Two-Qubits Entanglement Induced by a Faster Data Bus

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by  P. Zhang

Two-Qubits Entanglement Induced by a Faster Data Bus
P. Zhang, Y. D. Wang and C. P. Sun a,b
Institute of Theoretical Physics, the Chinese Academy of Science,
Beijing, 100080, China
We propose a theoretical protocol to create the entanglement of two qubits via the Born-
Oppenheimer(BO)approximation. Inourscheme,eachqubitiscoupledtoafasterdatabuswhose
frequency is much larger than the energy spacing of the qubits and thus the BO approximation is
valid. Then the adiabatic separation of qubits from the data bus can induce an effective potential
to couple the two qubits, which can be utilized to create a quantum logic gate. We also discuss
the quantum decoherence caused by the adiabatic entanglement between the two qubits and the
external field.
4 PACS number: 03.67.-a, 71.36.+c, 03.65.Fd, 05.30.Ch, 42.50.Fx
0
0 I. INTRODUCTION eters Ri(t) is neglected. Of course, the QAA can only
2 work well when R (t) behave classically.
i
n In quantuminformationtheory,it is importantto cre- WhentheHamiltonianofthequantumsystemdepends
a ate a ”maximum entanglement” between two qubits. It on the variables of another slowly varying quantum sys-
J has been shown by Barenco et. al. [1] that, together tem, (e.g., the motion of an electron near nuclear), one
1 with single-bit operations, the entanglement based two needs a more exact approximation, in which the inter-
1
qubit gates except the classical SWAP gate (including, actions between the fast system and the slow one, espe-

1 theCNOT[2]andcontrolledphasegates)formsauniver- cially for the ”back action” of the fast one on the slow
v sal set of logic gates of quantum computation. In other one, are considered. To this end, in 1930 [10], Born and
8 words, any unitary transformation of single or many- Oppenheimer suggested a high order approximation (we
5 qubit system can be decomposed into the quantum net- now call it Born-Oppenheimer (BO) approximation) to
0
works of single-bit gates and non-trivial two-qubit gates tackle this problem. In the view of the BO approxima-
1
0 . Inthelastfewyears,manyschemestocreatetwo-qubit tion, to solve the dynamics of a composite system with
4 interaction have been suggested and experimentally im- fast and slowly changing parts, we can first solve the
0 plemented [3-8]. In this paper, we will propose a new Schro¨dinger equation of the fast part for each fixed slow
h/ scenario to induce a kind of two qubit interaction cre- variable q. Then the obtained q-dependent eigen-value
p ating efficient quantum entanglement. In our proposal, Vn(q) provides the slow part with an effective potential
- each qubit interacts with a common quantum field as a whenthemotionoftheslowpartcannotexcitethetran-
t
n data bus. If the frequency of the quantum field is much sitions among those fast states, i.e., the adiabatic condi-
a largerthantheenergyspacingofthequbit,thedegreeof tion holds. In this sense, the time dependent wave func-
u
freedomofthequantumfieldandthevariablesofthetwo tion of the composite system can be written as
q
: qubit system can be separated adiabatically. In this sit-
v Ψ(q,x;t)=ψ (q,x)φ(q,t). (1)
uation, aneffective two-qubitinteractioncanbe induced n
i
X bymeansoftheBorn-Oppenheimer(BO)approximation.
Here, x is the variable of the fast part, ψ (q,x) the
n
r To explain the main idea in our proposal in details, we
a q-dependent eigen-state of the fast part corresponding
will first briefly review the relevant quantum adiabatic
to eigenvalue V (q); and the slow part φ(q,t) evolves
n
theorem and the BO approximationas follows. governed by the effective Hamiltonian Hˆ +V (qˆ), i.e.
f n
Adiabatic approximation [9] is widely used in many
fields of physics including classical physics and quan- φ(q,t) = e−i[Hˆs+Vn(qˆ)]tφ(q,0), where Hˆs is the free
Hamiltonian of the slow part.
tum physics. In the latter, one has a rich applications
Inthispaperwemakeacrucialobservationforthecre-
of quantum adiabatic approximation (QAA). We con-
ation of the nontrivial two qubits interaction of logical
sider a time-dependent quantum system. Its Hamilto-
gate in quantum information as a result from the appli-
nian H(t) = H[R (t)] is controlled by some adiabati-
i
cationof the BO approximaitonto a three body system.
cally changing parameters R (t)s. If the system is ini-
i
According to above discussion, it is very interesting that
tially prepared in the n-th instantaneous eigen-state of
the slow part can be divided into two subsystems with
the Hamiltonian H(0) at t = 0, then during the evo-
variablesq andq withoutinteractionbetweenthem. In
lution the quantum system will still persist in the n-th 1 2
this situation, the Hamiltonian of the composite system
eigen-stateoftheinstantaneousHamiltonianH(t)atan-
can be generally written as
other instance. The evolution of the system under the
controlofR (t)scanbeevaluatedbyQAA,buttheback
i H =H (q )+H (q )+W (q ,x)+W (q ,x)+H (x)
1 1 2 2 1 1 2 2 f
action of the quantum system on the controlling param-
(2)
1
where H (q ), H (q ) and H (x) are the free Hamilto- condition under which the decoherence effect can be ne-
1 1 2 2 f
nians of the two slow subsystems ( we can note them as glected and the two-qubit interaction model works well.
q ,q ) and the fast subsystem. W (q ,x) (W (q ,x)) There are some conclusions and discussions about the
1 2 1 1 2 2
are the interaction Hamiltonians between q (q ) and x. physical realization of our protocol in the last section.
