Decoherence from spin environment: Role of the Dzyaloshinsky-Moriya interaction

Decoherence from spin environment: Role of the Dzyaloshinsky-Moriya interaction

W. W. Cheng

National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China and Department of Physics, Hubei Normal University, Huangshi 435002, China

J.-M. Liu

*

National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China;

School of Physics, South China Normal University, Guangzhou 510006, China;

and International Center for Materials Physics, Chinese Academy of Sciences, Shenyang 110016, China 共Received 4 April 2009; published 15 May 2009兲

We study the time evolution of decoherence factor of two spin qubits and qutrits coupled to anXYspin chain with Dzyaloshinsky-Moriya 共DM兲 interaction environment. The dynamical process of the decoherence is numerically and analytically investigated. We find that the DM interaction can enhance slightly the decay of the decoherence factor in the weak-coupling region. However, in the strong-coupling region, the decoherence factor is very sensitive to the DM interaction.

DOI:10.1103/PhysRevA.79.052320 PACS number共s兲: 03.67.Mn, 03.65.Yz

I. INTRODUCTION

As a fundamental concept in quantum theory, quantum entanglement plays a key role as a potential resource for quantum information processing共QIP兲and attracts much at- tention in many branches of physics both in theoretical as- pects and experimental ones关1–3兴. On the other hand, a re- alistic system is surrounded by an environment, and the unavoidable coupling between them will lead to decoherence of the entanglement 关4,5兴, which is a major obstacle for building a practical quantum computer. This complex quan- tum many-body phenomenon is an interesting problem and many works have been dealt with various models关6–19兴. Yu and Eberly关7兴found that two entangled qubits become com- pletely disentangled in a finite time under the influence of pure vacuum noise. Dodd and Halliwell关8兴studied the com- peting effects of environmental noise and interparticle cou- pling on disentanglement by solving the dynamics of two harmonically coupled oscillators. Tsomokos et al.关16兴stud- ied the dynamics of entanglement in qubit chains influenced by noise and static disorder and they found that the amount of disorder that is typically present in experiments does not affect the entanglement dynamics dramatically, while the presence of noise can have a significant influence on the generation and propagation of entanglement. Oxtoby et al.

关17兴 investigated the effects of the spin-boson environment on a double-qubit register initially prepared in a maximally entangled state and revealed that the presence of TLF-TLF 共two-level fluctuators兲 coupling reduced the performance of entangling gate operations performed on the register. Re- cently, there is a growing interest in the study of decoherence due to spin baths关20–22兴. Cucchiettiet al.关21兴examined the decoherence of a central spin interacting with a collection of environment spins. They revealed that the decoherence factor will generically have a Gaussian decay when there is no self-Hamiltonian for the system and the main feature of the

decoherence process can be dramatically changed by adding a self-Hamiltonian for the system. For a correlated spin en- vironment, Rossini et al.关22兴analyzed a model of environ- ment consisting of an interacting one-dimensional quantum spin-1/2 chain and they revealed that at short time the Loschmidt echo 共LE兲 decays as a Gaussian, while for long time it approaches an asymptotic value which strongly de- pends on the transverse field strength. Furthermore, some researchers investigated the decoherence of a system coupled to a spin environment with quantum phase transition共QPT兲 关23–29兴, which happens at zero temperature and is driven only by quantum fluctuation关30兴. At the critical point there usually exists degeneracy between the energy levels of the system. For a quantum two-level system coupled to an Ising chain, Quan et al.关23兴found that the decaying behavior of the LE is best enhanced by the QPT of the surrounding sys- tem. Cucchietti et al. 关24兴 discovered that the decoherence induced by the critical environment possesses some universal behavior. For a two-qubit case under a transverse Ising model, Sun et al. 关26兴 found that the concurrence decays exponentially with fourth power of time in the vicinity of the critical point of the environment system. The multiqubit case was studied by Ma et al.关28兴.

