Download Non-Markovian entanglement dynamics between two coupled

arXiv:0910.5793v1 [quant-ph] 30 Oct 2009
Non-Markovian entanglement dynamics between
two coupled qubits in the same environment
Wei Cui1,2 , Zairong Xi1∗ and Yu Pan1,2
1
Key Laboratory of Systems and Control, Institute of Systems Science, Academy of
Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190,
People’s Republic of China
2
Graduate University of Chinese Academy of Sciences, Beijing 100039, People’s
Republic of China
E-mail: [email protected]
Abstract. We analyze the dynamics of the entanglement in two independent nonMarkovian channels. In particular, we focus on the entanglement dynamics as a
function of the initial states and the channel parameters like the temperature and
the ratio r between ω0 the characteristic frequency of the quantum system of interest,
and ωc the cut-off frequency of Ohmic reservoir. We give a stationary analysis of the
concurrence and find that the dynamic of non-markovian entanglement concurrence
Cρ (t) at temperature kB T = 0 is different from the kB T > 0 case. We find that
“entanglement sudden death” (ESD) depends on the initial state when kB T = 0,
otherwise the concurrence always disappear at finite time when kB T > 0, which
means that ESD must happen. The main result of this paper is that the nonMarkovian entanglement dynamic is fundamentally different from the Markovian
one. In the Markovian channel, entanglement decays exponentially and vanishes only
asymptotically, but in the non-Markovian channel the concurrence Cρ (t) oscillates,
especially in the high temperature case. Then an open-loop controller adjusted by the
temperature is proposed to control the entanglement and prolong the ESD time.
PACS numbers: 03.65.Ud, 03.65.Yz, 03.67.Mn, 05.40.Ca
Non-Markovian entanglement dynamics between two coupled qubits in the same environment2
1. Introduction
Entanglement is a remarkable feature of quantum mechanics, and its investigation is
both of practical and theoretical significance. It is viewed as a basic resource for quantum
information processing (QIP) [1], like realizing high-speed quantum computation [2] and
high-security quantum communication [3]. It is also a basic issue in understanding the
nature of nonlocality in quantum mechanics [4, 5, 6]. However, a quantum system
used in quantum information processing inevitably interacts with the surrounding
environmental system (or the thermal reservoir), which induces the quantum world into
classical world [7, 14, 21]. Thus, it is an important subject to analyze the entanglement
decay induced by the unavoidable interaction with the environment [8, 9, 10, 11, 12]. In
one-party quantum system, this process is called decoherence [13, 14, 15, 16, 17, 18, 19].
In this paper, we will analyze the entanglement dynamics of bipartite non-Markovian
quantum system. As well known, the system can only couple to a few environmental
degrees of freedom for short times. These will act as memory. In short time scales
environmental memory effects always appear in experiments [20]. The characteristic
time scales become comparable with the reservoir correlation time in various cases,
especially in high-speed communication. Then an exactly analytic description of the
open quantum system dynamic is needed, such as quantum Brownian motion(QBM)
[21], a two-level atom interacting with a thermal reservoir with Lorentzian spectral
density [22], and the devices based on solid state [23] where memory effects are
typically non-negligible. Due to its fundamental importance in quantum information
processing and quantum computation, non-Markovian quantum dissipative systems have
attracted much attention in recent years [7, 24, 25, 26, 27, 28]. Recently, researches
on quantum coherence and entanglement influenced and degraded by the external
environment become more and more popular, most of the works contributed to extend
the open quantum theory beyond the Markovian approximation [29, 30, 31]. In [29],
two harmonic oscillators in the quantum domain were studied and their entanglement
evolution investigated with the influence of thermal environments. In [30], the dynamics
of bipartite Gaussian states in a non-Markovian noisy channel were analyzed. All in
all, non-Markovian features of system-reservoirs interaction have made great progress,
but the theory is far from completion, especially how the non-Markovian environmental
influence the system and what the difference is between Markovian and non-Markovian
system evolution are not clear.
In this paper we will compare the non-Markovian entanglement dynamics with
the Markovian one [32] in Ohmic reservoir with Lorentz-Drude regularization in the
following three conditions: ω0 ≪ ωc , ω0 ≈ ωc and ω0 ≫ ωc , where ω0 is the
characteristic frequency of the quantum system of interest and ωc the cut-off frequency
of Ohmic reservoir. Thus, ωc ≪ ω0 implies that the spectrum of the reservoir does not
completely overlap with the frequency of the system oscillator and ω0 ≫ ωc implies
the converse case. Another point of the entanglement dynamics is the temperature.
