⇦Chinese Physics Letters, 2023, Vol. 40, No. 5, Article code 050501 Modulation of Steady-State Heat Transport in a Dissipative Multi-Mode Qubit-Photon System Ze-Huan Chen (陈泽欢)1, Fei-Yu Wang (王菲宇)1, Hua Chen (陈华)1, Jin-Cheng Lu (陆金成)2*, and Chen Wang (王晨)1* Affiliations 1Department of Physics, Zhejiang Normal University, Jinhua 321004, China 2Jiangsu Key Laboratory of Micro and Nano Heat Fluid Flow Technology and Energy Application, School of Physical Science and Technology, Suzhou University of Science and Technology, Suzhou 215009, China Received 16 February 2023; accepted manuscript online 29 March 2023; published online 17 April 2023 *Corresponding authors. Email: jinchenglu@usts.edu.cn; wangchen@zjnu.cn Citation Text: Chen Z H, Wang F Y, Chen H et al. 2023 Chin. Phys. Lett. 40 050501    Abstract Quantum heat transport is considered as an indispensable branch of quantum thermodynamics to potentially improve performance of thermodynamic devices. We theoretically propose a dissipative qubit-photon system composed of multiple coupled resonators interacting with a single two-level qubit, to explore the steady-state heat transport by tuning both the inter-resonator photon hopping and the qubit-photon coupling. Specifically in the three-mode case, the dramatic enhancement and suppression of the heat current into the central resonator can be modulated by the corresponding frequency, compared to the currents into two edge resonators. Moreover, fruitful cycle current components are unraveled at weak qubit-photon coupling, which are crucial to exhibit the nonmonotonic feature with increase of the reservoir temperature bias. In the one-dimensional case under the mean-field framework, the influence of the photon hopping on heat transport is analyzed. The steady-state heat current is comparatively enhanced to the single-mode limit at weak qubit-photon coupling, stemming from the nonvanishing mean-field photon excitation parameter and the additional cycle current component. We hope these obtained results may have possible applications in quantum thermodynamic manipulation and energy harvesting.
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 DOI:10.1088/0256-307X/40/5/050501 © 2023 Chinese Physics Society Article Text Nonequilibrium thermodynamics, one active frontier of thermodynamics, mainly deals with irreversible processes driven by external sources, which is tightly related with nonequilibrium transport.[1-3] Nonequilibrium quantum thermodynamics, an assembly of nonequilibrium thermodynamics and quantum mechanics, has attracted dramatic interest, attributed to the dramatic advances of quantum technology down to the nanoscale.[4-9] This may provide rich avenues to improve performance of quantum thermodynamic devices, ranging from quantum heat engines and refrigerators,[10-16] geometric heat pump,[17-20] and even multitasking quantum thermal machines.[21] In quantum thermodynamics, systems are traditionally modeled by time-dependent quantum Hamiltonian or under the finite thermodynamic bias.[4] Meanwhile, nonequilibrium heat transport indispensable in thermodynamic devices, e.g., mediated via photons or phonons, has been extensively analyzed in hybrid quantum systems,[22,23] which is bounded by the thermodynamic laws. Recently, thermodynamic uncertainty relations[24-26] and efficiency fluctuations[27-29] were also explored to deepen the understanding of thermodynamic devices. Quantum heat transport in quantum thermodynamics is characterized via system-bath interactions from the perspective of open quantum systems.[4,29] One main route to conduct heat transport is based on dissipative light-matter interacting systems. Likewise, circuit quantum electrodynamics (cQED) devices nowadays are considered as the prominent platforms to investigate light-matter interaction, which are mainly specified as superconducting cQED[22] and quantum-dot cQED.[30-32] Experimentally, the heat valve[33] and diode operations[34] are demonstrated in superconducting cQED devices, where the qubit is simultaneously connected with two resonators, which become individually coupled to mesoscopic reservoirs with finite temperature bias. Theoretically, cQED systems are generically described as one two-level qubit interacting with a single-mode optical resonator, e.g., the seminal quantum Rabi model (QRM) with transverse qubit-photon interaction.[35,36] The QRM and its two-mode extension have already been employed to study time-dependent quantum thermodynamics and steady-state heat transport.[37-40] Concurrently, the longitudinal qubit-photon model, successfully elaborated in quantum circuit engineering,[41,42] has also been proposed as the alternative systems to fertilize functional operations in quantum heat transport.[43,44] Furthermore, the two-mode longitudinal qubit-resonator model was preliminarily analyzed to explore the effect of resonator-resonator coupling on heat flows.[45] Recently, the influence of multiple bosonic modes on quantum heat transport in hybrid quantum systems has attracted increasing attention. In particular, the chirality is included to drive the permanent current in dissipative optomechanic lattice[46] and quantum Rabi ring,[47,48] and the pump-dissipation protocol is applied to generate the polariton lasing.[49] Hence, we believe that it should be intriguing to investigate the role of the multi-mode configuration on steady-state heat transport and underlying mechanism in multi-mode qubit-resonator systems for functional operations. In this work, we mainly investigate steady-state heat transport in three-mode and one-dimensional qubit-photon cases by modulating both the resonator-resonator coupling and qubit-photon interaction. The eigen-mode structures contributed from multi-mode configuration are analytically demonstrated, which is of significant importance for the controllability of heat flows. (i) In the three-mode qubit-photon case, the enhancement and suppression of the heat current into the central resonator compared to two edge counterparts are exhibited by varying the frequency of middle resonator, which shows the spatial concentration and decentralization of the heat currents. We note that due to the special three-mode configuration, such effect is unavailable with the single-mode and two-mode qubit-photon models. The expression of cycle fluxes are analytically presented at weak qubit-photon interaction, which are contributed by both first and third eigen-mode channels. It directly leads to appearance of negative differential thermal conductance (NDTC), i.e., the heat current is reduced with the increasing reservoir temperature bias.