Quantum Maxwell's Demon in Thermodynamic Cycles

PHYSICAL REVIEW E 83, 061108 (2011)

Quantum Maxwell’s demon in thermodynamic cycles

H. Dong (), D. Z. Xu (), C. Y. Cai (), and C. P. Sun ()* Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190, China (Received 20 September 2010; revised manuscript received 1 March 2011; published 8 June 2011) We study the physical mechanism of Maxwell’s demon (MD), which helps do extra work in thermodynamic cycles with the heat engine. This is exemplified with one molecule confined in an infinitely deep square potential with a movable solid wall. The MD is modeled as a two-level system (TLS) for measuring and controlling the motion of the molecule. The processes in the cycle are described in a quantum fashion. It is discovered that a MD with quantum coherence or one at a temperature lower than the molecule’s heat bath can enhance the ability of the whole working substance, formed by the heat engine plus the MD, to do work outside. This observation reveals that the essential role of the MD is to drive the whole working substance off equilibrium, or equivalently, to work between two heat baths with different effective temperatures. The elaborate studies with this model explicitly reveal the effect of finite size off the classical limit or thermodynamic limit, which contradicts common sense on a Szilard heat engine (SHE). The quantum SHE’s efficiency is evaluated in detail to prove the validity of the second law of thermodynamics.

DOI: 10.1103/PhysRevE.83.061108 PACS number(s): 05.70.Ln, 05.30.−d, 03.65.Ta, 03.67.−a

I. INTRODUCTION (QHEs) [7,8] with the quantum coherent system serving as the working substance. The quantum working substance deviates Maxwell’s demon (MD) is notorious since its existence from the thermodynamic limit and preserves its quantum could violate the second law of thermodynamics (SLOT) [1,2]: coherence [9,10] to some extent, and thus it has tremendous the MD distinguishes the velocities of the gas molecules and influence on the properties of QHEs, especially when quantum then controls the motions of molecules to create a difference of MD is included in the thermodynamical cycle [11–13]. There temperatures between the two domains. In 1929, Leo Szilard have been many attempts to generalize the SHE by quantum proposed the “one molecular heat engine” (which we call the mechanically approaching the measurement process [11], the Szilard heat engine, SHE) [3] as an alternative version of a motion control [12], and the insertion and expansion processes heat engine assisted by the MD. The MD first measures which [14]. However, to our best knowledge, a fully quantum domain the single molecule stays in and then manipulates the approach for all processes in the SHE intrinsically integrated system to extract work according to the measurement result. with a quantum MD is still lacking. The quantum-classical In a thermodynamic cycle, the molecule seems to extract heat hybrid description of the SHE may result in some observations from a single heat bath at temperature T and thus do work about the MD-assisted thermodynamic process that seem to k T ln 2 without evoking other changes. This consequence B challenge common senses in physics. Therefore, we need a obviously violates the second law of thermodynamics (SLOT). fully quantum theory for MD-assisted thermodynamics. In the early years after the discovery of the MD, there In this paper, we propose a quantum SHE assisted by MD was insufficient attention to this topic. The first revival of the with a finite temperature difference from that of the system. studies of MD is due to the recognition of the tradeoff between In this model, we give a consistent quantum approach to information and entropy in the MD-controlled thermody- the measurement process without using the von Neumann namic cycles. The milestone discovery was the “Landauer projection [15]. Then we calculate the work done by the principle” [4], which reveals that erasing one bit of informa- insertion of the movable wall in the framework of quantum tion from the memory in the computing process inevitably mechanics. The controlled gas expansion is treated with accompanies an increasing entropy of the environment. In the quantum conditional dynamics. Furthermore, we also consider SHE, the erasing needs work k T ln 2 done by the external B the process of removing the wall to complete a thermody- agent. It gives a conceptual solution for the MD paradox [5]by namic cycles. With these necessary subtle considerations, the considering the MD as a part of the whole working substance, quantum approach for the MD-assisted thermodynamic cycle and thus erasing the information stored in the demon’s memory will go beyond the conventional theories about the SHE. is necessary to restart a new thermodynamic cycle. This We show that a system that deviates from the thermody- observation about erasing the information of the MD finally namic limit exhibits uncommon observable quantum effects saves the SLOT. The recent generalization of the Landaeur due to the finite size of system, which results in discrete principle, including error effect in the measurement process, energy levels that could be washed out by the heat fluctu- has been discussed in Ref. [6]. ation. Quantum coherence can assist the MD in extracting The recent revival of studies of MD is due to the devel- more work by reducing effective temperature, while thermal opment of quantum information science. The corresponding excitation of the MD at a finite temperature would reduce quantum thermodynamics concerns quantum heat engines its quantum measurement and conditional control of the expansion. It means that the MD could help the molecule do maximum work outside only when cooled to absolute zero *[email protected]; [http://power.itp.ac.cn/suncp/index.html] temperature.

