March 15, 2014 / Vol. 39, No. 6 / OPTICS LETTERS
1489
One-step implementation of a multiqubit phase gate with one control qubit and multiple target qubits in coupled cavities Hong-Fu Wang,* Ai-Dong Zhu, and Shou Zhang Department of Physics, College of Science, Yanbian University, Yanji 133002, China *Corresponding author: [email protected] Received November 29, 2013; revised January 16, 2014; accepted January 31, 2014; posted February 3, 2014 (Doc. ID 202273); published March 11, 2014 We propose a one-step scheme to implement a multiqubit controlled phase gate with one qubit simultaneously controlling multiple qubits with three-level atoms at distant nodes in coupled cavity arrays. Selective qubit–qubit couplings are achieved by adiabatically eliminating the atomic excited states and photonic states, and the required phase shifts between the control qubit and any target qubit can be realized through suitable choices of the parameters of the external fields. Moreover, the effective model is robust against decoherence because neither the atoms nor the field modes are excited during the gate operation, leading to a useful step toward scalable quantum computing networks. © 2014 Optical Society of America OCIS codes: (270.5580) Quantum electrodynamics; (270.5585) Quantum information and processing. http://dx.doi.org/10.1364/OL.39.001489
It has been shown that any multiqubit gate can be decomposed into two classes of elementary quantum gates, namely, a universal two-qubit controlled phase gate and a one-qubit unitary gate, which are the basic building blocks of a quantum computer. However, the procedure of decomposing multiqubit gates into the elementary gates usually becomes very complicated as the number of qubits increases when using the conventional gate decomposition method. To reduce the complexity of the physical realization of practical quantum computing and quantum information processing, direct implementation of multiqubit logic gates with multiple control qubits [1–9] or multiple target qubits [10,11] is, thus, very important. In recent years, considerable theoretical effort has been devoted to a class of coupled cavity models, which typically describe a series of optical cavities, each containing one or more atoms, where photons are permitted to hop between cavities. The coupled cavity arrays show promise in overcoming the problem of individual addressability and have several interesting potential applications, including quantum information processing and simulations of quantum strongly correlated manybody systems. Theoretical studies on quantum computing and quantum information processing have been put forward for use of atom–light interactions in coupled microcavity arrays [12–21]. In this Letter, we propose a scheme for implementing a multiqubit phase gate with one control qubit and multiple target qubits with threelevel atoms at distant nodes in coupled cavity arrays. This type of multiqubit controlled phase gate is essentially equivalent to n two-qubit controlled phase gates, each having a shared control qubit (qubit 1) but a different target qubit (qubits 2; 3; …; N). In the scheme, the selective off-resonant qubit–qubit coupling between the ground states of two atoms, induced by the external fields and the cavity modes, can be used to realize an effective phase shift between the control qubit and any target qubit by choosing the parameters of the external fields appropriately. The scheme has the following merits: (1) it can be accomplished in only one step, which greatly simplifies the experimental realization and reduces the total 0146-9592/14/061489-04$15.00/0
gate time; (2) it does not require individual addressing of the trapped atoms; and (3) multiqubit gate operation among spatially separated quantum nodes is very meaningful for distributed quantum computing and quantum communication networks. Moreover, it would be an important step toward efficiently constructing quantum circuits and quantum algorithms. We consider an array of cavities that are coupled via exchange of photons, with one three-level atom in each cavity, as sketched in Fig. 1(a). Such a model can be constructed in several kinds of physical systems such as superconducting stripline resonators [22], photonic crystal defects [23], and microtoroidal cavity arrays [24]. Each atom has one excited state, jei, and two ground states, jgi and jai. The transition jgii → jeii (i 1; 2; …; N) is coupled to the cavity mode with coupling strength gi and detuning Δc i . On the other hand, the transition jgi1 → jei1 for atom 1 is driven by N − 1 classical laser fields with Rabi frequencies Ωk 1 and detunings Δk 1 (k 1; 2; …; N − 1) [Fig. 1(b)]. The transition jgij → jeij for atom j (j 2; 3; …; N) is driven by a classical laser field with Rabi frequency Ωj and detuning Δj [Fig. 1(c)]. In the interaction picture, the Hamiltonian describing the atom–field interaction is given by ˆ int J c H
N X
(" aˆ †j aˆ j1
aˆ j aˆ †j1
j1
× jeijj hgj
c
gj aˆ j eiΔj
t
j1 N −1 X m1
N X
N X
m
iΔ1 t Ωm jei11 hgj 1 e
Ωn eiΔn t jeinn hgj
#
) H:c: ;
(1)
n2
where J c is the cavity–cavity hopping strength. Considering the periodic boundary conditions aˆ N1 aˆ 1 and taking advantage of Fourier transformation to diagonalize ˆ int can the photon coupling terms, the Hamiltonian H be rewritten as © 2014 Optical Society of America
1490
OPTICS LETTERS / Vol. 39, No. 6 / March 15, 2014
ηm
2 Ωm 1
Δm 1
ζ n;k Fig. 1. (a) One-dimensional (1D) array of coupled cavities, with one three-level atom in each cavity. (b) Level configuration and excitation scheme of atom 1. (c) Level configuration and excitation scheme of atom j (j 2; 3; …; N). N X
ˆ int H
(" ωk bˆ †k bˆ k
k1 c
× eiΔj t jeijj hgj
N X N 1 X 2π p gj bˆ k e−i N jk N j1 k1
N −1 X
m
m1
N X
iΔ1 t Ωm jei11 hgj 1 e
#
iΔn t
Ωn e
)
jeinn hgj H:c: ;
(2)
n2
; NΔc l − ωk g1 Ωm 1 1 1 p m ; 2 N Δc Δ1 1 − ωk gn Ωn 1 1 : p Δn 2 N Δc n − ωk
k1
N X
c
× eiΔn
jeinn hgj H:c:
−
(3)
m ≫ Ωm Under the conditions jΔc j − ωk j ≫ gj ∕N, Δ1 1 , and Δn ≫ Ωn , the upper level jeij can be eliminated adiabatically, leading to
" N N −1 X X k1
ξm;k bˆ k e−i N k e 2π
iΔc −ωk −Δm t 1 1
m1 N X n2
c 2π ζn;k bˆ k e−i N nk eiΔn −ωk −Δn t
#
×jginn hgj H:c: −
X N −1 m1
N X n2
where
N X
N X
n2
k1
N X N X
jgi11 hgj ⊗ jginn hgj ! N X θm;k − ηm jgi11 hgj k1
!
