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Thermodynamics of Quantum Coherence Thermodynamics of quantum coherence The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Rodríguez-Rosario, César A., Thomas Frauenheim, and Alán Aspuru-Guzik. 2013. "Thermodynamics of quantum coherence." Working paper, Harvard University, August 6, 2013. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:17578537 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA Thermodynamics of quantum coherence 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 bej characterizedih j j ih j simplyj byih j 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 fj ig − 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!1 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 f g 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 <t< t = thermodynamics[19]. To go beyond this, we instead con- a 1 1 sider the decoherence operator Ld and the Hamiltonian P H(τ) = k Ek E(τ)k E(τ)k that at time τ it is con- trolled such thatj it isih not on thej preferred basis. Then, the system coherence leads to an energy exchange with the decoherence bath. This heat rate is driven by the change of quantum coherence due to the decoherence bath, a bath for which no temperature can be defined. Such heat rates driven by decoherence can be used to de- fine a quantum Carnot engine whose efficiency depends on coherence [2, 3] and for temperature control [8] (see b Appendix C). 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 fp are in quantumj ih jg thermodynamic equilibrium with Onsager reciprocal relations f jg 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 hasf g its own, different, stationary set in η in ther- f bg 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.
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