Quantum coherence thermal transistors
QQuantum coherence thermal transistors
Shanhe Su, Yanchao Zhang, Bjarne Andresen, Jincan Chen ∗ Department of Physics and Jiujiang Research Institute,Xiamen University, Xiamen 361005, China. Niels Bohr Institute, University of Copenhagen,Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark. (Dated: November 7, 2018)Coherent control of self-contained quantum systems offers the possibility to fabricate smallestthermal transistors. The steady coherence created by the delocalization of electronic excited statesarouses 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 thermalresistances, making the thermal transistor suitable for thermal amplification. The results show thatquantum coherence facilitates efficient thermal signal processing and can open a new field in theapplication of quantum thermal management devices.
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A thermal transistor, like its electronic counterpart, iscapable of implementing heat flux switching and mod-ulating. The effects of negative differential thermal re-sistance (NDTR) play a key role in the development ofthermal transistors [1]. Classical dynamic descriptionsutilizing Frenkel-Kontorova lattices conclude that nonlin-ear lattices are the origin of NDTR [2, 3]. Ben-Abdallahet al. introduced a distinct type of thermal transistorsbased on the near-field radiative heat transfer by evanes-cent thermal photons between bodies [4]. Joulain et al.first proposed a quantum thermal transistor with strongcoupling between the interacting spins, where the com-petition between different decay channels makes the tem-perature dependence of the base flux slow enough to ob-tain a high amplification [5]. Zhang et al. predicted thatasymmetric Coulomb blockade in quantum-dot thermaltransistors would result in a NDTR [6]. Stochastic fluc-tuations in mesoscopic systems have been regarded as analternate resource for the fast switching of heat flows [7].Recent studies showed that quantum coherence ex-hibits the ability to enhance the efficiency of thermalconverters, such as quantum heat engines [8–10] andartificial light-harvesting systems [11, 12]. Interferencebetween multiple transitions in nonequilibrium environ-ments enables us to generate non-vanishing steady quan-tum coherence [13, 14]. Evidence is growing that long-lived coherence boosts the transport of energy from light-harvesting antennas to photosynthetic reaction centers[15, 16]. The question arises whether quantum interfer-ence and coherence effects could also induce nonlinearheat conduction and enhance the performance of a ther-mal transistor.Scovil and Schulz-DuBois originally proposed a three-level maser system as an example of a Carnot engineand applied detailed balance ideas to obtain the maserefficiency formula [17]. Because the controlled (output) ∗ [email protected] thermal flux is normally higher than the controlling (in-put) thermal flux, a thermal transistor is able to amplifyor switch a small signal. The amplification factor mustbe tailored to suit specific situations. The Scovil andSchulz-DuBois maser model is not applicable for fabricat-ing thermal transistors, owing to the fact that its ampli-fication factor is simply a constant defined by the maserfrequency relative to the pump frequency [18, 19]. How-ever, the coherent excitation-energy transfer created bythe delocalization of electronic excited states may aid inthe design of powerful thermal devices. Coherent con-trol of a three-level system (TLS) provides us a heuristicapproach to better understand the prime requirementsfor the occurrence of anomalous thermal conduction inquantum systems.In this paper we design a quantum thermal transis-tor consisting of a TLS coupled to three separate baths.The dynamics of the system is derived by considering thecoupling between the two excited states. Steady-statesolutions will be used to prove that the coherent transi-tions between the two excited states induce nonlinearityin nonequilibrium quantum systems. Further analysisshows that quantum coherence gives rise to a NDTR andhelps improve the thermal amplification.Figure 1 shows the TLS modeled by the Hamiltonian H S as H S = (cid:88) i =0 , , ε i | i (cid:105) (cid:104) i | + ∆( | (cid:105) (cid:104) | + | (cid:105) (cid:104) | ) , (1)where ε ( ε ) gives the energy level of the excited statesin the molecules | (cid:105) ( | (cid:105) ) , ε denotes the energy of theground state | (cid:105) and is set to zero, and ∆ describes theexcitonic coupling between states | (cid:105) and | (cid:105) . For themodels of biological light reactions, ∆ occurs naturallyas a consequence of the intermolecular forces betweentwo proximal optical dipoles [12, 20]. In the presence ofthe dipole-dipole interaction, the optically excited statesbecome coherently delocalized. | + (cid:105) = cos θ | (cid:105) + sin θ | (cid:105) and |−(cid:105) = sin θ | (cid:105) − cos θ | (cid:105) are the usual eigenstates a r X i v : . [ qu a n t - ph ] N ov ۧȁ𝟐 Emitter Base Collecter ∆ ۧȁ𝟏ۧȁ𝟎 Figure 1.
