Entangling two-level atoms by spontaneous emission

11 Pages·0.13 MB·English

by  L. Jakobczyk

Entangling two-level atoms by spontaneous
emission
L. Jak´obczyk
3 Institute of Theoretical Physics
0
University of Wroc law
0
Pl. M. Borna 9, 50-204 Wroc law, Poland
2

n
a
J

4 Abstract: It is shownthat the dissipationdue to spontaneous emissioncan entangle two closely

separated two-level atoms.
2
v
0
1 Introduction
4
1
4 Analysis of various aspects of spontaneous emission by a system of two-level atoms, initiated in
0
the classical paper of Dicke [1] was further developed by several authors (see e.g. [2, 3, 4]). In
2
particular,inthecaseofspontaneousemissionbytwotrappedatomsseparatedbyadistancesmall
0
/ comparedtotheradiationwavelength,whereisasubstantialprobabilitythataphotonemittedby
h
one atom will be absorbed by the other, there are states of the system in which photon exchange
p
can enhance or diminish spontaneous decay rates. The states with enhanced decay rate are called
-
t superradiant and analogously states with diminished decay rate are called subradiant [1]. It was
n
a alsoshownbyDicke,thatthesystemoftwocoupledtwo-levelatomscanbetreatedasasinglefour-
u levelsystemwithmodifieddecayrates. Notealsothatsuchmodelcanberealizedinalaboratoryby
q two laser-cooledtrapped ions, where the observation of superradiance and subradiance is possible
:
v [5].
i Another aspects of the model of the spontaneous emission are studied in the present paper.
X
When the compound system of two atoms is in an entangled state, the irreversible process of
r
a radiative decay usually destroys correlations and the state becomes unentangled. In the model
studied here, the photon exchange produces correlations between atoms which can partially over-
come decoherence caused by spontaneous radiation. As a result, some amount of entanglement
can survive, and moreover there is a possibility that this process can entangle separable states of
two atoms. The idea that dissipation can create entanglement in physical systems, was recently
developed in several papers [6, 7, 8, 9]. In the present paper we show that the dissipation due to
spontaneous emission can entangle two atoms that are initially prepared in a separable state. We
study the dynamics of this process. In the Markovianapproximationit is givenby the semi-group
T of completely positive linear mappings acting on density matrices [10]. We consider time
t
{ }
evolutionofinitialstateofthe systemaswellasthe evolutionofits entanglement,measuredbyso
called concurrence [11, 12], in the case when the photon exchange rate γ is close to spontaneous
emission rate of the single atom γ and we can use the approximation γ =γ (similar model was
0 0
alsoconsideredin[13]). Wecalculateasymptoticstationarystatesρ forthesemi-group T and
as t
{ }
show that they depend on initial conditions (i.e. T is relaxing but not uniquely relaxing). The
t
{ }
concurrence of ρ also depends on initial state and can be non zero for some of them. We discuss
as
in details some classesof initial states. In particular,we show that there are pure separable states
evolving to entangled mixed states and such which remain separable during evolution. The first
class contains physically interesting initial state when one atom in in excited state and the other
is in ground state. The relaxation process given by the semi-group T produces in this case the
t
{ }
stateswithentanglementmonotonicallyincreasingintimetothemaximalvalue. Theclassofpure
1
maximallyentangledinitialstatesisalsodiscussed. Similar”production”ofentanglementisshown
to be presentfor some classesofmixedstates. Onthe otherhand, when the photonexchangerate
γ is smaller then γ , the relaxation process brings all initial states to the unique asymptotic state
0
whenbothatomsareinitsgroundstates. Asweshow,eveninthatcasethedynamicscanentangle
two separable states, but the amount of entanglement is decreasing to zero.
2 Pair of two-level atoms
Consider two-level atom A with ground state 0 and excited state 1 . This quantum system can
be describedintermsofthe Hilbertspace |=iC2 andthealgebra|Ai of2 2complexmatrices.
A A
H ×
If we identify 1 and 0 with vectors 1 and 0 respectively, then the raising and lowering
| i | i 0 1
operators σ , σ defined by
+
− (cid:0) (cid:1) (cid:0) (cid:1)
σ = 1 0, σ = 0 1 (1)
+
| ih | − | ih |
can be expressed in terms of Pauli matrices σ , σ
1 2
1 1
σ = (σ +iσ ), σ = (σ iσ ) (2)
+ 1 2 1 2
2 − 2 −
For ajoint systemAB of twotwo-levelatomsA andB, the algebraA is equalto 4 4complex
AB
