Exact decoherence-free state of two distant quantum systems in a non-Markovian environment

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by  Chong Chen

Exact decoherence-free state of two distant quantum systems in a non-Markovian
environment
Chong Chen,1 Chun-Jie Yang,1 and Jun-Hong An1,
∗
1Center for Interdisciplinary Studies & Key Laboratory for Magnetism and
Magnetic Materials of the MoE, Lanzhou University, Lanzhou 730000, China
Decoherence-free state (DFS) encoding supplies a useful way to avoid the detrimental influence
of the environment on quantum information processing. The DFS was previously well established
ineitherthetwosubsystemslocatingatthesamespatialposition orthedynamicsundertheBorn–
Markovian approximation. Here,we investigate theexact DFS of two spatially separated quantum
systemsconsistingoftwo-levelsystemsorharmonicoscillatorscoupledtoacommonnon-Markovian
6 zero-temperaturebosonicenvironment. Theexactdistance-dependentDFSandtheexplicitcriterion
1 for forming the DFS are obtained analytically, which reveals that the DFS can arise only in one-
0 dimensional environment. It is remarkable to further find that the DFS is just the system-reduced
2 state of the famous bound state in the continuum (BIC) of the total system predicted by Wigner

n and von Neumann. On the one hand our result gives insight into the physical nature of the DFS,
u andontheotherhanditsuppliesanexperimentallyaccessibleschemetorealizethemathematically
J curious BIC in thestandard quantum optical systems.

4
PACSnumbers: 03.65.Yz,03.67.Bg,42.25.Hz,42.50.Dv
2

]
h I. INTRODUCTION 35], while some other works claimed that the DFS for
p distant quantum systems is still possible to be formed
t- Asaubiquitousphenomenoninmicroscopicworld,de- [24–28]. However, the physical nature of the DFS, es-
n coherence describes an inevitable loss of quantum coher- pecially its explicit form and its dependence on the spa-
a
u ence due to the interactions between quantum system tial distance, and when it is formed have seldom been
q and its environment. It is seen as a main obstacle to the touched in these works. The realistic significance of an-
[ realization of any applications utilizing quantum coher- swering these questions is that it could supply meaning-
ence,e.g.,quantumcomputation[1],quantumteleporta- ful message for designing practical devices to distribute
2
tion [2], and quantum metrology [3]. Therefore, how to long-distance entanglement.
v
3 control decoherence is a crucial issue in quantum engi- Another inspiration of our study is the famous bound
0 neering. Many active schemes, such as feedback control state in the continuum (BIC), which was proposed soon
3 [4] and dynamical decoupling [5], have been proposed to after the birth of quantum mechanics [36]. Being stable
2 beat this unwanted effect. On the other hand, people in space but with its energy lying in the continuous en-
0 found that decoherence can also be used for good pur- ergy band, such counterintuitive eigenstate of the quan-
.
1 pose [6–8]. It was found that the decoherence caused tumsystemwasregardedasamathematicalcuriositydue
0 by a common environment can play a constructive role to the inaccessible potentials for a long time [37]. That
6 in generatingstable entanglementbetween twoquantum situation changed when it was proposed that the BIC
1
systems [9–12]. The intrinsic physics is the existence of can arise naturally by virtue of the destructive interfer-
:
v the decoherence-free state (DFS) [13–15], which triggers ence between two resonance states in molecule system
i the enthusiasm of relearning the role of decoherence of [38]. Although the BIC has been extensively studied in
X
compositesystemcausedbyacommonenvironmentfrom the classical optical systems [39–45], the BIC was rarely
r
a differentsystemssuchasharmonicoscillators[16–20]and demonstrated in quantum systems.
spins [21–24], and different environments such as crystal In this work,we revealthat these two seemingly unre-
chains [25–27] and waveguides [23, 24, 28–31]. latedconceptsmergetogetherinopenquantumsystems.
