The Correlated Environments Depress Entanglement Decoherence in the Dimer System

The Correlated Environments Depress Entanglement Decoherence in the Dimer System

Qin-Sheng Zhu^a, Chuan-Ji Fu^a, and Wei Lai^b

aDepartment of Applied Physics, University of Electronic Science and Technology of China, Chengdu 610054, P.R.China

bExperimental Middle School of Chengdu Economic and Technology Development Zone, Chengdu 610054, China

Reprint requests to Q.-S. Z.; E-mail:zhuqinsheng@gmail.com Z. Naturforsch.68a,272 – 278 (2013) / DOI: 10.5560/ZNA.2012-0111

Received August 13, 2012 / revised October 7, 2012 / published online January 23, 2013

In this work, the decoherence properties of two independent dimer systems coupled to two corre- lated Fermi-spin environments is investigated under the non-Markovian condition. We demonstrate that the correlated spin bath can effectively depress the entanglement decoherence, and a steady en- tanglement can be achieved when the coupling parameterqexceeds a critical value. This result shows a possible method of the entanglement preservation in decohering environments.

Key words:Decoherence; Entanglement; Quantum Noise.

PACS numbers:03.65.Yz; 03.67.Bg; 42.50.Lc; 03.65.Ud 1. Introduction

In a realistic physical system, the decoherence phe- nomena of the entanglement arising from the in- evitable interaction between the environment and sys- tems always exist and play a coherence-destructive role [1,2]. Hence, how to depress the decoherence and how to instrumentally keep the system entangled under environmental noise is of paramount relevance for a number of applications in modern physics, espe- cially the quantum information and quantum computa- tion. Some works related to the non-equilibrium pro- cess have been carried out on this aspect in the frame- work of a non-Markovian dynamics, including sudden death and sudden birth of entanglement [3–8], non- Markovianity-assisted steady state entanglement [9], measure for non-Markovian behaviour of quantum processes [10], non-Markovian entanglement dynam- ics in the presence of system–bath coherence [11], and so on [12–22]. But these works were mainly restricted to the case of the independent environments. On the other hand, entanglement preservation for the case of correlated environments is still an open problem.

In the present work, we demonstrate that the de- coherence behaviour of a dimer system can be effec- tively depressed by increasing the coupling parameter qbetween the environments and that a steady entangle-

ment can be obtained when the parameterq exceeds a critical value. This result is of particular interest in understanding the mechanisms that depress the entan- glement decoherence and assist entanglement preser- vation in condensed matter and biomolecular systems, where non-Markovian dephasing is a dominant noise source [14,23–28].

2. Formulation of the Problem

We consider a two-qubit system with no mutual in- teraction. Every qubit is made of the simplest elec- tronic energy transfer system which can be formed by a dimer system (pseudo-spin-¹₂ particles). The Hamil- tonian of two independent dimer systems is given by [9,29,30]

H_d=H_d1+H_d2 (1)

whereH_d1 =ε₁|1ih1|+ε₂|2ih2|+J₁(|1ih2|+|2ih1|) andH_d2 =ε3|3ih3|+ε4|4ih4|+J2(|3ih4|+|4ih3|)de- note the Hamiltonian of two independent dimer sys- tems, respectively.ε_iand|ii(i=1,2,3,4)are the en- ergy levels and the energy states of the dimer system.

J₁andJ₂are the amplitude of transition.

After some calculations, the eigenvalues and eigen- vectors of H_d can be obtained in the absence of the environment:

© 2013 Verlag der Zeitschrift f¨ur Naturforschung, T¨ubingen·http://znaturforsch.com