Recently, several experiments evidenced that entangle- ment can exist in variety of macroscopic and strong corre- lated systems in solid-state matter 关31兴 and some magnetic compounds can be used as an entanglement bath 关32兴. The Dzyaloshinsky-Moriya interaction is often present in the models of many low-dimensional magnetic materials. De- spite being small, this interaction is known to generate many spectacular features关33–35兴. In the field of condensed matter physics, antisymmetric spin coupling was first suggested by Dzyaloshinsky关36兴to explain the mechanism of weak ferro- magnetism of antiferromagnetic crystals from purely sym- metry ground state and was later derived theoretically by Moriya关37兴. The Dzyaloshinsky-Moriya共DM兲exchange in- teraction is proportional to the vector product of interacting spins and is allowed by symmetry in noncentric crystal struc- tures. In this work, we consider two spins coupled to anXY spin chain environment with DM interaction in order to re-

*liujm@nju.edu.cn

1050-2947/2009/79共5兲/052320共7兲 052320-1 ©2009 The American Physical Society

veal the effect of the DM interaction on the dynamic evolu- tion of the two-spin entanglement. We will investigate the role of DM interaction in the decoherence, given different coupling intensity regions, and both the two-qubit and two- qutrit cases will be considered.

The paper is organized as follows. In Sec.II, we introduce two-spin model system coupled to XYspin chains with DM interaction. By exactly diagonalizing the Hamiltonian, we give an expression of the time evolution operator. In Sec.III, the analytical analysis and numerical computation are carried out to understand the behavior of the decoherence factor both in the weak-coupling region and the strong one. In Sec.IV, a similar analysis to two qutrits coupled the spin chain is given. Finally, we give a summary for our main results.

II. MODEL AND SOLUTION

We choose the environment to be a generalXYspin chain with DM interaction and the two spins are transversely coupled to the chain. The corresponding Hamiltonian reads

H=J

兺

^L_l

冋

^{1 +2␥␴lx␴l+1x +1 −2␥␴ly␴l+1y +␭␴lz}

册

+J

兺

^L_l ^{D共␴l⫻␴l+1兲+g}₂^{J共sAz +sBz兲}

兺

^L_l ^␴lz=HE+HI.

共1兲 Here, we set the spin coupling interaction in Eq. 共1兲, J= 1, and the parametersD,␭, andg are dimensionless values.HI

denoted by ^g₂J共sA z+s_B^z兲兺l

L␴l

z represents the interaction be- tween the system and the environment.Lis the total number of the sites of the environment. Apart from the interaction HamiltonianHI, the remaining part of the whole Hamiltonian is the self-HamiltonianHE of the environment consisting of an XY spin chain with DM interaction. ␭ characterizes the strength of the transverse field and g denotes the coupling strength between the chain and the two spinss_A^z ands_B^z.␴l␣

共␣^=x,y,z兲are the Pauli operators defined on thelth site and Ddenotes the intensity of the DM interaction imposed along the zdirection. Noticing that关sA

z+s_B^z,␴l␣兴= 0, Eq.共1兲can be rewritten as

H=

兺

␮=1

␨

兩␾␮典具␾␮兩丢H_E^共␭␮兲, 共2兲

where 兩␾_␮典 共␮= 1 , . . . ,␨兲 is the ␮th eigenstate of operator

g 2共sA

z+s_B^z兲corresponding to the␮th eigenvalue␰␮, and ␭_␮is given by␭_␮=␭+␰␮.H_E^␭␮is given fromHEby replacement of

␭ with␭_␮. To examine the effect of the environment on the dynamical decoherence of two-qubit state, we should obtain the time evolution operatorU共t兲= exp共−iHt兲. To get an ana- lytical result, we follow the standard procedure by defining the conventional Jordan-Wigner transformation which maps spins to one-dimensional spinless fermions with creation共an- nihilation兲operatorsc_l^† 共c_l兲 关30兴. After a straightforward de- duction, the dressed environmental Hamiltonian becomes

H_E^共␭␮兲=

兺

^L_l ^{关共1 + 2iD兲cl†cl+1+共1 − 2iD兲cl+1† cl}

+␥共cl

†c_l+1^† +c_l+1cl兲+␭_␮共1 − 2cl

†cl兲兴. 共3兲 Now by introducing the Fourier transforms of the fermionic operators described by dk=_冑¹_L兺lcle^−i2␲lk/L, Hamiltonian 共1兲 can be diagonalized by transforming the fermion operators to momentum space and then using the Bogoliubov transforma- tion. The result is 关30,34兴

