行政院國家科學委員會專題研究計畫 成果報告
量子系統之強韌控制
計畫類別： 個別型計畫 計畫編號： NSC93-2218-E-110-032- 執行期間： 93 年 08 月 01 日至 94 年 07 月 31 日 執行單位： 國立中山大學機械與機電工程學系(所) 計畫主持人： 趙健祥 計畫參與人員： 賴俊峰 報告類型： 精簡報告 報告附件： 出席國際會議研究心得報告及發表論文 處理方式： 本計畫可公開查詢中 華 民 國 94 年 12 月 28 日
行 政 院 國 家 科 學 委 員 會 補 助 專 題 研 究 計 畫 成 果 報 告
量子系統之強韌控制
C.H. Chao
1, C.F. Lay
11
Department of Mechanical and Electro-Mechanical Engineering National Sun Yat-Sen University, Kaohsiung 804, Taiwan, ROC
計畫編號：NSC
93-2218-E-110-032-執行期間：93 年 8 月 1 日至 94 年 7 月 31 日
計畫主持人：趙健祥副教授 國立中山大學機電工程學系，
計畫參與人員： 賴俊峰 碩士班研究生 國立中山大學機械工程研究所 Abstract
Quantum control of the entanglement swapping in the strong coupling process of two level atoms which a pair interacting with the photonic crystal microcavities fields of coherent states is considered in this research. We investigated the atomic level population and the entanglement degree of the system, and found that the atomic maximal entangled state can be transformed into the photonic crystal microcavity maximal coherent entangled state cavity field, whereas the photonic crystal microcavity maximal coherent entangled state cavity field also can be transformed into the atomic maximal entangled state cavity.
Keywords
量子控制、腔電動力學、糾纏態轉換
Cavity QED, Photonic Crystal Microcavity,
Entanglement Swapping, Quantum Control
1. Introduction
The cavity quantum electrodynamics has investigated the strong coupling interaction dynamics process between the microcavity field and the atom. The high quality cavity is a key to the realization of cavity quantum electrodynamics. Photonic crystal nanocavities with small mode volumes and large quality factors light are confined within in the nanocavity[1,2]. They can be
used for cavity QED experiments of Fabry-Perot. In the realm of quantum information, cavity QED is the most effective scheme that it can realize the quantum information [3]. We have provided a realization of a quantum entanglement method for quantum information. Therefore, we may control the atomic entanglement state and teleportation state using the cavity QED scheme in the quantum information. The entanglement swapping is another approach to obtaining entanglement which makes use of a projection of the state of two particles onto an entangled state [4,5,6].
Entanglement swapping means to entangle quantum systems that have never interacted before [6], which has found a number of applications in quantum information [7] such as constructing a quantum telephone exchange, speeding up the distribution of entanglement, correcting errors in Bell states, preparing entangled states of a higher number of particles, and secret sharing of classical information.
2. Photonic Crystal Microcavities for Strong Coupling
We consider the photonic crystal microcavities to have strong coupling between the cavity field and a single atom. A single atom cavity quantum electrodynamics in the strong coupling regime for photonic bandgap structures was
investigated[8]. The microcavity quality factor (Q) has to be as large as possible and the mode volume (Vmode) as small as possible. However, in a cavity for strong coupling, an atom must be trapped at the point where it interacts most strongly with the cavity field as shown in Fig.1.
Fig. 1 The interaction between the
atom and the photonic crystal
microcavity.
Let us consider a system consisting of a single atom positioned at location rGa within a photonic crystal microcavity. The atomic and cavity resonance frequencies are labeled as ωa and ωc, but we assume that
a c o
ω =ω =ω .
mod e
V is the cavity mode volume which is defined as
( ) ( )
( ) ( )
2 mod 2 max a a e r E r dV V r E r ε ε = ⎡ ⎤ ⎢ ⎥ ⎣ ⎦∫∫∫
K K K K K K (1)The time decay of electromagnetic field energy stored in the cavity is given in the following expression [9]: 0 4 ( ) (0) (0) t t Q U t U e U e ω πκ − − = = (2)
The cavity field decay rate
κ
is proportional to the ratio of the angularfrequency of the mode (ω0) and the mode quality factor (Q): 0
4 Q
ω
κ
π
=
(3)The coupling rate g r( )Ka can be described as
( ) ( )
( ) ( )
0 ( ) max a a a r E r g r g r E rε
ε
= ⎡ ⎤ ⎣ ⎦ K K K K K K K (4)where g0 denotes the vacuum Rabi frequency, which is defined as
0 0 mod e V g V γ⊥ = (5)
Let us introduce the critical atom (N0) and photon (m0) numbers as:
0 2 2 [ ( )]a N g r κγ⊥ = K (6) 2 0 2 ( )a m g r γ⊥ ⎡ ⎤ = ⎢ ⎥ ⎣ K ⎦ (7)
It follows that the strong coupling is
possible if both N0 and m0 are smaller than 1. Therefore, in order to predict whether the
strong coupling can occur, we must calculate upper limits of N0 and m0 and compare them to 1. They have also demonstrated theoretically that photonic crystal cavities could be designed for strong interaction with atoms trapped in one of the photonic crystal holes [9].