We characterize our system by low temperature, kB T = 0.03ω0 , medium temperature,
Non-Markovian entanglement dynamics between two coupled qubits in the same environment3
kB T = 3ω0 , and the high temperature kB T = 300ω0. We give stationary analysis of the
concurrence [9] and find that the dynamics of non-markovian entanglement concurrence
C at temperature kB T = 0 is fundamentally different from the kB T > 0. We find
that “entanglement sudden death” (ESD) depends on the initial state when kB T = 0,
otherwise the concurrence always disappear at finite time when kB T > 0, which means
that ESD must happen. Maniscalco S et.al studied the separability function S(τ ) in
[30], where the entanglement oscillation appears for twin-beam state in non-Markovian
channels for high temperature reservoirs. The main result of this paper is that the
non-Markovian entanglement dynamics is fundamentally different from the Markovian
one. In the Markovian channel, entanglement decays exponentially and vanishes only
asymptotically, but in the non-Markovian channel the concurrence Cρ (t) oscillates,
especially in the high temperature case.
The paper is organized as follows. We first introduce the open quantum system and
the non-Markovian quantum master equation for driven open quantum systems by the
noise and dissipation kernels. In Sec. III we introduce the Wootters’ concurrence and
the initial “X” states. By substituting the initial states into the master equation we get
the first order coupled differential equations, and give the stationary analysis. In Sec.
IV, we numerically analyze the Markovian and non-Markovian entanglement dynamics.
Then an open-loop controller adjusted by the temperature is proposed to control the
entanglement and prolong the ESD time. Conclusions and prospective views are given
in Sec. V.
2. The model
Our system consists of a pair of two-level atoms (two qubits) equally and resonantly,
coupled to a single cavity mode, with the same coupling strength. The master
equation for the reduced density matrix ρ(t) which describes its dynamics is given by
[7, 18, 30, 31, 33]
2
∆(t) + γ(t) X
dρ(t)
{2σj− ρσj+ − σj+ σj− ρ − ρσj+ σj− }
=
dt
2
j=1
2
∆(t) − γ(t) X
{2σj+ ρσj− − σj− σj+ ρ − ρσj− σj+ }.
+
2
j=1
(1)
where σ + = 21 (σ1 + iσ2 ), σ − = 21 (σ1 − iσ2 ), with σ1 , σ2 the Pauli matrices. The time
dependent coefficients appearing in the master equation can be written, to the second
order in the coupling strength, as follows
Rt
∆(t) = 0 dτ k(τ ) cos(ω0 τ ),
Rt
(2)
γ(t) = 0 dτ µ(τ ) sin(ω0 τ ),
with
R∞
k(τ ) = 2 0 dωJ(ω) coth[ω/2kB T ] cos(ωτ ),
R∞
µ(τ ) = 2 0 dωJ(ω) sin(ωτ ),
(3)
Non-Markovian entanglement dynamics between two coupled qubits in the same environment4
being the noise and the dissipation kernels, respectively. This master equation (1)
is valid for arbitrary temperature. The coefficient γ(t) gives rise to a time dependent
damping term, while ∆(t) the diffusive term. The non-Markovian character is contained
in the time-dependent coefficients, which contain all the information about the short
time system-reservoir correlations [7]. In the previous equations J(ω) is the spectral
density characterizing the bath,
π X ki
δ(ω − ωi)
(4)
J(ω) =
2 i mi ωi
and the index i labels the different field mode of the reservoir with frequency ωi . Let
the Ohmic spectral density with a Lorentz-Drude cutoff function,
J(ω) =
ω2
2
ω 2 c 2,
π ωc + ω
(5)
where ω is the frequency of the bath, and ωc is the high-frequency cutoff.