[50,51] (ii) We also study steady-state heat currents in the dissipative one-dimensional qubit-photon case under the mean-field framework, which is reduced to the effective single-mode qubit-resonator case. The heat current is enhanced at weak qubit-photon interaction compared to the single-mode counterpart, mainly originating from the emergence of an additional cycle flux component. In the following, first we present the dissipative qubit-photon model, general eigen-mode solution, quantum master equation, transition rates in the three-mode and mean-field one-dimensional cases, and the expression of heat currents. Then, we study the influence of inter-resonator photon hopping and qubit-photon coupling on steady-state heat currents. Finally, we present a brief summary. Dissipative Multi-Mode Qubit-Photon System. Here, we first introduce the general route to obtain the eigenvalues and eigenvectors of the multi-mode qubit-resonator model. Then, we analytically solve two distinct cases, i.e., three-mode and one-dimensional qubit-resonator cases. Model. The dissipative multi-mode qubit-photon model consists of multiple coupled resonators interacting with a single two-level qubit, of which each component individually is coupled to a bosonic thermal reservoir. The full Hamiltonian reads $\hat{H}=\hat{H}_s+\sum_{u=q,r_1,r_2,\ldots}(\hat{H}_{b,u}+\hat{V}_u)$. Specifically, The Hamiltonian of the qubit-photon model is expressed as ($\hbar=1$) \begin{align} \hat{H}_{s}=\,&\sum^{N}_{i=1}\omega_i\hat{a}^†_i\hat{a}_i -t\sum^{N-1}_{i=1}(\hat{a}^†_i\hat{a}_{i+1}+\hat{a}^†_{i+1}\hat{a}_i)\notag\\ &+\frac{\varepsilon }{2}\hat{\sigma}_z+\hat{\sigma}_z\sum^N_{i=1} \lambda_i(\hat{a}^†_i+\hat{a}_i),~~ \tag {1} \end{align} where $\hat{a}^†_i (\hat{a}_i)$ denotes the creation (annihilation) of one photon in the $i$-th photonic resonator with the frequency $\omega_i$, $t$ shows the interaction strength between nearest-neighboring resonators,[52] $\sigma_z$ and $\varepsilon$ are the Pauli operator and splitting energy of the two-level qubit, $\lambda_i$ is the longitudinal coupling strength between the qubit and the $i$-th resonator, and $N$ the total number of photonic resonators. The $u$-th bosonic thermal reservoir is expressed as $\hat{H}_{b,u}=\sum_{k}\omega_{k,u}\hat{b}^†_{k,u}\hat{b}_{k,u}$, where $\hat{b}^†_{k,u}~(\hat{c}_{k,u})$ creates (annihilates) one boson in the $u$-th reservoir with the frequency $\omega_{k,u}$. The system–bath interaction is described as $\hat{V}_u=\sum_{u,k}\hat{X}_u(g_{k,u}\hat{b}^†_{k,u}+g^{*}_{k,u}\hat{b}_{k,u})$, where $g_{k,u}$ is the coupling strength and component projectors are specified as $\hat{X}_q=\hat{\sigma}_x$ and $\hat{X}_{r_i}=(\hat{a}^†_i+\hat{a}_i)$. To explore the eigen-modes of coupled resonators, i.e., $\hat{H}_{a}=\sum^{N}_{i=1}\omega_i\hat{a}^†_i\hat{a}_i -t\sum^{N-1}_{i=1}(\hat{a}^†_i\hat{a}_{i+1}+\hat{a}^†_{i+1}\hat{a}_i)$, we include the general canonical transformation of bosonic operators to diagonalize $\hat{H}_{a}$ via \begin{align} {\hat{B}}={W}^{-1}{\hat{A}},~~ \tag {2} \end{align} with ${\hat{A}}=[\hat{a}_1,\hat{a}_2,\ldots,\hat{a}_N]^{\scriptscriptstyle{\rm T}}$ and ${\hat{B}}=[\hat{b}_1,\hat{b}_2,\ldots,\hat{b}_N]^{\scriptscriptstyle{\rm T}}$. Then, we obtain $\hat{H}_a=\sum_i \varOmega_i\hat{b}^†_i\hat{b}_i$, where the bosonic operators fulfill the commutating relation $[\hat{b}_i,\hat{b}^†_j]=\delta_{ij}$. Consequently, the system Hamiltonian (1) is re-expressed as \begin{align} \hat{H}_{s}=\sum^{N}_{j=1}\varOmega_j \hat{b}^†_{j} \hat{b}_{j} +\frac{\varepsilon }{2}\hat{\sigma}_z+\hat{\sigma}_z\sum^N_{j=1} \varLambda_j(\hat{b}^†_j+\hat{b}_j),~~ \tag {3} \end{align} where the renewed qubit-photon interaction becomes $\varLambda_j=\sum_i\lambda_i{W}_{ij}\sqrt{{\omega}_i/\varOmega_j}$. Hence, the eigenstates of $\hat{H}_s$ are obtained as \begin{align} \!|\psi^{\sigma}_{\boldsymbol{m}}{\rangle}=|\sigma{\rangle}{\otimes}{\varPi}_i\Big\{\exp[-(\varLambda_{\sigma,i}/\varOmega_i)(\hat{b}^†_i \!-\!\hat{b}_i)]\frac{(\hat{b}^†_i)^{n_i}}{\sqrt{n_i!}}|0{\rangle}_{b_i}\Big\},~~ \tag {4} \end{align} where the index is given by $\boldsymbol{m}=[m_1,m_2,\ldots]$ with $m_i=0,\,1,\,2,\,\ldots$, the qubit state is specified by $|\sigma{\rangle}=|\uparrow(\downarrow){\rangle}$, with $\hat{\sigma}_z|\uparrow{\rangle}=|\uparrow{\rangle}$ and $\hat{\sigma}_z|\downarrow{\rangle}=-|\downarrow{\rangle}$, the qubit-resonator coupling strengths are specified as $\varLambda_{\uparrow,i}=\varLambda_i$ and $\varLambda_{\downarrow,i}=-\varLambda_i$, and $|0{\rangle}_{b_i}$ is the vacuum state of the mode $\hat{b}_i$, characterized as $\hat{b}_i|0{\rangle}_{b_i}=0$. The corresponding eigenvalues are expressed as \begin{align} E_{\boldsymbol{m},\sigma}=\sum_i(m_i\varOmega_i-\varLambda^2_i/\varOmega_i)+\varepsilon_\sigma/2,~~ \tag {5} \end{align} with $\varepsilon_\uparrow=\varepsilon$ and $\varepsilon_\downarrow=-\varepsilon$. We should admit that it is generally difficult to analytically obtain the transformation matrix $V$ in Eq. (2), the renormalized frequency $E_i$, and coupling strength $\varLambda_i$ in Eq. (3) for large $N$. However, for the generic case, i.e., the three-mode resonators with $N=3$ in Fig. 1(a), it is fortunate for us to be able to analytically gain the connection between ${\hat{A}}$ and ${\hat{B}}$ via the matrix $W$. Specifically for distinct resonators, e.g., $\omega_1=\omega_3=\omega_a$ and $\omega_2=\omega_b$ with $\omega_a{\neq}\omega_b$, the relations between two bosonic operator sets are given by $\hat{b}_1=[\cos(\theta/2)\hat{a}_1/\sqrt{2}+\cos(\theta/2)\hat{a}_3/\sqrt{2}+\sin(\theta/2)\hat{a}_2]$, $\hat{b}_2=(\hat{a}_1-\hat{a}_3)/\sqrt{2}$, and $\hat{b}_3=[\sin(\theta/2)\hat{a}_1/\sqrt{2}+\sin(\theta/2)\hat{a}_3/\sqrt{2}-\cos(\theta/2)\hat{a}_2]$, where the tunable angle becomes $\theta=\theta_c$ for $\omega_b{\ge}\omega_a$ and $\theta=\pi-\theta_c$ for $\omega_b < \omega_a$, with \begin{align} \theta_c=\arctan(2\sqrt{2}\,t/|\omega_b-\omega_a|).~~ \tag {6} \end{align} The eigen-mode energies become \begin{align} &\varOmega_1=[(\omega_a+\omega_b)-\sqrt{(\omega_b-\omega_a)^2+8t^2}\,]/2,~~~~~~~~~~~~~\,\,\,\, \tag{7a}\\ &\varOmega_2={\omega}_a, \tag{7b}\\ &\varOmega_3=[(\omega_a+\omega_b)+\sqrt{(\omega_b-\omega_a)^2+8t^2}\,]/2. \tag{7c} \end{align} The renewed qubit-photon coupling strengths are given by \begin{align} \varLambda_1=\,&\lambda[\sqrt{2}\cos(\theta/2)\sqrt{\omega_a/\varOmega_1} +\sin(\theta/2)\sqrt{\omega_b/\varOmega_1}\,],~~~~ \tag{8a}\\ \varLambda_2=\,&0, \tag{8b}\\ \varLambda_3=\,&\lambda[\sqrt{2}\cos(\theta/2)\sqrt{\omega_a/\varOmega_3} -\sin(\theta/2)\sqrt{\omega_b/\varOmega_3}\,],~~~~ \tag{8c} \end{align} also as shown in Fig. 1(b). It is interesting to find that $\varLambda_2=0$ always holds, regardless of the selection of resonator frequencies. Hence, $\hat{b}_2$ is the stable dark mode in such a three-mode configuration.
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Fig. 1. (a) Description of the dissipative three-mode qubit-resonator model. The harmonics with $\hat{a}_i$ shows the $i$-th mode optical resonator, the black springs with $t$ mean the inter-resonator photon hopping, the blue circle with $\varepsilon$ denotes the two-level qubit, the double-arrowed lines with $\lambda$ describes the qubit-photon interactions, the upper and bottom arc shapes describe bosonic thermal reservoirs with the temperatures $T_{r}$ and $T_{q}$, and the arrowed lines with $J_{r_{i}}$ and $J_{q}$ show the heat currents into the $r_i$-th and $q$-th thermal reservoirs, respectively. (b) schematic illustration of the dissipative three-mode qubit-resonator model in the eigen-mode picture of resonators, characterized as the eigen-mode $\hat{b}_i$ and the corresponding qubit-photon interaction strength $\varLambda_i$.
