Dynamical Symmetry Method Investigates the Dissipation and Decoherence of the two-level Jaynes–Cummings Model

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Dynamical Symmetry Method Investigates the Dissipation and Decoherence of the two-level Jaynes–Cummings Model

Qin-Sheng Zhu^a, Wei Lai^b, and Da-Lin Wu^b

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

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

Reprint requests to Z. Q.-S.; E-mail:zhuqinsheng@gmail.com Z. Naturforsch.67a,559 – 566 (2012) / DOI: 10.5560/ZNA.2012-0058

Received January 10, 2012 / revised May 17, 2012 / published online August 20, 2012

Based on the quantum master equation which describes the dynamical evolution of a two-level atom interacting with one mode of the quantized photon field in a cavity, we have investigated the systemic dissipative and decoherence properties by use of the development algebraic dynamic method for the automatic and non-automatic condition. Furthermore, using the exact solution of the quantum master equation, we have investigated the asymptotic behaviours of the system.

Key words:Algebraic Dynamics; Master Equation; Jaynes–Cummings Model.

PACS numbers:03.65.Fd; 03.65.Yz; 42.50.Lc

1. Introduction

The Jaynes–Cummings (JC) model [1] is the fundamental model for the quantum description of the one- photon exchange process between the two-level atom and the mono-mode quantum radiation field in the rotating-wave approximation (RWA). This model has been extensively studied in the past three decades [2, 3]. Based on this model, many purely quantum ef- fects have been predicted, such as Rabi oscillations and collapses and revivals of the atomic inversion operator, have been observed in the experiments both with micro-cavities [4,5] and with trapped ion systems [6].

However, for a realistic description about the JC systems in cavity quantum electrodynamics, it must be taken into account the dissipation of the cavity losses and incoherent decay mechanisms for an atom. These processes have been described by a master equation of the structure in the interaction picture [7]:

∂ ρ_I

∂t =Lˆ_aρ_I+Lˆ_σρ_I, (1) whereρIdenotes the density matrix of the atom–cavity system (put}=1).

The operator ˆLadescribes the coupling of the field to a thermal reservoir at a temperatureT:

LˆaρI=−A

2(ν+1)[a^†aρI+ρ_Ia^†a−2aρ_Ia^†]

−A

2ν[aa^†ρI+ρ_Iaa^†−2a^†ρIa]

(2)

with constantsA,ν ≥0; it describes the coupling of the field to a thermal reservoir at a temperature which corresponds to a mean numberνof thermal photons in the cavity or a laser.

The operator ˆL_σ describes incoherent pumping and decay processes of the atom at a temperatureT: Lˆ_σρI=−B

8(1−s₁)[σˆ+σˆ−ρI+ρIσˆ+σˆ−−2 ˆσ−ρIσˆ+]

−B

8s₁[σˆ−σˆ+ρ_I+ρ_Iσˆ−σˆ+−2 ˆσ+ρ_Iσˆ−]

−2C−B

8 [σˆ_z,[σˆ_z,ρ_I]]

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with constants 2C≥B≥0 and 0≤s₁≤1; it models incoherent pumping and decay processes of the atom.

Using the relationρ ≡exp[−i ˆH_JCt]ρ_Iexp[i ˆH_JCt], we can obtain the density matrix of the Schr¨odinger picture [8] (the ˆH_JC is the Jaynes–Cummings Hamilto- nian).

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In our present work, by analyzing the quantum master equation, we found that the common characteris- tic of this equation is the existence of sandwich terms of the Liouville operators. This sandwich structure of the quantum master equation is treated by the so-called thermal Lie algebra [9–15] in thermal field theory.

