The concept of time delayed creation of entanglement by the dissipative
process of spontaneous emission is investigated. A threshold effect for the
creation of entanglement is found that the initially unentangled qubits can be
entangled after a finite time despite the fact that the coherence between the
qubits exists for all times. This delayed creation of entanglement, that we
call sudden birth of entanglement, is opposite to the currently extensively
discussed sudden death of entanglement and is characteristic for transient
dynamics of one-photon entangled states of the system. We determine the
threshold time for the creation of entanglement and find that it is related to
time at which the antisymmetric state remains the only excited state being
populated. It is shown that the threshold time can be controlled by the
distance between the qubits and the direction of initial excitation relative to
the interatomic axis. This effect suggests a new alternative for the study of
entanglement and provides an interesting resource for creation on demand of
entanglement between two qubits.Comment: References added, version accepted for publication in PR

Ficek, Z.

Tanas, R.

English

arXiv

The concept of time delayed creation of entanglement by the dissipative process of spontaneous emission is investigated. A threshold effect for the creation of entanglement is found where the initially unentangled qubits can be entangled after a finite time despite the fact that the coherence between the qubits exists for all times. This delayed creation of entanglement, which we call sudden birth of entanglement, is opposite to the currently extensively discussed sudden death of entanglement and is characteristic for transient dynamics of one-photon entangled states of the system. We determine the threshold time for the creation of entanglement and find it is related to time at which the antisymmetric state remains the only excited state being populated. It is shown that the threshold time can be controlled by the distance between the qubits and the direction of initial excitation relative to the interatomic axis. This effect suggests an alternative for the study of entanglement and provides an interesting resource for creation on demand of entanglement between two qubits

