A multifunctional quantum thermal device: with and without inner coupling
aa r X i v : . [ qu a n t - ph ] A ug A multifunctional quantum thermal device: with and without inner coupling
Yong Huangfu, Shifan Qi, and Jun Jing ∗ Department of Physics, Zhejiang University, Hangzhou 310027, Zhejiang, China (Dated: August 31, 2020)Quantum thermal devices have attracted growing attentions since they can be used to manipulatethe heat currents of macroscopic thermal baths in a microscopic scale. In the present work, a three-level system attached to three thermal baths with different temperatures is exploited to serve asa valve, a refrigerator, an amplifier, and a thermometer in the quantum regime, via tuning theinner coupling strength of the system and the temperatures of the external baths. We discuss theconnections among these thermal functions in the steady state of this microscopic device with orwithout the inner coupling using the Redfield master equation under a partial secular approximation.It is found that the functions as valve, refrigerator and amplifier in this quantum model can berealized in a large parametric space. And a valve-associated thermometer of low-temperature canbe established without the assistance from the inner coupling of the system as well as the quantumcoherence in the steady state. A high sensitivity of the indirect measurement of the low temperatureof the sample can be obtained due to its nonlinear dependence on the temperatures of both the hotand the work terminals. Our study of such a multifunctional thermal device allows to provide adeeper insight to the underlying quantum thermodynamics mechanism associated with the quantumcoherence, which is induced by the inner coupling and demonstrated in the steady state of the three-level system. Also, our model is a concise instance for integrating multiple thermal functions into asingle microscopic system.
I. INTRODUCTION
Classical thermodynamics based on the macroscopicstatistics has been developed over two hundreds of yearsand has a mature theoretical framework. However itis always an interesting research project to incorporatequantum mechanics into thermodynamics, which stemsfrom a microscopic theory about discrete levels and quan-tum coherence. Many literatures have made an activeexploration about the role of quantumness in thermo-dynamics during the past two decades. These workson quantum thermodynamics can be roughly categorizedinto two divisions focused on the quantum effects fromenvironment [1–4] and those from system [5–8], respec-tively. The present work is devoted to investigate therole of the steady-state coherence determined by the in-ner coupling of the system utilized as a multifunctionalthermal device.Various quantum thermodynamical functions, suchas quantum valve [9–11], quantum refrigerator [12–14],quantum amplifier [15, 16], and quantum clock [17], haveraised growing attentions in recent years. These modelsprovide not only the fundamental physical platforms totest the macroscopic thermodynamic laws down to thelevel of quantum mechanics, but also the valuable ref-erences to design the microscopic quantum devices withcertain thermal functions to actively and precisely tunethe heat currents via the external macroscopic parame-ters, such as temperatures.The measurement on the temperatures, especially onthe low-temperatures of a microscopic system, is an open ∗ Email address: [email protected] problem in quantum thermodynamics. Precisely measur-ing the sample temperature in a very low region is par-ticularly an important issue in the development of mi-croscopic technologies. Thermometry at the microscopicscale has attracted a considerable amount of works in thefields of modern physics [18–21]. The smallest possiblethermometer that composed merely of a single qubit wasinvestigated in Ref. [22]. The authors studied the per-formance of the initial-state quantum coherence on thethermometer to discriminate a cold bath from a hot one.But it is necessary for that thermometer to find out thetemperatures of both baths in advance. Thus the single-qubit thermometry determines whether a bath was hotor cold rather than obtains exactly the unknown temper-ature of the colder sample. Schemes of a true thermome-ter in both theoretical and experimental aspects havebeen proposed in a variety of experimental platforms.For examples, optical thermometers have been tested onthe nitrogen-vacancy centres in diamond [19, 23] andquantum-dot system [24, 25]. Meanwhile, electronic ther-mometers have been available in the quantum-dot sys-tem [26–28] and in the superconducting qubits [29]. Inparticular, the low-temperature thermometry proposedin Ref. [29] is performed by a quantum Otto engine cou-pled to a hot reservoir with a known temperature T h anda cold one with an unkown temperature T c . A lineardependence of T c on T h was established when the heatengine is working with the Carnot efficiency. This resultimplies an instance to integrate multiple thermal func-tions into a single system.We endeavor to study a multifunctional thermal devicewithin the framework of the open-quantum-system the-ory. For a microscopic open system immersed in an en-vironment, master equations allow to track the relevantdegrees of freedom in both dynamics and steady-statebehavior, which are under the influence of all the otherdegrees of freedom that are not of the immediate inter-est or out of microscopic control [30–35]. Particularly, aperturbative expansion with respect to the system-bathcoupling strength is employed to obtain a master equa-tion for the system part. During an ordinary derivation, aMarkovian approximation is first applied to obtain a Red-field master equation, and then a further secular approx-imation leads to the master equation in the well-knownLindblad form [30, 36]. The Redfield master equationretains the steady-state coherence of the system undera non-equilibrium environment [31–33], yet which mayloss the positivity of the reduced density matrix in theshort-time scale. In contrast, the Lindblad master equa-tion [37, 38] always gives rise to a valid reduced densitymatrix upon the full secular approximation, yet fails totrack the system coherence in the long-time limit. Itwas recently reported that the positivity of the Redfieldmaster equation can be recovered by a partial secularapproximation [39, 40].In the present work, we investigate the steady-statequantum coherence induced by the inner-coupling withina microscopic model that is composed by a three-levelsystem and three external thermal-baths with differenttemperatures. This model can be regarded as a ther-mal device with three terminals. Non-vanishing quantumcoherence in the steady state is totally determined bythe inner coupling of the system as demonstrated in thesteady state. Certain thermal functions, such as valve,refrigerator and amplifier can be realized in this quantummodel with the inner coupling. Based on the functionsof valve and refrigerator, we further propose a quantumthermometer to measure the temperature of the coldestthermal bath. In particular, in the parametric space ofthe atomic-level spacings and the terminal temperatures,a sensitive thermometer can be realized at the workingpoints of a quantum valve (also the onset of a quantumrefrigerator).The rest of this work is organized as following. InSec. II, we present the total Hamiltonian of the model,including the interaction between the system and en-vironments (the three thermal baths). Then we intro-duce the Redfield master equation with a partial secularapproximation for this model with inner coupling. InSec. III, it is demonstrated that the quantum coherencein the steady state is induced by the inner coupling. Cer-tain multiple thermal functions are consequently realized,such as the valve and the amplifier for control the tar-get heat current and the refrigerator to cool down thecoldest bath. In Sec. IV, we present a theoretical schemeof a quantum thermometer for the coldest bath withoutthe assistance from the inner-coupling of the system andalso an accessible simulation proposal based on a doublequantum-dot system. In Sec. V, we summarize the wholework. II. MODEL
FIG. 1. Diagram of a three-level system coupled to threeterminals. The three terminals can be regarded as three bathswith different temperatures labelled as the hot ( h ), cold ( c ),and work ( w ) bath, respectively. In a typical thermodynamics model demonstrated inFig. 1, a three-level microscopic system consisting oflevel-1, b and a by the increasing order of energy, is cou-pled to three terminals in their respective thermal equi-librium states. They are labelled respectively by h , c ,and w indicating their functions and temperatures. Thetotal Hamiltonian for this model can be decomposed intothe system Hamiltonian, the bath Hamiltonian and theinteraction Hamiltonian: H = H S + H B + H SB = H S + X µ = h,c,w H µ + X µ = h,c,w S µ ⊗ B µ . (1)In the system energy basis (also named the bare basis),the Hamiltonian can be written as ( ~ ≡ H S = X l = a,b ω l | l ih l | + g ( | a ih b | + | b ih a | ) , (2)where the ground energy has been set as ω = 0 and g is the inner coupling strength between the two excitedstates [41]. The degenerate situation for ω a = ω b hasbeen investigated in Ref. [5] about the application of aquantum refrigerator. It can be covered by our model.Each thermal bath is assumed to be a collection of de-coupled harmonic oscillators with bosonic creation andannihilation operators b † q,µ and b q,µ for the mode q withfrequency ω q,µ in the bath µ : H µ = X q ω q,µ b † q,µ b q,µ . (3)The collective bath operators B µ in Eq. (1) describe theinstantaneous displacement of bath oscillators from theequilibrium state, B µ = X q λ q,µ (cid:0) b † q,µ + b q,µ (cid:1) . (4)The system operators S µ describe the interaction be-tween the three-level system and the environments: S h = | ih a | + | a ih | ,S c = | ih b | + | b ih | ,S w = ( | a ih a | − | b ih b | ) sin φ + ( | a ih b | + | b ih a | ) cos φ, (5)where φ ≡ arctan(2 g/ ∆) with ∆ ≡ ω a − ω b . The operator S w depends explicitly on the inner coupling-strength g of the system. Energy current can be transferred amongthe hot, cold and work baths through the microscopicsystem. III. THREE-LEVEL SYSTEM WITH THEINNER COUPLING
The derivation onset of a microscopic master equa-tion [36] for the open-quantum-system dynamics requiresall the system operators to be expressed in eigenbasis | λ i of the system Hamiltonian. In the present model, theHamiltonian (2) of the three-level system with the innercoupling should be diagonalized as H S = X λ =2 , ω λ | λ ih λ | ,ω , = 12 (cid:16) ω a + ω b ∓ p g + ∆ (cid:17) , | i = cos φ | a i − sin φ | b i , | i = sin φ | a i + cos φ | b i . Consequently, the coupling operators S µ , µ = h, c, w , inthe interaction Hamiltonian H SB are rewritten as S h = (cid:18) cos φ | ih | + sin φ | ih | (cid:19) + h.c.,S c = (cid:18) cos φ | ih | − sin φ | ih | (cid:19) + h.c.,S w = | ih | + | ih | . (6)The operators S h and S c in the eigenbasis indicate thatthe two excited states | i and | i are simultaneouslycoupled to the ground state | i via both the hot andthe cold baths. These interaction channels as shown inFig. 3 would induce quantum coherence in the densitymatrix spanned by {| i , | i , | i} . The bath- w does workto the system through the operator S w , which has beenapplied to the thermal rectification and the heat ampli-fication [35]. A. Master equation and the steady state withquantum coherence