It is clear in principle that the DFS is present when Bystudyingthedecoherenceoftwodistantquantumsys-
the two quantum systems are at same spatial position temsconsistingofeithertwo-levelsystems(TLSs)orhar-
[13–15]. While when they are spatially separated, there monic oscillators embedded in a common bosonic envi-
are still controversies on whether the DFS exists or not ronment, we derive analytically the exact DFS and the
or, equivalently, whether the common environment can physical criterion for forming the DFS without resort-
create stable entanglement distribution. Some works ing to the Born–Markovianapproximation (BMA). It is
pointedoutthattheentanglementwoulddisappearwhen foundthatthedistance-dependentDFScanonlyexistin
thespatialdistancesarelargerthanthewavelengthasso- aone-dimensionalenvironment. Thisisinsharpcontrast
ciated with the environmental cutoff frequency [18, 32– withthe caseinwhichthe twosubsystemsarelocatedin
the same spatialposition, where the DFS is presentirre-
spective of the environmental dimension. Further study
reveals that the DFS, which scales as 1/R with increas-
∗ [email protected] ing system distance R, corresponds exactly to the sys-
2
tem reduced state of the BIC of the total system. The III. DECOHERENCE FREE STATE
emergence of such a BIC can be physically attributed
to the destructive interference between the two indepen- A. DFS under BMA
dentinteractionchannelsofthe twosubsystemswiththe
common environment, which can be seen as a direct re-
To describe the DFS of two spatially separated quan-
alizationofFriedrichandWintgen’sideaontheBIC[38]
tumsystemsinfluencedbythecommonzero-temperature
in quantum optical system. Our conclusions are verified
environment, we consider first its decoherence dynamics
in the models of two TLSs interacting with a coupled
under the BMA. The master equation reads [47]
cavity chain acting as an environment. Our study gives
a realizable scheme to detect the BIC by observing the ρ˙(t)=−i[X(ω0+Ωii)Oˆi†Oˆi+(Ω12Oˆ1†Oˆ2+H.c.),ρ(t)]
decoherence dynamics of open quantum systems.
i
preTshenetpoaupremroidselo.rgTahneizDedFSausnfdoellroawnsd: bIenyoSnedc.theIBI,MwAe +Xγ2ijh2Oˆjρ(t)Oˆi†−{Oˆi†Oˆj,ρ(t)}i≡Lˇρ(t), (2)
i,j
is derived in Sec. III. The correspondence between the
DFS andthe BIC is alsoestablishedhere. By twoexam- where ρ(t) is the reduced density matrix of the systems,
ples of two TLSs interacting with the nearest-neighbor
andthe next-nearest-neighborcoupled-cavityarraysact- Ω = gk2eik·(ri−rj)
ing as environments, our conclusions are verified in Sec. ij PXk ωk−ω0
IV. In Sec. V, a summary is given.
with the Cauchy principal value, i = j denoting the
P
frequency shift, and i = j denoting the dipole-dipole in-
6
teractionstrengthinducedbytheenvironment,thedecay
rate reads
II. THE MODEL
γij =2πXgk2eik·(ri−rj)δ(ω0−ωk). (3)
Consider two spatially separated quantum systems k
coupled to a common dissipative bosonic environment. Generally, Ω is real due to the parity symmetry Ω =
12 12
The Hamiltonian reads Hˆ =Hˆ +Hˆ +Hˆ with Ω =Ω . ItcanbeseenfromEq. (2)thattheenviron-
S E I 21 ∗12
ment can not only induce individual spontaneous emis-
HˆS = X ω0Oˆj†Oˆj, HˆE =Xωkaˆ†kaˆk, siniodnucγejjtahnedcforrerqeulaetnecdyssphoifnttΩanjjeotuoseaecmhisssyiostnemγ,bu=t aγlso
j=1,2 k 12 21
(1) and coherent dipole-dipole interaction Ω between the
HˆI =Xgk(eik·rjOˆj+aˆk+H.c.), twoquantumsystemsbytheexchangeofv1i2rtualphotons.