matrices and the Hilbert space = = C4. Let be the set of all s×tates of the
AB A B AB
H H ⊗H E
compound system i.e.
= ρ A : ρ 0 and trρ=1 (3)
AB AB
E { ∈ ≥ }
The state ρ is separable [14], if it has the form
AB
∈E
ρ= λ ρA ρB, ρA , ρB , λ 0 and λ =1 (4)
k k ⊗ k k ∈EA k ∈EB k ≥ k
k k
X X
The set sep of all separable states forms a convex subset of . When ρ is not separable, it is
EAB EAB
called inseparable or entangled. Thus
ent = sep (5)
EAB EAB \EAB
If P is a pure state i.e. P is one-dimensional projector, then P is separable iff partial
AB
∈ E
traces tr P and tr P are also projectors. For mixed states, the separability problem is much
A B
more involved. Fortunately, in the case of 4 – level compound system there is a simple necessary
and sufficient condition for separability: ρ is separable iff its partial transposition ρTA is also a
state [15]. Another interesting question is how to measure the amount of entanglement a given
quantum state contains. For a pure state P, the entropy of entanglement
E(P)= tr[(tr P) log (tr P)] (6)
− A 2 A
is essentailly a unique measure of entanglement [16]. For mixed state ρ it seems that the basic
measure of entanglement is the entanglement of formation [17]
E(ρ)=min λ E(P ) (7)
k k
k
X
where the minimum is taken over all possible decompositions
ρ= λ P (8)
k k
k
X
Again,inthecaseof4–levelsystem,E(ρ)canbeexplicitelycomputedanditturnsoutthatE(ρ)
is the function of another useful quantity C(ρ) called concurrence, which also can be taken as a
2
measureofentanglement[11,12]. Sinceinthepaperweuseconcurrencetoquantifyentanglement,
now we discuss its definition. Let
ρ =(σ σ )ρ(σ σ ) (9)
† 2 2 2 2
⊗ ⊗
where ρ is the complex conjugation of the matrix ρ. Define also
ρ=(ρ1/2ρ†ρ1/2)1/2 (10)
Then the concurrence C(ρ) is given by [11, 12]
b
C(ρ)=max(0,2p (ρ) trρ) (11)
max
−
where p (ρ) denotes the maximal eigenvalue of ρ. The value of the number C(ρ) varies from 0
max b b
for separable states, to 1 for maximally entangled pure states.
b b
3 Decay in a system of closely separated atoms
We study the spontaneous emission of two atoms separated by a distance R small compared to
the radiation wavelength . At such distances there is a substantial probability that the photon
emitted by one atom will be absorbed by the other. Thus the dynamics of the system is given by
the master equation [18]
dρ
=Lρ, ρ (12)
AB
dt ∈E
with the following generator L
γ
Lρ= 0 [2σA ρσA+2σB ρσB (σA σA+σB σB)ρ ρ(σA σA+σB σB)]+
2 − + − + − + − + − − + − + −
(13)
γ
[2σA ρσB +2σB ρσA (σA σB +σB σA)ρ ρ(σA σB +σB σA)]
2 − + − + − + − + − − + − + −
where
1
σA =σ I, σB =I σ , σ = (σ iσ ) (14)
1 2
± ±⊗ ± ⊗ ± ± 2 ±
Here γ is the single atom spontaneous emission rate, and γ = gγ is a relaxation constant of
0 0
photon exchange. In the model, g is the function of the distance R between atoms and g 1
→
whenR 0. Inthis sectionweinvestigatethe time evolutionofthe initialdensitymatrixρ ofthe
→
compound system, governed by the semi - group T generated by L. In particular, we will
t t 0
{ } ≥
study the time development of entanglement of ρ, measured by concurrence.
Assumethatthe distance betweenatomsis sosmallthatthe exchangerateγ iscloseto γ and
0
we can use the approximation g = 1. Under this condition we study evolution of the system and
in particular we consider asymptotic states. Direct calculations show that the semi - group T
t
{ }
generated by L with g =1 is relaxing but not uniquely relaxing i.e. there are as many stationary
states as there are initial conditions. More precisely, for a given initial state ρ = (ρ ), the state
jk
3
ρ(t) at time t has the following matrix elements
ρ (t)=e 2γ0tρ
11 − 11
1
ρ12(t)= [e−2γ0t(ρ12+ρ13)+e−γ0t(ρ12 ρ13)]
2 −
1
ρ (t)= [e 2γ0t(ρ +ρ )+e γ0t(ρ ρ )]
13 − 12 13 − 13 12
2 −
ρ (t)=e γ0tρ
14 − 14
1 1 1
ρ22(t)= e−2γ0t(ρ22+ρ33+2Reρ23)+ e−γ0t(ρ22 ρ33)+γ0te−2γ0tρ11+ (ρ22+ρ33 2Reρ23)
4 2 − 4 −
1 1 1
ρ (t)= e 2γ0t(ρ +ρ +2Reρ )+ e γ0t(ρ ρ )+γ te 2γ0tρ (ρ +ρ 2Reρ )
23 − 22 33 23 − 23 32 0 − 11 22 33 23
4 2 − − 4 −
1 1
ρ (t)= e 2γ0t(ρ +ρ )+ e γ0t(2ρ +2ρ +ρ +ρ )+ (ρ ρ )
24 − 12 13 − 12 13 24 34 24 34
− 2 2 −
1 1 1
ρ (t)= e 2γ0t(ρ +ρ +2Reρ ) e γ0t(ρ ρ )+γ te 2γ0tρ + (ρ +ρ 2Reρ )
33 − 22 33 23 − 22 33 0 − 11 22 33 23
4 − 2 − 4 −
1 1
ρ (t)= e 2γ0t(ρ +ρ )+ e γ0t(2ρ +2ρ +ρ +ρ ) (ρ ρ )
34 − 12 13 − 12 13 24 34 24 34
− 2 − 2 −
1 1
ρ44(t)= e−2γ0t(1+ρ11 ρ44+2Reρ23) 2γ0te−2γ0tρ11+ (1+ρ11+ρ44+2Reρ23)
−2 − − 2
and remaining matrix elements can be obtained by hermiticity condition ρ = ρ . In the limit
kj jk
t we obtain asymptotic (stationary) states parametrized as follows
→∞
0 0 0 0
0 α α β
ρas =0 α −α β  (15)
− −
0 β β 1 2α
 − − 
 