E₁=∆1+∆3+ q

∆₂²+J₁²+ q

∆₄²+J₂², Ψ₁=





∆₂+q

∆₂²+J₁² J1

|1i+|2i









∆₄+q

∆₄²+J₂² J2

|3i+|4i



,

E₂=∆1+∆3+ q

∆₂²+J₁²−q

∆₄²+J₂², Ψ₂=





∆2+q

∆₂²+J₁² J1

|1i+|2i









∆4−q

∆₄²+J₂² J2

|3i+|4i



,

E₃=∆1+∆3− q

∆₂²+J₁²+ q

∆₄²+J₂², Ψ₃=





∆2−q

∆₂²+J₁² J1

|1i+|2i









∆4+q

∆₄²+J₂² J2

|3i+|4i



,

E₄=∆1+∆3− q

∆₂²+J₁²− q

∆₄²+J₂², Ψ₄=





∆2−q

∆₂²+J₁²

J₁ |1i+|2i









∆4−q

∆₄²+J₂²

J₂ |3i+|4i



, (2)

where∆1=^ε1+ε₂ ²,∆2=^ε1−ε₂ ²,∆3=^ε3+ε₂ ⁴, and∆4=

ε₃−ε₄ 2 .

The aim of this paper is to show, in general condi- tionsε₁6=ε₂andε₃6=ε₄, the dynamic evolution of the entanglement for the two dimers in contact with the same two spin environments. The Hamiltonian of the total system has the following form [5,9,29,30]:

H=H_d+

∑

i=1,2

H_Bi+H_d1B1+H_d1B2+H_d2B1+H_d2B2

+q

N1

∑

k=1 N2

∑

k⁰=1

σ^k,1_z 2

σ^k_z^0,2 2 .

(3)

Here, each environment Bi consists ofNi particles (i=1,2)with spin ¹₂:

H_Bi=α_i

N_i k=1

∑

σ^k,i_z

2 , (4)

where σ^k,i_z are the Pauli matrices, and α_i is the fre- quency ofσ^k,i_z .

Because a dimer system is a system made of two subsystems (for example, two spin-like particles or two qubits) by the typically bound to each other, we con- sider two subsystems (every subsystem is a pseudo- spin-¹₂ particles) in different environments [31], re- spectively. The interaction between the subsystem and the environment can be described byγ^σ₂^z∑^N_k=1^{i σ}

k,iz

2 [5]

(whereγis the coupling constant, andσzis the pseudo- spin Pauli operator which describes the pseudo-spin-¹₂ particles). Here we only consider the interaction be- tween the upper level and the environment [9,29–31],

so the interaction between the dimer and the environ- ment is described by [9,29–31]

H_d1B1=

N₁ k=1

∑

γ1|1ih1|σ^k,1_z

2 , H_d1B2=

N₂ k=1

∑

γ2|2ih2|σ^k,2_z 2 , H_d2B1=

N₁ k=1

∑

γ3|3ih3|σ^k,1_z

2 , H_d2B2=

N₂ k=1

∑

γ4|4ih4|σ^k,2_z 2 .

(5)

The last termq∑^N_k=1¹ ∑^N_k0²=1 σ^k,1z

2 σ^kz^0,2

2 of (3) describes an Ising-type correlation between the environments with strengthq. The casesq=0 and q6=0 describe independent and correlated spin bath, respectively.

In order to obtain the exact solution of the sys- tem, we define collective spin operatorsS^z_i=∑^N_k=1^{i σ}

k,i z

2 . Thus, the total Hamiltonian can be written as

H=H_d⁰

1+H_d⁰

2+

∑

i=1,2

α_iS^z_i+qS^z₁S^z₂,

H_d⁰

1= (ε₁+γ1S^z₁)|1ih1|+ (ε₂+γ2S₂^z)|2ih2|

+J₁(|1ih2|+|2ih1|), H_d⁰

2= (ε₃+γ₃S^z₁)|3ih3|+ (ε₄+γ₄S₂^z)|4ih4|

+J₂(|3ih4|+|4ih3|).

(6)

We introduce an orthonormal basis in the bath Hilbert spaceH_Bconsisting of states|j,mi[5]. These states are defined as eigenstates ofS_zand ofS²and

S²|j,mi=j(j+1)|j,mi,

S_z|j,mi=m|j,mi, S²=S²_x+S²_y+S²_z (7) withj=0, . . . ,^N₂,m=j, . . . ,−j.

The formal solution of the von Neumann equation d

dtρ(t) =Lρ(t) =−i[H,ρ(t)] (8) can then be written as

ρ(t) =e^Ltρ(0), (9) whereρ(t)denotes the density matrix of the total sys- tem.

Our main goal is to derive the dynamics of the re- duced density matrixρd(t).

