Thermodynamics of quantum coherence
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Citation Rodríguez-Rosario, César A., Thomas Frauenheim, and Alán Aspuru-Guzik. 2013. "Thermodynamics of quantum coherence." Working paper, Harvard University, August 6, 2013.
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C´esarA. Rodr´ıguez-Rosario1, Thomas Frauenheim1 Al´anAspuru-Guzik2 1 Bremen Center for Computational Materials Science, University of Bremen, Am Fallturm 1, D-28359, Bremen, Germany 2 Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138, USA
Quantum decoherence is seen as an undesired source of irreversibility that destroys quantum re- sources [1]. Quantum coherences seem to be a property that vanishes at thermodynamic equilibrium. Away from equilibrium, quantum coherences challenge the classical notions of a thermodynamic bath in a Carnot engines [2, 3], affect the efficiency of quantum transport [4–6], lead to violations of Fourier’s law [7], and can be used to dynamically control the temperature of a state [8]. However, the role of quantum coherence in thermodynamics [9] is not fully understood. Here we show that the relative entropy of a state with quantum coherence with respect to its decohered state captures its deviation from thermodynamic equilibrium. As a result, changes in quantum coherence can lead to a heat flow with no associated temperature, and affect the entropy production rate [10]. From this, we derive a quantum version of the Onsager reciprocal relations [11] that shows that there is a reciprocal relation between thermodynamic forces from coherence and quantum transport. Quan- tum decoherence can be useful and offers new possibilities of thermodynamic control for quantum transport [12, 13].
Introduction set of classical probability vectors on the preferred basis, P P ρ Bd (ρ) = j j j ρ j j = j pj j j . A decoher- ence→ bath can be| characterizedih | | ih | simply| byih | the preferred The evolution of a system state ρ can be described by basis j of the stationary set. a master equationρ ˙ = i [H, ρ] + L (ρ) where H is the {| i} − Hamiltonian of the system, and L describes the coupling to a Markovian bath [14]. The solution of this equation is Zeroth law of thermodynamics the dynamical map ρ(t) = B(0,t) ( ρ(0) ) [15]. Determin- ing L is experimentally demanding, requiring quantum The zeroth law of thermodynamics is a statement process tomography [16]. To overcome this difficulty, we about how systems can act like ‘thermometers’ such that will focus instead on quantum thermodynamic properties they are stationary upon coupling to a bath. To quan- that depend on equilibrium and deviations from it. We tify the surprise of a system with respect to a bath, we provide a description of the role of decoherence in terms will use the concept of relative entropy[18]. The relative of the change in energy, entropy and entropy production. entropy of a system ρ with respect to its corresponding Finally, we introduce a quantum version of the Onsager stationary state for process B is reciprocal relations between decoherence and transport. R [ρ B ] = Tr [ρ log ρ] Tr [ρ log B (ρ)] , (1) To understand the thermodynamics of quantum co- k − herence, we must go beyond characterizing thermody- This quantity captures that the state ρ is far from the sta- namic equilibrium simply by a temperature parameter. tionary set of the process B. We now express the zeroth As our starting point, we consider the stationary states law of quantum thermodynamics as: a state ρ is in quan- of a quantum process. These are reached when a sys- tum thermodynamic equilibrium with a bath B when tem is coupled to a bath for long enough such that (0,t) R [ρ B ] = 0. Although the states on quantum ther- ρ B (ρ) limt→∞ B (ρ). The state B (ρ) is sta- k → ≡ modynamic equilibrium might not be unique, the system arXiv:1308.1245v1 [quant-ph] 6 Aug 2013 tionary because L ( B (ρ) ) = 0 [17]. All stationary states acts like a thermometer in the sense that there is no sur- η with the property that L (η) = 0 (or equivalently { } prise from being coupled to the bath. Since R [ρ B] can- B (η) = η) form the stationary set. We propose to not increase in time (see Appendix A), a state ink contact use the stationary set of the quantum process as the with a bath tends to evolve towards quantum thermo- way to characterize a quantum thermodynamic bath. dynamic equilibrium. When considering relaxation to a This captures classical thermodynamics