Then the closed analytic expressions for ∆(t) and γ(t) are [18, 31]
ω0 r 2
γ(t) =
[1 − e−rω0 t cos(ω0 t) − re−rω0 t sin(ω0 t)],
2
1+r
∆(t) = ω0
r2
{coth(πr0 ) − cot(πrc )e−ωc t [r cos(ω0 t) − sin(ω0 t)]
1 + r2
1
+
cos(ω0 t)[F̄ (−rc , t) + F̄ (rc , t) − F̄ (ir0 , t) − F̄ (−ir0 , t)]
πr0
1
e−ν1 t
− sin(ω0 t)[
[(r0 − i)Ḡ(−r0 , t) + (r0 + i)Ḡ(r0 , t)]
π
2r0 (1 + r02 )
1
+ [F̄ (−rc , t) − F̄ (rc , t)]]},
2rc
(6)
(7)
where r0 = ω0 /2πkB T , rc = ωc /2πkB T , r = ωc /ω0 , and
F̄ (x, t) ≡2 F1 (x, 1, 1 + x, e−ν1 t ),
(8)
Ḡ(x, t) ≡2 F1 (2, 1 + x, 2 + x, e−ν1 t ).
(9)
ν1 = 2πkB T , and 2 F1 (a, b, c, z) is the hypergeometric function. Note that, for time t
large enough, the coefficients ∆(t) and γ(t) can be approximated by their Markovian
stationary values ∆M = ∆(t → ∞) and γM = γ(t → ∞). From Eqs. (6) and (7) we
have
ω0 r 2
,
(10)
γM =
1 + r2
and
r2
coth(πr0 ).
(11)
∆M = ω0
1 + r2
Note that γ(t) has nothing to do with the temperature [33]. In Fig.1 we plot the time
evolution of non-Markovian coefficients ∆(t) and γ(t) in different channel temperatures.
Non-Markovian entanglement dynamics between two coupled qubits in the same environment5
−4
−3
(a)
x 10
x 10
1.8
(c)
(b)
0.2
2
∆(t)
γ(t)
1.6
∆(t)
γ(t)
1.5
∆(t)
γ(t)
0.15
1.4
1
0.1
0.5
0.05
1
0.8
∆(t),γ(t)
∆(t),γ(t)
∆(t),γ(t)
1.2
0
0.6
0
−0.5
−0.05
−1
−0.1
0.4
0.2
0
0
20
40
ω0t
60
−1.5
0
20
ω0t
40
−0.15
0
50
ω0t
Figure 1. (Color online) Dynamics of non-Markovian coefficients ∆(t) (blue solid line)
and γ(t) (red dotted line) at different temperatures: (a)low temperature kB T = 0.01,
(b)medium temperature kB T = 1, and (c)high temperature kB (t) = 100, respectively.
The other coefficients are chosen as r = 0.1 ω0 = 1, and α2 = 0.01.
In Fig. 1(a), the temperature is kB T = 0.01. There are two important main points
embodied in the Figure, the first is that the coefficient γ(t) has dominated the system
dissipation at low temperature, the other ∆M = γM in the long time limit. Fig. 1(b)
and (c) are the evolution at the medium temperature and high temperature respectively.
The Figure shows that the larger the temperature, the more important the coefficient
∆(t).
3. Concurrence and initial states
In order to describe the entanglement dynamics of the bipartite system, we use the
Wootters concurrence [9, 34]. For a system described by the density matrix ρ, the
concurrence C(ρ) is
p
p
p
p
(12)
C(ρ) = max(0, λ1 − λ2 − λ3 − λ4 ),
where λ1 , λ2 , λ3 , and λ4 are the eigenvalues (with λ1 the largest one) of the “spin-flipped”
density operator ζ, which is defined by
ζ = ρ(σyA ⊗ σyB )ρ∗ (σyA ⊗ σyB ),
(13)
where ρ∗ denotes the complex conjugate of ρ and σy is the Pauli matrix. C ranges in
magnitude from 0 for a disentanglement state to 1 for a maximally entanglement state.
The concurrence is related to the entanglement of formation Ef (ρ) by the following
relation [34]
Ef (ρ) = ε[C(ρ)],
(14)
Non-Markovian entanglement dynamics between two coupled qubits in the same environment6
where
ε[C(ρ)] = h[
1+
p
1 − C 2 (ρ)
],
2
(15)
and
h(x) = −x log2 x − (1 − x) log2 (1 − x).
(16)
Assume that the system is initially an “X” state, which has non-zero elements only
along the main diagonal and anti-diagonal. The general structure of an “X” density
matrix is as follows


ρ11 0
0 ρ14


 0 ρ22 ρ23 0 
ρ̂ = 
(17)
.
