Dynamics of entanglement in Two-Qubit Open System Interacting with a Squeezed Thermal Bath via Dissipative interaction

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by  Subhashish Banerjee

Dynamics of entanglement in Two-Qubit Open System Interacting with a Squeezed
Thermal Bath via Dissipative interaction
Subhashish Banerjee,1,2,∗ V. Ravishankar,1,3,† and R. Srikanth4,1,‡
1Raman Research Institute, Bangalore- 560080, India
2Chennai Mathematical Institute, Padur PO, Siruseri- 603103, India
3Indian Institute of Technology, Kanpur, India
4Poornaprajna Institute for Scientific Research, Bangalore- 560080, India
Westudythedynamicsofentanglementinatwo-qubitsysteminteractingwithasqueezedthermal
bathviaadissipativesystem-reservoirinteractionwiththesystemandreservoirassumedtobeina
separableinitialstate. Theresultingentanglementisstudiedbymakinguseofarecentlyintroduced
measureofmixedstateentanglementviaaprobabilitydensityfunctionwhichgivesastatisticaland
9 geometrical characterization of entanglement by exploring the entanglement content in the various
0 subspaces spanning the two-qubit Hilbert space. We also make an application of the two-qubit
0 dissipative dynamics toa simplified model of quantumrepeaters.
2
PACSnumbers: 03.65.Yz,03.67.Mn,03.67.Bg,03.67.Hk
n
a
J

4

I. INTRODUCTION
]
h
p
Openquantumsystemstakeintoaccounttheeffectoftheenvironment(reservoirorbath)onthedynamicalevolution
-
t of the system of interest thereby providing a natural route for discussing damping and dephasing. One of the first
n
testinggroundsforopensystemideaswasinquantumoptics[1]. Itsapplicationtootherareasgainedmomentumfrom
a
the works of Caldeira and Leggett [2], and Zurek [3], among others. The total Hamiltonian is H =H +H +H ,
u S R SR
q where S stands for the system, R for the reservoir and SR for the system-reservoir interaction. Depending upon
[ the system-reservoir (S R) interaction, open systems can be broadly classified into two categories, viz., quantum
−
non-demolition(QND) or dissipative. A particular type ofquantum nondemolition (QND) S R interactionis given
1 −
by a class of energy-preserving measurements in which dephasing occurs without damping the system, i.e., where
v
4 [HS,HSR] = 0 while the dissipative systems correspond to the case where [HS,HSR] = 0 resulting in decoherence
6
0 along with dissipation [4].
4 A prototype of dissipative open quantum systems, having many applications, is the quantum Brownian motion
0 of harmonic oscillators. This model was studied by Caldeira and Leggett [2] for the case where the system and its
.
1 environment were initially separable. The above treatment of the quantum Brownian motion was generalized to the
0 physicallyreasonableinitial conditionofa mixedstate ofthe systemandits environmentby HakimandAmbegaokar
9 [5],SmithandCaldeira[6],Grabert,SchrammandIngold[7],andforthecaseofasysteminaStern-Gerlachpotential
0 [8], and also for the quantum Brownian motion with nonlinear system-environment couplings [9], among others.
:
v The interest in the relevance of open system ideas to quantum information has increased in recent times because
i of the impressive progress made on the experimental front in the manipulation of quantum states of matter towards
X
quantum information processing and quantum communication. Myatt et al. [10] and Turchette et al. [11] have
r performed a series of experiments in which they induced decoherence and decay by coupling the atom (their system-
a
S) to engineered reservoirs,in which the coupling to, and the state of, the environment are controllable.
Quantum entanglement is the inherent property of a system to exhibit correlations, the physical basis being the
non-local nature of quantum mechanics [12], and hence is a property that is exclusively quantum in nature. Entan-
glement plays a central role in quantum information theory [13] as in interesting non-classical applications such as
quantum computation[14] andquantum errorcorrection[15]. A number ofmethods have been proposedfor creating
entanglement involving trapped atoms [16, 17, 18].
An important issue is to study how quantum entanglement is affected by noise, which can be thought of as a
manifestation of an open system effect [19]. In [20] entanglement of a two-mode squeezed state in a phase-sensitive
Gaussian environment was studied and the criteria for the necessary and sufficient condition for separability of
Gaussiancontinuous-variablestates[21]wasemployedasameasureofentanglement. In[22]theentanglementbetween
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
‡Electronicaddress: [email protected]
2
charge qubits induced by a common dissipative environment was analyzed using concurrence as the measure. Some
recent experimental investigations on the influence of decoherence on the dynamics of entanglement have been made
in [23, 24]. In a related work [25], this issue was taken up with the noise coming from the effect of the environment
modelled by a QND S R interaction. Here we complement this program by studying the effect of noise, modelled
−
by a dissipative S R interaction with the reservoirin an initial squeezed-thermal state [4, 26], on the entanglement
−
evolution between two spatially separated (and initially uncorrelated) qubits, brought out by interaction with the
bath. This would be of relevance to evaluate the performance of two-qubit gates in practical quantum information
processing systems.
