Quantum coherence thermal transistors Shanhe Su,1 Yanchao Zhang,1 Bjarne Andresen,2 Jincan Chen1∗ 1Department of Physics and Jiujiang Research Institute, Xiamen University, Xiamen 361005, China. 2Niels Bohr Institute, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark. (Dated: May 31, 2021) Coherent control of self-contained quantum systems offers the possibility to fabricate smallest thermal transistors. The steady coherence created by the delocalization of electronic excited states arouses nonlinear heat transports in non-equilibrium environment. Applying this result to a three- level quantum system, we show that quantum coherence gives rise to negative differential thermal resistances, making the thermal transistor suitable for thermal amplification. The results show that quantum coherence facilitates efficient thermal signal processing and can open a new field in the application of quantum thermal management devices. PACS numbers: 05.90. +m, 05.70. –a, 03.65.–w, 51.30. +i A thermal transistor, like its electronic counterpart, is thermal flux is normally higher than the controlling (in- capable of implementing heat flux switching and mod- put) thermal flux, a thermal transistor is able to amplify ulating. The effects of negative differential thermal re- or switch a small signal. The amplification factor must sistance (NDTR) play a key role in the development of be tailored to suit specific situations. The Scovil and thermal transistors [1]. Classical dynamic descriptions Schulz-DuBois maser model is not applicable for fabricat- utilizing Frenkel-Kontorova lattices conclude that nonlin- ing thermal transistors, owing to the fact that its ampli- ear lattices are the origin of NDTR [2, 3]. Ben-Abdallah fication factor is simply a constant defined by the maser et al. introduced a distinct type of thermal transistors frequency relative to the pump frequency [18, 19]. How- based on the near-field radiative heat transfer by evanes- ever, the coherent excitation-energy transfer created by cent thermal photons between bodies [4]. Joulain et al. the delocalization of electronic excited states may aid in first proposed a quantum thermal transistor with strong the design of powerful thermal devices. Coherent con- coupling between the interacting spins, where the com- trol of a three-level system (TLS) provides us a heuristic petition between different decay channels makes the tem- approach to better understand the prime requirements perature dependence of the base flux slow enough to ob- for the occurrence of anomalous thermal conduction in tain a high amplification [5]. Zhang et al. predicted that quantum systems. asymmetric Coulomb blockade in quantum-dot thermal In this paper we design a quantum thermal transis- transistors would result in a NDTR [6]. Stochastic fluc- tor consisting of a TLS coupled to three separate baths. tuations in mesoscopic systems have been regarded as an The dynamics of the system is derived by considering the alternate resource for the fast switching of heat flows [7]. coupling between the two excited states. Steady-state Recent studies showed that quantum coherence ex- solutions will be used to prove that the coherent transi- hibits the ability to enhance the efficiency of thermal tions between the two excited states induce nonlinearity converters, such as quantum heat engines [8–10] and in nonequilibrium quantum systems. Further analysis artificial light-harvesting systems [11, 12]. Interference shows that quantum coherence gives rise to a NDTR and between multiple transitions in nonequilibrium environ- helps improve the thermal amplification. ments enables us to generate non-vanishing steady quan- Figure 1 shows the TLS modeled by the Hamiltonian tum coherence [13, 14]. Evidence is growing that long- HS as arXiv:1811.02400v1 [quant-ph] 6 Nov 2018 lived coherence boosts the transport of energy from light- X harvesting antennas to photosynthetic reaction centers HS = "i jii hij + ∆(j1i h2j + j2i h1j); (1) [15, 16]. The question arises whether quantum interfer- i=0;1;2 ence and coherence effects could also induce nonlinear heat conduction and enhance the performance of a ther- where "1 ("2) gives the energy level of the excited states mal transistor. in the molecules j1i (j2i), "0 denotes the energy of the Scovil and Schulz-DuBois originally proposed a three- ground state j0i and is set to zero, and ∆ describes the level maser system as an example of a Carnot engine excitonic coupling between states j1i and j2i. For the and applied detailed balance ideas to obtain the maser models of biological light reactions, ∆ occurs naturally efficiency formula [17]. Because the controlled (output) as a consequence of the intermolecular forces between two proximal optical