 0 ρ∗23 ρ33 0 
ρ∗14 0
0 ρ44
Such states are general enough to include states such as the Werner states, the maximally
entangled mixed states (MEMSs) and the Bell states; and it also arises in a wide variety
of physical situations [35, 36, 37]. This particular form of the density matrix allows us
to analytically express the concurrence as [38]
Cρ̂X = 2 max{0, K1, K2 },
where
(18)
√
K1 = |ρ23 | − ρ11 ρ44 ,
√
K2 = |ρ14 | − ρ22 ρ33 .
(19)
A remarkable aspect of the “X” states is that the time evolution of the master equation
(1) is maintained during the evolution. Substituting (17) into (1), the non-markovian
master equation of two-qubits system, we obtain the following first-order coupled
differential equations:
ρ̇11 (t)
ρ̇22 (t)
ρ̇33 (t)
ρ̇44 (t)
ρ̇23 (t)
ρ̇14 (t)
=
=
=
=
=
=
−2(∆(t) + γ(t))ρ11 (t) + (∆(t) − γ(t))ρ22 (t) + (∆(t) − γ(t))ρ33 (t),
(∆(t) + γ(t))ρ11 (t) − 2∆(t)ρ22 (t) + (∆(t) − γ(t))ρ44 (t),
(∆(t) + γ(t))ρ11 (t) − 2∆(t)ρ33 (t) + (∆(t) − γ(t))ρ44 (t),
(∆(t) + γ(t))ρ22 (t) + (∆(t) + γ(t))ρ33 (t) − 2(∆(t) − γ(t))ρ44 (t),
−2∆(t)ρ23 (t),
−2∆(t)ρ14 (t).
(20)
From Eq. (18) the concurrence C is dependent on the coefficients ∆(t → ∞) and
γ(t → ∞) in the asymptotic long time limit. Eqs. (10) and (11) give the stationary
value of γ(t) and ∆(t), the Markovian limit
γM
ω0 r 2
,
≡ γ(t → ∞) =
1 + r2
and
∆M ≡ ∆(t → ∞) = ω0
ω0
r2
coth(
).
2
1+r
2kB T
Non-Markovian entanglement dynamics between two coupled qubits in the same environment7
γM doesn’t depend on temperature, but ∆M is monotonically increasing with respect
ω0 r 2
1
≃ 2kωB0T , at
to temperature T . When T → 0, ∆M → 1+r
2 . Noting coth(πr0 ) ≃ 1 + πr
0
high temperature
r2
.
(21)
1 + r2
So ∆M > γM is noticeable when temperature kB T > 0. From Eqs. (20) we can get the
stationary solution
∆HT
M = 2kB T
M −γM
ρ (t → ∞),
ρ11 (t → ∞) = ∆
∆M +γM 33
ρ22 (t → ∞) = ρ33 (t → ∞),
2
∆2M −γM
ρ33 (t → ∞) =
,
4∆2
(22)
M
ρ44 (t → ∞) =
∆M +γM
∆M −γM
ρ33 (t → ∞).
and
ρ23 (t → ∞) = 0,
ρ14 (t → ∞) = 0.
(23)
According to Eqs. (18, 19),
2
∆2M − γM
< 0.
K1,2 (t → ∞) = 0 −
4∆2M
(24)
This means that entanglement must disappear in a finite time period, i.e. the ESD must
happen.
When temperature kB T = 0, ∆M ≈ γM . From Eqs. (20) we can also get the
stationary solution
and
ρ11 (t → ∞)
ρ22 (t → ∞)
ρ33 (t → ∞)
ρ44 (t → ∞)
=
=
=
=
0,
0,
0,
1.
(25)
ρ23 (t → ∞) = 0,
ρ14 (t → ∞) = 0.
(26)
From Eqs. (18, 19),
K1,2 (t → ∞) = 0.
(27)
This means that entanglement maybe disappear asymptotically, or oscillates, or other
complex behaviors. In the following, we use the numerical methods to demonstrate the
concurrence evolution for a special kind of “X” state, the ρY E state.
