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arXiv:0808.2409v1 [quant-ph] 18 Aug 2008

Entangled quantum heat engines based on two two-spin systems

with Dzyaloshinski-Moriya anisotropic antisymmetric interaction

Guo-Feng Zhang∗†

Department of physics, School of sciences,

Beijing University of Aeronautics and Astronautics,

Xueyuan Road No. 37, Beijing 100083, People’s Republic of China

Abstract

We construct an entangled quantum heat engine (EQHE) based on two two-spin systems with

Dzyaloshinski-Moriya (DM) anisotropic antisymmetric interaction. By applying the explanations

of heat transferred and work performed at the quantum level in Kieus work [PRL, 93, 140403

(2004)], the basic thermodynamic quantities, i.e., heat transferred, net work done in a cycle and

efficiency of EQHE are investigated in terms of DM interaction and concurrence. The validity of

the second law of thermodynamics is confirmed in the entangled system. It is found that there

is a same efficiency for both antiferromagnetic and ferromagnetic cases, and the efficiency can be

controlled in two manners: 1. only by spin-spin interaction J and DM interaction D; 2. only

by the temperature T and concurrence C. In order to obtain a positive net work, we need not

entangle all qubits in two two-spin systems and we only require the entanglement between qubits

in a two-spin system not be zero. As the ratio of entanglement between qubits in two two-spin

systems increases, the efficiency will approach infinitely the classical Carnot one. An interesting

phenomenon is an abrupt transition of the efficiency when the entanglements between qubits in

two two-spin systems are equal.

PACS numbers: 03.67.Hk; 03.65.Ud; 75.10.Jm

∗Corresponding author.

†Email: gf1978zhang@buaa.edu.cn

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I.INTRODUCTION

The dominating feature of an industrial society is its ability to utilize, whether for wise

or unwise ends, sources of energy other than the muscles of men or animals. Except for

waterpower, where mechanical energy is directly available, most energy supplies are in the

form of potential energy of molecular or nuclear aggregations. In chemical or nuclear reac-

tions, some of this potential energy is released and converted to random molecular kinetic

energy. Heat can be withdrawn and utilized for heating buildings, for cooking, or for main-

taining a furnace at a high temperature in order to carry out other chemical or physical

processes. But to operate a machine, one of the problems of the mechanical engineers is to

withdraw heat from a high-temperature source and convert as large as a fraction as pos-

sible to mechanical energy. We can solve the problems by using heat engines which can

extract energy from its environment in the form of heat and do useful work. We consider

for simplicity a heat engine in which the so-called “working substance” is carried through a

cycle process, that is, a sequence of processes in which it eventually returns to its original

state. All the heat engines absorb heat from a source at a high temperature, perform some

mechanical work, and reject heat at a lower temperature. Consider a heat engine operating

in a cycle over and over again and let Qhand Qlstand for the heats absorbed and rejected

by the working substance per cycle. The net heat absorbed is Q = Qh+ Ql. The useful

output of the engine is the net work W done by the working substance, and from the first

law: W = Q = Qh+ Ql. The heat absorbed is usually obtained from the combustion fuel.

The heat rejected ordinarily has no economic value. The thermal efficiency of a cycle is

defined as the ratio of the useful work to the heat absorbed: ηT= W/Qh= (Qh+ Ql)/Qh.

Because of friction losses, the useful work delivered by an engine is less than the work W,

and the overall efficiency is less than the thermal efficiency. A Carnot cycle, in which all

the heat input is supplied at a single high temperature and all the heat output is rejected

at a single lower temperature, has an efficiency less than or equal to the Carnot efficiency

ηc= (Qh+Ql)/Qh= 1−Tl/Th, where Thand Tlare the temperature of the high-temperature

energy source and the low-temperature energy sink, respectively. This is supported by the

second law of thermodynamics and numerous experimental evidences.

Quantum heat engines (QHEs), in contrast, operated by passing quantum matter through

a closed series of quantum adiabatic processes and energy exchanges with heat baths, re-

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spectively, at different stages of a cycle. Kieu[1] constructed a quantum heat engine (QHE)

which is a two-level quantum system and undergoes quantum adiabatic process and energy

exchanges with heat baths at different stages in a work cycle. Armed with this class of

QHE and the interpretations of heat transferred and work performed at the quantum level,

he clarified some important aspects of the second law of thermodynamics. Recently, the

physics of semiconductors with a spin-orbit interaction has attracted a lot of attention, as

it plays an important role for the emerging field of semiconductor spintronics [2]. Sun et

al. investigate the equilibrium property of a mesoscopic ring with a spin-orbit interaction,

persistent spin current is studied [3].

However, none of the QHEs mentioned above involve the most extraordinary phenomenon

in quantum mechanics, i.e., quantum entanglement [4, 5, 6, 7, 8]. To enrich research in

QHEs, an entangled QHE and the investigation about the influence of entanglement on

its thermodynamic characteristics should be considered.The entanglement of quantum

spin systems at finite temperatures has been extensively studied for various interaction

styles[9, 10, 11, 12]. Recently, Zhang et al.[13] extended Kieu’s work by considering the

quantum engine with a two-qubit (isotropic) Heisenberg XXX spin system as the working

substance. The spin is subject to a constant external magnetic field. The purpose of their

paper is to analyze the effect of quantum entanglement on the efficiency of the quantum

engine, but only a constant external magnetic field and entanglement was considered in that

study. Also, they only investigated the antiferromagnetic coupling case, that is incomplete.

For a ferromagnetic coupling spin chain, whether quantum heat engine is available or not

deserve investigation. Moreover in the theoretical analysis we think it is very interesting

to investigate the effects of spin-orbit coupling on the basic thermodynamic quantities and

should be included. In other words, quantum heat engine is a new physics concept, spin-orbit

coupling in spin system plays an important role in the preparation of quantum apparatus.