for the case of Gibbs state, this fully captures classical thermodynam- a relaxation process (subscript r) where any state ρ ics. For relaxation dynamics, the Gibbs state is the only evolves asρ ˙ = Lr (ρ) becoming the Gibbs stationary state stationary state and can be described solely in terms of e−βH ρ Br (ρ) = −βH . However, focusing on stationary → Tr(e ) the temperature β and the system Hamiltonian. Any sets also allows us to consider decoherence as a new ther- change in energy or temperature of the system will take modynamic process. Decoherence is a quantum process it away from thermodynamic equilibrium (see Fig. 1a) that, by means of a master equation Ld, destroys the off- with respect to the Gibbs state [9]. diagonal elements (coherences) of a density matrix. The The definition of equilibrium given by Eq. (1) goes stationary set of decoherence can be described with a beyond the Gibbs formalism. We now use the zeroth 2
t =0 0
Second law of thermodynamics
Irreversibility due to decoherence also plays an impor- tant role. To study it, we take the time derivative of Eq. (1) to obtain the entropy rate equation, d Tr [ρ ˙ log ρ] = Tr [ρ ˙ log B (ρ)] R [ρ B], − −dt k (2) | {z } | {z } | {z } FIG. 1: Approach to equilibrium. The solid ball painted as S˙ = Φ +P, Earth represents the Bloch sphere of possible density matri- − ces for a two-level system. Cartoon thermometers represent where S = Tr [ρ log ρ] is the von Neumann entropy of equilibrium. a, A relaxation bath squeezes all possible states the system,−Φ is the entropy flux due to the bath, and into a unique point that corresponds to the Gibbs state. A P is the entropy production rate. The second law of ‘thermometer’ state would contain information about the sys- quantum thermodynamics can be written therefore as tem Hamiltonian and the unique temperature of this bath. b, P = d R [ρ B] 0, which states that irreversibility A decoherence bath squeezes the states into a line along the − dt k ≥ preferred basis that corresponds to the set of states with- cannot decrease the quantum entropy production (see out quantum coherences. Any ‘thermometer’ state in the Appendix B). Classical non-equilibrium thermodynam- preferred basis is in equilibrium, but the temperature is not ics corresponds to the special case of a relaxation bath uniquely defined. Br [10]. Decoherence baths not only follow the second law [20, 21], but also provide an additional quantum con- tribution to entropy production. law to look at decoherence. The set of decohered states P ηd = j pj j j for all possible probability vectors {p are in quantum| ih |} thermodynamic equilibrium with Onsager reciprocal relations { j} respect to decoherence because R [ηd Bd ] = 0. Equilib- k rium of a decoherence process is not given by a unique Now, we examine the implications of an additional Gibbs state, but by the entirety of all states in the pre- source of irreversibility due to decoherence. This corre- ferred basis. For decoherence, a quantum state acts as sponds to a scenario where the system is coupled to many a ‘coherence thermometer’ when it has lost its coherence baths b , each with an operator Lb. Independently, each such that it is not surprised by the (classical) stationary bath has{ } its own, different, stationary set in η in ther- { b} distribution. The information that it obtains from the modynamic equilibrium, R [ηb Bb ] = 0. However, the decoherence bath is the preferred basis (see Fig. 1b). interplay between all the baths keepsk the system in a non- equilibrium steady state ν that is not in thermodynamic equilibrium with any of the baths [22]. The entropy flux First law of thermodynamics to each bath is then Φb = Tr [Lb(ν) log Bb (ν)]. It fol- lows that the entropy production− rate from the interplay A decoherence bath can also create a heat flow. To amongst multiple baths is show this, we start with the first law of quantum ther- d X modynamics, E = W˙ + Q˙ , that expresses the change P = Tr [Lb(ν) (log Bb (ν) log ν)] , dt − of energy d E = Tr [Hρ] in terms of the work rate b dt X W˙ = Tr[Hρ˙ ] and the heat rate Q˙ = Tr [H L(ρ)]. = Tr [Jb Xb] , (3) Coupling to a relaxation bath Lr recovers classical b 3 where Jb = Lb(ν) represents a flow and Xb = log Bb (ν) log ν is the corresponding thermodynamic force. This− a b decoherence is the quantum generalization of the entropy produc- Md,l tion rate for a non-equilibrium steady state. The rate Xd of change represented by Jb says how, away from