Our paper is organized as follows: In Sec. II we first give a brief review of the classical version of SHE and then present our model in a quantum version with MD included intrinsically. The role of quantum coherence of MD is emphasized with the definition of the effective temperature for an arbitrary two-level system. In Sec. III, we consider the quantum SHE with a quantum MD at finite temperature, measuring the molecular position in a one-dimensional infinitely deep well. The whole cycle consists of four steps: insertion, measurement, expansion, and removal. Detailed descriptions are given subsequently. We calculate the work done and heat exchange in every step. In Sec. IV, we discuss quantum SHE’s operation efficiency in comparison with the Carnot heat engine. We restore the well-known results in the classical case by tuning the parameters in the quantum version, such as the width of the potential well. Conclusions and remarks are given in Sec. VI. FIG. 1. (Color online) Classical and quantum Szilard’s single- II. QUANTUM MAXWELL’S DEMON IN SZILARD molecule heat engine. (a) Classical version: (i), (ii) A piston is inserted HEAT ENGINE in the center of a chamber. (ii), (iii) The demon finds which domain the single molecule stays in. (iii), (iv) The demon controls the system to In this section, we first revisit Szilard’s single-molecule do work according to its memory. (b) Quantum version: The demon is heat engine in brief. As illustrated in Fig. 1(a), the whole modeled as a two-level system with two energy levels |g and |e and thermodynamic cycle consists of three steps: insertion [(i) and energy spacing . The chamber is quantum mechanically described (ii)], measurement [(ii) and (iii)], and controlled expansion as an infinite potential with width L. (I), (II) An impenetrable wall [(iii) and (iv)] by the MD. The outside agent first inserts a piston is inserted at an arbitrary position in the potential. (II), (III) The isothermally in the center of the chamber. Then, it finds which demon measures the state of the system and then records the results domain the single molecule stays in and changes its own state in its memory by flipping its own state or not taking action. The to register the information. Without losing generality, we label measurement may result in the wrong results, illustrated in the green the initial state of the demon as 0. Finding that the molecule dot-dashed rectangle. (III), (IV) The demon controls the expansion is on the right, namely L/2

061108-2 QUANTUM MAXWELL’s DEMON IN THERMODYNAMIC CYCLES PHYSICAL REVIEW E 83, 061108 (2011) apparent efficiency of the heat engine [9,10]. For the demon by the MD. The MD may have quantum coherence as in Eq. (2) with coherence, the density matrix usually reads as or equivalently may possess a temperature TD = 1/βD lower than T = 1/β of the confined molecule’s heat bath. In each p F D = g ρ0 ∗ , (2) step, we evaluate the work done by the outside agent and the F pe heat exchange in detail. In order to concentrate on the physical where the off-diagonal element F measures the quantum properties, we put the calculations in Appendix A. coherence. The eigenvalues of the reduced density matrix represent two effective population probabilities as A. Step 1: Quantum insertion [(I) and (II)] 2 In the first process, the system is in contact with the heat p+ (F ) p − coth β |F | , e 2 D bath β, and then a piston is inserted isothermally into the (3) potential at position l. The potential is then divided into two − 2 domains, denoted simply as L and R, with lengths l and L l p− (F ) p + coth β |F | . g 2 D respectively. The eigenstates change into the following two = sets as We can define an effective inverse temperature βeff ⎧ − ⎨ 2 nπ(x−l) ln [p+ (F ) /p− (F )] / for the two-level MD, namely, − sin − l x L xψR(L − l) = L l L l , (8) 4 |F |2 n ⎩ β = β + cosh2 β coth β . (4) 00 x l eff D 2 D 2 D and The effective temperature T = 1/β here is lower than the eff eff 0 l x L bath temperature T . As shown in what follows, it is the D xψL(l) = , lowering of the effective temperature of the MD that results in n 2 l sin(nπx/l)0 x l an increase of the heat engine efficiency. Modeling the chamber as an infinite square potential well, with the corresponding eigenvalues En (L − l) and En (l). the eigenfunctions of the confined single molecule are In the following discussions, we use the free Hamiltonian HT = H + HD, where 2 x |ψn (L) = sin [nπx/L] , (5) = | | L H [En (l) ψn (l) ψn (l) n with the corresponding eigenenergies E (L) = n + E (L − l) |ψ (L − l)ψ (L − l)|] (¯hnπ)2/(2mL2), where the quantum number n ranges n n n ∞ from 1 to . The initial Hamiltonian can be written as for 0 l L and HD = |ee| . Here, we take the MD’s = | | H0 n En (L) ψn (L) ψn (L) . ground-state energy as the zero point of energy of atom. On this set of bases, the initial state of the total system is At the end of the insertion process, the system is still in expressed as a product state the thermal state with the temperature β and the MD is on its 1 own state without any change. With respect to the above split −βEn(L) D ρ0 = e |ψn (L)ψn (L)| ⊗ ρ , (6) bases, the state of the whole system is rewritten in terms of the Z (L) 0 n new bases as where = L + R − ⊗ D ρins [PL (l) ρ (l) PR (l) ρ (L l)] ρ0 , (9) = − Z (L) exp [ βEn (L)] (7) where n − e βEn(l) is the partition function of the system. ρL (l) = ψL(l) ψL(l) (10) Z (l) n n With the above model, the MD-assisted thermodynamic n cycle for the quantum SHE is divided into the four steps and illustrated in Fig. 1(b): (I), (II) the insertion of a mobile solid − − e βEn(L l) wall into the potential well at a position x = l (the origin is ρR (L − l) = ψR(L − l) ψR(L − l) (11) = Z (L − l) n n x 0); (II), (III) the measurement done by the MD to create n the quantum entanglement between its two internal states and refer to the system localized in the left and right domains the spatial wave functions of the confined molecule; (III), (IV) respectively. With respect to the their sum Z (l) = Z (l) + quantum control of the mobile wall to move the according Z (L − l) , the temperature-dependent ratios to the record in the demon’s memory; and (IV-V) the removal of the wall so that the next thermodynamic cycle can be started. PL (l) = Z (l) /Z (l) Their descriptions are discussed in the next section. and

