Maximum Power Output of Quantum Heat Engine. with Energy Bath

Transcription

1 Maximum Power Outut of Quantum Heat Engine with Energy Bath Shengnan Liu, Congjie Ou, College of Information Science and Engineering, Huaqiao University, Xiamen 360, China; Corresondence: el.: Abstract: he difference between quantum isoenergetic rocess and quantum isothermal rocess comes from the violation of the law of equiartition of energy in the quantum regime. o reveal an imortant hysical meaning of this fact, here we study a secial tye of quantum heat engine consisting of three rocesses: isoenergetic, isothermal and adiabatic rocesses. herefore, this engine works between the energy and heat baths. Combining two engines of this kind, it is ossible to realize the quantum Carnot engine. Furthermore, considering finite velocity of change of the otential shae, here an infinite square well with moving walls, the ower outut of the engine is discussed. It is found that the efficiency and ower outut are both closely deendent on the initial and final states of the quantum isothermal rocess. he erformance of the engine cycle is shown to be otimized by control of the occuation robability of the ground state, which is determined by the temerature and the otential width. he relation between the efficiency and ower outut is also discussed. Keywords: Quantum heat engine, two-state system, erformance otimization

2 . Introduction Quantum thermodynamics introduces the interdiscilinary field that combined classical thermodynamics and quantum mechanics since the concet of quantum heat engine aeared in the 960s [,]. Insired by the roerties of the classical thermodynamic rocesses and cycles, the quantum analogues of the rocesses and cycles have been develoed and discussed in more and more different quantum systems [3-4]. Recently, some micro sized heat engines with single Brownian article induced by otical laser tra [5,6] and single ion held within a modified linear Paul tra [7] have been exerimentally realized, which resents significant insight into the energy conversion on a microscoic level and would be exected to shed light on the exerimental investigation in quantum thermodynamic characteristics of small systems. herefore, it is of great interest to adot a single-article quantum system as the working substance to investigate the roerties of quantum thermodynamic rocesses and quantum engine cycles [5,,,5,6,0,,4]. A central concern of quantum thermodynamics is to understand the basic relationshis between classical thermodynamics and quantum mechanics [5,3,4,5]. he quantum analog of the classical engine cycles can be set u by emloying a single-article quantum system with two energy levels [,4,5,0] because of its simlicity. According to the first law of thermodynamics, the quantum analogue of mechanical work and heat transfer can be defined in a natural way [5,,4]. hus, the basic thermodynamic rocesses, such as adiabatic, isochoric, isobaric ones, can be well deicted in a quantum two-state system. Nevertheless, the quantum roerties of the two-state system determine the inherent difference of the thermodynamic rocesses. In classical thermostatistics, the law of equiartition of energy is crucial for the link between the energy and temerature [8]. However, it is violated in the quantum regime even for non-interacting articles confined in a box. In a two-state quantum system, the exectation value of the Hamiltonian deends not only on the temerature, but also on the quantum state of the system [3,4,5]. herefore, the quantum isothermal rocess (to fix the temerature) and the quantum isoenergetic rocess (to fix the exectation value of the Hamiltonian) are totally different from each other. During the quantum isoenergetic rocess, the mechanical exansion/comression and the quantum state engineering are controlled simultaneously by environmental system, which is considered as energy bath [,7,0,4]. It is worth noting that such kind of energy bath ensures the validation of

3 the second law of thermodynamics in quantum regime [,9,0]. herefore, by couling the quantum two-state system with a heat bath and an energy bath, it is ossible to construct an engine cycle, which is helful to understand the influence of quantum roerties on energy conversion for a small system.. wo-state Quantum System Couled to a Heat and an Energy Bath he model we consider here is a single article confined in an one-dimensional infinite square well otential with movable walls, which is a simlification of a iston. he corresonding stationary Schrödinger equation is given by H un ε n un = ( n =,,3,...), where n u and ε n reresent the n-th eigenstate and corresonding energy eigenvalue, resectively. Since we are interested in genuine quantum effects, here we assume that the temerature is low and the system size is small. In this aroximation, the ground ( n = ) and first excited ( n = ) states are dominantly relevant [4,0,]. herefore, the occuation robabilities of the ground state and excited state can be written as and. he exectation value of the Hamiltonian can be written as E = ε + ( ) ε. If the system is in thermal equilibrium with the heat bath at temerature, the robability of finding the system in a state with the energy ε is given by the Boltzmann factor ex( ε / k ) [3,3-5], where k is the Boltzmann constant. he energy eigenvalues of the ground and first excited states are given by = π ħ and ε / ml = π ħ ml, resectively, where m is the mass of ε / the article and L is the width of the square well otential. hus, the exectation value of the Hamiltonian is π ħ (4 3 ) E = ml. () For convenience, we set π ħ / m = below. he ratio between the robabilities of the ground state and the first excited state can be written as ex( / kl ) = ex( / kl ) () From Eq. () the robability that the system is in the ground state can be exressed as 3