Based on dynamical algebraic structures of the master equations, we have solved the equations by use of a novel algebraic dynamical method [12–15]. This novel algebraic dynamical method is just a general- ization [16–18] and development of the algebraic dynamical method and extends the algebraic dynamical method from quantum mechanical systems to quantum statistical systems. Using the adjoint representa- tion of the dynamical algebraic, namely the right and left representations [16–18], we can find the Lie algebra structure of the master equation and treat it, such as the sympathetic cooling of the Bose–Einstein conden- sate system in the mean field approximation [19], the dissipative two-level system [20], and so on [21,22].

In this paper, we apply this novel algebraic dynamical method to solve the quantum master equation which describes the dynamical evolution of a two- level atom interacting with one mode of the quantized photon field in a cavity, and to study the dissipative and decoherence properties for the non-autonomous case.

This paper is organized as follows: In Section2, by introducing the new composite algebras, we found theSU(1,1)⊕SU(2)⊕U(1)algebraic structure of the master equation which is constructed from the right and left adjoint representations. In Section3, using the algebraic dynamical method, we resolved the quantum master equation for the autonomous and the non- autonomous case. In Section4, we analyzed the dissipation and decoherence of the system by use of the analytic solution of the quantum master equation. Con- clusions are given in the final section.

2. The Algebraic Structure and Dynamical Symmetry of the Master Equation

Following the idea of [16–19], we introduce the right and left algebras and define the density matrix ρIas a super vector in the von Neumann space:

ρI=

∑

n,n⁰,s,s⁰

A^s,s_n,n⁰0|n,si s⁰,n⁰

, (4)

The operatorsa^†(σˆ+)anda(σˆ−)can operate on the right ket state |ni(|si) and the left ket state hn|(hs|), which form the right and left representations of the hω(4)(SU(2))algebra as follow:

hω(4)_R={n^r=a^r†a^r,a^r†,a^r,1}, hω(4)_L={n^l=a^l†a^l,a^l†,a^l,1}, SU(2)_R={σˆ_z^r,σˆ₊^r,σˆ₋^r}, SU(2)_L={σˆ_z^l,σˆ₊^l,σˆ₋^l}.

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They satisfy the commutation rules respectively as follows:

hω(4)_R:[n^ra^r†] =a^r†,[n^r,a^r] =−a^r,[a^r,a^r†] =1, hω(4)_L:[n^l,a^l†] =−a^l†,[n^l,a^l] =a^l,[a^l,a^l†] =−1, SU(2)_R:[σˆ_z^r,σˆ_±^r] =±2 ˆσ_±^r,[σˆ₊^r,σˆ₋^r] =σˆ_z^r,

SU(2)_L:[σˆ_z^l,σˆ±^l] =∓2 ˆσ±^l,[σˆ₊^l,σˆ−^l] =−σˆ_z^l.

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From the group theory, we know: (i) hω(4)_Ris isomorphic tohω(4), whilehω(4)_Lis anti-isomorphic to hω(4). Because hω(4)_R andhω(4)_L operate on different spaces (the right ket state |niand the left ket statehn|), they commute each other. (ii)SU(2)_Ris isomorphic to SU(2), while SU(2)_L is anti-isomorphic to SU(2). Because SU(2)_R and SU(2)_L operate on different spaces (the right ket state |si and the left ket statehs|), they commute each other. (iii) Because hω(4)_R(hω(4)_L)andSU(2)_R(SU(2)_L)operate on the spaces of the Fock state (|niandhn|) and the state of the two-level atom (|siandhs|), respectively, they commute each other:

[hω(4)_R,hω(4)_L] =0, [SU(2)_R,SU(2)_L] =0, [hω(4)_R,SU(2)_R] =0, [hω(4)_L,SU(2)_R] =0, [hω(4)_R,SU(2)_L] =0, [hω(4)_L,SU(2)_L] =0.

