Physica A 391 (2012) 6594–6600

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Thermal entangled quantum heat engineXian He, Jizhou He ∗, Jie ZhengDepartment of Physics, Nanchang University, Nanchang, 330031, China

a r t i c l e i n f o

Article history:Received 28 February 2012Received in revised form 13 July 2012Available online 22 July 2012

Keywords:Nonequilibrium and irreversiblethermodynamics

Quantum entangled heat engineThermal entanglement

a b s t r a c t

Based on a two-qubit Heisenberg XY model, we construct a four-level entangled quantumheat engine (QHE). It is an interesting quantum Otto cycle where the exchange constantis fixed and only the magnetic field is varied during the adiabatic steps. The expressionsfor several thermodynamic quantities such as the heat transferred, the work and theefficiency are derived. Moreover, the influence of the entanglement on the thermodynamicquantities is investigated numerically. Several interesting features of the variation ofthe heat transferred, the work and the efficiency with the concurrences of the thermalentanglement of different thermal equilibrium states are obtained. Finally, we discussedthe maximum efficiency of the QHE.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

A known feature of technological progress is the increase of human ability to control and design the microscopic world.One aspect is in manipulating simple quantum systems, for example, in the context of quantum computing or quantumchemistry. Another aspect is the field of quantum thermodynamics, whose main objective of quantum thermodynamics isin designing and studying new thermodynamic processes in the domain where quantum features of matter are relevant.The current activity in quantum thermodynamics includes quantum heat engines or refrigerators [1–21], work-extractionprocess from quantum systems [22–24], conditions of positive work [25], and so on. Quantum heat engines produce workusing a quantum system as the working substance, such as a spin system [2,8–13], harmonic oscillator system [3–7], two-level or multilevel system [15,17,19], cavity quantum electrodynamics system [16,18], coupled two-level system [20], et al..Because of the quantum features of the working substance, many unusual and exotic properties have been found. Forexample, QHE canproduce a finite power at an efficiency close to the Carnot bound [22] and also surpass the Carnot efficiencybound in some condition [18].

It is well known that many interesting phenomena in quantum systems are attributable to the existence of theentanglement. Quantum entanglement is studied intensely due to its potential applications in quantum communicationand information processing [26–30]. Recently, the quantum heat engines working with quantum entangled systems havebeen investigated. For example, Zhang et al. investigated a four-level entangled quantum heat engine based on a two-qubit isotropic Heisenberg XXX model in a constant external magnetic field [31]. Wang et al. studied the performancecharacteristics of an entangled quantum Otto engine using two two-level identical atoms coupled to single-mode opticalcavity as the working substance [32]. Zhang analyzed entangled quantum heat engines based on two two-spin systemswithDzyaloshinski–Moriya anisotropic antisymmetric interaction [33]. In the above studies, during the adiabatic processes theexchange constant J was altered between two chosen values (J1 → J2 → J1), while keeping the external magnetic field B ata fixed value. From practical and experimental points of view, it is interesting to investigate the performance of the quantumOtto heat engines where the exchange constant is fixed and only the magnetic field is varied during the adiabatic processes.

∗ Corresponding author.E-mail address: hjzhou@ncu.edu.cn (J. He).

0378-4371/$ – see front matter© 2012 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2012.07.050

X. He et al. / Physica A 391 (2012) 6594–6600 6595

Based on this view, Thomas and Johal studied the one-dimensional isotropic Heisenberg model of two spin-1/2 systemsas a coupled quantum Otto cycle [34]. Following Thomas and Johal’s work, we further construct the QHE with a two-qubitisotropic Heisenberg XY model in a constant exchange constant. We will show some interesting connections between theentanglement and basic thermodynamic quantities graphically.