j,k When the decay rates γij satisfy γ12 = γ21 = γ11 =
±
γ , there is a DFS
22
±
where Oˆ and ω are the annihilation operators and fre-
j 0 ρ = Ψ Ψ (4)
DFS DFS DFS
quency of the jth quantum system located at rj, aˆ†k and | i∓h |
vaˆikroanrme tehnetaclrekatthiomnoadnedwaintnhihfrileaqtuioenncoypωerka,toarnsdogfkthisetehne- Cwoitmhb|iΨneDdFSwi∓ith=E√q12.(Oˆ(31†)∓, tOhˆ2e†)|e0x,p0liicdituecrtioterLˇioρnDFfSor=th0e.
coupling strength between the systems and the environ- presence of the DFS (4) is
ment. OursystemcanbetwoTLSswhenOˆ =σˆ [23]or
two harmonic oscillators when Oˆ = ˆb [16, 18]. H−ere the k(ω0)·R=lπ, l ∈Z, (5)
rotating-waveapproximationisusedinHˆI,whichisvalid with R = r1 r2 being the relative coordinate of the
−
in the weak-coupling limit. Under this approximation, quantumsystems. ThesigninEq. (4)is“ ”(“+”)when
tthhee tsoytsatelmexcisitactoinosnernvuemdbseinrcNeˆ[=ˆ,PHˆj],k=(Oˆ0j†.OˆjT+heaˆ†kHaˆilkb)eortf ldiirsecetvieonn(oofdRd),.thEeqeuxaitsitoennc(e5o)filtlhuestDraFtSesr−tehqauti,regsivthenattahlel
space of the whole system is thNus divided into indepen- the degenerate wave vectors k with the same frequency
dent subspaces with definite excitation number . ω0 mustsatisfyEq. (5)simultaneously. Itstronglylimits
N the existence of the DFS in a multidegenerate environ-
Previously, it was found that when the two systems
ment as in two- and three-dimensional cases, where the
are placed in the same position, there is the DFS [15]
degeneracy of k is generally infinite. This condition is
|ΨDFSi = √12(Oˆ1† −Oˆ2†)|0,0i with |0i being the ground possible only for the one-dimensional-environment case.
state of the systems due to Hˆ Ψ = 0. This DFS For example, when the one-dimensional environment is
I DFS
| i
physically originates from the permutation symmetry of formedby the electromagneticfield, the wavevectorsfor
the quantum systems [46]. We here are interested in ex- ω can only take k. If k satisfies Eq. (5), then k
0
± −
ploringwhethertheDFSstillexistswhenthetwosystems satisfies it naturally. This explains well why all of the
are placed in different positions such that the permuta- works [23, 25–27, 48] on the DFS are in one-dimensional
tion symmetry is broken. environments.
3
B. Exact DFS beyond BMA is an integration identity for any function f(k):
f(k) f(k)
The above Markovian theory reveals that the DFS = iπ f(k)δ(E ω ). (8)
XE ω PXE ω − X − k
for distant quantum systems only exists in the one- k − k k − k k
dimensional environment case. A natural question is
whether it is still valid in the non-Markovian dynam- The imaginarypart inEq. (8) entering into the eigenen-
ics. The exploration to this issue is meaningful be- ergy E contributes to the dynamics a damping rate. Us-
causethenon-Markovianeffectisnon-negligibleinaone- ing this identity in Eq. (7), we can conclude that, to en-
dimensionalenvironment,especially forcomposite quan- sure the existence of the DFS, this imaginary part must
tumsystems. Besidestheweaksystem-environmentcou- vanish for one eigenenergy E0 of Eq. (7), i.e.,
pling, the validity of the BMA also requires that the en-
1 cos[k(E ) R]=0 k(E ) R=lπ, l Z. (9)
vironmental correlation timescale is much shorter than ± 0 · ⇒ 0 · ∈
the characteristic time of the system. For the compos-
This criterion is almost the same as Eq. (5) under the
ite quantum system as considered in Eq. (1), a new
BMA except that the argument E differs from ω . The
0 0
timescale characterizingthe communicationbetween the
eigenstate with the real E under Eq. (9) is an isolated
0
subsystems via the common environment is involved.