III. QUANTUM THERMODYNAMIC CYCLES PR (l) = Z (L − l) /Z (l) WITH MEASUREMENT are the probabilities of finding the single molecule on the In this section, we analyze in detail the thermodynamic left and the right sides, respectively. For simplicity, we cycle of the molecule confined in an infinite square potential denote PL (l) and PR (l) by PL and PR, respectively, in the well. The molecule’s position is monitored and then controlled following discussions. We emphasize that the probabilities

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FIG. 3. (Color online) Work done by the outside agent. (a) Wins vs l for different system inverse temperatures β = 1, 0.5, and 0.1. Here, we choose the same parameter as in Fig. 2.(b)Wins vs L for different insertion positions l = 0.1L,0.3L,and0.5L with β = 1.

FIG. 2. (Color online) Probability PL and the corresponding diverges when the temperature approaches infinity, namely, C classical one PL vs temperature 1/β for different piston positions = = l 1/3andl 1/4. Without losing generality, we set the parameters lim Wins =∞. (13) as L = 1, m = π 2/2, andh ¯ = 1. T →∞ This conclusion is mathematically strictly proved in c = Appendix B. This is different from the conclusion are different from the classical probabilities, PL l/L and c = − limT →∞ Wins = 0 obtained in Ref. [14], which can be PR (L l) /L, of finding the single molecule on the left and right sides. We numerically illustrate this discrepancy directly obtained from the result in Eq. (5)inRef.[11] with between this classical result and ours in Fig. 2 for different d = 0. Actually, in Ref. [11], Zurek took the high-temperature insertion positions l = 1/3 and l = 1/4. It is clear in Fig. 2 limit at the beginning of the calculation of partition function − that the probability PL approaches the corresponding classical and obtained the result as kBT ln [L/ (L d)]. We should c remark that the final result kBT ln2inbothRefs.[11] and [14] one PL as the temperature increases to the high-temperature limit. However, a large discrepancy is observed at low is correct since the infinity in whole cycle could be canceled temperature. This deviation from the classical one is due to with each other. To explain this inconsistency, we notice Z = − the discreteness of the energy levels of the potential well with that limT →∞ ln [Z (L) / (l)] ln [L/ (L d)].Wehave − = = finite width, which disappears as level spacing diminishes limd→0 ln [L/ (L d)] limd→0 limT →∞ ln [Zb/Zins] 0, =∞ = with L →∞. In this case, the heat excitation will erase all while limT →∞ kB T . Thus, the limit limT →∞ Wins 0 the quantum feature of the system and the classical limit is cannot determined with the approximate formula approached. ln [L/ (L − d)]. This is similar to the generic problem: Know- = =∞ In this step, work should be done to the system. In the ing limx→∞ f (x) 0 and limx→∞ g (x) , one could not = ∞ isothermal process, the work done by the outside agent can be simply determine limx→∞ f (x) g (x) 0or according expressed as Wins = Uins − TSins, with the internal energy to some approximation. The right answer depends on the change mathematical details rather than some qualitative argument. The discrete property of the system due to the finite width U = Tr [ρ H − ρ (H + H )] of the potential well results in a typical quantum effect, even ins ins T 0 0 D at a high temperature. We can understand this divergence by and the total entropy change some consideration in physics. The work Wins to insert a wall is proportional to two factors: the probability of finding a single molecule at the insertion position and the energy of the S = Tr (−ρ ln ρ + ρ ln ρ ) . ins ins ins 0 0 single molecule. Taking insertion position, L/2 for example, the probability decreases as temperature increases and finally During this isothermal process, the work done outside just reaches a stable with a single molecule staying anywhere in compensates the change of the free energy as the potential at equal probability 1/L. However, the average energy for the single molecule keeps increasing. This results Wins = T [ln Z (L) − ln Z (l)] . (12) in a divergence of Wins for high temperature. This finite-size- induced quantum effect is typical for a mesoscopic system. To The same result has been obtained in Ref. [14]. By taking restore the classical results, we simply take the limit L →∞to β = L = inverse temperature 1 and 1, we illustrate the work make the spectrum continuous, rather than T →∞. Under this needed for the insertion of the piston into the potential in Fig. 3. limit L →∞,wehaveZ (l) /Z (L) → 1, which just recovers It is shown that to insert the piston at the center of the the classical result that potential, one needs the maximum work to be done. Another reasonable fact is that no work is needed to insert the piston at lim W = 0, (14) →∞ ins positions l = 0 and l = L. Classically, it is well known that no L work should be done to insert the piston at any position. The as illustrated in Fig. 3(b) for different insertion positions l = classical√ limit for partition function means that L λ, where 0.1L,0.3L, and 0.5L. λ = h/ 2πmkBT is the mean thermal wavelength. However, After the insertion of the piston, the entropy of the system for a fixed L, it follows the numerical calculation that Wins changes. The system exchanges heat with the heat bath during

