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Chinese Physics B
Vol. 29, Issue 8, (2020)

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

Chen Wang^1,† and Da-Zhi Xu²

Author Affiliations

¹Department of Physics, Zhejiang Normal University, Jinhua 32004, China

²School of Physics and Center for Quantum Technology Research, Beijing Institute of Technology, Beijing 100081, China

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DOI: 10.1088/1674-1056/ab973b Cite this Article

Chen Wang, Da-Zhi Xu. A polaron theory of quantum thermal transistor in nonequilibrium three-level systems[J]. Chinese Physics B, 2020, 29(8): Copy Citation Text

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Abstract

We investigate the quantum thermal transistor effect in nonequilibrium three-level systems by applying the polaron-transformed Redfield equation combined with full counting statistics. The steady state heat currents are obtained via this unified approach over a wide region of system–bath coupling, and can be analytically reduced to the Redfield and nonequilibrium noninteracting blip approximation results in the weak and strong coupling limits, respectively. A giant heat amplification phenomenon emerges in the strong system–bath coupling limit, where transitions mediated by the middle thermal bath are found to be crucial to unravel the underlying mechanism. Moreover, the heat amplification is also exhibited with moderate coupling strength, which can be properly explained within the polaron framework.

Keywords

nonequilibrium and irreversible thermodynamics open systems phonons or vibrational states in low-dimensional structures and nanoscale materials quantum transport

\begin{array}{r} {\hat{H}}_{s} = ε_{0} | 0 ⟩ ⟨ 0 | + \sum_{u = l, r} ε_{u} | u ⟩ ⟨ u | + Δ (| l ⟩ ⟨ r | + H . c .), \end{array}

(1)

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\begin{array}{r} {\hat{V}}_{u} = ({\hat{S}}_{u}^{†} + H . c .) \sum_{k} (g_{k, u} {\hat{b}}_{k, u}^{†} + H . c .), u = l, r, \end{array}

(2)

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\begin{array}{r} {\hat{V}}_{m} = (| l ⟩ ⟨ l | - | r ⟩ ⟨ r |) \sum_{k} (g_{k, m} {\hat{b}}_{k, m}^{†} + H . c .), \end{array}

(3)

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\begin{array}{r} {\hat{H}}_{s}^{^{'}} = \bar{ε} \hat{N} + δ ε {\hat{σ}}_{z} + η Δ {\hat{σ}}_{x}, \end{array}

(4)

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\begin{array}{r} {\hat{V}}_{m}^{^{'}} = Δ [\cos (2 \hat{B}) - η] {\hat{σ}}_{x} + Δ \sin (2 \hat{B}) {\hat{σ}}_{y}, \end{array}

(5a)

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\begin{array}{r} {\hat{V}}_{u}^{^{'}} = (e^{- i {\hat{B}}_{u}} {\hat{S}}_{u}^{†} + e^{i {\hat{B}}_{u}} {\hat{S}}_{u}) \sum_{k} (g_{k, u} {\hat{b}}_{k, u}^{†} + g_{k, u}^{*} {\hat{b}}_{k, u}) u = l, r, \end{array}

(5b)

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\begin{array}{r} \frac{d}{d t} {\hat{ρ}}_{s} = - i [{\hat{H}}_{s}^{^{'}}, {\hat{ρ}}_{s}] + \sum_{u = l, m, r} L_{u} [{\hat{ρ}}_{s}], \end{array}

(6)

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\begin{array}{r} \begin{array}{ll} L_{m} [{\hat{ρ}}_{s}] = & \sum_{α = x, y; ω, ω^{^{'}}} γ_{α} (ω^{^{'}}) [{\hat{P}}_{α} (ω^{^{'}}) {\hat{ρ}}_{s} {\hat{P}}_{α} (ω) \\ - {\hat{P}}_{α} (ω) {\hat{P}}_{α} (ω^{^{'}}) {\hat{ρ}}_{s}] + H . c ., \end{array} \end{array}

(7)

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\begin{array}{r} γ_{x} (ω) = η^{2} Δ^{2} \int_{0}^{\infty} d τ e^{i ω τ} [\cosh ϕ_{m} (τ) - 1], \end{array}

(8a)

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\begin{array}{r} γ_{y} (ω) = η^{2} Δ^{2} \int_{0}^{\infty} d τ e^{i ω τ} \sinh ϕ_{m} (τ), \end{array}

(8b)

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\begin{array}{r} ϕ_{m} (τ) = 4 \sum_{k} | \frac{g_{k, m}}{ω_{k}} |^{2} {\cos (ω_{k} τ) [2 n_{m} (ω_{k}) + 1] - i \sin (ω_{k} τ)} . \end{array}

