Table Of ContentAnalytical solutions and genuine multipartite entanglement of the three-qubit Dicke
model
Yu-Yu Zhang1, , Xiang-You Chen1, Shu He2, Qing-Hu Chen2,3,
∗ †
1Department of Physics, Chongqing University, Chongqing 401331, P. R. China
2Department of Physics, Zhejiang University, Hangzhou 310027, P. R. China
3 Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China
(Dated: July 14, 2016)
Wepresentanalytical solutionstothreequbitsandasingle-mode cavitycouplingsystembeyond
the rotating-wave approximation (RWA). The zeroth-order approximation, equivalent to the adi-
abatic approximation, works well for arbitrary coupling strength for small qubit frequency. The
first-order approximation, called the generalized rotating-wave approximation (GRWA), produces
6
an effective solvable Hamiltonian with the same form as the ordinary RWA one and exhibits sub-
1
stantial improvements of energy levels overthe RWA even on resonance. Based on these analytical
0
eigen-solutions, we study both the bipartite entanglement and genuine multipartite entanglement
2
(GME). The dynamics of these two kinds of entanglements using the GRWA are consistent with
l thenumericalexactones. Interestingly,thewell-knownsuddendeathofentanglementoccursinthe
u
bipartite entanglement dynamics but not in theGME dynamics.
J
3 PACSnumbers: 42.50.Pq,42.50.Lc,64.70.Tg
1
] I. INTRODUCTION strong-coupling regime [24]. We will present an ana-
h
p lytical solution to a three-qubit Dicke model. However,
- explicit analytic solutions to the three- and more-qubit
t The interaction between qubits and a cavity is ubiq-
n Dicke model have not been extensively studied. De-
uitous in several branches of physics ranging from quan-
a spite the fact that the exact solution to the three-qubit
tumoptics[1],toquantuminformation[2]tocondensed-
u Dicke model has been given by a Bargmann space tech-
q matterphysics[3]. Inearlyworkoncavityquantumelec-
nique [25] where a numerical search for the zeros of very
[ trodynamics (QED), the qubit-cavity coupling strength
complicated transcendental functions is needed, an effi-
was much smaller than the cavity transition frequency,
2 cient, easy-to-implement theoretical treatment remains
the rotating-waveapproximation(RWA) can be applied,
v elusive. In this paper, we extend the previous GRWA
and an analyticalexact solutioncan be derivedstraight-
7 in the one-qubit Rabi model by Irish [19] to the three-
8 forwardly [4]. With recent advances in the circuit QED
qubit Dicke model. Including the CRW interactions, we
3 using superconducting qubits, it is possible to engineer
successfullyderive asolvable Hamiltonianwiththe same
0 systems for which the qubits are so far detuned from
0 the cavity, or are coupled to the cavity in a ultra-strong form as the ordinary RWA term. Therefore all eigenval-
. uesandeigenstatescanbeapproximatelysolvedandcan
1 couplingregimewherethe couplingstrengthiscompara-
be implemented with great ease by experimentalists.
0 ble to the cavity transition frequency, that the RWA is
6 demonstratedto fail to describe the system correctly[5– There is on going interest in the genuine multipar-
1 10]. The counter-rotating-wave (CRW) interactions in tite entanglement (GME) of the Dicke states for mul-
:
v thequbit-cavitysystemsarethereforeexpectedtoplaya tiple qubits systems [26, 27]. Most of the existing stud-
i crucial role. ies of entanglement focus on bipartite entanglement in
X
Under the RWA, the ground state is simply a direct the reduced state of two parties of a multipartite sys-
r
a product of the low state of the qubit and the vacuum tem [28–31], which can be quantified through the von
cavity. TheCRWinteractionsleadtoasqueezedvacuum Neumann entropy [32, 33] and the concurrence charac-
state containing virtual photons [11, 12]. The analytical terizing qubit-qubit entanglement [34–36]. However, bi-
exact study in the full model is highly nontrivial. There partiteentanglementcanonlygiveapartialcharacteriza-
havebeen numeroustheoreticalstudies onone-andtwo- tion. Multipartite entanglement is known to be different
qubit and cavity coupling systems, including the adia- from entanglement between all bipartitions [37–39]. Re-
batic approximation [13, 14], a Bargmann space tech- cently the bipartite entanglement decoherence has been
nique [15, 16], an extended coherent-state method [17, studied in connectionwith a phenomenontermed entan-
18], and a generalized RWA (GRWA) [19, 20]. Recently glement sudden death, indicating that the bipartite en-
there have been interesting applications of the Dicke tanglement can decay to zero abruptly during a finite
model[21]withthreequbits inthe quantuminformation periodoftime[40]. Whetherthispropertyoccursforthe
technology, such as the application of the Greenberger- dynamics of GME remains unexplored. So it is highly
Horne-Zeilinger states [22]. And the circuit QED has desirable to study both the bipartite entanglement and
entered the deep-strong-coupling regime [23], so it is the GME for the multipartite entanglement in the more
experimentally possible to realize the three-qubit Dicke than two qubits system, where the three qubits and cav-
model in circuit QED in the ultra-strong- and deep- ity coupling system can be served as the most simple
2
paradigm. where a and a are, respectively, the annihilation and
†
The paper is outlined as follows. In Sec. II, we map creationoperatorsoftheharmoniccavitywithfrequency
the three-qubit Dicke model with the CRW interactions ω, Ji(i=z, ) is the angular momentum operator, de-
±
intoasolvableHamiltonianbythezeroth-andfirst-order scribing the three qubits of level-splitting ∆ in terms of
approximation, giving an analytical expression of eigen- a pseudospin of length J = 3/2, and g denotes the col-
values and eigenstates. In Sec.III, we discuss dynam- lective qubit-cavity coupling strength.
