Embed Size (px)
Citation preview
Optics Communications 283 (2010) 1979–1983
Contents lists available at ScienceDirect
Optics Communications
journal homepage: www.elsevier .com/locate /optcom
One-dimensional cluster state generated in one step via one cavity
Yong He, Nian-Quan Jiang *, Yong-Yun JiCollege of Physics and Electric Information, Wenzhou University, Wenzhou 325035, China
a r t i c l e i n f o
Article history:Received 26 October 2009Received in revised form 31 December 2009Accepted 1 January 2010
Keywords:Cluster stateMicrowave cavityGHZ state
0030-4018/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.optcom.2010.01.003
* Corresponding author.E-mail address: [email protected] (N.-Q. Jiang).
a b s t r a c t
A scheme to generate multi-atom one-dimensional cluster state via one microwave cavity with anadditional driven classical field is proposed. According to the scheme, one-dimensional cluster state with2k-atom can be prepared in one step via one cavity, one-dimensional cluster state with ð2k� 1Þ-atom canbe generated by measuring the2kth-atom of an 2k-atom cluster state in a certain basis. This schemeavoids cavity-field induced decay and may achieve one-dimensional cluster states with ideal successprobability.
� 2010 Elsevier B.V. All rights reserved.
1. Introduction
The cluster states, which were firstly introduced in spin-chainsystems by Briegel and Raussendorf [1], are more immune to deco-herence than Greenberg–Horn–Zeilinger (GHZ) states [2]. Thesehighly entangled states are the key ingredient in one-way quantumcomputing [3,4] and have been studied in different physical sys-tems because of both conceptual and practical importances. Sev-eral schemes of generating cluster states in an ion-trap system[5], leaky cavities, resonant microwave cavities, and linear opticalsystems [6–8] have been proposed recently. It has also been foundthat the cluster states can be effectively generated in superconduc-tor charge qubit [9–11], semiconductor quantum dots [12–14] andatom cavity systems [15,16]. Moreover, for the four-qubit clusterstate, several experimental implementations have also been dem-onstrated [17–19].
However, most existing methods of generating cluster states re-quire multiple steps. Ideally, it is desirable to produce a clusterstate in just one step. There are some schemes preparing clusterstates with ions in just one step [10,11,20–22], but most of the pre-pared ‘‘cluster states” in these proposals are in fact the GHZ states(see Appendix A). So, it is interesting to propose a scheme for gen-erating cluster states in just one step. In this article we willdemonstrate such a scheme based on a microwave cavity withN identical two-level atoms simultaneously interacting with a sin-gle-mode cavity-field driven by a classical field.
ll rights reserved.
2. Evolution of atoms
We consider N identical two-level atoms simultaneously inter-acting with a single mode vacuum cavity-field and driven by astrong classical field. In the rotating-wave approximation, theHamiltonian of the whole system is given by (assuming �h ¼ 1)[23,24]
H ¼ 12
wXN
j¼1
rz;j þ ðdþwÞaþaþXXN
j¼1
rþj e�iw0t þ r�j eiw0t� �
þ gXN
j¼1
rþj aþ r�j aþ� �
; ð1Þ