where
1 1
α= (ρ +ρ 2Reρ ), β = (ρ ρ ) (16)
22 33 23 24 34
4 − 2 −
We can also compute concurrence of the asymptotic state and the result is:
Concurrence of asymptotic state of the semi - group T generated by L with g =1 equals to
t
{ }
1
C(ρ )=2α = ρ +ρ 2Reρ (17)
as 22 33 23
| | 2| − |
where ρ are the matrix elements of the initial state.
jk
4 Some examples
In this section we consider examples of initial states and its evolution.
I. Pure separable states
Let
ρ=P =P P (18)
Ψ Φ Ψ Φ
⊗ ⊗
where
Ψ Φ
Ψ= 1 , Φ= 1
Ψ2 ∈HA Φ2 ∈HB
(cid:18) (cid:19) (cid:18) (cid:19)
4
are normalized. Then one can check that
1
α= (1 Ψ, Φ 2)
4 −|h i|
(19)
1
β = (Φ 2Ψ Ψ Ψ 2Φ Φ )
2 1 2 2 1 2
2 | | −| |
where , is the inner product in C2. So
h· ·i
1
C(ρ )= (1 Ψ, Φ 2) (20)
as
2 −|h i|
From the formula (21) we see that there are separable initial states for which asymptotic states
are entangled. In particular, the asymptotic state has a maximal concurrence if vectors Ψ and Φ
are orthogonaland its concurrence is zero (the state remains separable) if Ψ, Φ =1.
|h i|
Now we discuss some special cases.
a.When one atom is in excited state and the other is in ground state
Ψ= 1 , Φ= 0
| i | i
the asymptotic (mixed) state is given by
0 0 0 0
0 1 1 0
ρ = 4 −4
as
0 1 1 0
 −4 4 
 