ics of the GME for the multi-qubit entanglement and In the RWA, the CRW terms a†J+ and aJ are ne-
the concurrencefor the qubit-qubit entanglementby our glected, and the Hamiltonian becomes −
method. Finally, a brief summary is given in Sec. IV.
g
HRWA = ∆Jz+ωa†a+ (a†J +aJ+),
− 2 −
which is restricted to relatively weak-coupling strength
II. AN ANALYTICAL TREATMENT TO THE
g ω, and to the qubit-cavity near resonance, ∆
THREE-QUBIT CAVITY SYSTEM ω.≪Now, the interaction couples only 3 n+2 , ≈
|− 2i| i |−
1 n+1 , 1 n , and 3 n 1 foreachn,whicharethe
The Hamiltonian of the three-qubit Dicke model, 2i| i |2i| i |2i| − i
eigenstates of the noninteracting Hamiltonian ∆J +
z
which describes three identical qubits coupled to a com- −
ωa a. The whole Hilbert space can then be decomposed
mon harmonic cavity, is written as (~=1) †
into the subspaces formed by these states which can be
diagonalizedanalytically. Itiseasytowritethefollowing
g
H = ∆J +ωa a+ (a +a)(J +J ), (1) tri-diagonalmatrix form:
z † † +
− 2 −
ω(n+2)+ 3∆ T 0 0
2 n+1,n+2
H = Tn+1,n+2 ω(n+1)+ ∆2 Tn,n+1 0 . (2)
RWA 0 T ωn ∆ T
0 n,0n+1 T − 2 ω(n n−1)1,n 3∆
n−1,n − − 2
where G a a + G a a a 2 + a2G a a + ... and
0 † 1 † † 1 †
sinh(cid:0) g (cid:1)a a (cid:0) =(cid:1)(cid:0) (cid:1)F a a a (cid:0) (cid:1)aF a a +
Tn+1,n+2=g 3(n+2)/4,Tn,n+1=g√n+1/4, F a(cid:2)ωa(cid:0) a†−3 (cid:1)(cid:3) a3F a a 1+(cid:0) †...,(cid:1) w†he−re G1(a(cid:0) a†)((cid:1)i =
T =g√p3n/4. 2 † † − 2 † i †
n 1,n 0,1(cid:0),...)(cid:1)a(cid:0)nd(cid:1)F (a a)(j(cid:0)= (cid:1)1,2,...) are coefficients that
− j †
depend on the cavity number operator n = a a and
If CRW terms a J andaJ areincluded, the Hilbert †
† +
space cannot be decomposed−into the finite dimensional the dimensionless parameter g/ω. A different order of
approximations can then be performed bby neglecting
spaces, because the total excitation number N = a a+
†
some terms in the expansions.