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\begin{array}{r} \begin{array}{ll} L_{u} [{\hat{ρ}}_{s}] = & \sum_{ω, ω^{^{'}}} [κ_{u, -} (ω^{^{'}}) {\hat{Q}}_{u} (ω^{^{'}}) {\hat{ρ}}_{s} {\hat{Q}}_{u}^{†} (ω) \\ + κ_{u, +} (ω^{^{'}}) {\hat{Q}}_{u}^{†} (ω^{^{'}}) {\hat{ρ}}_{s} {\hat{Q}}_{u} (ω) \\ - κ_{u, +} (ω^{^{'}}) {\hat{Q}}_{u} (ω) {\hat{Q}}_{u}^{†} (ω^{^{'}}) {\hat{ρ}}_{s} \\ - κ_{u, -} (ω^{^{'}}) {\hat{Q}}_{u}^{†} (ω) {\hat{Q}}_{u} (ω^{^{'}}) {\hat{ρ}}_{s}] + H . c ., \end{array} \end{array}

(9)

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\begin{array}{r} κ_{u, +} (ω) = \int_{- \infty}^{\infty} \frac{d ω_{1}}{4 π} Λ_{u} (ω_{1}) n_{u} (ω_{1}) C_{u} (ω_{1} - ω), \end{array}

(10a)

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\begin{array}{r} κ_{u, -} (ω) = \int_{- \infty}^{\infty} \frac{d ω_{1}}{4 π} Λ_{u} (ω_{1}) [1 + n_{u} (ω_{1})] C_{u} (- ω_{1} + ω) . \end{array}

(10b)

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\begin{array}{r} \begin{array}{ll} J_{l, N I B A} = & \frac{1}{A} [(G_{l}^{+} G_{m}^{+} G_{r}^{-} {⟨ ω ⟩}_{l, +} - G_{l}^{-} G_{m}^{-} G_{r}^{+} {⟨ ω ⟩}_{l, -}) \\ + (G_{m}^{-} + G_{r}^{-}) G_{l}^{-} G_{l}^{+} ({⟨ ω ⟩}_{l, +} - {⟨ ω ⟩}_{l, -})], \end{array} \end{array}

(11)

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\begin{array}{r} \begin{array}{ll} J_{r, N I B A} = & \frac{1}{A} [(G_{l}^{-} G_{m}^{-} G_{r}^{+} {⟨ ω ⟩}_{r, +} - G_{l}^{+} G_{m}^{+} G_{r}^{-} {⟨ ω ⟩}_{r, -}) \\ + (G_{m}^{+} + G_{l}^{-}) G_{r}^{+} G_{r}^{-} ({⟨ ω ⟩}_{r, +} - {⟨ ω ⟩}_{r, -})], \end{array} \end{array}

(12)

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\begin{array}{r} G_{m}^{\pm} = \int_{- \infty}^{\infty} d τ e^{\pm i (ε_{l} - ε_{r}) τ} η^{2} e^{ϕ_{m} (τ)} \end{array}

(13a)

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\begin{array}{r} \begin{array}{ll} G_{u}^{+} = & \frac{1}{4 π} \int_{- \infty}^{\infty} d ω_{1} Λ_{u} (ω_{1}) [1 + n_{u} (ω_{1})] \\ \times [C_{u} (- E_{u} - ω_{1}) + H . c .] \end{array} \end{array}

(13b)

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\begin{array}{r} G_{u}^{-} = \frac{1}{4 π} \int_{- \infty}^{\infty} d ω_{1} Λ_{u} (ω_{1}) n_{u} (ω_{1}) [C_{u} (ω_{1} + E_{u}) + H . c .] \end{array}

(13c)

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\begin{array}{r} \begin{array}{ll} {⟨ ω ⟩}_{u, +} = & \frac{1}{4 π G_{u}^{+}} \int_{- \infty}^{\infty} d ω_{1} ω_{1} Λ_{u} (ω_{1}) [1 + n_{u} (ω_{1})] \\ \times [C_{u} (- E_{u} - ω_{1}) + H . c .] \end{array} \end{array}

(14a)

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\begin{array}{r} \begin{array}{ll} {⟨ ω ⟩}_{u, -} = & \frac{1}{4 π G_{u}^{-}} \int_{- \infty}^{\infty} d ω_{1} ω_{1} Λ_{u} (ω_{1}) n_{u} (ω_{1}) \\ \times [C_{u} (ω_{1} + E_{u}) + H . c .] . \end{array} \end{array}