boundstate,whileotheroneswithEq. (9)unsatisfiedare
When this timescale is comparable with the environ-
called resonant states playing a significant role in Fano
mental correlationtime, the non-Markovianeffect would
effect[53,54]. Theeigenenergyoftheboundstatefallsin
dominate the dynamics even in the weak-coupling limit.
the environmentalcontinuous energyband [37], it thus is
This non-Markovian effect is especially important in a
aBIC.Equation(9)describesthedestructiveinterference
one-dimensional environment [24, 28, 49].
of the two interaction channels of the quantum systems
Based on the observation that the DFS must be a
withthe environment. The BIChere hasa closeanalogy
system-reduced state of the whole-system eigenstate,
with that predictedin molecule systems [38]. After trac-
only under which it is unchanged by the action of Hˆ,
ing over the environmental degrees of freedom from the
we here calculate the exact eigenstate of Eq. (1). The
BIC, we obtain the exact DFS as
DFS derived in this way can efficiently avoid the BMA
used in the preceding section. The eigenstate in the ρ = C 2 Ψ Ψ +(1 C 2)0,0 0,0, (10)
DFS DFS DFS
single-excitation subspace can be expanded as ψ = | | | i±h | −| | | ih |
| i
t[Phe2i=en1vciirOˆoi†nm+ePntkaldkvaaˆc†ku]|u0m,0,s{ta0tke}.i,Fwrohmereth|{e0Skc}hird¨oedninogteesr wseheenreth±atd,edpiffenerdenotnftrhoemptahreityreosufllti(n4)Equ.nd(9er).thItecBaMn bAe,
equation, we can obtain (i=j) theexactDFSisaclassicalmixtureof ΨDFS and 0,0 .
6 The above analysis reveal that, to m| ake tih±e DFS| (10i)
gk2 gk2eik·(ri−rj) exist, Eq. (9) must be satisfied simultaneously by all
ci(cid:16)E−ω0−XE ω (cid:17)=cjX E ω , (6) the degenerate modes k having the common eigenen-
k − k k − k ergy E . This again is possible for the one-dimensional-
0
environmentcasewhenkiseitherparallelorantiparallel
and
toR. Sowehave d 2 =g2[1 cos(kR)]C 2/(E ω )2.
dk =gk cje−ik·rj Withdth2e=h1e,lpweoof|bktth|aeinnorkma±lization co|nd|ition0−|C|2k +
X E ωk Pk| k|
j=1,2 −
J(ω)[1 cos(k(ω)R)] 1
C 2 = 1+ dω ± − , (11)
where E is the eigenenergy [50, 51]. After eliminating c1 | | h ˆ (E0 ω)2 i
and c , we have E satisfying −
2
wherethesummationoverkhasbeenreplacedbythein-
g2[1 cos(k R)] tegrationoverω intheinfinitelimitoftheenvironmental
k
E =ω0+Xk ±E−ωk· , (7) mtaoldspesecatnradlJde(ωns)it=y.PAcktgink2gδ(aωs−thωekw)eiisghthteofenΨvironmenin-
DFS
ρ [Eq. (10)], C 2 determines the entangle|mentaiv±ail-
where we used the environmental spatial reflection sym- DFS | |
metry, i.e., the modes k aredegenerate in their ω and ableintheDFS.Wenoticethatthemaincontributionto
k
± the integration in Eq. (11) comes from the modes with
gk, hasbeenused. Substituting Eq. (7)into Eq. (6)and
ω E . Making a Taylor expansion near E , it can be
using again the spatial reflection symmetry, we obtain ≈ 0 0
recast into
c = c C/√2. Absentinthesingle-quantum-system
1 2
± ≡
case[50,52], thecosineterminEq. (7)manifests thein- [1 cos(αRδ)] 1
C 2 1+J(E ) dδ − −
terferenceofthetwointeractionchannelsofthequantum | | ≈h 0 ˆ δ2 i
systems with the common environment. As seen in the =[1+J(E )π αR] 1 [J(E )π αR] 1, (12)
0 − 0 −
following,itisjustthisinterferencetermwhichproduces | | ≈ | |