0 0 0 0
 
 
Itcanalsobeshownthatinthiscasetherelaxationprocessproducesthestateρ withconcurrence
t
1 e γ0t
−
C(ρ )= −
t
2
increasing to the maximal value equal to 1/2. Thus two atoms initially in separable state become
entangledforalltandtheasymptotic(steady)stateattainsthemaximalamountofentanglement.
b. When two atoms are in excited states
Ψ=Φ= 1
| i
the asymptotic state equals to
0 0
| i⊗| i
Thus the relaxation process brings two atoms into ground states.
c. The state 0 0 is stationary state for semi - group T .
t
| i⊗| i { }
II. Pure maximally entangled states
Let
a2 a√1 a2e iθ1 a√1 a2e iθ2 a2e i(θ1+θ2)
2 2− − 2− − − 2 −
 a√1 a2eiθ1 1 a2 1 a2ei(θ1 θ2) a√1 a2e iθ2
ρ=Q(a,θ ,θ )= 2− −2 −2 − − 2− −
1 2  a√12−a2eiθ2 1−2a2e−i(θ1−θ2) 1−2a2 −a√12−a2e−iθ1
 a2ei(θ1+θ2) √1 a2eiθ2 a√1 a2eiθ1 a2 
− 2 − 2− − 2− 2 
 
5
where a [0,1], θ ,θ [0,2π]. Pure states Q(a,θ ,θ ) are maximally entangled and form a
1 2 1 2
∈ ∈
familyofallmaximallyentangledstatesofthe 4-levelsystem[19]. Itturnsoutthatρ isdefined
as
by
1
α= (1 a2)(1 cos(θ θ ))
1 2
4 − − −
(21)
1
β = a 1 a2(e iθ1 e iθ2)
− −
4 − −
p
and
1
C(ρ )= (1 a2)(1 cos(θ θ )) (22)
as 1 2
2 − − −
Fromtheformula(23)weseethatthereareinitialmaximallyentangledstateswhichasymptotically
become separable (a = 1 or θ θ = 2kπ) and such that the asymptotic concurrence is greater
1 2
−
then 0. States with a=0 and θ θ =(2k+1)π remain maximally entangled. For example the
1 2
−
state
1
(0 1 1 0 ) (23)
√2 | i⊗| i−| i⊗| i
is stable. On the other hand, the concurrence of
1
(0 1 + 1 0 ) (24)
√2 | i⊗| i | i⊗| i
goes to zero faster then the concurrence of
1
(0 0 + 1 1 ) (25)
√2 | i⊗| i | i⊗| i
as shown on Fig. 1. below. In Dicke’s theory of spontaneous radiation processes the state (24)
is called subradiant whereas the state (25) has half the lifetime of a single atom and therefore is
called superradiant [1]. We see that the time-dependence of concurrence reflects the relaxation
properties of those states.
6
C(ρ )
t
1
0.8
0.6
0.4
0.2
1 2 3 4 5 6 7 8 γ0t
Fig. 1. Concurrence as the function of time for initial states: 1 (0 0 + 1 1 )
√2 | i⊗| i | i⊗| i
(dotted line) and 1 (0 1 + 1 0 ) (solid line).
√2 | i⊗| i | i⊗| i
III. Some classes of mixed states
a. Bell - diagonal states. Let
ρ =p Φ+ Φ+ +p Φ Φ +p Ψ+ Ψ+ +p Ψ Ψ (26)
B 1 2 − − 3 4 − −
| ih | | ih | | ih | | ih |
where Bell states Φ and Ψ are given by
± ±
1 1
Φ = (0 0 1 1 ), Ψ = (1 0 0 1 ) (27)
± ±
√2 | i⊗| i ± | i⊗| i √2 | i⊗| i ± | i⊗| i
It is known that all p [0,1/2], ρ is separable, while for p > 1/2, ρ is entangled with
i B 1 B
∈
concurrence equalto 2p 1 (similarly for p , p p ) [20]. Now the asymptotic state has the form
1 2 3 4
−
0 0 0 0
0 p4 p4 0 
ρ = 2 − 2 (28)
as 0 p4 p4 0 
 − 2 2 
 