Alternatively, for the one-dimensional homogeneous qubit-photon system we consider $\omega_i=\omega_a$ and $\lambda_i=\lambda$. Within the mean-field framework, the photon hopping term can be effectively decoupled as $\hat{a}^†_i\hat{a}_{j}\,{\approx}\,a(\hat{a}^†_i+\hat{a}_{j})-a^2$,[52] with the mean-field photon excitation parameter \begin{align} a={\langle}\hat{a}_i{\rangle}.~~ \tag {9} \end{align} Then, the mean-field qubit-photon Hamiltonian is reduced to \begin{align} \frac{1}{N}\hat{H}^{\scriptscriptstyle{\rm MF}}_s=\,&\omega_a\hat{a}^†\hat{a}-2at(\hat{a}^†+\hat{a})+ \frac{\varepsilon }{2}\hat{\sigma}_z\notag\\ &+\lambda\hat{\sigma}_z(\hat{a}^†+\hat{a}),~~ \tag {10} \end{align} where the subindex $i$ and the parameter $a^2$ are ignored for safety. It is straightforward to find that $\hat{H}^{\scriptscriptstyle{\rm MF}}_s/N$ becomes the effective single-mode qubit-resonator model. In the limit of $a=0$, $\hat{H}^{\scriptscriptstyle{\rm MF}}_s/N$ is completely reduced to the single-mode case.[53] The corresponding eigenvalues and eigenvectors are given by \begin{align} E_{m,\sigma}=\omega_am-\lambda^2_\sigma/\omega_a+\varepsilon_\sigma/2,~~ \tag {11} \end{align} and \begin{align} |\psi^\sigma_m{\rangle}=|\sigma{\rangle}{\otimes}\exp[-{(\lambda_\sigma/\omega_a)(\hat{a}^†-\hat{a})}]\frac{(\hat{a}^†)^m} {\sqrt{m!}}|0{\rangle}_a,~~ \tag {12} \end{align} where the coefficients are $\lambda_\uparrow=\lambda-2at$, $\lambda_\downarrow=-\lambda-2at$, $\varepsilon_\uparrow=\varepsilon$, and $\varepsilon_\downarrow=-\varepsilon$. Master Equation. We apply the quantum master equation to analyze the steady-state heat transport in the multi-mode qubit-photon model, where the hybrid quantum system is driven by the finite temperature bias. Moreover, if the qubit-photon coupling strength goes beyond weak, the system components, i.e., qubit and resonators, should not be considered as separately being interacting with thermal reservoirs. Hence, the quantum master equation needs derive under the eigenstate basis of $\hat{H}_s$, i.e., $\big\{|\psi^\sigma_{\boldsymbol{m}}{\rangle}\big\}$ in Eq. (4). Under the Born–Markov approximation, one can individually perturb $\hat{V}_u$ to obtain the dressed master equation as[54,55] \begin{align} \frac{d\hat{\rho}_s}{dt}=-i[\hat{H}_s,\hat{\rho}_s]+\sum_{u=q,r_1,r_2,\ldots}\mathcal{L}_u[\hat{\rho}_s],~~ \tag {13} \end{align} where $\hat{\rho}_s$ is the density operator of the qubit-resonator model. The dissipators are shown as \begin{align} \mathcal{L}_u[\hat{\rho}_s]=\,& \sum_{\boldsymbol{m},\boldsymbol{m}^\prime,\sigma,\sigma^\prime}( \varGamma^-_{u}(\varDelta^{\sigma\sigma^\prime}_{\boldsymbol{m}\boldsymbol{m}^\prime})\mathcal{D}_u[|\phi^{\sigma^\prime}_{\boldsymbol{m}^\prime} {\rangle}{\langle}\phi^\sigma_{\boldsymbol{m}}|]\hat{\rho}_s\notag\\ &+\varGamma^+_{u}(\varDelta^{\sigma\sigma^\prime}_{\boldsymbol{m}\boldsymbol{m}^\prime})\mathcal{D}_u[|\phi^\sigma_{\boldsymbol{m}}{\rangle}{\langle}\phi^{\sigma^\prime}_{\boldsymbol{m}^\prime}|]\hat{\rho}_s),~~ \tag {14} \end{align} where the energy gap is $\varDelta^{\sigma\sigma^\prime}_{\boldsymbol{m}\boldsymbol{m}^\prime}=E_{\boldsymbol{m},\sigma}-E_{\boldsymbol{m}^\prime,\sigma^\prime}$, the eigenstate-dependent dissipators are given by $\mathcal{D}_u[|\phi^\sigma_{\boldsymbol{m}}{\rangle}{\langle}\phi^{\sigma^\prime}_{\boldsymbol{m}^\prime}|]\hat{\rho}_s= |\phi^\sigma_{\boldsymbol{m}}{\rangle}{\langle}\phi^{\sigma^\prime}_{\boldsymbol{m}^\prime}|\hat{\rho}_s|\phi^{\sigma^\prime}_{\boldsymbol{m}^\prime}{\rangle} {\langle}\phi^\sigma_{\boldsymbol{m}}| -(|\phi^{\sigma^\prime}_{\boldsymbol{m}^\prime}{\rangle}{\langle}\phi^{\sigma^\prime}_{\boldsymbol{m}^\prime} |\hat{\rho}_s +\hat{\rho}_s|\phi^{\sigma^\prime}_{\boldsymbol{m}^\prime}{\rangle}{\langle} \phi^{\sigma^\prime}_{\boldsymbol{m}^\prime}|)/2$, the transition rates are described as $\varGamma^{\pm}_{u}(\varDelta^{\sigma\sigma^\prime}_{\boldsymbol{m}\boldsymbol{m}^\prime}) =\gamma_u(\pm\varDelta^{\sigma\sigma^\prime}_{\boldsymbol{m}\boldsymbol{m}^\prime}) n_u(\pm\varDelta^{\sigma\sigma^\prime}_{\boldsymbol{m}\boldsymbol{m}^\prime})|{\langle} \phi^\sigma_{\boldsymbol{m}}|\hat{X}_u|\phi^{\sigma^\prime}_{\boldsymbol{m}^\prime} {\rangle}|^2$, the spectral density of the $u$-th reservoir is defined as $\gamma_u(\omega)=2\pi\sum_k|g_{k,u}|^2\delta(\omega-\omega_{k,u})$, which is specified as the Ohmic case $\gamma_u(\omega)=\alpha_u\omega\exp(-|\omega|/\omega_c)$[56] with $\alpha_u$ being the dissipation strength and $\omega_c$ the cutoff frequency of the reservoir, and the Bose–Einstein distribution function is given by $n_u(\omega)=1/[\exp(\omega/k_{\scriptscriptstyle{\rm B}}T_u)-1]$, with $k_{\scriptscriptstyle{\rm B}}$ being the Boltzmann constant and $T_u$ the temperature of the $u$-th reservoir. In the photonic coherent state basis (4), the rates are specified as \begin{align} \varGamma^{\pm}_{r_i}(\varDelta^{\sigma{\sigma}}_{\boldsymbol{m}^\prime\boldsymbol{m}})=\,& \sum_jV^2_{ij}\frac{\omega_i}{\varOmega_j}\delta_{{\boldsymbol{m},\boldsymbol{m}^\prime-\boldsymbol{I}_j}}m^\prime_j \gamma_{r_i}({\pm}\varOmega_j) n_{r_i}({\pm}\varOmega_j), \tag{15a} \\ \varGamma^{\pm}_q(\varDelta^{\overline{\sigma}\sigma}_{\boldsymbol{m}^\prime\boldsymbol{m}})=\,&\varTheta(\varDelta^{\overline{\sigma}\sigma}_{\boldsymbol{m}^\prime\boldsymbol{m}}) \gamma_q({\pm}\varDelta^{\overline{\sigma}\sigma}_{\boldsymbol{m}^\prime\boldsymbol{m}}) n_q({\pm}\varDelta^{\overline{\sigma}\sigma}_{\boldsymbol{m}^\prime\boldsymbol{m}})\notag\\ &{\times}\prod_iD^2_{m_i,m^\prime_i}(2\varLambda_i/\varOmega_i). \tag{15b} \end{align} where the step function is $\varTheta(\omega{>}0)=1$ and $\varTheta(\omega{\leq}0)=0$, the vector is $\boldsymbol{I}_i=[0_1,0_2,\ldots,1_i,\ldots]$ with the nonzero component only at $i$-th element, and the coherent state overlap coefficient becomes[57,58] \begin{align} D_{nm}(x)=e^{-x^2/2}\sum^{\min[n,m]}_{l=0}\frac{(-1)^l\sqrt{n!m!