ϑn;k − μn jginn hgj
χ l;k bˆ †k bˆ k jgill hgj;
with θm;k
ξ2m;k
; m
Δc 1 − ωk − Δ1
ϑn;k
ζ2n;k Δc n − ω k − Δn
;
2π
× jgi11 hgj
N −1 X
l1 k1
n2
ˆ 00 − H
#)
m1
#
Ωn e
Λm;n;k
−Δc −Δn Δm t 1 1
H:c:
iΔm t 1 Ωm jei11 hgj 1 e
iΔn t
N −1 X N X m1 n2
N X N X c 2π ˆ 0 p1 gj bˆ k e−i N jk eiΔj −ωk t jeijj hgj H N j1 k1
m1
(5)
p2 q2;q≠p
× jgipp hgj ⊗ jgiqq hgj
"
Ω2n ; Δn
The first two terms in Eq. (4) are the coupling between the bosonic modes bˆ k and the classical field assisted by the atoms, respectively. The last three terms are the Stark shifts for level jgij that are induced by the bosonic modes bˆ k and the classical pulse. In the case of jΔc 1 − ωk − c Δm j ≫ ξ and jΔ − ω − Δ j ≫ ζ , the bosonic n m;k k n n;k 1 modes do not exchange quanta with the atomic system: the bosonic modes are excited only virtually and any two atoms interfere with each other during the whole interaction process. The effective Hamiltonian is then given by (" N N N X X X c c ˆ H eff Γp;q;k eiΔp −Δq −Δp Δq t
p P −i2π∕Njk ˆ bk and ωkP where aˆ j 1∕ N N 2J c cos k1 e N ˆ† ˆ i 2π∕Nk. Performing the transformation e k1 ωk bk bk t , we obtain
N −1 X
μn
g2l
χ l;k ξm;k
;
μn jginn hgj
Γp;q;k
ζ p;k ζq;k e−i N p−qk 1 c 2 Δp − ω k − Δp ! 1 c ; Δq − ωk − Δq 2π
ηm jgi11 hgj
N X N X l1 k1
χ l;k bˆ †k bˆ k jgill hgj ; (4)
Λm;n;k
ξm;k ζn;k e−i N n−1k 1 c 2 Δ1 − ωk − Δm 1 ! 1 c ; Δn − ωk − Δn
(7)
where ζ ν;k ν p; q; n in Eq. (7) is denoted by Eq. (5). As the quantum number of the bosonic modes is conserved
March 15, 2014 / Vol. 39, No. 6 / OPTICS LETTERS
during the interaction, they will remain in the vacuum state if they are initially so. Choosing the detunings suitably so that c m Δc 0 n − Δ1 − Δn Δ1
m ≠ n − 1;
k1
−
Δc q
N X − Δp Δq j ≫ Γp;q;k
p; q ∈ f2; 3; …; Ng; p ≠ q;
(8)
then the effective Hamiltonian reduces to N X j2
ζ 01;j jgi11 hgj ξ0j jgijj hgj
Λ01;j jgi11 hgj ⊗ jgijj hgj;
(9)
with ζ 01;j
N X
θj−1;k − ηj−1 ;
k1
Λ01;j
N X
ξj−1;k ζj;k
1
N X
ϑj;k − μj ;
j−1
Δc 1 − ω k − Δ1
1 Δc j − ω k − Δj
;
(10)
where ηj−1 , μj , ξj−1;k , and ζ j;k are denoted by Eq. (5) and θj−1;k and ϑj;k are denoted by Eq. (7), respectively. The last term in Eq. (9) describes the coupling between atoms 1 and j (j 2; 3; …; N) mediated by the bosonic modes and the classical pulses. To implement quantum computation, the two longlived levels jai and jgi represent the states of the qubits, which correspond the logical 0 and 1 states, respectively, j0i ≡ jai and j1i ≡ jgi. Due to virtual excitation of the atoms in the interaction, atom 1 and any atom j will undergo an energy shift, leading to the evolution jai1 jaij → jai1 jaij ; jai1 jgij → e−iφj jai1 jgij ; jgi1 jaij → e−iψ 1;j jgi1 jaij ; jgi1 jgij → e−iφj ψ 1;j ϕ1;j jgi1 jgij ;
(11)
where φj ξ0j t, ψ 1;j ζ 01;j t, and ϕ1;j Λ01;j t. After the generation of the one-qubit phase shifts jgi1 → eiψ 1;j jgi1 and jgij → eiφj jgij , a conditional phase shift ϕ1;j , which is controllable via effective interaction time t and the corresponding parameters Λ01;j , is produced if and only if atoms 1 and j are in state jgi. While for state jgim jgin of any other pair of atoms m and n (m; n 2; 3; …; N), no conditional phase shift is generated. Therefore, for the computational basis states fjs1 i1 js2 i2 js3 i3 ⊗ ⊗ jsn−1 iN−1 jsn iN g (si a; g; i 1; 2; …; n) consisting of N
(12)
