Maximum Power Output of Quantum Heat Engine ... - Semantic Scholar

entropy Article

Maximum Power Output of Quantum Heat Engine with Energy Bath Shengnan Liu and Congjie Ou * Fujian Provincial Key Laboratory of Light Propagation and Transformation, College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China; [email protected] * Correspondence: [email protected]; Tel.: +86-592-616-2388 Academic Editor: Antonio M. Scarfone Received: 8 April 2016; Accepted: 23 May 2016; Published: 25 May 2016

Abstract: The difference between quantum isoenergetic process and quantum isothermal process comes from the violation of the law of equipartition of energy in the quantum regime. To reveal an important physical meaning of this fact, here we study a special type of quantum heat engine consisting of three processes: isoenergetic, isothermal and adiabatic processes. Therefore, this engine works between the energy and heat baths. Combining two engines of this kind, it is possible to realize the quantum Carnot engine. Furthermore, considering finite velocity of change of the potential shape, here an infinite square well with moving walls, the power output of the engine is discussed. It is found that the efficiency and power output are both closely dependent on the initial and final states of the quantum isothermal process. The performance of the engine cycle is shown to be optimized by control of the occupation probability of the ground state, which is determined by the temperature and the potential width. The relation between the efficiency and power output is also discussed. Keywords: quantum heat engine; two-state system; performance optimization

1. Introduction Quantum thermodynamics introduces the interdisciplinary field that combined classical thermodynamics and quantum mechanics since the concept of quantum heat engine appeared in the 1960s [1,2]. Inspired by the properties of the classical thermodynamic processes and cycles, the quantum analogs of the processes and cycles have been developed and discussed in more and more different quantum systems [3–24]. Recently, some micro sized heat engines with single Brownian particle induced by optical laser trap [25,26] and single ion held within a modified linear Paul trap [27] have been experimentally realized, which presents significant insight into the energy conversion on a microscopic level and would be expected to shed light on the experimental investigation in quantum thermodynamic characteristics of small systems. Therefore, it is of great interest to adopt a single-particle quantum system as the working substance to investigate the properties of quantum thermodynamic processes and quantum engine cycles [5,11,12,15,16,20,21,24]. A central concern of quantum thermodynamics is to understand the basic relationships between classical thermodynamics and quantum mechanics [5,13–15]. The quantum analog of the classical engine cycles can be set up by employing a single-particle quantum system with two energy levels [12,14,15,20] because of its simplicity. According to the first law of thermodynamics, the quantum analog of mechanical work and heat transfer can be defined in a natural way [5,12,14]. Thus, the basic thermodynamic processes, such as adiabatic, isochoric, isobaric ones, can be well depicted in a quantum two-state system. Nevertheless, the quantum properties of the two-state system determine the inherent difference of the thermodynamic processes. In classical thermostatistics, the law of equipartition of energy is crucial for the link between the energy and temperature [28]. However, it is violated in the quantum Entropy 2016, 18, 205; doi:10.3390/e18060205

www.mdpi.com/journal/entropy

Entropy 2016, 18, 205

2 of 10

regime even for non-interacting particles confined in a box. In a two-state quantum system, the expectation value of the Hamiltonian depends not only on the temperature, but also on the quantum state of the system [13–15]. Therefore, the quantum isothermal process (to fix the temperature) and the quantum isoenergetic process (to fix the expectation value of the Hamiltonian) are totally different from each other. During the quantum isoenergetic process, the mechanical expansion/compression and the quantum state engineering are controlled simultaneously by environmental system, which is considered as energy bath [12,17,20,24]. It is worth noting that such kind of energy bath ensures the validation of the second law of thermodynamics in quantum regime [12,19,20]. Therefore, by coupling the quantum two-state system with a heat bath and an energy bath, it is possible to construct an engine cycle, which is helpful to understand the influence of quantum properties on energy conversion for a small system. 2. Two-State Quantum System Coupled to a Heat and an Energy Bath The model we consider here is a single particle confined in an one-dimensional infinite square well potential with movable walls, which is a simplification of a piston. The corresponding stationary Schrödinger equation is given by H |un y “ εn |un y pn “ 1, 2, 3, ...q, where |un y and εn represent the n-th eigenstate and corresponding energy eigenvalue, respectively. Since we are interested in genuine quantum effects, here we assume that the temperature is low and the system size is small. In this approximation, the ground pn “ 1q and first excited pn “ 2q states are dominantly relevant [14,20,21]. Therefore, the occupation probabilities of the ground state and excited state can be written as p and 1 ´ p. The expectation value of the Hamiltonian can be written as E “ pε1 ` p1 ´ pqε2 . If the system is in thermal equilibrium with the heat bath at temperature T, the probability of finding the system in a state with the energy ε is given by the Boltzmann factor p9expp´ε{kTq [3,13–15], where k is the Boltzmann constant. The energy eigenvalues of the ground and first excited states are given by ε1 “ π2 h¯ 2 {2mL2 and ε2 “ 2π2 h¯ 2 {mL2 , respectively, where m is the mass of the particle and L is the width of the square well potential. Thus, the expectation value of the Hamiltonian is E“