the BIC in our bipartite quantum systems. where δ = ω E , α = ∂ k(ω) , and Eq. (9) have
− 0 ω |ω=E0
In the infinite limit of the environmentalmodes, there been used. It demonstrates that the weight of Ψ
DFS
| i±
4
in the exact DFS scales as 1/R with increasing distance ξ g g ξ
between the twoquantum systems. In practice,one gen-
erally is interested in realizing distant entanglement dis- m m
1 2
tribution by using the DFS [23, 25–27]. Our exactresult
implies that the available entanglement decays in power FIG. 1. Schematic diagram of a one-dimension nearest-
law as 1/R with the increase of the distance between neighbor coupled cavity array with two TLSs embedded in
quantum systems. This sets a practical bound on the cavities m1th and m2th, respectively. The distance between
performance of the scheme. two nearest-neighbor cavities is x0.
with ∆m=m m . The obtained result E =ω con-
IV. ILLUSTRATIONAL EXAMPLES 1 2 0 0
−
firmsthattheDFSisaBICbecauseω iswithintheenvi-
0
ronmental energyband. Equation (16) reveals that C 2
We first consider two TLSs embedded in a one- | |
scales as 1/∆m with the increase of the TLS distance,
dimensional environment formed by a nearest-neighbor
which is consistent with the conclusion in last section.
coupled-cavity array (see Fig. 1) with the Hamiltonian
To check whether the BIC is free of decoherence,
we resort to the dynamics under the initial condition
N
HˆE =X[ωcaˆ†jaˆj +ξ(aˆ†j+1aˆj +H.c.)], (13) |Φ(0)i=|1,0,{0k}i. Its evolved state takes the form
j=1
|Φ(t)i=[X αi(t)σˆi++Xβk(t)aˆ†k]|0,0,{0k}i, (17)
where aˆj and aˆ†j are the annihilation and creation op- i=1,2 k
erators of the jth cavity with frequency ω , and ξ is
c
where α (t) are governed by
the coupling strength between the nearest-neighbor cav- i
ities separated in distance x . By a Fourier trans-
0 t
ifnotromaHˆtiEon=aˆjP=kωPkaˆk†kaˆaˆkkewikjitxh0/√diNsp,erEsiqo.n r(e1la3t)ioins rωekcas=t α˙i(t)+iω0αi(t)+jX=1,2ˆ0 dτfij(t−τ)αj(τ)=0. (18)
ω +2ξcos(kx ). Thus the coupled cavity array defines
c 0
an environment with finite bandwidth 4ξ centered at its with fij = (g2/N)Pke−iωk(t−τ)+ik(mi−mj)x0. The con-
eigenmode ω . The two TLSs are embedded in cavities volution in Eq. (18) keeps all the non-Markovian effect
c
m th and m th, respectively. The interactions are induced by the backactions of the memory environment.
1 2
By the Laplace transform F˜(s) = 0∞e−stF(t)dt, it is
Hˆ =g (σˆ+aˆ +H.c.), (14) straightforwardto show that ´
I X j mj
j=1,2 1/2
α˜ (s)= ,(19)
which, in the Fourier space, takes the form HˆI = 1 jX=0,1s+iω0+ gN2 Pk 1+(−1)js+coiωs(kk∆mx0)
o(gu/s√lyNfo)uPndj=t1h,a2tPifkt[ehiekmTjLx0Saˆkfrσˆej+qu+enHcy.cf.a].llsItinwtahse pbraenvdi-- α˜2(s)=−(gs2/+Niω)P+kegi2k∆mx(0s(s++iωiω)k)1−1α˜1(s), (20)
gap regime of the environmental spectrum, i.e., ω0 < 0 N Pk k −
ω 2ξ or ω > ω +2ξ, a bound state in the band-gap
c 0 c Setting s = iE, one can find that the pole of Eq. (19)
−
forms [52, 55], which has been found to play a construc- −
satisfies Eq. (7) and thus corresponds exactly to the
tive role in decoherence suppression [56, 57], entangle-
eigenenergy of the BIC. There is no further pole in Eq.