Ficek, Zbigniew

Tanas, Ryszard

University of Queensland eSpace

Delayed sudden birth of entanglement
Zbigniew Ficek1,* and Ryszard Tanaś2
1Department of Physics, The University of Queensland, Brisbane 4072, Australia
2Nonlinear Optics Division, Institute of Physics, Adam Mickiewicz University, Poznań, Poland
Received 29 February 2008; published 8 May 2008
The concept of time delayed creation of entanglement by the dissipative process of spontaneous emission is
investigated. A threshold effect for the creation of entanglement is found where the initially unentangled qubits
can be entangled after a finite time despite the fact that the coherence between the qubits exists for all times.
This delayed creation of entanglement, which we call sudden birth of entanglement, is opposite to the currently
extensively discussed sudden death of entanglement and is characteristic for transient dynamics of one-photon
entangled states of the system. We determine the threshold time for the creation of entanglement and find it is
related to time at which the antisymmetric state remains the only excited state being populated. It is shown that
the threshold time can be controlled by the distance between the qubits and the direction of initial excitation
relative to the interatomic axis. This effect suggests an alternative for the study of entanglement and provides
an interesting resource for creation on demand of entanglement between two qubits.
DOI: 10.1103/PhysRevA.77.054301 PACS numbers: 03.67.Mn, 42.50.Nn, 42.50.Dv
Dynamical creation of entanglement in the presence of a
noisy environment and its disentangled properties are prob-
lems of fundamental interest in quantum computation and
quantum information processing. They have attracted a great
deal of attention, especially in connection with the phenom-
enon of decoherence induced by spontaneous emission re-
sulting from the interaction with the environment which
leads to irreversible loss of information encoded in the inter-
nal states of the system and thus is regarded as the main
obstacle in practical implementations of entanglement. Con-
trary to intuition that spontaneous emission should have a
destructive effect on entanglement, it has been shown that
under certain circumstances this irreversible process can in
fact entangle initially unentangled qubits 1, thus implying a
kind of quantum coherence induced in the emission. This
effect has been studied for identical qubits coupled to a com-
mon multimode vacuum field or coupled to a damped single-
mode cavity field and has a simple explanation in terms of
the collective nature of the spontaneous emission from a sys-
tem of qubits being located within a transition wavelength of
each other or coupled to a single-mode cavity field. In this
terminology, the system can be represented in terms of the
collective Dicke symmetric and antisymmetric states which
decay with significantly different rates 2,3. Both states are
maximally entangled states, but the entanglement results
solely from the trapping properties of the antisymmetric state
of the system. More precisely, with the initially only one
qubit excited, a part of the initial population is trapped in the
antisymmetric state from which it cannot decay or may de-
cay much slower than the populations of the remaining
states. This is the reason why the system decays to an en-
tangled long-living mixed state involving only the antisym-
metric and the ground states of the system. In this way, an
entanglement persisted over a long time is obtained dynami-
cally via spontaneous emission. The degree of the entangle-
ment such generated is determined by the population of the
antisymmetric state that with the initially only one atom ex-
cited approaches a steady state value of one-half.
Apart from the constructive effect of spontaneous emis-
sion on entanglement, it has been shown that some entangled
states of two qubits can have interesting decoherence prop-
erties that two initially entangled qubits can reach separabil-
ity abruptly in a finite time which is much shorter than the
exponential decoherence time of spontaneous emission 4.
This drastic nonasymptotic feature of entanglement has been
termed as the “entanglement sudden death,” and is character-
