this investigation to the case with stronger temperature gradients, where the applicability of the master equation and the possible influence of non-Markovian effects may offer interesting avenues. The revealed modification of heat current induced by the Lamb shift suggests new strategies for controlling heat current through quantum engineering of system-environment interactions. This work lays the foundation for future studies of quantum heat engines, quantum batteries, and the development of quantum materials with tailored thermal properties. Furthermore, the broader interest of the Lamb shift could also be cast to quantum thermodynamics and open quantum systems, such as the quantum description of Otto cycles [67,68,69,70,71] and q-deformation in heat engines [72,73,74]. In addition, one could explore the implications of the Lamb shift in cavity quantum electrodynamics (QED) [75,76,77,78], particularly in micromaser systems. When a cavity mode repeatedly interacts with a stream of atomic ensembles, the Lamb shift could significantly influence the system’s dynamics, the onset of maser action, and the resulting photon statistics. A detailed investigation into this effect could provide deeper insight into the role of vacuum fluctuations and field-atom interactions in non-equilibrium quantum systems, thereby bridging our findings with ongoing experimental efforts in cavity-based quantum information processing.