Since we are dealing here with a two qubit system which very rapidly evolves into a mixed state, a study of
entanglement would necessarily involve a measure of entanglement for mixed states. Entanglement of a bipartite
system in a pure state is unambigious and well defined. However, mixed state entanglement (MSE) is not so well
defined. Thus,althoughanumberofcriteriasuchasentanglementofformation[27,28,29]andseparability[30]exist,
there is a realization[27] that a single quantity is inadequate to describe MSE. This was the principal motivation for
the development of a new prescriptionof MSE [31] in which it is characterizednot as a function, but as a probability
densityfunction(PDF).Theknownprescriptionssuchasconcurrenceandnegativityemergeasparticularparameters
that characterize the probability density. We will principally make use of this measure in our study of entanglement
in the two-qubit system.
Theplanofthepaperisasfollows. InSectionII,werecapitulateforconsistency,therecentlydevelopedentanglement
measure ofMSE [31]. In SectionIII, the master equationdescribing the dynamicalevolutionof the two-qubitsystem
interacting with a squeezed thermal bath, via a dissipative S R interaction, is given which is then used in Section
−
IV,to study indetailthe dynamicsofthe systeminteractingwithavacuumbathwith zerobathsqueezinginSection
IV(A) and with a generalsqueezedthermal bath in Section IV(B). SectionV deals with the entanglement analysisof
the two-qubit open system using the PDF as a measure of entanglement. We compare it with the usual measure of
MSE,concurrence. WedwellonthescenarioswherethetwoqubitseffectivelyinteractvialocalizedS Rinteractions,
−
called the independent decoherence model, as also when they interact collectively with the bath, called the collective
decoherence model. In Section VI, we make a brief applicationof the model to practicalquantum communication, in
the form of a quantum repeater [32, 33]. In Section VII, we make our conclusions.
II. CHARACTERIZATION OF MIXED STATE ENTANGLEMENT THROUGH A PROBABILITY
DENSITY FUNCTION
Here we briefly recapitulate the characterizationof mixed state entanglement (MSE) through a PDF as developed
in [31]. As pointed out in the Introduction, the above criterion was evolved from the motivation that for the charac-
terizationof MSE, a single parameteris inadequate. The basic idea is to express the PDF of entanglement ofa given
system density matrix (in this case, a two-qubit) in terms of a weighted sum over the PDF’s of projection operators
spanning the full Hilbert space of the system density matrix. The PDF of a system in a state which is a projection
operator ρ= 1 Π of rank M is defined as:
M M
d δ( )
( )= HΠM Eψ −E , (1)
PΠM E R d
HΠM
R
where d is the volume measure for , which is the subspace spanned by Π . The volume measure is
HΠM HΠM M
determRined by the invariant Haar measure associated with the group of automorphisms of dHΠM, modulo the
stabilizer group of the reference state generating HΠM. Thus for a one dimensional projection oRperator, representing
a pure state, the group of automorphisms consists of only the identity element and the PDF is simply given by the
Diracdelta. Indeed, ifρ=Π ψ ψ , the PDFhasthe form ( )=δ( )therebyresultinginthe description
1 ρ ψ
≡| ih | P E E−E
ofpurestateentanglement,asexpected,byasinglenumber. Theentanglementdensityofasysteminageneralmixed
state ρ is given by resolving it in terms of nested projection operators with appropriate weights as
ρ = (λ λ )Π +(λ λ )Π +.......(λ λ )Π +λ Π
1 2 1 2 3 2 N−1 N N−1 N N
− − −
N
Λ Π , (2)
M M
≡
MX=1
where the projections are Π = M ψ ψ , with M = 1,2,...,N and the eigenvalues λ λ ...., i.e., the
M j=1| jih j| 1 ≥ 2 ≥
eigenvalues are arrangedin a non-iPncreasing fashion. Thus the PDF for the entanglement of ρ is given by
N
( )= ω ( ), (3)
Pρ E MPΠM E
MX=1
3
wheretheweightsoftherespectiveprojections ( )aregivenbyω =Λ /λ . Foratwoqubitsystem,thedensity
PΠM E M M 1
matrix would be represented as a nested sum over four projection operators, Π , Π , Π , Π corresponding to one,
1 2 3 4
two,threeandfourdimensionalprojections,respectively,withΠ correspondingtoapurestateandΠ corresponding
1 4
to a a uniformly mixed state, is a multiple of the identity operator. The most interesting structure is present in Π ,
2
the two-dimensional projection, which is characterized by three parameters, viz. , the entanglement at which
cusp
E
the PDF diverges, , the maximum entanglement allowed and ( ), the PDF corresponding to . The
max 2 max max
E P E E
three dimensional projection Π is characterized by the parameter , which parametrizes a discontunity in the
3 ⊥
E
entanglement density function curve. By virtue of the convexity of the sum over the nested projections (2), it can
be seen that the concurrence of any state ρ is given by the inequality (λ λ ) +(λ λ ) . Thus
Cρ ≤ 1 − 2 CΠ1 2 − 3 CΠ2
while the concurrence for a three and four dimensional projection is identically zero, through the PDF one is able
to make a statement about the entanglement content of these spaces. The fact that the PDF (3) enables us to
study entanglement of a physical state by exploiting the richness inherent in the subspaces spanned by the system
Hilbert space makes it an attractive statistical and geometric characterization of entanglement, of which an explicit
illustration is made in Section V.