equilib- EL V rium, there is a flow to bath b. The force Xb represents ER left how far the non-equilibrium steady state is from the sta- right Ml,d tionary set for bath b. Since the interplay between co- E0 Xl herences in the steady state and decoherence can lead to a thermodynamic force Xd, decoherence plays an im- portant role in the entropy production even under the presence of other (classical) relaxation baths. FIG. 2: Interplay of transport and decoherence. a, A de- The discovery of the Onsager reciprocal relations were vice used for quantum transport is represented by the three a turning point in thermodynamics by providing general energy levels. The interplay between the left and the right nonequilibrium results that applied without any specific relaxation baths leads to a flow that, with the coupling V , details of the model studied[11]. We now derive more creates a coherence hEL|V |ERi. The decoherence bath cre- ates a flow that destroys this coherence. b, The interplay general quantum relations that can be applied to study between the baths can be approximated using quantum re- the nonequilibrium role of coherence. The use of Eq. (3) ciprocal relations. The left bath creates a force (spring Xl) requires knowledge of the details of the non-equilibrium that pulls the device towards the stationary Gibbs state. The dynamics of each bath Lb. To simplify this, we approx- decoherence bath creates a force (spring Xd) that pulls the imate the current linearly in terms of the forces [11]. device towards a state with no coherence. This in turn create flows (arrows and ) that are reciprocally related. Since Jb and Xb are matrices, the linearization corre- Ml,d Md,l sponds to a super-operator Mb,a acting on the forces: P Jb a Mb,a(Xa). In this regime, the quantum entropy productions≈ can be written in terms of the forces as: to the other through the device is mediated in part by this quantum coherence. Decoherence can be seen as X X † an additional bath D that changes the non-equilibrium P = Tr Mb,a(Xa) Xb = Tr Ma,b(Xb) Xa , (4) a,b a,b steady state, and in turn, the quantum transport (see Fig. 2). Using Eq. (15) we conclude that the flow of † quantum coherence into the decoherence bath has a re- with the quantum reciprocal relations Mb,a( ) = Ma,b( ) (see Appendix D). The quantum reciprocal relations· give· ciprocal relation with the quantum transport between L us a phenomenological way to understand the interplay and R. The coherence coming from the flow through the between different quantum baths in terms of the de- device affects the amount of decoherence. Reciprocally, viations from equilibrium with respect to each bath. the amount of decoherence affects the quantum transport The quantum reciprocal relations apply for any quantum between L and R (see Appendix E). This effect could be Markovian bath and give a relationship that is indepen- experimentally verified on a molecular junction [13] by dent of the specific microscopic details of the dynamics. controlling decoherence [24]. We now use this to show how quantum coherence is a thermodynamic force in quantum transport. Conclusions
Thermodynamic role of coherence in transport We have shown how quantum coherences lead to new thermodynamic flows and forces. For this, we defined Coherence can have an enhancing effect on quantum quantum equilibrium in terms of relative entropy. This transport. It helps in energy transfer in photosyn- permitted us to write the laws of quantum thermodynam- thetic complexes [4–6] and affects quantum transport in ics in a way that we can apply them to study decoherence. nanoscale devices [12, 13, 23]. These suggest that there We showed how decoherence can lead to heat flows and is a general effect, which we now explain using the quan- change the entropy production. We used these to gener- tum reciprocal relations. We construct a simple model alize the Onsager relations to the quantum regime, which that has all the essential features by considering a cen- lead to a simple explanation of the role of coherence in tral quantum system, the device, between two relaxation quantum transport. This work prompts further studies baths L and R. The system has a Hamiltonian term on how to use coherence as a thermodynamic resource, V that can create quantum coherence between the left such as in the most recent experimental studies on heat and the right parts of the device. Although the baths dissipation in atomic-scale junctions [25] and in experi- are classical, the non-equilibrium steady state sustains a ments that use thermodynamic baths to created quantum quantum coherence. Quantum transport from one bath information resources at steady-state [26]. 4