}x^{n+m-2l}}{(n-l)!(m-l)!l!}.~~ \tag {16} \end{align} The transition rate $\varGamma^{\pm}_{r_i}(\varDelta^{\sigma{\sigma}}_{\boldsymbol{m}^\prime\boldsymbol{m}})$ involved with the reservoir connecting $i$-th resonator consists of multiple eigen-mode channels, which are incoherent and constructed via eigen-mode transformation coefficients, i.e., ${W}_{ij}$. For such a transition between $|\psi^\sigma_{\boldsymbol{m}}{\rangle}$ and $|\psi^\sigma_{\boldsymbol{m}^\prime}{\rangle}$, the qubit state ($\sigma p$) is unchanged. Likewise, the rate $\varGamma^{\pm}_q(\varDelta^{\overline{\sigma}\sigma}_{\boldsymbol{m}^\prime\boldsymbol{m}})$ describes the transition between eigenstates $|\psi^\sigma_{\boldsymbol{m}}{\rangle}$ and $|\psi^{\sigma^\prime}_{\boldsymbol{m}^\prime}{\rangle}$, which is accompanied by the qubit flip process. The rate is strongly affected by qubit-photon scattering, which is characterized as the coherent-state coefficient $D_{m_im^\prime_i}(2\varLambda_i/\varOmega_i)$. Transition Rates. We specify the expression of transition rates in the three-mode configuration of coupled resonators. Considering the case of distinct resonators, e.g., $\omega_1=\omega_3=\omega_a$ and $\omega_2=\omega_b$, the transition rates in Eq. (15a) are specified as \begin{align} \varGamma^{\pm}_{r_1}(\varDelta^{\sigma\sigma}_{\boldsymbol{m}^\prime\boldsymbol{m}}) =\,&\frac{\cos^2(\theta/2)}{2}\frac{\omega_a}{\varOmega_1}\delta_{{\boldsymbol{m},\boldsymbol{m}^\prime-\boldsymbol{I}_1}}m^\prime_1\gamma_{r_1}(\pm{\varOmega_1})\notag\\ &\cdot n_{r_1}(\pm{\varOmega_1}) +\frac{1}{2}\frac{\omega_a}{\varOmega_2}\delta_{{e {\boldsymbol{m},\boldsymbol{m}^\prime-\boldsymbol{I}_2},m^\prime_2-1}}n^\prime_2\gamma_{r_1}\notag\\ &\cdot(\pm{\varOmega_2})n_{r_1}(\pm{\varOmega_2}) +\frac{\sin^2(\theta/2)}{2}\frac{\omega_a}{\varOmega_3}\notag\\ &\cdot\delta_{{\boldsymbol{m},\boldsymbol{m}^\prime-\boldsymbol{I}_3}}m^\prime_3\gamma_{r_1}(\pm{\varOmega_3})n_{r_1}(\pm{\varOmega_3})\tag{17a}\\ \varGamma^{\pm}_{r_2}(\varDelta^{\sigma\sigma}_{\boldsymbol{m}^\prime\boldsymbol{m}})=\,&\sin^2(\theta/2)\frac{\omega_b}{\varOmega_1}\delta_{\boldsymbol{m},\boldsymbol{m}^\prime -\boldsymbol{I}_1}m^\prime_1\gamma_{r_2}(\pm{\varOmega_1})\notag\\ &\cdot n_{r_2}(\pm{\varOmega_1})+\cos^2(\theta/2)\frac{\omega_b}{\varOmega_3}\delta_{{\boldsymbol{m},\boldsymbol{m}^\prime-\boldsymbol{I}_3}}m^\prime_3\gamma_{r_2}\notag\\ &\cdot(\pm{\varOmega_3})n_{r_2}(\pm{\varOmega_3}), \tag{17b}\\ \varGamma^{\pm}_{r_3}(\varDelta^{\sigma\sigma}_{\boldsymbol{m}^\prime\boldsymbol{m}})=\,& \frac{\cos^2(\theta/2)}{2}\frac{\omega_a}{\varOmega_1}\delta_{{\boldsymbol{m},\boldsymbol{m}^\prime-\boldsymbol{I}_1}}m^\prime_1\gamma_{r_3}(\pm{\varOmega_1})\notag\\ &\cdot n_{r_3}(\pm{\varOmega_1})+\frac{1}{2}\frac{\omega_a}{\varOmega_2}\delta_{{\boldsymbol{m},\boldsymbol{m}^\prime-\boldsymbol{I}_2}}m^\prime_2\gamma_{r_3}(\pm{\varOmega_2})\notag\\ &\cdot n_{r_3}(\pm{\varOmega_2}) +\frac{\sin^2(\theta/2)}{2}\frac{\omega_a}{\varOmega_3}\delta_{{\boldsymbol{m},\boldsymbol{m}^\prime-\boldsymbol{I}_3}}m^\prime_3\gamma_{r_3}\notag\\ &\cdot(\pm{\varOmega_3})n_{r_3}(\pm{\varOmega_3}). \tag{17c} \end{align} Though the rates $\varGamma^{\pm}_{r_1}(\varDelta^{\sigma\sigma}_{\boldsymbol{m}^\prime\boldsymbol{m}})$ and $\varGamma^{\pm}_{r_3}(\varDelta^{\sigma\sigma}_{\boldsymbol{m}^\prime\boldsymbol{m}})$ are contributed by three eigen-modes, it should be noted that no energy exchange is allowed between $\hat{b}_2$ and the qubit, because the second eigen-mode, i.e. $\hat{b}_2$, is decoupled from the qubit, quantified by $\varLambda_2=0$ in Eq. (8b). Hence, transition terms $\frac{\omega_a}{2\varOmega_2}\delta_{{\boldsymbol{m},\boldsymbol{m}^\prime-\boldsymbol{I}_2}}m^\prime_2\gamma_{r_{1(3)}}(\pm{\varOmega_2})n_{r_{1(3)}}(\pm{\varOmega_2})$ mainly contribute to the thermalization of the boson mode $\hat{b}_2$. Noticeably, in the one-dimensional case under the framework of mean-field theory, the transition rates are given by \begin{align} \varGamma^{\pm}_r(\varDelta^{\sigma\sigma}_{m^\prime{m}})=\,&\delta_{m,m^\prime-1}\gamma_r(\pm\omega_a)n_r(\pm\omega_a)m^\prime,~~~~~~~ \tag{18a}\\ \varGamma^{\pm}_q(\varDelta^{\overline{\sigma}\sigma}_{m^\prime{m}})=\,&\varTheta(\varDelta^{\overline{\sigma}\sigma}_{m^\prime{m}}) \gamma_q(\pm\varDelta^{\overline{\sigma}\sigma}_{m^\prime{m}})n_q(\pm\varDelta^{\overline{\sigma}\sigma}_{m^\prime{m}})\notag\\ &\cdot D^{2}_{m^\prime{m}}\Big(\frac{2\lambda}{\omega_a}\Big).