3. Entanglement Swapping for Cavity QED
Assume that the two atoms with the maximally entangled states in 2-qubit systems of the Bell states or EPR states [10]:
(0) 1 ( 1 2 1 2 ) 2
a e g g e
Ψ = +
(8)
and the initial of two photonic crystal mocrocavities in the coherent states,
1 2
α α is a coherent states defined as [11]： 1 2 1 2 1 2 1 2 0 0 (0) f n n n n F F n n α α ∞ ∞ = = Ψ = =
∑ ∑
(9) where 2 1 1 1 1 2 1! n n F e n α α − = 2 2 2 2 2 2 2! n n F e n α α − =Fig. 2 The joint system consists of
two photonic crystal
microcavities and two
movable atoms, the arrows
show the direction of atomic
motion.
Consider the Fig. 2 of the physic system containing the atoms and the cavities. In order to simplify the calculation, we assume that the interaction between two cavities fields and two atoms are resonant, so in the interaction picture the Hamiltonian of the system gives [90]： 2 1 ˆ ( )ˆ ˆ ( )ˆ ˆ I j a j j j a j j j i g r a+σ− g r a∗ σ+ = ⎡ ⎤ Η = =
∑
⎣ K − K ⎦ (10)where the operators aˆj and aˆ+j are the
annihilation and creation operators for the
jth single mode of the resonator under
consideration, while σˆ+j and σˆ−j are the Pauli operators for the raising and lowering
of the
jth
atom located in the position rKa . The coherent coupling between the jthatom and the cavity mode is g rj( )Ka . Initially, the atoms and the photonic crystal microcavities are the Ψa(0) state
and the Ψf(0) state, so in the interaction
picture the state of the whole system at arbitrary time t is given by Ψa f, ( )t
:
1 2 1 2 , 0 0 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 ( ) 2 {[ sin(( 1) ) cos( ) , 1, cos( ) sin(( 1) ) , , 1 ] +[ sin(( 1) ) sin( ) , a f n n n n t F F i n t n t g g n n i n t n t g g n n n t n t g e n ∞ ∞ = = Ψ = ⋅ − + + − + + − + +
∑ ∑
2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1, 1 cos( ) cos(( 1) ) , , ] +[ cos(( 1) ) cos( ) , , sin( ) sin(( 1) ) , 1, 1 ] n n t n t g e n n n t n t e g n n n t n t e g n n − + + + − + − + 1 2 1 2 1 2 1 2 1 2 1 2 +[ cos(( 1) ) sin( ) , , 1 sin( ) cos(( 1) ) , 1, ]} i n t n t e e n n i n t n t e e n n − + − − + −(11)
After an interaction time t=π 2 (s), we
measure the two photonic crystal microcavities of two side atoms with A and B sates. If the measurement through of the photonic crystal microcavities by the atom A
and B are in g1 and g2 states, then the
system is then collapsed to the state:
1 2 1 2 1 2 , 2 2 0 0 1 2 1 2 1 2 1 2 1 2 1 ( 2) ( 1) 2 , 2 1, 2 , 2 , 2 1 n n a f n n n n F F N N g g n n g g n n π ψ ∞ ∞ + = = = − ⋅ ⎡⎣ + + + ⎤⎦
∑ ∑
(12) where1 1 2 1 2 0 n n N F ∞ = =
∑
2 2 2 2 2 0 n n N F ∞ = =
∑
On the other hand, if the measurement through of the photonic crystal microcavities
by the atom A and B are in e1 and e2
states, then the system is then collapsed to the state: 1 2 1 2 1 2 1 ' , ' ' 2 1 2 1 0 0 1 2 1 2 1 2 1 2 1 2 1 ( ) ( 1) 2 2 , 2 1, 2 , 2 , 2 1 n n a f n n n n F F N N e e n n e e n n π ψ ∞ ∞ + + + + = = = − ⋅⎡⎣ + + + ⎤⎦
∑∑
(13)where
1 1 2 ' 1 2 1 0 n n N F ∞ + = =∑
2 2 2 ' 2 2 1 0 n n N F ∞ + = =
∑
So the probability of the atoms in e1 g2
states and g1 e2 states is zero. Only
1 2
e e states and g1 g2 states are
possible, the probability of each situation is 50%. Eqs. (12) and (13) are the expressions of the two different kinds of distributions of the photonic crystal microcavity maximal coherent entangled states cavity field. This is called the entanglement swapping. The atomic maximal entangled state can be transformed into the photonic crystal microcavity maximal coherent entangled state cavity field.