istic of the dynamics of a special class of initial two-photon
entangled states. In fact, the effect shows up only if specific
initial two-photon coherences are created between the qubits.
A recent experiment by Almeida et al. 5 with correlated
horizontally and vertically polarized photons has shown evi-
dence of the sudden death of entanglement under the influ-
ence of independent environments. The required initial two-
photon coherence was created by the parametric down-
conversion process. Considering the present interest in
understanding of the decoherence in entangled qubits, it pre-
sents a fascinating example of a dynamical process in which
spontaneous emission affects entanglement and coherences
in very different ways. Although the sudden death feature is
concerned with the disentangled properties of spontaneous
emission there can be interesting “sudden” features in the
temporal creation of entanglement from initially independent
qubits. If such features exist, they would provide an interest-
ing resource for creation on demand of entanglement be-
tween two qubits.
In this paper we show that a “sudden” feature in the tem-
poral creation of entanglement exists in a dissipative time
evolution of interacting qubits. We term this feature as de-
layed sudden birth of entanglement, as it is opposite to the
sudden death of entanglement, and show the feature arises
dynamically with initially separable qubits. The delayed cre-
ation of entanglement is not found in the small sample Dicke
model which ignores the evolution of the antisymmetric
state. It is also not found in a system with initially only one
qubit excited. For this, an initial entanglement and the ex-
perimentally difficult individual addressing of qubits are not*ficek@physics.uq.edu.au
PHYSICAL REVIEW A 77, 054301 2008
1050-2947/2008/775/0543014 ©2008 The American Physical Society054301-1
required. The initial conditions considered here include both
qubits inverted which can be done using a standard technique
of a short  pulse excitation. The second initial condition
considered here involves excitation by a short  /2 pulse
which leaves qubits separable but simultaneously prepared in
the superposition of their energy states. We carry our consid-
erations in the context of concurrence and two-level atoms
interacting through the vacuum field and analyze how the
concurrence evolves in time. We determine the threshold
time for creation of entanglement and discuss the depen-
dence of the magnitude of the entanglement on the distance
between the qubits and direction of excitation relative to the
interqubit axis. Related calculations have appeared involving
entanglement creation via spontaneous emission 1. How-
ever, these calculations studied a limited set of initial condi-
tions and as such these calculations miss the feature of de-
layed birth of entanglement which depends on specific initial
conditions of both qubits. The sudden birth of entanglement
deserves more careful study, especially in view of its funda-
mental importance in a controled creation of entanglement
on demand in the presence of a dissipative environment.
The usual way to identify entanglement between two qu-
bits in a mixed state is to examine the concurrence, an en-
tanglement measure that relates entangled properties to the
coherence properties of the qubits 6. For a system de-
scribed by the density matrix , the concurrence C is defined
as
Ct = max0,1t − 2t − 3t − 4t , 1
where it are the square roots of the eigenvalues of the
non-Hermitian matrix t˜t with
˜t = y  y*ty  y , 2
and y is the Pauli matrix. The range of concurrence is from
0 to 1. For unentangled separated atoms Ct=0, whereas
Ct=1 for the maximally entangled atoms.
The density matrix, which is needed to compute Ct and
written in the basis of the separable product states 1
= g1g2, 2= e1g2, 3= g1e2, 4= e1e2 is in general com-
posed of sixteen nonzero density matrix elements. However,
in the case of the simple dissipative evolution of the system
without any initial coherences between the qubits and with-
out the presence of coherent excitations, the density matrix
takes a simple block diagonal form
t =	
11t 0 0 0
0 22t 23t 0
0 32t 33t 0
0 0 0 44t