J +3/2 is non-conserved and the subspace for different
z
zeroth-order approximation: In the zeroth-order ap-
index n defined above is highly correlated. So analytical
proximation, we only keep the first term G a a in
solutions in this case should be highly non-trivial. 0 †
The Hamiltonian (1) including the CRW terms with cosh ωg a†−a , and the Hamiltonian is appro(cid:0)xima(cid:1)ted
a rotation around the y axis by an angle π/2 can be as (cid:2) (cid:0) (cid:1)(cid:3)
rewritten as H0th =ωa†a− gω2Jz2+∆JxG0 a†a . (6)
H =∆Jx+ωa†a+g(a†+a)Jz. (3) (cid:0) (cid:1)
In the basis of the oscillatorstate n , the termG a a
0 †
| i
Introducing a unitary transformation U = only has non-vanishing diagonal element (cid:0) (cid:1)
exp gJ a a , one can obtain the transformed
Ham(cid:2)iωltozn(cid:0)ian†−HgS2′B(cid:1)(cid:3)=H0+H1, consisting of G0(n)=hn|coshhωg (cid:0)a†−a(cid:1)i|ni=e−2gω22Ln(ωg22), (7)
HH10 ==∆ωan†aJx−coωshJhz2ωg, (cid:0)a†−a(cid:1)i+iJysinhhωg (cid:0)a†−a(cid:1)i((o45)). Pwospshcaime=iclirlen0ea{tmcoa,rnn}Ln(au−bgme1u)bedner−errceio(ommp−epmripo)!ax!so(tnenlo−yd−rnii)oin!nmi!t.oaiaplpsdNeiffaoertrseLe,nmnstto−hanntt(hxme)onaHlnyiilfbotelhdr=est
spanned by the spin and cbavity basis of 3 n ,
| − 2i| i
Then We can expand the even and odd functions 1 n , 1 n and 3 n . In the subspace containing
cosh(y) and sinh(y), respectively, as cosh ωg a†−a = o|n−ly2it|hein|-t2hi|mianifold|2,it|heiHamiltonian takes the form
(cid:2) (cid:0) (cid:1)(cid:3)
3
ωn 9g2 √3∆G (n) 0 0
H0th = √23∆−0G004(ωn) 2ω∆nG−00(0n4gω2) √ω∆3n∆G−G0(n4g(ω2)n) √ω23n∆0G09(gn2). (8)
2 0 − 4ω
The corresponding eigenvalues and eigenvectors are The zeroth-order energy spectrum is plotted in Fig. 1
straightforwardlygiven by respectively with dash-dotted lines. In large detuning regime ∆/ω =
0.1,thezeroth-orderresultsagreewellwiththenumerical
5g2 1
ε =ωn B 2χ , ones from weak to strong coupling regimes in Fig. 1(a).
1,n n 1,n
− 4ω − 2 − ButtheRWAfailstogivecorrectenergiesasthecoupling
5g2 1 strength g/ω increases. Because of the coupling of the
ε =ωn + B 2χ ,
2,n n 2,n
− 4ω 2 − qubitandtheoriginaloscillator,thelattershouldbedis-
5g2 1 placed. Thus,thedisplacedoscillatorstateinthezeroth-
ε3,n =ωn− 4ω − 2Bn+2χ1,n, order approximation, |nij = exp[jωg(a† − a)]|ni(j =
5g2 1 3, 1), plays a more important role than the original
ε4,n =ωn + Bn+2χ2,n, (9) o±s2cil±lat2or state n in the RWA, resulting in more ac-
− 4ω 2 | i
curate eigen-energies in Eq. ( 9) and eigenfunctions in
and
Eq. ( 10). However, there is a noticeable deviation of
1 1 the zeroth-order approximated results for the resonance
−
K K case ∆/ω = 1, indicating that the higher-order terms in
|ϕ1,ni∝ K11,n,n ,|ϕ2,ni∝−K22,,nn , Eq. (5) should be taken into account. Physically, qubit
− 1 − 1 states with different n manifolds should be coupled by
the interactions.
1 1
−
K K
|ϕ3,ni∝ K33,n,n ,|ϕ4,ni∝−K44,,nn , (10)
− −
1 1
First-order approximation: Keeping the linear terms
where in a and a† and neglecting all higher order terms in the
interaction Hamiltonian H (5) gives
1
1 [ 2g2 ( 1)iB +4χ ],(i=1,2)
Ki,n ={ √√31B3Bnn[−−2ωg2ω−−(−−1)iBnn−4χi−i,2n,n],(i=3,4), H1 =∆{JxG0(cid:0)a†a(cid:1)+iJy[F1(cid:0)a†a(cid:1)a†−aF1(cid:0)a†a(cid:1)](}1.2)
(11)
The term F a a a describes the photon hopping
1 † †
and
from state n(cid:0) to(cid:1) n+1 . It is reasonable to set
| i | i
n+1 R a n = n+1 F a a a n by
χi,n =r4gω42 +(−1)i4gω2Bn+ B4n2(i=1,2) h | n+1,n †| i h g| 1(cid:0) † (cid:1) †| i
Rn+1,n = n+1 sinh a† a n /√n+1
ptwhrieothxtiwmBoan-tqi=uobn∆itisGssy0ism(tnei)lm.arI[t1no4te]t,rheweshtaiendrgiealybt,haetthicteraazpnepsrriototixhoin-morbadetetiorwneaepinn- =hn+1 1ω|ge−2gωh22ωL(cid:0)1n(ωg−22).(cid:1)i| i (13)
different manifolds is not considered, and the nth state
Similarly, the term aF a a only has non-vanishing el-
1 †
is only limited to the same n-th manifold.