where rz;j ¼ j1jih1jj � j0jih0jj, rþj ¼ j1jih0jj;r�j ¼ j0jih1jj, j1ji and j0jiare excited and ground states of the jth-atom. g is the atom–cavitycoupling strength. X is the Rabi frequency of the classical field. wand w0 are the atomic transition frequency and classic field fre-quency, respectively. d is the detuning between the atomic transi-tion frequency and cavity frequency. aþ and a are the creationand annihilation operations of the cavity mode. Assuming w ¼ w0,in a frame that rotates with the classical wave frequency w0, theHamiltonian is found as
Hi ¼ XSx þ daþaþ gðaSþ þ aþS�Þ ¼ H0i þ H1
i ; ð2Þ
where Sþ ¼PN
j¼1rþj ; S� ¼
PNj¼1r�j , Sx ¼ Sþ þ S�, H0
i ¼ XSx, andH1
i ¼ daþaþ gðaSþ þ aþS�Þ.Then, the time evolution of the system is decided by
idjwðtÞi
dt¼ ðH0
i þ H1i ÞjwðtÞi: ð3Þ
Performing the unitary transformations jwðtÞi ¼ e�iH0i tj/ðtÞi and
HI ¼ eiH0i tH1
i e�iH0i t , then the Eq. (3) becomes
1980 Y. He et al. / Optics Communications 283 (2010) 1979–1983
idj/ðtÞi
dt¼ HIj/ðtÞi: ð4Þ
Choosing the Rabi frequency to satisfy the condition 2X� d; g, un-der the rotating-wave approximation we can obtain the effectiveHamiltonian as follows:
Heff ¼ daþaþ g2ðaþ aþÞSx: ð5Þ
So, for the initial state j/ðt¼0Þi, the evolution state is given byj/ðtÞi ¼ e�iHeff tj/ðt¼0Þi, and then jwðtÞi ¼ e�iH0
i tj/ðtÞi ¼ e�iHeff te�iH0i t j
/ðt¼0Þi. Obviously, one can see that jwðt¼0Þi ¼ j/ðt¼0Þi, and then
jwðtÞi ¼ e�iHeff te�iH0i t jwðt¼0Þi. Thus, the evolution operator of the sys-
tem in the interaction picture is
UðtÞ ¼ e�iHeff te�iH0i t
¼ exp �i daþaþ g2ðaþ aþÞSx
h it
n oexpð�iXSxtÞ: ð6Þ
After operator calculation, we can rewrite it as
UðtÞ ¼ expð�idaþatÞ exp � gSx
2d½aðe�idt � 1Þ � aþðeidt � 1Þ�
� �
� exp iS2x
g2t4d� g2 sin dt
4d2
� �� expð�iXSxtÞ: ð7Þ
If the conditions X ¼ nd (n is an integer), d ¼ 2g, t ¼ s and ds ¼ 2pare satisfied, the evaluation operator in Eq. (7) turns into
UðsÞ ¼ exp ip8
S2x
� �: ð8Þ
Thus, if an initial state is given as jwðt¼0Þi, it will be evolved intojwðsÞi ¼ UðsÞjwðt¼0Þi after a period of time s.
If N atoms are initially in the ground state, then the state is
jwð0Þi ¼ j010203; . . . ;0Ni ¼1ffiffiffiffiffiffi2N
p �Nj¼1ðjþji þ j�jiÞ: ð9Þ
where jþji ¼ 1ffiffi2p ðj0ji þ j1jiÞ; j�ji ¼ 1ffiffi
2p ðj0ji � j1jiÞ, j ¼ 1;2; . . . ;N. We
find that jwð0Þi can be rewritten as
jwð0Þi ¼1ffiffiffiffiffiffi2N
p XN
l¼0
julNi; ð10Þ
where julNi ¼
PPPj�1�2 � � � �lþlþ1þlþ2 � � � þNi, P denotes the opera-
tion of permutation for � and +, for example,ju2
3i ¼P
PPj�1�2þ3i ¼ j�1�2þ3i þ j�1þ2�3i þ jþ1�2�3i (one canalso see the similar examples shown in the Appendix A). Usingthe relation Sxjul
Ni ¼ ðN � 2lÞjulNi and Eq. (8), we derive
UðsÞjulNi ¼ exp i
p8ðN � 2lÞ2
h ijul
Ni; l ¼ 0;1;2; . . . ;N: ð11Þ
From the relations exp i pl2
2
� �¼ 1ffiffi
2p eip4 þ ð�1Þle�ip4
� �and exp �i pNl
2
� �¼
ðð�iÞNÞl, we obtain exp i p8 ðN � 2lÞ2
h i¼ 1ffiffi
2p exp i pN2
8
� �eip4ðð�iÞNÞl�
þð�ð�iNÞÞle�ip4Þ. Then, using Eqs. (10) and (11) and relationsQNj¼1ðjþji þ ð�iÞNj�jiÞ ¼
PNl¼0ðð�iÞNÞlj/l
Ni andQN
j¼1ðjþji � ð�iÞNj�jiÞ¼PN
l¼0ð�ð�iÞNÞlj/lNi, we can express the state jwðsÞi ¼ UðsÞjwð0Þi as
jwðsÞi ¼ UðsÞjwð0Þi
¼ 1ffiffiffiffiffiffiffiffiffiffi2Nþ1
p �Nj¼1ðjþji þ ð�iÞN j�jiÞ � i�N
j¼1ðjþji � ð�iÞNj�jiÞh i
;
ð12Þ
where a common phase factor exp i pðN2þ2Þ8
� �is omitted.