III. TWO-QUBIT DISSIPATIVE INTERACTION WITH A SQUEEZED THERMAL BATH
We considerthe Hamiltonian, describing the dissipative interactionofN qubits (two-levelatomic system) with the
bath (modelled as a 3-D electromagnetic field (EMF)) via the dipole interaction as [34]
H = H +H +H
S R SR
N N
= ¯hω Sz + ¯hω (b† b +1/2) i¯h [~µ .~g (~r )(S++S−)b h.c.]. (4)
n n k ~ks ~ks − n ~ks n n n ~ks−
nX=1 X~ks X~ks nX=1
Here ~µ are the transition dipole moments, dependent on the different atomic positions ~r and
n n
S+ = e g , S− = g e , (5)
n | nih n| n | nih n|
are the dipole raising and lowering operators satisfying the usual commutation relations and
1
Sz = (e e g g ), (6)
n 2 | nih n|−| nih n|
is the energy operator of the nth atom, while b† , b are the creation and annihilation operators of the field mode
~ks ~ks
(bath) ~ks with the wave vector ~k, frequency ω and polarization index s = 1,2 with the system-reservoir (S-R)
k
coupling constant
ω
~g (~r )=( k )1/2~e ei~k.rn. (7)
~ks n 2ε¯hV ~ks
Here V is the normalization volume and ~e is the unit polarization vector of the field. It can be seen from Eq.
~ks
(7) that the S-R coupling constant is dependent on the atomic position r . This leads to a number of interesting
n
dynamicalaspects,asseenbelow. Fromnowwewillconcentrateonthecaseoftwoqubits. Assumingseparableinitial
conditions,andtaking atrace overthe bath the reduceddensity matrixofthe qubitsystemin the interactionpicture
and in the usual Born-Markov,rotating wave approximation (RWA) is obtained as [34]
2
dρ i 1
= [H ,ρ] Γ [1+N˜](ρS+S−+S+S−ρ 2S−ρS+)
dt −¯h S˜ − 2 ij i j i j − j i
iX,j=1
2 2
1 1
Γ N˜(ρS−S++S−S+ρ 2S+ρS−)+ Γ M˜(ρS+S++S+S+ρ 2S+ρS+)
− 2 ij i j i j − j i 2 ij i j i j − j i
iX,j=1 iX,j=1
2
1
+ Γ M˜∗(ρS−S−+S−S−ρ 2S−ρS−). (8)
2 ij i j i j − j i
iX,j=1
In Eq. (8)
N˜ =N (cosh2(r)+sinh2(r))+sinh2(r), (9)
th
4
1
M˜ = sinh(2r)eiΦ(2N +1) ReiΦ(ω0), (10)
th
−2 ≡
with
ω +ω
1 2
ω = , (11)
0
2
and
1
N = . (12)
th h¯ω
ekBT 1
−
Here N is the Planck distribution giving the number of thermal photons at the frequency ω and r, Φ are squeezing
th
parameters. The analogous case of a thermal bath without squeezing can be obtained from the above expressions
by setting these squeezing parameters to zero, while setting the temperature (T) to zero one recovers the case of the
vacuum bath. Eq. (8), for a single qubit case, canbe solvedusing the Blochvector formalism(cf. [19], [35]) and also
in the superoperator formalism [36]. Here the assumption of perfect matching of the squeezed modes to the modes
of the EMF is made along with, the squeezing bandwidth being much larger than the atomic linewidths. Also, the
squeezing carrier frequency is taken to be tuned in resonance with the atomic frequencies.
In Eq. (8),
2 2
H =h¯ ω Sz +h¯ Ω S+S−, (13)
S˜ n n ij i j
nX=1 Xi,j
(i6=j)
where
3 cos(k r )
Ω = Γ Γ [1 (µˆ.rˆ )2] 0 ij +[1 3(µˆ.rˆ )2]
ij i j ij ij
4 (cid:20)− − k r −
p 0 ij
sin(k r ) cos(k r )
0 ij 0 ij
[ + ] . (14)
× (k r )2 (k r )3 (cid:21)
0 ij 0 ij
Here µˆ =µˆ =µˆ andrˆ are unit vectorsalong the atomic transitiondipole moments and~r =~r ~r , respectively.