ement n aF a a n+(cid:0)1 .(cid:1) It follows that the term
The validity of the zeroth-order approximation is re- h 1 † |
F a a a(cid:12) crea(cid:0)tesa(cid:1)ndaF (cid:11)a a eliminatesasinglepho-
stricted to the large detuning regime ∆/ω 1. In the 1 † (cid:12)† 1 †
zero detuning limiting, ∆=0, within the sa≪me manifold ton(cid:0)of t(cid:1)he cavity. The phys(cid:0)ics p(cid:1)rocess is similar to that
n, 3 n and 1 n are nearly degenerate. For a describedintheRWAmodel,whichfacilitatesthefurther
|± 2i| i |± 2i| i analytic treatment.
large detuning ∆/ω 1, it is reasonable to consider the
≪ The Hamiltonian now is H1st =H′ +H′:
qubit states with the same n manifold coupled by the 0 1
ign/tωera≫ctio∆n/.ω,Etshpeecdiailalgy,onfoarl taersmtrsonogf tchoeupalpinpgrosxtirmenagtethd H0′ =ωa†a− gω2Jz2+∆βJx, (14)
Hamiltonian in Eq.( 8) play a more dominant role than
′
the off-diagonal terms dependent on ∆. And the high H1 =∆Jx[G0 a†a −β]+iJy∆[F1 a†a a†−aF1 a†a ],
order terms in Eq.( 5) still can be neglected even in the (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
g2
strongcouplingregimes. Hence,thezeroth-orderapprox- where β =G0(0)=e−2ω2.
imation is expected to work well from weak to strong Since the qubit and cavity in the noninteracting part
′
coupling regimes for the large detuning case ∆/ω 1. H are decoupled, we apply a unitary transformation S
≪ 0
4
numerical GRWA zero-th order RWA
2 2
1 1
0 0
E/-1 -E/1
-2 -2
-3 -3
-4 -4
-5 (a) -5 (b)
0.0 0.5 g/ 1.0 1.5 0.0 0.5 g/ 1.0 1.5
FIG.1. (Color online) Energy levelsobtainedbytheGRWA (dashedlines) fordifferent∆/ω=0.1 (a),and∆/ω=1(b). The
energiesbythenumericallyexactdiagonalization (solid lines),resultsofRWA(shortdottedlines) andresultsobtainedbythe
zeroth-order approximation (dashed dotted lines) are plotted for comparison.
′
to diagonalize the qubit part in H where K has been defined in Eq.( 11) for n = 0, and
0 i
the normalized parameter is C = 2+2K2. The cor-
i i
responding eigenvalues are εi,0 in Epq.( 9).
′
1 1 1 1 In terms of the transformation S†H1S, the Hamilto-
−KC11 CK22 −KC33 CK44 nianH1st ofthe three-qubitDickemodelcanbe approx-
S = C1 −C2 C3 −C4 , (15) imated as
K1 K2 K3 K4
−C1 −C2 −C3 −C4
1 1 1 1
C1 C2 C3 C4
3 3 1 1 1 1 3 3
H =ωa a+µ (a a) +µ (a a) +µ (a a) +µ (a a)
GRWA † 1 † 2 † 3 † 4 †
|− 2ih−2| |− 2ih−2| |2ih2| |2ih2|
√3K +K (√3+2K ) 1 3
2 1 2
+∆F1 a†a [− (a +h.c)
C C |− 2ih−2|
(cid:0) (cid:1) 1 2
√3K +K (√3 2K ) 1 1
3 2 3
+− − (a +h.c)
C C |2ih−2|
2 3
√3K +K (√3+2K ) 3 1
4 3 4
+− (a +h.c)], (16)
C C |2ih2|
3 4
where µi(a†a) = εi,0 ∆[G0 a†a β]2Ki[√3−(−1)iKi]. formwithrenormalizedcoefficients,thepresentapproach
− (cid:0) (cid:1)− Ci2 essentiallyborrowsthebasicideaoftheGRWAproposed
Thereareonlytheenergy-conservingterms(a 1 3 +
|−2ih−2| by Irish for the one-qubit model [19].