H_k^共␭␮兲=

兺

_k ^{⌳k共␭␮兲}

冉

^{␤k,†␭␮␤k,␭␮−12}

冊

^{, 共4兲}

where the energy spectrum⌳k共␭␮兲 is expressed by关35兴

⌳k共␭_␮兲= 2

冋

^{␧k+ 2Dsin}

^冉

^2␲Lk

^冊 册

^共5兲

with␧k=

冑

关cos共^2␲_L^k兲−␭_␮兴²+␥^2sin2共^2␲_L^k兲and the correspond- ing Bogoliubov transformed fermion operators are defined by

␤_k,␭␮= cos␪k共␭_␮兲

2 dk−isin␪k共␭_␮兲

2 d_−k^† 共6兲

with angles␪k共␭_␮兲 satisfying

␪k共␭_␮兲= arctan

冋

^{␭␮␥− cossin 2␲kL2␲Lk}

册

^{. 共7兲}

Here it should be mentioned that the energy spectrum de- rived in Ref. 关34兴may not be correct. In terms of these no- tations, we can drive the time evolution of quantum states and obtain the decoherence dynamics induced by the inter- action between the system and the environment. Let the ini- tial state be 兩⌿共0兲典=兩⌽共0兲典AB丢兩⌽共0兲典E, where 兩⌽共0兲典AB

and 兩⌽共0兲典E are the initial states of the two central spins and the environment spin chain ones, respectively. So 兩⌿共t兲典=U共t兲兩⌿共0兲典, where U共t兲= exp共−iHt兲. Then the re- duced density matrix can be obtained with following equa- tion:

␳A,B共t兲= TrE兩⌿共t兲典具⌿共t兲兩

=

兺

␮,␯=1

␨

c_␮c_␯^ⴱ具⌽E兩UE

†共␭_␯兲共t兲UE共␭_␮兲共t兲兩⌽E典兩␮典具␯兩, 共8兲 where c_␮=具␾␮兩⌽AB典 and U_E^共␭␮兲共t兲= exp共−iHE共␭_␮兲t兲 is the pro- jected time evolution operator for the spin chain dressed by the system-environment interaction parameter␭_␮. We define the decoherence factor兩F共t兲兩as follows:

F共t兲=具⌽E兩U_E^{†共␭␯兲}共t兲U_E^共␭␮兲共t兲兩⌽E典. 共9兲 Obviously, when ␮=␯^, 兩F共t兲_␮␯兩 equals to 1. 兩F共t兲_␮␯兩 can be considered as the amplitude of the overlap of different bases of the system under the defined environment.

III. DYNAMICAL DECOHERENCE OF TWO QUBITS First, we investigate the dynamic evolution of decoher- ence factor for a two-qubit case, so ␭1=␭+g, ␭2=␭−g,

␭3=␭, and ␭4=␭. We assume that the two qubits initially stem from a maximally entangled state,

兩⌽典AB= 1

冑

2共兩00典AB+兩11典AB兲, 共10兲 where兩0典and兩1典denote the spins up and down, respectively.

From Eq.共8兲, in the basis spanned by兵兩00典,兩11典,兩01典,兩10典其, the reduced density matrix of the two-qubit system is ob- tained as关26兴

␳A,B=1

2

冉

^Fⴱ1共t兲 ^F共t兲1

冊

^{丣Z2⫻2, 共11兲}

whereF共t兲is the decoherence factor defined by Eq.共9兲and Z_2⫻2 denotes the 2⫻2 zero matrix.