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Using the left and right algebras in (5), we can con- struct the composite algebras as

SU(1,1):

Kˆ₀=(n^r+n^l)

2 ,Kˆ₊=a^r†a^l,Kˆ−=a^ra^l†, SU(2):

Jˆ0=σˆ_z^r+σˆ_z^l

2 ,Jˆ+=σˆ₊^rσˆ₋^l,Jˆ−=σˆ₋^rσˆ₊^l , U(1):

Uˆ0=σˆ_z^r−σˆ_z^l

2 .

(8)

(3)

According to (6) – (8), it is easy to obtain the following commutation relations:

[Kˆ₀,Kˆ±] =±Kˆ±, [Kˆ₊,Kˆ−] =2 ˆK₀, [Kˆ_i,Jˆ_j] =0 (i,j=0,±),

[Kˆ_i,Uˆ0] =0 (i=0,±),

[Jˆ₀,Jˆ±] =±2 ˆJ±, [Jˆ₊,Jˆ−] =Jˆ₀, [Uˆ₀,Jˆ_i] =0 (i∈(0,1,±)).

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The action of the composite algebras on the bases of von Neumann space is

Kˆ0|n,si hs⁰,n⁰|=(n+n⁰+1) 2 |n,si

s⁰,n⁰ , Kˆ₊|n,si hs⁰,n⁰|=p

(n+1)(n⁰+1)|n,si hs⁰,n⁰+1|, Kˆ−|n,si hs⁰,n⁰|=√

nn⁰|n,si hs⁰,n⁰−1|, Jˆ₀|n,si hs⁰,n⁰|=s+s⁰

2 |n,si s⁰,n⁰

,

s⁰,n⁰

=s−s⁰ 2 |n,si

s⁰,n⁰

, (10) wheresands⁰are equal to−1 or+1.

Noticing the identities σˆ+σˆ−=1+σˆz

2 , σˆ−σˆ+=1−σˆz

2 , (11)

and the composite algebra equation (8), the quantum master equation can be rewritten as

∂ ρ_I

∂t =Γ ρˆ _I, (12)

where ˆΓ is the rate operator Γˆ=

AνKˆ₊+A(ν+1)Kˆ−−A(2ν+1)Kˆ₀+A 2

+ B

4s1Jˆ++B

4(1−s₁)Jˆ−−B

8(1−2s₁)Jˆ0−B 8

−2C−B 2

Uˆ₀². (13)

In (12), the rate operator ˆΓ plays the role of the Hamil- tonian, and the reduced density matrix ρI plays the role of the wave function. Because the rate operator Γˆ has SU(1,1)⊕SU(2)⊕U(1) dynamical symmetry, it can be solved by use of the algebraic dynamics method [12–15].

3. The Exact Solution of the Quantum Master Equation

3.1. The Eigensolutions of the Autonomous Case In order to better study the prosperities of the non- autonomous quantum master equation, we firstly study the autonomous case, namelyA,ν,B,s, andCare not dependent on time.

Equation (12) can be solved by use of a gauge transformation

Uˆ =exp[χ₊Kˆ+]exp[χ₋Kˆ−]exp[α₊Jˆ+]

·exp[α−Jˆ−],

Uˆ⁻¹=exp[−χ−Kˆ−]exp[−χ+Kˆ+]exp[−α−Jˆ−]

·exp[−α₊Jˆ+],

(14)

where χ± and α± are parameters and time- independent. After some calculations, (14) can be rewritten as the following eigenequation:

Γˆ¯ρ¯_I(n,n⁰,s,s⁰) =β(n,n⁰,s,s⁰)ρ¯_I(n,n⁰,s,s⁰), (15) where

Γˆ¯=Uˆ⁻¹ΓˆUˆ =

[2A(ν+1)χ₊−A(2ν+1)]Kˆ0+A 2

− B

8 +2C−B 2 Uˆ₀²

+ B

4s₁α−−B

4(1−s₁)(α₊+α₊²α−)

−B

8(1−2s₁)(1+2α₊α−)

Jˆ0

ρI(n,n⁰,s,s⁰) =Uˆρ¯I(n,n⁰,s,s⁰).