1 2
As we have emphasized, Eq. (2) implies that there is
no direct interaction between q and q . Under adia-
1 2
batic condition, the eigenvalue V (q ,q ) of the Hamil- II. EFFECTIVE INTERACTION BETWEEN TWO
n 1 2
tonian W (q ,x)+ W (q ,x)+ H (x) of fast part in- QUBITS
1 1 2 2 f
duces an effective potential on the slow variables q and
1
q . In usual case, V (q ,q ) can not be decomposed into Wegenerallyconsiderasystemconsistingoftwoqubits
2 n 1 2
the sum of two independent potentials of q1 and q2 i.e. with the same energy spacing ωa and an external field
V (q ,q ) = V (q )+V (q ). Therefore, an effective which is described as a dynamic data bus. We can treat
n 1 2 1n 1 2n 2
6
interaction between q and q is induced via the BO ap- theexternalfieldasaharmonicoscillatorwithfrequency
1 2
proximation and then one can regardthe fast variable x ω (see Fig. 1).
asadatabuswhichcanberemoved. Indeed,inquantum
Qubit1 Data Bus Qubit2
information process, the inter-qubit interaction is very
crucial to create the nontrivial interaction between two
qubits q and q if we can regard the above mentioned
1 2
two subsystems of q1 and q2 as two qubits. w w
a a
We should point out that, in many previous scenar-
iosofquantumlogicalgate,anexternalfieldwasusually
w
used as a data bus where the quantum information car-
ried by a qubit is storedand then transferredto another
qubit. Mostof those proposals rely onthe rotationwave
FIG.1. Theinteractionbetweentwoqubitsandaexternal
approximation(RWA).Theeffectiveinteractionbetween
field. Whenω>>ωa,theexternalworksasafasterdatabus
the two qubit system can be obtained through the adia-
batic elimination of the external field [8].
The Hamiltonian of the composite system reads
However,to adiabatically eliminate the degree of free-
dom of the external field, the coupling strength g be-
tween the qubit and the external field ought to be weak H =ωa σˆz(1)+σˆz(2) +g aˆ†+aˆ σˆx(1)+σˆx(2) +ωaˆ†aˆ
enough so that the ratio g is very small. Here, ∆ is the (cid:16) (cid:17) (cid:0) (cid:1)(cid:16) (cid:17) (3)
∆
detuning i.e. ∆ = ω ω where ω is the transition
a a
| − |
frequency of the qubit and ω is the frequency of the ex- whereσˆ(i) = e e g g ,σˆ(i) = e g + g e , e
ternal field. On the other hand, since the RWA requires and g zare |thieihex|−cit|ediihan|d gxroun|disithat|es |ofiithhe| i|-tihi
∆≤ωa, the above large detuning condition implies that qubit|;iaˆi† andaˆ the creationandannihilationoperatorof
g << ωa. Then the effective coupling strength between the external field, and g the coupling strength between
the two qubit system g2 is much smaller than ω . This the qubits andthis field. This modelsystemcanbe real-
∆ a
weak coupling may reduce the ”quality factor” Q of the izedinmanypracticalphysicalsituationssuchasthetwo
logic gate. Here Q can be understood as the ratio be- trappedionwith the spatialoscillationmode, twoatoms
tween the atomic life time and the logic operation time in a single mode cavity and the two Josephson charge
τ g2 −1 since the Rabi transition frequency can be qubits coupling to a large Josephson or nonamechanical
l ∼ ∆ resonator[12]. We assume ω is much largerthan ω . In
a
define(cid:16)d a(cid:17)s an upper limit of the operation frequency.
this case, the RWA could not be applied since the con-
In our proposal based on the BO approximaiton , the
diton for the validity of RWA is ω+ω >> ω ω .
a a
frequency of the external field can be significantly larger | | | − |
Thus,theusualadiabaticeliminationtechniquedoesnot
than the energy spacing of the qubits. Then g may
work well for this case. We note that the above Hamil-
be larger than ω or, at least, have the value in the
a toniandepicts a typicalspin-boson(continuousvariable)
sameorderasω . Theratiobetweeneffectiveinteraction
a system.
strength g2 andω canbemuchstrongerthanthatinthe
∆ a In the following parts of this paper, with the gener-
previousscenariosmentionedabove. Inthe nextsection,
alized BO approximation [11], and perturbation theory,
there is a brief introduction of our quantum logic gate
we will prove that under the conditions ω >> ω and
a
model. The precise analysis of our model based on the
ω >>g,theevolutionofthetwoqubitsystemisapprox-
perturbationtheoryis giveninsectionIII. InsectionIV,
imately governed by the effective Hamiltonian
the adiabatic decoherenceeffectis analyzeddetailedly in
association with the idea of the adiabatic entanglements g2
hˆ =ω σˆ(1)+σˆ(2) 2 σˆ(1)σˆ(2) (4)
presentedrecentlybySunet. al.[11]. Wealsoobtainthe a z z − ω x x
(cid:16) (cid:17)
2
andthe decoherenceeffect causedby adiabatic entangle- III. GENERALIZED BORN-OPPENHEIMER
ment [13] between the two qubits system and the data APPROXIMATION FOR DISCRETE SYSTEM
bus can be neglected.
Under the adiabatic condition that the external field
does not excite the transitions among the internal states In the above section, we obtain the effective Hamil-
of qubits, the BO approximaiton determines a formally tonian (4) for two qubit system through a intutitional
-factorized eigen-state of H for the total composite sys- analysis from the BO approximation directly. However,
tem. the BO approximation is usually used to seperate two
continuous quantum systems (e.g. a melocular and a
λ1,λ2 Ψ =φnα(λ1,λ2) n[λ1,λ2] (5) electron) or a continuous system and a discrete one (e.g.
h | i | i
the spatial motion and spin of a neutral particle [13]),
where λ ,λ is the common eigen-state λ ,λ
| 1 2i | 1 2i ratherthanthetwodescretesubsystems. Inthis section,
of σˆx(1),σˆx(2) with eigen-values λ1,λ2 = 1 and however,wecangeneralizetheBOapproximaitontodeal
{ ± }
n[λn1,λ2] theoeigen-state of the effective Hamiltonian withtheseperationoftwodiscretequantumsystems,i.e.
| i
the twoqubitsystemandtheexternalfield. We nowcan
Hf[λ1,λ2]=ωaˆ†aˆ+g aˆ†+aˆ (λ1+λ2) (6) give a rigorousproof to the intuituional argumentin the
above section based on the perturbation theory. In this
oftheexternalfieldforthegi(cid:0)venstat(cid:1)e λ ,λ ofthetwo
| 1 2i way, we obtain the effective Hamiltonian (9) naturally.