As derived in appendix A, the Redfield master equa-tion with a partial secular approximation about the present model can be explicitly expressed by˙ ρ S = − i [ H S , ρ S ] + X µ = h,c,w D µ [ ρ S ] , (7) D h [ ρ S ] = Γ + h ( ω ) L τ ( ρ S ) + Γ − h ( ω ) L τ ( ρ S )+ Γ + h ( ω ) L τ ( ρ S ) + Γ − h ( ω ) L τ ( ρ S )+ X m =2 , (cid:26) Γ − h ( ω m )[ τ m ρ S τ m + τ ¯ m ρ S τ m ]+ Γ + h ( ω m )[ τ m ρ S , τ ¯ m ] + [ τ m , ρ S τ m ] (cid:27) , D c [ ρ S ] = Γ + c ( ω ) L τ ( ρ S ) + Γ − c ( ω ) L τ ( ρ S )+ Γ + c ( ω ) L τ ( ρ S ) + Γ − c ( ω ) L τ ( ρ S ) − X m =2 , (cid:26) Γ − c ( ω m )[ τ m ρ S τ m + τ ¯ m ρ S τ m ]+ Γ + c ( ω m )([ τ m ρ S , τ ¯ m ] + [ τ m , ρ S τ m ]) (cid:27) , D w [ ρ S ] = Γ + w (Ω) L τ ( ρ S ) + Γ − w (Ω) L τ ( ρ S ) , where the Lindblad superoperator is defined as L X ( ρ S ) ≡ Xρ S X † − X † Xρ S − ρ S X † X , with X an arbitrary sys-tem operator τ nm ≡ | n ih m | ( ¯ m ≡ − m ) and Ω ≡ ω − ω = p g + ∆ . The transition rates can be fac-tored as Γ ± µl ( ω ) = f l Γ ± µ ( ω ), l = 1 , , µ = h, c, w ,where f = sin φ cos φ , f = cos φ , f = sin φ . AndΓ ± µ ( ω ) can be further factored as the product of G µ ( ω )and n µ ( ∓ ω ), where G µ ( ω ) ≡ π P q | λ q,µ | δ ( ω q,µ − ω ) isthe spectral density function of the bath- µ and n µ ( ω ) =( e β µ ω − − with β µ = 1 /T µ ( k B ≡
1) is the averagepopulation determined by the temperature of bath- µ . Inthis work, the three baths are assumed to be Ohmic,i.e., G µ ( ω ) = γ µ ωe − ω/ω c , where γ µ is a dimensionlesssystem-bath coupling strength and ω c is the cutoff fre-quency characterizing the largest energy scale.In comparison to the conventional Redfield masterequation under the full secular approximation, there isan extra summation on the last two lines for both D h and D c in Eq. (7), which is the slowly-oscillating termretained by the partial secular approximation.The master equation (7) yields a steady state ρ SS inthe following form: ρ SS = ρ ρ ρ ρ ρ , (8)where the non-vanishing off-diagonal elements are ob-tained by letting ˙ ρ S = 0 in Eq. (A11). In Fig. 2, theabsolute value of coherence | ρ | measuring the quantum-ness of the three-level system demonstrates a nontrivialdependence on the inner coupling strength g and the tem-perature of the work bath T w . A remarkable steady-statecoherence | ρ | turns out with a moderate magnitude of g and roughly decreases with an increasing T w . It is shownthat a too small or a too big g and a high T w are not FIG. 2. The steady-state coherence | ρ | versus the innercoupling strength g and the temperature of the work bath T w .The other parameters are ω b /ω a = 0 . γ h,c,w /ω a = 0 . ω c /ω a = 50, T h /ω a = 1 and T c /ω a = 0 . favourable to the residue quantumness of the three-levelsystem under the three thermal baths after a long-timeevolution. The maximal value of | ρ | is achieved when g/ω a = 0 .
02, a weak inner-coupling strength that is nu-merically obtained under the parametric setting of Fig. 2.
B. Thermal functions
A quantum system can be regarded as a microscopicthermal device when it can be used to control the heatcurrents back and forth from the system to the baths. Inthis subsection, it is shown that the open three-level sys-tem with inner coupling can be utilized as an integratedmultifunctional thermal device with functions as valve,refrigerator and amplifier within certain parametric re-gions.
FIG. 3. Diagram of the heat currents J µ ( µ = h, c, w ) betweenthe three-level system with inner coupling and the three ther-mal baths. J c is described as the heat current between thethree-level system and the cold bath, which is composed by J (12) c and J (13) c as displayed in Eq. (10). The rate of the energy transfer between the systemand the bath- µ ( µ = h, c, w ), Tr[ H S D µ ( ρ S )], defines the corresponding heat current. In the long-time limit, ρ S takes the steady-state solution ρ SS given by Eq. (8). Thestandard formula for the (steady-state) rate of the heatexchange with bath- µ [5] is J µ ≡ Tr [ H S D µ ( ρ SS )] . (9)Note a positive J µ means the heat current follows the di-rection from bath- µ to the central three-level system. Bythe master equation (7) and the definition about currentin Eq. (9), the steady-state energy current between thethree-level system and the cold bath can be decomposedinto two parts, each in charge of the coupling betweenthe energy-level pair {| i , | l i} and the cold bath: J c = X l =2 , J (1 l ) c = X l =2 , ω l (cid:2) Γ − c ¯ l ( ω l ) ρ − Γ + c ¯ l ( ω l ) ρ ll + Γ + c ( ω ¯ l )( ρ + ρ ) (cid:3) , (10)where ¯ l ≡ − l . Similarly, the heat current between thethree-level system and the hot bath can be expressed by J h = X l =2 , ω l (cid:2) Γ − hl ( ω l ) ρ − Γ + hl ( ω l ) ρ ll − + h ( ω ¯ l )( ρ + ρ ) (cid:3) . (11)The steady-state quantum coherences, i.e., the terms ρ and ρ , contribute to both J c and J h . While the heatcurrent between the three-level system and the work bathinvolves only with the populations on the two excitedlevels {| i , | i} of the system, J w = 2Ω (cid:2) Γ − w (Ω) ρ − Γ + w (Ω) ρ (cid:3) . (12)These currents in Eqs. (10), (11) and (12) are draftedin Fig. 3. One can check that they follow the law ofenergy conservation (the first law of thermodynamics): J c + J h + J w = 0. More detailed while cumbersome ana-lytical expressions for the three currents can be directlyobtained by the definitions of the decay rates Γ ± µ,l and thematrix elements ρ jk of the steady state in Eq. (8). Thefollowing results about the steady-state currents undercontrol are numerically obtained.Here the work bath is used as a control terminal tomanipulate the three currents embodying various ther-mal functions in the quantum regime. In analogy to aclassical valve, a quantum thermal valve can precisely cutoff the heat current from any one of the terminals whileleaving the currents to flow through the other two ter-minals under special conditions. The three heat currentswith respect to the temperature of control terminal T w are demonstrated in Fig. 4(a), where the inner couplingstrength g/ω a is set as 0 .
02. When T w /ω a approaches acritical value, which is about 3 .
42 with the parameters wechosen in the plot, the energy current J h vanishes whilethe other two currents do not. And then when T w /ω a approaches another critical value (about 3 . J c disappears. These critical values are fully de-termined by the settings of the system energy structure -5 FIG. 4. The three-level system in the steady-state can beregarded as a valve for the current J c or J h . (a) The threeheat currents J µ ( µ = c, h, w ) versus the temperature of workbath T w with inner coupling strength g/ω a = 0 .