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β(n,n⁰,s,s⁰) and ρ_I(n,n⁰,s,s⁰) are the eigenvalues and eigenstates of the rate operator ˆΓ, respectively.

(n,n⁰,s,s⁰) label the eigenstates in the von Newman space. From (16), we know that the operator ˆ¯Γ is a combination of the commuting invariant operators Kˆ0, ˆU0, and ˆJ0.

The gauge transformation satisfy the following conditions:

Aν+A(ν+1)χ₊²−A(2ν+1)χ₊=0,

A(ν+1)(1−2χ₊χ−) +A(2ν+1)χ−=0, (17) B

4s₁−B

4(1−s₁)α₊²−B

4(1−2s₁)α₊=0,

−B

4s₁α−²+B

4(1−s₁)(α₊α−+1)² +B

4(1−2s1)(α₋+α₊α₋²) =0.

(18)

(4)

Using (12), (15), (17), and (18), we can obtain four sets of solutions:

χ₊= ν

ν+1, χ−=−(ν+1), α+=−1, α−=1−s₁, β¹(n,n⁰,s,s⁰) =−n+n⁰

2 A+B 8

s+s⁰ 2

− (B

8 +2C−B 2

s−s⁰ 2

2)

;

(19a)

χ+= ν

ν+1, χ−=−(ν+1), α+=_1−s^s¹

1, α−=−(1−s₁), β²(n,n⁰,s,s⁰) =−n+n⁰

2 A−B 8

s+s⁰ 2

− (B

8 +2C−B 2

s−s⁰ 2

2)

;

(19b)

χ+=1, χ−= (ν+1), α+=−1, α−=1−s1, β³(n,n⁰,s,s⁰) =n+n⁰

2 A+A+B 8

s+s⁰ 2

− (B

8 +2C−B 2

s−s⁰ 2

2)

;

(19c)

χ+=1, χ−= (ν+1), α+= s₁

1−s1

, α−=−(1−s₁), β²(n,n⁰,s,s⁰) =n+n⁰

2 A+A−B 8

s+s⁰ 2

− (B

8 +2C−B 2

s−s⁰ 2

2) .

(19d)

Observing the above solutions, it is surprising that why four similarity gauge transformation lead to four sets of eigenvalues of the diagonalization of the same rate operator. The surprising results stem from the following two aspects:

(i) The solution (19c) and (19d) is unphysical because it does not contain the zero eigenvalue corresponding to the stationary solution (the equilibrium state of the system) to ˆL_aand the positive ofⁿ⁺ⁿ₂⁰A+A make the solution divergent. On the other hand, the solution (19a) and (19b) is the needed physical solution because it has a unique state with the zero eigenvalue corresponding to the equilibrium state of ˆLa and the other nonzero−ⁿ⁺ⁿ₂⁰Aare all negative, which guaran- tee that the system approaches its equilibrium physical state irrespectively of its initial state.

(ii) For the solution (19a) and (19b), the surprising results stem from the structure of the rate operator

Lˆσ: (a) It contains a vector (from the fifths to the sev- enth terms of (13)) in the linear space spanned by the SU(2)generators ( ˆJ+, ˆJ−, and ˆJ₀), and we can choose two gauge transformations to rotate this vector along the ˆJ₀ and −Jˆ₀ directions. (b) −^B₈ is a scalar in the SU(2)space and make the above diagonalzing transformations asymmetric.

However, as inAppendix Ashowed, after returning to the physical frame, the two sets of eigensolutions coincide.

3.2. The Eigensolutions of the Rate OperatorsΓˆin the Non-Autonomous Case

After obtaining the solutions of the autonomous quantum master equation, we can further study the solutions of the non-autonomous case, namelyA,ν,B,s, andCare dependent on time.

