Central European Journal of PhysicsCoherence-controlled stationary entanglement between two atoms embedded in a bad cavity injected with squeezed vacuum Research Article 1 College of Sci
Trang 1Central European Journal of Physics
Coherence-controlled stationary entanglement
between two atoms embedded in a bad cavity injected with squeezed vacuum
Research Article
1 College of Science, Hunan University of Technology,
Zhuzhou, Hunan 412007, China
2 Department of Physics and Information Science, Hunan Normal University,
Changsha, 410081, Hunan, China
Received 07 November 2012; accepted 04 November 2013
Abstract: We investigate the entanglement between two atoms in an overdamped cavity injected with squeezed
vacuum when these two atoms are initially prepared in coherent states It is shown that the stationary entanglement exhibits a strong dependence on the initial state of the two atoms when the spontaneous emission rate of each atom is equal to the collective spontaneous emission rate, corresponding to the case where the two atoms are close together It is found that the stationary entanglement of two atoms increases with decreasing effective atomic cooperativity parameter The squeezed vacuum can enhance the entanglement of two atoms when the atoms are initially in coherent states Valuably, this provides us with a feasible way to manipulate and control the entanglement, by changing the relative phases and the amplitudes of the polarized atoms and by varying the effective atomic cooperativity parameter of the system, even though the cavity is a bad one When the spontaneous emission rate of each atom is not equal to the collective spontaneous emission rate, the steady-state entanglement of two atoms always maintains the same value, as the amplitudes of the polarized atoms varies Moreover, the larger the degree of two-photon correlation, the stronger the steady-state entanglement between the atoms.
PACS (2008): 03.65.Ud; 03.67.Mn; 03.65.Yz
Keywords: quantum entanglement • the squeezed vacuum • coherent states • quantum control
© Versita sp z o.o.
1 Introduction
Entanglement is a kind of quantum correlation that has
played a central role in quantum information It has been
∗
found to be an indispensable resource in various quantum information processes [1 5], but the inevitability of the in-teraction between the system of interest and its environ-ment may cause decoherence and disentangleenviron-ment Many authors have shown that the collective interaction with a common thermal environment can cause the entanglement
of qubits [6 9]
Trang 2considerable importance to the prospects of maintaining
quantum information [10–16] Model systems that
the-oretically exhibit the rebirth of entanglement have been
proposed and discussed in several cases [17–19] Clark
and Parkins [20] have put forward a scheme to
control-lably entangle the internal states of two atoms trapped in
a high-finesse optical cavity by using quantum-reservoir
engineering Duan and Kimble have proposed an efficient
scheme to engineer multi-atom entanglement by detecting
cavity decay through single-photon detectors for
generat-ing multipartite entanglement [21] In Ref [22, 23], it
has been shown that white noise may play a positive role
in generating the controllable entanglement in some
spe-cific conditions Malinovsy and Sola [24] have proposed a
method of controlling entanglement in a two-qubit system
by changing a relative phase of the pulses In a recent
paper, Yu and Eberly [25] have shown that two
entan-gled qubits can become completely disentanentan-gled in a
fi-nite time under the influence of pure vacuum noise In
Ref [26], Ficek and Tanaś have investigated the concept
of time-delayed creation of entanglement by the
dissipa-tive process of spontaneous emission They have found a
threshold effect for the creation of entanglement, whereby
the initially unentangled qubits can be entangled after a
finite time despite the fact that the coherence between
the qubits exists for all times In Ref [27], theauthor
has studied the entanglement and the nonlocality of two
qubits interacting with a thermal reservoir The results
