Entangled quantum heat engine based on two-qubit Heisenberg XY model

Transcription

1 Chn. Phys. B Vol. 1, No Entangled quantum heat engne based on two-qubt Hesenberg XY model He J-Zhou 何济洲, He Xan 何弦, and Zheng Je 郑洁 Department of Physcs, Nanchang Unversty, Nanchang 3331, Chna Receved 14 Aprl 11; revsed manuscrpt receved 3 November 11 Based on a two-qubt sotropc Hesenberg XY model under a constant external magnetc feld, we construct a four-level entangled quantum heat engne QHE. The expressons for the heat transferred, the work, and the effcency are derved. Moreover, the nfluence of the entanglement on the thermodynamc quanttes s nvestgated analytcally and numercally. Several nterestng features of the varatons of the heat transferred, the work, and the effcency wth the concurrences of the thermal entanglement of two dfferent thermal equlbrum states n zero and nonzero magnetc felds are obtaned. Keywords: sotropc Hesenberg XY model, thermal entanglement, quantum heat engne, performance characterstcs PACS: 3.65.Ud, 5.7. a, 7..Pe DOI: 1.188/ /1/5/ Introducton The man objectve of quantum thermodynamcs s to desgn and study new thermodynamc processes n the doman where quantum features of matter are relevant. The current actve felds n quantum thermodynamcs nclude quantum heat engnes, [1 4 work-extracton processes from quantum systems, [5 7 condtons of postve work, [8 and so on. Quantum heat engnes QHEs produce work usng quantum systems, such as the spn system, [,9 17 harmonc oscllator system, [3 8 twolevel or multlevel system, [18,, cavty quantum electrodynamcs system [19,1 and coupled two-level system [3 as the workng substances. Because of the quantum features of the workng substance, many unusual and exotc propertes have been found. For example, the QHE can produce fnte power at an effcency close to [5 and even surpassng [1 the Carnot effcency bound at some condtons. It s well known that many nterestng phenomena n quantum systems are attrbutable to the exstence of the entanglement. Quantum entanglement s studed ntensely due to ts potental applcatons n quantum communcaton and nformaton processng. [9 34 However, most of the QHEs mentoned above only use general quantum systems as the workng substance. QHEs workng wth quantum entangled systems have been rarely nvestgated. Recently, Zhang et al. nvestgated a four-level entangled quantum heat engne based on a two-qubt sotropc Hesenberg XXX model n a constant external magnetc feld. [35 Wang et al. studed the performance characterstcs of an entangled quantum Otto engne usng two two-level dentcal atoms coupled to a sngle-mode optcal cavty as the workng substance. [36 Zhang analyzed entangled QHEs based on two two-spn systems wth the Dzyaloshnsk Morya ansotropc antsymmetrc nteracton. [37 Some novel results were obtaned. To enrch the theory of the entangled quantum heat engnes, we construct QHEs wth a two-qubt sotropc Hesenberg XY model n a constant external magnetc feld. We wll show some nterestng connectons between the entanglement and the basc thermodynamc quanttes.. Revew of two-qubt sotropc Hesenberg XY model The Hamltonan of a two-qubt sotropc Hesenberg XY model n a constant external magnetc feld B s gven by [3 H = B σz 1 + σ z + Jσ 1 + σ + σ+ σ 1, 1 where σ + = 1 σx + σ y and σ = 1 σx σ y are the rasng and the lowerng operators, σ x, σ y, and σ z are Project supported by the Natonal Natural Scence Foundaton of Chna Grant No Correspondng author. E-mal: hjzhou@ncu.edu.cn c 1 Chnese Physcal Socety and IOP Publshng Ltd