061108-4 QUANTUM MAXWELL’s DEMON IN THERMODYNAMIC CYCLES PHYSICAL REVIEW E 83, 061108 (2011) this isothermal reversible process. The heat is obtained by this information can be expressed in the form of mutual Q =−TS as information [18]: ins ins ∂ Q = T − [ln Z (L) − ln Z (l)] . (15) I = pe ln pe + pg ln pg − (PLpg + PRpe)ln(PLpg + PRpe) ins ∂β − (PLpe + PRpg)ln(PLpe + PRpg). Similar to the asymptotic properties of the work in Eq. (14), →∞ Qins approaches to zero when L . For the case TD = 0, we have maximum mutual information Ieff =−(PL ln PL + PR ln PR), while it reaches minimum B. Step 2: Quantum measurement [(II) and (III)] Imin = 0 with = 0. The worst case is that when we first let the MD become In the second step, the system is isolated from the heat degenerate, that is, = 0, the temperature approaches zero. bath. The MD finds which domain the single molecule stays In this sense, the demon is prepared in a mixing state in and registers the result into its own memory. In the classical way, the memory can also be imaged as a chamber ρD ( = 0) = 1 (|gg| + |ee|) containing a single molecule. The classical states of the single 0 2 molecule on the right and left sides are denoted as the states and the state of the whole system after the measurement reads 0 and 1. The memory is designed always as two bistable states with no energy difference = 0, and no energy is ρ = 1 [ρL(l) + ρR(L − l)] ⊗ ρD ( = 0) . (19) needed in the measurement process. This setup based on mea 2 0 the “chamber” argument seems to exclude the possibility for Thus, no information is obtained by the MD. There exists quantum coherence in a straightforward way. Therefore, we another limiting process that the nondegenerate MD is first adopt the TLS as the memory to allow the quantum coherence prepared in the zero-temperature environment, and then to take the role, as discussed in Sec. II. In the scheme here, approaches zero. Thus, the state of the MD is on state |gg|. the demon performs the controlled-NOT operation [12]. If the In this case, we get a more accurate MD, as mentioned above. molecule is on the left side, no operation is done, whereas the The physical essence of the difference between the two limit demon flips its state when finding the molecule on the right. processes is in the symmetry breaking [19] (we discuss this This operation is realized by the following unitary operator: again later). With such symmetry breaking, the degenerate U = ψL (l) ψL (l) ⊗ (|gg| + |ee|) MD could also make an ideal measurement. An intuitive n n understanding for the zero-temperature MD helping to do work n is that a more calm MD can see the states of the molecule were + R − R − ⊗ | | + ψn (L l) ψn (L l) ( e g H.c) . (16) clearly and control the molecule more effectively. We remark that the operation is unitary [17], which is different Next, we calculate the work done in the measurement from the previous discussion [6] of MD with the positive process by assuming that the total system is isolated from the operator valued measure (POVM). After the measurement, the heat bath of the molecule. The heat exchange here is exactly = MD and the system are correlated. This correlation enables the zero, namely Qmea 0, since the operation is unitary and the MD to control the system to perform work to the outside agent. total entropy is not changed during this process. However, The density matrix of the whole system after the measurement the total internal energy changes, which merely results from is the work done by the outside agent L R ρmea = [PLpgρ (l) + PRpeρ (L − l)] ⊗ |gg| Wmea = PR(pg − pe) L R +[PLpeρ (l) + PRpgρ (L − l)] ⊗ |ee| . (17) to register the MD’s memory. The work needed is actually a If the temperature of the demon is zero, namely T = 0, the D monotonous function of the demon’s bath temperature TD.If measurement actually results in a perfect correlation between the temperature of the demon is zero (the MD is prepared in a the system and the MD, pure state), namely TD = 0, the work reaches its maximum W max = P . The demon can distinguish exactly which ρ = P ρL (l) ⊗ |gg| + P ρR (L − l) ⊗ |ee| . (18) mea R mea L R domain the single molecule stays in, state |g representing Then the demon can distinguish exactly the domain where the molecule on the left and vice versa. As discussed as follows, single molecule stays, for example, state |g representing the the work done by the outside agent here is the main factor molecule on left side and vice versa. At a finite temperature, to slow down the efficiency of the heat engine. However, the this correlation gets ambiguous. As illustrated in the dot- low temperature results in a more perfect quantum correlation dashed green box in Fig. 1(b), the demon actually gets the between the MD and the system, thus enables the MD to wrong information about the which domain the molecule stays extract more work. Requirement of the work done in the in. For example, the demon thinks the molecule is on the left measurement and the ability of controlling free expansion are with memory registering |g, while the molecule is actually two competing factors of the QHE. It is clear that less work is on the right. The MD loses a certain amount of information needed if the insertion position is closer to the right boundary about the system and lowers its ability to extract work. For case of the potential. The work needed in the measurement process = 0 at finite temperature, the above imperfect correlation approaches zero, namely Wmea → 0, when l → L. Thus, the leads to a condition for the MD’s temperature under which efficiency is promoted to reach the corresponding Carnot the total system could extract positive work. Quantitatively, efficiency when l = L for this measurement.