show that the common thermal resevoir can enhance the
entanglement of two qubits when two qubits are initially
in coherent states
In this paper, we investigate the entanglement between
two atoms in an overdamped cavity injected with squeezed
vacuum when these two atoms are initially prepared in
coherent states It isshown thatthe stationary
entan-glement exhibits a strong dependence on the initial state
of the two atoms when the spontaneous emission rate of
each atom is equal to the collective spontaneous emission
rate Conversely, when the spontaneous emission rate of
each atom is not equal to the collective spontaneous
emis-sion rate, the steady-state entanglement of two atoms
al-ways maintain the same value as the amplitudes of the
polarized atoms varies Moreover, the larger the degree
of two-photon correlation, the stronger the steady-state
entanglement between the atoms
2 The master equation
We study the dynamics of entanglement between two
two-level atoms embedded in an overdamped cavity injected
identical atoms are located in a single-mode cavity The Hamiltonian of this system is
H = ωa+a + ωS z + g(a
+S − + aS+), (1)
where a and a+ are annihilation and creation operators
for the cavity field, and S z and S ±are the collective
pseu-dospin operators, which are defined as S z= Σ2j=1S (j )
S ± = Σ2
j=1S (j )
±. The squeezing of all modes seen by the
atom is difficult to realize experimentally Instead the cav-ity environment, where only those modes centred around the privileged cavity mode need be squeezed, provides a much more realistic scenario for experimental investiga-tion The simplest situation to examine is the bad cavity limit Moreover, it is experimentally easy to realize a cav-ity with the squeezed vacuum input on the one side Thus the broadband squeezed vacuum is injected into the cav-ity via its lossy mirror (the other mirror is assumed to be perfect) Taking the spontaneous emission into account, the time evolution of the system of atom-field interaction
is given by the following master equation [28–30]
d
dt ρ = −i [H , ρ] + L a ρ + L c ρ, (2)
L a ρ = γ (2S − ρS+− S+S − ρ − ρS+S −)
+ (γ12− γ )(2S(1)
− ρS(2) + + 2S
(2)
− ρS(1) + − S(1) +S(2)
− ρ
− S(2) +S(1)
− ρ − ρS(1)
+S(2)
− − ρS(2) +S(1)
L c ρ = k (N + 1)(2aρa+− a+aρ − ρa+a ) + k N (2a+ρa
− aa+ρ − ρaa+
) + k M e
iθ
(2a
+ρa+− a+2ρ − ρa+2
)
+ kMe −iθ
(2aρa − a
2ρ − ρa2
where γ is the spontaneous emission rate of each atom,
and γ12 is the collective spontaneous emission rate
stem-ming from the coupling between the atoms through the vacuum field, which is dependent on the separation of the atoms If the atomic separation is much larger than the
resonant wavelength, then γ12 ≈ 0; if it is much smaller
than the resonant wavelength, then γ12≈ γ. The
param-eter k denotes the cavity decay constant. The parameter
Nis the mean photon number of the broadband squeezed
vacuum field M measures the strength of two-photon
correlations They obey the relation M = η pN (N + 1),
(0 ≤ η ≤ 1). θ is the phase of the squeezed vacuum.
We term η the degree of two-photon correlation. The
Trang 3in-a nonidein-al one when η 6= 1; if η = 0 then this implies no
squeezing and our cavity field is then equivalently damped
by a chaotic field
Here, we are interested in the bad-cavity limit; that is,
k g γ , but with C1= g2/kγfinite. C1 is the
effec-tive cooperativity parameter of a single atom familiar from
optical bistability To ensure the validity of the broadband
squeezing assumption, the bandwidth of squeezing must
also be large compared to k In the following, it will be
convenient for us to use the basis of the collective states
[31–33] B = {|ei, |si, |ai, |gi}, where
|ei = |e1i|e2i,
|si = √1
2
(|e1i|g2i + |g1i|e2i ),
|ai = √1
2
(|e1i|g2i − |g1i|e2i ),
|gi = |g1i|g2i. (5)
The most important property of the collective states is
that the symmetric state |si and antisymmetric state |ai
are maximally entangled states
We take the initial coherent state of the two atoms as
ψ (0) = cos(α )|e1g2i + sin(α )e iβ |g1e2i, which can be
gen-erated by controlling the relative phase of the external fields [34] Here β is a relative phase Using the Born-Markoff approximation and tracing over the field state [35–