where g⊕g 0 and g⊕a 1. Therefore, a multiqubit controlled phase gate of one qubit simultaneously controlling multiple qubits is achieved if and only if atom 1 is in state jgi1 . We now give a brief analysis and discussion for some practical issues in relation to the experimental feasibility of the proposed scheme. For simplicity, we consider the case with N 3 and set J c 0.5g, g1 g2 g3 g, c c 2 Ω1 Δc 1 Δ2 Δ3 20g, 1 Ω1 Ω2 Ω3 g, 1 2 Δ1 Δ2 18g, and Δ1 Δ3 21.2842g. Then we have Λ01;2 Λ01;3 Λ03 1.225 × 10−3 g
k1
2π j − 1k cos N
k1
×
ξ0j
→ jai1 js2 i2 js3 i3 ⊗ ⊗ jsn−1 iN−1 jsn iN ; PN −i 1−g⊕sj π j2 jgi1 js2 i2 js3 i3 ⊗ ⊗ jsn−1 iN−1 jsn iN → e ×jgi1 js2 i2 js3 i3 ⊗ ⊗ jsn−1 iN−1 jsn iN ;
k1
ˆ 0eff H
atoms, after the qubit–qubit couplings and a series of one-qubit phase shift operations on state jgik (k 1; 2; …; N) and with the choice of ϕ1;j Λ01;j t π, one can obtain jai1 js2 i2 js3 i3 ⊗ ⊗ jsn−1 iN−1 jsn iN
m n − 1; n 2; 3; …; N; N X c c m Λm;n;k jΔn − Δ1 − Δn Δ1 j ≫ jΔc p
1491
(13)
and the time needed to complete the three-qubit controlled phase gate operation is t3 π∕Λ03 2.56457 × 103 ∕g, and all restrictive conditions in the cases of large detunings are satisfied well. The probability that the atoms undergo transition to the excited state due to qubit–qubit coupling with the classical fields is " 2 2 Ω2 1 3 Ω1 13 Ω2 2 1 pe × × 1 × 2 2 8 4 8 Δc Δc 1 2 # 3 Ω3 2 × c 2 1.09375 × 10−3 : 8 Δ3
(14)
Meanwhile, the probability that the field mode is excited due to qubit–qubit coupling is " 2 3 X ξ1;k 3 pc × c 1 4 k1 Δ1 − ωk − Δ1 2 # 3 X ξ2;k c 2 k1 Δ1 − ωk − Δ1 " 2 # 3 X 13 ζ2;k × c 8 k1 Δ2 − ωk − Δ2 " 2 # 3 X ζ 3;k 3 × c 8 k1 Δ3 − ωk − Δ3 5.98033 × 10−3 :
(15)
ˆ 0eff in Eq. (9) is Therefore, the effective Hamiltonian H valid. Furthermore, the effective decoherence rates due to the atomic spontaneous emission and field decay
1492
OPTICS LETTERS / Vol. 39, No. 6 / March 15, 2014
are γ e pe γ and κ c pc κ, with γ and κ being the decay rates for the atomic excited state and the field modes, respectively. If we consider the parameters γ ∼ κ ∼ 3 × 10−3 g, which were predicted to be available [24,25], the corresponding gate fidelity is about F ≃ 1 − γ e κ c t ≃ 95%. When we set N 4, with the choice of 2 3 J c 0.5g, g1 g2 g3 g4 g, Ω1 1 Ω1 Ω1 c c c 1 Ω2 Ω3 Ω4 g, Δc 1 Δ2 Δ3 Δ4 20g, Δ1 2 3 Δ2 18g, Δ1 Δ3 18.34g, and Δ1 Δ4 21.7492g, we have Λ01;2 Λ01;3 Λ01;4 Λ04 1.0195 × 10−3 g
(16)
and the time needed to complete the four-qubit controlled phase gate operation is t4 π∕Λ04 3.0815 × 103 ∕g. As reported in the cavity QED experiment in Ref. [26], a coupling strength of g 2π × 34 MHz can be achieved. Therefore, the time required to implement the three-qubit controlled phase gate is of the order of t3 1.20048 × 10−5 s and t4 1.44246 × 10−5 s for the four-qubit phase gate. In the experiments, the decay time of the cavity is tc 3.0 × 10−2 s [27,28], which is longer than the time periods required. Even when N 200, by setting J c 0.5g, gl g, Δc l 20g (l 1; 2; …; N), Ω g (m 1; 2; …; N − 1; n 2; 3; …; N), and Ωm n 1 1 0 Δ1 Δ2 18g, we obtain Λ200 9.45658 × 10−4 g and the time required to achieve the phase gate operation is t200 π∕Λ0200 1.5551 × 10−5 s. Therefore, based on the current cavity QED techniques, the proposed scheme might be experimentally realizable. In conclusion, we have proposed an efficient scheme to realize selective coherent coupling between the control qubit