Schematic illustration of the quantum thermal transis-tor composed of a three-level system (TLS) interacting with threebaths: its ground state | (cid:105) and excited state | (cid:105) ( | (cid:105) ) are cou-pled with the emitter (collector); the excited states | (cid:105) and | (cid:105) arediagonal-coupled with the base; and the coupling strength between | (cid:105) and | (cid:105) is characterized by ∆ . diagonalizing the subspace spanned by | (cid:105) and | (cid:105) with tan 2 θ = 2∆ / ( ε − ε ) .The absorption of a photon from the emitter (E) causesan excitation transfer from the ground state | (cid:105) to thestate | (cid:105) , whereas phonons are emitted into the base (B)by the transitions between | (cid:105) and | (cid:105) . The cycle is closedby the transition between | (cid:105) and | (cid:105) , and the rest ofthe energy is released as a photon to the collector (C).The Hamiltonians of the emitter, collector, and base are H i = (cid:80) k ω ik a † ik a ik ( i = E , C , and B ), where a † ik ( a ik )refers to the creation (annihilation) operator of the bathmode ω ik . The TLS couples to the emitter and the col-lector, each constituted of harmonic oscillators, via cou-pling constants g Ek and g Ck in the rotating wave approx-imation, where the corresponding Hamiltonians are for-mally written as H SE = (cid:80) k (cid:16) g † Ek a Ek | (cid:105) (cid:104) | + h.c. (cid:17) and H SC = (cid:80) k (cid:16) g † Ck a Ck | (cid:105) (cid:104) | + h.c. (cid:17) , respectively. Theoutput of the Scovil–Schulz-DuBois maser is a radiationfield with a particular frequency, provided there is pop-ulation inversion between levels ε and ε . In this study,the two excited states are coupled with a thermal reser-voir, namely, the base. The interaction Hamiltonian ofthe system with the base is described by H SB = ( | (cid:105) (cid:104) | − | (cid:105) (cid:104) | ) (cid:88) k g Bk (cid:16) a Bk + a † Bk (cid:17) . (2)For a finite coupling ∆ , the base modeled by Eq. (2) in-duces not only decoherence but also relaxation [21]. Thecounterintuitive effect of the energy exchange betweenthe two excited states and the dephasing bath becomesevident when the system operator coupled to the base isreplaced by | (cid:105) (cid:104) | = cos θ cos θ | + (cid:105) (cid:104) + | + sin θ sin θ |−(cid:105) (cid:104)−| + sin θ cos θ ( | + (cid:105) (cid:104)−| + |−(cid:105) (cid:104) + | ) (3) and | (cid:105) (cid:104) | = sin θ sin θ | + (cid:105) (cid:104) + | + cos θ cos θ |−(cid:105) (cid:104)−|− cos θ sin θ ( | + (cid:105) (cid:104)−| + |−(cid:105) (cid:104) + | ) . (4)The first two operators in | (cid:105) (cid:104) | and | (cid:105) (cid:104) | describe thepure dephasing of a two-level system, whereas the thirdterm leads to the energy exchange between the systemand the base with an effective coupling proportional tothe product sin θ cos θ , i.e., H SB − eff = 2 sin θ cos θ ( | + (cid:105) (cid:104)−| + |−(cid:105) (cid:104) + | ) (cid:88) k g k (cid:16) a k + a † k (cid:17) . (5)In reality, the TLS can be realized in the photosynthesisprocess. The pumping light, taking the sunlight photonsfor example, is considered the high temperature emitter.The collector is formed by the surrounding electromag-netic environment which models energy transfer to thereaction center. The base provides the phonon modescoupled with the excited states.The TLS becomes irreversible due to the interactionwith its surrounding environment. Using the Born-Markov approximation, which involves the assumptionsthat the environment is time independent and the envi-ronment correlations decay rapidly in comparison to thetypical time scale of the system evolution [22], we get thequantum dynamics