37], and using the atomic basis as B = {|ei, |si, |ai, |gi},
we can write down the time evolution equations of the density matrix elements for the atoms
˙
ρ ee = [− 4g
2
k (N + 1) − 4γ ]ρ ee+4g
2
k Nρ ss −
2g2
k η
p
N (N + 1) exp(−iθ )ρ eg
− 2g
2
k η
p
N (N + 1) exp(iθ )ρ ge ,
˙
ρ ss = [4g
2
k (N + 1) + 2(γ + γ12]ρ ee −[4g2
k (2N + 1) + 2(γ + γ12)]ρ ss+4g
2
k η
p
N (N + 1) exp(−iθ )ρ eg
+
4g
2
k η
p
N (N + 1) exp(iθ )ρ ge+4g
2
k Nρ gg ,
˙
ρ aa = 2(γ − γ12)ρ ee − 2(γ − γ12)ρ aa ,
˙
ρ eg = − 2g
2
k η
p
N (N + 1) exp(iθ )ρ ee+ 4g
2
k η
p
N (N + 1) exp(iθ )ρ ss + [− 2g
2
k (2N + 1) − 2γ ]ρ eg
− 2g2
k η
p
N (N + 1) exp(iθ )ρ gg ,
˙
ρ ge = − 2g
2
k η
p
N (N + 1) exp(−iθ )ρ ee+4g
2
k η
p
N (N + 1) exp(−iθ )ρ ss + [− 2g
2
k (2N + 1) − 2γ ]ρ ge
− 2g2
k η
p
N (N + 1) exp(−iθ )ρ gg ,
˙
ρ gg = [4g
2
k (N + 1) + 2(γ + γ12)]ρ ss + 2(γ − γ12)ρ aa − 2g
2
k η
p
N (N + 1) exp(−iθ )ρ eg
− 2g
2
k η
p
N (N + 1) exp(iθ )ρ ge − 4g
2
k Nρ gg ,
˙
ρ as = [−2γ − 2(2N + 1)
g ]ρ as ,
˙
ρ sa = [−2γ − 2(2N + 1)
g ]ρ sa ,
(6)
with the condition ρ gg + ρ ee + ρ ss + ρ aa = 1. It is
ev-ident that the other matrix elements retain their initial
zero values, and only the set of eight equations (6) can
the above coherent state It is difficult to obtain analyti-cal solutions to the Eqs (6) We use fourth-order Runge-Kutta method to solve these equations with the relevant
Trang 4we next transform them into the original basis |e1i ⊗ |e2i,
|e1i⊗|g2i , |g1i⊗|e2i , |g1i⊗|g2i In section 3, we will
ex-plore quantum entanglement between two two-level atoms
embedded in a bad cavity injected with a squeezed
vac-uum
3 Entanglement between two
atoms
In order to quantify the degree of entanglement, we choose
the Wootters concurrence C [38,39], defined as
C = max (0,
p
λ1−pλ2−pλ3−pλ4), (7)
where λ1, , λ4are the eigenvalues of the non-Hermition
matrix ˜ρ = ρ(σ y ⊗ σ y )ρ ∗ (σ y ⊗ σ y ) ρ is the density matrix
which represents the quantum state The matrix elements
are taken with respect to the ‘standard’ eigenbasis |e1i ⊗
|e2i , |e1i ⊗ |g2i , |g1i ⊗ |e2i , |g1i ⊗ |g2i. The concurrence
varies from C = 0 for unentangled atoms to C = 1 for the
maximally entangled atoms
First, we consider the case of γ = γ12= 0.1,
correspond-ing to the case where the two atoms are close together;
i.e., the atomic separation is much smaller than the
reso-nant wavelength
In Fig.1, we plot the time evolution of the entanglement
between two atoms for γ = γ12= 0.1, η = 1,
k
g2 = 10 and
θ = π , with (a) N = 0.05, (b)N = 0.5, (c)N = 2, when
two atoms are initially in different states with the same
phase β = 2π /3 From bottom to top, the lines correspond
to α = π /2, α = π /2.08, α = π /2.1363, α = π /2.4813,
α = π /3 and α = π /4. This figure shows the
dynami-cal entanglement as α varies and phase remains constant
at β = 2π /3. It can be seen that the stationary
entan-glement of two atoms increases as the amplitude of the
polarized atoms cos(α ) increases (ie α decreases) It
reaches its maximum when α = π /4. We also see from
Fig1(a), (b) and (c) that the stationary entanglement
de-creases with increasing average photon number N
How-ever, these steady entangled states are very robust at high
temperature (see Fig.1(c))
Figure2displays the time evolution of the entanglement
between two atoms for N = 0.001, γ = γ12= 0.1, η = 0.1,
k
g2 = 3 and θ = π , for different phases β (from bottom to
top, the lines correspond to β = 0, β = π /3, β = π /2,
β = 2π /3, β = 5π /6 and β = π ) and for given α : (a)
α = π/ 2.2941, (b) α = π/10, (c) α = π/4. This
fig-ure displays the dynamical entanglement as the relative
phase β varies and α remains unchanged, and depicts
how the entanglement of the two atoms depends on the
β
0 0.2 0.4 0.6 0.8 1
t
(a)
0 0.2 0.4 0.6 0.8 1
t
(b)
0 0.2 0.4 0.6 0.8 1
t
(c)
Figure 1. The entanglement of two atoms versus t for γ=γ12=
0.1 with (a)N= 0.05, (b)N= 0.5, (c)N= 2 when two qubits are initially in different states with the same phase
β = 2π /3 From bottom to top, the lines correspond to α = π/ 2, α = π /2.08, α = π /2.1363, α = π /2.4813, α = π /3,
α = π /4.