and any target qubit in a 1D quantum network. We consider the scheme in a 1D coupled cavity system with three-level atoms at distant nodes trapped in separated cavities. It is unnecessary to utlize a single-photon source or inject a single photon into an optical cavity. With suitable choice of system parameters, a multiqubit phase gate with one control qubit and multiple target qubits can be achieved directly, which greatly simplifies the experimental realization and reduces the total gate time. Furthermore, the interaction among the atoms that we use to implement the multiqubit phase gate can also be realized in other systems with three-level configuration. This work is supported by the National Natural Science Foundation of China under Grant Nos. 11264042, 61068001, and 11165015; the China Postdoctoral Science Foundation under Grant No. 2012M520612; the Program for Chun Miao Excellent Talents of Jilin Provincial Department of Education under Grant No. 201316; and the Talent Program of Yanbian University of China under Grant No. 950010001.
References 1. H. Goto and K. Ichimura, Phys. Rev. A 70, 012305 (2004). 2. L. M. Duan, B. Wang, and H. J. Kimble, Phys. Rev. A 72, 032333 (2005). 3. A. Gábris and G. S. Agarwal, Phys. Rev. A 71, 052316 (2005). 4. X. M. Lin, Z. W. Zhou, M. Y. Ye, Y. F. Xiao, and G. C. Guo, Phys. Rev. A 73, 012323 (2006). 5. C. Y. Chen, M. Feng, and K. L. Gao, Phys. Rev. A 73, 064304 (2006). 6. X. Zou, Y. Dong, and G. Guo, Phys. Rev. A 74, 032325 (2006). 7. Y. F. Xiao, X. B. Zou, and G. C. Guo, Phys. Rev. A 75, 054303 (2007). 8. S. Fujiwara, H. Osaki, I. M. Buluta, and S. Hasegawa, Phys. Rev. A 75, 012301 (2007). 9. S. B. Zheng, Phys. Rev. A 87, 042318 (2013). 10. C. P. Yang, Y. X. Liu, and F. Nori, Phys. Rev. A 81, 062323 (2010). 11. C. P. Yang, S. B. Zheng, and F. Nori, Phys. Rev. A 82, 062326 (2010). 12. M. J. Hartmann, F. G. S. L. Brandão, and M. B. Plenio, Phys. Rev. Lett. 99, 160501 (2007). 13. D. G. Angelakis, M. F. Santos, and S. Bose, Phys. Rev. A 76, 031805(R) (2007). 14. D. G. Angelakis and A. Kay, New J. Phys. 10, 023012 (2008). 15. C. D. Ogden, E. K. Irish, and M. S. Kim, Phys. Rev. A 78, 063805 (2008). 16. E. K. Irish, C. D. Ogden, and M. S. Kim, Phys. Rev. A 77, 033801 (2008). 17. J. Cho, D. G. Angelakis, and S. Bose, Phys. Rev. A 78, 062338 (2008). 18. P. B. Li, Y. Gu, Q. H. Gong, and G. C. Guo, Phys. Rev. A 79, 042339 (2009). 19. J. Song, Y. Xia, and H. S. Song, Appl. Phys. Lett. 96, 071102 (2010). 20. Z. X. Chen, Z. W. Zhou, X. Zhou, X. F. Zhou, and G. C. Guo, Phys. Rev. A 81, 022303 (2010). 21. L. T. Shen, X. Y. Chen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, Phys. Rev. A 84, 064302 (2011). 22. A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, Nature 431, 162 (2004). 23. B. S. Song, S. Noda, T. Asano, and Y. Akahane, Nat. Mater. 4, 207 (2005). 24. S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, Phys. Rev. A 71, 013817 (2005). 25. S. B. Zheng, C. P. Yang, and F. Nori, Phys. Rev. A 82, 042327 (2010). 26. A. Boca, R. Miller, K. M. Birnbaum, A. D. Boozer, J. McKeever, and H. J. Kimble, Phys. Rev. Lett. 93, 233603 (2004). 27. A. Rauschenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M. Brune, J. M. Raimond, and S. Haroche, Science 288, 2024 (2000). 28. P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. 89, 200402 (2002).