4 (, L) = 3 kl + e (3) On the other hand, the otential width L can be considered as the volume of this kind of one-dimensional system. herefore, the force (i.e., ressure in dimension) on the otential wall is [3,9], dε dε 4 3 f = + ( ) = 3 dl dl L (4) Form Eq. (4) one can see that the force varies with the otential width L so it is ossible to adot a curve on the f-l lane to describe a thermal-like quantum rocess. It is in fact a one dimensional analogue of the ressure-volume lane of classical thermodynamics. If the two-state system is couled to a thermal bath at temerature, Eq. (3) can be substituted into Eq. (4) and yields f = + 4e 3 kl 3 3 kl L ( + e ) (5) According to Eq. (5), the sloe of an isothermal quantum rocess curve on f-l lane can be obtained as f 3(4 3 ) 6 ( )ln[ /( )] = 4 L L (6) If the two-state system is couled to an energy bath to fix the exectation of Hamiltonian. From Eqs. () and (4) one can obtain f (4 3 ) = 4 L E L (7) Obviously, the isothermal curve on the f-l lane is different from the isoenergetic one originating from the quantum roerties. 3. Quantum Engine Cycle Based on wo-state System As mentioned above, the difference between quantum isoenergetic rocess and quantum isothermal rocess can be illustrated by their curves on the f-l lane. According to the quantum 4

5 adiabatic theorem [5,30-3], which should not be confused with the thermodynamic adiabaticity, if the time scale of the change of the Hamiltonian or the otential width is much larger than the tyical dynamical one, ħ / E, then the stationary Schrödinger equation for the energy eigenstate holds instantaneously [7]. he sloe of the curve of the adiabatic rocess, during which the state remains unchanged (i.e., is fixed) can be directly derived from Eq. (4) as follows: f 3(4 3 ) = 4 L L (8) It is worth noting that for the ositive temerature, > 0, Eq. (3) indicate that / < <. In this case, the sloes of quantum isothermal rocess, isoenergetic rocess and adiabatic rocess can be comared at the same otential width L and yields [9], f f f > >, (/ < < ) L L L (9) E According to Eq. (9), if the ositive temerature area is concerned, we can construct the ossible three-rocess cycles on the f-l lane as it is shown in Figure. Figure. he diagram of the constructed quantum cycle on the f-l lane, where ie, ad and it reresent the isoenergetic, adiabatic and isothermal quantum rocesses, resectively. is an isothermal quantum rocess, the system is couled to a heat bath with temerature H. During the exansion of the otential width, one wall of the otential acts as a iston to erform work [7] and the energy is transferred from the heat bath to the system. 3 is an isoenergetic quantum rocess, which means that the two-state system exchanges energy with an energy bath to kee 5

6 its exectation value of the Hamiltonian constant. 3 is an adiabatic quantum rocess to connect the first two rocesses so that a closed cycle on the f-l lane can be realized. 4. Performance of he Quantum Engine Cycle During the isothermal rocess, the heat absorbed from the heat bath is, = = Qin H S H( S S ), (0) where S is the entroy of the two-state system and it is given by, i Si = k i ln ln( i ) ( i =, ) i, () where i is the occuation robability of the ground state when the system is at oint i of the f-l lane. Substitution of Eq. (3) into Eq. () yields 3( ) Si = k ln i ( i =, ) L () i i i Since oints and are connected by an isothermal rocess with temerature H in the f-l lane, one has = = H in Eq. (). herefore, Eq. (0) can be rewritten as, 3 Qin = H ( S S) = ln kh L L (3) During the isoenergetic comression rocess 3, the exectation of Hamiltonian is fixed. From the first law of thermodynamic [5,3], the heat released from the system to the surroundings is comensated by the work, i.e., L Q = W = f dl = < out ln 0 L L (4) On the other hand, during the adiabatic rocess 3, the quantum state is fixed (i.e., no transitions between the states), that is, ' d Q = 0. herefore, the work erformed during one full cycle is Wtot = Qin + Qout and accordingly the efficiency of the cycle can be obtained as 6