ρ_d(t) =Tr_B(e^Ltρ(0)) (10) where Tr_B denotes the partial trace taken over the Hilbert space of the spin bath.

For the initial stateρ(0) =ρd(0)⊗ρB(0), the den- sity matrixρ_d(t)of the dimer system is

ρ_d(t) =Tr_B

U(t)ρ_d(0)ρ_B(0)U^†(t)

(11)

=1 Z

N1/2

∑

j1=0 j₁

∑

m1=−j₁ N2/2

∑

j2=0 j₂

∑

m2=−j₂

ν(N₁,J1)ν(N₂,J2) e^{β α1m1}e^{β α2m2}e^βqm1m2

×A^†DA

Here, the bath is given as the canonical distribution

ρB(0) =1 Ze^qβSz1Sz2

2

∏

i=1

e^{−β αiSzi} (12)

withβ =_K¹

BT (K_Bis Boltzmann constant,T is temper- ature) and the initial conditionρd(0) =A^†GA, where A,D, and Gdenote the matrices which are given in theAppendix.

The partition functionZof the bath is

Z=

N₁/2

∑

j1=0 j1

∑

m1=−j₁ N₂/2

∑

j2=0 j2

∑

m2=−j₂

ν(N₁,J₁)ν(N₂,J₂) e^{β α1m1}e^{β α2m2}e^βqm1m2 , (13) where ν(N_i,J_i) denotes the degeneracy of the spin bath [5,32,33].

To quantify the entanglement, we use the Woot- ters concurrence [34], defined asC(ρ) =max{0,λ₁− λ2−λ3−λ4}, whereλl(l =1,2,3,4)are the square roots of the eigenvalues of the matrixR=ρ^1/2(σ_y⊗ σy)ρ^∗(σ_y⊗σ_y)ρ^1/2in decrease order, andρis the den- sity matrix expressed in the standard basis| ↑↑i,| ↑↓i,

| ↓↑i,| ↓↓i.σyis the normal Pauli matrix, and the as- terisk indicates complex conjugation.

3. Results

After some calculation, we can obtain the evolu- tion of the entanglement states, as shown in Figure1 and Figure2.

In Figure1, we show the entanglement dynamics of two qubits for the initial stateA(disentanglement state) with parameter q and time t. From Figure1, for the smaller parameterq, the entanglement does not exhibit sudden birth. When we adjust the coupling strength be- tween the spin baths, namely the larger parameterq, the entanglement appears. The time evolution of en- tanglement presents the phenomenon of sudden birth and sudden death and shows the non-periodic oscilla- tion evolution.

In Figure2, it is shown that the entanglement dy- namics of two qubits for the entanglement initial state changes with parameterqand timet. The time evolu- tion of entanglement presents the phenomenon of sud- den birth and sudden death even if the initial state is the entanglement state, and shows the oscillation evo- lution. The most important and interesting nature is ef- fectively depressing the entanglement decoherence by increasing the parameterq. Especially, for the smaller concurrence [34] C(ρ) of the initial state, we must use largerqto depress the decoherence. The time evo-

0 2

4 6

8 10

0 1 2 3 4 5 60 0.02 0.04 0.06 0.08 0.1

q t

C

Fig. 1 (colour online). Change of the concurrence for the initial state φ_d(0) =

√2 4 |1i|3i+

√2 4 |2i|3i+

√6 4 |1i|4i+

√ 6

4 |2i|4i. The parameters are:γ1=0.2 ps⁻¹,γ2=0.3 ps⁻¹, γ3=0.4 ps⁻¹,γ4=0.15 ps⁻¹,J1=10 ps⁻¹,J2=12 ps⁻¹, N1 =22, N2 =20, α1 =250 ps⁻¹, α2 =10 ps⁻¹, ∆1 = 20 ps⁻¹,∆2=10 ps⁻¹,∆3=22 ps⁻¹,∆4=12 ps⁻¹, and the temperature of the baths is 300 K.