Non-Markovian entanglement dynamics between two coupled qubits in the same environment8
r=1 low temperature, non−Markovian
r=0.1 low temperature, non−Markovian
0.7
r=10 low temperature, non−Markovian
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.3
0.2
Cρ(t)
0.4
Cρ(t)
Cρ(t)
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.1
0
0
0
0
0
0
10
5
0.4
20
0.4
2000
ω0t
10
0.4
15
0.8
40
0.8
3000
1
0.2
0.6
30
0.6
ω0t
0
0.2
0.2
1000
0
0
0.6
20
a
0.8
a
ω0t
50
1
25
1
a
Figure 2. (Color online)Time evolution of non-Markovian concurrence as a function
of parameter “a” in the low temperature reservoirs.
r=0.1 medium temperature, non−Markovian
r=1 medium temperature, non−Markovian
r=10 medium temperature, non−Markovian
0.7
0.7
0.6
0.6
0.5
0.5
0.7
0.6
0.5
0.4
0.4
Cρ(t)
Cρ(t)
Cρ(t)
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
0
0
0
0
2
200
0
0
4
2
0
6
400
ω0t
0.5
8
ω0t
600
1
a
10
1
0
4
0.5
a
ω0t
0.5
1
6
a
Figure 3. (Color online)Time evolution of non-Markovian concurrence as a function
of parameter “a” in the medium temperature reservoirs.
Non-Markovian entanglement dynamics between two coupled qubits in the same environment9
non−Markovian entanglement dynamic at very low temperature with different initial state
0.7
a=0
a=0.3
a=0.5
a=0.6
a=0.8
a=1.0
0.6
0.5
K(t)
0.4
0.3
0.2
0.1
0
−0.1
0
0.5
1
1.5
2
ω0t
2.5
3
3.5
4
4
x 10
Figure 4. (Color online)Time evolution of K(t) for temperature kB T = 0.000001ω0,
r = 0.1, and initial state ρ̂Y E for the cases a = 0 (black dotted), a = 0.3 (cyan
dash-dotted line), a = 0.5 (red dash line), a = 0.6 (green dotted-dotted line), a = 0.8
(magenta asterisk), and a = 1.0 (blue solid line).
4. Non-Markovian vs. Markovian entanglement dynamics
In this section, we use the formalism of the preceding section to determine the
disentanglement. As an example, let us consider an important class of mixed states
with a single parameter a like the following [27, 39, 40]


a 0 0
0

1
0 
 0 1 1
(28)
ρ̂Y E = 
.
3 0 1 1
0 
0 0 0 1−a
Apparently,
the concurrence of ρY E is Cρ (t)p= max{0, K(t)}, and K(t) = |ρ23 (t)| −
p
ρ11 (t)ρ44 (t). Initially, C(ρ(0)) = 32 [1 − a(1 − a)]. In our simulations, ω0 = 1
are chosen as the norm unit, and we regard the temperature as a key factor in
disentanglement process, for high temperature kB T = 300ω0 , intermediate temperature
kB T = 3ω0 , and low temperature kB T = 0.03ω0 , respectively. Another reservoir
parameter playing a key role in the dynamics of the system is the ratio r = ωc /ω0
between the reservoir cutoff frequency ωc and the system oscillator frequency ω0 . As we
will see in this section, by varying these two parameters kB T and r = ωc /ω0, the time
evolution of the open system varies prominently from Markovian to non-Markovian.
In Fig.2, the time evolutions of the non-Markovian concurrence for various values
of the parameter a in low temperature is plotted. From Fig.2 we can see that the
Non-Markovian entanglement dynamics between two coupled qubits in the same environment10
r=1 high temperature, non−Markovian
r=0.1 high temperature, non−Markovian
r=10 high temperature, non−Markovian
0.6
0.6
0.8
0.4
0.6
Cρ(t)
0.4
Cρ(t)
0.8
Cρ(t)
0.8
0.2
0.2
0.5
20
ω0t
0.2
0.5
1
a
40
1
0
0
0
0
0
0
2
ω0t
0
0
a
1
0
0.4
0.5
0.2
0.4
0.6
ω0t
1
0.8
a
r=0.1 high temperature, Markovian evolution r=1 high temperature, Markovian evolution r=10 high temperature, Markovian evolution
0.8
0.8
0.8
0.6
0.6
0.2
Cρ(t)
0.4
Cρ(t)
Cρ(t)
0.6
0.4
0.2
0
0
0
0.5
20
40
ω0t
60
1
a
0.4
0 0.2
0
0
0
0.5
0.5
ω0t
1
1
a
0
0
0.2
0.5
0.4
ω0t
0.6
1
Figure 5. (Color online)Comparing the non-Markovian entanglement dynamics with
the Markovian one by the time evolution of concurrence as a function of parameter
“a” in high temperature reservoirs, at r = 0.1, r = 1, r = 10 respectively.