1 2 ij ij i j
−
Also k = ω /c, with ω being as in Eq. (11), r = ~r . The wavevector k = 2π/λ , λ being the resonant
0 0 0 ij ij 0 0 0
| |
wavelength, occuring in the term k r sets up a length scale into the problem depending upon the ratio r /λ .
0 ij ij 0
This is thus the ratio between the interatomic distance and the resonant wavelength, allowing for a discussion of the
dynamics intworegimes: (A). independentdecoherencewhere k .r rij 1and(B). collectivedecoherencewhere
0 ij ∼ λ0 ≥
k .r rij 0. The case (B) of collective decoherence would arise when the qubits are close enough for them
0 ij ∼ λ0 →
to feel the bath collectively or when the bath has a long correlation length (set by the resonant wavelength λ ) in
0
comparisonto the interqubit separationr . Ω (14) is a collective coherent effect due to the multi-qubit interaction
ij ij
and is mediated via the bath through the terms
ω3µ2
Γ = i i . (15)
i 3πε¯hc3
ThetermΓ ispresenteveninthecaseofsingle-qubitdissipativesystembathinteraction[35,36]andisthespontaneous
i
emission rate, while
Γ =Γ = Γ Γ F(k r ), (16)
ij ji i j 0 ij
p
where i=j with
6
3 sin(k r )
F(k r ) = [1 (µˆ.rˆ )2] 0 ij +[1 3(µˆ.rˆ )2]
0 ij ij ij
2(cid:20) − k r −
0 ij
cos(k r ) sin(k r )
0 ij 0 ij
[ ] . (17)
× (k r )2 − (k r )3 (cid:21)
0 ij 0 ij
Γ (16) is the collective incoherent effect due to the dissipative multi-qubit interaction with the bath. For the case
ij
of identical qubits, as considered here, Ω =Ω , Γ =Γ and Γ =Γ =Γ.
12 21 12 21 1 2
5
IV. DYNAMICS OF THE TWO-QUBIT DISSIPATIVE INTERACTION WITH A VACUUM AND
SQUEEZED THERMAL BATH
Herewe presentthe solutionsofthe density matrixequation(8)for the caseofatwo-qubitsysteminteractingwith
a (A). vacuum bathand(B). squeezedthermalbath. These results willbe ofuse inthe investigationofthe dynamics
of entanglement subsequently.
A. Vacuum bath
Here we proceed as in [34] and obtain the reduced density matrix from Eq. (8), setting T and bath squeezing to
zero,andbygoingovertothedressedstatebasis,ofthecollectivetwo-qubitdynamics,obtainedfromtheHamiltonian
H (13) as
S˜
g = g g ,
1 2
| i | i| i
1
s = (e g + g e ),
1 2 1 2
| i √2 | i| i | i| i
1
a = (e g g e ),
1 2 1 2
| i √2 | i| i−| i| i
e = e e , (18)
1 2
| i | i| i
with the corresponding eigenvalues being E = ¯hω , E =h¯Ω , E = ¯hΩ and E =h¯ω . The reduced density
g 0 s 12 a 12 g 0
− −
matrix in the dressed state basis (18) can be obtained from Eq. (8) as
dρ i dρ dρ
= [H ,ρ]+( ) +( ) , (19)
as s a
dt −¯h dt dt
where
H =h¯[ω (e e g g )+Ω (s s a a)], (20)
as 0 12
| ih |−| ih | | ih |−| ih |
dρ 1
( ) = (Γ+Γ )[(e e + s s)ρ+ρ(e e + s s)
s 12
dt −2 | ih | | ih | | ih | | ih |
2(s e + g s)ρ(e s + s g )], (21)
− | ih | | ih | | ih | | ih |
and
dρ 1
( ) = (Γ Γ )[(e e + a a)ρ+ρ(e e + a a)
a 12
dt −2 − | ih | | ih | | ih | | ih |
2(a e g a)ρ(e a a g )]. (22)
− | ih |−| ih | | ih |−| ih |
From the dressedstate basis (18), it can be seen that the two-qubitproblem canbe thought of as an equivalent four-
level system. For the case where r /λ 0, i.e., when the interatomic separation is much smaller than the resonant
ij 0
→
wavelength,constitutingtheDickemodel[37],(dρ/dt) =0andtheproblemreducestoaneffectivethree-levelsystem.