Acknowledgements When a state is in the stationary set of the dynamics, its relative entropy with respect to the bath is zero, and This work (A.A.-G. and C.A.R.R.) was supported by therefore we say it is in thermodynamic equilibrium. the Center of Excitonics, an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Office of Basic Energy Sciences under Award B. Entropy production rate for quantum processes Number DESC0001088. A.A.-G. also thanks the Corning foundation for their generous support. C.A.R.R. thanks Using the property that B(ρ) is stationary, we take the Peter Love, Stephanie Wehner and Kavan Modi for help- time derivative of Eq. (7) and obtain the entropy rate ful discussions. equation:
d APPENDICES Tr [ρ ˙ log ρ] = Tr [ρ ˙ log B (ρ)] R [ρ B], − −dt k (8) | {z } | {z } | {z } S˙ = Φ +P, A. Relative entropy and equilibrium − where S˙ = Tr [ρ ˙ log ρ] is the rate of the von Neumann The relative entropy of the density matrix ρ with re- − spect to σ is defined as entropy of the system, Φ is the entropy flux due to the bath, and P is the entropy production rate due to ir- S [ ρ σ ] = Tr [ρ log ρ] Tr [ρ log σ] . (5) reversibility (see Fig. (3)). The second law of quantum k − thermodynamics is P = d R [ρ B] 0, which means This expression characterizes the surprise of gaining the − dt k ≥ state σ when having the state ρ and is a measure of the that irreversibility cannot decrease the quantum entropy information loss when trying to approximate σ with ρ. production. This has already been proven for the special For further details on the relative entropy, we refer the case where the dynamics are relaxation Br to a single reader to the review by Vedral [18]. Relative entropy Gibbs state [10]. is well-defined for states that are not pure, but there are techniques [27] that can be applied to overcome this Bath limitation [35]. Relative entropy never increases when the System states evolve under the dynamics of a completely positive map A, such that
h (0,t) (0,t) i S [ ρ σ ] S A (ρ) A (σ) . (6) k ≥ k S˙ Here we consider the special case of the relative en- tropy of a state ρ with respect to its stationary state (0,t) P limt→∞ B (ρ) = B(ρ): R [ρ B ] = S [ ρ B(ρ) ] = Tr [ρ ( log ρ log B (ρ) )] , (7) k k − which captures the surprise that a state ρ is not station- ary under the process B. The quantity R [ρ B ] cap- FIG. 3: Quantum entropy rate equation. The entropy rate S˙ tures the approach to quantum thermodynamick equilib- depends on the entropy flux to the bath Φ and the irreversibil- rium because it never increases in time. The proof for this ity of the dynamics characterized by the entropy production (0,t) rate P. The second law of thermodynamics is simply P ≥ 0 uses the semigroup property of map B and Eq. (6) to and is satisfied by all quantum Markovian processes, including obtain: decoherence. h (0,t) (0,t) i h (0,t) (0,t) i R B (ρ) B (B) = S B (ρ) B ( B(ρ)) k k S [ ρ B(ρ) ] = R [ρ B] . ≤ k k We are interesting in showing how irreversibility due to more general quantum processes, such as decoherence, Using the property that B(ρ) is a stationary state of B(0,t) contributes to the entropy production. For this, we gen- such that B(0,t)( B(ρ) ) = B(ρ), we complete the proof: eralize the proof to all possible B. We start by using h (0,t) i R B (ρ) B R [ρ B] . the result from Eq. (8) that relative entropy with respect k ≤ k to the process can never increase. Therefore, its time This quantity captures the approach to thermodynamic derivative, equilibrium because R (0,t)(ρ) R [ρ ] h (0,t) i d B B B lim R B (ρ) B = +0. R [ρ B] = lim k − k 0, t→∞ k dt k t→+0 t ≤ 5 is never positive. It follows that P 0 and that the the form j Ek . From the first law of quantum ther- second law of thermodynamics is satisfied≥ for any quan- modynamics,h | thei heat rate due to a dephasing process tum process. Since this is true for any quantum process, is this shows how decoherence contributes to the entropy ˙ production. Interestingly, quantum contributions to the Qd = Tr [ H Ld(ρ)] , entropy production lead to novel effects that cannot be X X = γ E 2 j ρ j j E E j described classically. We discuss these effects in the next k h | | i h | kih k| i sections. k j j ρ E E j E ρ j j E . (10) − h | | ki h k| i − h k| | i h | ki