If N atoms are initially in the excited states, then it can be ex-pressed as jw0ð0Þi ¼ j111213; . . . ;1ni ¼ 1ffiffiffiffi
2Np PN
l¼0ð�1ÞljulNi. In the
same theory as above, we can derive
jw0ðsÞi ¼ UðsÞjw0ð0Þi
¼ 1ffiffiffiffiffiffiffiffiffiffi2Nþ1
p �Nj¼1ðjþji � ð�iÞN j�jiÞ � i�N
j¼1ðjþji þ ð�iÞNj�jiÞh i
:
ð13Þ
3. Schemes to generate cluster state
Now, based on the above analysis, we demonstrate our methodsto generate the cluster states. Firstly, we show the scheme to gen-erate 2k-atom cluster state. We let 2k (k is an integer) identicaltwo-level atoms be initially prepared in the ground states in a de-vice, then let these atoms effuse from the device in pairs at propertime with velocity v � 30 m=s [25], and then they pass though asingle mode vacuum cavity of high quality factor with a strongclassical driving field. The effusing time of the atoms is properlycontrolled so that the distances between every two atoms satisfyL2;3 ¼ L4;5 ¼ � � � ¼ L2k�2;2k�1 ¼ L and L
v ¼ s ¼ 2pd . The classical driving
field is not switched on until the atoms 1 and 2 arrive at the pointwhich is L from the exit of the cavity. During the period withoutclassical driving field, the interaction Hamiltonian for the atomsand the cavity is Hint ¼ g
PNj¼1ðarþj eidt þ aþr�j e�idtÞ [26], hence any
atoms, which are initially in the ground states, cannot be excited.Once all the atoms enter the cavity and the atoms 1 and 2 reachthe point which is L from the exit, the classical driving field willbe switched on. Then the atoms 1 and 2 leave the cavity at times, 3 and 4 leave the cavity at time 2s, . . ., 2k� 1 and 2k leave thecavity at time ks. Thus, using Eqs. (12) and (13), we achieve the fi-nal evolution state of 2k atoms after all the atoms leave the cavity,it can be expressed as following (see Appendix B),
jw2ki ¼1ffiffiffiffiffi2k
p �k�1j¼1 ðj02j�102jirð2j�2Þ
z � j12j�112jiÞh i
� ðj02k�102kirð2k�2Þz þ ij12k�112kiÞ: ð14Þ
where rð2j�2Þz ¼ j02j�2ih02j�2j � j12j�2ih12j�2j, ðj ¼ 1;2; . . . ; kÞ, and
rð0Þz ¼ 1. This formula should be understood as an iteration, withthe operator rð2j�2Þ
z acting on the (2j� 2)th qubit which is left fromrð2j�2Þ
z and is direct product with it. For example, jw4i ¼ 12
ðj01020304i þ j11120304i þ ij01021314i � ij11121314iÞ. One can alsosee the example in [27]. Implementing local unitary transformationon atoms 2j ðj ¼ 1;2; . . . ; kÞ separately, the above state jw2ki be-comes 1
2k �2kh¼1ðj0hirðh�1Þ
z þ j1hiÞ (This formula is also an iteration,with the operator rðh�1Þ
z acting on the (h� 1)th qubit which is leftfrom rðh�1Þ
z and is direct product with it. Thus, the expression12k �2k
h¼1ðj0hirðh�1Þz þ j1hi is equivalent to the Eq. (2) (let a ¼ 2k) in
[1] (where the operator rðaþ1Þz acts on the qubit aþ 1 which is right
from it) with the changes of the numbers of particles:1;2; . . . ;2k! 2k; . . . ;2;1Þ. Hence, we have jw2ki¼00l:u:00 1
2k �2kh¼1
ðj0hirðh�1Þz þ j1hiÞ, where ‘‘l.u.” indicates that the equality holds up
to a local unitary transformation on one or more of qubits [1].Therefore, one can see that the statejw2ki in Eq. (14) is an 2k-atomone-dimensional cluster state [1]. For example, when k ¼ 2, Eq.(14) reduces to jw4i ¼ 1
2 ðj01020304i þ j11120304i þ ij01021314i�ij11121314iÞ, obviously, it is a four-atom cluster state.
