Geometric entangling gates for coupled cavity system in decoherence-free subspaces
aa r X i v : . [ qu a n t - ph ] D ec Geometric entangling gates for coupled cavity system in decoherence-free subspaces
Yue-Yue Chen, Xun-Li Feng,
1, 2 and C.H. Oh Laboratory of Photonic Information Technology, LQIT & SIPSE,South China Normal University, Guangzhou 510006, China Department of Physics and Centre for Quantum Technologies,National University of Singapore, 2 Science Drive 3, Singapore 117542
Abstract.
We propose a scheme to implement geometric entangling gates for two logical qubitsin a coupled cavity system in decoherence-free subspaces. Each logical qubit is encoded with twoatoms trapped in a single cavity and the geometric entangling gates are achieved by cavity couplingand controlling the external classical laser fields. Based on the coupled cavity system, the schemeallows the scalability for quantum computing and relaxes the requirement for individually addressingatoms.
PACS numbers: 03.67.Lx, 03.67.Pp, 03.65.Vf, 42.50.Pq
Exploiting appropriate coherent dynamics to generate entangling gates between separate systems is of crucialimportance to quantum computing and quantum communication. Several schemes have been proposed to engineerentangling gates [1–3] between atoms trapped in spatially separated cavities. It is feasible and commonly used tomediate the distant optical cavities by optical fiber [4–6]. However, decoherence resulted from uncontrollable couplingto environment will collapse the state and impair the performance for quantum process. Thus, decoherence is themain obstacle for realizing quantum computing and quantum information processing. In order to protect the fragilequantum information and realize the promised speedup compared with classical counterpart, a wealth of strategieshave been proposed to deal with decoherence. One efficient way is to construct a decoherence-free subspace (DFS)if the interaction between quantum system and its environment possesses some symmetry [9]. Keeping a systeminside a DFS is regarded as a “passive” error-prevention approach while error-correcting code, which is comprisedof encoding information in a redundant way, is regarded as an active approach [10]. Another promising strategy tocope with decoherence is based on the mechanism of geometric phase [11]. Geometric phases depend only on someglobal geometric features of the evolution path and are insensitive to local inaccuracies and fluctuations. However,the total phases acquired during the evolution often consist of geometric phases and the concomitant dynamic phases.Dynamic phases may ruin the potential robustness of the scheme and should be removed according to conventionalwisdom. Literatures [12] and [13] proposed two simple methods to remove dynamic phases. In contrast, the so-called unconventional geometric gates, in which dynamic phases are not zero but proportional to the geometric ones,were proposed [14, 15]. The unconventional geometric gates were suggested to be realized in cavity QED systemssubsequently [16, 17].Schemes which combine the robust advantages of both DFS and the geometric phase have been presented [18, 19].Reference [18] exploits the spin-dependent laser-ion coupling in the presence of Coulomb interactions, and thenconstructs a universal set of unconventional geometric quantum gates in encoded subspaces. Reference [19] proposesto implement the geometric entangling gates in DFS by using a dispersive atom-cavity interaction in a single cavity.As is well known, the collective decoherence is often regarded