A polaron theory of quantum thermal transistor in nonequilibrium three-level systems

Chen Wang; Da-Zhi Xu

doi:10.1088/1674-1056/ab973b

1. Introduction

According to the Clausius statement, [1] the heat flow from the hot source to the cold drain naturally occurs driven by the thermodynamic bias. Without violating this fundamental law of thermodynamics, great efforts have been paid to find other ways to conduct the heat flow. [2 - 8] Accompanying with the rapid progress in quantum technology, the control of heat flow becomes an increasingly important issue in quantum computation [9] and quantum measurement. [10, 11]

The thermal transistor, one of the novel phenomena in quantum thermal transport, was initially proposed by B. Li and coworkers. [12, 13] In particular, heat amplification and negative differential thermal conductance (NDTC) are considered as two main components of the thermal transistor. Heat amplification describes an effect within the three-terminal setup, that the tiny modification of the base current will dramatically change the current at the collector and emitter, which enables the efficient energy transport. [12] While the NDTC effect is characterized by the suppression of the heat flow with increase of temperature bias within the two-terminal setup. [14 - 16]

Later, the spin-based fully quantum thermal transistor was proposed by K. Joulain et al., [17] which is composed by three coupled-qubits, each interacting with one individual thermal bath. The qubit-qubit interaction within the system is found to be crucial to exhibit the heat amplification. Consequently, the importance of the system nonlinearity and long-range interaction on the transistor effect has also been unraveled in various coupled-qubits systems. [18 - 20] Simultaneously, the strong qubit-bath interaction is revealed to be another key factor to cause giant heat amplification in the two- and three-qubits systems, [21, 22] which also stems from the NDTC effect. Hence, NDTC is widely accepted as the crucial intergradient to realize the giant heat amplification. However, J. H. Jiang et al. pointed out that heat amplification can be realized via the inelastic transport process independent of NDTC. [23] Hence, basic questions are raised: Can these two types of heat amplification coexist in one quantum system? What is the underlying microscopic mechanism?

Very recently, a three-level quantum heat transistor was preliminarily investigated by S. H. Su et al., which stressed the significant influence of quantum coherence on the heat amplification. [24] However, the calculations are based on the phenomenological Lindblad equation, which cannot be generalized to describe the heat transport beyond the weak system-bath coupling. Considering the scientific importance and extensive application of the nonequilibrium three-level systems, [25 - 33] it is intriguing to give a comprehensive picture of the heat transistor behavior. In particular, it is worthwhile to give a unified analysis on the effect of the system-bath interaction from the weak to strong coupling regimes on the heat amplification and NDTC.

In this paper, we devote to investigating quantum thermal transport in a quantum thermal transistor, which is composed by a three-level quantum system within the three-terminal setup (Fig. 1(a)) by applying the polaron-transformed Redfield equation (PTRE), detailed in Section 2 . In Section 3, the heat currents are obtained from PTRE combined with full counting statistics (FCS), [34, 35] and are reduced to the Redfield and nonequilibrium noninteracting blip approximation (NIBA) schemes. [36 - 39] In Subsection 4.1, the giant heat amplification is explored both with moderate and strong system-middle bath interaction strengths, and the finite heat amplification with inelastic-like process is analytically estimated in the weak system-middle bath coupling regime. The corresponding microscopic mechanisms are proposed. In Subsection 4.2, the negative differential thermal conductance is analyzed based on the two-terminal setup, where the cooperative contribution from thermal baths to exhibit NDTC is revealed to be crucial. Moreover, the NDTC is found at moderate system-middle bath coupling, which can not be explained by the Redfield equation. Finally, we give a concise summary in Section 5 .

Figure 1.Schematics of the nonequilbrium V-type three-level system in (a) the original framework and (b) the polaron framework. The three horizontal solid black lines represent the central three-level model (|u〉, u = l,r,0), and the double-arrowed solid brown line shows the coherent tunneling between two excited states |l〉 and |r〉; the rectangular left red, top-middle purple, and right blue boxes describe three thermal baths, which are characterized by temperatures T_l, T_m, and T_r, respectively; the double-arrowed solid red, purple, and blue curves describe interactions between the system and thermal baths, and the double-arrowed dashed black lines describe transitions between different states assisted by phonons in the corresponding thermal bath.

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2. Model and method

We first introduce the nonequilibrium three-level system and the framework of polaron transformation. Then, the polaron-transformed Redfield equation is applied to obtain the reduced system density matrix.

2.1. Nonequilibrium three-level system

\hat{H} = {\hat{H}}_{s} + \sum_{u = l, r, m} ({\hat{H}}_{b}^{u} + {\hat{V}}_{u})

$$ \begin{eqnarray}{\hat{H}}_{s}={\varepsilon }_{0}|0\rangle \langle 0|+\displaystyle \sum _{u=l,r}{\varepsilon }_{u}|u\rangle \langle u|+\varDelta (|l\rangle \langle r|+{\rm{H}}.{\rm{c}}.),\end{eqnarray}$$ (1)

where |l(r)〉 is the left (right) excited state with the occupation energy ε_l(r), and |0〉 is the ground state with ε₀ = 0 for simplicity. Specifically, for ε_l(r) > 0, the three-level system corresponds to a V-type configuration. Whereas for ε_l(r) < 0, it corresponds to a λ-type configuration with excited state |0〉 and ground states |l〉 and |r〉. It should be noted that the notions of V-type and λ-type have already been used in quantum optics and quantum thermodynamics.^[30,40,41] In the following, our work is based on the V-type system without losing any generality.