We then show the scheme to generate ð2k� 1Þ-atom one-dimensional cluster state. We rewrite the above 2k-atom clusterstate jw2ki as
jw2ki ¼1ffiffiffiffiffiffiffiffiffi2kþ1
p �k�1j¼1 ðj02j�102jirð2j�2Þ
z � j12j�112jiÞh i
� ðj02k�1irð2k�2Þz þ ij12k�1iÞ � jþ2ki
þðj02k�1irð2k�2Þ
z � ij12k�1iÞ � j�2ki�; ð15Þ
Measuring the 2kth-atom in the basis jþ2ki (The impact of a mea-surement in a cluster state in a Z basis was explicitly shown in[28]), then the other 2k� 1 atoms are prepared in the state as
Y. He et al. / Optics Communications 283 (2010) 1979–1983 1981
jw2k�1i ¼1ffiffiffiffiffi2k
p ½�k�1j¼1 ðj02j�102jirð2j�2Þ
z � j12j�112ji�
� ðj02k�1irð2k�2Þz þ ij12k�1iÞ ð16Þ
In the same theory as above, we can obtain that jw2k�1i¼00l:u:00 1ffiffiffiffiffiffiffiffiffi22k�1p
�2k�1h¼1 j0hirðh�1Þ
z þ j1hi� �
. So, the resulting state jw2k�1i in Eq. (16) is
an ð2k� 1Þ-atom one-dimensional cluster state.To show the experimental feasibility of the proposed scheme,
we give a brief evaluation of the experimental parameters. Forthe Rydberg atoms with principal quantum numbers 50 and 51,the radiative time is about Tr ¼ 3� 10�2 s, and the coupling con-stant of the Rydberg atoms and the high-Q cavity isg=2p � 50 kHz [25,29–32], and then s ¼ p=g � 4� 10�5 s. For2k � 102 atoms, the required time of operation in order to generatethe cluster states is t ¼ ks � 2� 10�3 s, which is much shorter thanthe atomic radioactive time Tr. In a practical situation, the couplingstrength between the atoms and the cavity depends on the atomicpositions g ¼ g0e�r2=x2 , where g0 is the coupling strength at thecavity center, x is the waist of the cavity mode, and r is the dis-tance between the atom and the cavity center [33,34]. If we as-sume that all the atoms (or ions) go along a nearly circular orbitthrough the cavity (This is possible, e.g. Rydberg-type ions passthrough a steady magnetic field), then the atom–cavity couplingstrength for each atom is identical. Moreover, the diameter of thecavity is d � 3� 10�2 m [25], and then pd
L ¼ pdvs � 75, which shows
that one can prepare one-dimensional cluster state with tens ofthe atoms (ions) via one cavity. We then show the effect of thespontaneous emission of light from the excited atomic state.Assuming the atomic decay rate is c, we find that the evaluationoperator in Eq. (8) should be change into
UðsÞ ¼ exp � Ncs4
� �exp i p
8 S2x
� �. Obviously, once N; c; s are given,
the factor exp � Ncs4
� �is a constant number, and then the evolution
of the state is the same as that caused by the evaluation operator inEq. (8). So, our scheme is not affected by the spontaneous emission.
4. Conclusion
We have proposed a scheme to generate one-dimensional clus-ter states. According to our scheme, an 2k-atom one-dimensionalcluster state can be prepared in one step by using only one cavitydriven by a classical field. Measuring the 2kth-atom in the gener-ated cluster state in a basis, one can achieve (2k� 1)-atom one-dimensional cluster state. Since the cavity-field acts as a databus, the effective long-range coupling among atoms, which isinsensitive to the cavity mode, can be achieved without the helpof auxiliary devices. Thus the scheme is insensitive to the cavity-field induced decay, and then may approach the ideal success prob-ability to generate one-dimensional cluster states.
Acknowledgment
The work was supported by National Natural Science Founda-tion of China under Grant No. 10947017/A05.