as a strict requirement for DFS strategy to overcomethe decoherence, however, such a requirement is largely relaxed in [19] because only two neighboring physical qubits,which encode a logical qubit, are required to undergo collective dephasing. With this merit, in this paper we extendthe idea of [19] to a coupled cavity system where each cavity contains two atoms which encode one logical qubit.In contrast to [19], the extension to the coupled cavity system in this work allows the realization of scalability ofcavity QED based quantum computing by using the idea of the distributed quantum computing [20] and relaxes therequirement for individually addressing atoms.Now let us describe our scheme more specifically. Considering two coupled cavities which are linked with an opticalfiber. We suppose each cavity contains two Λ-type three-level atoms. For convenience, we label the two cavities with j and k , respectively, the atoms in cavity j ( k ) are denoted by j , j ( k , k ). The atomic level configuration withcouplings to the cavity modes and the driving laser fields is shown in Fig. 1: | e i is an excited state and | i and | i aretwo stable ground states, the latter two constitute the basis of a physical qubit. Both transitions | i ↔ | e i and | i ↔| e i are supposed to dispersively couple to the cavity mode and be driven by two classical laser fields with oppositedetunings. One of the classical laser field acts on transitions | i ↔ | e i and | i ↔ | e i has a frequency ω closed to thecavity frequency ω c . Note that, ω − ω c = δ , where δ is a small quantity. The detuning of this classical field from thetransition | m i ↔ | e i is ∆ m = ω m − ω ( m = 0 , ω m is the energy difference between ground state | m i and | e i . The corresponding detuning for the cavity mode is ∆ m + δ (see Fig. 1). Similarly, the other laser with frequency ω ′ is tuned to satisfy the relation ω m − ω ′ = − ∆ m .To overcome the collective dephasing, we encode the logical qubit in the cavity j with a pair of physical qubits in (cid:39) (cid:58) e (cid:58) g g (cid:39) '0 (cid:58) - (cid:39) (cid:39) + (cid:71) (cid:39) + (cid:71) - (cid:39) '1 (cid:58) FIG. 1: Atomic level structure and couplings. The transition | m i ( m = 0 , ↔ | e i is coupled to the cavity mode with strength g m and driven by classical field lasers with Rabi frequency Ω m / Ω ′ m . a form | j i L = | j j i , | j i L = | j j i . The subspace C j = n | j i L , | j i L o constitutes a DFS for the single logicalqubit j . Similarly, the logical qubit k is encoded by the two physical qubits k , k in the cavity k .The coupling between the cavity fields and the fiber modes can be written as the interaction Hamiltonian [1] H cf = ∞ X i =1 ν i h b i (cid:16) a † + ( − e iϕ a † (cid:17) + H.c. i , (1)where ν i is the coupling strength between fiber mode i and the cavity mode, b i is the annihilation operator for thefiber mode i while a † (cid:16) a † (cid:17) is the creation operator for the cavity mode j ( k ), and ϕ is the phase induced by thepropagation of the field through the fiber. In the short fiber limit, only resonant mode b of the fiber interacts withthe cavity mode. In this case, the Hamiltonian H cf can be approximately written as [1] H cf = ν h b (cid:16) a † + a † (cid:17) + H.c. i , (2)where the phase ( − e iϕ in H cf has been absorbed into a † and a [2].To implement the geometric entangling gate, we let the classical laser fields plotted in Fig. 1 individually act onboth atoms j and k . In the interaction picture, the Hamiltonian describing the atom-field interaction takes the form H AC = X l = j ,k X m =0 , Ω ′ m e − i ∆ m t | e i l h m | + Ω m e i ∆ m t | e i l h m | + X l = j ,j X m =0 , g m | e i l h m | a e i (∆ m + δ ) t + X l = k ,k X m =0 , g m | e i l h m | a e i (∆ m + δ ) t + H.c. (3)Following Ref. [1], we define three bosonic modes c = √ ( a − a ), c = (cid:0) a + a + √ b (cid:1) , c = (cid:0) a + a − √ b (cid:1) , c n ( n = 0 , ,