2 Chn. Phys. B Vol. 1, No the Paul operators, and J s the exchange constant, J > and J < correspond to the antferromagnetc and the ferromagnetc cases, respectvely. The egenvalues and the egenvectors of Hamltonan H are H = B, H 11 = B 11, H ψ ± = ±J ψ ±, where ψ ± = 1 1 ± 1 are the maxmally entangled states. A state s descrbed by densty operator ρt = exp H/kT /Z when the system s n the equlbrum state at temperature T, where Z = Tr[exp H/kT s the partton functon, and k s the Boltzmann constant. In the standard bass {, 1, 1, 11 }, densty matrx ρt s wrtten as 1 ρt = cosh J T + cosh B T e B T cosh J T snh J T. snh J T cosh J T 3 e B T For convenence, we assume k = 1. The concurrence of the thermal entanglement s gven by snh J T c = max 1 cosh J T + cosh B,. 4 T 3. Model of a thermal entangled QHE The Hamltonan of an arbtrary quantum system wth a fnte number of energy levels may be expressed as H = E, 5 where s the -th egenstate of the system, and E s the correspondng egenenergy. The expectaton value of the Hamltonan s U = H = p E, 6 where p s the occupaton probablty n the -th egenstate. From Eq. 6, we have du = E dp + p de. 7 Comparng Eq. 7 wth the frst law of thermodynamcs du = dq + dw, 8 we can make the followng dentfcatons for nfntesmal heat transferred dq and work done dw : dq := E dp, dw := p de. 9 Thus, equaton 7 s the frst law of thermodynamcs n the quantum system. Equaton 9 mples that the work done on or by a system can be obtaned only through a change n the generalzed coordnates of the system, whch n turn gves rse to a change n the dstrbuton of the energy levels. [5 The cycle of the entangled quantum heat engne consdered here s composed of two sothermal and two adabatc processes, and the workng substance s descrbed by a two-qubt sotropc Hesenberg XY model. The schematc dagram of the cycle s shown n Fg. 1. J 1 J 1 p1 stage stage 1 stage 3 stage 4 p = p J p hot bath T 1 cold bath T Fg. 1. Schematc dagram of the entangled quantum heat engne. At stage 1, the workng substance s coupled to bath 1 at temperature T 1. At the begnnng, the probablty of each egenstate s p = 1,, 3, 4. After some contact tme, the probablty of the egenstate changes from p to p 1, whle the egenenergy s fxed at E 1, and the exchange constant s mantaned at J 1. Thus, only heat s transferred and no work s performed at ths stage. At stage, the workng substance s decoupled from bath 1 and undergoes a quantum adabatc expanson to decrease the egenenergy from E 1 to a smaller value E, whle the probablty of each egenstate s mantaned at p 1 accordng to the quantum adabatc theorem. [38 The exchange constant changes from J 1 to J. Thus, some amount of work s performed and no heat s transferred at ths stage. At stage 3, the workng substance s coupled to bath at temperature T. After some contact tme, the probablty of each egenstate changes from p 1 to p, whle the egenenergy s fxed at E, and the exchange constant s mantaned at J. Thus, only heat s transferred and no work s performed at ths stage. J p