The initial state of environment兩⌽共0兲典E is assumed to be 兩G典_␭=兿_k=1^M 共cos^␪₂^k␭兩0典k兩0典−k+isin^␪₂^k␭兩1典k兩1典−k兲, where 兩0典k and

兩1典kdenote the vacuum and single excitation of thekth mode dk, respectively. Noticing that

␤k,␭_␮=共cos⌰k共␭_␮兲兲␤k,␭−i共sin⌰k共␭_␮兲兲␤_−k,␭^{† ,} 共12兲 where ⌰k共␭_␮兲=共␪k共␭_␮兲−␪k共␭兲兲/2 and ␤_k,␭␮^,␤_k,␭ are the normal mode dressed by the system-environment interaction and the purely environment, respectively, the ground state兩G典_␭of the HamiltonianHEcan be obtained from the ground state兩G典_␭␮ of the qubit-dressed HamiltonianH_E^␭␮by the following rela- tion 关23–25兴:

兩G典_␭=

兿

k=1 M

共cos⌰k共␭_␮兲+isin⌰k共␭_␮兲␤_k,␭^† _␮␤_−k,␭^† _␮兲兩G典_␭␮. 共13兲 Then the decoherence factor兩F共t兲兩can be obtained from Eqs.

共9兲and共13兲as follows 关29兴:

兩F共t兲兩=兩_␭具G兩UE

†共␭2兲共t兲UE共␭1兲共t兲兩G典_␭兩=

兿

k⬎0兵1 + 2 sin共2⌰k共␭1兲兲sin共2⌰k共␭2兲兲sin共⌳k共␭1兲t兲sin共⌳k共␭2兲t兲cos共⌳k共␭1兲t−⌳k共␭2兲t兲

− 4 sin共2⌰k共␭1兲兲sin共2⌰k共␭2兲兲sin²共⌰k共␭1兲−⌰k共␭2兲兲sin²共⌳k共␭1兲t兲sin²共⌳k共␭2兲t兲

− sin²共2⌰k共␭1兲兲sin²共⌳k共␭1兲t兲− sin²共2⌰k共␭2兲兲sin²共⌳k共␭2兲t兲其^1/2

⬅_k

兿

_⬎0^{Fk共t兲. 共14兲}

A. Weak-coupling case (g™1)

Before proceeding with the detailed calculation, we first consult to earlier work on relatively simpler system. Yuanet al.关29兴studied the decoherence factor of two qubits coupled to an Ising chain under the weak-coupling case. They found that apart from the critical point␭c,兩F共t兲兩in the time domain is characterized by an oscillatory localization behavior.

When␭approaches␭c,兩F共t兲兩evolves from unity to zero in a very short time, which implies that the disentanglement of the two central spins is best enhanced by a QPT of the envi- ronmental spin chain. Now, we consider the effect of DM interaction on the decay of the decoherence factor for the two-qubit interacting with the environment in the weak- coupling region. First, we investigate the dynamical property of F共t兲 by numerical calculation from the exact expression 共14兲. In Fig.1,兩F共t兲兩is plotted as a function of parameterD and timet with parameters L= 801, g= 0.05, and␥^{= 1.0. At} point␭= 1, we vary the value ofDfrom 0 to 1.0. It can be found that the decoherence factor decays more sharply with increasing the intensity of DM interaction in a short-time regime. However, the plot also indicates that the effect of DM interaction on the decay of decoherence factor is not remarkable. To check this phenomenon, we carry out an

analysis similar to Ref. 关23兴. We introduce a cutoff number Kcand define the partial product for the decoherence factor,

兩F共t兲兩c=_k

兿

⬎0 K_c

Fk共t兲ⱖ兩F共t兲兩, 共15兲

from which the corresponding partial sum is obtained as

FIG. 1. Decoherence factor兩F共t兲兩as a function ofDand timet for two qubits coupled to an Ising spin chain with DM interaction in the weak-coupling region under parameters ␭= 1.0, ␥= 1.0, g

= 0.05, andL= 801. The unit of quantitytisប/J.