a
The Eq. (19) can be solved to yield the various density matrix elements as follows:
ρ (t)=e−2Γtρ (0), (23)
ee ee
(Γ+Γ )
ρ (t)=e−(Γ+Γ12)tρ (0)+ 12 (1 e−(Γ−Γ12)t)e−(Γ+Γ12)tρ (0), (24)
ss ss ee
(Γ Γ ) −
12
−
(Γ Γ )
ρ (t)=e−(Γ−Γ12)tρ (0)+ − 12 (1 e−(Γ+Γ12)t)e−(Γ−Γ12)tρ (0), (25)
aa aa ee
(Γ+Γ ) −
12
ρ (t) = ρ (0)+(1 e−(Γ+Γ12)t)ρ (0)+(1 e−(Γ−Γ12)t)ρ (0)
gg gg ss aa
− −
(Γ+Γ ) 2 (Γ+Γ ) (Γ Γ )
+ 12 1 [ 12 (1 e−(Γ−Γ12)t)+ − 12 ]e−(Γ+Γ12)t
(cid:20) 2Γ n − (Γ Γ12) 2 − 2 o
−
(Γ Γ ) (Γ Γ )
+ − 12 (1 e−(Γ−Γ12)t) − 12 (1 e−2Γt) ρ (0). (26)
ee
(Γ+Γ12)n − − 2Γ − o(cid:21)
6
The Eqs. (23) to (26) give the dynamics of the population of the two-qubit system interacting with a vacuum bath
andρ (t)+ρ (t)+ρ (t)+ρ (t)=ρ (0)+ρ (0)+ρ (0)+ρ (0). The off- diagonalterms ofthe density matrix
ee ss aa gg ee ss aa gg
are:
ρes(t) = e−i(ω0−Ω12)te−12(3Γ+Γ12)tρes(0),
ρ (t) = ρ∗ (t), (27)
se es
ρ (t) = e−i2ω0te−Γtρ (0),
eg eg
ρ (t) = ρ∗ (t), (28)
ge eg
ρea(t) = e−i(ω0+Ω12)te−21(3Γ−Γ12)tρea(0),
ρ (t) = ρ∗ (t), (29)
ae ea
ρ (t) = e−i2Ω12te−Γtρ (0),
sa sa
ρ (t) = ρ∗ (t), (30)
as sa
ρag(t) = e−i(ω0−Ω12)te−21(Γ−Γ12)tρag(0)− (Γ(Γ2+−4ΓΩ122))e−i(ω0−Ω12)t
12
e−12(Γ−Γ12)t 2Ω12e−Γtsin(2Ω12t)+Γ(1 e−Γtcos(2Ω12t)) ρea(0)
× −
+ i(Γ(Γ2+−4ΓΩ122)(cid:2))e−i(ω0−Ω12)te−12(Γ−Γ12)t 2Ω12(1−e−Γtcos(2Ω(cid:3)12t))
12 (cid:2)
Γe−Γtsin(2Ω t) ρ (0),
12 ea
−
ρ (t) = ρ∗ (t), (cid:3) (31)
ga ag
ρsg(t) = e−i(ω0+Ω12)te−12(Γ+Γ12)tρsg(0)+ (Γ(Γ2++4ΓΩ122))e−i(ω0+Ω12)t
12
e−12(Γ+Γ12)t 2Ω12e−Γtsin(2Ω12t)+Γ(1 e−Γtcos(2Ω12t)) ρes(0)
× −
+ i(Γ(Γ2++4ΓΩ122)(cid:2))e−i(ω0+Ω12)te−21(Γ+Γ12)t 2Ω12(1−e−Γtcos(2Ω(cid:3)12t))
12 (cid:2)
Γe−Γtsin(2Ω t) ρ (0),
12 es
−
ρ (t) = ρ∗ (t). (cid:3) (32)
gs sg
The sixteen density matrix elements given by Eqs. (23) to (32) completely solve the master equation (19) describing
the dynamics of the two-qubit system interacting with the vacuum bath. For the case of the Dicke model, where
r /λ 0,i.e.,whentheinteratomicseparationis muchsmallerthantheresonantwavelength,(dρ/dt) =0andthe
ij 0 a
→
solution of the effective three-level system can be extracted out of the above equations. The conditions under which
the Dicke model is obtained is analogous to the case of collective decoherence for the case of two-qubit interaction
with a bath via a quantum nondemolitioninteraction(QND) [25]. There it was found that for the case of interaction
with a thermal (and also a vacuum) bath, the subspace spanned by e ,g , g ,e is a decoherence-free subspace,
1 2 1 2
{| i | i}
implying that the matrix elements ρ ,ρ ρ and ρ remain invariant inspite of the
e1,g2;e1,g2 g1,e2;g1,e2 e1,g2;g1,e2 g1,e2;e1,g2
system’s interaction with a bath. However, from Eqs. (23)–(32) it is clear that none of the matrix elements is
invariant as a function of time, reflecting the greater complexity of the dissipative interaction.