The closed loop system time evolution is reversible. We are necessary to discuss the photonic crystal microcavities maximal coherent entangled states cavity field with the non-entanglement states atoms on strong coupling interaction of the entanglement by two atoms.
Assume the photonic crystal microcavities are in the maximal coherent entanglement state in t=0, and it is given by:
1 2 1 2 1 2 2 2 1 2 1 2 1 2 1 ( ) ( 1) 2 2 1, 2 2 , 2 1 n n f n n n n t F F N N n n n n ∞ ∞ + Ψ = − ⋅ ⎡⎣ + + + ⎤⎦
∑∑
(14) where 1 1 2 1 2 0 n n N F ∞ = =∑
2 2 2 2 2 0 n n N F ∞ = =
∑
The two atoms are in ground state g1 and
2
g of initial time, so the system of atom -
1 2 1 2 1 2 , 2 2 1 2 1 2 1 2 1 2 1 2 1 (0) ( 1) 2 , 2 1, 2 , 2 , 2 1 n n a f n n n n F F N N g g n n g g n n ∞ ∞ + Ψ = − ⋅ ⎡⎣ + + + ⎤⎦
∑∑
(15)After the interaction of the atoms with the cavity fields, and the state of the system combining the atom and the cavity is given by: 1 2 1 2 1 2 , 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 ( ) = ( 1) 2 {[cos((2 1) ) cos(2 ) , 2 1, 2 cos(2 ) cos((2 1) ) , 2 , 2 1 ] +[ cos((2 1) ) sin(2 n n a f n n n n t F F N N n t n t g g n n n t n t g g n n i n t n ∞ ∞ + Ψ − ⋅ + + + + + − +
∑∑
2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 ) , 2 1, 2 1 cos(2 ) sin((2 1) ) , 2 , 2 ] +[ sin((2 1) ) cos(2 ) , 2 , 2 sin(2 ) cos((2 1) ) , 2 t g e n n i n t n t g e n n i n t n t e g n n i n t n t e g n + − − + − + − + − 2 1 2 1 2 1 2 1 2 1 2 1 2 1, 2 1 ] +[ sin((2 1) ) sin(2 ) , 2 , 2 1 sin(2 ) sin((2 1) ) , 2 1, 2 ]} n n t n t e e n n n t n t e e n n + − + − − + − (16)In order to explain the interaction process of the atom and the photonic crystal microcavity field, the time evolution operator is the atomic entangled characteristic. The system density operator expression is given by:
, , 1 1 2 1 2 2 1 2 1 2 3 1 2 1 2 4 1 2 1 2 5 1 2 1 2 6 1 2 1 2 7 1 2 1 2 8 1 ˆ ( ) ( ( ) ( ) ) ˆ ˆ ( ) , , ( ) , , ˆ ˆ + ( ) , , ( ) , , ˆ ˆ + ( ) , , ( ) , , ˆ ˆ + ( ) , , ( ) , a t Trf a f t a f t t e e e e t e e g g t e g e g t e g g e t g e e g t g e g e t g g e e t g g ρ ρ ρ ρ ρ ρ ρ ρ ρ = Ψ Ψ = + + + + 2 g g1, 2
(17) where 1 2 1 2 2 2 1 2 2 2 2 0 0 1 2 2 1 2 2 1 2 1 ˆ ( ) 2 {[sin((2 1) ) sin(2( 1) )] +[sin(2( 1) ) sin((2 1) )] } n n n n t F F N N n t n t n t n t ρ ∞ ∞ = = = ⋅ + + + +
∑ ∑
1 2 1 2 2 2 2 7 2 2 2 2 0 0 1 2 1 2 1 2 1 2 1 2 1 ˆ ( ) ˆ ( ) 2{[sin((2 1) )sin(2( 1) ) cos(2 ) cos((2 1) )] +[sin(2( 1) )sin((2 1) ) cos((2 1) ) cos(2 )]}
n n n n t t F F N N n t n t n t n t n t n t n t n t ρ ρ ∞ ∞ = = − = = ⋅ + + + + + + ∑ ∑ 1 2 1 2 2 2 3 2 2 2 2 0 0 1 2 2 1 2 2 1 2 1 ˆ ( ) 2 {[sin((2 1) ) cos(2 )] +[sin(2( 1) ) cos((2 1) )] } n n n n t F F N N n t n t n t n t ρ ∞ ∞ = = = ⋅ + + +
∑ ∑
1 2 1 2 2 2 4 5 2 2 2 2 0 0 1 2 1 2 1 2 1 2 1 2 1 ˆ( ) ˆ( ) 2{[sin((2 1) )cos(2 )cos(2 )sin((2 1) )] +[sin(2( 1) )cos((2 1) )cos((2 1) )sin(2( 1) )]}
n n n n t t F F N N n t n t nt n t n t n t n t n t ρ ρ ∞ ∞ = = = = ⋅ + + + + + +
∑∑