C. Heat rate from decoherence This heat rate depends on the quantum coherence j Ek . When the coherence vanishes, so does the heat rate.h | Fori The irreversible loss of quantum coherences to a deco- many states ρ there is a heat rate, even though a deco- herence bath can lead to heat rates. To show this, we now herence bath has no unique temperature associated with consider a master equation for decoherence dynamics of it. This heat rate is an example of how quantum pro- the form: cesses can created thermodynamic flows that depend on quantum variables. X Ld(ρ) = γ 2 j j ρ j j j j ρ ρ j j , (9) | ih | | ih | − | ih | − | ih | To be more concrete, we now consider a two level sys- j 1 tem as an example, ρ = 2 (I + xσx + zσz), where I = 1 1 + 0 0 , σ = 1 1 0 0 and σ = 1 0 0 1 . which is equivalent to a continuous measurement along z x |Weih also| | assumeih | that| theih |−| preferredih | basis of the| ih dephasing|−| ih | the basis j . Equilibrium in a decoherence bath can be bath is 0 , 1 . The system parameter x is thus the fully characterized{| i} by this preferred basis. Such a bath amount{| of coherencei | i} of the system. has no temperature associated with it, but can create The Hamiltonian is at a different basis H = E E E a heat rate by changing quantum coherences. For this, | ih | where we chose E = √1 ( 0 + 1 ). This choice makes we will assume that, at some time, the Hamiltonian of | i 2 | i | i the system is H = P E E E , where E is not quantum coherences to simply be 1 E = 0 E = √1 . k k k k k h | i h | i 2 in the preferred basis, leading| ih to quantum| coherences{| i} of The decoherence heat rate for this example is
h i Q˙ = γE 1 √1 1 ρ E + 0 ρ E + E ρ 1 + E ρ 0 = γEx. (11) d − 2 h | | i h | | i h | | i h | | i
The quantum heat rate is proportional to amount of of the state. quantum coherence of the system x. This is an exam- ple of how a decoherence bath can create a heat rate that depends on quantum coherence. D. Quantum reciprocal relations Previous publications showed the dependence of a quantum Carnot engine in terms of quantum coherences The classical Onsager reciprocal relations [11, 28] use [2, 3]. In those cases, the quantum coherences are subject a linear approximation of the flows in terms of the forces to decoherence. We suggest the equivalent but alterna- to study non-equilibrium thermodynamics. They can tive interpretation that such decoherence produces a heat be derived using classical stochastic processes and con- rate, that has no temperature associated with it, which sidering deviations from the stationary state of each affects the efficiency of this engine. bath [29]. This method has been very successful to un- Another publication has suggested how continuous derstand how many classical irreversible processes affect quantum measurements can be used for thermodynamic each other [22]. Important work showed that the classi- control [8]. In their model they considered the tempera- cal Onsager relations were recovered for quantum relax- ture relaxation of a state while also under the influence of ation processes [30, 31]. We are interested in using them frequent quantum measurements. We suggest the inter- to study the effects of decoherence on other thermody- pretation that since such measurements can be modeled namic variables. For this, we must extend these relations as an additional decoherence bath, the act of measuring to more general quantum processes. the state introduces another source of heat. This heat When a system is coupled to many baths, the interplay rate from decoherence serves to control the temperature between them can lead to a non-equilibrium steady state 6
† ν. This state creates an entropy flux to each bath of The Heisenberg picture of the Mb,a is Ma,b( ) = † · the form Φb = Tr [Lb(ν) log Bb (ν)]. The total entropy C ( ( ) ), which leads to the quantum reciprocal re- − La Lb production rate is lations: ·
X † P = Tr Lb(ν) log Bb (ν) log ν b,a( ) = ( ). (18) − M Ma,b b · · X The quantum reciprocal relations give us a phe- = Tr [J X ] , (12) b b nomenological way to understand the non-equilibrium b interplay between different quantum thermodynamic where baths. They are useful because they do not require knowledge of the master equation of each bath. We only Jb = Lb(ν) (13) need to know the deviations from equilibrium for each bath, given by the matrix Eq (14). These can account represents a quantum flow and for thermodynamic effects due to decoherence.