B. Squeezed thermal bath
Hereweconsiderthetwo-qubitdynamicsresultingfromaninteractionwithasqueezedthermalbath,i.e.,makeuse
of Eq. (8). The equations of the reduced density matrix (8) taken in the two-qubit dressed state basis (18) are not
allmutuallycoupled, butdivide into fourirreducible blocksA,B,C,D,thereby reducingthe task fromanevaluation
of fifteen coupled linear differential equations to that of a maximum of four coupled equations. Thus we have:
7
Block A:
ρ˙ (t) = 2Γ(N˜ +1)ρ (t)+N˜ (Γ+Γ )ρ (t)+(Γ Γ )ρ (t) +Γ M˜ ρ (t),
ee ee 12 ss 12 aa 12 u
− n − o | |
ρ˙ (t) = (Γ+Γ ) N˜ +(1+3N˜)ρ (t) ρ (t)+N˜ρ (t)+ M˜ ρ (t) ,
ss 12 ss ee aa u
− n− − | | o
ρ˙ (t) = (Γ Γ ) N˜ (1+3N˜)ρ (t)+ρ (t) N˜ρ (t)+ M˜ ρ (t) ,
aa 12 aa ee ss u
− n − − | | o
ρ˙ (t) = 2M˜ Γ 4ω ρ sin(Φ+χ) (2N˜ +1)Γρ 2M˜ (Γ+2Γ )ρ (t) (Γ 2Γ )ρ (t) . (33)
u 12 0 ge u 12 ss 12 aa
| | − | | − − | |n − − o
Here
ρ (t)=eiΦρ (t)+h.c., (34)
u ge
ρ = ρ eiχ and Φ is as in Eq. (10). Also ρ (t)=1 ρ (t) ρ (t) ρ (t). The Eqs. (33) give the dynamics of
ge ge gg aa ss ee
| | − − −
the population of the two-qubit system interacting with a squeezed thermal bath while the off-diagonal terms, given
by the Blocks B, C and D are:
Block B:
1
ρ˙ (t) = i(ω Ω ))ρ (t) (3Γ+Γ )+2N˜(2Γ+Γ ) ρ (t)+N˜(Γ+Γ )ρ (t)
es 0 12 es 12 12 es 12 sg
− − − 2n o
+ M˜Γ ρ (t) M˜(Γ+Γ )ρ (t),
12 gs 12 se
−
ρ˙ (t) = ρ˙∗ (t),
se es
1
ρ˙ (t) = i(ω Ω ))ρ (t) (Γ+Γ )+2N˜(2Γ+Γ ) ρ (t)+(1+N˜)(Γ+Γ )ρ (t)
gs 0 12 gs 12 12 gs 12 se
− − − 2n o
+ M˜∗Γ ρ (t) M˜∗(Γ+Γ )ρ (t),
12 es 12 sg
−
ρ˙ (t) = ρ˙∗ (t). (35)
sg gs
Block C:
ρ˙ (t) = i2Ω ρ (t) Γ(1+2N˜)ρ (t),
as 12 as as
−
ρ˙ (t) = ρ˙∗ (t). (36)
sa as
Block D:
1
ρ˙ (t) = i(ω +Ω ))ρ (t) (3Γ Γ )+2N˜(2Γ Γ ) ρ (t) N˜(Γ Γ )ρ (t)
ea 0 12 ea 12 12 ea 12 ag
− − 2n − − o − −
+ M˜Γ ρ (t) M˜(Γ Γ )ρ (t),
12 ga 12 ae
− −
ρ˙ (t) = ρ˙∗ (t),
ae ea
1
ρ˙ (t) = i(ω Ω ))ρ (t) (Γ Γ )+2N˜(2Γ Γ ) ρ (t) (1+N˜)(Γ Γ )ρ (t)
ga 0 12 ga 12 12 ga 12 ae
− − 2n − − o − −
+ M˜∗Γ ρ (t) M˜∗(Γ Γ )ρ (t),
12 ea 12 ag
− −
ρ˙ (t) = ρ˙∗ (t). (37)
ag ga
The Eqs. (36) can be trivially solved to yield
ρ (t) = e[i2Ω12−Γ(1+2N˜)]tρ (0)
as as
ρ (t) = ρ∗ (t), (38)
sa as
while the blocks A, B and D consist of four linear coupled differential equations which can be written in matrix form
as:
P˙(t)= QP(t)+W, (39)
−
where Q is a time-independent 4 4 matrix and P(t), W are 4 1 column vectors. The Eq. (39) has the general
× ×
solution
P(t)=(Ve−DtV−1)P(0)+VD−1(1 e−Dt)V−1W. (40)
−
8
purity purity
1 1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
T T
2 4 6 8 10 12 14 2 4 6 8 10 12 14
(a) (b)
FIG.1: Purityasafunction oftemperatureT (inunitswhere¯h≡kB =1)for (a)theindependentdecoherencemodel, where
rij/λ0 (14)is≥1and(b)thecollectivedecoherencemodel,whererij/λ0 (14)is≈0. Herewithr12 istheinter-qubitdistance.
Thelarge-dashed,boldanddottedcurvescorrespondtoevolutiontimet=1.0andbathsqueezingparameter(9,10)r=−0.5,
1.0 and 1.5, respectively. Here and in all the subsequent figures, the squeezing parameter Φ (10) is set equal to zero. Also ω0
(11) and the bath parameter Γ (15), are set equal to 1.0 and 0.05, respectively. All the inter-qubit distances are defined on
the scale of the resonant wavelength coming from the wavevector k (7) as a result of the position dependent couplings of the
qubits with the bath. In figure (a) related to the independent decoherence model, kr12 is set equal to 1.5 while in figure (b)
related to thecollective decoherence model, kr12 is set equal to 0.08.