of the system in (cid:126) = 1 units, i.e., dρdt = − i [ H S , ρ ] + D E [ ρ ] + D B [ ρ ] + D C [ ρ ] . (6)The operators D i [ ρ ] ( i = E , B , and C ) denote the dissi-pative Lindblad superoperators associated with the emit-ter, base, and collector (Supplementary Eq. (S-1)), whichtake the form D i [ ρ ] = (cid:88) v γ i ( v ) (cid:20) A i ( v ) ρA † i ( v ) − (cid:110) ρ, A † i ( v ) A i ( v ) (cid:111)(cid:21) , (7)where v = ε − ε (cid:48) is the energy difference between twoarbitrary eigenvalues of H S , and A i ( v ) is the jump op-erator associated with the interaction between the sys-tem and bath i . Considering a quantum bath consistingof harmonic oscillators, we have the decay rate γ i ( v ) = Γ i ( v ) n i ( v ) for v < and γ i ( v ) = Γ i ( v ) [1 + n i ( v )] for v > , where Γ i ( v ) labels the decoherence rate and isrelated to the spectral density of the bath, and T i is thetemperature of bath i . The thermal occupation numberin a mode is written as n i ( v ) = 1 / (cid:2) e v/ ( k B T i ) − (cid:3) . TheBoltzmann constant k B is set to unity in the following.The steady-state populations and coherence of theopen quantum system are obtained by setting the left-hand side of Eq. (6) equal zero. Then the steady stateenergy fluxes are determined by the average energy goingthrough the TLS, i.e., . E ( ∞ ) = (cid:88) i = E,C,B Tr { H S D i [ ρ ( ∞ )] } = J E + J C + J B = 0 (8)which complies with the 1st law of thermodynamics. Theheat fluxes J E , J C , and J B are defined with respect totheir own dissipative operators. Thus, J E = − Γ E ( ε ) ( n E + 1) (cid:20) ε (cid:18) ρ − n E n E + 1 ρ (cid:19) + ∆ (cid:60) ( ρ ) (cid:21) = J E + J E , (9) J C = − Γ C ( ε ) ( n C + 1) (cid:20) ε (cid:18) ρ − n C n C + 1 ρ (cid:19) + ∆ (cid:60) ( ρ ) (cid:21) = J C + J C , (10)and J B = − Γ B ( ω ) sin θ (2 n B + 1)[ ε − ε ρ − ρ )+ (cid:113) ( ε − ε ) / ∆ n B + 1 + 2 ∆ (cid:60) ( ρ )] = J B + J B . (11)The three heat fluxes are no longer linear functions ofthe rate of the spontaneous emission, indicating that thesymmetric property is closely related to the base inducedcoherence of the excited states. In Eqs. (9) − (11) ,each heat flux is divided into two categories. The terms J i ( i = E, C, B ) are connected to the coherence in thelocal basis, i.e., (cid:60) ( ρ ) (the real part of ρ ). J i is theremainder components depending on the populations ofthe TLS.The thermodynamics of a TLS was originally proposedby Scovil and Schulz-DuBois [17]. Boukobza et al. ob-tained the Scovil–Schulz-DuBois maser efficiency formulawhen the TLS was operated as an amplifier [18, 23, 24].The efficiency of the amplifier is defined as the ratio of theoutput energy to the energy extracted from the hot reser-voir [25]. In a nonequilibrium steady state, the efficiencyis a fixed value which equals − ( ε − ε ) / ( ε − ε ) , be-cause all heat fluxes are linear functions of the same rateof excitation. However, a thermal transistor is a thermaldevice used to amplify or switch the thermal currents atthe collector and the emitter via a small change in thebase heat flux or the base temperature. Nonlinearity isthe essential element needed to give rise to such ther-mal amplification. For the purpose of flexible control ofthe thermal currents, the characteristic functions of theTLS should not entirely depend upon the energy levelstructure of the TLS.A thermal amplifier requires a transistor with a highamplification factor α E/C , which is defined as the instan-taneous rate of change of the emitter or collector heat fluxto the heat flux applied at the base. The quantum ther-mal transistor has fixed emitter and collector tempera-tures T E and T C ( T E > T C ), respectively. The fluxes