dipoles [12, 20]. In the presence of the dipole-dipole interaction, the optically excited states become coherently delocalized. j+i = cos θ j1i + sin θ j2i ∗ [email protected] and |−i = sin θ j1i − cos θ j2i are the usual eigenstates 2 and j2i h2j = sin θ sin θ j+i h+j + cos θ cos θ |−i h−| − cos θ sin θ (j+i h−| + |−i h+j) : (4) ȁퟏۧ Base Emitter The first two operators in j1i h1j and j2i h2j describe the ∆ pure dephasing of a two-level system, whereas the third ȁퟐۧ term leads to the energy exchange between the system and the base with an effective coupling proportional to ȁퟎۧ Collecter the product sin θ cos θ, i.e., X y HSB−eff = 2 sin θ cos θ (j+i h−| + |−i h+j) gk ak + ak : k (5) Figure 1. Schematic illustration of the quantum thermal transis- In reality, the TLS can be realized in the photosynthesis tor composed of a three-level system (TLS) interacting with three process. The pumping light, taking the sunlight photons baths: its ground state j0i and excited state j1i (j2i) are cou- for example, is considered the high temperature emitter. pled with the emitter (collector); the excited states j1i and j2i are The collector is formed by the surrounding electromag- diagonal-coupled with the base; and the coupling strength between j1i and j2i is characterized by ∆. netic environment which models energy transfer to the reaction center. The base provides the phonon modes coupled with the excited states. The TLS becomes irreversible due to the interaction diagonalizing the subspace spanned by j1i and j2i with with its surrounding environment. Using the Born- tan 2θ = 2∆= ("1 − "2). Markov approximation, which involves the assumptions The absorption of a photon from the emitter (E) causes that the environment is time independent and the envi- an excitation transfer from the ground state j0i to the ronment correlations decay rapidly in comparison to the state j1i, whereas phonons are emitted into the base (B) typical time scale of the system evolution [22], we get the by the transitions between j1i and j2i. The cycle is closed quantum dynamics of the system in = 1 units, i.e., by the transition between j2i and j0i, and the rest of ~ the energy is released as a photon to the collector (C). dρ The Hamiltonians of the emitter, collector, and base are = −i[HS; ρ] + DE [ρ] + DB [ρ] + DC [ρ]: (6) P y y dt Hi = k !ikaikaik (i = E, C, and B), where aik (aik) refers to the creation (annihilation) operator of the bath The operators Di [ρ] (i = E, B, and C) denote the dissi- mode !ik. The TLS couples to the emitter and the col- pative Lindblad superoperators associated with the emit- lector, each constituted of harmonic oscillators, via cou- ter, base, and collector (Supplementary Eq. (S-1)), which pling constants gEk and gCk in the rotating wave approx- take the form imation, where the corresponding Hamiltonians are for- P y mally written as HSE = k gEkaEk j0i h1j + h:c: and X y 1 n y o Di [ρ] = γi (v) Ai (v) ρAi (v) − ρ, Ai (v) Ai (v) ; y 2 P v HSC = k gCkaCk j0i h2j + h:c: , respectively. The (7) output of the Scovil–Schulz-DuBois maser is a radiation 0 where v = " − " is the energy difference between two field with a particular frequency, provided there is pop- arbitrary eigenvalues of HS, and Ai (v) is the jump op- ulation inversion between levels "1 and "2. In this study, erator associated with the interaction between the sys- the two excited states are coupled with a thermal reser- tem and bath i. Considering a quantum bath consisting voir, namely, the base. The interaction Hamiltonian of of harmonic oscillators, we have the decay rate γi (v) = the system with the base is described by Γ i (v) ni (v) for v < 0 and γi (v) = Γi (v) [1 + ni(v)] for v > 0 , where Γi (v) labels the decoherence rate and is X y related to the spectral density of the bath, and Ti is the HSB = (j1i h1j − j2i h2j) gBk aBk + a : (2) Bk temperature of bath i. The thermal occupation number k v=(kB Ti) in a mode is written as ni(v) = 1= e −1 . The For a finite coupling ∆, the base modeled by Eq. (2) in- Boltzmann constant kB is set to unity in the following. duces not only decoherence but also relaxation [21]. The The steady-state populations and coherence of the counterintuitive effect of the energy exchange between open quantum system are obtained by setting the left- the two excited states and the dephasing bath becomes hand side of Eq. (6) equal zero. Then the steady state evident when the system operator coupled to the base is energy fluxes are determined by the average energy going replaced by through the TLS, i.e., : X j1i h1j = cos θ cos θ j+i h+j + sin θ sin θ |−i h−| E(1) = TrfHSDi [ρ (1)]g = JE + JC +JB = 0 i=E;C;B + sin θ cos θ (j+i h−| + |−i h+j) (3) (8) 3 which complies with the 1st law of thermodynamics.