1489
One-step implementation of a multiqubit phase gate with one control qubit and multiple target qubits in coupled cavities Hong-Fu Wang,* Ai-Dong Zhu, and Shou Zhang Department of Physics, College of Science, Yanbian University, Yanji 133002, China *Corresponding author: [email protected] Received November 29, 2013; revised January 16, 2014; accepted January 31, 2014; posted February 3, 2014 (Doc. ID 202273); published March 11, 2014 We propose a one-step scheme to implement a multiqubit controlled phase gate with one qubit simultaneously controlling multiple qubits with three-level atoms at distant nodes in coupled cavity arrays. Selective qubit–qubit couplings are achieved by adiabatically eliminating the atomic excited states and photonic states, and the required phase shifts between the control qubit and any target qubit can be realized through suitable choices of the parameters of the external fields. Moreover, the effective model is robust against decoherence because neither the atoms nor the field modes are excited during the gate operation, leading to a useful step toward scalable quantum computing networks. © 2014 Optical Society of America OCIS codes: (270.5580) Quantum electrodynamics; (270.5585) Quantum information and processing. http://dx.doi.org/10.1364/OL.39.001489
It has been shown that any multiqubit gate can be decomposed into two classes of elementary quantum gates, namely, a universal two-qubit controlled phase gate and a one-qubit unitary gate, which are the basic building blocks of a quantum computer. However, the procedure of decomposing multiqubit gates into the elementary gates usually becomes very complicated as the number of qubits increases when using the conventional gate decomposition method. To reduce the complexity of the physical realization of practical quantum computing and quantum information processing, direct implementation of multiqubit logic gates with multiple control qubits [1–9] or multiple target qubits [10,11] is, thus, very important. In recent years, considerable theoretical effort has been devoted to a class of coupled cavity models, which typically describe a series of optical cavities, each containing one or more atoms, where photons are permitted to hop between cavities. The coupled cavity arrays show promise in overcoming the problem of individual addressability and have several interesting potential applications, including quantum information processing and simulations of quantum strongly correlated manybody systems. Theoretical studies on quantum computing and quantum information processing have been put forward for use of atom–light interactions in coupled microcavity arrays [12–21]. In this Letter, we propose a scheme for implementing a multiqubit phase gate with one control qubit and multiple target qubits with threelevel atoms at distant nodes in coupled cavity arrays. This type of multiqubit controlled phase gate is essentially equivalent to n two-qubit controlled phase gates, each having a shared control qubit (qubit 1) but a different target qubit (qubits 2; 3; …; N). In the scheme, the selective off-resonant qubit–qubit coupling between the ground states of two atoms, induced by the external fields and the cavity modes, can be used to realize an effective phase shift between the control qubit and any target qubit by choosing the parameters of the external fields appropriately. The scheme has the following merits: (1) it can be accomplished in only one step, which greatly simplifies the experimental realization and reduces the total 0146-9592/14/061489-04$15.00/0
gate time; (2) it does not require individual addressing of the trapped atoms; and (3) multiqubit gate operation among spatially separated quantum nodes is very meaningful for distributed quantum computing and quantum communication networks. Moreover, it would be an important step toward efficiently constructing quantum circuits and quantum algorithms. We consider an array of cavities that are coupled via exchange of photons, with one three-level atom in each cavity, as sketched in Fig. 1(a). Such a model can be constructed in several kinds of physical systems such as superconducting stripline resonators [22], photonic crystal defects [23], and microtoroidal cavity arrays [24]. Each atom has one excited state, jei, and two ground states, jgi and jai. The transition jgii → jeii (i 1; 2; …; N) is coupled to the cavity mode with coupling strength gi and detuning Δc i . On the other hand, the transition jgi1 → jei1 for atom 1 is driven by N − 1 classical laser fields with Rabi frequencies Ωk 1 and detunings Δk 1 (k 1; 2; …; N − 1) [Fig. 1(b)]. The transition jgij → jeij for atom j (j 2; 3; …; N) is driven by a classical laser field with Rabi frequency Ωj and detuning Δj [Fig. 1(c)]. In the interaction picture, the Hamiltonian describing the atom–field interaction is given by ˆ int J c H
N X
(" aˆ †j aˆ j1
aˆ j aˆ †j1
j1
× jeijj hgj
c
gj aˆ j eiΔj
t
j1 N −1 X m1
N X
N X
m
iΔ1 t Ωm jei11 hgj 1 e
Ωn eiΔn t jeinn hgj
#
) H:c: ;
(1)
n2
where J c is the cavity–cavity hopping strength. Considering the periodic boundary conditions aˆ N1 aˆ 1 and taking advantage of Fourier transformation to diagonalize ˆ int can the photon coupling terms, the Hamiltonian H be rewritten as © 2014 Optical Society of America
1490
OPTICS LETTERS / Vol. 39, No. 6 / March 15, 2014
ηm
2 Ωm 1
Δm 1
ζ n;k Fig. 1. (a) One-dimensional (1D) array of coupled cavities, with one three-level atom in each cavity. (b) Level configuration and excitation scheme of atom 1. (c) Level configuration and excitation scheme of atom j (j 2; 3; …; N). N X
ˆ int H
(" ωk bˆ †k bˆ k
k1 c
× eiΔj t jeijj hgj
N X N 1 X 2π p gj bˆ k e−i N jk N j1 k1
N −1 X
m
m1
N X
iΔ1 t Ωm jei11 hgj 1 e
#
iΔn t
Ωn e
)
jeinn hgj H:c: ;
(2)
n2
; NΔc l − ωk g1 Ωm 1 1 1 p m ; 2 N Δc Δ1 1 − ωk gn Ωn 1 1 : p Δn 2 N Δc n − ωk
k1
N X
c
× eiΔn
jeinn hgj H:c:
−
(3)
m ≫ Ωm Under the conditions jΔc j − ωk j ≫ gj ∕N, Δ1 1 , and Δn ≫ Ωn , the upper level jeij can be eliminated adiabatically, leading to
" N N −1 X X k1
ξm;k bˆ k e−i N k e 2π
iΔc −ωk −Δm t 1 1
m1 N X n2
c 2π ζn;k bˆ k e−i N nk eiΔn −ωk −Δn t
#
×jginn hgj H:c: −
X N −1 m1
N X n2
where
N X
N X
n2
k1
N X N X
jgi11 hgj ⊗ jginn hgj ! N X θm;k − ηm jgi11 hgj k1
!