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C. Step 3: Controlled expansion [(III) and (IV)] D. Step 4: Removal [(IV) and (V)] In the third step, the system is brought into contact with To complete the thermodynamic cycle, the system and the the heat bath with temperature β. Then the expansion is MD should be reset to their own initial states. As for the performed slowly enough to enable the process to be reversible system, the piston inserted in the ﬁrst step should be removed. and isothermal. The expansion is controlled by the demon In the previous studies, this process is neglected, since the according to its memory. Finding its state on |g, the outside measurements are always ideal and the piston is moved to an agent allows the piston to move right and the single molecule end boundary of the chamber. Thus, no work is required to performs work to the outside. However, the agent pays some remove the piston. However, in an arbitrary process, we can work to move the piston to the right if the MD’s memory is show the importance of removing the piston in the whole inaccurate, for example, the situation in the green dot-dashed cycle. During this process, the system keeps contact with box in Fig. 1(b). If in state |e, the piston is allowed to move to the heat bath with inverse temperature β and the removal is the left side. Under this description, we avoid the conventional performed isothermally. The density matrix of the total system heuristic discussion with adding an object in the classical after removing the piston reads version of SHE. Here, we choose two arbitrary ﬁnal positions − e βEn(L) of the controlled expansion as lg and le for the corresponding ρ = |ψ (L)ψ (L)| |g |e rev Z(L) n n MD’s state and . We prove later that the total work n extracted is independent of the expansion position chosen here. ⊗ [(P p + P p )|gg|+(P p + P p )|ee|]. After the expansion process, the density matrix of the whole L g R e L e R g system is expressed as (23) In this process, the work done and the heat absorbed by the = L + R − ⊗| | ρexp [PLpgρ (lg) PRpeρ (L lg)] g g outside are + [P p ρL(l ) + P p ρR(L − l )] ⊗|ee|. (20) L e e R g e Wrev = Tr[ρrev(H + HD) − ρexpHT ] − T Tr[−ρrev ln ρrev]

+T Tr[−ρexp ln ρexp] (24) During the expansion, the system performs work −Wexp 0 to the outside agent, and

Qrev =−T Tr[−ρrev ln ρrev] + T Tr[−ρexp ln ρexp], (25) Wexp = T [ln Z(l) + PL ln PL + PR ln PR.

−PLpg ln Z(lg) − PRpe ln Z(L − lg) respectively. We refer to Appendix A for the exact expression of those two formulas. The MD now is no longer entangled − − − PLpe ln Z(le) PRpg ln Z(L le)]. (21) with the system, and the density matrix of the demon is factorized out as For a perfect correlation (pg = 1), with the piston moved to = = D = + | |+ + | | the side of the potential, namely lg L and le 0, the work ρrev (PLpg PRpe) g g (PLpe PRpg) e e . (26) is simply In the ideal case TD = 0, the demon is on the state

Wexp = T (PL ln PL + PR ln PR) − Wins, D = | |+ | | ρrev PL g g Pe e e which is the maximum work that can be extracted in this with entropy process. In the classical limit L →∞,theworkis D =− − Srev PL ln PL PR ln PR. = + Wexp T (PL ln PL PR ln PR) . According to Landauer’s principle, erasing the memory of the D = MD dissipates at least TDSrev 0 work into the environment. We restore the well-known result W =−k T ln 2, when the exp B In this sense, we can extracted kBT ln 2 work with MD’s help. piston is inserted in the center of the potential. If the demon is However, we does not violate the SLOT, since the whole not perfectly correlated to the position of the single molecule system functionalizes as a heat engine working between a − (pg < 1), the work extracted Wexp would be less. Therefore, high-temperature bath and a zero-temperature bath. Actually, it is clear that the ability of MD to extract work closely depends the increase of entropy in the zero-temperature bath is exactly on the accuracy of the measurement. D Srev. Therefore, the energy dissipated actually depends on In this step, the heat exchange is related to the change of the temperature of the environment where the information is entropy as erased. In the previous studies, people always set the same temperature for the system and the MD. Thus, the exact ∂ Q = P T − [ln Z(l)−p ln Z(l ) − p ln Z(L − l )] mechanism of MD was not clear to a certain extent, especially exp L ∂β g g e e for SHE. Let us consider another special case, = 0, which directly results in pe = pg = 1/2. MD is prepared on its ∂ D = + PR T − [ln Z(L − l) − pe ln Z(L − lg) maximum entropy state ρ0 ( 0). At the end of the cycle, ∂β D = D = MD actually is on the same state, namely ρrev ρ0 ( 0). − pg ln Z(le)]. (22) Thus, no work is paid to erase the memory.