 , 3
in which we put all the coherences, except the one-photon
coherences 23t and 32t, equal to zero. As we will see the
zeroth coherences remain zero for all time, they cannot be
created by spontaneous decay. However, the coherences
23t and 32t can be created by spontaneous emission
even if they are initially zero.
For a system described by the density matrix 3, the con-
currence has a simple analytical form
Ct = max0,C˜t , 4
with
C˜t = 223t − 211t44t . 5
It is evident there is a threshold for the coherence at which
the system becomes entangled. Thus, the nonzero coherence
23t is the necessary condition for entanglement, but not in
general a sufficient one since there is also a rather subtle
condition of a minimum coherence between the qubits.
Alternatively, we may study conditions for entanglement
by writing the concurrence 5 in terms of the maximally
entangled Dicke symmetric s= 2+ 3 /2 and antisym-
metric a= 2− 3 /2 states
C˜t = sst − aat2 − sat − ast2 − 211t44t ,
6
which shows the threshold for entanglement depends on the
distribution of the population between the entangled and
separable states. Notice that the threshold depends on the
population of the upper state 4. Thus, no threshold features
can be observed in entanglement creation by spontaneous
emission for qubits initially prepared in a single photon state.
In addition to the threshold phenomenon, there is also an
evident competition between the symmetric and antisymmet-
ric states in creation of entanglement. We see that the best for
creation of entanglement through the one-photon states is to
populate either symmetric or antisymmetric states, but not
both simultaneously. Thus, one could expect a large en-
tanglement can be created when one of the two entangled
states is excluded from the dynamics and remains unpopu-
lated for all times.
However, we demonstrate a somewhat surprising result in
which entanglement cannot be created by spontaneous emis-
sion in the Dicke model which excludes the dynamics of the
antisymmetric state. The Dicke model is composed of three
states in a ladder configuration: the upper state 4, the inter-
mediate symmetric state s, and the ground state 1. Physi-
cally, the Dicke model represents two qubits confined to a
region much smaller than the resonant wavelength 2. In
this case, the time evolution of the density matrix elements
under the spontaneous emission is determined by the follow-
ing density matrix elements 3
44t = 440e−2t,
sst = ss0e−2t + 2t440e−2t,
aat = aa0 , 7
which shows that the antisymmetric state does not participate
in the spontaneous dynamics of the system. The population
of the antisymmetric state remains constant in time. As a
result, if the system is prepared in the antisymmetric state it
stays there for all times. We are, however, interested in the
dynamical creation of entanglement by spontaneous emis-
sion from separable states to entangled states.
In the Dicke model, the only entangled state which par-
ticipates in the spontaneous dynamics is the symmetric state
BRIEF REPORTS PHYSICAL REVIEW A 77, 054301 2008
054301-2
s, so let us see if one can create entanglement by sponta-
neous emission that can populate the symmetric state from
the upper state 4. Figure 1 shows the time evolution of the
population sst and the threshold factor 211t44t for
the initially fully inverted qubits. We see that the threshold
factor overweights the population sst for all times, which
indicates that despite a large population of the symmetric
state, no entanglement is created. Thus, we may conclude
that spontaneous emission cannot create entanglement in the
Dicke model where only the symmetric state participates in
the atomic dynamics.
We now turn to the system of two qubits which are sepa-
rated by distances comparable to the resonant wavelength. In
this case, the antisymmetric states fully participates in the
spontaneous dynamics and the time evolution of the density
matrix elements under the spontaneous emission and for an
arbitrary initial state is given by 3
44t = 440e−2t,
sst = ss0e−+12t + 440e−2t
 + 12
 − 12
e−12t − 1 ,
aat = aa0e−−12t + 440e−2t
 − 12
 + 12
e+12t − 1 ,
sat = sa0e−+2i12t, 8
and 11t=1−44t−sst−aat. Note that the full solu-
tion for the density matrix elements exhibits the effect of the
cooperative damping 12 and the dipole-dipole interaction
12.
We consider spontaneous creation of entanglement in the
system initially prepared in a separable state. The entangle-
ment depends, of course, on the initial state of the system.
We consider two examples of initial states. As the first ex-
ample, consider a state which covers a broad class of initial
states in which the qubits are prepared in the superposition of
their energy states
 0 =
1
2
g1 + ieik