For (12), we also introduce the time-dependent gauge transformation

Uˆ_g=exp[χ₊(t)Kˆ₊]exp[χ−(t)Kˆ−]exp[α₊(t)Jˆ₊]

·exp[α−(t)Jˆ−],

Uˆ_g⁻¹=exp[−χ−(t)Kˆ−]exp[−χ₊(t)Kˆ₊]

·exp[−α₋(t)Jˆ−]exp[−α₊(t)Jˆ+].

(20)

Under the gauge conditions dχ+(t)

dt =Aν+A(ν+1)χ₊²(t)−A(2ν+1)χ₊(t), dχ−(t)

dt =A(ν+1)(1−2χ₊(t)χ−(t)) +A(2ν+1)χ−(t),

dα+(t)

dt=−B

4(1−s₁)[α₊(t) +1]

·

α+(t)− s₁ 1−s1

, dα−(t)

dt=B

4(1−s₁)[1+2α+(t)α₋(t)]

+B

4(1−2s₁)α−(t),

(21)

the transformed operator ˆΓ becomes simple and inte- gral:

∂ρ¯_I

∂t =Γˆ¯ρ¯I,

(5)

Γˆ¯=Uˆ_g⁻¹ΓUˆ_g−Uˆ_g⁻¹d ˆU_g dt

=

[λ(t)Kˆ₀+A 2]

+

−B

8(1−2s1)

2(1−s1)

(1−2s₁)α+(t) +1

Jˆ0−B 8

−2C−B 2

Uˆ₀²,

(22)

whereλ(t) =2A(ν+1)χ+(t)−A(2ν+1).

The equations (22) have the solution ρ¯I(t) =exp

Z t 0

Γˆ¯(τ)dτ

ρ¯(0). (23) Under the initial conditions χ±(0) =α±(0) =0 and ρ(0) =ρ(0) =¯ ∑nn0

ss0 pⁿⁿ_ss0⁰|n,si hs⁰,n⁰|, we obtain the solution

ρI(t) =Uˆ_gexp Z _t

0

Γˆ¯(τ)dτ

ρ(0) (24)

=

∑

nn0 ss0

pⁿⁿ_ss0⁰exp{φ(t)}Uˆ_g|in,s s⁰,n⁰

,

φ(t) = Z _t

0

λ(τ)n+n⁰+1

2 +A

2

+

−B

8(1−2s₁)

2(1−s₁)

(1−2s₁)α+(τ) +1 s+s⁰

2 −B 8

−2C−B 2

s−s⁰ 2

2# dτ.

4. The Asymptotic Behaviour of the Analytic Solution for the Non-Autonomous Master Equation

Because the JC model plays an important role in the study of quantum information, our interest mainly fo- cuses on the asymptotically behaviour of the solutions and whether the solutions approaches the steady solutions or not.

Noticing (9), we can rewrite (24) as ρ_I(t) =

∑

nn⁰ ss⁰

pⁿⁿ_ss0⁰ρⁿⁿ

0

I (t)ρ_I^ss⁰(t), (25)

where ρⁿⁿ

0

I (t) =η₁exp[χ+(t)Kˆ+]exp[χ−(t)Kˆ−]|ni n⁰

,

ρ^ss

0

I (t) =η2exp[α₊(t)Jˆ+]exp[α₋(t)Jˆ−]|si s⁰

, η1=exp

_Z _t

0

λ(τ)n+n⁰+1

2 +A

2

dτ

,

η2=exp Z _t

0

−B

8(1−2s₁)

2(1−s₁)

(1−2s₁)α+(τ) +1

·s+s⁰ 2 −B

8

−2C−B 2

s−s⁰ 2

2# dτ

) .

In order to investigate the asymptotically behaviours of the solutions, we can study the asymptotically behaviour forρ_Iⁿⁿ⁰(t)andρ_I^ss⁰(t), respectively.