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After this procedure, the MD is decoupled from the system the same as those of the classical one, namely PL = PR = 1/2. and brought into contact with its own thermal bath with inverse The total work extracted here can be written in a simple form, temperature βD. Since Wtot = T (ln 2 + pe ln pe + pg ln pg) − (pg − pe)/2. (31) + PLpe PRpg pe, (27) In this special case, to make the system capable of doing work to the outside, there is a requirement for the temperature of the the MD releases energy into its heat bath. We do not discuss demon (low-temperature bath). For example, when we choose this thermalization process here in detail. The MD and the β = 1 and = 0.5, the PWC is βD 2.09. This requirement system are reset to their own initial states ρ0, which allows a is more strict than that of the Carnot heat engine, βD > 1. The new cycle to start. efficiency of this heat engine reads − = − (pg pe) IV. EFFICIENCY OF SZILARD HEAT ENGINE η 1 , (32) 2T (ln 2 + pe ln pe + pg ln pg) For the quantum version of the SHE, the quantum coherent which is lower than the corresponding Carnot efficiency, based on the finite size of the chamber results in different properties from the classical one. Work is required during the TD ηCarnot = 1 − . insertion and removal processes, while the same process can T be done freely in the classical version. The microscopic model Here, the efficiency is a monotonic function of the energy here relates the efficiency of the measurement carried out by spacing and reaches its maximum the MD to the temperature of the heat bath. In the whole 2TD thermodynamic cycle, the work done by the system to outside ηmax = 1 − ηCarnot is the sum of all the work done in each process, T with = 0. In the general case, we show the work done by the system Wtot =−(Wins + Wmea + Wexp + Wrev) and the efficiency of the heat engine versus the position = T [(pe ln pe + pg ln pg) of the wall l and the temperature of demon βD in Figs. 4 − (PLpg + PRpe)ln(PLpg + PRpe) and 5. As illustrated in Fig. 4(c), for small insertion positions, = − (PLpe + PRpg)ln(PLpe + PRpg)] for example, l 0.16 and 0.36, the system cannot extract positive work. There exists a critical insertion position l to −P (p − p ). (28) cri R g e extract positive work, namely, To enable the system to do work outside, the temperature of cri cri + cri cri + cri = T PL ln PL PR ln PR PR 0, (33) the MD should be low enough to make sure Wtot 0, which is cri = cri = known as the positive-work condition (PWC) [8]. To evaluate where PR PR (lcri) and PL PL (lcri). This critical value the efficiency of QHE, we need to obtain the heat absorbed of insertion position is lcri = 0.447 for the typical parameter from the high-temperature heat bath. In contrast to the classical chosen here. Due to the requirement of work in the measure- one, the exchange of heat with the high-temperature source ment process, the work extracted is not a symmetric function of persists in each step. The total heat absorbed from the high- the insertion piston l, namely Wtot (0.5 − l) = Wtot (0.5 + l), temperature source is the sum over all four steps, as illustrated in Figs. 4(b) and 4(c). Since the high-energy

Qtot =−(Qins + Qmea + Qexp + Qrev)

= T [(pe ln pe + pg ln pg)

− (PLpg + PRpe)ln(PLpg + PRpe)

− (PLpe + PRpg)ln(PLpe + PRpg)]. (29)

It is obviously that Qtot = TI. Here, the absorbed energy is used to perform work to the outside, while only the measurement process wastes Wmea, which is released to the low-temperature heat bath. It is very interesting to notice that Wmea → 0as → 0, while the total heat Qtot → 0 and Wtot → 0. To check the validity of SLOT, one should determine the efﬁciency of this heat engine in a cycle, P (p − p ) η = 1 − R g e . (30) Q tot FIG. 4. (Color online) Work vs insertion position l and MD’s inverse temperature βD. (a) Total work as a function of βD for l = 0.2, As an example, we consider the special case l = L/2, which 0.5, and 0.8. (b) Total work as a function of insertion position l for is similar to the case of the ordinary SHE with the piston βD = 2.0, 3.0, and 4.0. (c) Contour plot for total work as function of inserted in the center of the chamber. In this special case, the l and βD. The position for maximum work extracted is denoted by probabilities for the single molecule staying at the two sides are the white dashed line.

061108-7 H. DONG, D. Z. XU, C. Y. CAI, AND C. P. SUN PHYSICAL REVIEW E 83, 061108 (2011)

for the MD’s initial state results in different work extracted, namely,

lim lim Wtot = 0, (37) βD →+∞ →0

lim lim Wtot = kB T ln 2. (38) →0 βD →+∞ The former one means that MD actually gets no information about the position of the molecule and extracts no work, while the latter one shows that MD obtains the exact information on the position of the molecule and enables the system to perform maximum work to the outside agent. The same phenomenon has also been revealed in the process of dynamic thermalization [19].

FIG. 5. (Color online) Efficiency vs insertion position l and V. CONCLUSIONS = inverse temperature βD. (a) Efficiency as a function of βD for l 0.4, In summary, we have studied a quantum version of SHE = 0.5, and 0.6. (b) Efficiency as a function l for βD 2, 3, and 4. (c) with a quantum MD at a finite temperature lower than that Contour plot of efficiency vs l and βD. of the system. Overall, we simplify the MD as a two-level system, which carries out the measurement in quantum state |e of the demon is utilized to register the right side fashion and controls the system to do work to the outside for a single molecule, more work is need when l

061108-8 QUANTUM MAXWELL’s DEMON IN THERMODYNAMIC CYCLES PHYSICAL REVIEW E 83, 061108 (2011) problems in thermodynamics, such as the relationship between APPENDIX A: DETAILED CALCULATION quantum unitarity and SLOT [20]. Also, the nonequilibrium IN EACH STEP properties related to Jarzynski and Crooks’s theorem [21] can In this appendix, we present a detailed calculation for the be discussed in this model. work done and efﬁciency of SHE. Following the calculations for the four steps, the reader can deeply understand the physical ACKNOWLEDGMENTS essences of the MD in some subtle fashion. H.D. thank S J. N. Zhang for helpful discussionS. This Step 1: Insertion. In this process, the changes of work was supported by NSFC through Grants No. 10974209 internal energy Uint = Tr [ρinsHT − ρ0 (H0 + HD)] and to- and No. 10935010 and by the National 973 Program (Grant tal entropy Sins = Tr [−ρins ln ρins] − Tr [−ρ0 ln (ρ0)] are No. 2006CB921205). explicitly given by