h.c), (a 1 1 +h.c), and (a 3 1 +h.c) with renor- Note that the individual bosonic creation (annihila-
|2ih−2| |2ih2|
malized coefficients, originating from the CRW terms tion)operatora (a)appearsintheGRWA,sothequbits
†
iJy[F1 a†a a† aF1 a†a ]. Thedominatedeffectofthe stateswithdifferentoscillatornumber n, n 1andn+2
origina(cid:0)lCR(cid:1)W t−erms i(cid:0)s con(cid:1)sideredhere. Becauseit is the arecoupledwitheachother. Inthe basisof± 3 n+2 ,
three-qubit Dicke model Hamiltonian in the same RWA 1 n+1 , 1 n and 3 n 1 (n>0), t|h−e2Hi|amiltoi-
|− 2i| i |2i| i |2i| − i
nian H can be written in the matrix form as
GRWA
ω(n+2)+µ (n+2) ∆R 0 0
1 n′+1,n+2
H = ∆Rn′+1,n+2 ω(n+1)+µ2(n+1) ∆Rn′,n+1 0 , (17)
GRWA 0 ∆R ωn+µ (n) ∆R
0 n′0,n+1 ∆Rn′−31,n ω(n−1)+n′−µ14,n(n−1)
5
with Rn′+1,n+2 = −√3K2+CK11C(2√3+2K2)Rn+1,n+2√n+2, III. QUANTUM ENTANGLEMENT
Rn′,n+1 = −√3K3+CK22C(3√3−2K3)Rn,n+1√n+1 and In the present three-qubit system, we study the GME
Rn′−1,n = −√3K4+CK33C(4√3+2K4)Rn−1,n√n. for the multipartite entanglement and the concurrence
To this end, the GRWA can be also performed an- for the bipartite entanglement. A fully separable three-
alytically without more efforts than those in the origi- particlestatemustcontainno entanglement. Ifthe state
nalHamiltonian H in Eq.(2). The displacedoscillator is not fully separable, then it contains some entangle-
RWA
states n , n 1 and n+2 dependupontheDicke ment, but it might be still separable with respect to
m m m
| i | ± i | i
state j,m , and are definitely different from both the two-party configurations. For genuine multiparticle en-
| i
RWA ones and the zeroth-order approximations where tangled states, all particles are entangled and therefore
only the state n is considered. Hence, as ∆/ω in- GME is very important among various definition of en-
m
| i
creases, the first-order correction provides an efficient, tanglements.
yet accurate analytical solution. We review the basic definitions of GME for the three
The ground-stateenergy for the groundstate 3 0 qubits A, B, and C. A separable state is a mixture of
is |−2i| i product states with respect to a bipartition ABC, that
is ρsep = p ϕj ϕj ϕj ϕj , whe|re p is a
ABC j j| Aih A|⊗| BCih BC| j
5g2 ∆ g2 coeffi|cient. PSimilarly, we denote other separable states
E0 =−4ω − 2e−2ω2 −2χ1,0. (18) for the two other bipartitions as ρsep and ρsep . A
BAC CAB
biseparable state is a mixture of sep|arable stat|es, and
The first and second excited energies Ek (k = 1,2) combines the separable states ρsep , ρsep , and ρsep
{ 0} ABC BAC CAB
can be given by expanding the GRWA Hamiltonian in with respect to all possible bipar|titions.|Any state t|hat
the basis 3 1 and 1 0 is nota biseparablestate is calledgenuinely multipartite
|− 2i| i |− 2i| i
entangled.
ω+µ (1) ∆R Recently, a powerful technique has been advanced
HGRWA =(cid:18) ∆R0′1,1 µ2(0′0,)1 (cid:19). (19) to characterize multipartite entanglement using positive
partial transpose (PPT) mixtures [41]. It is well known
that a separable state is PPT, implying that its partial
Similarly, H is given in terms of 3 2 , 1 1 ,
GRWA |− 2i| i |− 2i| i transposeispositivesemidefinite. WedenoteaPPTmix-
1 0 as tureofatripartitestateasaconvexcombinationofPPT
|2i| i
states ρPPT , ρPPT and ρPPT with respect to different
ABC BAC CAB
2ω+µ (2) ∆R 0 bipartitio|ns. The|setofPPT|mixturescontainsthesetof
1 1′,2
HGRWA = ∆R1′,2 ω+µ2(1) ∆R0′,1 , (20) biseparablestates. TheadvantageofusingPPTmixtures
0 ∆R µ (0) instead of biseparable states is that the set of PPT mix-
0′,1 3
turescanbefullycharacterizedbythelinearsemidefinite
programming(SDP)[42],whichisastandardproblemof
which provides three analytical excited energies Ek
{ 0} constrained convex optimization theory.