Firstly, usingAppendix B, we obtained

ρⁿⁿ

0

I (t) =exp[χ₊(t)Kˆ+]exp



 1 2 Zt

0

dτ(λ(τ) +A)





·Re(t)ⁿ⁺ⁿ

0 2

n+n0 2

∑

q=

n−n0 2

χ−(t)^−q

·

n(n−1). . .

q+n−n⁰ 2 +1

n⁰(n⁰−1). . .

·

q−n−n⁰ 2 +1

¹₂ n+n⁰

2 −q

! −1

·

q+n−n⁰

2 q−n−n⁰ 2

,

(26)

and only ifn=n⁰ andq=0,ρ_Iⁿⁿ⁰(t)does not vanish fort→∞. So we further obtained

ρⁿⁿ

0

I (t)

_t→∞→exp[ξKˆ+]exp



 1 2

∞ Z

0

dτ(λ(τ) +A)





·Reⁿ₀δ_nn⁰|0i h0|=Cexp[ξKˆ₊]δ_nn0|0i h0|. (27) Because trρI(t) and the mean value of the number operator remain finite for t → ∞, exp[¹₂^R₀^∞dτ(λ(τ) +A)] must be bounded and C= exp[¹₂^R₀^∞dτ(λ(τ) +A)]Reⁿ₀is a constant.

Secondly, for the asymptotically behaviour of ρ^ss

0

I (t), we are only interested in the zero-mode eigensolutions of ˆL_σ, namely the parametersα+andα−of the steady solutions obeying

dα₊

dt =0, dα−

dt =0. (28)

(6)

Using the results ofAppendix C, we obtained the following results:

ρ_I^1,−1(t)

_t→∞=e{^R₀^t[⁻^B₈⁻^2C−B₂ ]dτ}e^[α⁺^(t)^J^ˆ⁺^]

·e^[α⁻^(t)^J^ˆ⁻^]|1i h−1|=e{^R0^t[⁻^B₈⁻^2C−B₂ ]dτ} |1i h−1| →0, ρ_I^−1,1(t)

_t→∞=e{^R0^t[⁻^B₈⁻^2C−B₂ ]dτ}e^[α⁺^(t)^J^ˆ⁺^]

·e^[α⁻^(t)^J^ˆ⁻^]|−1i h1|=e{^R0^t[−^B₈−^2C−B₂ ]dτ} |−1i h1| →0, ρ_I^1,1(t)

_t→∞=e

nR_t 0

n

−^B₈(1−2s₁)h_2(1−s

1)

(1−2s1)α+(τ)+1i

−^B₈o dτo

·e^[α⁺^(t)^J^ˆ⁺^]e^[α⁻^(t)^J^ˆ⁻^]|1i h1|

=e

nR_t 0

n−^B₈(1−2s₁)h_2(1−s

1) (1−2s1)α+(τ)+1i

−^B₈o dτo

·[|1i h1|+α−(t) (|i h−1|+α+(t)|1i h1|)]

→const×

|−1i h−1|+ s₁

(1−s₁)|1i h1|

,

ρ_I^−1,−1(t) =e

nR_t 0

nB

8(1−2s₁)h_2(1−s

1) (1−2s1)α+(τ)+1i

−^B₈o dτo

·e^[α⁺^(t)^J^ˆ⁺^]e^[α⁻^(t)^J^ˆ⁻^]|i h−1|

=e

nR_t 0

nB

8(1−2s₁)h_2(1−s

1)

(1−2s1)α+(τ)+1i

−^B₈o dτo

·[|−1i h−1|+α₊(t)|1i h1|]

→const×

|−1i h−1|+ s₁

(1−s₁)|1i h1|

.

(29)

In order to obtain above derivation, we used 2C ≥ B ≥ 0, 0 ≤ s₁ ≤ 1 and noticed

e⁻

R_t 0B

4(1−2s₁)[_1−2s^1−s¹

1α+(τ)+_1−2s^s¹

1]dτ

|t→∞ →1, which can be proved by (21).