J E and J C are controlled by J B , which can be adjusted by the base temperature T B . Then the amplification factor α E/C explicitly reads α E/C = ∂J E/C ∂J B . (12)Comparison of the slopes of the thermal currents is thekey parameter to find out whether the amplification ef-fect exists. When (cid:12)(cid:12) α E/C (cid:12)(cid:12) > , a small change in J B stimulates a large variation in J E or J C and the ther-mal transistor effect appears. This implies that a smallchange of the heat flux signal of the base would leadto noticeable changes of the energy flowing through theemitter and collector.We consider heat fluxes from the baths into the TLSas positive. As T E and T C are fixed values and T B is ad-justable, the thermal conductances of the three terminalsare defined as σ i = − ∂J i ∂T B = σ i + σ i , (13)where σ ij = − ∂J ij ∂T B ( i = E, C, B ; j = 1 , , σ i are thethermal conductances with respect to the spontaneousemission, and σ i are the thermal conductances relyingon the coherence (cid:60) ( ρ ) . Using Eq. (13), the amplifica-tion factor in Eq. (12) can be recast in terms of σ E and σ C , i.e., α E/C = − σ E/C σ C + σ E . (14)The absolute value of the amplification factor (cid:12)(cid:12) α E/C (cid:12)(cid:12) > implies that one of the thermal conductances is negative,i.e., σ C < or σ E < . This means that there existsa NDTR, and consequently, the TLS can behave as athermal transistor by controlling the heat flow in analogyto the usual electric transistor.In the following section, we need to explore the extentto which the quantum nature of the TLS affects the ther-mal transistor. The formalism obtained here will allowus to access how coherences can lead to a NDTR andan enhancement of the amplification factor. To do so,the thermal conductances and temperatures of the threebaths are recast in units of ∆ . In the wide-band ap-proximation, we write the decoherence rates of the threeterminals as Γ i ( v ) = Γ i and the dephasing rate of thebase as γ B (0) = γ .Figure 2(a) shows the thermal conductances σ i of eachterminal as functions of the base temperature T B . | σ E | , σ C , and σ B decrease with T B at low temperature andbecome constant as T B approaches T E . As expected, σ B remains lower than | σ E | and σ C over the whole range.A tiny change of the base heat flux J B or temperature T B is able to dramatically change the emitter and col-lector thermal flows J E and J C , leading to a noticeableamplification effect. Similar to the decomposition of thethermal fluxes, each thermal conductance can be divided 𝒂 𝒃𝒄 𝒅 × 𝟏𝟎 −𝟑 × 𝟏𝟎 −𝟑 × 𝟏𝟎 −𝟑 × 𝟏𝟎 −𝟑 Figure 2. (a) The overall thermal conductances σ i ; (b) the ther-mal conductances σ i ; (c) the thermal conductances σ i ; and (d)the real part of the coherence (cid:60) [ ρ ] versus the base temperature T B . We choose the parameters in units of ∆ : Γ E /∆ = Γ C /∆ = Γ B /∆ = γ /∆ ≡ , ε /∆ = 10 , ε /∆ = 7 , T E = ∆/ . , and T C = ∆/ . . into two separate parts. Figures 2 (b) and (c) display thethermal conductances σ i pertaining to the populationdistributions and to the coherence contributed thermalconductances σ i varying with the base temperature T B . σ E , σ C , and σ B share a magnitude close to each other,indicating that it is unlikely to create an autonomousthermal amplifier without coherence. Quantum coher-ence (cid:60) ( ρ ) exists [Fig. 2(d)], allowing us to modify thethermodynamic behavior through the quantum control.For the two thermal conductances σ B and σ B of thebase, σ B >0 [Fig. 2 (b)], whereas σ B originating fromthe coherence is negative [Fig. 2(c)], ensuring that weachieve a vanishing σ B [Fig. 2(a)]. Such a phenomenonmakes large thermal amplifications possible.The curves of the amplification factors α E and α C asfunctions of the base temperature T B are illustrated inFig. 