entanglement dynamic relies on the different values of r = ωc /ω0 . If the spectrum of
the reservoir does not completely overlap with the frequency of the system oscillator,
r ≪ 1, we can see from Fig.2 that the ESD time is considerably long. As increases
the ratio r, the ESD time becomes shorter and shorter. With different initial state
we can see that the concurrence varies prominently. When the initial state a = 0,
the non-Markovian entanglement decay slowly, as increasing a, the entanglement decay
intensely, which means that we can prepare certain initial entanglement states and use
this fact to control the system environment in order to prolong the entanglement time.
Fig.3 is the medium temperature case. Like Fig.2, under different systems, different
entanglement initial states, corresponding to different values of a, and different r, some
decay faster, some slower. But there are some fundamental difference between Fig.2 and
Fig.3. In Section III, we get the concurrence in the long time limit, and we affirmed that
when temperature kB T = 0, the dynamics of non-markovian entanglement concurrence
C is fundamentally different from the case of kB T > 0. As we can see from Fig.3,
for “ρY E ” states, as soon as the temperature larger than zero, the concurrence always
a
Non-Markovian entanglement dynamics between two coupled qubits in the same environment11
r=0.1 a=0, Markovian entanglement evolution
0.8
0.8
0.6
0.6
Crho(t)
0.4
ρ
C (t)
r=0.1 a=0, non−Markovian entanglement evolution
0.2
0.4
0.2
0
0
0
0
20
20
40
40
60
60
80
100
0
kBT
5
10
15
ω0t
20
25
30
80
100
0
kBT
5
10
15
20
25
ω0t
Figure 6. (Color online)Comparing the non-Markovian entanglement dynamics with
the Markovian one by the time evolution of Cρ (t) as a function of “kB T ” for initial
state a = 0 and r = 0.1.
disappear at finite time and there were no long-lived entanglement for any value of a,
which means that ESD must happen. The theoretical proof is K(t → ∞) < 0. But
when kB T = 0, the stationary value of K(t → ∞) equals zero. So, whether or not
and when the ESD will happen are not sure in kB T = 0. In Fig.4 we give a numerical
analysis of entanglement dynamic with different initial states and find that there exists
a ξ ∈ (0, 1), for almost all values a > ξ, the concurrence is completely vanished at a
finite time, which is the effect of ESD. However, for 0 ≤ a ≤ ξ, the entanglement of
this state decays exponentially. But when t → ∞, for all initial state, i.e. a ∈ [0, 1] the
concurrence will tend to be 0.
Fig.5 is the high temperature case. One of the remarkable phenomenon in this
figure is that the ESD time is short. In typical experimental conditions, quantum
dots are subjected to an external magnetic field B ∼ 1 − 10T [46], the ESD time
tESD ∼ (3 × 10−1 − 3)/kB T . Another obvious phenomenon is in high temperature the
Markovian quantum system decays exponentially and vanish only asymptotically, but in
the non-Markovian system the concurrence Cρ (t) oscillates, which is evidently different
from the Markovian. In this case the non-Markovian property becomes evidently. This
oscillatory phenomenon is induced by the memory effects, which allows the two qubit
entanglement to reappear after a dark period of time. This phenomenon of revival of
entanglement after finite periods of “entanglement death” appears to be linked to the
environment single qubit non-Markovian dynamics, in particular, the ∆(t) < 0 at some
times in some environment [31]. The physical conditions examined here are, moreover,
more similar to those typically considered in quantum computation, where qubits are
30
Non-Markovian entanglement dynamics between two coupled qubits in the same environment12
taken to be independent and where qubits interact with non-Markovian environments
typical of solid state microdevices [41].