π2 h¯ 2 p4 ´ 3pq 2mL2

(1)

For convenience, we set π2 h¯ 2 {m “ 1 below. The ratio between the probabilities of the ground state and the first excited state can be written as p expp´1{2kTL2 q “ 1´ p expp´2{kTL2 q

(2)

From Equation (2) the probability that the system is in the ground state can be expressed as ppT, Lq “

1 ´3

(3)

1 ` e 2kTL2 On the other hand, the potential width L can be considered as the volume of this kind of one-dimensional system. Therefore, the force (i.e., pressure in 1 dimension) on the potential wall is [13,29], „  dε1 dε2 4 ´ 3p f “´ p ` p1 ´ pq “ (4) dL dL L3 Form Equation (4), one can see that the force varies with the potential width L so it is possible to adopt a curve on the f–L plane to describe a thermal-like quantum process. It is in fact a one dimensional analog of the pressure–volume plane of classical thermodynamics.

Entropy 2016, 18, 205

3 of 10

If the two-state system is coupled to a thermal bath at temperature T, Equation (3) can be substituted into Equation (4) and yields ´3

f “

1 ` 4e 2kTL2 ´3

(5)

L3 p1 ` e 2kTL2 q According to Equation (5), the slope of an isothermal quantum process curve on f–L plane can be obtained as ˇˆ ˙ ˇ ˇ Bf ˇ 3p4 ´ 3pq ´ 6pp1 ´ pqlnrp{p1 ´ pqs ˇ ˇ (6) ˇ BL ˇ “ L4 T If the two-state system is coupled to an energy bath to fix the expectation of Hamiltonian. From Equations (1) and (4) one can obtain ˇˆ ˙ ˇ ˇ p4 ´ 3pq ˇ Bf ˇ ˇ ˇ BL ˇ “ L4 E

(7)

Obviously, the isothermal curve on the f–L plane is different from the isoenergetic one originating from the quantum properties. 3. Quantum Engine Cycle Based on Two-State System As mentioned above, the difference between quantum isoenergetic process and quantum isothermal process can be illustrated by their curves on the f–L plane. According to the quantum adiabatic theorem [5,30–32], which should not be confused with the thermodynamic adiabaticity, if the time scale of the change of the Hamiltonian or the potential width is much larger than the typical dynamical one, h¯ {E, then the stationary Schrödinger equation for the energy eigenstate holds instantaneously [17]. The slope of the curve of the adiabatic process, during which the state remains unchanged (i.e., p is fixed) can be directly derived from Equation (4) as follows: ˇˆ ˙ ˇ ˇ Bf ˇ 3p4 ´ 3pq ˇ ˇ ˇ“ ˇ ˇ BL p ˇ L4

(8)

It is worth noting that for the positive temperature, T ą 0, Equation (3) indicate that 1{2 ă p ă 1. In this case, the slopes of quantum isothermal process, isoenergetic process and adiabatic process can be compared at the same potential width L and yields [29], ˇˆ ˙ ˇ ˇˆ ˙ ˇ ˇˆ ˙ ˇ ˇ ˇ Bf ˇ Bf ˇ ˇ Bf ˇ ˇ ˇ ˇ ˇąˇ ˇ , p1{2 ă p ă 1q ˇ ˇą ˇ BL p ˇ ˇ BL T ˇ ˇ BL E ˇ

(9)

According to Equation (9), if the positive temperature area is concerned, we can construct the possible three-process cycles on the f–L plane as it is shown in Figure 1. 1 Ñ 2 is an isothermal quantum process, the system is coupled to a heat bath with temperature TH . During the expansion of the potential width, one wall of the potential acts as a piston to perform work [17] and the energy is transferred from the heat bath to the system. 2 Ñ 3 is an isoenergetic quantum process, which means that the two-state system exchanges energy with an energy bath to keep its expectation value of the Hamiltonian constant. 3 Ñ 1 is an adiabatic quantum process to connect the first two processes so that a closed cycle on the f–L plane can be realized.

Entropy 2016, 18, 205 Entropy 2016, 18, 205

4 of 10

f

4 of 10

L Figure 1.1. The Thediagram diagram constructed quantum f–L where plane, ie, where and it of of thethe constructed quantum cyclecycle on theonf–Lthe plane, ad andie,it ad represent represent the isoenergetic, adiabatic and isothermal quantum processes, respectively. the isoenergetic, adiabatic and isothermal quantum processes, respectively.