Here V is the vector composed of the eigenvectors of the matrix Q, while D is composed of its eigenvalues. We solve
Eq. (40) by numerically obtaining the eigenvalues and eigenvectors of the matrix Q for the Blocks A, B, C, and D.
Figures(1(a)),(b)depictthebehaviorofpurity,definedhereasTr(ρ2(t))forρ(t)asobtainedinthissubsectionfor
the independent(k .r =0)andcollective(k .r 0)decoherencemodel,respectively,asafunctionoftemperature
0 ij 0 ij
6 →
T for an evolution time t and bath squeezing r (9, 10). In all the figures in this article, we consider the initial state
of one qubit in the excited state e and the other in the ground state g , i.e., e g and µˆ.rˆ (17) is equal to
1 2 1 2 ij
| i | i | i| i
zero. It can be seen that with the increase in temperature, as also evolution time t and bath squeezing r, the system
becomes more mixed and hence loses its purity.
V. ENTANGLEMENT ANALYSIS
In this section, we will study the development of entanglement in the two qubit system, both for the independent
as well as the collective decoherence model interacting with a squeezed thermal bath. A well known measure of MSE
is the concurrence [28] defined as
=max(0, λ λ λ λ ), (41)
1 2 3 4
C − − −
p p p p
where λ are the eigenvalues of the matrix
i
R=ρρ˜, (42)
with ρ˜=σ σ ρ∗σ σ and σ is the usual Pauli matrix. is zero for unentangled states and one for maximally
y y y y y
⊗ ⊗ C
entangled states. Since the reduced dynamics of the two-qubit system was obtained in the dressed state basis (18),
whichcontainsentangledstates,inordertoremovespuriousentanglementcomingfromthebasis,fortheentanglement
analysis we rotate the density operator back to a separable basis by means of a Hadamard transformation acting in
the subspace spanned by a , s , i.e., the tensor sum of a Hadamard in this subspace and an identity operation in
{| i | i}
the subspace spanned by e , g , i.e., H I .
(as) (eg)
{| i | i} ⊕
Here we study concurrence for the two-qubit system interacting with a squeezed vacuum bath. In figure (2)
concurrenceis plotted for the initial state e g for both the independent as wellas collective dynamics. Figure (3)
1 2
| i| i
depicts the behaviorofconcurrencefor the same initial state with respectto the inter-qubitdistancer . It is clearly
12
seenthat the buildup of entanglementis greaterfor the collective dynamics whencomparedto the independent one.
Now we take up the issue of entanglement from the perspective of the PDF as in Eq. (3). In figures (4 (a))
and (b), we plot the weights ω , ω , ω and ω (3) of the entanglement densities of the projection operators of the
1 2 3 4
various subspaces which span the two qubit Hilbert space with respect to the evolution time t for the independent
andcollectivedecoherencemodels,respectively,forthecaseofaninteractionwithanunsqueezedvacuumbath. Since
9
C C
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
t t
10 20 30 40 50 2 4 6 8 10
(a) (b)
FIG. 2: Concurrence C (41) as a function of time of evolution t. Figure (a) deals with the case of vacuum bath (T =r = 0),
while figure (b) considers concurrence in the two-qubit system interacting with a squeezed thermal bath, for a temperature
T = 1 and and bath squeezing parameter r (9, 10) equal to 0.1. In both the figures the bold curve depicts the collective
decoherencemodel(kr12 =0.05), whilethedashedcurverepresentstheindependentdecoherencemodel(kr12 =1.1). Infigure
(b) for the given settings, theconcurrence for theindependentdecoherence model is negligible and is thusnot seen.
C C
1
0.7
0.8 0.6
0.5
0.6
0.4
0.4 0.3
0.2
0.2
0.1
2 4 6 8 10 r12 2 4 6 8 10 r12
(a) (b)
FIG.3: ConcurrenceC (41)withrespecttointer-qubitdistancer12. Figure(a)dealswiththecaseofvacuumbath(T =r=0),
while figure (b) considers concurrence in the two-qubit system interacting with a squeezed thermal bath, for T =1, evolution
timet=1andbathsqueezingparameterr (9,10)equalto0.1. Infigure(a)theoscillatory behaviorofconcurrenceisstronger
in the collective decoherence regime, in comparison with the independent decoherence regime (kr12 ≥ 1). In figure (b), the
effectoffinitebathsqueezingandT hastheeffectofdiminishingtheconcurrencetoagreatextentincomparisontothevacuum
bath case. Heretheconcurrence for theindependentdecoherence regime is negligible, in agreement with theprevious figure.