3 Chn. Phys. B Vol. 1, No At stage 4, the workng substance s decoupled from bath and undergoes a quantum adabatc compresson to ncrease the egenenergy from E back to the larger value E 1, whle the probablty of each egenstate s mantaned at p, and the exchange constant changes from J back to J 1. Thus, some amount of work s performed and no heat s transferred at ths stage. Due to the cyclcty of the heat engne, there s a constrant on the probablty,.e., p = p. Accordng to Eq., the egenvalues n baths 1 and are gven by and E 11 = J 1, E 1 = B, E 31 = B, E 41 = J 1, E 1 = J, E = B, E 3 = B, E 4 = J, 1a 1b respectvely. If the system contacts wth the two baths suffcently at stages 1 and 3, they wll reach the thermal equlbrum states. Then probabltes p 1 and p only depend on the temperatures of the baths and the egenenergy,.e., e Ej/kTj where = 1,, 3, 4 and j = 1,. p j =, 11 e Ej/kTj Accordng to Eq. 9, the amount of heat absorbed from bath 1 at stage 1 s Q 1 = E 1 p 1 p, 1 and the amount of heat released to bath at stage 3 s Q = E p p 1, 13 where Q 1 > and Q < must be satsfed. The work performed by the QHE s W = Q 1 + Q = E 1 E p 1 p, 14 where T 1 > T and W >. The effcency can be defned as η = W = 1 + Q. Q 1 Q Basc thermodynamc quanttes and thermal entanglement Accordng to Eq. 4, the concurrences of the thermal entanglement n the two thermal equlbrum states are = max snh J1 T 1 1 cosh J 1 T 1 + cosh B T 1,, 16a snh J T = max 1 cosh J T + cosh B,. 16b T The entangled state means > and >. From Eq. 16, we may obtan [ e B/T e B/T e J 1 = B + T 1 ln B/T 1 + 4c1 e 3B/T 1 + c 1 1 eb/t 1 + c 1 e 4B/T 1 1 [ e B/T + + e B/T + 8 e J = B + T ln B/T + 4c e 3B/T + c eb/t + c e 4B/T 1, 17a. 17b Substtutng Eq. 17 nto Eqs , we obtan the expressons for Q 1, Q, W, and η as functons of,, T 1, T, and B. We frst nvestgate the smplest case wthout magnetc feld. When B =, equaton 17 becomes [ 1 + c1 + + c 1 J 1 = T 1 ln, 18a 1 [ 1 + c + + J = T ln 1. 18b The amount of heat transferred, the work, and the effcency may be smplfed as e J 1/T 1 e J 1/T 1 e J /T e J /T Q 1 = J 1, 19a e J 1/T 1 + e J 1 /T e J /T + e J /T e J1/T1 e J1/T1 e J/T e J/T Q = J, 19b e J1/T1 + e J1/T e J/T + e J/T e J 1/T 1 e J 1/T 1 e J /T e J /T W = J 1 J, e J 1/T 1 + e J 1 /T e J /T + e J /T 533-3

4 Chn. Phys. B Vol. 1, No [ 1 + c + + c η = 1 J T ln 1 c = 1 J [ c1 + + c, 1 1 T 1 ln 1 where Q 1 and Q obvously have contrary sgns. In order to analyze how the entanglement affects the basc thermodynamc quanttes of the heat engne n detal, we plot an solne map to show the varaton of effcency wth and, as shown n Fg.. It s found from Fg. that all of the solnes are above the dagonal = lne. It means that only when < can the QHE generate postve work. The Carnot effcency, defned as η c = 1 T /T 1, cannot be acheved. The effcency ncreases monotoncally wth when s fxed, and decreases monotoncally wth when s fxed. When =, the effcency of the QHE s equal to the Carnot effcency,.e., η = η c. Next we study the complcated case of nonzero magnetc feld. In ths case, t s too complcated to gve analytcal expressons for the basc thermodynamc quanttes, so we show them graphcally. We choose three representatve B values to plot the varatons of basc thermodynamc quanttes wth and, as shown n Fgs It s found from Fgs. 3 5 that the solnes of the amount of heat transferred are open lnes. And Q 1 > Q > s always true. The amount of heat transferred decreases monotoncally wth when s fxed and ncreases monotoncally wth when s fxed. But the solnes of the work and the effcency are loops. Thus, they no longer change monotoncally wth or. The loop becomes smaller as the magnetc feld ncreases. The magnetc feld not only affects the shape of the effcency but also affects ts value. The acceptable ranges of and vary wth the magnetc feld. In a larger magnetc feld, the loops also appear when >, whereas n a smaller magnetc feld, t seems that only < s relevant = Fg.. colour onlne Varaton of effcency η wth and n an solne map of effcency for kt 1 =, kt = 1, and B = a = b = c d = = Fg. 3. colour onlne Varatons of a Q 1, b Q, c W, and d η wth and n solne maps of effcency for kt 1 =, kt = 1, and B =