the stationary entanglement increases with the
increas-ing of the relative phase β This is particularly valuable
in that it provides us with a feasible way to manipulate and control the entanglement by changing the relative phases In particular, when β = π , the entanglement of
< π/
Trang 50 10 20 30 40 50 60
0
0.2
0.4
0.6
0.8
1
t
(a)
0
0.2
0.4
0.6
0.8
1
t
(b)
0
0.2
0.4
0.6
0.8
1
t
(c)
Figure 2. The entanglement of two atoms versus t for γ = γ12= 0.1,
N = 0.001, for different phases β (from bottom to top, the
lines correspond to β = 0, β = π /3, β = π /2, β = 2π /3,
β = 5π /6, β = π ) and given α :(a)α = π /2.2941, (b)α=
π/ 10, (c)α = π /4.
values; this indicates the possibility of obtaining steady
entangled states with a larger amount of entanglement
originating from entangled states with a smaller amount
of entanglement It should be noted that the initial state
with α = π /4, β = π always maintains maximal
entan-glement, regardless of time and temperature, as shown in
to interact with the squeezed vacuum all the time This result can be explained as follows: in an interaction
pic-ture with respect to H = ωa
+a + ωS z, after some lengthy
algebra and tracing over the field state, from the master equation (2) we find the following master equation for the
atoms ρ a:
dρ a
dt = − g
2
k(S˜+S˜− ρ a + ρ a S˜+S˜− −2S˜− ρ a S˜+)
L a ρ a = γ (2S − ρ a S+− S+S − ρ a − ρ a S+S −)
+ (γ12− γ )(2S
(1)
− ρ a S(2) + + 2S
(2)
− ρ a S(1) + − S(1) +S(2)
− ρ a
− S(2) +S(1)
− ρ a − ρ a S(1)
+S(2)
− − ρ a S(2)
+S(1)
where
˜
S+ = µS++ ν ∗ S − ,
˜
S − = νS++ µS − ,
µ = √ N + 1,
ν = √ N exp(iθ ). (10)
One can see that the (Bell) state |ψ − i =
1
√
2(|10i −
| 01i)(corresponding to the initial condition α = π/4,
β = π) is in fact a “dark" state of the system, i.e.
S − |ψ − i = S+|ψ − i = 0, implying that this state is not
influenced by coupling to the squeezed vacuum reser-voir (hence the straight line in Fig.2(c)) However, when
β= 0, Fig.2(a), (b) and (c) show that entanglement can
fall abruptly to zero before entanglement recovers to a sta-tionary state value The time at which the entanglement falls to zero is dependent upon the degree of entanglement
of the initial state The bigger the initial degree of en-tanglement, the later the entanglement vanishes This im-plies that two-atom entanglement may terminate abruptly
in a finite time under the influence of the squeezed vac-uum This phenomenon is referred to as “sudden death"
of entanglement [25]and it has elucidated a numberof new characteristics of entanglement evolution in systems
of two qubits
Figure3displays the entanglement of two atoms versus t for N = 0.05, γ = γ12= 0.1, η = 1, α = π /10 and θ = π ,
for different phases β (from bottom to top, the lines corre-spond to β = 0, β = π /3, β = π /2, β = 2π /3, β = 5π /6,
β = π ) and given k
g2: (a)
k
g2 = 4, (b)
k
g2 = 10, (c)
k
g2 = 100, (d)
k
g2 = 1000 This figure displays the dynamical entan-glement as
k
g2 (cf the effective atomic cooperativity
pa-rameter C1= g
2/kγ) varies It is shown that the stationary
k
Trang 60 10 20 30 40 50 60
0
0.2
0.4
0.6
0.8
1
t
(a)
0 0.2 0.4 0.6 0.8 1
t
(b)
0
0.2
0.4
0.6
0.8
1
t
(c)
0 0.2 0.4 0.6 0.8 1
t
(d)
Figure 3. The entanglement of two atoms versus t for γ = γ12= 0.1, N = 0.05, α = π /10, for different phases β (from bottom to top, the lines
correspond to β = 0, β = π /3, β = π /2, β = 2π /3, β = 5π /6, β = π ) and given k
g2: (a)k g2 = 4, (b)
k g2 = 10, (c)
k g2 = 100,(d)
k g2 = 1000.