2) are linearly relative to the field modes of the cavities and fiber. Then we canrewrite the whole Hamiltonian in the interaction picture as H = H + H i , (4)where H = √ νc † c − √ νc † c , (5)and H i = X l = j ,k m =0 , Ω ′ m e − i ∆ m t | e i l h m | + Ω m e i ∆ m t | e i l h m | + X l = j ,j m =0 , g m | e i l h m | (cid:16) c + c + √ c (cid:17) e i (∆ m + δ ) t + X l = k ,k m =0 , g m | e i l h m | (cid:16) c + c − √ c (cid:17) e i (∆ m + δ ) t + H.c. (6)We now perform the unitary transformation e iH t , and obtain [3] H i = X l = j ,k m =0 , (cid:18) Ω ′ m e − i ∆ m t | e i l h m | + Ω m e i ∆ m t | e i l h m | (cid:19) + X l = j ,j m =0 , g m | e i l h m | (cid:16) c e − i √ νt + c e i √ νt + √ c (cid:17) e i (∆ m + δ ) t + X l = k ,k m =0 , g m | e i l h m | (cid:16) c e − i √ νt + c e i √ νt − √ c (cid:17) e i (∆ m + δ ) t + H.c. (7)Here we assume that ∆ m ≫ √ ν, δ, g m and Ω m to make sure that atoms cannot exchange energy with the fiber mode,cavity modes, and classical fields on account of the large detuning. In this case, we may adiabatically eliminate theexcited atomic state considering no population transferred to the this state. In order to cancel the Stark shifts causedby classical laser fields, we set | Ω m | = | Ω ′ m | . Assuming further g m ≪ Ω m , we can neglect the terms of g m , whichindicate the Stark shifts caused by bosonic modes. From the above, we obtain an effective Hamiltonian describingthe coupling between the atoms and bosonic modes assisted by the classical fields [21] H eff = (cid:16) | i j h | + | i k h | (cid:17) λ e − i ( δ −√ ν ) t c † + (cid:16) | i j h | + | i k h | (cid:17) λ e − i ( δ + √ ν ) t c † + (cid:16) | i j h | − | i k h | (cid:17) λ e − iδt c † + (cid:16) | i j h | + | i k h | (cid:17) λ ′ e − i ( δ −√ ν ) t c † + (cid:16) | i j h | + | i k h | (cid:17) λ ′ e − i ( δ + √ ν ) t c † + (cid:16) | i j h | − | i k h | (cid:17) λ ′ e − iδt c † + H.c., (8)where λ = − √ g ∗ (cid:16) + + δ (cid:17) , λ = − Ω g ∗ (cid:16) + + δ −√ ν (cid:17) , λ = − Ω g ∗ (cid:16) + + δ + √ ν (cid:17) , λ ′ = − √ g ∗ (cid:16) + + δ (cid:17) , λ ′ = − Ω g ∗ (cid:16) + + δ −√ ν (cid:17) , λ = − Ω g ∗ (cid:16) + + δ + √ ν (cid:17) .Because the logical qubits j and k are located at different cavities, the available DFS for the whole system isconstructed by C jk ≡ C j ⊗ C k = (cid:8)(cid:12)(cid:12) Lj Lk (cid:11) , (cid:12)(cid:12) Lj Lk (cid:11) , (cid:12)(cid:12) Lj Lk (cid:11) , (cid:12)(cid:12) Lj Lk (cid:11)(cid:9) , and in this DFS the Hamiltonian H eff isdiagonal and takes the form H eff = diag (cid:2) H j k , H j k , H j k , H j k (cid:3) , (9)where the diagonal matrix elements H µ j ν k ( µ, ν = 0 ,