4. Performance 4. Performanceof ofthe theQuantum QuantumEngine EngineCycle Cycle During the the isothermal isothermal process process 11Ñ the heat heat absorbed absorbed from from the the heat heat bath bath is, is,  22,, the During

Qin  TH S  TH ( S2  S1 ) (10) Qin “ TH ∆S “ TH pS2 ´ S1 q (10) where S is the entropy of the two-state system and it is given by, where S is the entropy of the two-state system and it is given by,   1  pi Si  „k  pi ln  ln(1  pi )   (i  1, 2) (11) 1 ´ ppi i Si “ k pi ln ´ lnp1 ´ piq pi “ 1, 2q (11) pi where pi is the occupation probability of the ground state when the system is at point “i” of the where pi isSubstitution the occupation probability of the ground(11) stateyields when the system is at point “i” of the f–L f–L plane. of Equation (3) into Equation plane. Substitution of Equation (3) into Equation (11) yields 3(1  pi )  k ln pi (i  1, 2) Si  (12) 3p1 ´ 2p q (12) Si “ 2Ti Li2 i ´ klnpi pi “ 1, 2q 2Ti Li Since points 1 and 2 are connected by an isothermal process with temperature TH in the f–L Since connected by an isothermal process with temperature TH inas, the f–L plane, T2 2Tare plane, one points has T11and H in Equation (12). Therefore, Equation (10) can be rewritten one has T1 “ T2 “ TH in Equation (12). Therefore, Equation (10) can be rewritten as, p 3 1  p 1  p  Qin  TH ( S 2  S1 )  « 2 2  2 1   ff kTH ln 2 (13) 32  1 ´ L2 p2 L11´ p1 p1 p2 Qin “ TH pS2 ´ S1 q “ ´ ´ kT ln (13) H 2 p1 L22 L21 During the isoenergetic compression process 2  3 , the expectation of Hamiltonian is fixed. FromDuring the firstthe law of thermodynamic [5,32],process the heat released the system to the surroundings isoenergetic compression 2Ñ 3 , the from expectation of Hamiltonian is fixed. is compensated by the work, i.e., From the first law of thermodynamic [5,32], the heat released from the system to the surroundings is compensated by the work, i.e.,

3 4  3 p2 L3 0 Qout  W23   f 23 dL  ln (14) L2 L22 ż32 4 ´ 3p2 L3 Qout the “ Wadiabatic f 23process dL “ ln ă0 (14) 23 “ On the other hand, during L2 quantum state is fixed (i.e., no L322 1 , the 2  0 . Therefore, the work performed during one full transitions between the states); that is, d 'Q

Qin  hand, Qout and accordingly the efficiency be obtained cycleOn is W tot  the other during the adiabatic processof3the Ñ cycle 1 , thecan quantum stateasis fixed (i.e., no 1 transitions between the states); that is, d Q “ 0. Therefore, performed during one full cycle is 4  3 p2 theLwork 2 ln can Wtot “ Qin ` Qout and accordingly the efficiency of the2cycle be obtained as W L3 L2 (15)   tot  1  Qin TH4(´S3p  S ) L 2 2 ln1 L23 Wtot L22 η “ “ 1 ´ (15) From Equation (3), one can also obtain Qin TH pS2 ´ S1 q 3 2 From Equation (3), one can also Tobtain (i  1, 2, 3) i Li  p (16) 2k ln i 3  p 1 i Ti L2i “ pi “ 1, 2, 3q (16) p 2kln 1´ip 2 2 i During the isoenergetic process 2  3 , one has to yield E  (4  3 p2 ) / 2 L2  (4  3 p3 ) / 2 L3

Entropy 2016, 18, 205

5 of 10

During the isoenergetic process 2 Ñ 3 , one has E “ p4 ´ 3p2 q{2L22 “ p4 ´ 3p3 q{2L23 to yield L22 4 ´ 3p2 “ 2 4 ´ 3p3 L3

(17)

Substituting Equation (17) into Equation (15), and considering that the quantum state is fixed during the adiabatic process 3 Ñ 1 (i.e., p3 “ p1 ), one can have, p

η “ 1´

4´3p

kp4 ´ 3p2 qln 1´2p2 ln 4´3p2 1

3pS2 ´ S1 q

(18)

From Equation (18), one can see that the efficiency of such three-process quantum engine cycle depends on p1 and p2 . It means that the properties of quantum state are crucial for performance of the quantum engine of this kind. In the classical point of view, the efficiency of engine cycle is described in terms of the thermodynamic variables, such as pressure, temperature, volume, etc., whereas the concept of quantum states is also relevant in the quantum regime. In fact, the probabilities of ground states, pi , are functions of temperature Ti and volume Li , as indicated in Equation (3). By this relationship, we can also analyze the behavior of Carnot efficiency in a similar way. From Equation (16), one can have T“

3

(19)

4´2EL2 2kL2 ln 2EL 2 ´1

and, consequently, obtain the variation of temperature with respect of potential width during the isoenergetic process [29], ˆ