qubit system with corresponding eigen-value
g2
V (λ ,λ ) =nω (λ +λ )2 (7) A. Matrix Representation of Motion Equation
n 1 2 − ω 1 2
g2
=nω 2 λ λ . In section II, we define n(λ ,λ ) as the eigen-state
− ω 1 2 | 1 2 i
of the Hamiltonian H [λ ,λ ] of the external field and
f 1 2
Here, we have neglected a constant term. For different λ ,λ the eigen-states of σˆ(1),σˆ(2) . With the com-
1 2 x x
n, V (λ ,λ ) contribute different effective potentials for | i
n 1 2
pleteness relation [11] n o
the slow part so that the wave function φ of the two
nα
qubit system is determined by
n(λ ,λ ) n(λ ,λ ) λ ,λ λ ,λ =1,
1 2 1 2 1 2 1 2
| ih |⊗| ih |
Hˆeff(n) φnα =Enα φnα (8) λX1,λ2Xn
| i | i
(10)
where the effective Hamiltonian is
the quantum state of the composite system can be ex-
Hˆeff(n)≡Hˆeff(n;σˆx(1),σˆx(2)) (9) panded as
=ω σˆ(1)+σˆ(2) +V σˆ(1),σˆ(2)
a z z n x x Ψ = C (n) n[λ ,λ ]) (11)
| i λ1,λ2 | 1 2
=hˆ+(cid:16)nω (cid:17) (cid:16) (cid:17) {n,Xλ1,λ2}
where hˆ defined in Eq. (4) is the n-indepedent part of where
the effective Hamiltonian. n[λ ,λ ]) n(λ ,λ ) λ ,λ .
1 2 1 2 1 2
So far, we have formally obtained a X X-type of | ≡| i| i
−
interbit coupling basedon the BOapproximation,which Substituting Eq. (11) into the Schro¨dinger equation
canbeconvenientlyusedtoformulateanefficientscheme H Ψ = E Ψ , we obtain the effective equations of the
| i | i
for quantum-computing. Recently, a similar interaction coefficients Cλ1,λ2(n):
generating the entanglement of two Josephson junction
charge qubits was proposed by Averin [14]. In ref. [14], Hλλ′11,,λλ′22(n)Cλ′1,λ′2(n)+ Fλλ′11,,λλ′22(n)Cλ′1,λ′2(n) (12)
two Josephson junction charge qubits are assumed to be λX′1,λ′2 λX′1,λ′2
cdoautaplebdusw. itWh iatnhotthheerBjuOncatpiopnrowxihmicahtiowno,rkthseasenaerfgaysteorf + Oλλ′11,,λλ′22(n,m)Cλ′1,λ′2(m)=ECλ1,λ2(n)
λ′,λ′ m=n
the lowest band of the latter junction can be considered X1 2 X6
as the effective interaction between the two Josephson where we have defined some complicated notations:
junction charge qubits.
In the next section, we give a detailed derivation of Hλλ′11,,λλ′22(n)=hλ1,λ2|Hˆeff(n)|λ′1,λ′2i (13)
the effective Hamiltonian (9) based on the perturbation
tqhueboirtys.yTsthemediencdouhceerdenbcyetohfetahdeiaqbuaatnitcuemntsatnagtleemofetnhtewtiwtho Fλλ′11,,λλ′22(n) =hλ1,λ2|ωa σˆz(1)+σˆz(2) λ′1,λ2′ (14)
the external field is analyzed in section IV. ×[1−hn(λ(cid:16)1,λ2)|n(λ′1,(cid:17)λ(cid:12)(cid:12)′2)i] (cid:11)
3
Oλλ′11,,λλ′22(n,m) =hλ1,λ2|ωa σˆz(1)+σˆz(2) |λ′1,λ′2i (15) H¯ ≃H¯(0)+ ωgH¯(1)+ ωg 2 V¯(2)+O¯(2) (20)
n(λ ,λ(cid:16)) m(λ ,λ )(cid:17). (cid:16) (cid:17) (cid:16) (cid:17)
×h 1 2 | ′1 ′2 i where
Obviously, only when Fλ1,λ2(n) and Oλ1,λ2 (n,m)
λ′1,λ′2 λ′1,λ2′ H¯(0) =ω 0, 0 , (21)
are negligible, can we obtain the effective Hamilto- 0, I
nian Hˆ (n). With a straightforward calculation (see (cid:20) (cid:21)
eff
the Appendix A), n(λ ,λ ) m(λ ,λ ) , Fλ1,λ2(n) and
Oλλ′1,,λλ′2(n,m) can bhe exp1res2sed| as p′1ow′2eriserλie′1s,λo′2f the pa- H¯(1) =ω ωga σˆz(1)+σˆz(2) , 0 , (22)
ram1et2er ωg:  (cid:16) 0, (cid:17) ωga σˆz(1)+σˆz(2) 
 (cid:16) (cid:17)
m(λ ,λ ) n(λ ,λ ) (16)
h 1 2 | ′1 ′2 i
=exp(cid:20)−12(cid:12)ωgΣ¯λλ1′1,,λλ2′2(cid:12)2(cid:21)× V¯(2) =ω" σˆx(1)0σˆ,x(2), σˆx(1)0σˆx(2) #, (23)
m (−1)n(cid:12)(cid:12)−m ωgΣ¯(cid:12)(cid:12)λλ1′1,,λλ2′2 2l+(n−m) and
l!((cid:16)l+n m(cid:17))! ×
l=0 − 0, ωaΣ¯
X O¯(2) = ω g . (24)
n(n 1)...(m l+1) m(m 1)..(m l+1), − ωaΣ¯, 0
− − − − (cid:20) g (cid:21)
p p Here,I istheunit4 4matrixandΣ¯ a4 4matrixwhose
Fλλ′11,,λλ′22(n)=ωa(cid:20)(n+1)ωg22Σ¯λλ′11,,λλ′222+...(cid:21) (17) seelecmonedntosradreeriΣ¯nλλ′11g,,λλa′22×r.eOcnonlysitdheeretderimnsEoqf.×z(e2r0o)t.hT,hfierstteramnds
ω
of higher order are neglected. According to Eq. (17),
Oλ1,λ2(n,m)=ω √ngΣ¯λ1,λ2δ +... (18) F¯(i)(i=0,1)doesnotcontainthe termsofzeroth,first
λ′1,λ′2 a − ω λ′1,λ′2 n,m±1 and second order. Therefore, F¯(i) are neglected in Eq.
h i (20).
where we have assumed n m 0 and Σ¯λ1,λ2 = λ +
− ≥ λ′1,λ′2 1
λ λ λ . According to Eq. (17), Fλ1,λ2(n) can be
ne2g−lect′1ed−if′2n and g are small enough. λC′1o,λn′2sidering the B. Perturbation Solution of Motion Equation
ω
condition ω << ω mentioned in the above section, in
a To apply the perturbation theory, we split H¯ into two
the following discussion, we assume n .1 , g <<ω and
ω <<ω. Forsimplicity,wealsoassumeg˜ω and g˜ωa parts:
a a ω ω
is a small parameter.