{\hat{H}}_{b}^{u} = \sum_{k} ω_{k} {\hat{b}}_{k, u}^{†} {\hat{b}}_{k, u}

$$ \begin{eqnarray}{\hat{V}}_{u}=({\hat{S}}_{u}^{\dagger }+{\rm{H}}.{\rm{c}}.)\displaystyle \sum _{k}({g}_{k,u}{\hat{b}}_{k,u}^{\dagger }+{\rm{H}}.{\rm{c}}.),\,\,u=l,r,\end{eqnarray}$$ (2)

with

{\hat{S}}_{u}^{†} = | u 〉 〈 0 |

. Here, the state |0〉 only interacts with the left and right baths. While the quantum dissipation of the two excited states induced by the middle bath is modeled by a diagonal interaction,^[24,28,29] which reads

$$ \begin{eqnarray}{\hat{V}}_{m}=(|l\rangle \langle l|-|r\rangle \langle r|)\displaystyle \sum _{k}({g}_{k,m}{\hat{b}}_{k,m}^{\dagger }+{\rm{H}}.{\rm{c}}.),\end{eqnarray}$$ (3)

where g_k,u is the system–bath coupling strength. The u-th thermal bath here is characterized by the spectral function Λ_u(x) = 4πΣ_k|g_k,u|²δ(x – ω_k), Here, the spectral functions are selected to have the super-Ohmic form

Λ_{l (r)} (x) = π γ_{l (r)} \frac{x^{3}}{ω_{c}^{2}} e^{- | x | / ω_{c}}

and

Λ_{m} (x) = π α_{m} \frac{x^{3}}{ω_{c}^{2}} e^{- | x | / ω_{c}}

, where γ_l(r) and α_m are the coupling strengths corresponding to the l(r)-th bath and the m-th bath, and the cut-off frequency is ω_c. The super-Ohmic bath has been extensively included to investigate quantum dissipation^[42,43] and quantum energy transport.^[44–46]

{\hat{H}}^{'} = {\hat{U}}^{†} \hat{H} \hat{U}

$$ \begin{eqnarray}{\hat{H}}_{s}^{^{\prime} }=\bar{\varepsilon }\hat{N}+\delta \varepsilon {\hat{\sigma }}_{z}+\eta \varDelta {\hat{\sigma }}_{x},\end{eqnarray}$$ (4)

with the average occupation energy

\bar{ε} = (ε_{l} + ε_{r}) / 2 - \sum_{k} | g_{k, m} |^{2} / ω_{k}

, the energy bias δε = (ε_l – ε_r)/2, the excitation number operator

\hat{N} = | l 〉 〈 l | + | r 〉 〈 r |

, the bias operator

{\hat{σ}}_{z} = | l 〉 〈 l | - | r 〉 〈 r |

, and the tunneling operator

{\hat{σ}}_{x} = | l 〉 〈 r | + | r 〉 〈 l |

. Here

\hat{N}

can be re-expressed as

\hat{N} = \hat{I} - | 0 〉 〈 0 |

with

\hat{I}

the unit operator in the space of the three-level system. It should be noted that the operators

{\hat{σ}}_{a} (a = x, y, z)

are not Pauli operators for

{\hat{σ}}_{a}^{2} \neq \hat{I}

. The renormalization factor

η = {Tr}_{m} {{\hat{ρ}}_{b}^{m} e^{\pm 2 i \hat{B}}} / {Tr}_{m} {{\hat{ρ}}_{b}^{m}}

, with

{Tr}_{m} {}

the trace over the middle bath,

{\hat{ρ}}_{b}^{m} = e^{- {\hat{H}}_{b}^{m} / k_{B} T_{m}}

thermal equilibrium state, and T_m the temperature of the middle bath. Then, the transformed system Hamiltonian

{\hat{H}}_{s}^{'}

can be exactly solved as

{\hat{H}}_{s}^{'} | \pm 〉_{η} = E_{\pm} | \pm 〉_{η}

, where the eigenstates are

| + 〉_{η} = \cos \frac{θ}{2} | l 〉 + \sin \frac{θ}{2} | r 〉

and

| - 〉_{η} = - \sin \frac{θ}{2} | l 〉 + \cos \frac{θ}{2} | r 〉

, with

\tan θ = η Δ / δ ε

and the eigenvalues

E_{\pm} = \bar{ε} \pm \sqrt{{(δ ε)}^{2} + {(η Δ)}^{2}}

. The modified system-bath interactions are given by^[34,35]

$$ \begin{eqnarray}{\hat{V}}_{m}^{^{\prime} }=\varDelta [\cos (2\hat{B})-\eta ]{\hat{\sigma }}_{x}+\varDelta \sin (2\hat{B}){\hat{\sigma }}_{y},\,\end{eqnarray}$$ (5a)

$$ \begin{eqnarray}{\hat{V}}_{u}^{^{\prime} }=({{\rm{e}}}^{-{\rm{i}}{\hat{B}}_{u}}{\hat{S}}_{u}^{\dagger }+{{\rm{e}}}^{{\rm{i}}{\hat{B}}_{u}}{\hat{S}}_{u})\displaystyle \sum _{k}({g}_{k,u}{\hat{b}}_{k,u}^{\dagger }+{g}_{k,u}^{* }{\hat{b}}_{k,u})\,\,\,u=l,r,\end{eqnarray}$$ (5b)

with

{\hat{σ}}_{y} = - i (| l 〉 〈 r | - | r 〉 〈 l |)

{\hat{B}}_{l} = \hat{B}

, and

{\hat{B}}_{r} = - \hat{B}

. It should be noted that the factor η results from the renormalization of the dressed coherent tunneling between the two excited states, i.e.,

Δ {\hat{U}}^{†} {\hat{σ}}_{x} \hat{U} = η Δ {\hat{σ}}_{x} + {\hat{V}}_{m}^{'}

, which makes the thermal average of

{\hat{V}}_{m}^{'}

vanish for arbitrary system–middle bath coupling strength.^[34,43]

{\hat{V}}_{u}^{'} (u = l, m, r)