02. (b) Theworking spots ( J h = 0 and J c = 0) of quantum valve by T w and g . The other parameters are set as ω b /ω a = 0 . γ h,c,w /ω a = 0 . ω c /ω a = 50, T h /ω a = 1 and T c /ω a = 0 . and the temperatures of the other two terminals. Theyare the working spots for the function as a quantum valve,which can be tuned by the inner coupling strength g asshown in Fig. 4(b). It is found that as long as T w is overa critical value, the quantum thermal valve for J c = 0can be realized by increasing g . A higher T w correspondsto a larger g . In comparison, when T w has been properlydetermined by the rest parameters, the choice of g for J h = 0 is almost irrelevant to the value of T w . -5 -3 -2 -1 FIG. 5. The three-level system under the steady state canalso be regarded as a refrigerator for the coldest bath. (a)The current J c versus the temperature of work bath T w withinner coupling strength g/ω a = 0 .
02. (b) The coefficient ofcooling performance (COP) under various g and T w . (c) COPversus T w with g/ω a = 0 .
02. The other parameters are set as ω b /ω a = 0 . γ h,c,w /ω a = 0 . ω c /ω a = 50, T h /ω a = 1 and T c /ω a = 0 . When focusing on the direction or signal of the currentflows between the system and the cold bath, one can findthat our three-level system under the steady state can be used as a quantum refrigerator characterized by observ-ing a net current flows out of the cold bath, i.e. J c > J c switches from negativeto positive when T w /ω a is over the critical point 3 .
53 ofthe quantum valve. According to Eq. (10), J c can bedecomposed of J (12) c and J (13) c (also shown in Fig. 3).The two sub-currents are the results by the coupling be-tween respective energy-level pairs of the system and thecold bath. One can see that the heat current J (13) c stem-ming from the top level | i is always negative, i.e., thetop level | i always conveys energy to the cold bath. Itcan be understood that the effective temperature for theenergy-level pairs of {| i , | i} is always higher than thetemperature of bath- c . That is supported by the exis-tence of inner coupling within the three-level system. Incontrast, J (12) c could become positive by increasing T w .In the refrigerator regime depending on the working bath, J c > J w >
0. According to the first law of thermo-dynamics, the heat current J h follows the direction fromthe central three-level system to the bath- h , i.e., J h < J c >
0, which is defined as η ≡ J c /J w [5], presents in Fig. 5(b). It is shown that thecooling efficiency of our quantum refrigerator will begradually degraded with an increasing inner-coupling. InFig. 5(c), we also compare the COP of our refrigeratorwith the Carnot limit η C : η C ≡ β h − β w β c − β h , (13)which is the upper bound of the cooling efficiency forthe thermal-machine. Meanwhile, due to the Clausiustheorem for the second law of thermodynamics, we have˙ Q c T c + ˙ Q h T h + ˙ Q w T w = J c T c + J h T h + J w T w ≤ , (14)where ˙ Q µ , µ = h, c, w , is the heat current transferredbetween the system and bath- µ . Based on the first andsecond laws of thermodynamics, one can obtain η ≡ J c J w ≤ T h − T w T c − T h · T c T w = β h − β w β c − β h = η C . (15)It is shown in Fig. 5(c) that η is always less than η C .An overall picture for the thermal functions of valveand refrigerator can be illustrated by a phase diagramin Fig. 6 for the heat current J c with the temperatureof the work bath T w and the inner coupling g . The redbelt consisting of the working spots with J c = 0 for thequantum valve of the heat current between the systemand the cold bath separates the refrigerator regime (the FIG. 6. The phase diagram of the thermal functions of valveand refrigerator in the parametric space of the temperatureof the work bath T w and the inner coupling strength g . Theother parameters are set as ω b /ω a = 0 . γ h,c,w /ω a = 0 . ω c /ω a = 50, T h /ω a = 1 and T c /ω a = 0 . yellow region) with J c > FIG. 7. The amplification factor α J under various temper-atures of the work bath T w and the inner-coupling strength g . The other parameters are set as ω b /ω a = 0 . γ h,c,w /ω a =0 . ω c /ω a = 50, T h /ω a = 1 and T c /ω a = 0 . In the same parametric space as in Fig. 6, we can alsoshow the region of a quantum thermal amplifier set up byour model. Here we focus on the variation of the heat cur-rent between the work bath and the system J w upon thatof the current between the cold bath and the system J c ,i.e., the amplification factor [15] α J ≡ | ∂J c /∂J w | . Due tothis definition, an amplifier can be realized when α J > J w . The regionof the amplification factor α J > α J <
1. It is clear thatthe three-level microscopic system with a small inner- coupling can be utilized as an amplifier. By observingthe overlap between the region of valve and refrigeratorin Fig. 6 and that of the amplifier in Fig. 7, one can findthat all of these thermal functions can be realized in ourmodel with a finite inner-coupling strength g under thesame parametric setting. IV. THREE-LEVEL SYSTEM WITHOUT THEINNER COUPLING