1) are of the form H µ j ν k = X n =0 c † n χ nµ j ν k e − iη n t + H.c., (10)where χ j k = 0, χ j k = 2 λ , χ j k = 2 λ ; χ j k = λ − λ ′ , χ j k = λ + λ ′ , χ j k = λ + λ ′ ; χ j k = λ ′ − λ , χ j k = λ + λ ′ , χ j k = λ + λ ′ ; χ j k = 0, χ j k = 2 λ ′ , χ j k = 2 λ ′ .and η = δ , η = δ − √ ν , η = δ + √ ν . Obviously, in the DFS C jk , time evolution matrix U ( t ) also takes a diagonalform, U ( t ) = diag (cid:2) U j k , U j k , U j k , U j k (cid:3) . (11)The corresponding diagonal matrix elements U µ j ν k ( t ) can be derived from Eq. (10) and they are in terms ofdisplacement operator U µ j ν k ( t ) = ˆ T exp (cid:20) − i Z t H µ j ν k ( τ ) dτ (cid:21) = Y n =0 exp (cid:16) iφ nµ j ν k (cid:17) D (cid:18)Z c dα nµ j ν k (cid:19) = exp (cid:2) iφ µ j ν k (cid:3) Y n =0 D (cid:18)Z c dα nµ j ν k (cid:19) , (12)with ˆ T being the time ordering operator, and φ µ j ν k = X n =0 φ nµ j ν k = X n =0 Im (cid:20)Z c (cid:16) α nµ j ν k (cid:17) ∗ dα nµ j ν k (cid:21) , (13) dα nµ j ν k = − iχ nµ j ν k e − iη n τ dτ (14)Considering the situation, where each bosonic mode is assumed initially in vacuum state, the state of each bosonicmode evolves to coherent state at time t n >
0. The corresponding amplitude R c dα nµ j ν k is dependent on the logiccomputational basis state (cid:12)(cid:12) µ Lj ν Lk (cid:11) . It is not difficult to obtain α nµ j ν k by integrating Eq. (14) α nµ j ν k = χ nµ j ν k η n (cid:0) e − iη n t − (cid:1) . (15)The above equation indicates that there is a time period T fulfilling the relation T = 2 πl n /η n , where l n is a positiveinteger and n = 0 , ,
2, in which the bosonic mode c n completes l n evolutions and returns to its initial vacuum state.During this process the system accumulates the following total phase γ µ j ν k ( T ) = φ µ j ν k ( T ) = − X n =0 πl n η n (cid:12)(cid:12)(cid:12) χ nµ j ν k (cid:12)(cid:12)(cid:12) = γ gµ j ν k + γ dµ j ν k , (16)where γ dµ j ν k and γ gµ j ν k stand for the dynamical and geometric phases respectively, and can be calculated by using thecoherent state path integral method [22] γ dµ j ν k = X n =0 − Z T H nµ j ν k (cid:16)(cid:16) α nµ j ν k (cid:17) ∗ , α nµ j ν k ; t (cid:17) dt = − X n =0 πl n η n (cid:12)(cid:12)(cid:12) χ nµ j ν k (cid:12)(cid:12)(cid:12) , (17) γ gµ j ν k = γ µ j ν k − γ dµ j ν k = X n =0 πl n η n (cid:12)(cid:12)(cid:12) χ nµ j ν k (cid:12)(cid:12)(cid:12) , (18)we find γ µ j ν k = − γ gµ j ν k = γ dµ j ν k . Thus the total phase γ µ j ν k and dynamical phase γ dµ j ν k possess global geometricfeatures as does the geometric phase γ gµ j ν k . Therefore at time t = T = 2 πl n /η n the time evolution matrix takes theform U ( T ) = diag h e iγ j k , e iγ j k , e iγ j k , e iγ j k i . (19) U ( T ) is actually the geometric entangling gate operation we are targeting at and U ( T ) is a nontrivial entangling gatewhen the condition γ j k + γ j k = γ j k + γ j k is fulfilled [19].We now give a brief discussion about the decoherence mechanisms of our scheme: atomic spontaneous emission,cavity decay and fiber loss. Considering none of the atoms are initially populated in the excited state since thequantum information is encoded in ground states, and atoms cannot exchange energy with the fiber mode, cavitymodes and classical fields due to the large detuning, thus no population is transferred to the excited atomic state. Inthis sense, the spontaneous emission of the atomic excited state can be ignored.Regarding the cavity decay and the fiber loss, the fidelity of the resulting gates will be greatly impaired by thembecause the geometric phases are acquired by the evolution of the optical modes. So, strictly speaking, our schemerequires ideal good cavities and fiber. However, if the mean number of photons of the optical fields is sufficiently small,the cavities and fiber are normally not excited and the moderate cavity decay and fiber loss can thus be tolerated. Fora coherent state the mean number of photons is equal to the square of the amplitude of the state which is determinedby Eq.(15). Thus when the condition χ nµjνk η n ≪ ν/ π = 26 .