2.2. Polaron-transformed Redfield equation

{Tr}_{m} {{\hat{ρ}}_{b}^{m} {\hat{V}}_{m}^{'}} / {Tr}_{m} {{\hat{ρ}}_{b}^{m}}

$$ \begin{eqnarray}\displaystyle \frac{{\rm{d}}}{{\rm{d}}t}{\hat{\rho }}_{s}=-{\rm{i}}[{\hat{H}}_{s}^{^{\prime} },{\hat{\rho }}_{s}]+\displaystyle \sum _{u=l,m,r}{ {\mathcal L} }_{u}[{\hat{\rho }}_{s}],\end{eqnarray}$$ (6)

where

{\hat{ρ}}_{s}

is the density operator of the three-level system. The m-th dissipator is specified as^[34,35]

$$ \begin{eqnarray}\begin{array}{ll}{ {\mathcal L} }_{m}[{\hat{\rho }}_{s}]= & \displaystyle \sum _{\alpha =x,y;\omega,{\omega }^{^{\prime} }}{\gamma }_{\alpha }({\omega }^{^{\prime} })[{\hat{P}}_{\alpha }({\omega }^{^{\prime} }){\hat{\rho }}_{s}{\hat{P}}_{\alpha }(\omega )\\ & -{\hat{P}}_{\alpha }(\omega ){\hat{P}}_{\alpha }({\omega }^{^{\prime} }){\hat{\rho }}_{s}]+{\rm{H}}.{\rm{c}}.,\end{array}\end{eqnarray}$$ (7)

where the dissipation rates between the two excited eigenstates are

$$ \begin{eqnarray}{\gamma }_{x}(\omega )={\eta }^{2}{\varDelta }^{2}\displaystyle {\int }_{0}^{\infty }{\rm{d}}\tau {{\rm{e}}}^{{\rm{i}}\omega \tau }[\cosh {\phi }_{m}(\tau )-1],\,\end{eqnarray}$$ (8a)

$$ \begin{eqnarray}{\gamma }_{y}(\omega )={\eta }^{2}{\varDelta }^{2}\displaystyle {\int }_{0}^{\infty }{\rm{d}}\tau {{\rm{e}}}^{{\rm{i}}\omega \tau }\sinh {\phi }_{m}(\tau ),\,\end{eqnarray}$$ (8b)

with the correlation phase

$$ \begin{eqnarray}{\phi }_{m}(\tau )=4\displaystyle \sum _{k}|\displaystyle \frac{{g}_{k,m}}{{\omega }_{k}}{|}^{2}\{\cos ({\omega }_{k}\tau )[2{n}_{m}({\omega }_{k})+1]-{\rm{i}}\sin ({\omega }_{k}\tau )\}.\end{eqnarray}$$ ()

The operators

{\hat{P}}_{α} (ω) (α = x, y)

are the projective operators of the system eigenbasis, which are defined by

{\hat{σ}}_{α} (- τ) = \sum_{ω} {\hat{P}}_{α} (ω) e^{i ω τ}

with

{\hat{P}}_{α} (- ω) = {\hat{P}}_{α}^{†} (ω)

.^[47] The rate γ_y(ω) describes the transition between the two excited eigenstates |±〉_η involving odd number of phonons from the middle thermal bath. The bath average phonon number is n_u(ω) = 1/[exp(ω/T_u) – 1], u = r,l,m, with T_u the temperature of the u-th bath. The approximated expression of the real part of γ_y(ω) to the first-order of ϕ_m reads

Re [γ_{y} (ω)] \approx 4 π η^{2} Δ^{2} \sum_{k} | \frac{g_{k, m}}{ω_{k}} |^{2} [n_{m} (ω_{k}) + 1] δ (ω - ω_{k})

, which contains the sequential process of creating one phonon with frequency ω = ω_k in the m-th bath. A direct consequence of the polaron transformation is that the dissipative rates γ_x(ω) and γ_y(ω) contain all the high-order terms of ϕ_m, which can be understood as the contribution of the multiple-phonon correlation. In the strong system–bath coupling strength regime, such high-order correlations should be properly incorporated in the evolution of the open quantum system.

Moreover, the dissipators associated with the left and right baths are given by

$$ \begin{eqnarray}\begin{array}{ll}{ {\mathcal L} }_{u}[{\hat{\rho }}_{s}]= & \displaystyle \sum _{\omega,{\omega }^{^{\prime} }}[{\kappa }_{u,-}({\omega }^{^{\prime} }){\hat{Q}}_{u}({\omega }^{^{\prime} }){\hat{\rho }}_{s}{\hat{Q}}_{u}^{\dagger }(\omega )\\ & +{\kappa }_{u,+}({\omega }^{^{\prime} }){\hat{Q}}_{u}^{\dagger }({\omega }^{^{\prime} }){\hat{\rho }}_{s}{\hat{Q}}_{u}(\omega )\\ & -{\kappa }_{u,+}({\omega }^{^{\prime} }){\hat{Q}}_{u}(\omega ){\hat{Q}}_{u}^{\dagger }({\omega }^{^{\prime} }){\hat{\rho }}_{s}\\ & -{\kappa }_{u,-}({\omega }^{^{\prime} }){\hat{Q}}_{u}^{\dagger }(\omega ){\hat{Q}}_{u}({\omega }^{^{\prime} }){\hat{\rho }}_{s}]+{\rm{H}}.{\rm{c}}.,\end{array}\,\end{eqnarray}$$ (9)

where the system part operators are defined by

{\hat{S}}_{u} (- τ) = \sum_{ω} {\hat{Q}}_{u} (ω) e^{i ω τ}

,^[operatorsu] and the dissipation rates are

$$ \begin{eqnarray}{\kappa }_{u,+}(\omega )=\displaystyle {\int }_{-\infty }^{\infty }\displaystyle \frac{{\rm{d}}{\omega }_{1}}{4\pi }{\varLambda }_{u}({\omega }_{1}){n}_{u}({\omega }_{1}){C}_{u}({\omega }_{1}-\omega ),\,\end{eqnarray}$$ (10a)

$$ \begin{eqnarray}{\kappa }_{u,-}(\omega )=\displaystyle {\int }_{-\infty }^{\infty }\displaystyle \frac{{\rm{d}}{\omega }_{1}}{4\pi }{\varLambda }_{u}({\omega }_{1})[1+{n}_{u}({\omega }_{1})]{C}_{u}(-{\omega }_{1}+\omega ).\,\end{eqnarray}$$ (10b)

The phonon correlation function above is shown as

C_{u} (ω) = η_{u}^{2} \int_{0}^{\infty} d τ e^{i ω τ} e^{ϕ_{m} (τ) / 4}

, where

η_{u} = {Tr}_{u} {{\hat{ρ}}_{u} e^{i \hat{B}}} / {Tr}_{u} {{\hat{ρ}}_{u}}

, with Tr_u{} the trace over the space of the u-th bath and

{\hat{ρ}}_{u} = e^{- {\hat{H}}_{b}^{u} / k_{B} T_{u}}

. The transition rates in Eqs. (10a) and (10b) demonstrate the joint contribution of the left (right) and the middle thermal baths on the nonequilibrium energy exchange. Specifically, κ_l(r),+(ω) describes the process that one phonon with the frequency ω₁ is emitted from the l(r)-th thermal bath to assist the excitation from |0〉 to the eigenstate with energy gap ω, and the resultant energy ω₁ – ω > 0(ω₁ – ω < 0) is released (absorbed) into (from) the m-th bath. While the rate κ_l(r), – (ω) shows the transition that one phonon with frequency ω₁ is absorbed by the l(r)-th thermal bath, and the three-level system is relaxed from the excited state with energy ω into |0〉.