In this section, we assume a vanishing inner-couplingstrength within the system, i.e., g = 0. In this condi-tion, the system Hamiltonian is diagonal in the bare basis {| i , | a i , | b i} . The coupling operators S h and S c are thesame as those in Eq. (5), and the operator S w reduces to S w = | a ih b | + | b ih a | . (16)Now the interaction with the work bath merely inducesthe energy exchange between the two excited states with-out affecting their energy levels. A. The steady state without quantum coherence
The absence of the inner-coupling g = 0 renders φ = 0.Thus the master equation (7) reduces to˙ ρ S = − i [ H S , ρ S ] + X µ = h,c,w D µ [ ρ S ] , (17) D h [ ρ S ]= Γ + h ( ω a ) L τ a ( ρ S ) + Γ − h ( ω a ) L τ a ( ρ S ) , D c [ ρ S ]= Γ + c ( ω b ) L τ b ( ρ S ) + Γ − c ( ω b ) L τ b ( ρ S ) , D w [ ρ S ]= Γ + w (∆) L τ ba ( ρ S ) + Γ − w (∆) L τ ab ( ρ S ) . In the long-time limit, the master equation (17) gives riseto a diagonal steady-state of the system: ρ SS = ρ ρ bb
00 0 ρ aa , (18)where ρ = Γ + c ( ω b )[Γ + h ( ω a ) + Γ + w (∆)] + Γ + h ( ω a )Γ − w (∆)Λ ,ρ bb = Γ − c ( ω b )[Γ + h ( ω a ) + Γ + w (∆)] + Γ − h ( ω a )Γ + w (∆)Λ ,ρ aa = Γ − h ( ω a )[Γ + c ( ω b ) + Γ − w (∆)] + Γ − c ( ω b )Γ − w (∆)Λ , (19)with the normalization factor Λ = Γ + h ( ω a )Γ + c ( ω b ) +Γ + c ( ω b )Γ + w (∆) + Γ + h ( ω a )Γ − w (∆) + Γ + h ( ω a )Γ − c ( ω b ) +Γ − c ( ω b )Γ + w (∆) + Γ − h ( ω a )Γ + w (∆) + Γ − h ( ω a )Γ + c ( ω b ) +Γ − c ( ω b )Γ − w (∆) + Γ − h ( ω a )Γ − w (∆). Physically, the threequantum channels connecting the thermal baths and thesystem are separable in the absence of the inner-coupling.The dynamics of the populations and coherence in thismodel are therefore naturally decoupled from each other.Consequently the quantum coherence vanishes in thelong-time limit.The steady-state solution certainly satisfies the prin-ciple of detailed balance. Note the rates of Γ + µ ( ω ) andΓ − µ ( ω ) represent the probabilities of the decay and exci-tation transitions, respectively. Due to the conservationof the total population in the steady state, the proba-bilities of the population gain must be equivalent to thepopulation loss for each level. In particular, we haveΓ + c ( ω b ) ρ bb + Γ + h ( ω a ) ρ aa = [Γ − c ( ω b ) + Γ − h ( ω a )] ρ , Γ − c ( ω b ) ρ + Γ + w (∆) ρ aa = [Γ − w (∆) + Γ + c ( ω b )] ρ bb , Γ − h ( ω a ) ρ + Γ − w (∆) ρ bb = [Γ + h ( ω a ) + Γ + w (∆)] ρ aa . (20)The result in Eq. (19) can be determined by the solu-tion of Eq. (20) and the normalization condition withoutinvoking the master equation. B. Microscopic Thermometer
While temperature is an intuitive notion deeply rootedin the daily life as well as the classical world, yet it issubtle and surprisingly difficult to formalise in quantummechanics, especially in the field of low temperature re-gion [18, 42]. In this subsection, we introduce a quantumthermometer to estimate the temperature of the coldestbath in the model of a three-level system without inner-coupling. Here baths- c , h , and w are respectively labelledas the sample, conductor, and control terminals in liter-atures and it is assumed that the sample temperaturecannot be directly measured, in contrast to the temper-atures of the conductor and the control terminals.Due to the definition in Eq. (9), the steady-state energycurrent between the sample and the system is given by J c = 2 ω b [Γ − c ( ω b ) ρ − Γ + c ( ω b ) ρ bb ] . (21)In comparison with Eq. (10), J c is now equivalent to J (12) c having no contribution from quantum coherence. Simi-larly, the heat currents between the conductor and con-trol terminals and the system are respectively given by J h = 2 ω a [Γ − h ( ω a ) ρ − Γ + h ( ω a ) ρ aa ] , (22) J w = 2∆[Γ − w (∆) ρ bb − Γ + w (∆) ρ aa ] . (23)In Figs. 8(a) and (b), the heat currents are describedby Eqs. (21), (22), and (23) with respect to the temper-ature of the control terminal T w . It is shown that ourthree-level system in the absence of the inner couplingcan work as a special thermal valve for both J c and J h when J w is switched off. The particular working pointsof the valve depend on the choice about the other param-eters. The parameters including the temperatures of thethree baths would determine the population distributionson the three levels of our system. One can then definetwo effective temperatures for the energy-level pairs of -4 -4 FIG. 8. The thermal behaviors vary with temperature of thecontrol terminal in the model without the inner coupling. (a)and (b): the heat currents towards the three terminals with T c /ω a = 0 . T c /ω a = 0 .
7, respectively. (c) and (d): theeffective temperatures under the same conditions about T c for (a) and (b), respectively. The other parameters are set as ω b /ω a = 0 . T h /ω a = 1, γ c,h,w /ω a = 0 .