The effect of spin-orbit coupling on quantum heat engine is a very interesting topic and

deserves studying. In this paper, We construct an entangled quantum heat engine (EQHE)

based on two two-spin systems with a kind of spin-orbit coupling interaction. The positive

net work condition in terms of spin-orbit interaction and concurrence will be given. The

validity of the second law of thermodynamics at the quantum level is again confirmed.

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II. ENTANGLED QUANTUM HEAT ENGINE BASED ON TWO TWO-SPIN

SYSTEMS WITH DM INTERACTION

Our EQHE is based on the Heisenberg model with DM interaction, which can be described

by

HDM=J

2[(σ1xσ2x+ σ1yσ2y) +− →

D · (− →

σ1×− →σ2)],(1)

here J is the real coupling coefficient and− →

D is the DM vector coupling. The DM anisotropic

antisymmetric interaction arises from spin-orbit coupling [14, 15]. The coupling constant

J > 0 corresponds to the antiferromagnetic case and J < 0 to the ferromagnetic case. For

simplicity, we choose− →

D = D− →z , then the Hamiltonian HDMbecomes

HDM =

J

2[σ1xσ2x+ σ1yσ2y+ D(σ1xσ2y− σ1yσ2x)]

= J[(1 + iD)σ1+σ2−+ (1 − iD)σ1−σ2+]. (2)

Without loose of generality, we define |0? (|1?) as the ground (excited) state of a two-level

particle. The eigenvalues and eigenvectors of HDMare given by

|Ψ1? = |00?,|Ψ2? = |11?,|Ψ3? =

with E1= E2= 0, E3= J√1 + D2= −E4and θ = arctanD. As the thermal fluctuation

(temperature T) is introducing into the system, the state of a typical solid-state system at

1

√2

?|01? + eiθ|10??,|Ψ4? =

1

√2

?|01? − eiθ|10??,(3)

thermal equilibrium (temperature T) is ρ(T) =

Z = tre−βHDMis the partition function, β = 1/(kT) and k is Boltzmann’s constant, for

1

Ze−βHDM, where HDMis the Hamiltonian,

simplicity, we write k = 1. The entanglement between two qubits in this model is known as

[16]:

C =

sinh(|J|√1+D2/T)−1

cosh(|J|√1+D2/T)+1, if |J|√1 + D2> Tarcsinh(1);

0,if |J|√1 + D2≤ Tarcsinh(1).

(4)

Here, we use concurrence directly as the measurement of entanglement, since there is a

one-to-one correspondence between the entanglement of formation and concurrence[17, 18].

The concurrence C = 0 indicates the vanishing entanglement. The critical temperature

|J|√

Tc=

1+D2/T

arcsinh(1)≈ 1.1346|J|√1 + D2above which the concurrence is zero.

Based on the Kieus explanations of heat transferred and work performed at the quantum

level [1]: The expectation value of the measured energy of a quantum system with discrete

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energy levels is: U =< E >=

?

ipiEi, where Ei are the energy levels and pi are the

corresponding occupation probabilities.

which we can obtain the following identifications for infinitesimal heat transferred¯dQ and

work done¯dW

¯dQ :=

?

Mathematically speaking, these are not total differentials but are path dependent. These

Infinitesimally, dU =

?

i{pidEi+ Eidpi}, from

i

Eidpi,¯dW :=

?

i

pidEi. (5)

expressions interpret heat as a change in the occupation probabilities but not in the dis-

tribution of the energy eigenvalues themselves; and work done as the redistribution of the

energy eigenvalues but not of the occupation probabilities of each energy level [19]. Thus,

dU =¯dQ +¯dW, which is an expression of the first law of thermodynamics.

To set the stage, we describe the four quantum thermodynamic processes of the quan-

tum cycle. In the following, we consider as working substance a bipartite quantum system

consisting of two subsystems, A and B, with the above Hamiltonian (2). A cycle of the

quantum heat engine consists of four stages:

?1? The system has the probability pi0(i =1,2,3,4) to be in each of its four eigenstates,

respectively. By contacting with a heat bath at temperature Thfor some time, the occupation

probability of each eigenstate becomes pi1. The quantum state of the system is given by

the density operator ρ1

AB=?

ipi1|Ψi1??Ψi1|,(i = 1,2,3,4) with pi1= exp(−Ei1/kTh)/Z1

iexp(−Ei1/kTh). In other words, we assume the system is initially in thermal

equilibrium with a heat bath at temperature Th. In our model, we have Ei1= Ei, |Ψi1? =

|Ψi? with J = J1and D = D1. Only heat is transferred in this process due to the change in

the occupation probability. ?2? The system is then isolated from the heat bath and undergoes

a quantum adiabatic process, with J changing from J1to J2and D changing from D1to D2.

and Z1=?

Provided the rate of change is sufficiently slow, pi1’s are maintained throughout according to

the quantum adiabatic theorem. At the end of process ?2?, the system has the probability

pi1in the eigenstate |Ψi1?. An amount of work is performed by the system, but no heat is

transferred. ?3? The system is next brought into some kind of contact with another heat bath

at temperature Tl(Tl< Th) for some time. After the irreversible thermalization process, the

quantum state of the system is given by the density operator ρ2

AB=?

ipi2|Ψi2??Ψi2|,(i =

1,2,3,4) with pi2= exp(−Ei2/kTl)/Z2and Z2=?

|Ψi2? = |Ψi? with J = J2and D = D2. Only heat is transferred in this process to yield a

iexp(−Ei2/kTl). Here, we have Ei2= Ei,

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