H¯ =H¯ +W¯ (25)
Inorderto solveEq. (12)withthe standardperturba- 0
tion theory and thus give a proof for the generalizedBO
where
approximation for the discrete case, we rewrite Eq. (12)
g
in matrix-value form as H¯ =H¯(0)+ H¯(1) (26)
0 ω
H¯C =EC (19)
is treated as the unperturbated Hamiltonian and
where the total Hamiltonian H¯ and the eigen-vector C g 2
are defined as W¯ = V¯(2)+O¯(2) (27)
ω
H¯ (0)+F¯(0), O¯(0,1) C (cid:16) (cid:17) (cid:16) (cid:17)
H¯ = ,C = 1 . as the perturbation term.
O¯(1,0), H¯ (1)+F¯(1) C2 It is pointed out that we can also choose H¯(0) as the
(cid:20) (cid:21) (cid:18) (cid:19)
unperturbated Hamiltonian, gH¯(1) the first order per-
Here, the vector C contains two sub-vectors ω
turbation term and g 2 V¯(2)+O¯(2) the second order
ω
C = C (i), C (i), C (i), C (i) T ,i=0,1. one. It is easy to prove that, if gH¯(1) is regarded as
i 11 1 1 11 1 1 (cid:0) (cid:1) (cid:0) ω (cid:1)
− − − − thefirstorderperturbationHamiltonian,theeigenenergy
Each(cid:0)of them is a 4-dimension vector; H¯ (i), (cid:1)F¯(i) and andeigenstatetothefirstordergivenbytheperturbation
O¯(i,j)(i,j =0,1)are4 4matriceswhoseelementsare theory is just the same as the ones we obtain by diago-
×
Hλλ′1,,λλ′2(i), Fλλ′1,,λλ′2(i) and Oλλ′1,,λλ′2(i,j). nalizingthematrixH¯(0)+ωgH¯(1) exactly. Therefore,our
1 2 1 2 1 2
Making use of Eq. (17) and Eq. (18), we express the finalresultis notrelevanttothe choiceofunperturbated
matrix H¯ as a power series of g: Hamiltonian and perturbation terms.
ω
4
T
The eigen-energy and eigen-state of unperturbated C = 0, 1, 0, 0
1
Hamiltonian H¯ in Eq. (26) are easy to obtain. The
0 C =(cid:0)0, 0, 1, 0(cid:1)T .
eigen-energy can be expressed as: 2
(cid:0) (cid:1)
E(0) =ε +nω (28) By making use of the perturbation theory, the eigen-
ni i energiesandeigen-statestothefirstordercanbewritten
as
where n=0,1 and i=0,+1, 1. Here, we have defined
−
ε0 =0, ε 1 = 2ωa. (29) Eni =En(0i)+En(1i)
± ±
The energy level structure of the compositive system and
is illustrated in Fig. 2. Since ω is much smaller
a E = E(0) + E(1)
than ω, it follows from Eq. (28) and Eq. (29) that k n±1i n±1 n±1
E1(0i)−E0(0j) ∼ ω. Noting En(0i)−En(0j) = |εi−εj| ∼ ωa, kEn0;ki =(cid:13)(cid:13)(cid:13)En(00);kE+(cid:13)(cid:13)(cid:13)En(10);kE.
weregyhasvpeec(cid:12)(cid:12)tErnu(0im)−ofEHn(0aj)m(cid:12)(cid:12)i<lt<on(cid:12)(cid:12)(cid:12)Eia1(n0i)H−¯(E0)0(0pj)o.s(cid:12)(cid:12)(cid:12)Tsehsseersefaorqeu,stih-beaennd- Aelsc,cothrdeiznegrot-oththoerdpeerrteuigr(cid:13)(cid:13)(cid:13)ebna-tsiotantEetsheoE(cid:13)(cid:13)(cid:13)ry(0)forEdaergeelnineeraartecolemv--
structure(cid:12): it consists(cid:12)of two qusi-bands corresponding to n0;k
n=0 and 1. binations of kEn0;1i and kEn0;2(cid:13)(cid:13)i. ForEany k and k′,
E O¯(2) E = 0, E(0) (cid:13) can be obtained by
h n0;1k k n0;2i n0;k
diagonalizing the 2 2 mat(cid:13)rices E
× (cid:13)
(cid:13)
n = 1 E V¯(2) E , E V¯(2) E
hEn0;1kV¯(2)kEn0;1i, hEn0;1kV¯(2)kEn0;2i . (32)
(cid:20) h n0;2k k n0;1i h n0;2k k n0;2i(cid:21)
E(0)1i-E(0)0j~w The first order corrections to the eigen-energies Eni
andeigen-statescanbecalculateddirectly. Sincewehave
E O¯(2) E =0 foreverygivenn,itis obviously
n 1 n 1
hthat±thke contkribu±tioins from O¯(2) to the correctionof the
e -e ~w eigen-energiesare zero. On another hand, the first order
i j a eigen-vector E(1) can be expressed as:
n = 0 ni
(cid:13) E
(cid:13)
(cid:13) E(0) V¯(2) E(0)
E1(F0i)IG−.E2.0(0jT) h>e>enεeir−gyεlje.veTlshoefreH¯fo(r0e),. Sthinecreeωar>e>twωoas,uwbebahnadves (cid:13)En(1i)E =(cid:16)ωg(cid:17)2 D n−iε(cid:13)(cid:13)(cid:13)i−ε−(cid:13)(cid:13)(cid:13)i ni E(cid:13)En(1−)iE (33)
with respect to n=0 and n=1. (cid:13) E(0) V¯(2)(cid:13)E(0)
(cid:13) + g 2 n0;k (cid:13) ni E(1)
The eigen-states corresponding to En± (cid:16)ω(cid:17) kX=1,2D (cid:13)(cid:13)(cid:13) εi (cid:13)(cid:13)(cid:13) E(cid:13) n0;kE
(cid:13)
can be written(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)iEEnC00((t00±±e))11r=EEms==o1(cid:0)(cid:0)f,tC00h,±,eC,0u±,n0i0t(cid:1)(cid:1)TTvTe,c,tors (30) ++(cid:16)(cid:16)ωωgg(cid:17)(cid:17)22kjXX==±1,12DD(EE−|(|(n1n00(−))−)−n11+|1|j01)(cid:13)(cid:13)(cid:13);nkωO+(cid:13)(cid:13)(cid:13)¯1+O(¯2ω)(ε2(cid:13)(cid:13)(cid:13)i+)E−(cid:13)(cid:13)(cid:13)εnE(i0εi)nj(E0i(cid:13))E(cid:13)(cid:13)(cid:13)E(cid:13)(cid:13)|E(n1−)|(n11−)|j1E|0;kE.