008 and ω c /ω a = 50. {| i , | a i} and {| i , | b i} due to the Boltzmann distribu-tion. They read, T s = ω s / ln (cid:18) ρ ss ρ (cid:19) , s = a, b. (24)In general situations for the non-equilibrium steady state,the nonzero currents J c and J h will lead to T h = T a and T c = T b as demonstrated in Figs. 8(c) and 8(d). Thusat the working points of quantum valve, for instances T w /ω a = 1 . T h /ω a = 1, T c /ω a = 0 . T w /ω a = 2 . T h /ω a = 1, T c /ω a = 0 . T a = T h and T b = T c and the self-consistencyabout the effective temperatures. In another word, thepopulations of the three-level system in the thermal equi-librium state [see the two instances in Figs. 8(c) and 8(d)]satisfy ρ aa ρ = e − ωaTh , ρ aa ρ bb = e − ∆ Tw , ρ bb ρ = e − ωbTc . (25)These expressions yield ω a T h = ∆ T w + ω b T c . (26)Note again that the two T w ’s in Fig. 8 are twospecial cases in accordance to the choices of bath-temperatures, system energy configuration, and the in-teraction strengthes with the baths. In particular, aproper T w for an equilibrium state can always be ob-tained by sweeping over the parametric space when T c is settled and it increases with a decreasing T c . Efficientcooling a lower temperature sample requires more energyinput from the control terminal with a higher tempera-ture.The three-level system under the thermal equilibriumstate can be used to switch on the quantum functionsof both valve and refrigerator. In particular, the workingpoints ( J c = 0) for the quantum valve are the onset pointfor the quantum refrigerator ( J c > η ≡ J c J w = ω b ∆ , (27)which is obtained by Eqs. (20), (21) and (23). Comparingto the T w -dependent result (see Fig. 5) in the nonvanish-ing inner-coupling situation, now the COP η becomes aconstant ω b / ∆ = ω b / ( ω a − ω b ). Taking Eq. (26) intoaccount, it is interesting to find that the equivalence inEq. (15), i.e., the Carnot limit, can be achieved at theworking points of the quantum valve. With respect tothe quantum refrigerator, according to Eq. (21), J c ≥ − c ( ω b )Γ + h ( ω a )Γ − w (∆) ≥ Γ + c ( ω b )Γ − h ( ω a )Γ + w (∆) , (28)where Γ ± µ ( ω ) = G µ ( ω ) n µ ( ∓ ω ) with G µ ( ω ) the spectraldensity function and n µ ( ω ) the average occupation num-ber of bath- µ . Immediately we have e β h ω a ≥ e β c ω b e β w ∆ , (29)which is equivalent to Eq. (15). Thus the onset of thecooling windows as well as the refrigerator performanceof our multifunctional device in the absence of the innercoupling is always consistent with the Carnot limit. FIG. 9. Diagram of the directions of heat currents J µ ( µ = h, c, w ) between the system without the inner coupling andthe three thermal terminals in the refrigerator regime. In the refrigerator regime, J c > J w > J h <
0. Their directions are shown in Fig. 9. One cantherefore observe the heat current flows from the sampleto the conductor through the three-level system. Ac-cording to the definitions of the effective temperaturesin Eq. (24), the effective temperatures of the system andthe temperatures of terminal- h and - c can be ordered by T b < T c < T h < T a . More importantly, the sample temperature T c can beimmediately obtained as T c = T h T w ξT w − (1 − ξ ) T h , (30)where ξ ≡ ω b /ω a , by the thermal equilibrium condition.This relation between T c and the other two temperaturesindicates an indirect measurement approach for T c bymeasuring the control temperature T w when the systemmoves into a thermal-equilibrium state. In addition, theenergy-configuration parameter of the three-level system ξ and the temperature of conductor terminal T h are sup-posed to be determined in advance. FIG. 10. Under different conductor temperature T h , (a) thecontrol temperature T w as a function of the sample temper-ature T c ; (b) the measurement sensitivity for the quantumthermometer α T as a function of T c . The energy-configurationparameter is set as ξ = 0 . From Eq. (30), the lower bound of the sample tem-perature that can be measured by our thermometer isdetermined by lim T w →∞ T c = ξT h = ω b ω a T h , (31)which means that the measurement range for T c islinearly proportional to both T h and the energy-configuration parameter ξ . This result under an infinitehigh-temperature control terminal coincides with thatobtained by a previous proposal about the quantum ther-mometer connecting to a quantum thermal machine [29],in which the sample temperature T c has a linear rela-tionship with the conduct terminal T h . In the presentthermometer scheme, however, we have a non-linear re-lation between the sample temperature and the controltemperature: T c T w = ξξ − ξ ( ξ − T w T h + ξ ( ξ − (cid:18) T w T h (cid:19) + O (cid:18) T w T h (cid:19) . (32)Thus our proposal might provide a more sensitive mea-surement on T c than the existing one.In Fig. 10(a), we display the dependence of the con-trol temperature T w on the sample temperature T c un-der three different conductor temperatures T h and a fixedparameter ξ = 0 .
6. Note T c ≤ T h according to the defini-tion and T c ≥ ξT h due to the constraint by Eq. (31). Asalso shown in Fig. 10(a), the variation of T w with respectto T c is rapidly enhanced by decreasing T c . This be-havior is more clear in the measurement sensitivity [18] α T shown in Fig. 10(b), which is defined as the abso-lute value of the control temperature bias divided by thesample-temperature variation: α T ≡ (cid:12)(cid:12)(cid:12)(cid:12) ∂T w ∂T c (cid:12)(cid:12)(cid:12)(cid:12) = ξ (1 − ξ ) T h ( ξT h − T c ) . (33)Then for a prescribed critical value of the sensitivity α T , a high-precision measurement region for the sam-ple temperature can be confirmed as ( ξT h , T ′ c ], where theso-called critical sample temperature T ′ c is T ′ c = ξT h + s ξ (1 − ξ ) T h α T . (34)In Fig. 10(b), we set α T = 10 as displayed by the horizon-tal dashed line. Then the cross points of the vertical linesand the horizontal line are critical temperatures T ′ c ( α T ).For example, when T h /ω a = 2 and ξ = 0 .