3. The amplification factors α E and α C are clearlygreater than 1 over a large range of T B . As seen from Eq.(14), these effects result from σ E < , which is similar tothe property of some electrical circuits and devices wherean increase in voltage across the overall assembly resultsin a decline in electric current through it, i.e., negativedifferential conductance. Specifically, Fig. 3 shows thatthe amplification factors diverge at T B = 135 . due tothe fact that the thermal conductance of the base σ B = 0 ,induced by the quantum coherence. Under these condi-tions, an infinitesimal change in J B makes a considerabledifference in J E and J C .Figures 4 and 5 reveal the influences of the decoherencerate Γ B and the dephasing rate of the base γ on the per-formance of the thermal transistor. The base tempera-ture T B = ∆/ . , while the values of other parametersare the same as those used in Fig. 2. The amplifica-tion factor α C increases as Γ B increases in the small- Γ B regime ( Γ B < . Δ ) , but it decreases as Γ B increases Figure 3.
The amplification factors α E (solid line) and α C (dashedline) versus the base temperature T B . All parameters are the sameas those used in Fig. 2 𝒂 𝒄 ×𝟏𝟎 −𝟑 𝒃 Figure 4. (a) The overall thermal conductances σ i ; (b) the thermalconductances of the base σ Bj (inset); and (c) the amplificationfactors α E (solid line) and α C (dashed line) versus the decoherencerate of the base Γ B . in the large- Γ B regime ( Γ B > . Δ ) , while α C tendsto divergence for Γ B → . Δ . The amplification factor α E as a function of Γ B has opposite signs. The deco-herence rate Γ B is an important parameter for buildinga desirable amplifier. As illustrated in Figure 4(b), thethermal conductance σ B of the base is the sum of σ B and σ B . Once again, we observe that σ B is always positive,while the thermal conductance relevant to the coherenceeffect σ B <0 leading to a cancellation of the sum when Γ B → . Δ . For the same reason, the amplificationfactors diverge at Γ B → . Δ when σ B = 0 .Coherence is maintained in a nonequilibrium steadystate even in the presence of the dephasing bath. How-ever, a large dephasing rate has a deleterious effect on thecharacteristics of the TLS thermal transistor [Fig. 5(b)].Figure 5(a) shows that the absolute value | ρ | and thereal part (cid:60) [ ρ ] of coherence are monotonically decreas-ing functions of γ , the decoherence rate of the base. Thepure-dephasing bath acting on the TLS induces the lossof steady coherence, yielding smaller α E/C .In summary, we build a TLS to analyze the effects ofthe dipole–dipole interaction and the dephasing on theenergy transfer processes in a thermal transistor. Thecoupling between the two excited states of the TLS is ca-pable of generating steady coherence in a nonequilibriumenvironment, making the thermal fluxes behave nonlin-early. The coherence, at the same time, gives rise to 𝒂 𝒃
Figure 5. (a) The absolute value and the real part of coherence, | ρ | and (cid:60) [ ρ ] , versus the dephasing rate of the base γ . (b) Theamplification factors α E (solid line) and α C (dashed line) versusthe dephasing rate of the base γ . NTDR of the base. Quantum coherence enables the ther-mal flow through the collector and emitter to be con-trolled by a small change in the heat flux through thebase. Such a thermal transistor can amplify a small in-put signal as well as direct heat to flow preferentially inone direction. The thermal transistor effect can be sig-nificantly improved by optimizing the base temperatureand coherence rate or reducing the dephasing rate.