5 Chn. Phys. B Vol. 1, No a b = = c = d = Fg. 4. colour onlne Varatons of a Q 1, b Q, c W, and d η wth and n solne maps of effcency for kt 1 =, kt = 1, and B = 3..8 a b = = c = d = Fg. 5. colour onlne Varatons of a Q 1, b Q, c W, and d η wth and n solne maps of effcency for kt 1 =, kt = 1, and B =

6 Chn. Phys. B Vol. 1, No Concluson We have constructed a QHE wth the two-qubt sotropc Hesenberg XY model n a constant external magnetc feld n ths paper. Based on the thermal entanglement and the frst law of thermodynamcs n the quantum system, the expressons for the heat transferred, the work, and the effcency have been obtaned. It s found that the amount of heat transferred decreases ncreases monotoncally wth when s fxed. But the solnes of the work and the effcency are loops. The Carnot effcency cannot be acheved. References [1 Kosloff R 1984 J. Chem. Phys [ Geva E and Kosloff R 199 J. Chem. Phys [3 Geva E and Kosloff R 199 J. Chem. Phys [4 Ln B H and Chen J C 3 Phys. Rev. E [5 Ln B H and Chen J C 5 Phys. Scr [6 Wang J H, He J Z and Mao Z Y 7 Scence n Chna Seres G: Phys. Mech. Astro [7 He J Z, He X and Tang W 9 Scence n Chna Seres G: Phys. Mech. Astro [8 Rezek Y and Kosloff R 6 New J. Phys. 8 1 [9 He J Z, Chen J C and Hua B Phys. Rev. E [1 Wu F, Chen L G, Sun F R, Wu C and L Q 6 Phys. Rev. E [11 Feldmann T and Kosloff R Phys. Rev. E [1 Feldmann T and Kosloff R 3 Phys. Rev. E [13 Feldmann T and Kosloff R 4 Phys. Rev. E [14 Wu F, Chen L G, Wu S, Sun F R and Wu C 6 J. Chem. Phys [15 Wang J H, He J Z and Xn Y 7 Phys. Scr [16 Wu F, Chen L G, Wu S and Sun F R 6 J. Phys. D: Appl. Phys [17 He J Z, Xn Y and He X 7 Appled Energy [18 Quan H T, Lu Y X, Sun C P and Nor F 7 Phys. Rev. E [19 Quan H T, Zhang P and Sun C P 6 Phys. Rev. E [ Bender C M, Brody D C and Mester B K J. Phys. A: Math. Gen [1 Scully M O, Zubary M S, Agarwal G S and Walther H 3 Scence [ Scully M O 1 Phys. Rev. Lett [3 Henrch M J, Mahler G and Mchel M 7 Phys. Rev. E [4 Arnaud J, Chusseau L and Phlppe F 8 Phys. Rev. E [5 Keu T D 4 Phys. Rev. Lett [6 Allahverdyan A E, Serral G R and Neuwenhuzen T M 5 Phys. Rev. E [7 Allahverdyan A E, Johal R S and Mahler G 8 Phys. Rev. E [8 Quan H T, Zhang P and Sun C P 5 Phys. Rev. E [9 X X Q, Chen W X, Hao S R and Yue R H Phys. Lett. A [3 Wang X G 1 Phys. Rev. A [31 Wootters W K 1998 Phys. Rev. Lett [3 Arnesen M C, Bose S and Vedral V 1 Phys. Rev. Lett [33 Guo Z, Yan L S, Pan W, Luo B and Xu M F 11 Acta Phys. Sn n Chnese [34 Lu D M 11 Acta Phys. Sn n Chnese [35 Zhang T, Lu W T, Chen P X and L C Z 7 Phys. Rev. A [36 Wang H, Lu S Q and He J Z 9 Phys. Rev. E [37 Zhang G F 8 Eur. Phys. J. D [38 Messah A 1999 Quantum Mechancs New York: Dover 533-6