6, one canperform a high-precision indirect measurement for thesample temperatures in the range 1 . < T c /ω a ≤ . C. Experimental simulation of the microscopiclow-temperature thermometer
The quantum-dot systems as a kind of experimentalplatform have been widely applied in the field of quan-tum thermodynamics to simulate or realize various ther-mal functions, such as the heat engine [43–45], the re-frigerator [46–50], the novel energy carrier [26, 51, 52],and a Maxwell demon in the strong-coupling regime [53].Here we can show that a quantum-dot system can alsobe used to display the thermometer function in the pro-ceeding subsection, by simulating the heat currents withthe charge currents.A three-terminal quantum-dot thermometer is consid-ered in Fig. 11. The three-level system can be mimickedby a double-quantum-dot system which consists of twocapacitively coupled quantum dots (QD and QD ) op-erated in the Coulomb-blockade regime. The system iscoupled to three terminals characterized by the two metalleads and a radiation field, which are labelled respectivelyby h , c , and w [43]. The temperatures of the three termi-nals satisfy the same ordering set up by our theoreticalproposal: T c < T h < T w . The terminal- h ( c ) can onlyexchange electrons with the QD . The corresponding FIG. 11. An experimental scheme for the quantum thermome-ter established in our microscopic device, which can be real-ized by an open double-quantum-dot system having a singleelectron coupled to three independent baths consisting of twometal leads and a radiation field. The three levels | i , | b i ,and | a i of the central system in our theoretical model (seeFig. 9) can be mapped to the electron states | i , | i , and | i , respectively. Hamiltonian [43] is H = H + V M + V P , (35) H = X m =1 , ε m c † m c m + X µ = h,c ε µ b † µ b µ + X α ω α a † α a α , (36) V M = ( V h c b † h + V c c b † c ) + H . c ., (37) V P = X α ( V α a α c † c + H . c . ) . (38)The three terms in the free Hamiltonian H of Eq. (36)describe the isolated quantum dots, the free leads, andthe radiation field, respectively. c ( c † ), b ( b † ) and a ( a † )are the annihilation (creation) operators for the electronin QD, the electron along the lead, and the photons, re-spectively. The Hamiltonian V M in Eq. (37) describesthe coupling between the quantum dots and the metalleads. The Hamiltonian V P in Eq. (38) describes thedot-radiation-field inteaction, addressing both the driv-ing and the spontaneous light emission. Similar Hamil-tonians can also be found in Refs. [54, 55].A remarkable feature of the Coulomb-coupledquantum-dot system is that the electron transportthrough the system is forbidden but the capacitive cou-pling between the two dots allows electronic fluctua-tions to transmit heat between the terminals [26, 56].Coulomb interactions prevent two electrons from be-ing present at the same time in the Coulomb-blockaderegime. The Coulomb-coupled quantum-dot system inthe subspace spanned by {| i , | i , | i} constitutes anartificial three-level atom [26, 43, 47], where the energylevels | i and | i indicate the state of the quantum dotoccupied by one and zero electron, respectively. Then thethree states of the center system in our model could bemapped to the double-dot system in the single-electronsubspace by | i ↔ | i , | a i ↔ | i and | b i ↔ | i . Thedifferent dot energies ε and ε can be regarded as the0energy gaps ω a and ω b , respectively. The energy con-figuation of the quantum-dot system [ ξ in Eq. (30)] canbe determined by manipulating the gate voltage in ad-vance [46].The thermometer by the quantum-dot system can beimplemented by the following procedure. As indicatedby Fig. 8, one can start from T w = T h , and increase T w until J h = 0, which is in experiment the charge currentbetween the central system and the conductor terminal.At this working-point, the quantum-dot system actuallyreaches a thermal equilibrium state. Then the tempera-ture of sample terminal T c can be obtained by measuring T w according to Eq. (30). V. CONCLUSION
In conclusion, we have investigated the steady-statethermal functions realized in an open-quantum-systemmodel of a three-level system attached to three thermalbaths with or without inner coupling. The residue quan-tum coherence in steady-state is obtained by the Redfieldmaster equation with a partial secular approximation.It is found that a microscopic multifunctional thermaldevice can be established by varying the inner couplingwithin the system and the temperatures of the externalbaths. We identify the respective working regimes for thethermal functions as valve, refrigerator, and amplifier.We also demonstrate that the three-level system canbe utilized as a microscopic thermometer to determinethe temperature of coldest terminal in the absence of theinner coupling. The quantum thermometer is establishedwhen the three-level system reaches the thermal equilib-rium state under the three terminals, which also switcheson the functions of valve and refrigerator. We find a non-linear dependence of the sample-temperature on the hotand the control temperatures, which could be exploitedto perform a high-precision measurement. Our work pro-vides a deep insight to understand the roles of the systemquantumness and the inner-coupling in quantum thermo-dynamics.
ACKNOWLEDGEMENTS
We acknowledge grant support from the National Sci-ence Foundation of China (Grants No. 11974311 and No.U1801661), Zhejiang Provincial Natural Science Founda-tion of China under Grant No. LD18A040001, and theFundamental Research Funds for the Central Universities(No. 2018QNA3004).
Appendix A: The master equation under the partialsecular approximation