(14b)

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\begin{array}{r} \begin{array}{ll} J_{l} = & \sum_{ξ = \pm} \frac{(1 + ξ \cos θ)}{4 B} E_{ξ} (Γ_{+}^{e} + Γ_{-}^{e}) \\ \times {κ_{l, ξ}^{a} [Γ_{ξ}^{e} Γ_{\bar{ξ}}^{a} + (Γ_{+}^{e} + Γ_{-}^{e}) Γ_{p}^{ξ}] \\ - κ_{l, ξ}^{e} (Γ_{+}^{a} Γ_{-}^{a} + Γ_{+}^{a} Γ_{p}^{+} + Γ_{-}^{a} Γ_{p}^{-})}, \end{array} \end{array}

(15)

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\begin{array}{r} \begin{array}{ll} J_{r} = & \sum_{ξ = \pm} \frac{(1 - ξ \cos θ)}{4 B} E_{ξ} (Γ_{+}^{e} + Γ_{-}^{e}) \\ \times {κ_{r, ξ}^{a} [Γ_{ξ}^{e} Γ_{\bar{ξ}}^{a} + (Γ_{+}^{e} + Γ_{-}^{e}) Γ_{p}^{ξ}] \\ - κ_{r, ξ}^{e} (Γ_{+}^{a} Γ_{-}^{a} + Γ_{+}^{a} Γ_{p}^{+} + Γ_{-}^{a} Γ_{p}^{-})}, \end{array} \end{array}

(16)

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\begin{array}{r} J_{m} = - (E_{+} - E_{-}) \frac{Γ_{+}^{e} + Γ_{-}^{e}}{B} (Γ_{+}^{a} Γ_{-}^{e} Γ_{p}^{+} - Γ_{-}^{a} Γ_{+}^{e} Γ_{p}^{-}), \end{array}

(17)

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\begin{array}{r} β_{u} = | \partial J_{u} / \partial J_{m} |, u = l, r . \end{array}

(18)

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\begin{array}{r} J_{r, N I B A} \approx \frac{1}{A} G_{l}^{-} G_{m}^{-} G_{r}^{+} {⟨ ω ⟩}_{r, +}, \end{array}

(19)

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\begin{array}{r} J_{m, N I B A}^{(a)} = \frac{1}{A} G_{m}^{-} G_{l}^{-} G_{r}^{+} ({⟨ ω ⟩}_{l, -} - {⟨ ω ⟩}_{r, +}), \end{array}

(20a)

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\begin{array}{r} J_{m, N I B A}^{(b)} = \frac{1}{A} G_{m}^{-} G_{l}^{-} G_{l}^{+} ({⟨ ω ⟩}_{l, -} - {⟨ ω ⟩}_{l, +}) . \end{array}

(20b)

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\begin{array}{r} β_{r} \approx \frac{\sin^{2} θ}{16} | \frac{κ_{l, +}^{e} κ_{r, +}^{a} - κ_{l, +}^{a} κ_{r, +}^{e}}{Γ_{-}^{a} (Γ_{+}^{a} + Γ_{+}^{e}) + Γ_{+}^{a} Γ_{-}^{e}} |, \end{array}

(21)

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\begin{array}{r} \begin{array}{ll} κ_{u, +}^{(1)} (E_{\pm}) = & κ_{u, +}^{(0)} (E_{\pm}) + \underset{- \infty}{\int^{\infty}} \frac{d ω_{1}}{4 π} Λ_{u} (ω_{1}) n_{u} (ω_{1}) \\ \times R e [C_{u}^{(1)} (ω_{1} - E_{\pm})], \end{array} \end{array}

(22a)

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\begin{array}{r} \begin{array}{ll} κ_{u, -}^{(1)} (E_{\pm}) = & κ_{u, -}^{(0)} (E_{\pm}) + \underset{- \infty}{\int^{\infty}} \frac{d ω_{1}}{4 π} Λ_{u} (ω_{1}) [1 + n_{u} (ω_{1})] \\ \times R e [C_{u}^{(1)} (- ω_{1} + E_{\pm})], \end{array} \end{array}

(22b)

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\begin{array}{r} J_{l - m} = \frac{1}{A^{^{'}}} G_{m}^{-} G_{l}^{+} G_{l}^{-} ({⟨ ω ⟩}_{l, -} - {⟨ ω ⟩}_{l, +}), \end{array}

(23)

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Chen Wang, Da-Zhi Xu. A polaron theory of quantum thermal transistor in nonequilibrium three-level systems[J]. Chinese Physics B, 2020, 29(8):

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