Appendix A
In Ref. [20] (similarly in Refs. [11,18,19]), the authors intro-duced schemes to generate cluster state, the ‘‘cluster state” gener-ated by them is expressed as
jwiN ¼1ffiffiffiffiffiffi2N
p �Nj¼1 j0ijð�1ÞN�j
YNl¼jþ1
rzl þ j1ij
!; ðiÞ
where rzl ¼ j0il lh0j � j1il lh1j, rz
l j0il ¼ j0il and rzl j1il ¼ �j1il, here we
replace the symbol rxl in [20] by rz
l . (The ‘‘cluster state” shown in
Refs. [11,19] is expressed as jwiN ¼ 1ffiffiffiffi2Np �N
j¼1 j � ijð�1ÞN�jQNl¼jþ1
�rx
l þ j þ ijÞ, where rxl j � il ¼ j � il and rx
l j þ il ¼ �j þ il, obviously,it is equivalent to the expression in Eq. (i)). However, this state isnot a real cluster state, it is in fact a Greenberger–Horne–Zeilinger(GHZ) state. For example, when N ¼ 4, we can obtain from Eq. (i)that
jwi4 ¼1ffiffiffiffiffi24
p �4j¼1 j0ijð�1Þ4�j
Y4
l¼jþ1
rzl þ j1ij
!
¼ 1ffiffiffiffiffi24
p j11121314i þ j01121314i þ j11021314i þ j11120314ið
þj11121304i � j01021314i � j01120314i � j01121304i� j11020314i � j11021304i � j11120304i � j01020314i�j01021304i � j01120304i � j11020304i þ j01020304iÞ: ðiiÞ
Let
jv04i ¼
XP
Pj11121314i ¼ j11121314i;
jv14i ¼
XP
Pj01121314i
¼ j01121314i þ j11021314i þ j11120314i þ j11121304i;
jv24i ¼
XP
Pj01021314i
¼ j01021314i þ j01120314i þ j01121304i þ j11020314iþ j11021304i þ j11120304i;
jv34i ¼
XP
Pj01020314i
¼ j01020314i þ j01021304i þ j01120304i þ j11020304i;
jv44i ¼
XP
Pj01020304i ¼ j01020304i;
where P denotes the operation of permutation for 0 and 1. Then, wecan rewrite Eq. (ii) as
jwi4 ¼1ffiffiffiffiffi25
p X4
k¼0
ð�iÞkeip4 þ ðiÞke�ip4h i
jvk4i: ðiiiÞ
Noting the relations
1ffiffiffiffiffiffi2N
p �Nj¼1ðj1ji þ ij0jiÞ ¼
1ffiffiffiffiffiffi2N
p sumNk¼0ikjvk
Ni; ðivÞ
1ffiffiffiffiffiffi2N
p �Nj¼1ðj1ji � ij0jiÞ ¼
1ffiffiffiffiffiffi2N
p XN
k¼0
ð�iÞkjvkNi; ðvÞ
we can rewrite Eq. (iii) as
jwi4 ¼1ffiffiffiffiffi25
p eip4�4j¼1ðj1ji � ij0jiÞ þ e�ip4�4
j¼1ðj1ji þ ij0jiÞn
¼ 1ffiffiffi2p eip4ðj001002003004i � ij101102103104iÞ;
where j00li ¼ ðj1li � ij0liÞ=ffiffiffi2p
, j10li ¼ ðj1li þ ij0liÞ=ffiffiffi2p
ðl ¼ 1;2;3;4Þ,and h00lj1
0li ¼ 0.
So, we see that the state jwi4 is in fact a GHZ state.
1982 Y. He et al. / Optics Communications 283 (2010) 1979–1983
Generally, Eq. (i) can be rewritten as
jwiN ¼1ffiffiffiffiffiffiffiffiffiffi2Nþ1
p XN
k¼0
ð�iÞkeip4 þ ðiÞke�ip4h i
jvkNi; ðviÞ
where jvkNi ¼
PPPj0102; . . . ;0k1kþ11kþ2; . . . ;1Ni, P denotes the opera-
tion of permutation for 0 and 1. Using Eqs. (iv) and (v), we can ex-press Eq. (vi) as
jwiN ¼1ffiffiffiffiffiffiffiffiffiffi2Nþ1
p eip4�Nj¼1ðj1ji � ij0jiÞ þ e�ip4�N
j¼1ðj1ji þ ij0jiÞn
¼ 1ffiffiffi2p eip4ðj001002003; . . . ; 00Ni � ij101102103; . . . ;10NiÞ;
where j00li ¼ ðj1li � ij0liÞ=ffiffiffi2p
, j10li ¼ ðj1li þ ij0liÞ=ffiffiffi2p
, ðl ¼ 1;2; . . . ;NÞ,and h00lj1
0li ¼ 0.