As we indicated above, temperature is one of the key factor in the entanglement
dynamic. Figs. 2, 3, 4, 5 are plotted in the chosen temperature, while in Fig. 6
kB T ranges from 0 to 100. In Fig. 6 the concurrence vs “temperature kB T ” vs ω0 t in
r = 0.1, and the initial state is the “XY E ” state with a = 0. From Fig. 6 we can compare
the non-Markovian entanglement dynamics with the Markovian one clearly. The left
is the non-Markovian one from which we can see the oscillation of the concurrence.
Moreover, at the 0 temperature the non-Markovian effect is faint, as the temperature
rises, the non-Markovian becomes more and more obvious, while the Markovian one
decays exponentially. This phenomenon embodies the non-Markovian effect, which
is evidently different from the Markovian property. Maniscalco S et. al studied the
separability function S(τ ) in [30], where entanglement oscillation appears for twin-beam
state in non-Markovian channels in high temperature reservoirs. Both of them have the
same phenomenon. Ref. [31] gave a distribution curve when ∆(r, t) − γ(r, t) > 0 and
∆(r, t) − γ(r, t) < 0. We convince that due to the non-Markovian memory effect,
particularly ∆(t) < 0 in Eqs (20), the entanglement concurrence oscillates. With
∆(t) − γ(t) > 0 the concurrence descended whilst ∆(t) − γ(t) < 0 the concurrence
ascended, which guide us to adjust the temperature to control the entanglement
evolution. In order to show this and motivate the related research we design the open
loop controller
kB T = e−α|∆(t)−γ(t)| kB T0
(29)
where α is the modulation, and kB T0 is the initial temperature. In Fig. 7, we plot the
controlled entanglement evolution, where the initial temperature is chose as kB T0 = 30,
which oscillates and ESD occurs at t ≈ 19. According to Fig. 1, γ(t) can be neglected.
For different modulation α, different controlled entanglement evolution is plotted, and
the ESD time can be prolonged for considerable long time.
5. Conclusions
In this paper we have presented a procedure that allows to obtain the dynamic of a
system consisting of two identical independent qubits, each of them locally interacting
with a bosonic reservoir. A non-Markovian master equation between two qubit systems
in the same environment was obtained. We characterize our entanglement by the
temperature and the ratio r between ω0 the characteristic frequency of the quantum
system of interest, and ωc the cut-off frequency of Ohmic reservoir. For a broad class
of initially entangled states, “X” states, by useing Wootters’ concurrence, we analyze
the long time limit phenomenon of the entanglement dynamic. We find that the
dynamic of non-markovian entanglement concurrence Cρ (t) at temperature kB T = 0
is fundamentally different from kB T > 0. When kB T = 0, from our numerical
analysis, we find that “entanglement sudden death” (ESD) occurs depending on the
Non-Markovian entanglement dynamics between two coupled qubits in the same environment13
α=0
α=1
α=2
α=3
0.6
Cρ(t)
0.5
0.4
0.3
0.2
0.1
0
0
10
20
30
ω0t
40
50
60
Figure 7. (Color online)Controlled entanglement evolution with different modulation
α = 3 (green dotted line), α = 2 (red dashed line), α = 1 (blue dashed-dotted line),
and initially evolution (black solid line), respectively.
initial state, but if kB T > 0 the concurrence always disappear at finite time, which
means that ESD must happen. In the kB T = 0 case, we find that there exist a
ξ ∈ (0, 1), for all values a > ξ, the concurrence is completely vanished in a finite
time, which is the effect of ESD. However, for 0 ≤ a ≤ ξ, the entanglement of this
state decays exponentially. But when t → ∞, for all initial state, i.e. a ∈ [0, 1]
the concurrence will tend to be 0. From our numerical analysis we also find that the
entanglement dynamic relies on the different values of r = ωc /ω0 . If r ≪ 1, the ESD
time is considerably long. As increases the ratio r, the ESD time becomes shorter
and shorter. Moreover, when the initial state a = 0, the non-Markovian entanglement
decays slowly, as increases a, the entanglement decays intensely. Most of all, we have
shown that the non-Markovian dynamics of entanglement, described by concurrence,
presents oscillation even revivals after entanglement disappearance, typically for high
temperature non-Markovian system. At last, we design the open loop controller which
adjust the temperature to control the entanglement and prolong the ESD time.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (No.
60774099, No. 60221301), the Chinese Academy of Sciences (KJCX3-SYW-S01), and
by the CAS Special Grant for Postgraduate Research, Innovation and Practice.
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