ω is the weight of the one dimensional projection, representing a pure state, and ω that of the maximally mixed
1 4
state with ω and ω being intermediary, these plots depict the variationin the contribution of the various subspaces
2 3
to the entanglement of the two-qubit system as t increases. From the figures it can be seen that in the case of the
independent decoherencemodel, as depicted infigure (4 (a)), the weightω dominatesthe other weightsandremains
2
almost constant, while the remaining weights are much lower and their increase is very small compared to it. This
is in contrast to the collective decoherence model, wherein we find the weight ω decreases while ω increases with
1 2
time. Sincetheweightω isindicativeofpurestateentanglement,itisevidentfromthefiguresthattheentanglement
1
contentin the collective decoherencemodel is higherthan that inthe independent decoherencemodel. Also since the
weight ω , representing the weight of the two-dimensional projection operator corresponding to the PDF ( ), has
2 2
P E
arichentanglementstructure,wecometotheconclusionthatforthecaseofthe two-qubitinteractionwithavacuum
bath, the entanglement is preserved in the system for a long time. In both the figures, the weight ω , indicative of a
4
completely mixed state, is negligible over the time range considered.
In figures (5 (a)) and (b), we plot the weights ω , ω , ω andω (3) of the entanglementdensities of the projection
1 2 3 4
operatorsof the various subspaces which span the two qubit Hilbert space with respect to T for the independent and
collective decoherence models, respectively. These plots thus depict the variation in the contribution of the various
subspaces to the entanglement of the two qubit system as T increases. From figure (5 (b)), we can see that in the
collective decoherence regime,the weightω initially falls and then stabilizes arounda finite value while ω rises, but
1 4
remains wellbelow the value ofω , indicating that for the collectivedecoherencemodel, under the givensettings, the
1
10
weights weights
1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
t t
10 20 30 40 50 10 20 30 40 50
(a) (b)
FIG.4: Theweights(3)asafunctionofevolutiontimet,forthecaseofaninteractionwithanunsqueezedvacuumbath. Figure
(a) refers to the independent decoherence model, with kr12 =1.5 and (b) the collective decoherence model, with kr12 =0.08.
In both the figures, the bold curve corresponds to the weight ω1, while the large-dashed, small-dashed and dotted curves
correspond to theweights ω2, ω3 and ω4, respectively. In both the figures, theweight ω4 is negligible and henceis not seen.
weights weights
0.7
0.8
0.6
0.6 0.5
0.4
0.4
0.3
0.2
0.2
0.1
T T
2 4 6 8 10 2 4 6 8 10
(a) (b)
FIG. 5: The weights (3) as a function of temperature T, for the case of an interaction with a squeezed thermal bath for an
evolution time t = 5 and bath squeezing parameter r (9, 10) equal to 0.5. Figure (a) refers to the independent decoherence
model, with kr12 = 1.5 and (b) the collective decoherence model, with kr12 = 0.08. In both the figures, the bold curve
correspondstotheweightω1,whilethelarge-dashed,small-dashedanddottedcurvescorrespondtotheweightsω2,ω3 andω4,
respectively.
systemmaintainsa finite value ofentanglement. This feature is notobservedinthe figure(5 (a)), whereforthe same
settings, ω decreases to zero, while ω rises, thereby indicating a loss of purity and destruction of entanglement.
1 4
As explainedinSectionII,the characterizationofMSEfor atwo-qubitsysteminvolvesthe density function offour
projectionoperators,Π ,Π ,Π ,Π ,correspondingtoone,two,three,andfourdimensionalprojections,respectively.
1 2 3 4
These will be representedhere as ( ), ( ), ( ) and ( ), respectively. As also discussed above, ( ) would
1 2 3 4 4
P E P E P E P E P E
be universal for the two-qubit density matrices and would involve the Harr measure on SU(4) [38]. This is depicted
in figure (6) and is common to all the two-qubit PDF of entanglement.
Now we consider the ( ) and ( ) density functions for some representative states of the two qubit system,
2 3
P E P E
both for the independent as well as collective decoherence models. This enables us to compare the entanglement in
the respective subspaces of the system Hilbert space. We also plot the full entanglementdensity function curve ( )
P E
with respect to the entanglement , at a particular time t. This will enable us to look at the contribution to the
E
entanglement from the different projections.
Figures (7 (a)) and (b) depict the behavior of the density function ( ) for the bath evolution time t = 5.0
2
P E
and T = 0 for the independent and collective decoherence models, respectively. For these conditions, the value of
concurrence(41)is0.17forthecaseoftheindependentdecoherencemodeland0.42forthecollectivemodel,depicting
thegreaterentanglementcontentinthelatercomparedtotheformer. Thisisalsoborneoutbythesefigures. Asshown
in[31],theconcurrenceforatwodimensionalprojectionis =( )/2. Fromthefigures(7),itcanbeseen
CΠ2 Emax−Ecusp
that the value of is greater for the case of figure (7(b)) when compared to that of figure (7(a)). In all the figures
CΠ2
related to the probability density functions of entanglement for the two-qubit system interacting with an unsqueezed
vacuum bath, for the independent decoherence model, kr is set equal to 1.5, while for the collective decoherence
12