ment trapping [50], and entanglement generation [58].
(20). Using the residue theorem, we readily have
Here we exclude this situation from consideration and
concentrateonthepotentiallyformedBIC.Suchabound α (t)=Z e iω0t+ ′iǫ+∞ dEα˜ ( iE)e iEt, (21)
statehasbeenfoundinquantumHallinsulators[59]and j j − ˆ 2π j − −
iǫ
optical waveguide array structure [39, 40]. −∞
Weseefromthedispersionrelationthatthereisatwo- where the first term with residue Zj is contributed from
fold degeneracy with k for one explicit ωk. It can be the BIC with eigenenergy ω0, the second term contains
calculated exactly from±Eq. (7) that the eigenenergy of all the contributions from the continuous energyband,
theDFS(10)isE =ω (seeAppendixA),whichinturn and the prime in the integration represents the integra-
0 0
reduces criterion (9) for forming the DFS (10) to tion region excluding the eigenenergy of the BIC. It is
interesting to find that the two residues take exactly as
ω ω
0 c
∆marccos( 2−ξ )=lπ, (15) Z1 = Z2 = C2 /2. (22)
± | |
and the weight (11) to Oscillating with time in continuously changing frequen-
cies,the secondtermin Eq. (21)behavesasa decayand
C 2 = 1+ g2∆m −1, (16) approaches zero in the long-time limit due to the out-
| | (cid:2) 4ξ2 (ω ω )2(cid:3) of-phase interference. Therefore, the steady state of our
c 0
− −
5
FIG. 3. (a): Environmental dispersion relation reveals that
a four-fold degeneracy exists when ωk > 1.04ωc. (b) and
(c): Time evolution of Ct in different TLS separation ∆m
FIG. 2. Time evolution of the TLS excited-state population when the next-nearest-neighbor hopping of the cavity array
Pt in panels (a) and (b) and concurrence Ct in panels (c) is considered. The blue solid and green dashed lines denote
and (d) as thechange of the TLS separation ∆m when ω0 = thecaseswithandwithouttheDFSformed,respectively. The
1.0ωc in panels (a) and (c) and 1.2ωc in panels (b) and (d) orange dotted lines show Ct, although dramatically is slowed
obtained by numerically solving Eqs. (18). The blue solid down, finally decays to zero. The parameter ξ′ = 0.9ξ and
and green dashed lines denote the cases with and without theothers are the same as in Fig. 2(b).
theDFSformed,respectively. Thedotsconnectedbythered
dotdashed lines plot the results analytically evaluated from
Eq. (16). Other parameters are ξ = 0.2ωc, g = 0.05ωc, and spondence between the analytical results C 4/2 and the
N =1201. numericaldynamicstestifiesthedistinguis|hed| roleplayed
by the formed DFS in the steady-state behavior.
To further verify the validity of Eq. (23), we plot the
system after tracing over the environmental degrees of
evolutionofthe entanglementbetweenthe TLSs in Figs.
freedom from Φ( ) is
| ∞ i 2(c)and2(d). Theentanglementisquantifiedbyconcur-
C 2 C 2 rence [60], which for the state (17) is t = 2α1(t)α2(t).