S共t兲= ln兩F共t兲兩c= −_k

兺

⬎0 K_c

兩lnF_k兩. 共16兲

For the case of smallkand largeL, we have

⌳k共␭兲⬇2兩␭− 1兩+ 4Dsin

冉

^2␲L^k

冊

^,

⌳k共␭_␮兲⬇2兩␭_␮− 1兩+ 4Dsin

冉

^2␲L^k

冊

^{, 共17兲}

and then

sin共2⌰k共␭_␮兲兲 ⬇ ⫿2␥␲^kg L兩共␭_␮− 1兲共␭− 1兲兩,

sin共⌰k共␭1兲−⌰k共␭2兲兲 ⬇ − 2␥␲kg

L兩共␭1− 1兲共␭2− 1兲兩. 共18兲 As a result, ifL is large enough andkis small relatively, the approximation ofS共t兲 can be obtained as

S共t兲 ⬇−1 2

4␲²␥^2g2

L²共␭− 1兲²

兺

^K_k^{c k2}

再

^{sin共␭22共⌳− 1k共␭2兲兲2t兲+sin共␭21共⌳− 1k共␭1兲兲2t兲}

+ 2sin共⌳k共␭1兲t兲sin共⌳k共␭2兲t兲

兩共␭1− 1兲共␭2− 1兲兩 cos共4gt兲

冎

^{. 共19兲}

In the derivation of the above equation, we ignore the terms related to the sumk⁴/L⁴and employ the approximation for- mula ln共1 −x兲⬇−xfor smallx. Consequently, in a short time and when␭→1, one has

兩F共t兲兩 ⬇e^{−共␶1+␶2兲t2}, 共20兲 where

␶1= 8E共K兲␥²g²/共␭− 1兲², E共K兲= 4␲²

兺

k=1 K_c

k²/L²,

␶2= 32F共K兲␥²gD/共␭− 1兲², F共K兲= 8␲³

兺

k=1 K

k³/L³. 共21兲 From the above approximation analysis we may expect that when the parameter␭is adjusted to point␭= 1, the decoher- ence factor will exponentially decay with the second power of time. From Eqs. 共20兲 and 共21兲, it is easily seen that the decoherence factor is determined by␶1and␶2; but from Eq.

共21兲, we can see that E共Kc兲ⰇF共Kc兲. So F共t兲 is determined mainly by␶1, andDonly can change the decay slightly in the weak-coupling region. This is consistent with our numerical results shown in Fig. 1.

We give a brief explanation about the effect of DM inter- action on the decoherence factor under the case of ␥⫽1 which means a generalXYspin chain. In Figs. 2共a兲–2共c兲we plot the decoherence factor against time t under differentD with ␥ equal to 0.8, 0.6, and 0.4, respectively. We can see that the decay of the decoherence factor can be enhanced

with increasing D. Furthermore, Fig. 2 shows also that the smaller the values of ␥, the stronger the effect of the DM interaction on the decay of decoherence factor.

B. Strong-coupling case (gš1)

We now turn to discuss the effect of DM interaction on the decoherence factor in the strong-coupling regime gⰇ1.

In a previous work, Yuan et al.关29兴revealed that the decay of兩F共t兲兩is characterized by an oscillatory Gaussian envelope for a pure Ising spin chain environment. In Fig.3we display the time evolution of 兩F共t兲兩as a function of timet with the parameters ␭= 1, ␥= 1, L= 801, and g= 500. It is observed that the decay is characterized by an oscillatory Gaussian envelop, too. However, in contrast to the pure Ising chain environment, the width of the Gaussian envelop is very sen- sitive to the DM interactionD. In other words, the decay of the decoherence factor can be enhanced dramatically by the DM interaction. To explain this feature, we take a similar tactic used by Yuan et al.关29兴. WhengⰇ1, from the angle for Bogoliubov transformation, ␪k共␭_␮兲= arctan关 ^␥sin

2␲k L

␭_␮−cos2␲k L

兴, we have ␪k共␭1兲⬇0 and␪k共␭2兲⬇␲, so we get

FIG. 2.共Color online兲Decoherence factor兩F共t兲兩as a function of timetunder differentDfor two qubits coupled to a generalXYspin chain with DM interaction in the weak-coupling region under pa- rameters␭= 1.0,g= 0.05, andL= 801.共a兲␥= 0.8,共b兲␥= 0.6, and共c兲

␥= 0.4. The unit of quantitytisប/J.