For quantum heat transfer in the nonequilibrium spin-boson model, the polaron transformation and NIBA scheme was firstly proposed by D. Segal and A. Nitzan, [36, 37] where the NDTC was unraveled at strong system-bath coupling under the Marcus approximation. Moreover, D. Segal et al. combined the NIBA scheme with full counting statistics to investigate the heat current flucutuation. [38, 39] Later, inspired by these works, the Redfield equation and NIBA limits were unified by introducing the PTRE embedded with full counting statistics. [34, 35] It should be noted that in this work the perturbation treatment of the system-middle bath interaction by the polaron transformation is the same as done in previous works for the nonequilibrium spin-boson model, which makes it proper to arbitrarily tune the system-middle bath coupling strength.

However, for the dissipator in Eq. (9), it is found that though the system-left (right) bath coupling is weak, the energy exchange reflected by the transition rates in Eqs. (10a) and (10b) is cooperatively contributed by the left (right) bath and the middle bath. Such joint exchange process is apparently different from the sequential process in the nonequilibrium spin-boson model as treated by the Redfield method. [34, 37] Therefore, we nontrivially extend the application of the polaron-transformed Redfield equation from the nonequilibrium spin-boson model [34, 35] to the nonequilibrium three-level quantum system. This is the main technical point in the present work.

3. Steady state heat currents

We analyze steady state behaviors of heat currents via full counting statistics [33, 49] by tuning the system-middle bath interaction in Fig. 2 . The detail derivation of quantum master equation combined with full counting statistics and expressions of heat currents can be found in Appendix A. It is found that the currents are all significantly enhanced in the moderate coupling regime around α m \in(0.5,2), but are dramatically suppressed in both the strong and weak coupling limits. Moreover, the current flowing into the middle bath is more sensitive in response to the change of α m . It can be seen that J m begins to increase significantly when α m is around 0.01, which is one order smaller than those of the other two currents. It should be noted that though the heat currents are seemly negligible in the weak system-middle bath coupling regime, they are actually nonzero (e.g., J l / γ = -0.00295 with α m = 0.001).

Figure 2.Steady state heat currents (a) J_l/γ, (b) J_m/γ, and (c) J_r/γ as a function of the coupling strength α_m. The red circles are based on the Redfield scheme; the blue squares are based on the nonequilibrium noninteracting blip approximation (NIBA); the black solid line is calculated from the nonequilibrium polaron-transformed Redfield approach. The other parameters are given as ε_l = 1.0, ε_r = 0.6, Δ = 0.6, γ = 0.0002, ω_c = 10, T_l = 2, T_m = 1.2, and T_r = 0.4.

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The PTRE combined with FCS has been successfully introduced to investigate quantum thermal transport in the nonequilibrium spin-boson systems, [34, 35, 45] which is able to fully bridge the strong and weak system-bath coupling limits. While for the nonequilibrium three-level quantum system, it should admit that it is generally difficult to analytically obtain the expressions of steady state heat currents with arbitrary coupling strength α m between two excited states and middle thermal bath. However, in the strong and weak coupling limits the NIBA approach and Redfield equation can adequately describe the steady state of the system, respectively.

3.1. Strong coupling limit

In the strong coupling limit, the renormalization factor is dramatically suppressed, i.e., η ≪1. The eigenstates are reduced to the localized ones |+〉 η \approx | l 〉, |-〉 η \approx | r 〉, and the renormalized energies become E u = (ε u - Σ k | g k,m | 2 / ω k), u = l, r . As the renormalization energy Σ k | g k,m | 2 / ω k exceeds the bare energy ε u, the configuration of the transformed three-level system becomes the Λ -type as shown in Fig. 3(a) . Hence, the nonequilibrium PTRE is reduced to the NIBA scheme, which can be analytically solved (the details are given in Appendix B). Accordingly, the heat currents are explicitly given by

$$ \begin{eqnarray}\begin{array}{ll}{J}_{l,{\rm{NIBA}}}= & \displaystyle \frac{1}{{\mathscr{A}}}[({G}_{l}^{+}{G}_{m}^{+}{G}_{r}^{-}{\langle \omega \rangle }_{l,+}-{G}_{l}^{-}{G}_{m}^{-}{G}_{r}^{+}{\langle \omega \rangle }_{l,-})\\ & +({G}_{m}^{-}+{G}_{r}^{-}){G}_{l}^{-}{G}_{l}^{+}({\langle \omega \rangle }_{l,+}-{\langle \omega \rangle }_{l,-})],\end{array}\end{eqnarray}$$ (11)

$$ \begin{eqnarray}\begin{array}{ll}{J}_{r,{\rm{NIBA}}}= & \displaystyle \frac{1}{{\mathscr{A}}}[({G}_{l}^{-}{G}_{m}^{-}{G}_{r}^{+}{\langle \omega \rangle }_{r,+}-{G}_{l}^{+}{G}_{m}^{+}{G}_{r}^{-}{\langle \omega \rangle }_{r,-})\\ & +({G}_{m}^{+}+{G}_{l}^{-}){G}_{r}^{+}{G}_{r}^{-}({\langle \omega \rangle }_{r,+}-{\langle \omega \rangle }_{r,-})],\end{array}\end{eqnarray}$$ (12)

and J_m = –J_l – J_r, where the coefficient is

A = (G_{m}^{+} + G_{m}^{-}) (G_{l}^{+} + G_{r}^{+}) + G_{m}^{+} G_{r}^{-} + G_{m}^{-} G_{l}^{-} + G_{l}^{-} G_{r}^{+} + G_{r}^{-} (G_{l}^{+} + G_{l}^{-})