+ (cid:13)
C =(cid:0)0, 0, 0, 1(cid:1)T . The another eigenstate En(10);k have the similar expres-
−
sion and thus for simplic(cid:13)ity, weEneed not give its explicit
It is obvious from Eq. (cid:0)(21) and Eq.(cid:1)(22) that the eigen- expression here. (cid:13)(cid:13)
energiesE(0) aretwo-folddegenerate,whichcorresponds Substituting Eq. (29-32), into Eq. (33), we can cal-
n0
to the two degenerate eigen-vectors culate the contribution from O¯(2) and V¯(2) to the first
order correction of eigen-state. It is obvious that
T
E = C , 0 ,
00;k k
k i
kE10;ki =(cid:0)0, Ck (cid:1)T . (31) g 2 En(0−)i V¯(2) En(0i)
ω (cid:12)D ε(cid:13) ε (cid:13) E(cid:12)
where k =1,2 and (cid:0) (cid:1) (cid:16) (cid:17) (cid:12)(cid:12) (cid:13)(cid:13)i− −(cid:13)(cid:13)i (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
5
g 2 En(00);k V¯(2) En(0i) Thelatterhavethesameorderofωa. Thentheprobabil-
ities of the interband transitions are much smaller than
∼ ω (cid:12)D (cid:13) ε (cid:13) E(cid:12)
(cid:12) (cid:13) i (cid:13) (cid:12) the ones of the transitions in the same energy band.
(cid:16) (cid:17) (cid:12) (cid:13) (cid:13) (cid:12)
g 2(cid:12)ω g (cid:12) With the above discussions, the Hamiltonian matrix
∼ ω (cid:12)(cid:12)ω ∼ ω (cid:12)(cid:12) (34) H¯ can be written as
a
(cid:16) (cid:17) g g 2
and H¯ H¯(0)+ H¯(1)+ V¯(2) (36)
≈ ω ω
g 2 E(n0)1j O¯(2) En(0i) Which is just the effective Ham(cid:16)ilto(cid:17)nian for the BO ap-
| − |
ω (cid:12)(cid:12)D( 1)n+1(cid:13)(cid:13)ω+ε(cid:13)(cid:13)i εkE(cid:12)(cid:12) proximation. In fact, substituting Eqs. (23-21) into Eq.
(cid:16) (cid:17) (cid:12) − (cid:13) (cid:13)− (cid:12) (36), we can write the eigen equation (19) as
(cid:12) (cid:12)
(cid:12) E(0) O¯(2) E(0)(cid:12)
g 2(cid:12) |n−1|0,k ni (cid:12) ωa σˆz(1)+σˆz(2)
∼(cid:16)ω(cid:17) (cid:12)(cid:12)(cid:12)D (−1)n+(cid:13)(cid:13)(cid:13)1ω+(cid:13)(cid:13)(cid:13)εi E(cid:12)(cid:12)(cid:12)  +(cid:16)gω2σˆx(1)σˆx(2) (cid:17) , 0 C =EC.
g 2(cid:12)(cid:12) (cid:12)(cid:12) (35)  ωa σˆz(1)+σˆz(2) 
whereweha∼ve(cid:16)uωs(cid:17)ed(cid:12)g ˜1. Therefore,itcanb(cid:12)eseenfrom  0, +gω(cid:16)2σˆx(1)σˆx(2)+ω(cid:17) 
ωa This equation equals to
Eq. (33) that the contribution from O¯(2) to E(1) and
ni
En(10),k are much smaller than the ones from(cid:13)(cid:13)(cid:13)V¯(2)E. λ′,λ′ Hλλ′11,,λλ′22(n)Cλ′1,λ′2(n)=ECλ1,λ2(n) (37)
(cid:13) In suEmmary, we have shown that one can reasonably X1 2
n(cid:13)(cid:13)eglect the first order corrections from O¯(2) to both the which is approximated form of Eq. (12). It can be seen
eigen-values and the eigen-states. from Eq. (13) that Hλ1,λ2(n) are the matrix elements
λ′,λ′
1 2
of the effective Hamiltonian Hˆ (n) in Eq. (9). It is
eff
pointedoutthat,Eq. (37)isequivalentwithEq. (5)and
C. The Generalized BO Approximation From
Eq. (8) in the above section.