(k =3,4,5).
In order to characterize PPT mixtures, a multipartite
Energies obtained by the GRWA are presented in
state which is not a PPT mixture can be detected by a
dashed lines in Fig. 1. Especially, for the resonance case
decomposableentanglementwitnessW [26]. Thewitness
∆ = ω, the GRWA results are much better than the operatorisdefinedasW =P +QTM forallbipartitions
zeroth-orderresults(bluedottedlines)inFig.1(b). Itas- M M¯, where P , and Q aMre posMitive semidefinite op-
cribes to the effect of the coupling between states with | M M
erators, and T is the partial transpose with respect to
different manifolds. Our approach is basically a pertur- M
M. This observable W is positive on all PPT mixtures,
bativeexpansionintermsof∆/ω. As the increaseofthe
but has a negative expectation value on at least one en-
∆/ω , the high order terms in Eq.(5) still cannot be ne-
tangled state. To find a fully decomposable witness for
glected in the intermediate and strong coupling regimes.
a given state ρ, the convex optimization technique SDP
So the GRWA works reasonably well in the ultra-strong
becomesimportant,sinceitallowsustooptimizeoverall
coupling regime g/ω < 0.3 at resonance. Interestingly,
fully decomposable witnesses. Hence, a state ρ is a PPT
the level crossing is present in both the GRWA results
mixture only if the optimization problem [26],
and the exact ones. The RWA requires weak coupling
duetothecompleteneglectoftheCRWterms,whichare minimize:Tr(Wρ). (21)
qualitativelyincorrectasthecouplingstrengthincreases.
So the GRWA includes the dominantcontributionof the has a positive solution. If the minimum in Eq. ( 21) is
CRW terms, exhibiting substantial improvement of en- negative, ρ is not a PPT mixture and hence is genuinely
ergy levels over the RWA one. The RWA fails in partic- multipartite entangled. We denote the absolute value of
ular to describe the eigenstates, which should be more theaboveminimizationasE(ρ). ForsolvingtheSDPwe
sensitive in the quantum entanglement presented in the use the programs YALMIP and SDPT3 [43, 44], which
next section. are freely available.
6
and falls off to a nonzero minimum value, implying no
numerical GRWA zero-th RWA
0.14 death of the three-qubit entanglement. The GME dy-
(a)
namics obtained by the GRWA are consistent with the
0.13
numericalresults,whiletheRWAresultsarequalitatively
E()ρ0.12 incorrect for the off-resonance case ∆/ω = 0.1 in Fig. 2
(a). The zeroth-order approximation, where only states
0.11
within the same manifold are included, works well for
0.1 the off-resonance case ∆ = 0.1 in Fig. 2 (a) but not for
0 0.2 0.4 0.6 0.8 1 the on-resonance case in Fig. 2 (b). The validity of the
0.15 (b) GRWA ascribes to the inclusion of the CRW interaction
iJ F a a (a a).
y 1 † †
0.1 −
)ρ The(cid:0)ons(cid:1)et of the decay of the multipartite entangle-
E(
ment is due to the information loss of qubits dynamics
0.05
to the cavity. On the other hand, it is the interaction
0 with the cavity that leads to the entanglement resurrec-
0 5 10 15 20
∆t/(2π) tion. Thelostinformationwillbetransferredbacktothe
qubit subsystem after a finite time, which is associated
withtheratiobetweenthecouplingstrengthg/ωandthe
FIG.2. (Color online) Dynamicsof theGME for three-qubit
entanglement with the initial W state for the ultrastrong- level-splittingofqubits∆/ω. Astheratiog/∆increases,
coupling strength g/ω = 0.1 with the different detuning the contributions of the qubit-cavity interaction become
∆/ω=0.1(a)and∆/ω =1(b)bytheGRWAmethod(dash- dominant and the lost entanglement will be transferred
dotted lines), numerical method (solid lines), RWA (short- quickly from the cavity to qubits with less revivals time,
dotted lines), and the zeroth-order approximation (dashed as shown in Fig 2 (a).