BT BL

˙ E

¨ ˛2 ˆ ˙ 3 ˝ 6EL2 1 4 ´ 2EL2 ‚ “ 3 ´ ln ą0 ´2EL2 kL p4 ´ 2EL2 qp2EL2 ´ 1q 2EL2 ´ 1 ln 42EL 2 ´1

(20)

During the isoenergetic compression process, from Equation (20) one can easily find that the temperature decreases with the compression of the potential width. On the other hand, during the adiabatic compression process 3 Ñ 1 , the probability distribution of each energy level is fixed. From Equation (3), one can obtain TL2 “ const, which means that the temperature increases with the decreasing of potential width. Therefore, the lowest temperature TC is at point 3 on the f–L plane and the highest temperature TH is at the isothermal process 1 Ñ 2 . Suppose that there is a quantum Carnot cycle composed by two quantum isothermal processes and two quantum adiabatic processes, working between TH and TC . The efficiency of it coincides with the classical Carnot cycle [15], say, ηC “ 1 ´

TC TH

(21)

By using Equations (16) and (17), the quantum Carnot efficiency can be rewritten as, p

ηC “ 1 ´

p4 ´ 3p2 qln 1´2p2 p

p4 ´ 3p1 qln 1´1p

(22)

1

Equations (18) and (22) are both the functions of p1 and p2 . Therefore, we can compare η with ηC by varying p1 and p2 . It is worth noting that from Equation (3) one can obtain, ˆ

Bp BL

˙ “ T

2 1´ p pp1 ´ pqln L p

(23)

6 of 10 6 of 10

Entropy 2016, 18, 205 Entropy 2016, 18, 205

p 2 1 p  p   2 p(1  p ) ln 1  p  L T  L p(1  p ) ln p p  L T L

Entropy 2016, 18, 205

(23)

6(23) of 10

when the the positive positive temperature temperature is is considered, considered, i.e., i.e., Equation (23) (23) shows shows that that pp // LL T  00 when Equation  T Equation (23) shows that pB p{BLq ă 0 when the positive temperature is considered, i.e., 1{2 ă T 1/ 22  pp  11.. It It means means that that the the probability probability of find find the the system in in the the ground ground state state of of the the two-state two-state 1/ pă 1. It means that the probability of findofthe systemsystem in the ground state of the two-state system Therefore, the the 3D 3D plot plot system decreases decreases during during the the isothermal isothermal expansion, expansion, which which indicates indicates pp1  pp2 .. Therefore, system 1 2 decreases during the isothermal expansion, which indicates p1 ą p2 . Therefore, the 3D plot of η and and C varying varying with with p1 and and pp2 can can be be shown shown in in Figure Figure 2. 2. of  and of ηC varying with p1 and p2 can pbe 2. C 1 shown 2in Figure

Figure 2. Comparison between  and  at the positive temperature region, 1/ 2  p  p  1 Figure 2. Comparison between  and CC at the positive temperature region, 1/ 2  p22  p11  1 Figure 2. Comparison between η and ηC at the positive temperature region, 1{2 ă p2 ă p1 ă 1 where p1 and p2 are ground state probabilities of the two-state system at points 1 and 2 in f–L where and p2 ground are ground probabilities oftwo-state the two-state system at points 2 in f–L where pp statestate probabilities of the system at points 1 and12 and in f–L plane, 1 1and p2 are plane, respectively: (a)  varies with p1 and p2 ; (b) C varies with with p1 and p2 ; and (c) p2 ; (b) p1the p2 ; and (c) plane, respectively: (a)  varies with with and respectively: (a) η varies with p1 and p2 ; p(b) ηC varies withwith p1 and p2 ;with and (c) combination of 1 and C varies theand combination of (a) and (b). (a) (b). the combination of (a) and (b).

From Figure Figure 2, 2, one one can can see see that that for every every possible possible pair pair of pp1 and is always always smaller smaller and pp2 ,,  is From 2, one can see that forfor every possible pair of pof than ηC , 1 p2 , η is 2always smaller 1 and as Itexpected. expected. It is is worth worth noting that in inwork our previous previous work [29], another another three-process than as expected. is worth noting that innoting our previous [29], another three-process quantum engine C ,, as than It that our work [29], three-process C