0 2 4 6 8 10 0

2 4 60 0.02 0.04 0.06 0.08

q t

C

0 2 4 6 8 10

0 2 4 60 0.2 0.4 0.6 0.8

q t

C

(a)

(b)

Fig. 2 (colour online). Change of the concurrence for the entanglement initial state. In (a), the initial state φ_d(0) = 0.5|1i|3i+0.48|2i|3i+0.49|1i|4i+0.53|2i|4i. In (b), the initial state φ_d(0) =0.4|1i|3i+0.2|2i|3i+0.25|1i|4i+ 0.86|2i|4i. The parameters are:γ1=0.2 ps⁻¹,γ2=0.3 ps⁻¹, γ3=0.4 ps⁻¹,γ4=0.15 ps⁻¹,J1=10 ps⁻¹,J2=12 ps⁻¹, N1 =22, N2 =20, α1= 250 ps⁻¹, α2 =10 ps⁻¹, ∆1= 20 ps⁻¹,∆2=10 ps⁻¹,∆3=22 ps⁻¹,∆4=12 ps⁻¹, and the temperature of the baths is 300 K.

lution of the behaviour of the concurrence gradually changes from non-periodic evolution to periodic evo- lution and the amplitude of the concurrence becomes smaller. Finally, the fluctuation of the concurrence dis- appears when the coupling parameterqexceeds some value which depends on the initial state. For the initial states in Figure2, the threshold value of the coupling parameter q is about 5. This means that we can ob- tain a steady entanglement by adjusting the coupling strength between the environments. This nature may provide promising prospect for the study of quantum information, the design of quantum devices, and en- tanglement preservation.

From the view of the non-equilibrium statistical physics, we may understand this novel result, namely the information transfer or the entropy transfer be- tween the dimer system and the environments. Firstly, the coupling of the environments led to the exchange of information (entropy) between the environments, and this exchange has been gradually strengthened with in-

creasingq. Secondly, under the non-Markovian condi- tion, the interaction between each single dimer system and its own spin bath not only makes the information (entropy) of the dimer system flow into the spin bath, but also arouses the feedback of the information (en- tropy) from the spin bath to the dimer system [4,6,8], showing sudden birth and sudden death phenomena.

Finally, based on the above two reasons, there is a mu- tual influence between the information (entropy) of the two dimer systems, and this mutual influence strength- ens the coherence of the dimer systems with increas- ingq. Especially, for the initial state of entanglement, a steady entanglement can be obtained when the pa- rameterqexceeds some value depending on the initial state.

Further, we study an interesting special case, namely all the environments with zero temperature. Under this condition, all the spin baths will be in the ground states, and the ground states of the spin baths have the follow- ing forms:

ψ¹_B(0)

=

N₁ 2 ,−N₁

2

⊗

N₂ 2 ,−N₂

2

,

forq<q₀ ψ²_B(0)

=

N₁ 2 ,−N₁

2

⊗

N₂ 2 ,N₂

2

,

forq>q₀,α1>α2

ψ³_B(0)

=

N₁ 2 ,N₁

2

⊗

N₂ 2 ,−N₂

2

,

forq>q₀,α₁<α₂ q₀=2 min

α1

N2

,α2

N1

.

(14)

In the case of degeneracy of the parameters, for ex- ample,q0=qorα1=α2, the ground state is the linear combination of|ψ¹_B(0)i,|ψ²_B(0)i, and|ψ³_B(0)i.

Considering the reservoirsH_Bi commuting withH_d, H_d1Bi, and H_d2Bi, the states of the system will have the following form for the random initial condition φ_d(0) =a13|1i|3i+a23|2i|3i+a14|1i|4i+a24|2i|4i (satisfy|a₁₃|²+|a₂₃|²+|a₁₄|²+|a₂₄|²=1):

|ψ_Total(t)i=e^−iΘt

k=3,4

∑

i=1,2;`=1,2,3

f_ik(t)|ii|ki ⊗ |ψ^`_B(0)i, (15)

whereΘ =hψ^{`</sup><sub>B</sub>(0)|α1S<sub>1</sub><sup>z</sup>+α2S<sub>2</sub><sup>z</sup>+qS<sup>z</sup><sub>1</sub>S<sup>z</sup><sub>2</sub>|ψ<sup>`}_B(0)i, and the index` change according to the value ofq₀. The

f_ik(t)are given in theAppendix.