lines). Moreover, it is significant to study the different be-
havior of the multipartite entanglement and the bipar-
tite entanglement. The concurrence characterizes the
NowwediscussthedynamicsoftheGMEforthethree-
entanglement between two qubits. Due to the sym-
qubit entanglement. The initial entangled three-qubit
metric Dicke states in the three-qubit collective model,
state is chosen as the W state with only one excitation
the concurrence is evaluated in terms of the expecta-
1 tion values of the collective spin operators as C =
W = (100 + 010 + 001 ), (22) max 0,C ,C , where the quantity C is defined for
| i √3 | i | i | i a giv{en dyireczt}ion n(= y,z) as C =n 1 N2
n 2N(N 1){ −
which corresponds to the Dicke state |D3i=|− 21i. For 4hSn2i− [N(N −2)+4hSn2i]2−[4(N −1)hS−ni]2} [35].
the Hamiltonian(3)withrespectto therotationaround From thepdynamical wavefunction φ(t) , we can easily
| i
the y axis by the angle π/2, the initial Dicke state can evaluate the coefficients for the qubit to remain in the
be written as j,m state
| i
1 3 1 1 3
|D3i= √8(−√3|− 2i−|− 2i+|2i+√3|2i), (23) Pm0th = ∞ 4 fn(t)e−iEnkt, (25)
nX=0kX=1
and the initial cavity state is the vacuum state 0 .
Based on the eigenstates ϕ and eigenvalues E|ki in the zeroth-order approximationand
{| k,ni} n
in the GRWA and the zeroth-order approximation(cid:8), th(cid:9)e
4
wanv,ekfue−nicEtinkotn|ϕekv,noilhvϕeks,nfr|Dom3i.theAinnidtiatlhestatthereaes-q|uφb(itt)ire=- PmGRWA ≈X∞n Xk=1fnk(t)(e−iEnk−2t+e−iEnk−1t
ddPuegcreedesstoaftefreρe(dt)omcan be given by tracing out the cavity +e−iEnkt+e−iEnk+1t), (26)
intheGRWA. fk(t)isadynamicalparameterassociated
ρ(t)=Tr (φ(t) φ(t)). (24) n
cavity withthe initialstate andthe k-theigenstatesfor eachn.
| ih |
From PGRWA in Eq.( 26), we observe energy-level transi-
We then calculate the absolute value of the minimum m
tions among Ek , Ek and Ek in the GRWA, which
E(ρ) to detect the GME by solving the minimum in n 2 n 1 n
produce essentia−l impro±vement of the dynamics over the
Eq.( 21).
zeroth-orderones in Eq.( 25). Since the averagevalue of
Fig. 2 shows the E(ρ) plotted against parameter
collective spin operators can be expressed by P , such
∆t/(2π)fordifferentdetunings∆/ω fortheultra-strong- m
as 4 S2 = 4√3( n 2n P P + n 1n +
coupling strength g/ω = 0.1. For comparison, results h yi −32h − | i21 −23 12 −12h − |
from numerical exact diagonalization and RWA are also 1i32P−21P32) − 4(P−212 + P122) + 3, we calculate the con-
shown. Weobserveaquasi-periodicbehavioroftheGME currence C by the zeroth-order approximation and the
dynamics. E(ρ)decaysfromtheinitialentangledWstate GRWA, respectively.
7
Negativity(AB|C) 0.1
numerical GRW A zero-th RWA 1 Concurrence(AB)
0.7 GME(ABC) 0.05
0.8 0
2 4
(t)0.6 0.6
C
0.4
0.5
0.2
(a)
0.4 0
0.0 0.5 1.0 1.5 0 5 10 15 20
t/2 ∆t/2π
0.6
FIG. 4. (Color online) GME for the three qubits A, B,C
) (dash-dotted line), negativity for the entanglement with re-
(t0.4
C spect to the bipartition AB|C (solid line), and concurrence
between A and B qubits (dashed line) obtained by the nu-
0.2 merical method for g/ω=0.1 and ∆/ω =1.