, the transition rates are

$$ \begin{eqnarray}{G}_{m}^{\pm }=\displaystyle {\int }_{-\infty }^{\infty }{\rm{d}}\tau {{\rm{e}}}^{\pm {\rm{i}}({\varepsilon }_{l}-{\varepsilon }_{r})\tau }{\eta }^{2}{{\rm{e}}}^{{\phi }_{m}(\tau )}\,\end{eqnarray}$$ (13a)

$$ \begin{eqnarray}\begin{array}{ll}{G}_{u}^{+}= & \displaystyle \frac{1}{4\pi }\displaystyle {\int }_{-\infty }^{\infty }{\rm{d}}{\omega }_{1}{\varLambda }_{u}({\omega }_{1})[1+{n}_{u}({\omega }_{1})]\\ & \times [{C}_{u}(-{E}_{u}-{\omega }_{1})+{\rm{H}}.{\rm{c}}.]\,\end{array}\end{eqnarray}$$ (13b)

$$ \begin{eqnarray}{G}_{u}^{-}=\displaystyle \frac{1}{4\pi }\displaystyle {\int }_{-\infty }^{\infty }{\rm{d}}{\omega }_{1}{\varLambda }_{u}({\omega }_{1}){n}_{u}({\omega }_{1})[{C}_{u}({\omega }_{1}+{E}_{u})+{\rm{H}}.{\rm{c}}.]\,\end{eqnarray}$$ (13c)

and the average energies into the u-th thermal bath are

$$ \begin{eqnarray}\begin{array}{ll}{\langle \omega \rangle }_{u,+}= & \displaystyle \frac{1}{4\pi {G}_{u}^{+}}\displaystyle {\int }_{-\infty }^{\infty }{\rm{d}}{\omega }_{1}{\omega }_{1}{\Lambda }_{u}({\omega }_{1})[1+{n}_{u}({\omega }_{1})]\\ & \times [{C}_{u}(-{E}_{u}-{\omega }_{1})+{\rm{H}}.{\rm{c}}.]\,\end{array}\end{eqnarray}$$ (14a)

$$ \begin{eqnarray}\begin{array}{ll}{\langle \omega \rangle }_{u,-}= & \displaystyle \frac{1}{4\pi {G}_{u}^{-}}\displaystyle {\int }_{-\infty }^{\infty }{\rm{d}}{\omega }_{1}{\omega }_{1}{\Lambda }_{u}({\omega }_{1}){n}_{u}({\omega }_{1})\\ & \times [{C}_{u}({\omega }_{1}+{E}_{u})+{\rm{H}}.{\rm{c}}.]\,.\end{array}\end{eqnarray}$$ (14b)

In the following, the subscript u only represents l or r without further declaration. The rates

G_{u}^{\pm}

in Eqs. (13b) and (13c) are contributed by two physical processes. Take

G_{u}^{+}

for example: (i) resonant energy relaxation from the state |0〉 to |u〉, with the energy E_u absorbed by the u-th thermal bath; (ii) off-resonant transport process, where the thermal baths show non-additive cooperation. As the three-level system releases energy E_u, part of the heat ω₁ is absorbed by the u-th bath, whereas the left energy (–E_u – ω₁) is consumed by the middle bath. Similarly, the rate

G_{u}^{-}

describes the reversed process of

G_{u}^{+}

Figure 3.(a) The globally cyclic transition contributed by the $G_{l}^{\pm} G_{m}^{\pm} G_{r}^{\mp} / A$ , and the locally conditional transitions contributed by (b) $G_{m}^{-} G_{l}^{+} G_{l}^{-} / A$ , (c) $G_{m}^{+} G_{r}^{+} G_{r}^{-} / A$ , (d) $G_{r}^{-} G_{l}^{+} G_{l}^{-} / A$ , and (e) $G_{l}^{-} G_{r}^{+} G_{r}^{-} / A$ within the nonequilibrium NIBA scheme, respectively. The horizontal solid black line at top represents |0〉; two horizontal solid black lines at bottom describe renormalized energy levels |l(r)〉 with the energies E_l(r) = ε_l(r) – Σ_k|g_k,m|²/ω_k. The other symbols are the same as those in Fig. 1.

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G_{l}^{\pm} G_{m}^{\pm} G_{r}^{\mp} / A

We compare the heat currents calculated by the nonequilibrium NIBA with the ones calculated by the PTRE in Fig. 2 . It is found that J l(r) obtained by these two methods are consistent with each other in a wide regime of the coupling strength α m . While J m obtained from the nonequilibrium NIBA shows apparently disparity with the result of the PTRE, unless the coupling strength increases to the regime α m ≳2.

3.2. Weak coupling limit

Re [κ_{u, +} (ω^{'}, χ_{u})] \approx \frac{1}{4} Λ (ω^{'}) n_{u} (ω^{'}) e^{- i ω^{'} χ_{u}}

$$ \begin{eqnarray}\begin{array}{ll}{J}_{l}= & \displaystyle \sum _{\xi =\pm }\displaystyle \frac{(1+\xi \cos \theta )}{4 {\mathcal B} }{E}_{\xi }({\varGamma }_{+}^{e}+{\varGamma }_{-}^{e})\\ & \times \{{\kappa }_{l,\xi }^{a}[{\varGamma }_{\xi }^{e}{\varGamma }_{\bar{\xi }}^{a}+({\varGamma }_{+}^{e}+{\varGamma }_{-}^{e}){\varGamma }_{p}^{\xi }]\\ & -{\kappa }_{l,\xi }^{e}({\varGamma }_{+}^{a}{\varGamma }_{-}^{a}+{\varGamma }_{+}^{a}{\varGamma }_{p}^{+}+{\varGamma }_{-}^{a}{\varGamma }_{p}^{-})\},\end{array}\end{eqnarray}$$ (15)