(ie decreasing C1= g2/kγ) From figure 3(d), we can see
that, when the parameter C1 is very small, which
corre-sponds to the field’s inside cavity being more chaotic, the
stationary entanglement of two atoms is very large This is
possible, since dissipation plays a crucial role in the
gen-eration of the stationary entanglement From figure 3(a),
we have found an interesting phenomenon: when β = 0
and
k
g2 is not large, the entanglement can fall abruptly to
zero twice before entanglement recovers to a stationary
state value We see two time intervals (dark periods) at
which the entanglement vanishes and two time intervals at
which the entanglement revives And, with the increase of
β, though the phenomenon of “sudden death" of
entangle-ment does not occur, the rate of evolution of entangleentangle-ment
can suddenly change twice Meanwhile, with the increase
of
k
g2, the entanglement can fall abruptly to zero only once
before entanglement recovers to a stationary state value
when β = 0 (see Figure 3(b)–(d)) Furthermore, the
big-ger the parameter
k
g2, the shorter the state will stay in the disentangled separable state So, we can steer the
evo-effective atomic cooperativity parameter C1of the system
Next, we discuss the situation of γ 6 = γ12, which means that the separation between two atoms is not very small
In Fig.4, we plot the entanglement of two atoms versus t for γ = 0.1, γ12 = 0.06, N = 0.05,
k
g2 = 10 and θ = π , with (a) η = 1, (b)η = 0.7, (c)η = 0.2 when two atoms are initially in different states with the same phase β = 2π /3.
From bottom to top, the lines correspond to α = π/2,
α = π/ 2.08, α = π/ 2.1363, α = π/ 2.4813, α = π/3
and α = π /4 When the degree of two-photon correlation
η= 1, the injected field in the cavity is an ideal squeezed
vacuum, corresponding to the reservoir being in an ideal
or minimum uncertainty squeezed state When the degree
of two-photon correlation η 6= 1, the injected field in the
cavity is not ideal, which means that some of the photon pairs in the squeezed field are not correlated due to the cavity effect From this figure, which displays the
dynami-cal entanglement as α varies and phase remains constant
at β = 2π /3, it is discovered that the steady-state en-tanglement of two atoms always remains constant as the
Trang 70 20 40 60 80 100
0
0.2
0.4
0.6
0.8
1
t
(a)
0
0.2
0.4
0.6
0.8
1
t
(b)
0
0.2
0.4
0.6
0.8
1
t
(c)
Figure 4. The entanglement of two atoms versus t for γ = 0.1, γ12=
0.06 with (a)η = 1, (b)η = 0.7, (c)η = 0.2 when two qubits
are initially in different states with the same phase β =
2π /3 From bottom to top, the lines correspond to α =
π/ 2, α = π /2.08, α = π /2.1363, α = π /2.4813, α = π /3,
α = π /4.
be explained as follows: when γ 6= γ12, Eqs.(6) imply that
ρ aa = ρ ee Therefore, regardless of whether the
asymmet-ric state |ai is initially populated, in the long-time limit,
due to the interaction of the nonclassical field, the
asym-metric state will be equally as populated as the upper
to zero, and the symmetric state |si, ρ eg and ρ ge tend to certain values, regardless of the initial states of the atoms
In addition, from Fig3.(a), (b) and (c), we can see that the
larger the degree of two-photon correlation η, the stronger
the steady-state entanglement between the atoms Thus, the nonclassical two-photon correlations of the injected squeezed vacuum are significant for the stationary entan-glement in the system
4 Conclusion
In this paper, we have investigated the entanglement be-tween two atoms in an overdamped cavity injected with squeezed vacuum when these two atoms are initially pre-pared in coherent states It is shown that the stationary entanglement exhibits a strong dependence on the initial
state of the two atoms when γ = γ12, corresponding to the
case where the two atoms are close together It is found that the stationary entanglement of two atoms increases with decreasing effective atomic cooperativity parameter The squeezed vacuum can enhance the entanglement of two atoms when two atoms are initially in coherent states Valuably, this provides us with a feasible way to manip-ulate and control the entanglement by changing the rel-ative phases and the amplitudes of the polarized atoms, and by varying the the effective atomic cooperativity pa-rameter of the system even though the cavity is a bad
one When γ 6= γ12, the steady-state entanglement of two
atoms always remains constant as the amplitudes of the
polarized atoms α vary Moreover, the larger the degree
of two-photon correlation η, the stronger the steady-state
entanglement between the atoms Thus, the nonclassical two-photon correlations are significant for the entangle-ment in the system
Acknowledgement
This work is supported by the National Natural Sci-ence Foundation of China (Grant No 11074072 and No.61174075), Hunan Provincial Natural Science Foun-dation of china (Grant No 10JJ3088 and No.11JJ2038) and by the Major Program for the Research Foundation
of Education Bureau of Hunan Province of China (Grant
No 10A026)
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