We thank Dr. Dazhi Xu for helpful discussions. This workhas been supported by the National Natural Science Foundation ofChina (Grant No. 11805159) and the Fundamental Research Fundfor the Central Universities (No. 20720180011).[1] N. Li, J. Ren, L. Wang, G. Zhang, P. Hänggi, and B. Li, Rev.Mod. Phys. , 1045-1066 (2012).[2] B. Li, L. Wang, and G. Casati, Appl. Phys. Lett. , 143501(2006).[3] L. Wang and B. Li, Phys. Rev. Lett. , 177208 (2007).[4] P. Ben-Abdallah and S. Biehs, Phys. Rev. Lett. , 044301(2014).[5] K. Joulain, J. Drevillon, Y. Ezzahri, and J. Ordonez-Miranda,Phys. Rev. Lett. , 200601 (2016).[6] Y. Zhang, Z. Yang, X. Zhang, B. Lin, G. Lin, and J. Chen,EPL, , 17002 (2018).[7] R. Sánchez, H. Thierschmann, and L. W. Molenkamp, Phys.Rev. B , 241401(R) (2017).[8] M. O. Scully, K. R. Chapin, K. E. Dorfman, M. B. Kim, andA. Svidzinsky, Proc. Natl. Acad. Sci. USA , 15097 (2011).[9] A. Insinga, B. Andresen, and P. Salamon, Phys. Rev. E ,012119 (2016).[10] K. E. Dorfman, D. Xu, and J. Cao, Phys. Rev. E , 042120(2018).[11] E. Romero, V. I. Novoderezhkin, and R. van Grondelle, Nature , 355–365 (2017).[12] A. Fruchtman, R. Gómez-Bombarelli, B. W. Lovett, and E.M. Gauger, Phys. Rev. Lett. , 203603 (2016).[13] S. Su, C. Sun, S. Li, and J. Chen, Phys. Rev. E , 052103(2016). [14] S. W. Li, C. Y. Cai, and C. P. Sun, Ann. Phys. , 19-32(2015).[15] E. Romero, R. Augulis, V. I. Novoderezhkin, M. Ferretti, J.Thieme, D. Zigmantas, and R. van Grondelle. Nat. Phys. ,676–682 (2014).[16] J. Brédas, E. H. Sargent, and G. D. Scholes, Nat. Mater. ,35–44 (2017).[17] H. E. D Scovil and E. O. Schulz-DuBois, Phys. Rev. Lett. ,262-263 (1959).[18] E. Boukobza and D. J. Tannor, Phys. Rev. Lett. , 240601(2007).[19] D. Xu, C. Wang, Y. Zhao, and J. Cao, New J. Phys. ,023003 (2016).[20] N. Lambert, Y. N. Chen, Y. C. Cheng, C. M. Li, G. Y. Chen,and F. Nori, Nat. Phys., , 10-18 (2013).[21] T. Werlang and D. Valente, Phys. Rev. E , 012143 (2013).[22] H. P. Breuer and F. Petruccione, The Theory of Open Quan-tum Systems (Oxford University Press, Oxford, 2001).[23] E. Boukobza and D. J. Tannor, Phys. Rev. A , 063822(2006).[24] E. Boukobza and D. J. Tannor, Phys. Rev. A , 063823(2006).[25] R. Kosloff and A. Levy, Annu. Rev. Phys. Chem.65