$$ \begin{eqnarray}\begin{array}{ll}{J}_{r}= & \displaystyle \sum _{\xi =\pm }\displaystyle \frac{(1-\xi \cos \theta )}{4 {\mathcal B} }{E}_{\xi }({\varGamma }_{+}^{e}+{\varGamma }_{-}^{e})\\ & \times \{{\kappa }_{r,\xi }^{a}[{\varGamma }_{\xi }^{e}{\varGamma }_{\bar{\xi }}^{a}+({\varGamma }_{+}^{e}+{\varGamma }_{-}^{e}){\varGamma }_{p}^{\xi }]\\ & -{\kappa }_{r,\xi }^{e}({\varGamma }_{+}^{a}{\varGamma }_{-}^{a}+{\varGamma }_{+}^{a}{\varGamma }_{p}^{+}+{\varGamma }_{-}^{a}{\varGamma }_{p}^{-})\},\end{array}\end{eqnarray}$$ (16)

with

\bar{ξ} \equiv - ξ

. The current into the middle bath is

$$ \begin{eqnarray}\,{J}_{m}=-({E}_{+}-{E}_{-})\displaystyle \frac{{\varGamma }_{+}^{e}+{\varGamma }_{-}^{e}}{ {\mathcal B} }({\varGamma }_{+}^{a}{\varGamma }_{-}^{e}{\varGamma }_{p}^{+}-{\varGamma }_{-}^{a}{\varGamma }_{+}^{e}{\varGamma }_{p}^{-}),\end{eqnarray}$$ (17)

where the coefficient is

ℬ = \sum_{ξ = \pm} (Γ_{ξ}^{a} + Γ_{+}^{e} + Γ_{-}^{e}) [Γ_{ξ}^{e} Γ_{\bar{ξ}}^{a} + (Γ_{+}^{e} + Γ_{-}^{e}) Γ_{p}^{ξ}]

, the combined rates are

Γ_{+}^{e (a)} = \frac{1}{2} (κ_{l, +}^{e (a)} \cos^{2} \frac{θ}{2} + κ_{r, +}^{e (a)} \sin^{2} \frac{θ}{2})

Γ_{-}^{e (a)} = \frac{1}{2} (κ_{l, -}^{e (a)} \sin^{2} \frac{θ}{2} + κ_{r, -}^{e (a)} \cos^{2} \frac{θ}{2})

, and

Γ_{p}^{+ (-)} = \frac{\sin^{2} θ}{2} κ_{p}^{e (a)}

, with the local rates

κ_{u, \pm}^{e} = Λ_{u} (E_{\pm}) n_{u} (E_{\pm})

κ_{u, \pm}^{a} = Λ_{u} (E_{\pm}) [1 + n_{u} (E_{\pm})]

κ_{p}^{e} = Λ_{m} (E_{+} - E_{-}) n_{m} (E_{+} - E_{-})

, and

κ_{p}^{a} = Λ_{m} (E_{+} - E_{-}) [1 + n_{m} (E_{+} - E_{-})]

We plot the currents [Eqs. (15)-(17)] in Fig. 2 to analyze the valid regime of α m by comparing with the counterpart based on the PTRE. It is found that for J l and J r, the Redfield scheme is applicable even for the system-middle bath coupling strength α = 0.1. While for J m, the Redfield scheme becomes invalid as the system-middle bath coupling strength surpasses 0.01. This fact indicates that the influence of the phonons in the middle bath should be necessarily included to describe the transitions between |\pm〉 η and |0〉, which may enhance the energy flow into the middle bath accordingly.

In the following, based on the consistent analysis of the heat currents (particular for J m) we approximately classify the strength of the system-middle bath interaction into three regimes: (i) weak coupling regime α m < 0.01; (ii) moderate coupling regime 0.01 \leq α m \leq 2; (iii) strong coupling regime α m > 2.

4. Results and discussion

Heat amplification and negative differential thermal conductance are considered as two crucial components of the quantum thermal transistor. Particularly for heat amplification, the schematic which is shown in Fig. 1(a) has the ability to significantly enhance the heat flow into the left or right terminal by a tiny modulation of the middle terminal temperature. Formally, the amplification factor is defined as [12]

$$ \begin{eqnarray}{\beta }_{u}=|\partial {J}_{u}/\partial {J}_{m}|,\,\,u=l,r.\end{eqnarray}$$ (18)

Moreover, owning to the flux conservation of the three-level system J_l + J_m + J_r = 0, the amplification factors β_l and β_r are related by β_l = |β_r+(–1)^θ|, with θ = 0 when ∂J_r/∂J_m > 0, and θ = 1 when ∂J_r/∂J_m < 0. The thermal transistor is properly functioning under the condition β_l(r) > 1. Currently, it is known that the heat amplification can be realized mainly via two mechanisms: (i) one is driven by NDTC within the two-terminal setup, where the heat current is suppressed with the increase of the temperature bias;^[12,16] (ii) the other is driven by the inelastic transfer process without NDTC, which can be unraveled even in the linear response regime.^[23]

4.1. Transistor effect

We first analyze the effect of the system-middle bath interaction on the heat amplification of the nonequilibrium three-level system, shown in Fig. 4 . It is found that the giant amplification factor appears in the moderate and strong system-middle bath coupling regimes both for the low and high temperature biases between the left and right thermal baths. Moreover, the finite amplification factor can be observed at the high temperature bias with weak system-middle bath coupling strength. Hence, we mainly investigate the behavior and the underlying mechanism of the heat amplification at the high temperature bias.

Figure 4.Heat amplification factor as a function of system–middle bath coupling strength α_m in low (T_l = 0.5, T_r = 0.4) and high (T_l = 2, T_r = 0.4) temperature bias regimes, with $\max_{T_{r} < T_{m} < T_{l}} {β_{r}}$ the maximal value of β_r by tuning the temperature of the middle bath T_m between T_r and T_l. The other parameters are given by ε_l = 1.0, ε_r = 0.6, Δ = 0.6, γ = 0.0002, and ω_c = 10.