\tag{18b} \end{align} It is interesting to find that $\varGamma^{\pm}_r(\varDelta^{\sigma\sigma}_{m^\prime{m}})\delta_{m^\prime,m+1}$, denoting one photon excitation (annihilation) process, has the identical expression with the single-mode case, which is irrelevant with the mean field parameter ${\langle}\hat{a}{\rangle}$. $\varGamma^{\pm}_q(\varDelta^{\overline{\sigma},\sigma}_{m^\prime{m}})$ includes the multi-mode information in the energy gap $\varDelta^{\overline{\sigma}\sigma}_{m^\prime{m}}$. Expression of Heat Current. The heat current is considered as one main feature of quantum heat transport in nonequilibrium quantum systems.[4,5,20] In the following, we focus on the steady-state behaviors of the heat currents. Based on quantum master Eq. (13), the general expression of steady-state heat current flowing into the $u$-th thermal reservoir is expressed as \begin{align} J_{u}=\,&\sum_{\boldsymbol{m},\boldsymbol{m}^\prime,\sigma,\sigma^\prime} \Theta(\varDelta^{\sigma^\prime{\sigma}}_{\boldsymbol{m}^\prime\boldsymbol{m}}) \varDelta^{\sigma^\prime{\sigma}}_{\boldsymbol{m}^\prime\boldsymbol{m}} [\varGamma^-_u(\varDelta^{\sigma^\prime{\sigma}}_{\boldsymbol{m}^\prime\boldsymbol{m}})\rho^s_{\boldsymbol{m}^{\prime},\sigma^{\prime}}\notag\\ &-\varGamma^+_u(\varDelta^{\sigma^\prime{\sigma}}_{\boldsymbol{m}^\prime\boldsymbol{m}})\rho^s_{\boldsymbol{m},\sigma}],~~ \tag {19} \end{align} where the steady-state population is given by $\rho^s_{\boldsymbol{m},\sigma}={\langle}\psi^\sigma_{\boldsymbol{m}}|\hat{\rho}_s|\psi^\sigma_{\boldsymbol{m}}{\rangle}$, which is obtained from ${d}\rho^s_{\boldsymbol{m},\sigma}/{dt}=0$. Results and Discussions—Three-Mode Case. We first study behaviors of heat currents under the condition of three identical resonators, i.e., $\omega_{i}=\omega_a$ and $\lambda_i=\lambda$ with $i=1,\,2,\,3$. Hence, the angle $\theta_c$ in Eq. (6) becomes $\pi/2$. The relations between two kinds of bosonic operators are established as $\hat{b}_1=(\hat{a}_1+\hat{a}_3+\sqrt{2}\hat{a}_2)/2$, $\hat{b}_2=(\hat{a}_1-\hat{a}_3)/\sqrt{2}$, and $\hat{b}_3=(\hat{a}_1+\hat{a}_3-\sqrt{2}\hat{a}_2)/2$. This directly results in the eigen-mode energies (7a)–(7c) \begin{align} \varOmega_1=&{\omega}_a-\sqrt{2}t\tag{20a}\\ \varOmega_2=&{\omega}_a,\tag{20b}\\ \varOmega_3=&{\omega}_a+\sqrt{2}t, \tag{20c} \end{align} and the corresponding qubit-phonon interaction strengths (8a)–(8c) \begin{align} \varLambda_1=\,&(1+1/\sqrt{2})\lambda\sqrt{{\omega}_a/\varOmega_1}\tag{21a}\\ \varLambda_2=\,&0,\tag{21b}\\ \varLambda_3=\,&(1-1/\sqrt{2})\lambda\sqrt{{\omega}_a/\varOmega_3}.
\tag{21c} \end{align} Consequently, the transition rates involved with $r_i$-th reservoir are specified as \begin{align} \varGamma^{\pm}_{r_1}(\varDelta^{\sigma\sigma}_{\boldsymbol{m}^\prime\boldsymbol{m}})=\,&\frac{1}{4}\sum_{i=1,3} \frac{\omega_a}{\varOmega_i}\delta_{{\boldsymbol{m},\boldsymbol{m}-\boldsymbol{I}_i}}m^\prime_i\gamma_{r_1}(\pm{\varOmega_i})n_{r_1}(\pm{\varOmega_i})\notag\\ &+\frac{1}{2}\frac{\omega_a}{\varOmega_2}\delta_{{\boldsymbol{m},\boldsymbol{m}-\boldsymbol{I}_2}}m^\prime_2\gamma_{r_1}(\pm{\varOmega_2})\notag\\ &\cdot n_{r_1}(\pm{\varOmega_2}),
\tag{22a}\\ \varGamma^{\pm}_{r_2}(\varDelta^{\sigma\sigma}_{\boldsymbol{m}^\prime,\boldsymbol{m}})=\,& \frac{1}{2}\sum_{i=1,3}\frac{\omega_a}{\varOmega_i}\delta_{{\boldsymbol{m},\boldsymbol{m}-\boldsymbol{I}_i}} m^\prime_i\gamma_{r_2}(\pm{\varOmega_i})\notag\\ &\cdot n_{r_2}(\pm{\varOmega_i}),
\tag{22b}\\ \varGamma^{\pm}_{r_3}(\varDelta^{\sigma\sigma}_{\boldsymbol{m}^\prime\boldsymbol{m}})=\,& \frac{1}{4}\sum_{i=1,3}\frac{\omega_a}{\varOmega_i}\delta_{{\boldsymbol{m},\boldsymbol{m}-\boldsymbol{I}_i}} m^\prime_i\gamma_{r_3}(\pm{\varOmega_i})n_{r_3}(\pm{\varOmega_i})\notag\\ &+\frac{1}{2}\frac{\omega_a}{\varOmega_2}\delta_{{\boldsymbol{m},\boldsymbol{m}-\boldsymbol{I}_2}}m^\prime_2\gamma_{r_3}(\pm{\varOmega_2})\notag\\ &\cdot n_{r_3}(\pm{\varOmega_2}).\tag{22c} \end{align} In Fig. 2 we compare the steady-state currents through coupled resonators with the case of individual resonators ($t=0$). Specifically, in the absence of inter-resonator photon hopping it is expected that $J_{r_i}$ are identical (i.e., $J_{r_i}=J_q/3$) as shown in Fig. 2(a), due to the homogeneous arrangement of the resonators. However, as we tune on photon hopping in Fig. 2(b), it is intriguing to find that the current $J_{r_2}$ becomes twice the $J_{r_{1(3)}}$ (i.e., $J_{r_2}=J_q/2$ and $J_{r_{1(3)}}=J_q/4$), irrelevant with the qubit-photon interaction strength. This concentration of currents through the resonators can be exhibited even at weak photon hopping strength, e.g., $t=0.01\omega_a$. Note that such a feature is unique in the three-mode qubit-photon model due to the multi-mode spatial configuration, which is unavailable in the two-mode and single-mode cases.[45,53] Then, we try to explore the underlying mechanism. Since the second eigen-mode is always decoupled from the qubit ($\varLambda=0$), the second eigen-mode channel shows no contribution to the currents $J_{r_1}$ and $J_{r_3}$. Thus the effective rates to assist heat flow from/into the $r_{1(3)}$-th reservoirs can be reorganized as $\varGamma^{\pm}_{r_{1(3)},\,\rm{eff}}(\varDelta^{\sigma\sigma}_{\boldsymbol{m}^\prime\boldsymbol{m}}) =\sum_{i=1,\,3}\frac{\omega_a}{4\varOmega_i}\delta_{{ \boldsymbol{m},\,\boldsymbol{m}-\boldsymbol{I}_i}} m^\prime_i\gamma_{r_{1(3)}}(\pm{\varOmega_i})n_{r_{1(3)}}(\pm{\varOmega_i})$. Based on the expression of heat current in Eq. (19), we directly obtain the general result $J_{r_2}=2J_{r_1}=2J_{r_3}$. We also plot the 3D-view of the heat current $J_q$ by modulating both the resonator-resonator interaction and qubit-photon coupling strengths in Fig. 2(c). It is found that increasing $t$ is preferred to exhibit the optimal heat current $J_q$, whereas the corresponding qubit-photon interaction strength is reduced.