Xb = log Bb (ν) log ν (14) − is the corresponding quantum thermodynamic force. The E. Reciprocal relations between decoherence and transport rate of change represented by Jb says how far away from equilibrium there is a flow to bath b. The force Xb repre- sents deviations of the non-equilibrium steady state from We illustrate how the Onsager reciprocal relations can the corresponding stationary state for bath b. be used to understand the interplay between decoherence Quantum irreversible processes are more general than and transport. For this, we consider the simple device the classical irreversible processes and require density described in the main text, Fig. (2). matrices to express deviations from the stationary state. The Hamiltonian of the system is This is why Jb and Xb are matrices. The quantum On- V sager linearization has to be given by a super-operator H = EL L L + ER R R + L R + R L P | ih | | ih | 2 | ih | | ih | Mb,a such Jb Mb,a(Xa). The entropy production ≈ a rate in terms of quantum forces is: where, for simplicity, we set EL > ER, E0 = 0 and as- ∗ X sumed V = V . The system is subject to the interplay P = Tr [Mb,a(Xa) Xb] . (15) between three baths. The bath on the left is a relaxation a,b bath, described by the master equation: We now show that the relations between these forces are reciprocal. To do this, we start with Eq. (14), and lin- h i Ll(ρ) =(1 + nL) 2 L 0 ρ 0 L 0 0 ρ ρ 0 0 early approximate log Ba (ν) δLa (ν) to obtain from | ih | | ih | − | ih | − | ih | ≈ h i Eq. (14) the approximation δLa (ν) log ν + Xa. The + nL 2 0 L ρ L 0 L L ρ ρ L L . ≈ † | ih | | ih | − | ih | − | ih | Heisenberg picture representation of La, La, allows us to write where n = 1/(eβLEL 1). The parameter n character- L − L X † izes the stationary Gibbs distribution of this bath. The ν CL (log ν + Xa) , (16) ≈ a bath on the right is also a relaxation bath, with master a equation: where C is a constant. We expand to first order log ν P † † ≈ h i ν I, and Eq. (16) becomes ν a CLa (Xa)+CLa (ν), Lr(ρ) = nR 2 R 0 ρ 0 R 0 0 ρ ρ 0 0 − † ≈ | ih | | ih | − | ih | − | ih | because L (I) = 0. Recall that since ν is a non- a P h i equilibrium steady state, then (ν) = (ν) = 0. +(1 + nR) 2 0 R ρ R 0 R R ρ ρ R R . (19) a La L | ih | | ih | − | ih | − | ih | Assuming quantum detailed balance for the total dynam- βRER ics L [32–34], the Heisenberg picture operator also follows where nR = 1/(e 1), which characterizes the sta- † P † − L (ν) = a La (ν) = 0. With this, we approximate tionary distribution of the bath in the right. This con- the non-equilibrium steady state in the linear regime as stitute a simple quantum transport device [4–6, 12, 13]. P † ν a CLa (Xa). Its total evolution can be found by solving ≈This equation shows how we can express the non- equilibrium steady state ν linearly in terms of the devia- ρ˙ = i [H, ρ] + Ll(ρ) + Lr(ρ). (20) − tions from equilibrium Xa from each of the baths. From Eq. (12), the flow in the linear regime is At non-equilibrium steady state, these two baths create an energy flow Q˙ = Q˙ through the system by means r − l X X † of the coherence L H R = V . To find this flow, we Jb = Lb(ν) Mb,a(Xa) CLb L (Xa) , (17) ≈ ≡ a h | | i a a must solve the total master equation. 7
In practice, such a device is also subject to some incoherent transport, the efficiency can be maximized, quantum decoherence. When decoherence is strong, the which is important in energy transport in photosynthesis device operates in the incoherent regime, and classical [4, 6]. transport efficiencies are recovered [7]. It has been shown To study this intermediate regime, decoherence is in- that in the intermediate regime between coherent and troduced as a third bath. Its master equation is:
Ld(ρ) = γ 2 L L ρ L L L L ρ ρ L L + 2 R R ρ R R R R ρ ρ R R . (21) | ih | | ih | − | ih | − | ih | ih | | ih | − | ih | − | ih |