ϑn;k − μn jginn hgj
χ l;k bˆ †k bˆ k jgill hgj;
with θm;k
ξ2m;k
; m
Δc 1 − ωk − Δ1
ϑn;k
ζ2n;k Δc n − ω k − Δn
;
2π
× jgi11 hgj
N −1 X
l1 k1
n2
ˆ 00 − H
#)
m1
#
Ωn e
Λm;n;k
−Δc −Δn Δm t 1 1
H:c:
iΔm t 1 Ωm jei11 hgj 1 e
iΔn t
N −1 X N X m1 n2
N X N X c 2π ˆ 0 p1 gj bˆ k e−i N jk eiΔj −ωk t jeijj hgj H N j1 k1
m1
(5)
p2 q2;q≠p
× jgipp hgj ⊗ jgiqq hgj
"
Ω2n ; Δn
The first two terms in Eq. (4) are the coupling between the bosonic modes bˆ k and the classical field assisted by the atoms, respectively. The last three terms are the Stark shifts for level jgij that are induced by the bosonic modes bˆ k and the classical pulse. In the case of jΔc 1 − ωk − c Δm j ≫ ξ and jΔ − ω − Δ j ≫ ζ , the bosonic n m;k k n n;k 1 modes do not exchange quanta with the atomic system: the bosonic modes are excited only virtually and any two atoms interfere with each other during the whole interaction process. The effective Hamiltonian is then given by (" N N N X X X c c ˆ H eff Γp;q;k eiΔp −Δq −Δp Δq t
p P −i2π∕Njk ˆ bk and ωkP where aˆ j 1∕ N N 2J c cos k1 e N ˆ† ˆ i 2π∕Nk. Performing the transformation e k1 ωk bk bk t , we obtain
N −1 X
μn
g2l
χ l;k ξm;k
;
μn jginn hgj
Γp;q;k
ζ p;k ζq;k e−i N p−qk 1 c 2 Δp − ω k − Δp ! 1 c ; Δq − ωk − Δq 2π
ηm jgi11 hgj
N X N X l1 k1
χ l;k bˆ †k bˆ k jgill hgj ; (4)
Λm;n;k
ξm;k ζn;k e−i N n−1k 1 c 2 Δ1 − ωk − Δm 1 ! 1 c ; Δn − ωk − Δn
(7)
where ζ ν;k ν p; q; n in Eq. (7) is denoted by Eq. (5). As the quantum number of the bosonic modes is conserved
March 15, 2014 / Vol. 39, No. 6 / OPTICS LETTERS
during the interaction, they will remain in the vacuum state if they are initially so. Choosing the detunings suitably so that c m Δc 0 n − Δ1 − Δn Δ1
m ≠ n − 1;
k1
−
Δc q
N X − Δp Δq j ≫ Γp;q;k
p; q ∈ f2; 3; …; Ng; p ≠ q;
(8)
then the effective Hamiltonian reduces to N X j2
ζ 01;j jgi11 hgj ξ0j jgijj hgj
Λ01;j jgi11 hgj ⊗ jgijj hgj;
(9)
with ζ 01;j
N X
θj−1;k − ηj−1 ;
k1
Λ01;j
N X
ξj−1;k ζj;k
1
N X
ϑj;k − μj ;
j−1
Δc 1 − ω k − Δ1
1 Δc j − ω k − Δj
;
(10)
where ηj−1 , μj , ξj−1;k , and ζ j;k are denoted by Eq. (5) and θj−1;k and ϑj;k are denoted by Eq. (7), respectively. The last term in Eq. (9) describes the coupling between atoms 1 and j (j 2; 3; …; N) mediated by the bosonic modes and the classical pulses. To implement quantum computation, the two longlived levels jai and jgi represent the states of the qubits, which correspond the logical 0 and 1 states, respectively, j0i ≡ jai and j1i ≡ jgi. Due to virtual excitation of the atoms in the interaction, atom 1 and any atom j will undergo an energy shift, leading to the evolution jai1 jaij → jai1 jaij ; jai1 jgij → e−iφj jai1 jgij ; jgi1 jaij → e−iψ 1;j jgi1 jaij ; jgi1 jgij → e−iφj ψ 1;j ϕ1;j jgi1 jgij ;
(11)
where φj ξ0j t, ψ 1;j ζ 01;j t, and ϕ1;j Λ01;j t. After the generation of the one-qubit phase shifts jgi1 → eiψ 1;j jgi1 and jgij → eiφj jgij , a conditional phase shift ϕ1;j , which is controllable via effective interaction time t and the corresponding parameters Λ01;j , is produced if and only if atoms 1 and j are in state jgi. While for state jgim jgin of any other pair of atoms m and n (m; n 2; 3; …; N), no conditional phase shift is generated. Therefore, for the computational basis states fjs1 i1 js2 i2 js3 i3 ⊗ ⊗ jsn−1 iN−1 jsn iN g (si a; g; i 1; 2; …; n) consisting of N
(12)