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4.1.1. Heat amplification at strong coupling

We first investigate the influence of the strong system-middle bath interaction on the heat amplification by tuning the temperature T m of the middle bath. As shown in Fig. 5(a), the amplification factor is monotonically enhanced when the coupling strength increases from the moderate coupling regime (e.g., α m = 0.5). When the interaction strength enters the strong coupling regime (α m = 2), the amplification factor becomes large but finite in the low temperature regime of T m (see Appendix B3 for brief analysis), whereas it is strongly suppressed as T m reaches T m = T l = 2. Interestingly, as α m is further strengthened (e.g., up to 4), a giant heat amplification appears with a divergent point, which results form the turnover behavior of J m shown in the inset of Fig. 5(b) . Moreover, the heat currents into the left and right thermal baths in Fig. 5(b) corresponding to α m = 4 are much larger than J m, which ensures the validity of the heat amplification in the strong coupling regime.

Next, we give a comprehensive picture of the amplification factor by modulating the temperature T m and coupling strength α m in Fig. 5(c) . It is found that the divergent behavior of the heat amplification is generally robust in the strong coupling regime (α m ≳ 2.8). In summary, we conclude that the giant heat amplification feature favors the strong system-middle bath interaction.

Figure 5.(a) Heat amplification factor β_r as a function of the middle bath temperature T_m with various system–middle bath coupling strength α_m; (b) three steady state heat currents J_u/γ (u = l,m,r) as a function of T_m with the coupling strength α_m = 4, and the inset is the zoom in view of J_m/γ; (c) the 3D view of the heat amplification factor β_r by tuning T_m and α_m. The other parameters are given by ε_l = 1.0, ε_r = 0.6, Δ = 0.6, γ = 0.0002, ω_c = 10, T_l = 2, and T_r = 0.4.

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4.1.2. Mechanism of the giant heat amplification

G_{r}^{-}

$$ \begin{eqnarray}{J}_{r,{\rm{NIBA}}}\approx \displaystyle \frac{1}{{\mathscr{A}}}{G}_{l}^{-}{G}_{m}^{-}{G}_{r}^{+}{\langle \omega \rangle }_{r,+},\end{eqnarray}$$ (19)

with the coefficient reduced to

A \approx (G_{m}^{+} + G_{m}^{-}) (G_{l}^{+} + G_{r}^{+})

J_{r, NIBA}

is determined by the globally cyclic transition |0〉→|l〉→|r〉→|0〉, which is characterized by the cooperative rate

\frac{1}{A} G_{l}^{-} G_{m}^{-} G_{r}^{+}

. Moreover, it should be noted that though

G_{m}^{+}

is much larger than

G_{m}^{-}

, the ratio

G_{m}^{-} / G_{m}^{+} = \exp [- (ε_{l} - ε_{r}) / (k_{B} T_{m})]

shows monotonic increase as a function of T_m [see dashed lines with circles and up-triangles in Fig. 6(a)]. Then, by tuning up the temperature T_m from T_r = 0.4, the increase of

G_{m}^{-} / G_{m}^{+}

dominates the monotonic enhancement of

J_{r, NIBA}

, as the rates

G_{l}^{\pm}

G_{r}^{+}

and energy 〈ω〉_r,+ are nearly constant as shown in Figs. 6(a) and 6(b).

Figure 6.Steady state behaviors as a function of the middle bath temperature T_m within the nonequilibrium NIBA at strong coupling (α_m = 4): (a) transition rates $G_{u}^{\pm} (u = l, m, r)$ in Eqs. (13a)–(13c), and (b) average energy quanta 〈ω〉_u,± (u = l,r) in Eqs. (14a) and (14b); (c) heat current J_m,NIBA and its main components in Eqs. (20a) and(20b), and (d) comparison of the approximate amplification factor β_r,NIBA with β_r. (e) and (f) Schematic illustrations of flow components $J_{m, NIBA}^{(a)}$ and $J_{m, NIBA}^{(b)}$ . The other parameters are the same as those in Fig. 5.

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G_{r}^{-}

$$ \begin{eqnarray}{J}_{m,{\rm{NIBA}}}^{({\rm{a}})}=\displaystyle \frac{1}{{\mathscr{A}}}{G}_{m}^{-}{G}_{l}^{-}{G}_{r}^{+}({\langle \omega \rangle }_{l,-}-{\langle \omega \rangle }_{r,+}),\,\end{eqnarray}$$ (20a)

$$ \begin{eqnarray}{J}_{m,{\rm{NIBA}}}^{({\rm{b}})}=\displaystyle \frac{1}{{\mathscr{A}}}{G}_{m}^{-}{G}_{l}^{-}{G}_{l}^{+}({\langle \omega \rangle }_{l,-}-{\langle \omega \rangle }_{l,+}).\end{eqnarray}$$ (20b)

The approximate factor

β_{r, NIBA} = | \partial J_{r, NIBA} / \partial (J_{m, NIBA}^{(a)} + J_{m, NIBA}^{(b)}) |

is agreeable with the counterpart obtained by the PRTE, shown in Fig. 6(d). Specifically,

J_{m, NIBA}^{(a)}

describes a globally cyclic current with the loop rate

G_{m}^{-} G_{l}^{-} G_{r}^{+} / A

to extract energy 〈ω〉_l,– out of the l-th bath and input 〈ω〉_r,+ into the r-th bath, the resultant energy difference (〈ω〉_l,– – 〈ω〉_r,+) is absorbed by the middle bath. While

J_{m, NIBA}^{(b)}

is only associated with the local transition process between states |l〉 and |0〉, and each transition pumps energy (〈ω〉_l,– –〈ω〉_l,+) out of the left bath into the middle bath. These two currents are schematically illustrated in Figs. 6(e) and 6(f), respectively.

J_{m, NIBA}^{(a)}

4.1.3. Heat amplification at weak and moderate couplings

Re [κ_{u, +} (E_{\pm})] \approx κ_{u, \pm}^{e} / 2

$$ \begin{eqnarray}{\beta }_{r}\approx \displaystyle \frac{{\sin }^{2}\theta }{16}\left|\displaystyle \frac{{\kappa }_{l,+}^{e}{\kappa }_{r,+}^{a}-{\kappa }_{l,+}^{a}{\kappa }_{r,+}^{e}}{{\varGamma }_{-}^{a}({\varGamma }_{+}^{a}+{\varGamma }_{+}^{e})+{\varGamma }_{+}^{a}{\varGamma }_{-}^{e}}\right|,\end{eqnarray}$$ (21)

which is irrelevant to T_m and α_m. While in the low temperature bias limit (T_l ≈ T_r), due to

κ_{l, \pm}^{a (e)} \approx κ_{r, \pm}^{a (e)}

, the current into the right bath is determined by the current component

J_{r} \propto (κ_{r, +}^{a} Γ_{-}^{e} Γ_{p}^{+} - κ_{r, +}^{e} Γ_{-}^{a} Γ_{p}^{-})

. Then, the amplification factor is approximated as β_r≈(1 – cosθ)/2, which demonstrates that the amplification effect becomes weak. This is qualitatively consistent with the result shown in Fig. 4 with weak system–middle bath coupling strength (e.g., β_r≈1 with α_m = 0.001).