The introduction of decoherence changes the non- where xl = L log ν R + R log ν L is a force parameter equilibrium steady state of the device, and would require due to the coherenth | | couplingi h | driving| i the non-equilibrium the solution of a new master equation to compute the steady state away from equilibrium with respect to the ˙ ˙ ˙ new energy flow Ql = Tr [HLl(ν)]. left bath. Clearly, the flows Ql and Qd are related recip- Instead of solving this equation, we could use the recip- rocally by their rates because mld = mdl. rocal relations to approximate this calculation in terms The power of the quantum reciprocal relations lie in of deviations from equilibrium. We now use this to un- their generality. Instead of a lengthy calculation depend- derstand understand the relationship between decoher- ing on all the details of the master equation, to study ence and energy transport. Decoherence introduces a the role of decoherence on quantum transport, we can new heat rate into the system, as in Eq. (10). This heat estimate the relationship between the flow due to the rate is given by Q˙ = Tr [H (ν)]. Both Q˙ and Q˙ seem d Ld l d bath on the left and the flow due to decoherence sim- to have a complicated relationship because they both de- ply as phenomenological reciprocal relations. Even with- pend on ν, which requires in turn the calculation of the out knowledge of the master equations for each bath, or full dynamics. the parameters γ, V, n , we can still conclude that the However, in the linear regime, using Eq. (17), the heat l relationship between decoherence and transport depends rates can be approximated as: reciprocally between the transport force xl and the de- ˙ Ql Tr [HMl,l(Xl)] + Tr [HMl,d(Xd)] + Tr [HMl,r(Xl)] . coherence force xd by mld = mdl. That is, decoherence ≈ affects transport exactly as much as transport phenom- The force Xl (Xr) represents how for is the steady state from equilibrium with the Gibbs state that character- ena affects decoherence. izes the bath l (r). The force Xd captures how far from equilibrium is the steady state ν from decoherence. It depends on the coherence of the non-equilibrium steady state as log L ν R . This is a force due to the coherent h | | i [1] W. Zurek, Reviews of Modern Physics 75, 715 (2003). coupling between the left and the right bath. It cre- [2] M. Scully, M. Zubairy, G. Agarwal, and H. Walther, Sci- ates a thermodynamic flow that, by means of the ma- ence 299, 862 (2003). † ˙ [3] M. O. Scully, Physical Review Letters 104, 207701 trix Md,l(H), affects Ql. Using Eq. (17) with Eq. (19) and Eq. (21), and by explicit calculation, we obtain that: (2010). † † [4] M. Mohseni, P. Rebentrost, S. Lloyd, and A. Aspuru- M (H) L ( Ll (H) ) = γnl ( R L + L R ). Thus, d,l ≈ d | ih | | ih | Guzik, The Journal of Chemical Physics 129, 174106 the heat rate on the left Q˙ l depends on the coherence (2008). force parameter xd = log R ν L + log L ν R as: [5] M. B. Plenio and S. F. Huelga, New Journal of Physics h | | i h | | i 10, 113019 (2008). h † i Tr Ml,d(H)Xd (γnlV ) xd = mld xd, [6] P. Rebentrost, M. Mohseni, I. Kassal, S. Lloyd, and ∝ A. Aspuru-Guzik, New Journal of Physics 11, 033003 This quantity depends only on the equilibrium distribu- (2009). tion of the left bath, as given by parameter nl, the cou- [7] D. Manzano, M. Tiersch, A. Asadian, and H. J. Briegel, pling V , the decohernece rate γ and the quantum coher- Physical Review E 86, 061118 (2012). [8] N. Erez, G. Gordon, M. Nest, and G. Kurizki, Nature ent force xd. 452, 724 (2008). Reciprocally, we can also estimate how much the deco- [9] D. Kondepudi and I. Prigogine, Modern Thermodynam- herence heat rate depends on the the temperature of the ics (John Wiley and Sons Ltd, Sussex, England, 1998). left bath. A similar explicit calculation tells us that the [10] H. Spohn, Journal of Mathematical Physics 19, 1227 Q˙ d depends on the force due to the left bath by: (1978). h † i [11] L. Onsager, Physical Review 37, 405 (1931). Tr M (H)Xd (γnlV ) xl = mdl xl. l,d ∝ [12] G. C. Solomon, D. Q. Andrews, R. P. Van Duyne, and 8
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