Perturbation Theory
To solve the time-dependent Schro¨dinger equation
which describe the time evolution of the composite sys-
The above result can be obtained with intutitve anal-
tem, we can calculate the approximate eigen-energies E
ysis. It can be seen from Eq. (24) and Eq. (23) that the
matrix V¯(2) leads to the transitions in the same energy oftheHamiltonianofthecompositesystemwithEq. (9):
bandandO¯(2) leadstotheinterbandtransitions(seeFig. E =ǫ +nω (38)
ni i
3).
where ǫ is the i-th eigen-energy of the Hamiltonian hˆ
i
defined in Eq. (4). Eq. (13) is the time-independent
(2)
V
Schro¨dinger equationunder the BO approximation. The
n = 1 time-dependent Schro¨dinger equation can be written as:
d
O(2) idtCλ1,λ2(n)= Hλλ′11,,λλ′22(n)Cλ′1,λ′2(n). (39)
λ′,λ′
X1 2
MakinguseoftheaboveBOapproximationfordiscrete
system, we can calculate the evolution of the composite
system with Eq. (39). If the system is initially prepared
in the state
n = 0
Ψ(0) = C (n,0) n[λ ,λ ]),
| i λ1,λ2 | 1 2
n,(Xλ1,λ2)
FIG. 3. The transitions induced by W¯. We have shown
that, g 2O¯(2)onlyinducesthetransitionsbetweensubbands then with Eq. (38), Eq. (9) and Eq. (4), the quantum
ω
and g 2V¯(2) only induces the transitions in the same sub- state at any instance can be expressed as:
(cid:0)ω (cid:1)
band.
(cid:0) (cid:1) Ψ(t) = C (n,t)e inωt n[λ ,λ ])
| i λ1,λ2 − | 1 2
Aswehavementioned,theenergygapbetweenthetwo n,(Xλ1,λ2)
bands has the same order of ω and is much larger than = Cλ1,λ2(n,t)e−inωt|λ1,λ2i⊗|n(λ1,λ2)i (40)
theenergydifferencebetweenthelevelsinthesameband.
n,(Xλ1,λ2)
6
where where
C (n,t) C (n,0)= λ ,λ ψ n(λ ,λ ) φ . (42)
λ1,λ2 λ1,λ2 h 1 2| i·h 1 2 | i
= hλ1,λ2|e−ihˆt|λ′1,λ′2iCλ′1,λ′2(n,0). (41) istheprojectionofinitalstateontothecompletenessba-
λX′1,λ′2 sis n[λ1,λ2] . With the BO approximation generalized
| i
in the above section, the evolution of the composite sys-
Apparently,theevolutionofthetwoqubitsystemisgov-
erned by the Hamiltonian hˆ in Eq. (4) which contains tem is depicted by Eq. (40).
Substituting Eq. (42) into Eq. (40), we have the wave
the two qubit interaction 2 g σˆ(1)σˆ(2). However, it can
− ω2 x x function in the (λ1,λ2) picture at time t.
be seen from Eq. (40) that because of the dependence
ooff|tnh(eλc1o,mλ2p)oisoitne{sλy1s,tλem2},isthuesiunasltlaynatanneenotuasnsgtlaetest|aΨte(to)if hλ1,λ2|Ψ(t)i=
λ ,λ e ihˆt λ ,λ λ ,λ ψ φ(t) (43)
the two qubitsystemandthe externalfield. Thus,if one h 1 2| − | ′1 ′2ih ′1 ′2| i⊗| i
traceoverthevariableofthedatabus,thenthethestate λX′1,λ′2
of the two qubit system is not a pure sate but a mixed
where
one. This adiabatic decoherence effect will be analyzed
in next section.
|φ(t)i =|φ(t,λ1,λ2,λ′1,λ′2)i
= e−inωthn(λ′1,λ′2)|φi n(λ1,λ2)
IV. DECOHERENCE EFFECT FROM THE n
X (cid:12) (cid:11)
ADIABATIC ENTANGLEMENT (cid:12)
depends onboth coordinatesets (λ ,λ ) and(λ ,λ ). It
1 2 ′1 ′2
should be noted that the factor n(λ ,λ ) φ n(λ ,λ )
Intheabovesection,wehaveobtainedanapproximate h ′1 ′2 | i| 1 2 i
has different pairs of indices (λ ,λ ) and (λ ,λ ). If
Hamiltonian Hˆ (n) = hˆ +nω for the composite sys- 1 2 ′1 ′2
eff φ(t) = φ(t,λ ,λ ,λ ,λ ) were independent of λ ,λ
temwithtwoqubitsinteractingwithanexternalfield. As | i | 1 2 ′1 ′2 i 1 2
andλ ,λ ,thenwewouldhaveafactorizedevolutionde-
we have mentioned, there is quantum entanglement be- ′1 ′2
scribed by Ψ(t) = e ihˆt ψ φ(t) . In this case, the
tweenthestatesoftheexternalfieldandthoseofthetwo | i − | i⊗| i
two qubit system were just in the pure state controlled
qubitsystembecauseofthedependenceof n(λ ,λ ) on
| 1 2 i by the effective Hamiltonian hˆ and the decoherence phe-
λ ,λ . This is just the adiabatic quantum entangle-
{ 1 2} nomenon does not occur.
ment which has been studied in detail for the develop-
In the usual case, φ(t) = φ(t,λ ,λ ,λ ,λ ) does
mentofBOapproximation[11]. Itiseasytoseethatthe | i | 1 2 ′1 ′2 i
dependonλ ,λ ,λ andλ . Thenthedecoherenceeffect
adiabaticentanglementmaycausetheadiabaticdecoher- 1 2 ′1 ′2
must result from the dependence φ(t) to different sets
enceofthe quantumstateofthetwoqubitsystem. Only | i
of (λ ,λ ,λ ,λ ). In fact, we can calculate the elements
when this decoherence effect can be neglected, can our 1 2 ′1 ′2
effective Hamiltonian hˆ be used to create a two bit gate. of the reduced density matrix of the two qubit system
In this section, the adiabatic decoherence effect in our
problemwillbeanalyzedindetail. Itwillbeshownthat, ρλ1,λ2,λ′1,λ′2(t)=
in some practicalcases,this kind ofdecoherence effect is Kλ1,λ2;µ1,µ2[|ψi,t]Kλ∗′1,λ′2;µ′1,µ′2[|ψi,t]
negligibleandthenthefasterquantumentanglementcan µX1,µ2µX′1,µ′2
beWcreeaatsesudmveiaththeetwBoOqaupbpitrosyxsimteamtioann.dtheexternalfield ×hφ(t;λ′1,λ′2;µ′1,µ′2)|φ(t,λ1,λ2;µ1,µ2)i (44)
are initially prepared in a factorizable state: where the propagationfunctionals
|Ψ(0)i =|ψi⊗|φi Kλ1,λ2;µ1,µ2[|ψi,t]=hλ1,λ2|e−ihˆt|µ1,µ2ihµ1,µ2|ψi.