After we trace out the degrees of freedom of the spin bath, the dynamic evolution of the system is obtained.

Using the definition of concurrenceC(ρ)[34], we can obtain the following expression about concurrence and further study the properties of the entanglement:

C(Tr_E(|ψ_Total(t)i)hψ_Total(t)|)

=2|a₁₃a₂₄−a₁₄a₂₃|. (16) From above equation, we know that the concur- rence of the random initial states do not change with time under zero temperature. It can be understood from the view of the non-equilibrium statistical physics that the information or the entropy is transfered between the dimer system and the environments. For non- Markovian dynamics, the action of the environments do not make the information (entropy) of the dimer system flow into the environments, and the changing of the environments always feedbacks some informa- tion (entropy) to the dimer system. However, for zero temperature, because the environments do not change, there does not exist an information (entropy) transfer between the dimer system and the environments.

4. Conclusions

The possible relevance of the Fermi-spin environ- ment to the entanglement has been investigated. For the initial state of entanglement or disentanglement, we show that the correlations between the spin baths with coupling strength parameterqimpact on the evolution of the entanglement. Especially, for the entangling ini- tial state, we can effectively depress the entanglement decoherence by increasing the value of the parameter q, and a steady entanglement can be obtained whenq exceeds the critical value. This means that we can real- ize the preservation and control of entanglement by ad- justing the coupling strength between the environments in the dimer system. The results of this paper may be practically useful for further study of the controllable quantum device, entanglement preservation [17–22], the quantum information of the biological systems, and more complex environmental models.

Acknowledgements

The authors are grateful to Prof. S. J. Wang, Prof.

X. Y. Kuang, and Prof. S. Y. Wu for valuable sugges- tions. The work was supported by the Fundamental Re- search Funds for the Central Universities under Grants No. ZYGX2011J046.

Appendix

In this appendix, we did not only give the parameters A,D,G, and f_ik(t)in (11) and (15), respectively, but also the main process about the parameters calculation method is been given. Considering the commutation relations ofH_d⁰

1,H_d⁰

2,∑i=1,2αiS^z_i, andqS^z₁S^z₂, the main process has the following steps.

The eigenvalues ofH_d0 1+H_d0

2:

∆1,p=∆1+z_1,p, ∆2,p=∆2+z_2,p,

∆3,p=∆3+z_3,p, ∆4,p=∆4+z_4,p, (p=0,1) z_1,0=

ψ^`_B(0)

γ₁S^z₁+γ₂S^z₂ 2

ψ^`_B(0) , z_2,0=

ψ^`_B(0)

γ₂S^z₂−γ₁S^z₁ 2

ψ^`_B(0) , z_3,0=

ψ^`_B(0)

γ₃S^z₁+γ₄S^z₂ 2

ψ^`_B(0) , z4,0=

ψ^`_B(0)

γ₄S^z₂−γ₃S^z₁ 2

ψ^`_B(0) , z_1,1=γ₁m₁+γ₂m₂

2 , z_2,1=−γ₁m₁−γ₂m₂

2 ,

z3,1=γ₃m1+γ₄m2

2 , z4,1=−γ₃m1−γ₄m2

2 ,

E1,p=∆1,p+∆3,p+ q

(∆_2,p)²+J₁²+ q

(∆_4,p)²+J₂²,

E2,p=∆1,p+∆3,p+ q

(∆_2,p)²+J₁²−q

(∆_4,p)²+J₂²,

E3,p=∆1,p+∆3,p−q

(∆_2,p)²+J₁²+ q

(∆_4,p)²+J₂²,

E4,p=∆1,p+∆3,p−q

(∆_2,p)²+J₁²−q

(∆_4,p)²+J₂².