cpl-40-5-050501-fig2.png
Fig. 2. Steady-state heat currents $J_{r_i}$ of three coupled identical resonators by modulating qubit-photon coupling strength $\lambda$ at (a) $t=0$ and (b) $t=0.01\omega_a$. (c) The heat current into the $q$-th thermal reservoir by tuning $\lambda$ and $t$. Other system parameters are given as $\varepsilon=\omega_a$, $\alpha_{r_i}=\alpha_q=10^{-3}\omega_a$, $\omega_c=20\omega_a$, $k_{\scriptscriptstyle{\rm B}}T_{r}=1.5\omega_a$, and $k_{\scriptscriptstyle{\rm B}}T_q=0.5\omega_a$.
cpl-40-5-050501-fig3.png
Fig. 3. The ratio of heat currents $J_{r_2}/J_{r_1}$ of three coupled distinct resonators by tuning $\omega_b/\omega_a$ with $\omega_1=\omega_3=\omega_a$, and $\omega_2=\omega_b$. The inset shows the eigen-mode energies $\varOmega_1$ and $\varOmega_3$. Other system parameters are given by $\varepsilon=\omega_a$, $t=0.1\omega_a$, $\alpha_{r_i}=\alpha_q=10^{-3}\omega_a$, $\omega_c=20\omega_a$, $k_{\scriptscriptstyle{\rm B}}T_{r}=1.5\omega_a$, and $k_{\scriptscriptstyle{\rm B}}T_{q}=0.5\omega_a$.
In Fig. 3 we study the influence of the distinct resonators on the heat current. When the frequency $\omega_b$ decreases, it is found that the ratio $J_{r_2}/J_{r_1}$ becomes apparently enhanced at weak qubit-photon coupling (e.g., $\lambda=0.01\omega_a$). This makes the current highly concentrate on $J_{r_2}$. However, $J_{r_2}/J_{r_1}$ is monotonically suppressed with the increase of qubit-photon coupling strength. It can be understood that as $\omega_b{\ll}\omega_a$ (i.e., the angle $\theta{\rightarrow}\pi$), the effective rates involved with the $r_i$-th reservoir to contribute to the heat current are simplified to \begin{align} \varGamma^{\pm}_{r_1,\rm{eff}}(\varDelta^{\sigma\sigma}_{\boldsymbol{m}^\prime\boldsymbol{m}})=\,& \frac{\omega_a}{2\varOmega_3}\delta_{{\boldsymbol{m},\boldsymbol{m}^\prime-\boldsymbol{I}_3}} m^\prime_1\gamma_{r_1}(\pm{\varOmega_3})n_{r_1}(\pm{\varOmega_3}),\tag{23a}\\ \varGamma^{\pm}_{r_2,\rm{eff}}(\varDelta^{\sigma\sigma}_{\boldsymbol{m}^\prime\boldsymbol{m}})=\,& \frac{\omega_b}{\varOmega_1}\delta_{{\boldsymbol{m},\boldsymbol{m}^\prime-\boldsymbol{I}_1}} m^\prime_2\gamma_{r_2}(\pm{\varOmega_1})n_{r_2}(\pm{\varOmega_1}),\tag{23b}\\ \varGamma^{\pm}_{r_3,\rm{eff}}(\varDelta^{\sigma\sigma}_{\boldsymbol{m}^\prime\boldsymbol{m}})=\,& \frac{\omega_a}{2\varOmega_3}\delta_{{\boldsymbol{m},\boldsymbol{m}^\prime-\boldsymbol{I}_3}} m^\prime_3\gamma_{r_3}(\pm{\varOmega_3})n_{r_3}(\pm{\varOmega_3}).\tag{23c} \end{align} With weak qubit-photon coupling, the energy exchange between the qubit and the corresponding reservoir is dominated by individual processes, i.e., ${\varPi}_{i}D^2_{m_i,m^\prime_i}(2\varLambda_i/\varOmega_i)\,{\approx}\,[D^2_{m_1,m^\prime_1}(2\varLambda_1/\varOmega_1)\delta_{m_2,m^\prime_2}\delta_{m_3,m^\prime_3} +D^2_{m_2,m^\prime_2}(2\varLambda_2/\varOmega_2)\delta_{m_1,m^\prime_1}\delta_{m_3,m^\prime_3} +D^2_{m_3,m^\prime_3}(2\varLambda_3/\varOmega_3)$ $\delta_{m_1,m^\prime_1}\delta_{m_2,m^\prime_2}]$ in Eq. (15b) with $D_{m_i,m^\prime_i}(2\varLambda_i/\varOmega_i)$ specified with Eq. (S1) in the Supplementary Material. Hence, the transitions in the first eigen-mode channel are efficient compared to the third eigen-mode channel. It stems from the dramatic decrease of $\varOmega_1$ (shown in the inset of Fig. 3), which results in the enhancement of $J_{r_2}/J_{r_1}$. Noticeably, by tuning up the qubit-photon coupling strength to strong regime (e.g., $\lambda/\omega_a=0.5$), the intense multi-photon scattering processes significantly blockade transitions of the first eigen-mode channel, characterized as the embedded coefficient $(\varLambda_1/\omega_1)$ in $D_{m_1,m^\prime_1}(2\varLambda_1/\omega_1)$ with $\varLambda_1\,{\approx}\,\lambda\sqrt{\omega_b/\varOmega_1}$. Thus the ratio $J_{r_2}/J_{r_1}$ is expected to reduce. In contrast, in the regime $\omega_b{\gg}\omega_a$ (i.e., $\theta{\rightarrow}0$), the effective transition rates for the heat current are changed to \begin{align} \varGamma^{\pm}_{r_1,\rm{eff}}(\varDelta^{\sigma\sigma}_{\boldsymbol{m}^\prime\boldsymbol{m}})=\,& \frac{\omega_a}{2\varOmega_1}\delta_{{\boldsymbol{m},\boldsymbol{m}^\prime-\boldsymbol{I}_1}} m^\prime_1\gamma_{r_1}(\pm{\varOmega_1})n_{r_1}(\pm{\varOmega_1}),\tag{24a}\\ \varGamma^{\pm}_{r_2,\rm{eff}}(\varDelta^{\sigma\sigma}_{\boldsymbol{m}^\prime\boldsymbol{m}})=\,& \frac{\omega_b}{\varOmega_3}\delta_{{\boldsymbol{m},\boldsymbol{m}^\prime-\boldsymbol{I}_3}} m^\prime_2\gamma_{r_2}(\pm{\varOmega_3})n_{r_2}(\pm{\varOmega_3}),\tag{24b}\\ \varGamma^{\pm}_{r_3,\rm{eff}}(\varDelta^{\sigma\sigma}_{\boldsymbol{m}^\prime\boldsymbol{m}})=\,& \frac{\omega_a}{2\varOmega_1}\delta_{{\boldsymbol{m},\boldsymbol{m}^\prime-\boldsymbol{I}_1}} m^\prime_3\gamma_{r_3}(\pm{\varOmega_1})n_{r_3}(\pm{\varOmega_1}).\tag{24c} \end{align} At weak $\lambda/\omega_a$ the increasing frequency $\varOmega_3$ (see the inset of Fig. 3) reduces $J_{r_2}$, which dramatically suppresses the heat current ratio $J_{r_2}/J_{r_1}$. Also noticeably, at strong $\lambda/\omega_a$ the remarkable photon scattering processes effectively impede the energy exchange in the first eigen-mode channel, finally causing the increase of $J_{r_2}/J_{r_1}$. Therefore, we conclusion that distinct frequencies of coupled resonators is crucial to modulate the spatial concentration and decentralization of heat currents through resonators. Next, we analytically investigate cycle fluxes with weak qubit-photon interaction strength in Fig. 4(a). In brief, we re-express the quantum master equation as $d|\boldsymbol{P}{\rangle}/dt=\mathcal{L}|\boldsymbol{P}{\rangle}$ with $\mathcal{L}\,{\approx}\,\mathcal{L}^{(0)}+(\frac{\lambda}{\omega_a})^2\mathcal{L}^{(1)}$ and $|\boldsymbol{P}{\rangle}\,{\approx}\,|\boldsymbol{P}^{(0)}{\rangle}+(\frac{\lambda}{\omega_a})^2|\boldsymbol{P}^{(1)}{\rangle}$. Then, the zeroth-order steady state solution is given by $\mathcal{L}^{(0)}|\boldsymbol{P}^{(0)}{\rangle}=0$, and the first-order solution is given by $\mathcal{L}^{(0)}|\boldsymbol{P}^{(1)}{\rangle}+\mathcal{L}^{(1)}|\boldsymbol{P}^{(0)}{\rangle}=0$. Hence, the zeroth-order steady-state populations are described as \begin{align} P^{(0)}_{\boldsymbol{m},\uparrow}=\,&\frac{1}{e^{\beta_q\varepsilon}+1}\prod_i[(1-e^{-\beta_{r_i}\varOmega_i})e^{-n_i\beta_{r_i}\varOmega_i}]\tag{25a}\\ P^{(0)}_{\boldsymbol{m},\downarrow}=&\frac{e^{\beta_q\varepsilon}}{e^{\beta_q\varepsilon}+1} \prod_i[(1-e^{-\beta_{r_i}\varOmega_i})e^{-n_i\beta_{r_i}\varOmega_i}]. \tag{25b} \end{align} In particular at resonance $\varepsilon=\omega_a$, based on the expression of heat current in Eq. (19) $J_q$ is obtained as (see the Supplementary Material for the details) \begin{align} J_q\,{\approx}\,(C_{a}+C_{b}+C_{c}),~~ \tag {26} \end{align} where three cycle components are specified as \begin{align} C_{a}=\,&\sum_{i=1,3}(4\varLambda^2_i/\varOmega_i)\frac{\gamma_q(\varepsilon+\varOmega_i)}{2n_q(\varepsilon)+1} \big\{[1+n_q(\varepsilon+\varOmega_i)]n_q(\varepsilon)\notag\\ &\cdot n_{r_i}(\varOmega_i)-n_q(\varepsilon+\varOmega_i)[1+n_q(\varepsilon)]\notag\\ &\cdot[1+n_{r_i}(\varOmega_i)]\big\},
\tag{27a}\\ C_{b}=\,&(4\varLambda^2_1/\varOmega_1)\frac{\gamma_q(\varepsilon-\varOmega_1)}{2n_q(\varepsilon)+1} \big\{[1+n_q(\varepsilon)]n_{r_1}(\varOmega_1)\notag\\ &\cdot n_q(\varepsilon-\varOmega_1)-n_q(\varepsilon)[1+n_{r_1}(\varOmega_1)]\notag\\ &\cdot[1+n_q(\varepsilon-\varOmega_1)]\big\},
\tag{27b}\\ C_{c}=\,&(4\varLambda^2_3/\varOmega_3)\frac{\gamma_q(\varOmega_3-\varepsilon)}{2n_q(\varepsilon)+1} \big\{[1+n_q(\varepsilon)]n_{r_3}(\varOmega_3)\notag\\ &[1+n_q(\varOmega_3-\varepsilon)]-n_q(\varepsilon)[1+n_{r_3}(\varOmega_3)]\notag\\ &\cdot n_q(\varOmega_3-\varepsilon)\big\}.
\tag{27c} \end{align} Specifically, the component $C_a$ includes the transition loop $|\psi^{\uparrow}_{\boldsymbol{m}}{\rangle}{\rightarrow}| \psi^{\downarrow}_{{ \boldsymbol{m}-\boldsymbol{I}_i}}{\rangle}{\rightarrow} |\psi^{\uparrow}_{{\boldsymbol{m}-\boldsymbol{I}_i}}{\rangle}{\rightarrow}| \psi^\uparrow_{\boldsymbol{m}}{\rangle}$ and the counter loop $|\psi^{\uparrow}_{\boldsymbol{m}}{\rangle}{\rightarrow}|\psi^{\uparrow}_{{\boldsymbol{m}-\boldsymbol{I}_i}}{\rangle}{\rightarrow} |\psi^{\downarrow}_{{ \boldsymbol{m}-\boldsymbol{I}_i}}{\rangle}{\rightarrow}|\psi^{\uparrow}_{\boldsymbol{m}}{\rangle}$. $C_b$ is composed of one pair of cycle transitions involved with the $r_1$-th reservoir: $|\psi^{\uparrow}_{\boldsymbol{m}}{\rangle}{\rightarrow}|\psi^{\downarrow}_{\boldsymbol{m}}{\rangle}{\rightarrow} |\psi^{\downarrow}_{{ \boldsymbol{m}+\boldsymbol{I}_1}}{\rangle}{\rightarrow}|\psi^{\uparrow}_{\boldsymbol{m}}{\rangle}$ and the counter-clockwise case. Here, $C_c$ consists of two cycle transitions involved with the $r_3$-th reservoir: $|\psi^{\uparrow}_{\boldsymbol{m}}{\rangle}{\rightarrow}|\psi^{\downarrow}_{\boldsymbol{m}}{\rangle}{\rightarrow} |\psi^{\downarrow}_{{ \boldsymbol{m}+\boldsymbol{I}_3}}{\rangle}{\rightarrow}|\psi^{\uparrow}_{\boldsymbol{m}}{\rangle}$ and the counter-clockwise one. We note that the positive terms of these three current components denote energy transferred from the $r_i$-th reservoir into the $q$-th reservoir, whereas the negative terms represent the energy left away from the $q$-th reservoir. Due to the relation $\varLambda^2_1/{\varOmega_1}{\gg}\varLambda^2_3/{\varOmega_3}$, the contribution of the third eigen-mode channel to the heat current becomes negligible, compared to the first eigen-mode channel, which further reduces the current to $J_q\,{\approx}\,(C_a+C_b)$. Hence, as the bath temperature bias becomes dramatic, e.g., $T_r\,{\approx}\,2\omega_a$ and $T_q\,{\approx}\,0$, all of transition loops in current components $C_a$ and $C_b$ are significantly suppressed, which originates from the inability of qubit excitation assisted by the $q$-th reservoir, i.e., $n_q(\omega>0)\,{\approx}\,0$. Therefore, this directly leads to the emergence of negative differential thermal conductance,[50,51] where the heat current is reduced with the increase of the temperature bias. Moreover, we also investigate the influence of the temperature bias on heat current at strong qubit-photon coupling in Fig. 4(b). In contrast, the current $J_q$ exhibits monotonic enhancement with the increase of ${\Delta}T$. The coherent-state overlap coefficient beyond weak coupling is approximated as \begin{align} D_{m_im^\prime_i}(2\varLambda_i/\varOmega_i)\,{\approx}\,&(-1)^{m_i}\Big\{[1-(m_i+1/2) (2\varLambda_i/\varOmega_i)^2]\delta_{m_i,m^\prime_i}\\ &{}+(2\varLambda_i/\varOmega_i)(\sqrt{m_i+1}\delta_{m_i,m^\prime_i-1}-\sqrt{m_i} \delta_{m_i,m^\prime_i+1})\\ &{}+2(\varLambda_i/\varOmega_i)^2\big[\sqrt{m_i(m_i-1)} \delta_{m_i,m^\prime_i+2}+\sqrt{(m_i+1)(m_i+2)}\delta_{m_i,m^\prime_i-2}\big]\Big\}.\end{align} Thus, one additional transition channel is open, e.g., $|\psi^\downarrow_{{\boldsymbol{m}+2\boldsymbol{I}_1}}{\rangle}{\rightarrow}|\psi^\uparrow_{\boldsymbol{m}}{\rangle}$, which is robust even at large temperature bias. This may dramatically enhance the heat current with strong qubit-photon interaction strength. Therefore, weak qubit-photon interaction is important to exhibit the cycle heat currents and NDTC effect.
cpl-40-5-050501-fig4.png
Fig. 4. (a) The heat current $J_q$ and cycle current components $C_{i}$ $(i=a,b,c)$ at weak qubit-photon coupling strength $\lambda=0.01\omega_a$ by tuning the temperature bias $\Delta{T}=T_r-T_q$. (b) The current $J_q$ at strong qubit-photon coupling by both tuning $\Delta{T}$ and $\lambda$. Other system parameters are given by $\varepsilon=\omega_a$, $t=0.1\omega_a$, $\alpha_{r_i}=\alpha_q=10^{-3}\omega_a$, $\omega_c=20\omega_a$, $k_{\scriptscriptstyle{\rm B}}T_{r}=\omega_a+k_{\scriptscriptstyle{\rm B}}{\Delta}T/2$, and $k_{\scriptscriptstyle{\rm B}}T_{q}=\omega_a-k_{\scriptscriptstyle{\rm B}}{\Delta}T/2$.