兩F共t兲兩 ⬇

兿

_k ^{兩cos2⌰k共␭1兲exp共i⌳kt兲+ sin2⌰k共␭1兲exp共−i⌳kt兲兩,}

共22兲 where ⌳k=⌳k共␭1兲+⌳k共␭2兲. This expression is similar to that given in Refs.关21,24兴. We can follow the mathematical pro- cedure and the resultant approximate expression for兩F共t兲兩is as follows:

兩F共t兲兩 ⬇exp共−s_L²t²/2兲兩cos共⌳t兲兩^L/2, 共23兲 where⌳is the mean value of⌳k.

⌳= 1

M_k

兺

_⬎0^{共⌳k␭1+⌳k␭2兲}

⬇ 1

M

兺

k⬎0

2

冉

²

^冑

^{g2+␥2sin22␲Lk+ 4Dsin2␲Lk}

冊

⬇ 1

M_k

兺

_⬎0⁴

冉

^{g+12␥g2sin22␲Lk}

冊

⬇4g+␥²

g . 共24兲

In the deduction, the approximation formula 共1 −x兲^1/2⬇1

−共1/2兲xis employed for smallxand s_L²=_k

兺

⬎0

sin²共2⌰k共␭1兲兲␦k

2. 共25兲

Here the quantity ␦k describes the deviation of ⌳k from its mean values. Its value is

␦k=⌳k−⌳⬇2

冉

²

^冑

^{g2+␥2sin22␲}L^k+ 4Dsin2␲^k L

冊

^{− 4g}

−␥² g ⬇−␥²

g cos4␲^k

L + 8Dsin2␲^k

L . 共26兲

So one can see that the width of the Gaussian envelope is proportional to兵共^␥_g⁴2+ 64D²兲L其^−1/2. Figures3共a兲–3共c兲show the numerical results plotted by solid line and the Gaussian en- velope factor plotted by dashed ones for the intensity of DM interaction D is equal to 0.0, 0.01, and 0.1, respectively.

From the above discussion, we can conclude that the decay of the decoherence factor is determined mainly by Din the region ofgⰇ1.

IV. DYNAMICAL DECOHERENCE OF TWO QUTRITS Now, we investigate the dynamic evolution of the deco- herence factor for a two-qutrit case and assume that the two- qutrit case initially stems from a maximally entangled state,

兩⌽典AB= 1

冑

3共兩00典AB+兩11典AB+兩22典AB兲, 共27兲 where兩0典,兩1典, and兩2典denote the spin 1 with magnetic quan- tum numbers 1, 0, and −1, respectively. From Eq.共8兲, in the

basis spanned by

兵兩00典,兩11典,兩22典,兩01典,兩10典,兩02典,兩20典,兩12典,兩21典其, the reduced density matrix of the two-qutrit system is 关26兴

␳A,B=1

3

冢

^{FF12ⴱ13ⴱ1共共t兲t兲 FF1223ⴱ1共t兲共t兲 FF13231共t兲共t兲}

冣

^{丣Z2⫻2丣Z2⫻2丣Z2⫻2,}

共28兲 where the decoherence factors F₁₂共t兲, F₁₃共t兲, and F₂₃共t兲 are given by Eq.共14兲exactly and␭_␮共␮= 1 , 2 , 3兲in兩F_␮␯共t兲兩can be obtained by substituting ␭1=␭+g, ␭2=␭, and ␭3=␭−g.

From the previous discussion, it is obvious that the property of兩F13共t兲兩is completely similar to the two-qubit case both in the weak-coupling region and the strong one. In the follow- ing, we only discuss the role of DM interaction on the deco- herence factors 兩F₁₂共t兲兩and兩F₂₃共t兲兩.

A. Weak-coupling case (g™1)

We first investigate the effect of DM interaction on the decoherence factor 兩F_12共23兲共t兲兩 in the weak-coupling region.

From ⌰k共␭_␮兲=共␪k共␭_␮兲−␪k共␭兲兲/2, Eq.共14兲can be rewritten as 兩F12共23兲共t兲兩=

兿

k⬎0兵1 − sin²共2⌰_k^␭1共3兲兲sin²共⌳_k^␭1共3兲t兲其^1/2. 共29兲 So, one can get S共t兲as follows:

FIG. 3.共Color online兲Decoherence factor兩F共t兲兩as a function of the timetfor two central spin qubits coupled to an Ising spin chain under different D in the strong-coupling region under parameters

␭= 1.0,␥= 1.0,g= 500, andL= 801.共a兲D= 0.0,共b兲D= 0.01, and共c兲 D= 0.1. The unit of quantitytisប/J.