Obviously, it is a GHZ state. So, the ‘‘cluster state” prepared inRefs. [11,18–20] is not a cluster state, but a GHZ state.
Appendix B
The number of atoms is N ¼ 2k, if k is an even number, i.e.,k ¼ 2l ðl ¼ 1;2; . . .Þ and N ¼ 4l, then Eqs. (15) and (16) become
jwðsÞi ¼1ffiffiffiffiffiffiffiffiffiffi2Nþ1
p ½�Nj¼1ðjþji þ j�jiÞ � i�N
j¼1ðjþji � j�jiÞ�
¼ 1ffiffiffi2p ðj0102; . . . ;0Ni � ij1112; . . . ;1NiÞ; ðIÞ
jw0ðsÞi ¼1ffiffiffiffiffiffiffiffiffiffi2Nþ1
p ½�Nj¼1ðjþji � j�jiÞ � i�N
j¼1ðjþji þ j�jiÞ�
¼ 1ffiffiffi2p ðj1112; . . . ;1Ni � ij0102; . . . ;0NiÞ: ðIIÞ
If k is an odd number, i.e., k ¼ 2l� 1 ðl ¼ 1;2; . . .Þ and N ¼ 4l� 2,then Eqs. (15) and (16) become
jwðsÞi ¼1ffiffiffiffiffiffiffiffiffiffi2Nþ1
p ½�Nj¼1ðjþji � j�jiÞ � i�N
j¼1ðjþji þ j�jiÞ�
¼ 1ffiffiffi2p ðj1112; . . . ;1Ni � ij0102; . . . ;0NiÞ; ðIIIÞ
jw0ðsÞi ¼1ffiffiffiffiffiffiffiffiffiffi2Nþ1
p ½�Nj¼1ðjþji þ j�jiÞ � i�N
j¼1ðjþji � j�jiÞ�
¼ 1ffiffiffi2p ðj0102; . . . ;0Ni � ij1112; . . . ;1NiÞ: ðIVÞ
Firstly, we consider 2k-atom state in which k is an even number.When atoms 1 and 2 leave the cavity, all the atoms have evolveda period of time s in the cavity. One can see from Eq. (I) that thestate of N atoms, which is initially in the ground statej0102; . . . ;0Ni, is evolved into 1ffiffi
2p ðj0102; . . . ;0Ni � ij1112; . . . ;1NiÞ. It
can be rewritten as
jwðsÞi ¼1ffiffiffi2p ðj0102; . . . ;0Ni � ij1112; . . . ;1NiÞ
¼ 12ffiffiffi2p ½ðj0102i � j1112iÞðj0304; . . . ; 0Ni þ ij1314; . . . ;1NiÞ
þ ðj0102i þ j1112iÞðj0304; . . . ; 0Ni � ij1314; . . . ;1NiÞ�
¼ 12ffiffiffi2p ðj0102i � j1112iÞ ðj0304; . . . ; 0Ni � ij1314; . . . ;1NiÞrð2Þz
þðj0304; . . . ; 0Ni þ ij1314; . . . ;1NiÞ�;
where rð2Þz ¼ j02ih02j � j12ih12j, rð2Þz acts on the qubit 2 which is leftfrom rð2Þz and is direct product with it. During the next evolutiontime s, the N � 2 atoms in the cavity interact with the single modevacuum cavity-field and the classical field. When atoms 3 and 4
leave the cavity, one can see from Eqs. (III) and (IV) thatj0304; . . . ; 0Ni and j1314; . . . ;1Ni in above equation become1ffiffi2p ðj1314; . . . ;1Ni � ij0304; . . . ;0NiÞ and 1ffiffi
2p ðj0304; . . . ; 0Ni � ij13
14; . . . ;1NiÞ, respectively. Substituting them into above equation,one can see that the state of the atoms becomesjwð2sÞi ¼ 1