Compared to the two-mode qubit-photon model,[45] the cycle current components of three-mode case are richer due to multiple coupled resonators configuration. Particularly, one additional cycle current component, i.e., $C_b$ in Eq. (27b) is unraveled compared to the two-mode case, which is crucial to heat flows and dominated via the first eigen-mode channel. However, both of them exhibit one dark mode with eigen-mode energy identical with the bare resonator (7b) and the vanishing renewed qubit-photon coupling (8b).
cpl-40-5-050501-fig5.png
Fig. 5. (a) Mean-field photon excitation parameter $|a|=|{\langle}\hat{a}{\rangle}|$ in Eq. (10). (b) The steady-state heat current $J_q$ with various photon-photon hopping and qubit-photon coupling strengths. Other system parameters are given as $\varepsilon=\omega_a$, $\alpha_{r_i}=\alpha_q=10^{-3}\omega_a$, $\omega_c=20\omega_a$, $k_{\scriptscriptstyle{\rm B}}T_{r}=1.5\omega_a$, and $k_{\scriptscriptstyle{\rm B}}T_{q}=0.5\omega_a$.
One-Dimensional Case. We study the quantum heat transport in the one-dimensional qubit-photon case under the mean-field framework. In the absence of $t$ such a one-dimensional model is simplified to the single-mode qubit-photon case.[53] By tuning up $t$ one can find that the mean-field qubit-photon model is also analogous with the single-mode one, except for the effective qubit-photon coupling strength, which is drifted by mean-field photon excitation parameter $a$ (i.e., $\lambda_\uparrow=\lambda-2at$ and $\lambda_\downarrow=-\lambda-2at$). In Fig. 5(a) we first plot the mean-field parameter by tuning both the photon hopping and qubit-photon coupling strengths. It is found that $|a|$ grows dramatically with the increase of qubit-photon coupling strength, and can be further enhanced via raising the photon hopping strength. This parameter $a$ is expected to modify the behavior of the steady-state heat current. Then, we plot Fig. 5(b) to analyze $J_q$ with various photon hopping strengths. It is shown that at weak qubit-photon coupling the current with finite hopping strength (even with weak hopping case, e.g., $t=0.01\omega_a$) is comparative enhanced to the counterpart with $t=0$. This can be understood from the analytical expression of the heat current \begin{align} J_q=C_a+C_b,~~ \tag {28} \end{align} with two cycle components \begin{align} C_a=\,&\Big(\frac{2\lambda}{\omega_a}\Big)^2\frac{{\gamma_q}(\tilde{\varepsilon}+\omega_a)}{2n_q(\tilde{\varepsilon})+1} \big\{1+n_q(\tilde{\varepsilon}+\omega_a)n_q(\tilde{\varepsilon})n_r(\omega_a)\notag\\ &-n_q(\tilde{\varepsilon}+\omega_a)[1+n_q(\tilde{\varepsilon})][1+n_r(\omega_a)]\big\}\tag{29a}\\ C_b=\,&\Big(\frac{2\lambda}{\omega_a}\Big)^2\frac{\gamma_q(\tilde{\varepsilon}-\omega_a)}{2n_q(\tilde{\varepsilon})+1} \big\{n_q(\tilde{\varepsilon}-\omega_a)[1+n_q(\tilde{\varepsilon})]n_r(\omega_a)\notag\\ &-[1+n_q(\tilde{\varepsilon}-\omega_a)]n_q(\tilde{\varepsilon})[1+n_r(\omega_a)]\big\}, \tag{29b} \end{align} which can be analogously obtained with the single-mode qubit-photon model in Ref. [53], by including the effective qubit splitting energy $\tilde{\varepsilon}{\equiv}(\varepsilon/2-2\lambda^2_\uparrow/\omega_a)-(-\varepsilon/2-2\lambda^2_\downarrow/\omega_a)=\varepsilon+8at\lambda/\omega_a$ in Eq. (11). The inclusion of additional component $I_b$ strengthens $J_q$, compared to the single-mode limit $J_q(t=0)=C_a$ with $\tilde{\varepsilon}$ replaced by $\varepsilon$. Therefore, the photon hopping is preferred to optimize the heat current with weak qubit-photon coupling. However, at strong qubit-photon coupling it is shown that the inclusion of the photon hopping is detrimental to $J_q$, i.e., $J_q$ is monotonically suppressed by increasing $t$. From Eq. (18b), one knows that $\varGamma^{\pm}_q(\varDelta^{\overline{\sigma}\sigma}_{m^\prime{m}})$ tightly relies on $t$ and $\lambda$, which are characterized by the energy gap $\varDelta^{\overline{\sigma}\sigma}_{m^\prime{m}}$ and coherent-state overlap coefficient $D_{m^\prime{m}}(2\lambda/\omega)$. The existence of the photon hopping generally makes the transition between two photonic coherent states only allowed with higher photon number difference than the case in the absence of $t$, because the transition should fulfill the positive gap, e.g., $\varDelta^{\downarrow\uparrow}_{m^\prime{m}}=[\omega_a(m^\prime-m)-8at\lambda/\omega_a]>0$. Hence, this may dramatically inhibit the effective transitions, mainly controlled by $D_{m^\prime{m}}(2\lambda/\omega)$. In summary, we have investigated steady-state heat transport in multi-mode qubit-photon systems, which are exemplified as the three-mode and mean-field one-dimensional cases. For the three-mode case with identical resonators, the heat current through the second resonator is exactly as twice as the other two resonators by including the inter-resonator photon hopping, in sharp contrast with the individual resonators case. In the eigen-mode picture, the main mechanism stems from the negligible contribution of second eigen-mode channel to the heat current and the accurate match of the transition rates in Eqs. (22a)–(22c). Moreover, typical cycle transition components of the heat current, Eqs. (27a)–(27c), are analytically unraveled at weak qubit-photon coupling, which is crucial to the emergence of NDTC effect. The influence of distinct resonators on the heat current is also analyzed, which dramatically modulates the enhancement and suppression of the heat current into the central resonator, compared to the currents of the two edge resonators. The efficient transition and blockade in the first eigen-mode are the key ingredients to modulate heat transport at weak and strong qubit-photon couplings, respectively. We also investigate heat current in the one-dimensional case under the mean-field framework, where the mean-field photon excitation parameter is shown to be indispensable, particularly at qubit-photon coupling strength. Compared to the single-mode qubit-photon model, the heat current at weak qubit-photon coupling is dramatically enhanced, mainly resulting from the contribution of additional cycle flux component. Our study may fertilize possible routes for the realization of nonequilibrium heat transport and thermal operations in qubit-photon hybrid quantum systems. Acknowledgement. This work was supported by the National Natural Science Foundation of China (Grant No. 11704093), the Opening Project of Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology, the Jiangsu Key Disciplines of the Fourteenth Five-Year Plan (Grant No. 2021135), the National Natural Science Foundation of China (Grant No. 12174345), and Zhejiang Provincial Natural Science Foundation of China (Grant No. LZ22A040002). References Reciprocal Relations in Irreversible Processes. I.Reciprocal Relations in Irreversible Processes. 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