If we increase the coupling strength α m up to the moderate regime (e.g., α m = 0.02), the giant amplification factor appears in the comparatively low temperature regime [dashed line with up-triangle in Fig. 7(a)], which is due to the turnover behavior of J m in Fig. 7(b) . Such feature results from NDTC, which will be addressed in the following subsection. It should be emphasized that the heat amplification is purely explored by the PTRE, which however cannot be explained with the Redfield equation.

Figure 7.(a) Heat amplification factor β_r with various coupling strengthes α_m, and (b) steady state heat currents with α_m = 0.02 as a function of T_m, the inset is the zoom-in view of J_m/γ. The other parameters are the same as those in Fig. 5.

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4.2. Negative differential thermal conductance

κ_{u, +}^{(0)} (E_{\pm}) = η_{u}^{2} κ_{u, \pm}^{e} / 2

$$ \begin{eqnarray}\begin{array}{ll}{\kappa }_{u,+}^{(1)}({E}_{\pm })= & {\kappa }_{u,+}^{(0)}({E}_{\pm })+\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\displaystyle \frac{{\rm{d}}{\omega }_{1}}{4\pi }{\varLambda }_{u}({\omega }_{1}){n}_{u}({\omega }_{1})\\ & \times {\rm{Re}}[{C}_{u}^{(1)}({\omega }_{1}-{E}_{\pm })],\end{array}\end{eqnarray}$$ (22a)

$$ \begin{eqnarray}\begin{array}{ll}{\kappa }_{u,-}^{(1)}({E}_{\pm })= & {\kappa }_{u,-}^{(0)}({E}_{\pm })+\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\displaystyle \frac{{\rm{d}}{\omega }_{1}}{4\pi }{\varLambda }_{u}({\omega }_{1})[1+{n}_{u}({\omega }_{1})]\\ & \times {\rm{Re}}[{C}_{u}^{(1)}(-{\omega }_{1}+{E}_{\pm })],\end{array}\end{eqnarray}$$ (22b)

with the single phonon correlation function

C_{u}^{(1)} (\pm ω_{1} \mp E_{\pm}) = \frac{η_{u}^{2}}{4} \int_{0}^{\infty} d τ e^{i (\pm ω_{1} \mp E_{\pm}) τ} ϕ_{m} (τ)

. The heat current

J_{polaron}^{(1)}

up to the first order correction shows interesting NDTC feature, which is almost identical with the exact numerical solution from the PTRE J. Therefore, we conclude that the middle bath phonon induced transition between |0〉 and |±〉_η is crucial to the appearance of NDTC, as shown in Fig. 8(a), which cannot be found from the standard Redfield scheme.

Figure 8.(a) Schematic illustration of quantum thermal transport in the three-level system (|±〉_η and |0〉) contacting with the l-th and m-th thermal baths; (b) steady state heat currents by modulating the temperature bias T_l – T_m, which have different order approximations with α_m = 0.02; (c) steady state heat currents by tuning the temperature bias T_l – T_m with various system–middle bath coupling strengthes. The temperature of the left thermal bath is T_l = 2, and the other parameters are the same as those in Fig. 5.

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Moreover, we investigate the effect of the system-middle bath coupling strength on the steady state heat currents by modulating the temperature bias T l - T m, as shown in Fig. 8(c) . It is found that besides the emergence of NDTC with moderate interaction strength (e.g., α m = 0.05), the nonmonotonic behavior of the current is also clearly seen at strong system-middle bath coupling α m = 4. With certain approximation, such NDTC can be explained within the nonequilibrium NIBA, by which the current is simplified to

$$ \begin{eqnarray}\,{J}_{l-m}=\displaystyle \frac{1}{{{\mathscr{A}}}^{^{\prime} }}{G}_{m}^{-}{G}_{l}^{+}{G}_{l}^{-}({\langle \omega \rangle }_{l,-}-{\langle \omega \rangle }_{l,+}),\end{eqnarray}$$ (23)

with

A^{'} = (G_{m}^{+} + G_{m}^{-}) G_{l}^{+} + G_{m}^{-} G_{l}^{-}

. It should be noted that

G_{l (m)}^{\pm}

and

{〈 ω 〉}_{l, \pm}

in this two-terminal case have the identical expression as shown in Eqs. (13a)–(13c) and (14a)–(14b) within the three-terminal setup, respectively. Considering the dramatic suppression of the transition rate

G_{m}^{-}

with large temperature bias T_l – T_m, i.e., low temperature regime of T_m in Fig. 6(a), the transition from |r〉 to |l〉 is strongly blocked, which dramatically suppresses the flux rate in J_{l – m}. It finally leads to NDTC.

5. Conclusion

To summarize, we study the steady state heat currents in the nonequilibrium three-level system interacting with three individual thermal baths. We apply the PTRE combined with FCS to investigate the system density matrix and the resultant heat currents. In the weak and strong system-middle bath coupling limits, we obtain the analytical expression of heat currents with the Redfield scheme and nonequilibrium NIBA approach, which are consistent with the counterpart of the PTRE. This extends the application of the PTRE to the nonequilibrium three-level models from the previous nonequilibrium (coupled) spin-boson model.

G_{m}^{-} / G_{m}^{+}

We hope the analysis of the heat amplification and negative differential thermal conductance may provide some theoretical insight in design of the quantum thermal transistor.

Category: Phononics and phonon engineering

Received: Mar. 26, 2020

Accepted: --

Published Online: Apr. 29, 2021

The Author Email: Chen Wang (dzxu@bit.edu.cn)

DOI:10.1088/1674-1056/ab973b