Finally, by use of (25), (27), and (29), we can obtain the asymptotically behaviour ofρI(t)for the time t→∞(together with pⁿⁿ_ss, the constantCof (27), and the constant of (29), we denoted the product of these constants byDⁿⁿ_ss)

ρI(t)|_t→∞=

∑

n,s

Dⁿⁿ_ssexp[ξKˆ₊](|0i h0|)

·

|−1i h−1| + s₁

(1−s1)|1i h1|

.

(30)

Notice 0<ξ = ^ν(∞)

ν(∞)+1 <1, so we can expand the exp[ξKˆ₊]under some precision.

The above solution indicates that: (i) For random initial condition and long time evolution,ρI(t)asymptotically approaches the steady solutions and the non- diagonal elements ofρI(t)disappear. (ii) For some initial condition, the entanglement state which is formed

by spin does not lose its coherence even for long time evolution.

5. Conclusion

In this paper, we investigated dissipative and decoherence properties of a two-level atom interacting with one mode of the quantized photon field in a cavity and proved the integrability of the non-autonomous quantum master equation of this system. We did not only find the dynamicalSU(1,1)⊕SU(2)⊕U(1)algebraic structure of the quantum master equation by use of left and right algebras, but also obtained the exact solutions of the quantum master equation for the autonomous and the non-autonomous case by use of the algebraic dynamical method. Finally, for the non-autonomous case, the dissipative and decoherence properties of the system are investigated under the long time evolvement which is related to the quantum information, and found the solutions of the non-autonomous quantum master equation asymptotically approaching the steady solutions for a random initial condition. Be- cause the master equation plays an important role for the investigation of dissipative and decoherence problems and have some dynamical algebraic structure, this method [12,17–22] is a useful tool to treat dissipative and decoherence problems in quantum statistical physics. The results of this paper may be practi- cally useful for the analysis of the decoherence of two- level atom systems and the study of quantum information.

Appendix A

In this appendix, we prove that the solution (19a) and (19b) coincide under the physical frame.

Using (9) and (10), we can obtain the eigensolutions of the rate operator by use of (16):

ρ_I(n,n⁰,s,s⁰) =Uˆρ¯_I(n,n⁰,s,s⁰) =ρ_I(n,n⁰)ρ_I(s,s⁰);

ρI(n,n⁰) =exp[χ₊Kˆ₊]exp[χ−Kˆ−]|ni n⁰

, ρI(s,s⁰) =exp[α₊Jˆ₊]exp[α−Jˆ−]|si

s⁰

= (1+α₊Jˆ₊)(1+α−Jˆ−)|si s⁰

. After some calculation, we obtain

β₁¹(n,n⁰,−1,−1) =β₂²(n,n⁰,1,1) =−n+n⁰ 2 A−B

4,

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ρ₁¹(n,n⁰,−1,−1) =ρ₂²(n,n⁰,1,1)

=ρ(n,n⁰)(|−1i h−1| − |1i h1|), β₁²(n,n⁰,1,1) =β₂¹(n,n⁰,−1,−1) =−n+n⁰

2 A, ρ₁²(n,n⁰,1,1) =ρ₂¹(n,n⁰,−1,−1)

=ρ(n,n⁰){(1−s₁)|−1ih−1|+s₁|1ih1|}, β₁³(n,n⁰,1,−1) =β₂³(n,n⁰,1,−1)

=−n+n⁰

2 A+3B−8C

8 ,

ρ₁³(n,n⁰,1,−1) =ρ₂³(n,n⁰,1,−1) =ρ(n,n⁰)(|1i h−1|) β₁⁴(−1,1) =β₂⁴(−1,1) =−n+n⁰

2 A+3B−8C

8 ,

ρ₁⁴(−1,1) =ρ₂⁴(−1,1) =ρ(n,n⁰)(|−1i h1|).