adiabatic quantum engine cycle cycleby was constructed by following following sequence: “isoenergetic process  isothermal cycle was engine constructed following sequence: “isoenergetic process Ñ adiabaticprocess process Ñ adiabatic quantum was constructed by sequence: “isoenergetic  process isothermal process”. There exist a non-monotonic relationship between efficiency and  process”.There exist a non-monotonic between efficiency andbetween ∆T ” pTefficiency H ´ TC q when process isothermal process”. There relationship exist a non-monotonic relationship and  T  ( T  T ) when T is larger than the characteristic value of temperature . However, T ( E ) T the characteristic TH,C pEq. of However, in theTcycle described by H ,C HT is  (larger THH  TCCthan ) when THH is largervalue than of thetemperature characteristic value temperature H , C ( E ) . However, Figure 1, the non-monotonic relationship disappears. In fact, the cycle in Figure 1 and the onecycle in [29] in the the cycle cycle described described by by Figure Figure 1, 1, the the non-monotonic non-monotonic relationship relationship disappears. disappears. In In fact, fact, the the in in cycle in are two 1separate parts of a[29] quantum Carnot cycle parts [15], as shown in Figure 3. According to as Equation (1), Figure and the one in are two separate of a quantum Carnot cycle [15], shown in Figure 1 and the one in [29] are two separate parts of a quantum Carnot cycle [15], as shown in the expectation valueto ofEquation the Hamiltonian depends on potential width L and ground state probability p. Figure 3. According According (1), the the expectation expectation value of of the the Hamiltonian depends on potential potential Figure 3. to Equation (1), value Hamiltonian depends on It is possible to find astate set ofprobability pp2 , L2 , p3 , Lp3 q. It that satisfies to find a set of ( p , L , p , L ) that satisfies width and ground ground is possible width LL and state probability p . It is possible to find a set of ( p22, L22, p33, L33) that satisfies

(4 ´333p (4 p4 p4(4 ´3p pp3)3)q 33pp22q2))“ (4 2 223  2 2L LL233 22L2L22 222L 2

(24) (24) (24)

3

ff

which means meansthat thatthe the expectation value ofHamiltonian the Hamiltonian Hamiltonian at 2point point equals topoint that3.of ofTherefore, point 3. 3. which expectation value of the at point equals that ofto which means that the expectation value of the at 22 to equals that point Therefore, points 2 and 3 can be connected by an isoenergetic quantum process on the f–L plane. points 2 and 3 can2be connected an isoenergetic quantum process on process the f–L plane. Therefore, points and 3 can be by connected by an isoenergetic quantum on the f–L plane.

'

L L '

Figure 3. 3. A quantum Carnot cycle is composed composed of of two two isothermal isothermal processes ( 1  and 1' 33 )) Figure A quantum quantum Carnot Carnot cycle cycle is is Ñ 22 2 and 3) A composed of isothermal processes (11  and 111 Ñ ' 2 and two two quantumadiabatic adiabatic processes( 2(Ñ ). It aisquantum a quantum isoenergetic process ' and  and 11 11and 3Ñ 1 ).11It isoenergetic process that and two quantum quantum adiabaticprocesses processes ( 2  and ). is It is a quantum isoenergetic process 33  ' 1 1'  3 ” ” cycle is identical to Figure 1 and “Ñ3 2Ñ2 1 that connects points 2 and 3.Ñ “ 12 2  3  1 connects points 2 and 3. “ 1 Ñ 3 Ñ 1 ” cycle is identical to Figure 1 and “ 3 Ñ 3 ” that connects points 2 and 3. “ 1  2  3  1 ” cycle is identical to Figure 1 and “ 3  2  1 ”3is is another kind of three-process cycle discussed in [29]. another kind of of three-process cycle discussed in in [29]. is another kind three-process cycle discussed [29].

Entropy 2016, 18, 205

7 of 10

The efficiency of cycle “ 3 Ñ 2 Ñ 11 Ñ 3 ” in Figure 3 is given in [29], η1 “ 1 ´

3pS2 ´ S3 q p

(25)

4´3p

kp4 ´ 3p3 qln 1´3p3 ln 4´3p32

Since from point 3 to point 1 is a quantum adiabatic compression process, the quantum state of the two-state system does not change. Therefore, one can have p3 “ p1 as well as S3 “ S1 and then Equation (25) can be rewritten as, η1 “ 1 ´

3pS2 ´ S1 q p

(26)

4´3p

kp4 ´ 3p1 qln 1´1p ln 4´3p2 1 1

From Equations (18), (22) and (26), one can verify the following relationship, ηC “ η ` p1 ´ ηqη1

(27)

Equation (27) shows clearly that Carnot efficiency can be precisely reproduced by ideal coupling of the two three-process cycles indicated in Figure 3. We stress that, in the classical Carnot cycle, it is not possible to connect point 2 and 3 by a thermodynamic process because of the absence of the isoenergetic process. It shows again that the three-process quantum cycle discussed above has no counterpart in classical thermodynamics. Inspired by the finite-time thermodynamics [17], we can discuss the power output of the above mentioned three-process quantum engine cycle. As indicated in Figure 1, the potential wall moves from point 1 to point 2 and then moves back after one full cycle and the total movement of it can be expressed as 2pL2 ´ L1 q. Assuming that this velocity is small in order to avoid transition to higher excited states, but still with finite average speed v. The total cycle time can be expressed as τ “ 2 pL2 ´ L1 q {v. Therefore, the power output is given by, " „ P“