S共t兲 ⬇−1 2

4␲²␥²g²

L²共␭− 1兲²

兺

^K_k^c

再

^{k2sin共⌳共␭1共3兲k␭− 1兲1共3兲t兲}

冎

^{2. 共30兲}

Consequently, in a short time and when ␭→1, one has the similar previous result, 兩F_12共23兲兩⬇e^{−共␶1+␶2兲t2}, where ␶1

= 2E共K兲␥^2g2/共␭− 1兲²and␶2= 8F共K兲␥²gD/共␭− 1兲².E共K兲and F共K兲 are completely the previous ones. One can see that, when the parameter ␭ approaches point ␭= 1, the decoher- ence factor will exponentially decay with the second power of time, too. We can find that the decay of兩F_12共23兲共t兲兩can be enhanced slightly by the DM interaction, and the numerical results are shown in Fig.4.

B. Strong-coupling case (gš1)

Now we check the dynamical property of 兩F_12共23兲共t兲兩 by numerical analysis from the exact expression共14兲. In Fig.5, 兩F12共t兲兩is plotted as a function of timet andD. Panels 共a兲, 共b兲, and共c兲correspond to parameterDequal to 0.0, 1.0, and 2.0, respectively. We find that the decay of the decoherence factor approaches zero sharply in the strong-coupling region and characterized by an oscillatory Gaussian envelop, too.

Despite the DM interaction can enhance the decay of the decoherence factor, however, in contrast to the two-qubit case, the decays of the decoherence factors 兩F₁₂共t兲兩 and 兩F23共t兲兩are not significantly affected by the DM interaction.

V. CONCLUSION

In summary, we have studied the dynamic process of the decoherence factor of two spin qubits coupled to a general XY spin chain with DM interaction. In the weak-coupling region 共gⰆ1兲, the decoherence factor presents a Gaussian behavior in a short time, e.g.,兩F共t兲兩=e^{−共␶1+␶2兲t2}, and the DM interaction can enhance the decay of the decoherence factor slightly. However, in the strong-coupling region, we find that the decay of the decoherence factor is characterized by an

oscillatory Gaussian envelop, and the width of the Gaussian envelop is very sensitive to the DM interaction. It is mainly determined by the values of D and is proportional to 兵共^␥_g⁴2

+ 64D²兲L其^−1/2. The role of DM interaction on decoherence factor for a generalXY chain environment is also analyzed.

In the weak-coupling region, DM interaction can enhance the decay of the decoherence, too.

In such case, the decoherence factor兩F₁₃共t兲兩exhibits the same behavior as the two-qubit case. However, in the strong- coupling region, the decays of兩F12共t兲兩 and兩F23共t兲兩approach zero sharply, in contrast to the two-qubit case, which are not significantly affected by the DM interaction.

ACKNOWLEDGMENTS

The authors would like to acknowledge the financial sup- port from the National Natural Science Foundation of China 共Grant Nos. 10674061, 50832002, and 50572038兲, the Na- tional Key Projects for Basic Researches of China 共Grant Nos. 2006CB921802 and 2009CB623303兲, and the Key Projects of Hubei Province 共Grant No. D20092204兲. FIG. 4. Decoherence兩F₁₂共t兲兩as a function ofDand timetfor

two qutrits coupled to an Ising spin chain with DM interaction in the weak-coupling region under the parameters ␭= 1.0, ␥= 1.0, g

= 0.05, andL= 801. The unit of quantitytisប/J.

FIG. 5.共Color online兲Decoherence factors兩F₁₂共t兲兩as a function of the timet for two qutrits coupled to an Ising spin chain under different D in the strong-coupling region under the parameters ␭

= 1.0, ␥= 1.0,g= 500, andL= 201.共a兲D= 0.0,共b兲 D= 1.0, and 共c兲 D= 2.0. The unit of quantitytisប/J.

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