2. Review of a two-qubit isotropic Heisenberg XY model

The Heisenberg Hamiltonian for a chain of N spin-1/2 systems having nearest-neighbor interactions is [30]

H =

Nn=1

(JxSxnSxn+1 + JySynS

yn+1 + JzSznS

zn+1), (1)

where Sx = σ x/2, Sy = σ y/2 and Sz = σ z/2 are spin-1/2 operators (h = 1), σ x, σ y and σ z are Pauli operators. Weconsider the two-qubit XY model (Jz = 0) in a magnetic field,

H =B2(σ z

1 + σ z2 ) + J(σ+

1 σ _2 + σ+

2 σ−

1 ), (2)

where J = Jx = Jy is the exchange constant and B is the magnetic field along the z axis. While J > 0 and J < 0 correspondto the antiferromagnetic and the ferromagnetic cases, respectively. In this paper, we consider antiferromagnetic case only.σ±

=12 (σ

x± iσ y) are the raising and lowering operators. The four energy eigenvalues of the Hamiltonian H can be written

asE1 = −E4 = −J, E2 = −E3 = −B. (3)

In the standard basis, {|00⟩, |01⟩, |10⟩, |11⟩}, for a system in equilibrium at temperature T , the density operator is writtenas (k = 1)

ρ(T ) =1

2cosh

JT

+ cosh

BT

×

e−BT 0 0 0

0 cosh

JT

− sinh

JT

0

0 − sinh

JT

cosh

JT

0

0 0 0 eBT

. (4)

The occupation probabilities of the system in the thermal state at temperature T are given by

pi =e−Ei/Tie−Ei/T

, i = 1, 2, 3, 4. (5)

Then the concurrence of the thermal entanglement c is given by

c = max

sinh J

T − 1

cosh JT + cosh B

T

, 0

. (6)

From Eq. (6) we find that the concurrence or entanglement apparently vanishes in case J ≤ 0; therefore we focus on thecase J > 0 in the paper; and we find that the value of J should satisfied the inequality,

sinhJT

> 1. (7)

3. Model of a thermal entangled QHE

The Hamiltonian of an arbitrary quantum system with a finite number of energy levels may be expressed as

H =

i

Ei|i⟩ ⟨i|, (8)

where |i⟩ is the ith eigenstate of the system and Ei is its corresponding eigenenergy. The expectation value of theHamiltonianis

U = ⟨H⟩ =

pi

Ei, (9)

where pi is the occupation probabilities in the ith eigenstate. From Eq. (8) we have

dU =

i

(Eidpi + pidEi) . (10)

6596 X. He et al. / Physica A 391 (2012) 6594–6600

Fig. 1. Schematic diagram of the entangled quantum heat engine (QHE).

Comparing Eq. (10) with the first law of thermodynamics

dU = dQ + dW , (11)we can make the following identifications for the infinitesimal heat transferred dQ and work done dW ,

dQ :=

i

Eidpi, (12a)

dW :=

i

pidEi. (12b)

Thus, Eq. (10) is just an expression of the first law of thermodynamics in the quantum system. Eq. (12) implies that workdone on or by a system can be performed only through a change in the generalized coordinates of the system, which in turngives rise to a change in the distribution of the energy levels [22].

The cycle of the entangled quantum heat engine considered here is composed of the two isothermal and two adiabaticprocesses, inwhich theworking substance is a two-qubit Heisenberg XYmodel. The schematic diagramof the cycle is shownin Fig. 1. The four stages involved in our quantum Otto cycle are described below.

On stage 1, the working substance is coupled to the hot bath at temperature T1. In the beginning the probability of eacheigenstate is pi0 (i = 1, 2, 3, 4). After some contact time, the probability of each eigenstate changes from pi0 to pi1, while eacheigenenergy is kept fixed at the value Ei1 and the external magnetic field is maintained as B1. Thus, only heat is transferredand no work is performed in this stage.

On stage 2, the working substance is decoupled from the hot bath and undergoes a quantum adiabatic expansion todecrease each eigenenergy from Ei1 to the smaller value Ei2, while the probability of each eigenstate is maintained as pi1throughout according to the quantum adiabatic theorem [35]. The magnetic field changes from B1 to B2. Thus, an amount ofwork is performed and no heat is transferred in this stage.

On stage 3, the working substance is coupled to the cold bath at temperature T2. After some contact time, the probabilityof each eigenstate changes from pi1 to pi2, while each eigenenergy is kept fixed at the value Ei2 and the external magneticfield is maintained as B1. Thus, only heat is transferred and no work is performed in this stage.