Qin ` Qout “ τ

3 2

*

 1´ p 2 L22

´

1´ p 1 L21

p

´ kTH ln p2 ` 1

4´3p2 ln LL32 L22

v

2pL2 ´ L1 q

(28)

Substituting Equations (16), (17) and (24) into Equation (28) yields, ” ı p p 4´3p p p 3 p1 ´ p2 q ln 1´2p2 ´ p1 ´ p1 q ln 1´1p ´ 3ln p2 ` p4 ´ 3p2 q ln 1´2p2 ln 4´3p12 1 1 P“ ¸ ˜ˆ p ˙1{2 1 ln 1´p L31 p1 1 ´1 4 v ln 1´ p p2 ln 1´p

1

(29)

2

Equation (29) indicates that the power output is a function of p1 and p2 if the initial potential width L1 and average speed v are given. For the sake of convenience, we discuss the behavior of dimensionless power output, P˚ “ PL31 {v, below. With the positive temperature condition, 1{2 ă p2 ă p1 ă 1, the variation of P˚ with p1 and p2 can be shown in Figure 4. ˚ From Figure 4, one can find that there exist a global maximum value for P˚ . More precisely, Pmax can be obtained by solving the following coupled equations, $ B P˚ ’ & B p1 “ 0 (30) ’ % B P˚ “ 0 Bp 2

* Figure 4. Dimensionless power output P with respect of p1 and p2 .

From Figure 4, one can find that there exist a global maximum value for P* . More precisely, * 2016, 18, 205by solving the following coupled equations, 8 of 10 canEntropy be obtained Pmax Entropy 2016, 18, 205 8 of 10  P* 0   p1  *  P  0  p  2

(30)

* The numerical result shows that Pmax  0.052 when p1  0.86 and p2  0.62 . Thus, the power

output can be optimized by adjusting the probabilities of ground states at point 1 and 2 on the f–L plane. From Figure 4, it can also be seen that for any given value of p1 , the curve of P* vs. p2 is always concave to give the global maximum. From Equation (16) we can see that a given p1 *˚ with respect of p and p . Figure 4. Dimensionless power outputPPwidth 2 point TH if the potential at theofinitial indicates a given temperature 1 p and p2 . is set. During the Figure 4. Dimensionless power output with respect 1 to a heat bath with temperature TH , i.e., expansion process 1  2 , the system is coupled ˚

The numerical result shows that Pmax “ 0.052 when p1 “ 0.86 and p2 “ 0.62. Thus, the power precisely, From Figure 4, one can find that there exist a global maximum value for P* . More of ground states at point 1 and 2 on the f–L * output can be optimized by adjusting the probabilities 3 3 coupled Pmax can be obtained by solving the ˚ T following   equations,

plane. From Figure 4, it can also beH seen that for any given value of p1 , the curve of P vs. p2 is always p1 p2 (31) kL 2kL1 ln 2 ln concave to give the global maximum. From Equation we can see that a given p1 indicates a given * (16) 2   P  p2 the expansion process 1 Ñ 2 , 1  p1point  0 is set. 1During temperature TH if the potential width at the initial  p1  the system is coupled to a heat bath with temperature TH , i.e., (30) 

Equation (31) shows that L2 will tend to infinity if p2 is close to 1/ 2 , which indicates that a P* 

0 3output. On the other hand, if p is very 3 full cycle time will be very large and zero “ TH yields “ (31)2  pp1 2 power p 2kL1 ln 1´ p 2kL2 ln 1´2p2 1 close to p1 , the area of cycle 1  2  3* 1 on the f–L plane tends to zero. Vanishing work also

The numerical result shows that Pmax  0.052 when p1  0.86 and p2  0.62 . Thus, the power

shows that L2 will tendthe to infinity if p2output is close to 1{2, be which indicates that means zeroEquation power(31) output. Therefore, power can optimized ina full the region output can be optimized by adjusting the probabilities of ground states at point 1 and 2 on the f–L cycle time will be very large and yields zero power output. On the other hand, if p is very close 2 1/ 2 plane.  p2  From p1 . Figure 4, it can also be seen that for any given value of p , the curve of P* vs. top is 1 p1 , the area of cycle 1 Ñ 2 Ñ 3 Ñ 1 on the f–L plane tends to zero. Vanishing work also means zero

2

Furthermore, (18) andmaximum. (29) can show thatEquation theinefficiency and output always concave to give thethe global we can a givenarep1 both power output.Equations Therefore, power output beFrom optimized the (16) region 1{2 ă ppower ă pthat 2see 1. pEquations functions of Furthermore, and temperature we can generate the curves power output with respectthe to the (18)T and show that thewidth efficiency power output areisboth functions 2 . Therefore, if(29) the potential at and theof initial point set. During indicates ap1given H

of pby p2 . Therefore, we canp generate thethe curves of power output respect 5 to the efficiency by * 1 and p12 ,and condition p2with  pwith . ˚ Figure shows efficiency varying TH , i.e., the P vs.  the system is coupled to a heat bath expansion process 1 2 under 1 temperature varying p1 and p2 under the condition p2 ă p1 . Figure 5 shows the P vs. η relationship for some

relationship values of p1 . valuesfor of some p1 .