2 ðj0102i � j1112iÞ½�ij0304; . . . ;0Nirð2Þz þ j1314; . . . ;1Ni�. Itcan be rewritten as
jwð2sÞi ¼14ðj0102i � j1112iÞ½ðj0304irð2Þz � j1314iÞðj0506; . . . ;0Ni
� ij1516; . . . ;1NiÞ þ ðj0304irð2Þz þ j1314iÞðj0506; . . . ; 0Niþ ij1516; . . . ;1NiÞ�
¼ 14ðj0102i � j1112iÞðj0304irð2Þz � j1314iÞ½ðj0506; . . . ;0Ni
þ ij1516; . . . ;1NiÞrð4Þz þ ðj0506; . . . ; 0Ni � ij1516; . . . ;1NiÞ�;
where the common phase e�ip=2 is omitted, andrð4Þz ¼ j04ih04j � j14ih14j, rð4Þz acts on the qubit 4 which is left fromrð4Þz and is direct product with it. Similarly, when the atoms 5 and6 leave the cavity, one can see from Eqs. (I) and (II) thatj0506; . . . ; 0Ni and j1516; . . . ;1Ni in above equation become1ffiffi2p ðj0506; . . . ;0Ni � ij1516; . . . ;1NiÞ and 1ffiffi
2p ðj1516; . . . ;1Ni � ij0506;
. . . ; 0NiÞ, respectively. Then the state of the atoms becomes
jwð3sÞi ¼1
2ffiffiffi2p ðj0102i � j1112iÞðj0304irð2Þz � j1314iÞðj0506; . . . ;0Nirð4Þz
� ij1516; . . . ;1NiÞ ¼1
4ffiffiffi2p ðj0102i � j1112iÞðj0304irð2Þz
� j1314iÞðj0506irð4Þz � j1516iÞ � ðj0708; . . . ;0Ni½�ij1718; . . . ;1NiÞrð6Þz þ ðj0708; . . . ;0Ni þ ij1718; . . . ;1NiÞ
�where rð6Þz ¼ j06ih06j � j16ih16j, rð6Þz acts on the qubit 6 which is leftfrom rð6Þz and is direct product with it.
In the same theory as above, when the atoms (2k � 3) and(2k � 2) leave the cavity, the state is
jwððk�1ÞsÞi ¼1ffiffiffiffiffi2k
p ðj0102i � j1112iÞ j0304irð2Þz � j1314i� �
� � � ðj02k�302k�2irð2k�4Þz � j12k�312k�2iÞ � ðj02k�102ki½
�ij12k�112kiÞrð2k�2Þz þ ðj02k�102ki þ ij12k�112kiÞ
�;
where rð0Þz ¼ 1 and rð2j�2Þz ¼ j02j�2ih02j�2j � j12j�2ih12j�2j,
ðj ¼ 1;2;3; . . . ; kÞ, rð2j�2Þz acts on the (2j� 2)th qubit which is left
from rð2j�2Þz and is direct product with it.
Finally, when all the atoms leave the cavity, the evolution stateis
jwðksÞi ¼1ffiffiffiffiffi2k
p ðj0102i � j1112iÞðj0304irð2Þz
� j1314iÞ � � � ðj02k�302k�2irð2k�4Þz � j12k�312k�2iÞ
� ðj0N�10NirðN�2Þz þ ij1N�11NiÞ
¼ 1ffiffiffiffiffi2k
p �k�1j¼1 ðj02j�102jirð2j�2Þ
z � j12j�112jiÞ
� j02k�102kirð2k�2Þz þ ij12k�112ki
�For the 2k-atom state in which k is an odd number, using the sametheory as above, we can also achieve the final evolution state as
jwðksÞi ¼1ffiffiffiffiffi2k
p �k�1j¼1 ðj02j�102jirð2j�2Þ
z � j12j�112jiÞ
� j02k�102kirð2k�2Þz þ ij12k�112ki
�:
References
[1] H.-J. Briegel, Raussendorf, Phys. Rev. Lett. 86 (2001) 910.