7 4 3 L ln Wtot L L3 η = = Q ( S S ) in H (5) From Eq. (3) one can also obtain L = ( i =,,3) i k ln 3 i i i (6) During the isoenergetic rocess 3, one has E = (4 3 ) / L = (4 3 ) / L to yield 3 3 L 4 3 L (7) = Substituting Eq. (7) into Eq. (5), and considering that the quantum state is fixed during the adiabatic rocess 3 (i.e., 3 = ), one can have, 4 3 k(4 3 ) ln ln 4 3 η = 3( S S ) (8) From Eq. (8) one can see that the efficiency of such three-rocess quantum engine cycle deends on and. It means that the roerties of quantum state are crucial for erformance of the quantum engine of this kind. In the classical oint of view, the efficiency of engine cycle is described in terms of the thermodynamic variables, such as ressure, temerature, volume, etc., whereas the concet of quantum states is also relevant in the quantum regime. In fact, the robabilities of ground states, i, are functions of temerature i and volume L i, as indicated in Eq. (3). By this relationshi, we can also analyze the behavior of Carnot efficiency in a similar way. From Eq. (6) one can have = 3 4 EL kl ln EL (9) and consequently obtain the variation of temerature with resect of otential width during the isoenergetic rocess [9], 7

8 3 6EL 4 EL = ln 0 3 > L E kl 4 EL (4 )( ) ln EL EL EL EL (0) During the isoenergetic comression rocess, from Eq. (0) one can easily find that the temerature decreases with the comression of the otential width. On the other hand, during the adiabatic comression rocess 3, the robability distribution of each energy level is fixed. From Eq. (3) one can obtain L = const, which means that the temerature increases with the decreasing of otential width. herefore, the lowest temerature C is at oint 3 on the f-l lane and the highest temerature H is at the isothermal rocess. Suose that there is a quantum Carnot cycle comosed by two quantum isothermal rocesses and two quantum adiabatic rocesses, working between H and C. he efficiency of it coincides with the classical Carnot cycle [5], say, η C = C () H By using Eqs. (6) and (7), the quantum Carnot efficiency can be rewritten as, η C = (4 3 )ln (4 3 )ln () Eqs. (8) and () are both the functions of and. herefore, we can comare η with by varying and. It is worth noting that from Eq. (3) one can obtain, η C = ( ) ln L L (3) Eq. (3) shows that ( / L) < 0 when the ositive temerature is considered, i.e., / < < means that the robability of find the system in the ground state of the two-state system decreases during the isothermal exansion, which indicates. It >. herefore, the 3D lot of η and η C varying with and can be shown in Figure. 8

9 Figure. Comarison between η and η C at the ositive temerature region, / < < < where and are ground state robabilities of the two-state system at oints and in f-l lane, resectively. (a)η varies with and ; (b) η C varies with and ; (c) the combination of (a) and (b). From Figure one can see that for every ossible air of and, η is always smaller than η C, as exected. It is worth noting that in our revious work [9], another 3-rocess quantum engine cycle was constructed by following sequence: isoenergetic rocess adiabatic rocess isothermal rocess. here exist a non-monotonic relationshi between efficiency and ( ) when H is larger than the characteristic value of temerature, ( E ). However, in H C the cycle described by Fig., the non-monotonic relationshi disaears. In fact, the cycle in Fig. and the one in Ref. [9] are two searate arts of a quantum Carnot cycle [5], as shown in Fig. 3. According to Eq. (), the exectation value of the Hamiltonian deends on otential width L and ground state robability. It is ossible to find a set of (, L, 3, L 3) that satisfy H C (4 3 ) (4 3 3 ) = L L (4) 3 which means that the exectation value of the Hamiltonian at oint equals to that of oint 3. herefore, oints and 3 can be connected by an isoenergetic quantum rocess on the f-l lane. 9

10 Figure 3. A quantum Carnot cycle is comosed of two isothermal rocesses ( and ' and two quantum adiabatic rocesses ( and 3 ' 3) ). It is a quantum isoenergetic rocess that ' connects oints and 3. 3 cycle is identical to Fig. and 3 3 is another kind of 3-rocess cycle discussed in Ref. [9]. ' he efficiency of cycle 3 3 in Fig. 3 is given in [9], ' 3( S S3) η = 4 3 k(4 3 3)ln ln (5) Since from oint 3 to oint is a quantum adiabatic comression rocess, the quantum state of the two-state system does not change. herefore, one can have 3 = as well ass 3 = S and then Eq. (5) can be rewritten as, ' 3( S S) η = 4 3 k(4 3 )ln ln 4 3 (6) From Eqs. (8), () and (6) one can verify the following relationshi, = + ( ) ' η η η η C (7) Eq. (7) shows clearly that Carnot efficiency can be recisely reroduced by ideal couling of the two 3-rocess cycles indicated in Fig. 3. We stress that, in the classical Carnot cycle, it is not ossible to connect oint and 3 by a thermodynamic rocess because of the absence of the isoenergetic rocess. 0