In this appendix, we derive the master equation (7) inthe main text under the partial secular approximation. It is of a Redfield master equation that conserves thepositivity of the reduced density matrix even in the short-time scale.The interaction Hamiltonian describing the couplingbetween the three-level system and the three thermalbaths in Eq. (1) can be written as H SB = X µ = h,c,w H µSB = X µ = h,c,w S µ ⊗ B µ . (A1)In the interaction picture with respect to H S + H B , itbecomes H SB ( t ) = X µ = h,c,w H µSB ( t ) = X µ = h,c,w S µ ( t ) ⊗ B µ ( t ) , (A2)where S h ( t ) = cos φ (cid:0) τ e − iω t + τ e iω t (cid:1) + sin φ τ e − iω t + τ e iω t ) , (A3) S c ( t ) = cos φ (cid:0) τ e − iω t + τ e iω t (cid:1) − sin φ τ e − iω t + τ e iω t ) , (A4) S w ( t ) = τ e − i Ω t + τ e i Ω t , (A5)with τ mn ≡ | m ih n | and Ω ≡ ω − ω . The collective bathoperators are B µ ( t ) = X q λ q,µ (cid:0) b q,µ e − iω q,µ t + b † q,µ e iω q,µ t (cid:1) . (A6)Under the assumptions of a weak coupling betweenthe microscopic system and the thermal baths (Born-approximation) and an ignorable relaxation time-scalefor the thermal baths (Markovian approximation), onecan apply the Redfield master equation to investigatethe central-system dynamics to the second order of thecoupling strength:˙ ρ S = − Z ∞ ds Tr B [ H SB ( t ) , [ H SB ( t − s ) , ρ S ⊗ ρ B ]] . (A7)The commutator operators in Eq. (A7) expands as[ H SB ( t ) , [ H SB ( t − s ) , ρ S ⊗ ρ B ]] (A8)= H SB ( t ) H SB ( t − s ) ρ S ρ B − H SB ( t ) ρ S ρ B H SB ( t − s ) − H SB ( t − s ) ρ S ρ B H SB ( t ) + ρ S ρ B H SB ( t − s ) H SB ( t ) . Partial tracing over the degrees of freedom of the bathsupon substituting the interaction Hamiltonian (A2) intothe master equation (A7) turns out to be a summationover 12 terms. The derivation is tedious but straightfor-1ward. A typical term presents as following:Tr B (cid:2) H hSB ( t ) ρ S ρ B H hSB ( t − s ) (cid:3) (A9)= Γ + h ( ω )[ τ ρ S τ e − iω t + τ ρ S τ ]+ Γ + h ( ω )[ τ ρ S τ e − i ( ω + ω ) t + τ ρ S τ e − i ( ω − ω ) t ]+ Γ − h ( ω )[ τ ρ S τ + τ ρ S τ e iω t ]+ Γ − h ( ω )[ τ ρ S τ e i ( ω − ω ) t + τ ρ S τ e i ( ω + ω ) t ]+ Γ + h ( ω )[ τ ρ S τ e − iω t + τ ρ S τ ]+ Γ + h ( ω )[ τ ρ S τ e − i ( ω + ω ) t + τ ρ S τ e − i ( ω − ω ) t ]+ Γ − h ( ω )[ τ ρ S τ + τ ρ S τ e iω t ]+ Γ − h ( ω )[ τ ρ S τ e i ( ω − ω ) t + τ ρ S τ e i ( ω + ω ) t ] . Here the decay rates are defined as Γ ± hl ( ω ) = Γ ± h ( ω ) f l with the factors f = sin φ cos φ , f = cos φ , f = sin φ and Γ ± h ( ω ) = G h ( ω ) n h ( ∓ ω ). n h ( ω ) is the average occu-pation number characterized by the temperature of thebath- h , n h ( ω ) = ( e β h ω − − with β h = 1 /T h ( k B ≡ G h ( ω ) = γ h ωe −| ω | /ω c , where γ h is a dimensionless cou-pling strength and ω c is the cutoff frequency.Collecting all the 12 terms renders the Redfieldmaster equation. Many time-dependent terms (calledthe non-secular terms), such as τ ρ S τ e − iω t , τ ρ S τ e − i ( ω + ω ) t , and τ ρ S τ e − i ( ω − ω ) t , thenpresent in Eq. (A9).We denote by t S the timescale of the intrinsic evolutionof the system, which is in the same order of a typical valuefor | ω m − ω n | − , m = n , involving the energy-spacing of the system. If t S is much larger than the typicaltimescale of the relaxation time of the system t R ∼ γ − µ , µ = h, c, w , the non-secular terms of rapid oscillation maybe safely neglected [36]. Thus the condition of perform-ing a full secular approximation is γ µ ≪ | ω m − ω n | .While in many models as well as ours, although thedecay rates are set as γ h,c,w /ω a ∼ .
01 satisfying Bornapproximation, the energy level ω b is comparable to ω a in magnitude. Thus the actual condition is ω + ω ≫ ω − ω ∼ γ h,c,w . Consequently we have to employ apartial secular approximation by neglecting the rapid-oscillating terms, such as e − iω t and e − i ( ω + ω ) t , butkeeping the slow-oscillating terms, such as e − i ( ω − ω ) t .Under the partial secular approximation, Eq. (A9) isthus rewritten asTr B (cid:2) H hSB ( t ) ρ S ρ B H hSB ( t − s ) (cid:3) (A10)= Γ + h ( ω ) τ ρ S τ + Γ + h ( ω ) τ ρ S τ e − i ( ω − ω ) t + Γ − h ( ω ) τ ρ S τ + Γ − h ( ω ) τ ρ S τ e i ( ω − ω ) t + Γ + h ( ω ) τ ρ S τ + Γ + h ( ω ) τ ρ S τ e − i ( ω − ω ) t + Γ − h ( ω ) τ ρ S τ + Γ − h ( ω ) τ ρ S τ e i ( ω − ω ) t . So did all the remaining terms in Eq. (A8). Then afterrotating back to the Schr¨odinger picture, we attain theRedfield master equation with a partial secular approxi-mation as Eq. (7) in the main text.Following Eq. (7), the dynamical equations of thethree-level system with inner coupling in terms of ma-trix elements are given by˙ ρ = − − h ( ω ) + Γ − h ( ω ) + Γ − c ( ω ) + Γ − c ( ω )] ρ + 2[Γ + h ( ω ) + Γ + c ( ω )] ρ + 2[Γ + h ( ω ) + Γ + c ( ω )] ρ + +[Γ + h ( ω ) − Γ + c ( ω ) + Γ + h ( ω ) − Γ + c ( ω )]( ρ + ρ )˙ ρ = 2[Γ − h ( ω ) + Γ − c ( ω )] ρ − + h ( ω ) + Γ + c ( ω ) + Γ − w (Ω)] ρ + 2Γ + w (Ω) ρ − [Γ + h ( ω ) − Γ + c ( ω )]( ρ + ρ )˙ ρ = 2[Γ − h ( ω ) + Γ − c ( ω )] ρ + 2Γ − w (Ω) ρ − + h ( ω ) + Γ + c ( ω ) + Γ + w (Ω)] ρ − [Γ + h ( ω ) − Γ + c ( ω )]( ρ + ρ )˙ ρ = [2 i Ω − Γ + h ( ω ) − Γ + c ( ω ) − Γ + h ( ω ) − Γ + c ( ω ) − Γ + w (Ω) − Γ − w (Ω)] ρ + [Γ − h ( ω ) − Γ − c ( ω ) + Γ − h ( ω ) − Γ − c ( ω )] ρ − [Γ + h ( ω ) − Γ + c ( ω )] ρ − [Γ + h ( ω ) − Γ + c ( ω )] ρ ˙ ρ = [ − i Ω − Γ + h ( ω ) − Γ + c ( ω ) − Γ + h ( ω ) − Γ + c ( ω ) − Γ + w (Ω) − Γ − w (Ω)] ρ + [Γ − h ( ω ) − Γ − c ( ω ) + Γ − h ( ω ) − Γ − c ( ω )] ρ − [Γ + h ( ω ) − Γ + c ( ω )] ρ − [Γ + h ( ω ) − Γ + c ( ω )] ρ . (A11)As shown by Eq. (A11), the off-diagonal (coherence)terms ρ and ρ are closely associated with the diagonal terms ρ , ρ and ρ . Thus the steady state obtainedby ˙ ρ = 0 may have residue quantum coherence, which isa mark of quantumness. [1] M. O. Scully, M. S. Zubairy, G. S. Agarwal, andH. Walther, Extracting work from a single heat bath viavanishing quantum coherence,
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