·r1e1  g2 + ieik

·r2e2 , 9
where k is the wave vector of the excitation field. The initial
state  0 is separable and can be created in practice by an
incident  /2 pulse excitation of each qubit. In this case, the
initial values of the density matrix elements are
sa0 =
1
4
i sin k · r12, ss0 =
1
4
1 + cos k · r12 ,
aa0 =
1
4
1 − cos k · r12, 440 =
1
4
, 10
which shows that a particular initial state depends on the
distance between the qubits and the direction of excitation
relative to the interatomic axis.
Figure 2 shows the concurrence as a function of time and
the angle  between the excitation direction and the vector
r12 connecting the atoms. It is seen there is no entanglement
at earlier times independent of the direction of excitation,
and suddenly at some finite time an entanglement starts to
build up. However, no entanglement builds up if the system
is initially excited in the direction perpendicular to the inter-
atomic axis. One can see from Eq. 10 that in this case the
system is excited through the symmetric state. Thus, similar
to the Dicke model, entanglement in the system cannot be
created by an excitation of the system through the symmetric
state. A large entanglement is created only if the system is
excited in the direction of the interatomic axis. This means
that it is crucial for the entanglement creation by spontane-
ous emission to be accomplished an excitation through the
antisymmetric state.
The above conclusion is supported by the analysis of the
time evolution of the population of the excited states of the
system which is illustrated in Fig. 3. It is quite evident from
the figure that at the time t4 / when the entanglement
starts to build up, the antisymmetric state is the only excited
state of the system being populated. This effect is attributed
to the slow decay rate of the antisymmetric state. The state
0 1 2 3 4 5
0
0.1
0.3
0.5
0.7
γ t
ρ s
s(
t)
,2
(ρ
11
(t
)ρ
44
(t
))
1/
2
FIG. 1. The time evolution of the population sst solid line
and the threshold factor 211t44t dashed line for initially
both qubits inverted, 440=1.
0
5
10
15
20
0
0.2
0.4
0.6
0.8
1
0
0.05
0.1
γ t
θ/π
C
FIG. 2. The time evolution of the concurrence and its depen-
dence on the direction of excitation relative to the interatomic axis
for r12 /=0.25 and the polarization of the atomic dipole moments
	 r12.
BRIEF REPORTS PHYSICAL REVIEW A 77, 054301 2008
054301-3
decays on the time scale of −12−1 that is much shorter
than the decay time of the symmetric and the upper states.
In the second example, we consider the qubits initially
prepared in their excited states, which can be realized in
practice by a short  pulse excitation. In this case
440 = 1, sa0 = ss0 = aa0 = 0. 11
Figure 4 illustrate the concurrence as a function of time and
the distance between the qubits. Similar to the first example,
illustrated in Fig. 2, there is no entanglement at earlier times,
but suddenly at some finite time an entanglement starts to
build up. However, it happens only for a limited range of the
distances r12. It is easy to show that the “islands” of en-
tanglement seen in Fig. 4 appear at distances for which 12 is
different from zero.
One can easily show that similar to the first example, the
entanglement seen in Fig. 4 decays out on a time scale 
−12−1 which is the time scale of the population decay from
the antisymmetric state.
In summary, we have predicted an interesting phenom-
enon of delayed sudden birth of entanglement which ini-
tially separable qubits become entangled via spontaneous
emission after a finite time. In contrast to the sudden death
phenomenon that involves two-photon entangled states, the
sudden birth involves one-photon entangled symmetric and
antisymmetric states. We have demonstrated that the partici-
pation of the antisymmetric state in the dynamics is crucial
for creation of entanglement in the systems.
This work was supported by the Australian Research
Council and the University of Queensland.
1 A. Beige et al., J. Mod. Opt. 47, 2583 2000; D. Braun, Phys.
Rev. Lett. 89, 277901 2002; A. M. Basharov, JETP Lett. 75,
123 2002; L. Jakóbczyk, J. Phys. A 35, 6383 2002; Z.
Ficek and R. Tanaś, J. Mod. Opt. 50, 2765 2003; F. Benatti,
R. Floreanini, and M. Piani, Phys. Rev. Lett. 91, 070402
2003; M. Paternostro, M. S. Tame, G. M. Palma, and M. S.
Kim, Phys. Rev. A 74, 052317 2006; S. Natali and Z. Ficek,
ibid. 75, 042307 2007; L. Derkacz and L. Jakóbczyk, e-print
arXiv:0710.5048.
2 R. H. Dicke, Phys. Rev. 93, 99 1954.
3 R. H. Lehmberg, Phys. Rev. A 2, 883 1970; G. S. Agarwal,
Quantum Statistical Theories of Spontaneous Emission and
their Relation to Other Approaches, edited by G. Höhler,
Springer Tracts in Modern Physics Vol. 70 Springer-Verlag,
Berlin, 1974; Z. Ficek and R. Tanaś, Phys. Rep. 372, 369
2002.
4 T. Yu and J. H. Eberly, Phys. Rev. Lett. 93, 140404 2004; J.
H. Eberly and T. Yu, Science 316, 555 2007; L. Jakóbczyk
and A. Jamróz, Phys. Lett. A 333, 35 2004; A. Jamróz, J.
Phys. A 39, 7727 2006; Z. Ficek and R. Tanaś, Phys. Rev. A,
74, 024304 2006; H. T. Cui, K. Li, and X. X. Yi, Phys. Lett.
A 365, 44 2007; I. Sainz and G. Björk, Phys. Rev. A, 76,
042313 2007; X. Cao and H. Zheng, ibid. 77, 022320
2008; C. E. López et al., e-print arXiv:0802.1825.
5 M. P. Almeida et al., Science 316, 579 2007.
6 W. K. Wootters, Phys. Rev. Lett. 80, 2245 1998.
0 5 10 15
0
0.05
0.1
0.15
0.2
0.25
γ t
ρ 4
4(
t)
,ρ
ss
(t
),
ρ a
a(
t)
FIG. 3. The time evolution of the populations 44t solid line,
sst dashed line, and aat dashed-dotted line for =0, r12 /
=0.25, and 	 r12.
0
5
10
15
20
25
0
0.2
0.4
0.6
0.8
1
0
0.01
0.02
0.03
γ t
r
12
/λ
C
FIG. 4. The time evolution of the concurrence and its depen-
dence on the distance between two initially inverted qubits.
BRIEF REPORTS PHYSICAL REVIEW A 77, 054301 2008
054301-4


Delayed sudden birth of entanglement

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Delayed (sudden) birth of entanglement

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