Fig. 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.

Fig. 1. The transition rates (a) Gl+, (b) Gr+, (c) Gl− and the energy quanta (d) 〈ω〉_l,+, (e) 〈ω〉_r,+, (f) 〈ω〉_l,– within the nonequilibrium NIBA scheme. The solid black lines represent the full order calculation with expressions shown in Eqs. (13b), (13c), (14a), and (14b); the dashed red lines with circles represent the zeroth order approximation in Eqs. (B8a), (B8b) (b11a), and (B11b); the dashed blue lines with up-triangles represent the first order approximation in Eqs. (B9a), (B9b), (B10a), and (B10b). 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.

Fig. 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.

Fig. 2. (a) Average energy quanta 〈ω〉_l,±, 〈ω〉_r,+, and the flux rates Gl−Gm−Gr+/A and Gl+Gl−Gm−/A; (b) heat current into the middle bath J_m,NIBA/γ, the main components Jm,NIBA(a)/γ and Jm,NIBA(b)/γ; (c) heat current into the right bath J_r,NIBA/γ and the main component Jr,NIBA(a)/γ within the nonequilibrium NIBA scheme at strong coupling α_m = 2. 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.

Fig. 3. (a) The globally cyclic transition contributed by the Gl±Gm±Gr∓/A, and the locally conditional transitions contributed by (b) Gm−Gl+Gl−/A, (c) Gm+Gr+Gr−/A, (d) Gr−Gl+Gl−/A, and (e) Gl−Gr+Gr−/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.

Fig. 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 maxTr<Tm<Tl{βr} 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.

Fig. 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.

Fig. 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 Gu±(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 Jm,NIBA(a) and Jm,NIBA(b). The other parameters are the same as those in Fig. 5.

Fig. 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.

Fig. 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.