where g⊕g 0 and g⊕a 1. Therefore, a multiqubit controlled phase gate of one qubit simultaneously controlling multiple qubits is achieved if and only if atom 1 is in state jgi1 . We now give a brief analysis and discussion for some practical issues in relation to the experimental feasibility of the proposed scheme. For simplicity, we consider the case with N 3 and set J c 0.5g, g1 g2 g3 g, c c 2 Ω1 Δc 1 Δ2 Δ3 20g, 1 Ω1 Ω2 Ω3 g, 1 2 Δ1 Δ2 18g, and Δ1 Δ3 21.2842g. Then we have Λ01;2 Λ01;3 Λ03 1.225 × 10−3 g
k1
2π j − 1k cos N
k1
×
ξ0j
→ jai1 js2 i2 js3 i3 ⊗ ⊗ jsn−1 iN−1 jsn iN ; PN −i 1−g⊕sj π j2 jgi1 js2 i2 js3 i3 ⊗ ⊗ jsn−1 iN−1 jsn iN → e ×jgi1 js2 i2 js3 i3 ⊗ ⊗ jsn−1 iN−1 jsn iN ;
k1
ˆ 0eff H
atoms, after the qubit–qubit couplings and a series of one-qubit phase shift operations on state jgik (k 1; 2; …; N) and with the choice of ϕ1;j Λ01;j t π, one can obtain jai1 js2 i2 js3 i3 ⊗ ⊗ jsn−1 iN−1 jsn iN
m n − 1; n 2; 3; …; N; N X c c m Λm;n;k jΔn − Δ1 − Δn Δ1 j ≫ jΔc p
1491
(13)
and the time needed to complete the three-qubit controlled phase gate operation is t3 π∕Λ03 2.56457 × 103 ∕g, and all restrictive conditions in the cases of large detunings are satisfied well. The probability that the atoms undergo transition to the excited state due to qubit–qubit coupling with the classical fields is " 2 2 Ω2 1 3 Ω1 13 Ω2 2 1 pe × × 1 × 2 2 8 4 8 Δc Δc 1 2 # 3 Ω3 2 × c 2 1.09375 × 10−3 : 8 Δ3
(14)
Meanwhile, the probability that the field mode is excited due to qubit–qubit coupling is " 2 3 X ξ1;k 3 pc × c 1 4 k1 Δ1 − ωk − Δ1 2 # 3 X ξ2;k c 2 k1 Δ1 − ωk − Δ1 " 2 # 3 X 13 ζ2;k × c 8 k1 Δ2 − ωk − Δ2 " 2 # 3 X ζ 3;k 3 × c 8 k1 Δ3 − ωk − Δ3 5.98033 × 10−3 :
(15)
ˆ 0eff in Eq. (9) is Therefore, the effective Hamiltonian H valid. Furthermore, the effective decoherence rates due to the atomic spontaneous emission and field decay
1492
OPTICS LETTERS / Vol. 39, No. 6 / March 15, 2014
are γ e pe γ and κ c pc κ, with γ and κ being the decay rates for the atomic excited state and the field modes, respectively. If we consider the parameters γ ∼ κ ∼ 3 × 10−3 g, which were predicted to be available [24,25], the corresponding gate fidelity is about F ≃ 1 − γ e κ c t ≃ 95%. When we set N 4, with the choice of 2 3 J c 0.5g, g1 g2 g3 g4 g, Ω1 1 Ω1 Ω1 c c c 1 Ω2 Ω3 Ω4 g, Δc 1 Δ2 Δ3 Δ4 20g, Δ1 2 3 Δ2 18g, Δ1 Δ3 18.34g, and Δ1 Δ4 21.7492g, we have Λ01;2 Λ01;3 Λ01;4 Λ04 1.0195 × 10−3 g
(16)
and the time needed to complete the four-qubit controlled phase gate operation is t4 π∕Λ04 3.0815 × 103 ∕g. As reported in the cavity QED experiment in Ref. [26], a coupling strength of g 2π × 34 MHz can be achieved. Therefore, the time required to implement the three-qubit controlled phase gate is of the order of t3 1.20048 × 10−5 s and t4 1.44246 × 10−5 s for the four-qubit phase gate. In the experiments, the decay time of the cavity is tc 3.0 × 10−2 s [27,28], which is longer than the time periods required. Even when N 200, by setting J c 0.5g, gl g, Δc l 20g (l 1; 2; …; N), Ω g (m 1; 2; …; N − 1; n 2; 3; …; N), and Ωm n 1 1 0 Δ1 Δ2 18g, we obtain Λ200 9.45658 × 10−4 g and the time required to achieve the phase gate operation is t200 π∕Λ0200 1.5551 × 10−5 s. Therefore, based on the current cavity QED techniques, the proposed scheme might be experimentally realizable. In conclusion, we have proposed an efficient scheme to realize selective coherent coupling between the control qubit