where ψ is the initial state of the two qubits and φ (45)
| i | i
the state of the external field. Ψ(0) can be expanded
| i depend on the initial state ψ of the external field. On
with respect to the basis n[λ ,λ ]) as
1 2 | i
| theotherhand,withoutdecoherence,thereduceddensity
|Ψ(0)i= Cλ1,λ2(n,0)⊗|n[λ1,λ2]) meleamtreixntosfthe qubits is ρ′ =e−ihˆt|ψihψ|eihˆt with matrix
n,(Xλ1,λ2)
or equivalently in the |λ1,λ2i picutre ρ′λ1,λ2,λ′1,λ′2(t) (46)
hλ1,λ2 |Ψ(0)i= n Cλ1,λ2(n,0)⊗|n[λ1,λ2]i =µX1,µ2µX′1,µ′2Kλ1,λ2;µ1,µ2[|ψi,t]Kλ∗′1,λ′2;µ′1,µ′2[|ψi,t]
X
7
g
Comparing Eq. (46) with Eq. (44), we measure the Υ= (µ +µ µ µ ) e iωt
ω 1 2− ′1− ′2 −
extentofdecoherencebythesocalleddecoherencefactor
h g i
[15]. −ω (λ1+λ2−λ′1−λ′2). (53)
F ≡hφ(t;λ′1,λ′2;µ′1,µ′2)|φ(t,λ1,λ2;µ1,µ2)i (47) However,in the case with
As pointed in ref. [15], if F =1, there is no decoherence. Ω <<1, Υ2 <<1 (54)
| | | |
Itshouldbepointedoutthat,notonlythenormofF but
we have φ(t;λ ,λ ;µ ,µ ) φ(t,λ ,λ ;µ ,µ ) 1 and
also its phase are crucially important to the description h ′1 ′2 ′1 ′2 | 1 2 1 2 i ≃
the decoherence effect can be neglected.
of quantum decoherence process. For example, if F =
It is observed from the explicit expression for the de-
exp[iΘ] and Θ is a random phase with respect to the
coherence factor that Ω and Υ are all periodic functions
parameters λ ,λ ,λ ,λ ;µ ,µ ,;µ1,µ2, even if F = 1,
1 2 ′1 ′2 ′1 ′2 | | of t with frequency ω. The amplitudes of Ω and Υ do
ρ (t) still have significant difference from the
′λ1,λ2,λ′1,λ′2 not vary with time (see Fig. 4). Therefore, in case the
pure ρλ1,λ2,λ′1,λ′2(t) . amplitudes of Ω and Υ are much smaller than 1, the in-
We assume the initial state of the external field is in
equalities in Eq. (54) can be satisfied at any instance.
a coherence state, φ = α (noting the condition n . 1
| i | i
in the above section, we assume α < 0.5 here). By a
1
| |
straightforwardcalculation, we obtain
1
0.99 k=€€€€€€€
φ(t,λ ,λ ,λ ,λ ) 20
| 1 2 ′1 ′2 i
=eiΦ(t,λ1,λ2,λ′1,λ′2) 0.98
α+ ωg (λ′1+λ′2×) e−iωt− ωg (λ1+λ2) (48) ReF 0.97k=€1€€€€€€
10
(cid:12)h i E
(cid:12)
where t(cid:12)he time-varing phase
0.96
g
Φ= ω (λ′1+λ′2) Imα∗ 0.95
0 10 20 30 40 50
h g i
(λ +λ ) Im α eiωt t
− ω 1 2 ∗
h g 2 i (cid:0) (cid:1)
(λ +λ )(λ +λ ) sinωt (49) FIG.4. The real part of the decoherence factor ReF as a
−(cid:20)(cid:16)ω(cid:17) ′1 ′2 1 2 (cid:21) fµu1n+ctiµon2 −ofµt′. −Thµe2′un=itλof+t iλs ω−−1λ.1′W−eλa2s′su=me1.αT=he0raatnido
depends on the parameters λ′1,λ′2, λ1 and λ2 and g is denoted1as k. It ca1n be2seen from the figure that the
α+ g (λ +λ ) e iωt g (λ +λ ) is also a coher- ω
ω ′1 ′2 − − ω 1 2 decoherence factor is a periodic function of time.
ent state.
(cid:12)(cid:2) (cid:3) (cid:11)
(cid:12) Considering the inner product of coherent state satis- Because every λ or λ may be 1, the sufficient con-
1 2
fies dition of Υ2 << 1 is just 8g 2 <±< 1 and if g is small
| | ω ω
enough so that this condition can be satisfied, then the
hα|α+pi=exp −|p|2 exp[iIm(pα∗)], (50) lasttermofthe expressiono(cid:12)(cid:12)fΩ(cid:12)(cid:12)canbeneglectedandthe
h i sufficient condition of Ω <<1 becomes
we can express the decoherence factor | |
g
16 α <<1. (55)
F =exp[iΩ] exp Υ2 (51) ω | |
· −| |
h i Thisis arestrictionforthe strengthofthe externalfield.
intermsofthephasefactorΩanditsnormexp Υ2 . Therefore, we know that if the ratio g is small enough
ω
−| | (for example, g 0.01) and the external field is weak
Here,theexplicitexpressionofΩandΥcanbehobtaineid: ω ∼
enough (e.g. α << 1), the atomic state decoherence
g caused by the|ex|ternal field can be omitted and the two
Ω=2 (µ +µ µ µ ) Imα
ω 1 2− ′1− ′2 ∗ qubitsystemcanevolveunderthecontroloftheeffective
+h2 ωg (λ′1+λ′2−λ1−λ2i) Im α∗eiωt Hamiltonian hˆ.
hg 2 i (cid:0) (cid:1)
+ (µ +µ µ µ )
ω 1 2− ′1− ′2 V. CONCLUSION WITH REMARKS
(cid:16)(λ (cid:17)+λ +λ +λ )sinωt (52)
× ′1 ′2 1 2
Inthispaper,wehaveproposedanewprotocolforthe
and creation of quantum entanglement between two qubits
8
interacting with a high frequency data bus. Unlike the Acknowledgement: This work is supported by the
previousschemes,ourprotocaldoesnotrequiretherota- NSFC (grant No.90203018) and the knowledge Innova-
tionwaveapproximationfortheinteractionHamiltonian. tion Program (KIP) of the Chinese Academy of Sci-
We can first assume the external field is prepared in the ences and the National Fundamental Research Program
coherencestate α . Ifthe frequencyofthe externalfield ofChina withNo001GB309310. Wealsosincerelythank
| i
is large enough and the strength of it is weak enough so L. You for the helpful discussions with him.