0 0 0 1 p4
 − 
 
with concurrence C(ρ ) = p . So even when the initial state is separable, the asymptotic state
as 4
becomes entangled.
b. Werner states [21]. Let
I
ρ =(1 p) 4 +pΦ+ Φ+ (29)
W
− 4 | ih |
7
If p>1/3, ρ is entangled with concurrence equal to (3p 1)/2. On the other hand
W
−
0 0 0 0
0 1−p p−1 0 
8 8
ρ = (30)
as 0 p−81 1−8p 0 
 
0 0 0 3+p
 4 
 
 
has the concurrence C(ρas) = 1−4pI, so the asymptotic states are entangled for all p 6= 1. Notice
that even completely mixed state 4 evolves to entangled asymptotic state.
4
c. Maximally entangled mixed states . The states
h(δ) 0 0 δ/2
 0 1 2h(δ) 0 0  1/3 δ [0,2/3]
ρ = − , h(δ)= ∈ (31)
M
 0 0 0 0  (δ/2 δ [2/3,1]
  ∈
 
δ/2 0 0 h(δ)
 
 
are conjectured to be maximally entangled for a given degree of inpurity measured by trρ2 [22].
According to (18) the concurrence of the asymptotic state is given by
1
C(ρ )= (1 2h(δ)) (32)
as
2 −
C(ρ )
t
1
0.8
0.6
0.4
0.2
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 trρ2
M
Fig. 2. Concurrence of ρ (solid line) and ρ (dotted line) as the function of trρ2
M as M
Even in that case, there are initial states (for sufficiently small trρ2 ) such that the asymptotic
M
state is more entangled (see Fig. 2.).
8
5 Remarks on general case
In the case of arbitrary distance between the atoms i.e. when g [0,1), semi-group generated
∈
by L is uniquely relaxing, with the asymptotic state 0 0 . Thus, for any initial state ρ, the
| i⊗| i
concurrenceC(ρ )approaches0whent . Butitdoesnotmeanthatthefunctiont C(ρ )is
t t
→∞ →
always monotonic. The general form of C(ρ ) is rather involved, so we consider only some special
t
cases.
1. Let the initial state of the compound system be equal to 0 1 . This states evolves to
| i⊗| i
0 0 0 0
ρ =0 12e−γ0t(coshγt+1) −21e−γ0tsinhγt 0  (33)
t
0 −12e−γ0tsinhγt 21e−γ0t(coshγt−1) 0 
 
0 0 0 1 e−γ0tcoshγt
 − 
 
with concurrence
C(ρt)=e−γ0tsinhγt (34)
In the interval [0,t ], where
γ
1 γ +γ
0
t = ln
γ
2γ γ γ
0
−
the function (34) is increasing to its maximal value
γ +γ
0
γ γ0+γ − 2γ
C =
max
γ γ γ γ
0− (cid:18) 0− (cid:19)
whereas for t>t , C(ρ ) decreases to 0. Thus for any nonzero photon exchange rate γ, dynamics
γ t
given by the semi - group T produces some amount of entanglement between two atoms which
t
{ }
are initially in the ground state and excited state. Note that the maximal value of C(ρ ) depends
t
only on emission rates γ and γ.
0
2. For the initial states
1
Ψ± = (0 1 1 0 )
√2 | i⊗| i±| i⊗| i
the relaxation to the asymptotic state 0 0 is given by density matrices
| i⊗| i
0 0 0 0
ρ±t =00 −1221ee−−((γγ00±±γγ))tt) −2121ee−−(γ(γ0±0±γγ)t)t 00  (35)
 
0 0 0 1 e (γ0 γ)t
 − − ± 
 
with the corresponding concurrence
C(ρ±t )=e−(γ0±γ)t
ThestateΨ isnolongerstable(asinthecaseofγ =γ ),butduringtheevolutionitsconcurrence
− 0
goes to zero slower than C(ρ+) (Fig. 3.). For γ close to γ , Ψ is almost stable.
t 0 −
9
C(ρ )
t
1
0.8
0.6
0.4
0.2
2 4 6 8 10 γ0t
Fig. 3. C(ρ+t ) (dotted line) and C(ρ−t ) (solid line) for γ/γ0 =0.99
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