and any target qubit in a 1D quantum network. We consider the scheme in a 1D coupled cavity system with three-level atoms at distant nodes trapped in separated cavities. It is unnecessary to utlize a single-photon source or inject a single photon into an optical cavity. With suitable choice of system parameters, a multiqubit phase gate with one control qubit and multiple target qubits can be achieved directly, which greatly simplifies the experimental realization and reduces the total gate time. Furthermore, the interaction among the atoms that we use to implement the multiqubit phase gate can also be realized in other systems with three-level configuration. This work is supported by the National Natural Science Foundation of China under Grant Nos. 11264042, 61068001, and 11165015; the China Postdoctoral Science Foundation under Grant No. 2012M520612; the Program for Chun Miao Excellent Talents of Jilin Provincial Department of Education under Grant No. 201316; and the Talent Program of Yanbian University of China under Grant No. 950010001.
References 1. H. Goto and K. Ichimura, Phys. Rev. A 70, 012305 (2004). 2. L. M. Duan, B. Wang, and H. J. Kimble, Phys. Rev. A 72, 032333 (2005). 3. A. Gábris and G. S. Agarwal, Phys. Rev. A 71, 052316 (2005). 4. X. M. Lin, Z. W. Zhou, M. Y. Ye, Y. F. Xiao, and G. C. Guo, Phys. Rev. A 73, 012323 (2006). 5. C. Y. Chen, M. Feng, and K. L. Gao, Phys. Rev. A 73, 064304 (2006). 6. X. Zou, Y. Dong, and G. Guo, Phys. Rev. A 74, 032325 (2006). 7. Y. F. Xiao, X. B. Zou, and G. C. Guo, Phys. Rev. A 75, 054303 (2007). 8. S. Fujiwara, H. Osaki, I. M. Buluta, and S. Hasegawa, Phys. Rev. A 75, 012301 (2007). 9. S. B. Zheng, Phys. Rev. A 87, 042318 (2013). 10. C. P. Yang, Y. X. Liu, and F. Nori, Phys. Rev. A 81, 062323 (2010). 11. C. P. Yang, S. B. Zheng, and F. Nori, Phys. Rev. A 82, 062326 (2010). 12. M. J. Hartmann, F. G. S. L. Brandão, and M. B. Plenio, Phys. Rev. Lett. 99, 160501 (2007). 13. D. G. Angelakis, M. F. Santos, and S. Bose, Phys. Rev. A 76, 031805(R) (2007). 14. D. G. Angelakis and A. Kay, New J. Phys. 10, 023012 (2008). 15. C. D. Ogden, E. K. Irish, and M. S. Kim, Phys. Rev. A 78, 063805 (2008). 16. E. K. Irish, C. D. Ogden, and M. S. Kim, Phys. Rev. A 77, 033801 (2008). 17. J. Cho, D. G. Angelakis, and S. Bose, Phys. Rev. A 78, 062338 (2008). 18. P. B. Li, Y. Gu, Q. H. Gong, and G. C. Guo, Phys. Rev. A 79, 042339 (2009). 19. J. Song, Y. Xia, and H. S. Song, Appl. Phys. Lett. 96, 071102 (2010). 20. Z. X. Chen, Z. W. Zhou, X. Zhou, X. F. Zhou, and G. C. Guo, Phys. Rev. A 81, 022303 (2010). 21. L. T. Shen, X. Y. Chen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, Phys. Rev. A 84, 064302 (2011). 22. A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, Nature 431, 162 (2004). 23. B. S. Song, S. Noda, T. Asano, and Y. Akahane, Nat. Mater. 4, 207 (2005). 24. S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, Phys. Rev. A 71, 013817 (2005). 25. S. B. Zheng, C. P. Yang, and F. Nori, Phys. Rev. A 82, 042327 (2010). 26. A. Boca, R. Miller, K. M. Birnbaum, A. D. Boozer, J. McKeever, and H. J. Kimble, Phys. Rev. Lett. 93, 233603 (2004). 27. A. Rauschenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M. Brune, J. M. Raimond, and S. Haroche, Science 288, 2024 (2000). 28. P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. 89, 200402 (2002).