1 2 1 2 2 2 6 2 2 2 2 0 0 1 2 2 1 2 2 1 2 1 ˆ ( ) 2 {[cos(2 ) sin((2 1) )] +[cos((2 1) ) sin(2( 1) )] } n n n n t F F N N n t n t n t n t ρ ∞ ∞ = = = ⋅ + + +∑ ∑
1 2 1 2 2 2 8 2 2 2 2 0 0 1 2 2 2 1 2 1 2 1 ˆ ( ) 2{[cos(2 )cos((2 1) )] +[cos((2 1) )cos(2 )] }
n n n n t F F N N n t n t n t n t ρ ∞ ∞ = = = ⋅ + +
∑∑
In order to observe two atoms of entanglement the characteristic, it investigates the entanglement states measuring technique proposed by Kim [12]:
ˆ ( a) 2 i i ε ρ = −
∑
λ− (18) where λi −is a negative eigenvalue. When 0
ε = the two atoms are in independent, non-entanglement state; but when ε =1 the two atoms are in maximally entanglement
state, so λ1 −
and λ2 −
are given by:
[ ]2 2 1 1 8 1 8 1 8 4 1 ˆ( ) ˆ( ) ˆ( ) ˆ ( ) 4 ˆ( )ˆ ( ) ˆ ( ) 2 t t t t t t t λ−= ⎡ρ +ρ − ρ +ρ − ⎡ρ ρ −ρ ⎤⎤ ⎣ ⎦ ⎢ ⎥ ⎣ ⎦
[ ]2 2 2 3 6 3 6 3 6 2 1 ˆ ( ) ˆ ( ) ˆ ( ) ˆ ( ) 4 ˆ ( )ˆ ( ) ˆ ( ) 2 t t t t t t t λ−= ⎡ρ +ρ − ρ +ρ − ⎡ρ ρ −ρ ⎤⎤ ⎣ ⎦ ⎢ ⎥ ⎣ ⎦
When the average photon number （
2 2 1 2
α =α ） supposes is 10. The evolution
of the two atomic level populations and the atomic entanglement degree had shown in Fig. 3-4.
(a)
(b)
(c)
(d)
Fig. 3 Evolution of the atomic level
population.
(a)
e e1, 2 e e1, 2,
(b)
e g1, 2 e g1, 2,
(c)
g e1, 2 g e1, 2,
(d)
g g1, 2 g g1, 2Fig. 4 Evolution of the atomic
entanglement degree
4. Conclusions
In summary, as long as we control the time of interaction between the photonic crystal microcavities fields and the atoms, the atoms go cross the photonic crystal microcavities at a controlled speed. We have found that the atomic maximal entangled state can be transformed into the photonic crystal microcavity maximal coherent entangled state
cavity field, whereas the photonic crystal microcavity maximal coherent entangled state cavity field also can be transformed into the atomic maximal entangled state. These results will be useful in quantum control of photonic crystal microcavity related problems.
5. References
[1] J. Vuckovic, Y. Yamamoto, Appl. Phys. Lett. 82, 2374 (2003).
[2] A. Kiraz, A.Imamoglu, J. Opt. B: Quantum Semiclass. Opt. 5, 129 (2003).
[3] J. McKeever, H.J. Kimble, Science 303, 1992 (2004).
[4] A. Furusawa et al., Science 282, 706 (1998).
[5] Jian-Wei Pan, H. Weifurter, A. Zeilinger, Phys. Rev. Lett. 80, 3891 (1998).
[6] S. Bose, V. Verdral, P.L. Night, Phys. Rev. A 57, 822 (1998).
[7] S. Bose, V. Vedral, The Physics of Quantum Information, Springer, Berlin, 2000.
[8] H.J. Kimble, in Cavity Quantum Electrodynamics, edited by P.Berman ~Academic, San Diego, 1994.
[9] J. Vuckovic, A. Scherer, Phys. Rev. E 65, 016608 (2002).
[10] M. A. Nielsen, Quantum computation and quantum information, Cambridge: Cambridge University Press 2000. [11] Jin-Shen Peng, Introduction to
Modern Quantum Optics, World Scientific, 1998.
[12] M.S. Kim, Phys. Rev. Lett. 84, 4236 (2000). (六)計畫成果自評 本計畫探討了在腔體內的量子糾 纏態轉換以作為在量子資訊處理時的量子 控制之用。內容有原創性其結果在量子光 學、光子晶體、量子資訊有意義，但和原 始計畫構想的系統有別。控制部分仍可以 進一步探討。因為此成果比原計畫構想有 價值，所以以此內容作為計畫的成果報告。