72 MHz, g / π = g / π = 20 MHz, Ω / π = Ω / π = 120 MHz, ∆ / π = 3000 MHz,∆ / π = 600 MHz, δ/ π = 35 MHz. These parameters satisfy the requirement χ nµjνk η n ≪ U ( t ) = diag (cid:8) e . i , e . i , e . i , e iπ (cid:9) with the gate operation time t ≈ . µ s. Obviously the gate operation time ismuch shorter than the photon lifetime in optical cavities [24]. According to Eq. (15) the amplitude of the coherentstate is dependent on the atomic states, for the above parameters the amplitude corresponding to state (cid:12)(cid:12) Lj Lk (cid:11) takesthe maximal value, and the maximal mean number of photons is 0.1087. In this case, the optical modes are hardlyexcited and thus the moderate cavity decay and fiber loss can be tolerated.In conclusion, we have proposed a scheme to implement geometric entangling gates for two logical qubits in acoupled cavity system in DFS. Our scheme possesses both advantages of DFS and the geometric phase. Besides,in comparison with the scheme of Ref. [19] which works in a single cavity, the scheme proposed in this paper caneasily realize the scalability of cavity QED-based quantum computing by using the idea of the distributed quantumcomputing[20] and can relax the requirement for individually addressing atoms.The work is supported by the NSFC under Grant No. 11074079, the Ph.D. Programs Foundation of Ministry ofEducation of China, the Open Fund of the State Key Laboratory of High Field Laser Physics ( Shanghai Institute ofOptics and Fine Mechanics), and National Research Foundation and Ministry of Education, Singapore, under researchGrant No. WBS: R-710-000-008-271. [1] A. Serafini, S. Mancini, and S. Bose Phys. Rev. Lett. (2006) 010503.[2] L.-B. Chen, M.-Y. Ye, G.-W. Lin, Q.-H. Du, and X.-M. Lin, Phys. Rev. A (2007) 062304; J. Song, Y. Xia, H.-S. Song,J.-L. Guo, J. Nie Europhysics Lett. (2007) 60001.[3] S.-B. Zheng, Appl. Phys. Lett. (2009) 154101.[4] J. I. Cirac et al ., Phys. Rev. Lett. (1997) 3221; S. J.van Enk et al., ibid . (1997) 5178.[5] S. J. van Enk et al ., Phys. Rev. A (1999) 2659.[6] S. Clark et al ., Phys. Rev. Lett. (2003) 177901.[7] S. Bose et al ., Phys. Rev. Lett. (1999) 5158; S.Manciniand S. Bose, Phys. Rev. A (2001) 032308; D. E.Browne etal ., Phys. Rev. Lett. (2003) 067901.[8] L.M. Duan and H. J. Kimble, Phys. Rev. Lett. (2003) 253601.[9] G. M. Palma, K. Suominen, and A. K. Ekert, Proc. R. Soc. A (1996) 567; L.-M. Duan and G.-C. Guo, Phys. Rev.Lett. (1997) 1953; P. Zanardi and M. Rasetti, ibid. (1997) 3306; D. A. Lidar, I. L. Chuang, and K. B. Whaley, ibid. Phys. Rev. Lett. (1998) 2594.[10] P. W. Shor, Phys. Rev. A (1995) R2493.[11] P. Zanardi and M. Rasetti, Phys. Lett. A (1999) 94.[12] J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, Nature (London) (2000) 869; G. Falci, R. Fazio, G. M. Palma,J. Siewert, and V. Vedral, ibid (2000) 355; X.-B. Wang and M. Keiji, Phys. Rev. Lett. (2001) 097901.[13] L.M. Duan, J. I. Cirac, and P. Zoller, Science (2001) 1695.[14] X. Wang and P. Zanardi, Phys. Rev. A (2002) 032327.[15] S.-L. Zhu and Z.D. Wang, Phys. Rev. Lett. (2003) 187902.[16] S. B. Zheng, Phys. Rev. A (2004) 052320.[17] X.-L. Feng, Z. Wang, C. Wu, L.C. Kwek, C. H. Lai, and C. H. Oh, Phys. Rev. A (2007) 052312.[18] L.-X. Cen, Z. D. Wang, and S. J. Wang, Phys. Rev. A (2006) 032321.[19] X.-L. Feng, C. Wu, H. Sun, and C.H. Oh, Phys. Rev. Lett. (2009) 200501.[20] J. I. Cirac, A. K. Ekert, S. F. Huelga and C. Macchiavello, Phys. Rev. A (1999) 4249. [21] D. F. V. James, Fortschr. Phys. (2000) 823.[22] Mark Hillery, M. S. Zubairy, Phys. Rev. A (1982) 451.[23] A. D. Boozer, A. Boca, R. Miller, T. E. Northup, and H. J. Kimble, Phys. Rev. Lett. (2006) 083602.[24] A. A. Savchenkov, A. B.Matsko, V. S. Ilchenko, and L.Maleki, Opt. Express15