that the following conditions
g2 APPENDIX A: THE CALCULATION OF THE
ωa ∼g <<ω, 64ω2 <<1, OVERLAP FACTOR hM(λ1,λ2)|N(λ′1,λ′2)i
16g
α <0.5, α <<1 (56)
| | ω | | In this appendix, we calculate the explicit expression
of m(λ ,λ ) n(λ ,λ ) . It can be seen from Eq. (6)
can be satisfied, an effective coupling between the two h 1 2 | ′1 ′2 i
thatthe HamiltonianH (λ ,λ )isjusttheHamiltonian
qubit system can be obtained under the BO approxima- f 1 2
of a displaced harmonic oscillator and can be expressed
tion and the decoherence caused by the cavity field can
as
be neglected.
TheexplicitexpressionoftheHamiltonianofthecom- g2
positesystemisgiveninEq. (3). IftheconditionsbyEq. Hf(λ1,λ2)≃ωAˆ†λ1,λ2Aˆλ1,λ2 −2ω λ1λ2 (A1)
(56) are satisfied, the evolution of the two qubit system
wheretheconstanttermindependentonλ ,λ isomitted.
is governed by the effective Hamiltonian 1 2
Here,
g2
hˆ =ωa σˆz(1)+σˆz(2) −2ω σˆx(1)σˆx(2). (57) Aˆλ1,λ2 = aˆ+ ωg (λ1+λ2)
(cid:16) (cid:17)
In our previous calculation, we have assumed ωa is a displaced boson opera-
∼
geff<ec<tivωe.HaTmhiilstoisnitahne(5su7)ffi.cIitenistecaosnydtiotipornovteotohbattaiifnωthies tor that satisfies Aˆλ1,λ2,Aˆ†λ1,λ2 = 1. With Eq. (A1),
a
smaller than g and g << ω, the effective Hamiltonian the eigen-state ofhHf(λ1,λ2) canibe written as:
(57)canbealsoobtainedwiththedeductionintheabove
1
sections. To realize a logic gate with high efficiency, we |n(λ1,λ2)i = √n!Aˆ†λn1,λ2|0(λ1,λ2)i
suppose our proposal work under the condition ω <<
a g
g and g << ω so that the interaction strength 2gω2 is =Dˆ −ω (λ1+λ2) |ni (A2)
comparablewiththestrengthωaofthefreeHamiltonian. h i
byAtshewefahcatovresΩhoawnnd,|tΥh|e2dwechoichheraernecepeerffieocdticisfduentcetrimoninseodf ianndtertmhes onfatthueradlisFpolackcemsteantteo|pneira=tor√D1ˆn!(cid:2)aˆ−†nωg|0(iλ.1+Hλe2r)e(cid:3),
time. TheniftheconditionsinEq. (56)aresatisfied,the 0 is the realistic vacuum state satisfies aˆ 0 = 0
| i | i
decoherence effect can be neglected no matter how long and 0(λ ,λ ) the displaced vacuum state satisfies
1 2
the operation takes. Aˆ | 0(λ ,λi) =0. Itiseasytoproofthat 0(λ ,λ )
λ1,λ2| 1 2 i | 1 2 i
The scheme proposedin this paper can be generalized is actually a coherence state which can be defined as
tothesystemthatN qubitscoupledtoadatabus. With 0(λ ,λ ) =Dˆ g (λ +λ ) 0 .
| 1 2 i −ω 1 2 | i
thesamediscussionsasabove,itisapparentlythatinthe With
(cid:2) (cid:3)
large detuning and weak coupling case where ωa << ω, Eq. (A2), the explicit value of hm(λ1,λ2)|n(λ′1,λ′2)i
g << ω, and the number of the qubits is not very large can be obtained easily:
sothatthecondition2Nω <<ωissatisfied,aneffective
a g
coupling between the qubits m(λ ,λ ) n(λ ,λ ) = m D Σ¯λ1,λ2 n
h 1 2 | ′1 ′2 i h | ω λ′1,λ′2 | i
V ∼−gω2 σˆx(1)+σˆx(2)+...+σˆx(N) 2 =−gω2Jx2 =e−21(cid:12)(cid:12)ωgΣ¯λλ1′1,,λλ2′2(cid:12)(cid:12)2hm|eωgΣ¯λλ1′1,,λλ2′2a+e−ωghΣ¯λλ1′1,,λλ2′2a|nii
cpaonintbeed oobuttaitnhe(cid:16)adt wunitdherthtehe”JBx2Ointaeprparc(cid:17)otxioinm”atbioetnw.eeInt Nis =e−21(cid:12)(cid:12)(cid:12)(cid:12)ωgΣ¯λλ1′1,,λλ2′2(cid:12)(cid:12)(cid:12)(cid:12)2 m (−1)n−ml!((cid:16)lωg+Σ¯nλλ1′1,,λλ2′2m(cid:17))2!l+(n−m)
qubits, the N-qubit GHZ state can be created easily ( (cid:12)(cid:12) (cid:12)(cid:12) Xl=0 −
[16]). Then a physical qubit of the form α 0 +β 1 can n(n 1)...(m l+1) m(m 1)..(m l+1).
| i | i − − − −
be encoded into a logical qubit of the form α 000... +
β 111... ( [17]). It is also proved that with the| interiac- ThispisjustEq. (16)insectiopnIII.Substituting Eq. (16)
tio|n of tihis type, Shor code for error correction can be into Eq. (14) and Eq. (15), we can get the expressionof
easily realized. Fλλ′1,,λλ′2(n) and Oλλ′1,,λλ′2(n) i.e. Eq. (17) and Eq. (18).
1 2 1 2
9
a Electronic address: [email protected]
b Internetwww site: http:// www.itp.ac.cn/ suncp
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