Here β₁ⁱ(n,n⁰,s,s⁰) and ρ₁ⁱ(n,n⁰,s,s⁰) (i∈(1,2,3,4)) denote the eigenvalues and eigenvectors of solution (19a), respectively.β₂ⁱ(n,n⁰,s,s⁰)andρ₂ⁱ(n,n⁰,s,s⁰)(i∈ (1,2,3,4)) denote the eigenvalues and eigenvectors of solution (19b), respectively. The second term of the above solutions is the zero-mode solution corresponding to the steady state forn=n⁰=0.

Appendix B

In this appendix, we investigate the asymptotically behaviour ofχ±(t)by use of (21).

(i) The asymptotically behaviour ofχ+(t).

Notice A>0 and ν >0, we found _ν+1^ν <1 for all timest. Then, dχ₊(t)

dt>0{<0} if 0<χ+<

ν

ν+1{_ν+1^ν <χ+<1}. We see thatχ+(t)asymptotically approaches the value _ν(∞)+1^ν(∞) =ξ <1from below, and λ(∞) =−A<0 with the initial conditionχ+(0) =0.

(ii) The asymptotically behaviour ofχ−(t).

To study the asymptotically behaviour of χ−(t), we define Re(t) =χ−(t)exp[^R₀^tλ(τ)dτ] and found

dRe(t)

dt=A(ν+1)exp[^R₀^tλ(τ)dτ]. BecauseA(ν+ 1) is bounded and λ(t) negative for large t, we know that dRe(t)

dt tended to zero and Re(t) towards a constant Re₀. This implies thatχ−(t)diverges asymptotically, namelyχ−(∞) =∞.

Appendix C

Observing (28), we found that the solutions of equation (28) have the two same sets of solutions as given in (19a). So we must study the behaviours of the two

sets of solutions of α+ and α− with respect to their time evolution.

It is found from (21) that

(i) Ifα+=−1−ε, then ^dα_dt⁺^(t) <0, andα+ will move further away from−1 towards the negative direction; whereasα+=−1+ε, then ^dα_dt⁺^(t)>0 andα+

will move further away from−1 towards the positive direction. Thus the steady solutionα+=−1 is unsta- ble. Becauseα−(t)depend onα+(t), it is also unsta- ble. Therefore the solution of (19a) and (19c) are un- stable and can not be reached from the initial condition α+(0) =α−(0) =0.

(ii) If α+= _1−s^s¹

1 −ε, then ^dα_dt⁺^(t) >0; if α+=

s₁

1−s₁ +ε, then ^dα_dt⁺^(t) <0. Thus the steady solution α+= _1−s^s¹

1 is stable. We know that α+(t)asymptotically approaches the valueα+(∞) = _1−s^s¹

1 under the initial conditionα+(0) =0, butα−(t)can not reach its steady value−(1−s₁). In order to study the behaviour of the solutions, we need to define the function y(t) =α−(t)exp

− Z _t

0

B

4(1−s₁)(α₊(τ) +1)dτ

.

The time differential function ofy(t):

dy(t) dt =bexp

− Z t

0

B

4(1−s₁)(α+(τ) +1)dτ

, b= dα−(t)

dt +α−(t)

−B

4(1−s₁)(α₊(τ) +1)

. Because b → ^B₄(1−s₁) is bounded and [−^B₄(1− s1)(α₊(τ) +1)]is negative for larget,dy(t)

dt tends to zero andy(t)towards a constant. This implies that α−(t)diverges asymptotically. So we obtain

α+(∞) = s₁ 1−s₁, α−(t)exp

− Z t

0

B

4(1−s₁)(α₊(τ) +1)dτ

_t→∞

=const.

Acknowledgements

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

X. Y. Kuang, Prof. C. J. Fu, and Prof. S. Y. Wu for valuable suggestions. The work was supported by the Fundamental Research Funds for the Central Universi- ties under Grants No. ZYGX2011J046.

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