0.0(b)
0 5 10 15 20 inthethree-qubitsystem. Intuitively,wemaythinkthat
t/2
entanglementisstillstoredinthebipartitionAB C. Neg-
|
ativity is used to detect the entanglement for this bipar-
FIG. 3. (Color online) Dynamics of the concurrence for the tition[45],whichfallsofftoanonzerominimuminFig.4.
qubit-qubitentanglement with theinitial W state for the ul- It reveals that the state for the bipartition AB C is not
trastrong coupling strength g/ω = 0.1. The parameters are a separable state. Similarly, those states with| respect
thesame as in Fig. 2.
to other bipartitions AC B and BC A are not separa-
| |
ble. Therefore, the three-qubit state stays in an entan-
gled state and the GME for the three-qubit entangle-
We plot the dynamics of the concurrence for differ- mentneverdisappearsduringthe deathofthe two-qubit
ent detunings ∆/ω = 0.1 and 1 in Fig. 3. The initial W entanglement. The theory of the multipartite entangle-
stategivesthemaximumpairwiseentanglementC =2/3 ment is not fully developed and requires more insightful
of any Dicke states. Fig. 3 (a) shows that dynamics of investigations into more- than two-party systems. We
the concurrence by the zeroth-order approximation are highlight here the different features of the multipartite
similar to the numerical ones in the off-resonance case entanglement and bipartite entanglement in the more-
∆/ω = 0.1, in which the RWA results are invalid. The than two-qubit system, and have found that the GME
sudden death of the bipartite entanglement is observed is always robust at least in the qubits and single-mode
in the resonance case in Fig. 3 (b). The dynamics of the cavity system.
concurrenceobtained by the GRWA is similar to the nu-
merical results, exhibiting the disappearance of the en-
tanglement for a period of time. However, there is no IV. CONCLUSION
sudden death of the entanglement in the RWA case, in-
dicating that the CRW terms are not negligible.
In this work, we have extended the original GRWA
Very interestingly, as shown in Fig. 2, the GME for by Irish for the one-qubit Rabi model to the three-qubit
the three-qubit entanglement never vanishes, in sharp Dicke model by the unitary transformation. The zeroth-
contrast with bipartite entanglement. During the van- orderapproximation,equivalenttotheadiabaticapprox-
ishment of concurrence, the GME is generally small but imation,issuitedforarbitrarycouplingstrengthsforthe
still finite. It follows that the two-qubit state is separa- large detuning case. The first-order approximation, also
ble inthe system, but the three-qubit state stillcontains called GRWA, works well in a wide range of coupling
residual entanglement. This may be one advantage to strength even on resonance and much better than the
using GME as a quantum information resource. RWAones. IntheGRWA,theeffectiveHamiltonianwith
Finally, it is significant to clarify why the GME of the the CRW interactions is evaluated as the same form of
tripartiteentanglementbehavesdifferentlywiththe con- theordinaryRWAone,whichfacilitatesthederivationof
currence of the bipartite entanglement. The well-known theexplicitanalyticsolutions. Alleigenvaluesandeigen-
death of the concurrence is related to the disappearance states can be approximately given.
oftheentanglementinanarbitrarytwo-qubitsubsystem, By the proposed GRWA scheme, we have also calcu-
say A and B, while a deep understanding is associated latedthedynamicsofconcurrenceforthebipartiteentan-
with the question of whether there exists entanglement glementandtheGMEforthemultipartiteentanglement,
8
which are in quantitative agreement with the numerical In the end of the preparation of the present work, we
ones. The well-knownsudden death ofthe two-qubiten- noted a recent paper by Mao et al. [46] for the same
tanglement is observed by our analytic solution. An in- model. We should say that the approach used there is
teresting phenomenon of entanglement is that the GME theadiabaticapproximationofthepresentwork,i.e.,the
for the three-qubit entanglement decays to the nonzero zeroth-order approximation.
minimumduringthetimewindowinwhichthetwo-qubit
entanglement disappears, implying that three qubits re-
main entangled when the two-qubit state is separable.
OurresultsindicatethattheGMEisthepowerfulentan- V. ACKNOWLEDGEMENTS
glement to detect quantum correlations in multipartite
systems that cannot be described via bipartite entangle-
This work was supported by National Natural
mentinsubsystemsofsmallerparticles. Therestillexists
Science Foundation of China (Grants No.11547305,
many open problems to the theory of entanglement for
and No.11474256), Chongqing Research Program
multipartite systems due to much richer structure of the
of Basic Research and Frontier Technology (Grant
entanglement in a more- than two-party system. In par-
No.cstc2015jcyjA00043),andResearchFundfortheCen-
ticular, the dynamical behaviors for two kinds of entan-
tral Universities (Grant No.106112016CDJXY300005).
glement may be explored in the multi-qubit realized in
the recent circuit QED systems in the ultra-strong cou- ∗ Email:[email protected]
pling. Email:[email protected]
†
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