∂ U = p (l) E (l) + p (L )E (L ) − p (L) E (L) = [ln Z (L) − ln Z (l)] , (A1) int n n n n n n ∂β n n n where L = L − l and pn (l) En (l) ∂ S = [ln Z (l) − ln Z (L)] + β = 1 − β [ln Z (l) − ln Z (L)], (A2) ins + − ∂β n pn(L )En(L ) pn (L) En (L) where exp(−βE (y)) p (y) = n . n Z(y)

For the isothermal process, the work done by the outside agent and the heat exchange are simply Wins = Uint − TSins and Qins =−TSins, namely, = − Z Wins T[ln Z (L) ln (l)] , (A3) ∂ Q = T − [ln Z (L) − ln Z (l)] . (A4) ins ∂β

Step 2: Measurement. The measurement is realized by a controlled-NOT unitary operation, which has been illustrated clearly in Sec. II. After the measurement process, the density matrix for the total system is

L R L R ρmea = [PLpgρ (l) + PRpeρ (L )] ⊗ |gg| + [PLpeρ (l) + PRpgρ (L )] ⊗ |ee| . The entropy is not changed in this step. The work done by the outside agent is

Wmea = Umea = PR(pg − pe). (A5)

Step 3: Controlled expansion. At the ending of expansion, the state for the total system reads

L R L R ρexp = [PLpgρ (lg) + PRpeρ (Lg)] ⊗ |gg| + [PLpeρ (Le) + PRpgρ (le)] ⊗ |ee| . where Lg = L − lg and Le = L − le We move the wall isothermally. The work done by the outside agent can be obtained by the same methods used in the insertion process as W = Tr[ρ (H + H )] − Tr [ρ (H + H )] − T Tr[−ρ ln ρ ] + T Tr [−ρ ln ρ ] exp exp D mea D exp exp mea mea = [PLpgpn(lg)En(lg) + PRpepn(Lg)En(Lg) + PLpepn(Le)En(Le) + PRpgpn(le)En(le)] + (PLpe + PRpg) n

− (PLpn (l) En (l) + PRpn(L )En(L )) + (PLpe + PRpg) + T [PLpgpn(lg)lnPLpgpn(lg) n n + P p p (L )lnP p p (L ) + P p p (L )lnP p p (L ) + P p p (l )lnP p p (l )] Re n g R e n g L e n e L e n e R g n e R g n e

− T [PLpgpn(l)lnPLpgpn(l) + PRpepn(L )lnPRpepn(L ) n

+ PLpepn(l)lnPLpepn(l) + PRpgpn(L )lnPRpgpn(L )] (A6)

= PLT [ln Z (l) − pg ln Z(lg) − pe ln Z(Le)] + PRT [ln Z(L ) − pe ln Z(Lg) − pg ln Z(le)]. (A7)

061108-9 H. DONG, D. Z. XU, C. Y. CAI, AND C. P. SUN PHYSICAL REVIEW E 83, 061108 (2011)

The internal energy changes can be also evaluated as Uexp = [PLpgpn(lg)En(lg) + PRpepn(Lg)En(Lg) + PLpepn(Le)En(Le) + PRpgpn(le)En(le)] + (PLpe + PRpg) n

− (PLpn (l) En (l) + PRpn(L )En(L )) + (PLpe + PRpg) n = [PLpgpn(lg)En(lg) + PRpepn(Lg)En(Lg) + PLpepn(Le)En(Le) + PRpgpn(le)En(le)] n

− [PLpn (l) En (l) + PRpn(L )En(L )] n ∂ ∂ = P [ln Z (l) − p ln Z(l ) − p ln Z(L )] + P [ln Z(L ) − p ln Z(L ) − p ln Z(l )]. (A8) L ∂β g g e e R ∂β e g g e

Then, we obtain the heat exchanges in this process as Qexp =−TSexp = Wexp − Uexp or ∂ ∂ Q = P T − [ln Z l − p ln Z(l ) − p ln Z(L )] + P T − [ln Z(L ) − p ln Z(L ) − p ln Z(l )]. exp L ∂β g g e g R ∂β e g g e

Step 4: Removal. The piston is removed in this process. After that, the system returns to its initial state and is not entangled with MD. The last step would be to remove the wall in the trap. The system is on the state as exp [−βE (L)] ρ = n |ψ (L)ψ (L)| ⊗ [(P p + P p ) |gg| + (P p + P p ) |ee|]. (A9) rev Z(L) n n L g R e L e R g n Then, the work done and the heat absorbed are respectively

Wrev = Tr [ρrev (H + HD)] − Tr[ρexp(H + HD)] − T Tr [−ρrev ln ρrev] + T Tr[−ρexp ln ρexp] (A10) or Wrev = pn (L) En (L) + (PLpe + PRpg) n − [PLpgpn(lg)En(lg) + PRpepn(Lg)En(Lg) + PLpepn(Le)En(Le) + PRpgpn(le)En(le)] − (PLpe + PRpg) n + T pn (L) ln pn (L) + (PLpg + PRpe)ln(PLpg + PRpe) + (PLpe + PRpg)ln(PLpe + PRpg) n −T {PLpgpn(lg)ln[PLpgpn(lg)] + PRpepn(Lg)ln[PRpepn(Lg)] + PLpepn(Le)ln[PLpepn(Le)] n