Y. He et al. / Optics Communications 283 (2010) 1979–1983 1983
[2] W. Dür, H.-J. Briegel, Phys. Rev. Lett. 92 (2004) 180403.[3] R. Raussendorf, H.-J. Briegel, Phys. Rev. Lett. 86 (2001) 5188.[4] R. Raussendorf, D.E. Browne, H.-J. Briegel, Phys. Rev. A 68 (2003) 022312.[5] S.-B. Zheng, Phys. Rev. A 73 (2006) 065802.[6] X.B. Zou, K. Pahlke, W. Mathis, Phys. Rev. A 69 (2004) 052314.[7] X.B. Zou, W. Mathis, Phys. Rev. A 71 (2005) 032308.[8] X.B. Zou, W. Mathis, Phys. Rev. A 72 (2005) 013809.[9] T. Tanamoto, Y.X. Liu, S. Fujita, X. Hu, F. Nori, Phys. Rev. Lett. 97 (2006) 230501.
[10] J.Q. You, X.B. Wang, T. Tanamoto, F. Nori, Phys. Rev. A 75 (2007) 052319.[11] Z.Y. Xue, Z.D. Wang, Phys. Rev. A 75 (2007) 064303.[12] M. Borhani, D. Loss, Phys. Rev. A 71 (2005) 034308.[13] G.P. Guo, H. Zhang, T. Tu, G.C. Guo, Phys. Rev. A 75 (2007) 050301(R).[14] Y.S. Weinstein, C.S. Hellberg, J. Levy, Phys. Rev. A 72 (2005) 020304(R).[15] J. Metz, C. Schön, A. Beige, Phys. Rev. A 76 (2007) 052307.[16] Y.L. Lim, S.D. Barrett, A. Beige, P. Kok, L.C. Kwek, Phys. Rev. A 73 (2006) 012304.[17] P. Walther, K.J. Resch, T. Rudolph, E. Schenk, H. Weinfurter, V. Vedral, M.
Aspelmeyer, A. Zeilinger, Nature (London) 434 (2005) 169.[18] N. Kiesel, C. Schmid, U. Weber, G. Tóth, O. Gühne, R. Ursin, H. Weinfurter, Phys.
Rev. Lett. 95 (2005) 210502.[19] G. Vallone, E. Pomarico, P. Mataloni, F. De Martini, V. Berardi, Phys. Rev. Lett.
98 (2007) 180502.
[20] G. Chen, Z. Chen, L. Yu, J. Liang, Phys. Rev. A 76 (2007) 024301.[21] J.-Q. Li, G. Chen, J.-Q. Liang, Phys. Rev. A 77 (2008) 014304.[22] Z.-R. Lin, G.-P. Guo, T. Tu, F.-Y. Zhu, G.-C. Guo, Phys. Rev. Lett. 101 (2008)
230501.[23] S.B. Zheng, Phys. Rev. A 68 (2003) 035801.[24] X.-W. Wang, Z.-H. Peng, C.-X. Jia, Y.-H. Wang, X.-J. Liu, Opt. Commun. 282
(2009) 670.[25] E. Hagley, X. Maı̂tre, G. Nogues, C. Wunderlich, M. Brune, J.M. Raimond, S.
Haroche, Phys. Rev. Lett. 79 (1997) 1.[26] S.B. Zheng, G.C. Guo, Phys. Rev. Lett. 85 (2000) 2392.[27] O. Gühne, F. Bodoky, M. Blaauboer, Phys. Rev. A 78 (2008) 060301(R).[28] M. Hein, J. Eisert, H.J. Briegel, Phys. Rev. A 69 (2004) 062311.[29] L. Ye, L.B. Yu, G.C. Guo, Phys. Rev. A 72 (2005) 034304.[30] Y. He, N.-Q. Jiang, Opt. Commun. 283 (2010) 1558.[31] Z.-Q. Yin, Y.-B. Zhao, Y. Yang, C.-L. Zou, Z.-F. Han, G.-C. Guo, Opt. Commun. 283
(2010) 617.[32] W. Song, P. Zou, Opt. Commun. 282 (2009) 3190.[33] J.M. Raimond, M. Brune, S. Haroche, Rev. Mod. Phys. 73 (2001) 565.[34] M. Brune, P. Nussenzveig, F. Schmidt-Kaler, F. Bernardot, A. Maali, J.M.
Raimond, S. Haroche, Phys. Rev. Lett. 72 (1994) 3339.