C | |
ρ( )= | | ρDFS+(1 | | )0,0 0,0, (23) We can find that the parameter regimes in Figs. 2(a)
∞ 2 − 2 | ih |
and 2(b) where a nonzero P is achieved match exactly
t
whichdemonstratesthedominateroleoftheformedρ wellwiththeregimesinFigs. 2(c)and2(d)whereafinite
DFS
inthe long-timesteadystate. Onthe contrary,ifnoBIC concurrenceisobtained. Theconcurrenceapproachesthe
and DFS formed, then ρ( ) = 0,0 0,0. The results analyticalvalue C 4/2,whichisjusttheconcurrencecal-
∞ | ih | | |
verifyanalyticallyfromthepointofviewofthedynamics culated from the steady state (23). It demonstrates well
the validity of our expectation that the reduced state of the distinguished role played by the formed DFS in the
the formed BIC is a DFS of our decoherent system. dynamics and steady-state behavior.
We plot the time evolution of the total excited-state On the other hand, if the environment is not two-fold
population P = α (t)2 in different TLS separa- degeneracy, then the criterion (9) for forming the DFS
t Pi=1,2| i |
tion ∆m when ω = ω in Fig. 2(a) and 1.2ω in Fig. is hardto be satisfied even for the one-dimensionalenvi-
0 c c
2(b) calculated by numerically solving Eq. (18). Equa- ronment case. To verify this, we next consider another
tion (15) indicates that the DFS is formed when ∆m is situation where the next-nearest-neighbour coupling of
an even number for ω = ω . Figure 2(a) indeed shows the cavity arrayis involved. Then Eq. (13)is recastinto
0 c
that P tends to a finite value when∆m is anevennum-
t
ber, anddecaysto zerowhenever ∆m is anodd number. HˆE′ =X[ωcaˆ†jaˆj +(ξaˆ†j+1aˆj +ξ′aˆ†j+2aˆj +H.c.)], (24)
Thepreservedsteady-statepopulationmatcheswellwith j
fTrro[mρ(∞Eq).Pj(=161,)2,σˆwj+hσˆij−ch] =ver|iCfie|4s/2uncaamlcbuilgauteodusalynatlhyatitcatlhlye where ξ′ is the next-nearest-neighbor hopping rate. The
dispersion relation is derived to be
initial state evolves exclusively to the ρ( ) obatined in
∞
Eq. (23). When ω0 = 1.2ωc, the DFS is formed when ωk =ωc+2ξcos(kx0)+2ξ′cos(2kx0). (25)
∆m=3n with n an integer. As confirmed by Fig. 2(b),
P approaches C 4/2when ∆m=3n anddecaysto zero Compared with the two-fold degeneracy in the nearest-
t
| |
in other cases without any exception. The exact corre- neighborhoppingcase,themodesherecantakefour-fold
6
degeneracy k and k [see Fig. 3(a)]. The formation fies our prediction on the DFS. Our scheme supplies an
1 2
± ±
of the DFS require that the criterion (9) should be sat- implementation of Friedrich and Wintgen’s idea on the
isfied for k and k simultaneously. We numerically realizationofthe mathematically curious BIC in explicit
1 2
± ±
calculate the dynamics and plot the P obtained in Fig. quantum optical system [37]. By giving insight into the
t
3(b). It shows that the DFS previously formed in Fig. physical nature of the DFS, our exact DFS result is ex-
2(b) disappears with the next-nearest-neighbor hopping pectedtobehelpfultointerprettheenvironmentinduced
considered. Thus no stable concurrence can be estab- entanglement between two quantum systems.
lished in the long-time limit except for the trivial case
∆m = 0. Although in some cases the decay of tran-
t
C
siently formed is dramatically slowed down, it decays to ACKNOWLEDGMENTS
zero asymptotically [see Fig. 3(c)]. This gives a coun-
terexample to illustrate the validity of criterion (9). This work was supported by the Specialized Research
Fund for the Doctoral Programof Higher Education, by
the Program for New Century Excellent Talents in Uni-
versity,and by the National NaturalScience Foundation
V. CONCLUSIONS
of China (Grant Nos. 11175072and 11474139).