TH 

3 2kL1 ln

p1 1  p1



3 2kL2 ln

p2 1  p2

(31)

Equation (31) shows that L2 will tend to infinity if p2 is close to 1/ 2 , which indicates that a full cycle time will be very large and yields zero power output. On the other hand, if p2 is very close to p1 , the area of cycle 1  2  3  1 on the f–L plane tends to zero. Vanishing work also means zero power output. Therefore, the power output can be optimized in the region 1/ 2  p2  p1 . Furthermore, Equations (18) and (29) show that the efficiency and power output are both functions of p1 and p2 . Therefore, we can generate the curves of power output with respect to the * p1 and p2power under the psome . Figure 5 shows efficiency by5.varying PP* ˚condition Figure Dimensionless power output vs. some given Figure 5. Dimensionless output vs. efficiency efficiencypη2 for given values ofvalues p1 . theof Pp1 .vs.  1 for relationship for some values of p1 .

From Figure 5, one can find that all the P˚ vs. η curves are concave. Thus, there exists an efficiency ˚ 1 q that corresponding to the maximum power output Pmax pp1 q for each value of p1 . The physical meaning of each η˚ pp1 q is nontrivial. When 0 ă η ă η˚ , the power output increases with the increasing of efficiency. It means that the cycle is not working in optimal regions. Both efficiency and power output can be optimized towards positive direction. When η˚ ă η ă 1, the power output is decreasing with the increasing of η. It means that in order to improve the engine’s efficiency, the cost is to decrease η˚ pp

Entropy 2016, 18, 205

9 of 10

the engine’s power output, and vice versa. Therefore, this kind of trade-off between the efficiency and power output should be concerned when the engine is working at this region, and η˚ is the lower bound of the region. 5. Conclusions With the analysis of a two-state quantum particle trapped in an infinite square well, a three-process quantum cycle was proposed by coupling the system to a heat bath and an energy bath, respectively. Based on the difference between isothermal process and isoenergetic process in quantum thermodynamics, the heat transferred into quantum cycle and total work performed during one cycle were obtained to yield the efficiency η. Comparison between η and Carnot efficiency ηC showed that the quantum Carnot cycle can be constructed by the combination of two symmetrical three-process quantum cycles, in spite of the fact that the isoenergetic quantum process has no counterpart in classical thermodynamics. Furthermore, by considering the average speed of square potential wall, the power output of this kind of three-process cycle was shown. It was found that the probability distributions at the starting and ending points of the isothermal expansion process are crucial to optimize the cycle performances. It was also shown that there exists a region of preferable performance, where the efficiency is still high and the power output is not low. These features of the present engine may suggest experiments of a new kind. Acknowledgments: Project supported by the Natural Science Foundations of Fujian Province (Grant No. 2015J01016), the Program for prominent young Talents in Fujian Province University (Grant No. JA12001), Program for New Century Excellent Talents in Fujian Province University (Grant No. 2014FJ-NCET-ZR04), Scientific Research Foundation for the Returned Overseas Chinese Scholars (Grant No. 2010-1561), and Promotion Program for Young and Middle-Aged Teacher in Science and Technology Research of Huaqiao University (Grant No. ZQN-PY114). Author Contributions: Congjie Ou conceived the idea and formulated the theory. Shengnan Liu and Congjie Ou designed the model and carried out the research. Shengnan Liu and Congjie Ou wrote the paper. Both authors have read and approved the final manuscript. Conflicts of Interest: The authors declare no conflict of interest.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Geusic, J.E.; Schulz-DuBois, E.O.; Scovil, H.E.D. Quantum equivalent of the Carnot cycle. Phys. Rev. 1967, 156, 343–351. [CrossRef] Scovil, H.E.D.; Schulz-DuBois, E.O. Three-Level masers as heat engines. Phys. Rev. Lett. 1959, 2, 262–263. [CrossRef] Fialko, O.; Hallwood, D.W. Isolated quantum heat engine. Phys. Rev. Lett. 2012, 108, 085303. [CrossRef] [PubMed] Plastina, F.; Alecce, A.; Apollaro, T.J. Irreversible work and inner friction in quantum thermodynamic processes. Phys. Rev. Lett. 2014, 113, 260601. [CrossRef] [PubMed] Anders, J.; Giovannetti, V. Thermodynamics of discrete quantum processes. New J. Phys. 2013, 15, 033022. [CrossRef] Scully, M.O.; Zubairy, M.S.; Agarwal, G.S.; Walther, H. Extracting work from a single heat bath via vanishing quantum coherence. Science 2003, 299, 862–864. [CrossRef] [PubMed] Scully, M.O. Extacting work from a single thermal bath via quantum negentropy. Phys. Rev. Lett. 2001, 87, 220601. [CrossRef] [PubMed] Scully, M.O. Quantum afterburner: Improving the efficiency of an ideal heat engine. Phys. Rev. Lett. 2002, 88, 050602. [CrossRef] [PubMed] Harbola, U.; Rahav, S.; Mukamel, S. Quantum heat engines: A thermodynamic analysis of power and efficiency. Europhys. Lett. 2012, 99, 50005. [CrossRef] Huang, X.L.; Wang, L.C.; Yi, X.X. Quantum Brayton cycle with coupled systems as working substance. Phys. Rev. E 2013, 87, 012144. [CrossRef] [PubMed]

Entropy 2016, 18, 205

11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.