11 It shows again that the 3-rocess quantum cycle discussed above has no counterart in classical thermodynamics. Insired by the finite-time thermodynamics [7], we can discuss the ower outut of the above mentioned 3-rocess quantum engine cycle. As indicated in Fig., the otential wall moves from oint to oint and then moves back after one full cycle and the total movement of it can be exressed as ( ) L L. Assuming that this velocity is small in order to avoid transition to higher excited states, but still with finite average seed v. he total cycle time can be exressed as τ = ( ) herefore, the ower outut is given by, L L / v L 3 ln + ln kh v Qin + Q L L out L L P = = τ ( L L ) (8) Substituting Eqs. (6), (7) and (4) into Eq. (8) yields, P = ( ) ( ) ( ) / ln 3 L 4 ln v ln ln ln 3 ln ln ln 4 3. (9) Eq. (9) indicates that the ower outut is a function of and if the initial otential width L and average seed v are given. For the sake of convenience, we discuss the behavior of dimensionless ower outut, P = PL v, below. With the ositive temerature condition, 3 / / < < <, the variation of P with and can be shown in Fig. 4.

12 Figure 4. Dimensionless ower outut P with resect of and From Fig. 4 one can find that there exist a global maximum value for P. More recisely, P max can be obtained by solving the following couled equations, P P = 0 = 0 (30) he numerical result shows that P = when = 0.86 max 0.05 and = 0.6. hus, the ower outut can be otimized by adjusting the robabilities of ground states at oint and on the f-l lane. From Fig. 4 it can also be seen that for any given value of, the curve of P versus is always concave to give the global maximum. From Eq. (6) we can see that a given indicates a given temerature H if the otential width at the initial oint is set. During the exansion rocess, the system is couled to a heat bath with temerature H, i.e., H 3 3 = = kl ln kl ln (3) Eq. (3) shows that L will tend to infinite if is close to /, which indicates that a full cycle time will be very large and yields zero ower outut. On the other hand, if is very close to,

13 the area of cycle 3 on the f-l lane tends to zero. Vanishing work also means zero ower outut. herefore, the ower outut can be otimized in the region / < <. Furthermore, Eqs. (8) and (9) show that the efficiency and ower outut are both functions of and. herefore, we can generate the curves of ower outut with resect to the efficiency by varying and under the condition <. Fig. 5 shows the P vs. η relationshi for some values of. Figure 5. Dimensionless ower outut P versus efficiency η for some given values of From Fig. 5, one can find that all the P vs. η curves are concave. So there exists an efficiency ( ) η that corresonding to the maximum ower outut P ( ) for each value of max. he hysical meaning of each ( ) η is nontrivial. When 0 < η < η, the ower outut increases with the increasing of efficiency. It means that the cycle is not working in otimal regions. Both efficiency and ower outut can be otimized towards ositive direction. When η < η <, the ower outut is decreasing with the increasing of η. It means that in order to imrove the engine s efficiency, the cost is to decrease the engine s ower outut, and vice versa. herefore, this kind of trade-off between the efficiency and ower outut should be concerned when the engine is working at this region, and η is the lower bound of the region. 3

14 5. Conclusions With the analysis of a two-state quantum article traed in an infinite square well, a 3-rocess quantum cycle was roosed by couling the system to a heat bath and an energy bath, resectively. Based on the difference between isothermal rocess and isoenergetic rocess in quantum thermodynamics, the heat transferred into quantum cycle and total work erformed during one cycle were obtained to yield the efficiency η. Comarison between η and Carnot efficiency η showed C that the quantum Carnot cycle can be constructed by the combination of two symmetrical 3-rocess quantum cycles, in site of the fact that the isoenergetic quantum rocess has no counterart in classical thermodynamics. Furthermore, by considering the average seed of square otential wall, the ower outut of this kind of 3-rocess cycle was shown. It was found that the robability distributions at the starting and ending oints of the isothermal exansion rocess are crucial to otimize the cycle erformances. It was also shown that there exists a region of referable erformance, where the efficiency is still high and the ower outut is not low. hese features of the resent engine may suggest exeriments of a new kind. Acknowledgments: Project suorted by the Natural Science Foundations of Fujian Province (Grant No. 05J006), the Program for rominent young alents in Fujian Province University (Grant No. JA00), Program for New Century Excellent alents in Fujian Province University (Grant No. 04FJ-NCE-ZR04), Scientific Research Foundation for the Returned Overseas Chinese Scholars (Grant No ), and Promotion Program for Young and Middle-aged eacher in Science and echnology Research of Huaqiao University (Grant No. ZQN-PY4). Authors contributions: C.O. conceived the idea, formulated the theory. S.L. and C.O. designed the model, carried out the research. S.L. and C.O. wrote the aer. Cometing Interests: he authors declare that they have no cometing interests. References 4

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