Appendix A: Derivation of the eigenvalue of the
In summary, we investigated the DFS of two distant bound state in the continuum
quantum systems embedded in a common environment.
Going from a general model of dissipative systems, we TheeigenvalueoftheBICforthecoupledcavityarray
derived the criterion for forming the DFS for both of environment satisfies
the Markovian and non-Markovian decoherence dynam-
ics. It is interesting to find that the DFS may be formed g2 1 cos(kx0∆m)
E =ω + ± , (A1)
only in one-dimensionalenvironment case. We have also 0 0 N X E0 ωk
k −
revealed that the exact DFS for the non-Markovian dy-
namics is a reduced density matrix of the so-called BIC where are determined by the separation ∆m accord-
±
of the total system, which consists of a classicalmixture ing to Eq. (15). With the dispersion relation ω =
k
of the maximally entangled state Ψ and 0,0 . The ω + 2ξcos(kx ), Eq. (A1) in the continuous limit of
weight of the former scales as 1/R| wii±th the|incriease of thce environmen0tal modes is recast into
the system distance, which sets a bound on distribut-
ing entanglement over distant quantum systems via the g2x0 x2π0 1 cos(kx0∆m)
E =ω + dk ± . (A2)
common environment. The exact dynamics of two TLSs 0 0 2π ˆ E ω 2ξcos(kx )
0 0− c− 0
embedded in two types of coupled cavity array as the
common environment are studied explicitly, which veri- Setting z =eikx0, we have
E =ω + g2 dz 1± 12(z∆m+z−∆m)
0 0 2πi‰ z(E ω ) ξ(z2+1)
z=1 0− c −
g2 dz 1 z∆m g2 dz 1 1 z ∆m
=ω ± ( z2) − ± −
0− 2ξ ‰ 2πi(z a iǫ)(z a iǫ) − 2ξ ‰ − 2πi [z a iǫ][z a iǫ]
z=1 − − − ∗− z=1 − − − ∗−
g2 dz 1 z∆m g2 dz′ 1 z∆m
′
=ω ± + ± , (A3)
0− 2ξ ‰z=1 2πi(z−a−iǫ)(z−a∗−iǫ) 2ξ fiz′=1 2πi[z′−(a+iǫ)−1][z′−(a∗+iǫ)−1]
where a and a are the solutions of the equation z2 theorem, we can evaluate the integrations as
∗
−
z(E ω )/ξ +1 = 0 and satisfy a a = 1 and ǫ is an
0 c ∗
−
infinitesimal positive value. Here an integration relation g2 1 a∆m 1 (a ) ∆m
z=1dz = z−1=1dz−1 has been used. There are two E0 =ω0− 2ξh a± a + (a±) 1∗ −a 1i
s(cid:23)ingularitiesfffor each of the integrations in Eq. (A3). − ∗ ∗ − − −
However, only one of them falls within the circle z = g2(1 a∆m)
=ω ± . (A4)
0
1 and z = 1, respectively. According to the residue − ξ(a a )
′ − ∗
Because E , as the the eigenvalue of the BIC, must be
0
real, we readily have 1 a∆m = 0 and thus E = ω .
0 0
±
Substituting the form of a= ω02−ξωc −iq1−(ω02−ξωc)2 =
7
exp[−iarccos(ω02−ξωc)] into 1±a∆m =0, we readily have where 1±a∆m =0 has been used.
ω ω
0 c
∆marccos( − )=lπ, (A5)
2ξ
which is just the criterion Eq. (15) for forming the BIC.
With the similar procedure, the weight C 2 can also
| |
be evaluated as
C 2 = 1+ g2x0 x2π0 dk 1±cos(kx0∆m) −1
| | h 2π ˆ (E ω 2ξcos(kx ))2i
0 0− c− 0
g2∆ma∆m 1
= 1 −
h ± ξ2(a a )2i
− ∗
g2∆m 1
= 1+ − , (A6)
h 4ξ2 (ω ω )2i
0 c
− −
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