On the stage 4, the working substance is decoupled from the cold bath and undergoes a quantum adiabatic compressionto reduce each eigenenergy from Ei2 back to the larger value Ei1, while the probability of each eigenstate is maintained as pi2and the magnetic field change from B2 to B1. Thus, an amount of work is performed and no heat is transferred in this stage.Due to the cyclicity of the heat engine there is a constraint on the probability, i.e. pi2 = pi0.

According to Eq. (2), each eigenvalue in the hot bath and cold bath is given byE11 = −J, E21 = −B1, E31 = B1, E41 = J,E12 = −J, E22 = −B2, E32 = B2, E42 = J. (13)

When the working substance reaches the thermal equilibrium with the baths, the probabilities can be written as

P11 =eJ/T1

Z1, P21 =

eB1/T1

Z1, P31 =

e−B1/T1

Z1, P41 =

e−J/T1

Z1,

P12 =eJ/T2

Z2, P22 =

eB2/T2

Z2, P32 =

e−B2/T2

Z2, P42 =

e−J/T2

Z2(14)

X. He et al. / Physica A 391 (2012) 6594–6600 6597

where Zj = eJ/Tj + e−J/Tj + eBj/Tj + e−Bj/Tj and j = 1, 2. According to Eq. (12), the heat transferred in stages 1 and 3 is

Q1 =

i

Ei1(pi1 − pi2) = J(p41 − p42 + p12 − p11) + 2B1(p22 − p21 + p31 − p32), (15)

Q2 =

i

Ei2(pi2 − pi1) = −J(p41 − p42 + p12 − p11) − 2B2(p22 − p21 + p31 − p32), (16)

respectively. Q1 > 0 and Q2 < 0 correspond to absorb and release of heat from and to the heat baths, respectively. The network done by the QHE is

W = Q1 + Q2 = 2(B1 − B2)(p22 − p21 + p31 − p32). (17)

Note thatW > 0 corresponds to work performed by the QHE. The efficiency of the QHE can be defined as

η =WQ1

=2(B1 − B2)(p22 − p21 + p31 − p32)

J(p41 − p42 + p12 − p11) + 2B1(p22 − p21 + p31 − p32). (18)

The entanglement of the stage 1 and stage 3, denoted by c1 and c2, respectively, i.e.

c1 =

sinh JT1

− 1

cosh JT1

+ cosh B1T1

,

c2 =

sinh JT2

− 1

cosh JT2

+ cosh B2T2

. (19)

4. Basic thermodynamic quantities and the thermal entanglement

Now we start to explore the relation between entanglement and basic thermodynamics quantities and the efficiency ofthe QHE. From Eq. (19) we find

B1 = T1 cosh−1sinh(J/T1) − 1 − c1 cosh(J/T1)

c1

,

B2 = T2 cosh−1sinh(J/T2) − 1 − c2 cosh(J/T2)

c2

. (20)

From Eqs. (15) to (18), we find

Q1 = Je−J/T1 − eJ/T1

Z1+

eJ/T2 − e−J/T2

Z2

+ 2B1

e−B1/T1 − eB1/T1

Z1+

eB2/T2 − e−B2/T2

Z2

, (21)

Q2 = −Je−J/T1 − eJ/T1

Z1+

eJ/T2 − e−J/T2

Z2

− 2B2

e−B1/T1 − eB1/T1

Z1+

eB2/T2 − e−B2/T2

Z2

, (22)

W = 2(B1 − B2)

e−B1/T1 − eB1/T1

Z1+

eB2/T2 − e−B2/T2

Z2

, (23)

η =

2(B1 − B2)

e−B1/T1−eB1/T1Z1

+eB2/T2−e−B2/T2

Z2

J

e−J/T1−eJ/T1Z1

+eJ/T2−e−J/T2

Z2

+ 2B1

e−B1/T1−eB1/T1

Z1+

eB2/T2−e−B2/T2Z2

. (24)

By substituting Eq. (20) to Eqs. (21)–(24), the thermodynamic quantities Q1,Q2,W , and η become the function of c1, c2, T1,and T2 for given J .

We first consider the simplest unentanglement case, i.e. J = 0. Using Eq. (24), the efficiency of the QHE is η0 = 1 −B2B1.