The conversion factors between the states|ii|ki(i= 1,2;k=3,4)and the eigenvectors ofH_d0

1+H_d0 2: M_1,p=1

4

J₁J₂ q

(∆_2,p)²+J₁² q

(∆_4,p)²+J₂² ,

M_2,p=1 4



1− ∆2,p

q

(∆_2,p)²+J₁²









J₂ q

(∆_4,p)²+J₂²



,

M_3,p=1 4



1+ ∆2,p

q

(∆_2,p)²+J₁²









J₂ q

(∆_4,p)²+J₂²



,

M4,p=1 4





J₁ q

(∆_2,p)²+J₁²







1− ∆4,p

q

(∆_4,p)²+J₂²



,

M5,p=1 4





J1

q

(∆_2,p)²+J₁²







1+ ∆4,p

q

(∆_4,p)²+J₂²



,

M6,p=1 4



1− ∆2,p

q

(∆_2,p)²+J₁²







1− ∆4,p

q

(∆_4,p)²+J₂²



,

M7,p=1 4



1+ ∆2,p

q

(∆_2,p)²+J₁²







1− ∆4,p

q

(∆_4,p)²+J₂²



,

M_8,p=1 4



1− ∆2,p

q

(∆_2,p)²+J₁²







1+ ∆4,p

q

(∆_4,p)²+J₂²



,

M_9,p=1 4



1+ ∆_2,p q

(∆_2,p)²+J₁²







1+ ∆_4,p q

(∆_4,p)²+J₂²



,

Q_1,p=

∆2,p+ q

(∆_2,p)²+J₁²

J₁ ,

Q_2,p=

∆2,p−q

(∆_2,p)²+J₁² J1

,

Q3,p=

∆4,p+q

(∆_4,p)²+J₂²

J₂ ,

Q4,p=∆4,p−q

(∆_4,p)²+J₂²

J₂ .

The expression of the parameters f_ik(t):

R₁=

a₁₃M1,0+a₂₃M2,0+a₁₄M4,0+a₂₄M_6,0 , R₂=

−a₁₃M_1,0−a₂₃M_2,0+a₁₄M_5,0+a₂₄M_8,0 , R3=

−a13M1,0+a23M3,0−a14M4,0+a24M7,0 ,

R4=

a13M1,0−a23M3,0−a14M5,0+a24M9,0 , f₁₃(t) =R₁Q_1,0Q_3,0e^−iE1,0t+R₂Q_1,0Q_4,0e^−iE2,0t

+R₃Q2,0Q3,0e^−iE3,0t+R₄Q2,0Q4,0e^−iE4,0t, f₂₃(t) =R₁Q_3,0e^−iE1,0t+R₂Q_4,0e^−iE2,0t

+R₃Q3,0e^−iE3,0t+R4Q4,0e^−iE4,0t, f₁₄(t) =R₁Q_1,0e^−iE1,0t+R₂Q_1,0e^−iE2,0t

+R₃Q2,0e^−iE3,0t+R4Q2,0e^−iE4,0t, f₂₄(t) =R₁e^−iE1,0t+R₂e^−iE2,0t

+R₃e^−iE3,0t+R4e^−iE4,0t.

The expression of the parametersA,D, andG:

(A)^†= h3|h1| h3|h2| h4|h1| h4|h2|

G=









a₁₃a^∗₁₃ a₂₃a^∗₁₃ a₁₄a^∗₁₃ a₂₄a^∗₁₃ a₁₃a^∗₂₃ a₂₃a^∗₂₃ a₁₄a^∗₂₃ a₂₄a^∗₂₃ a13a^∗₁₄ a23a^∗₁₄ a14a^∗₁₄ a24a^∗₁₄ a₁₃a^∗₂₄ a₂₃a^∗₂₄ a₁₄a^∗₂₄ a₂₄a^∗₂₄







 ,

M=









M1,1 −M_1,1 −M_1,1 M1,1

M_2,1 −M_2,1 M_3,1 −M_4,1 M4,1 M_5,1 −M4,1 −M_5,1 M_6,1 M_8,1 M_7,1 M_9,1







 ,

B=









e^−iE1,1t 0 0 0

0 e^−iE2,1t 0 0

0 0 e^−iE3,1t 0

0 0 0 e^−iE4,1t







 ,

Q=









Q1,1Q3,1 Q3,1 Q1,1 1 Q_1,1Q_4,1 Q_4,1 Q_1,1 1 Q_2,1Q_3,1 Q_3,1 Q_2,1 1 Q2,1Q4,1 Q4,1 Q2,1 1







 ,

D= (Q)^†B^†(M)^†GMBQ.

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