10 of 10

Gelbwaser-Klimovsky, D.; Alicki, R.; Kurizki, G. Minimal universal quantum heat machine. Phys. Rev. E 2013, 87, 012140. [CrossRef] [PubMed] Bender, C.M.; Brody, D.C.; Meister, B.K. Quantum mechanical Carnot engine. J. Phys. A 2000, 33, 4427–4436. [CrossRef] Quan, H.T. Quantum thermodynamic cycles and quantum heat engines II. Phys. Rev. E 2009, 79, 041129. [CrossRef] [PubMed] Quan, H.T.; Liu, Y.X.; Sun, C.P.; Nori, F. Quantum thermodynamic cycles and quantum heat engines I. Phys. Rev. E 2007, 76, 031105. [CrossRef] [PubMed] Abe, S.; Okuyama, S. Similarity between quantum mechanics and thermodynamics: Entropy, temperature, and Carnot cycle. Phys. Rev. E 2011, 83, 021121. [CrossRef] [PubMed] Beretta, G.P. Quantum thermodynamic Carnot and Otto-like cycles for a two-level system. Europhys. Lett. 2012, 99, 20005. [CrossRef] Abe, S. Maximum-power quantum-mechanical Carnot engine. Phys. Rev. E 2011, 83, 041117. [CrossRef] [PubMed] Abe, S. General formula for the efficiency of Quantum-Mechanical analog of the Carnot engine. Entropy 2013, 15, 1408–1415. [CrossRef] Wang, J.H.; He, J.Z. Optimization on a three-level heat engine working with two noninteracting fermions in a one-dimensional box trap. J. Appl. Phys. 2012, 111, 043505. [CrossRef] Wang, J.H.; He, J.Z.; He, X. Performance analysis of a two-state quantum heat engine working with a single-mode radiation field in a cavity. Phys. Rev. E 2011, 84, 041127. [CrossRef] [PubMed] Wang, R.; Wang, J.H.; He, J.Z.; Ma, Y.L. Efficiency at maximum power of a heat engine working with a two-level atomic system. Phys. Rev. E 2013, 87, 042119. [CrossRef] [PubMed] Bergenfeldt, C.; Samuelsson, P.; Sothmann, B. Hybrid microwave-cavity heat engine. Phys. Rev. Lett. 2014, 112, 076803. [CrossRef] [PubMed] Zhuang, Z.; Liang, S.D. Quantum Szilard engines with arbitrary spin. Phys. Rev. E 2014, 90, 052117. [CrossRef] [PubMed] Ou, C.; Abe, S. Exotic properties and optimal control of quantum heat engine. Europhys. Lett. 2016, 113, 40009. [CrossRef] Blickle, V.; Bechinger, C. Realization of a micrometre-sized stochastic heat engine. Nat. Phys. 2012, 8, 143–146. [CrossRef] Martínez, I.A.; Roldán, É.; Dinis, L.; Petrov, D.; Parrondo, J.M.; Rica, R.A. Brownian carnot engine. Nat. Phys. 2016, 12, 67–70. [CrossRef] Roßnagel, J.; Dawkins, S.T.; Tolazzi, K.N.; Abah, O.; Lutz, E.; Schmidt-Kaler, F.; Singer, K. A single-atom heat engine. Science 2016, 352, 325–329. [CrossRef] [PubMed] Pathria, R.K. Statistical Mechanics; Pergamon Press: Oxford, UK, 1972. Ou, C.J.; Huang, Z.F.; Lin, B.H.; Chen, J.C. A three-process quantum engine cycle consisting of a two-level system. Sci. China Phys. Mech. Astron. 2014, 57, 1266–1273. [CrossRef] Born, M.; Fock, V. Beweis des adiabatensatzes. Z. Phys. 1928, 51, 165–180. (In German) [CrossRef] Deffner, S.; Lutz, E. Nonequilibrium work distribution of a quantum harmonic oscillator. Phys. Rev. E 2008, 77, 021128. [CrossRef] [PubMed] Gardas, B.; Deffner, S. Thermodynamic universality of quantum Carnot engines. Phys. Rev. E 2015, 92, 042126. [CrossRef] [PubMed] © 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).