In this case, the entanglement or c vanishes. But it is too complicated to give the analytical expressions for the basicthermodynamic quantities directly in entanglement case, i.e. J = 0. By numerical calculationwe can plot the isolinemaps ofthe variation of basic thermodynamic quantities with c1 and c2 at given three different exchange constant values, as shownin Figs. 2–4.We find some intriguing features of these figures. Firstly, the isolines of the amount of heat transferred are openlines. Q1 > −Q2 > 0 is always true. The amount of heat transferred changes monotonically with c1 or c2. Secondly, theisolines of the net work and the efficiency are open lines. This indicates thatW and η no longer changes monotonically withc1 (or c2) when c2 (or c1) is given. The loop lines become larger ones as the exchange constant J increases. Thus, the exchangeconstant J not only affects the shape of the net work and the efficiency but also affects its value. Thirdly, the acceptable rangeof c1 and c2 is c2 > c1 or c2 < c1 and varies with different exchange constant values in the isolines of the efficiency. In a small

6598 X. He et al. / Physica A 391 (2012) 6594–6600

Fig. 2. Lines of constant X (c1, c2) in the space of the concurrences c1, c2 for X = Q1 (Figure a), Q2 (Figure b),W (Figure c), and η (Figure d). The additionalparameters are T1 = 2, T2 = 1 and J = 4.

Fig. 3. Lines of constant X (c1, c2) in the space of the concurrences c1, c2 for X = Q1 (Figure a), Q2 (Figure b),W (Figure c), and η (Figure d). The additionalparameters are T1 = 2, T2 = 1 and J = 6.

exchange constant J it seems that there are the loop lines only in the smaller range of the concurrences c1 and c2, shown inFig. 2. In a larger exchange constant J , there are the loop lines either in the smaller or larger range of the concurrences c1and c2, as shown in Figs. 3 and 4. Lastly, we find that the Carnot efficiency ηc = 1 − T2/T1 = 0.5 is not achievable in thesethree figures. Therefore, the second law of thermodynamics holds all the while.

5. Maximum efficiency

In order to fully understand the performance of the QHE, we analyze the maximum efficiency. We choose the threerepresentative exchange constants J = 4, 6 and 8, respectively, and keep the temperature of the cold bath T2 = 1 unchanged.

X. He et al. / Physica A 391 (2012) 6594–6600 6599

Fig. 4. Lines of constant X (c1, c2) in the space of the concurrences c1, c2 for X = Q1 (Figure a), Q2 (Figure b),W (Figure c), and η (Figure d). The additionalparameters are T1 = 2, T2 = 1 and J = 8.

Fig. 5. Variation of the maximum efficiency and Carnot efficiency with the temperature of the hot bath for T2 = 1 and J = 4, 6, 8, respectively.

Based on the extremal conditions of ∂η

∂c1= 0 and ∂η

∂c2= 0 at given J, T1, and T2, the variation of themaximum efficiency with

the temperature of the hot bath T1 can be plotted, as shown in Fig. 5. It is found from Fig. 5 that the maximum efficiencyincreasesmonotonicallywith the temperature of the hot bath at given exchange constant J . On the other hand, themaximumefficiency decreases as the exchange constant increases at a given temperature of the hot bath. The maximum efficiency issmaller than the Carnot efficiency. When the exchange constant J is large enough, the maximum efficiency approaches theCarnot efficiency.

6. Conclusions

In summary, we have constructed QHE with the two-qubit isotropic Heisenberg XY model in a magnetic field in thispaper. Based on the thermal entanglement and the first law of thermodynamics in the quantum system, the expressionsfor the heat transferred, the work and the efficiency have been obtained. The major results are: (i) the heat transferredchanges monotonically with the concurrences c2 or c1, but the work and the efficiency do not change monotonically with

6600 X. He et al. / Physica A 391 (2012) 6594–6600

the concurrences c2 or c1; (ii) the exchange constant J not only affects the shape of the efficiency but also affects its value. Theacceptable range of the concurrences c1 and c2 varies with different exchange constant values; (iii) the maximum efficiencyincreases monotonically with the temperature of the hot bath at a given exchange constant. When the exchange constantis large enough, the maximum efficiency approaches but is not equal to the Carnot efficiency.

Acknowledgment

This work was supported by the National Natural Science Foundation (No. 11065008).

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