+ PRpgpn(le)ln[PRpgpn(le)]}

= T [− ln Z (L) + (PLpg + PRpe)ln(PLpg + PRpe) + (PLpe + PRpg)ln(PLpe + PRpg) − PL ln PL − PR ln PR

−pe ln pe − pg ln pg + PLpg ln Z(lg) + PRpe ln Z(Lg) + PLpe ln Z(Le) + PRpg ln Z(le)] (A11) and

Qrev =−T Tr [−ρrev ln ρrev] + T Tr[−ρexp ln ρexp] = T pn (L) ln pn (L) + (PLpg + PRpe)ln(PLpg + PRpe) + (PLpe + PRpg)ln(PLpe + PRpg) n −T {PLpgpn(lg)ln[PLpgpn(lg)] + PRpepn(Lg)ln[PRpepn(Lg)] n

+PLpepn(Le)ln[PLpepn(Le)] + PRpgpn(le)ln[PRpgpn(le)]}

= T {− ln Z (L) + (PLpg + PRpe)ln(PLpg + PRpe) + (PLpe + PRpg)ln(PLpe + PRpg)

−PL ln PL − PR ln PR − pe ln pe − pg ln pg + PLpg ln Z(lg) + PRpe ln Z(Lg) + PLpe ln Z(Le) + PRpg ln Z(le)} − [pn (L) En (L) − PLpgpn(lg)En(lg) − PRpepn(Lg)En(Lg) − PLpepn(Le)En(Le) − PRpgpn(le)En(le)] n 061108-10 QUANTUM MAXWELL’s DEMON IN THERMODYNAMIC CYCLES PHYSICAL REVIEW E 83, 061108 (2011)

= T [(P p + P p )ln(P p + P p ) + (P p + P p )ln(P p + P p ) − P ln P − P ln P − p ln p − p ln p ] L g R e L g R e L e R g L e Rg L L R R e e g g ∂ ∂ ∂ − T − ln Z (L) + P p T − ln Z(l ) + P p T − ln Z(L ) ∂β L g ∂β g R e ∂β g ∂ ∂ + P p T − ln Z(L ) + P p T − ln Z(l ). (A12) L e ∂β e R g ∂β e

The total work extracted by outside agent is the sum of work extracted in each step as

Wtot =−(Wins + Wmea + Wexp + Wrev) = T [(pe ln pe + pg ln pg) − (PLpg + PRpe)ln(PLpg + PRpe)

− (PLpe + PRpg)ln(PLpe + PRpg)] − PR(pg − pe). (A13) The total heat absorbed can also be obtained as (pe ln pe + pg ln pg) − (PLpg + PRpe)ln(PLpg + PRpe) Qtot =−(Qins + Qexp + Qrev) = T . (A14) −(PLpe + PRpg)ln(PLpe + PRpg)

APPENDIX B: PROOF OF THE LIMITATION Next, we use the above result in Eq. (B6)toestimate Wins(β). With the above definition, we rewrite Wins(β)as In this appendix, we mathematically consider the limit √ behavior of the insertion work 1 1 − 2c(z) z/π Wins(β) = ln √ , (B7) ∞ − 2 2 2 β 1 − 4c(4z) z/π 1 n=1 exp( βn α /L ) W (β) = ln ∞ (B1) ins − 2 2 2 = 2 2 β 2 n=1 exp( 4βn α /L ) where z βα /L .√ Now, let us consider√ the high-temperature as β → 0. To this end, we first estimate the series case, in which c(z) z/π and c(4z) z/π are much less than unity. A straightforward calculation gives ∞ 2 S(k) ≡ e−kn , (B2) 1 Wins = ln{[1 − 2c(z) z/π][1 + 4c(4z) z/π + O(β)]} n=1 β 1 where k is a positive real number. Actually, we can approx- = ln[1 + (4c(4z) − 2c(z)) z/π + O(β)] imate the series S(k) with integrals of f (x) = exp(−kx2) β over two domains (1,∞) and (0,∞) respectively. After re- 1 = [(4c(4z) − 2c(z)) z/π + O(β)] expressing the two integrals back to two sums approximately β in Figs. 6(a) and 6(b), we observe − = √2a 2c(4z√) c(z) + ∞ ∞ ∞ O(1). (B8) 2 2 2 πL β e−kx dx < e−kn < e−kx dx. 1 n=1 0 Finally, we mathematically strictly prove our claim that the √ work for insertion diverges as temperature tends to infinity, ∞ − 2 = When we noticed that 0 exp( kx )dx π/(4k), it is easy namely, to prove that the deviation a ∞ lim W = lim √ =∞. (B9) ∞ −→ ins −→ 2 2 β 0 β 0 πβL c(k) ≡ e−kx dx − e−kn 0 = n 1 π = − S(k)(B3) 4k satisfies 1 1 2 0

lim c(k) = 1 . (B5) k−→ 0 2 The detailed calculation can be found in Appendix. This result could be veriﬁed by the numerical simulation. Therefore, we (a) (b) conclude that FIG. 6. (Color online) Comparison between summation and π 1 ∞ ∞ lim S(k) → − . (B6) integral over two domains (a) (0, ) and (b) (1, ). The summation k−→ 0 4k 2 is marked as thee ﬁlled area in both panels.

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