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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 2 — Jan. 17, 2011
  • pp: 1207–1216
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Deterministic generation of multiparticle entanglement in a coupled cavity-fiber system

Peng-Bo Li and Fu-Li Li  »View Author Affiliations


Optics Express, Vol. 19, Issue 2, pp. 1207-1216 (2011)
http://dx.doi.org/10.1364/OE.19.001207


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Abstract

We develop a one-step scheme for generating multiparticle entangled states between two cold atomic clouds in distant cavities coupled by an optical fiber. We show that, through suitably choosing the intensities and detunings of the fields and precisely tuning the time evolution of the system, multiparticle entanglement between the separated atomic clouds can be engineered deterministically, in which quantum manipulations are insensitive to the states of the cavity and losses of the fiber. The experimental feasibility of this scheme is analyzed based on recent experimental advances in the realization of strong coupling between cold 87Rb clouds and fiber-based cavity. This scheme may open up promising perspectives for implementing quantum communication and networking with coupled cavities connected by optical fibers.

© 2011 Optical Society of America

1. Introduction

Multiparticle entangled states are indeed valuable resources, which not only can be employed to test quantum nonlocality against local hidden variable theory in fundamental physics [1

1. J. S. Bell, “On the einstein-podolsky-rosen paradox,” Phys. 1, 195–200 (1964).

, 2

2. D. M. Greenberger, M.A. Horne, A. Shimony, and A. Zeilinger, “Bell’s theorem without inequalities,” Am. J. Phys. 58, 1131–1143 (1990). [CrossRef]

], but also have practical applications in quantum information processing [3

3. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge2000).

], such as quantum communication and computation. Typical such entangled states are GHZ states [2

2. D. M. Greenberger, M.A. Horne, A. Shimony, and A. Zeilinger, “Bell’s theorem without inequalities,” Am. J. Phys. 58, 1131–1143 (1990). [CrossRef]

], W states [4

4. W. Dur, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62, 062314 (2000). [CrossRef]

] and cluster states [5

5. H. J. Briegel and R. Raussendorf, “Persistent entanglement in arrays of interacting particles,” Phys. Rev. Lett. 86, 910–913 (2001). [CrossRef] [PubMed]

]. It is known that entangled states become increasingly susceptible to environmental interactions if the number of particles increases. Therefore, an important practical challenge is the design of robust and most importantly decoherence-resistant mechanisms for its generation. Generating multiparticle entangled states has been proposed or even experimentally demonstrated in different physical systems, such as atomic ensembles in free space [6

6. For a review see, K. Hammerer, A. S. Sorensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041–1093 (2010), and references therein. [CrossRef]

], trapped ions [7

7. For a review see, L.-M. Duan and C. Monroe, “Colloquium: Quantum networks with trapped ions,” Rev. Mod. Phys. 82, 1209–1224 (2010), and references therein. [CrossRef]

, 8

8. R. Blatt and D. Wineland, “Entangled states of trapped atomic ions,” Nature (London)453, 1008–1015 (2008). [CrossRef]

], cold atoms in optical lattice [9

9. For a review see, D. Jaksch and P. Zoller, “The cold atom Hubbard toolbox,” Ann. Phys. 315, 52–79 (2005), and references therein. [CrossRef]

], and cavity QED [10

10. H. J. Kimble, “Strong interactions of single atoms and photons in cavity QED,” Phys. Scr. T76, 127–137 (1998). [CrossRef]

, 11

11. H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in Context,” Science 298, 1372–1377 (2002). [CrossRef] [PubMed]

]. Among various excellent systems, cavity QED [10

10. H. J. Kimble, “Strong interactions of single atoms and photons in cavity QED,” Phys. Scr. T76, 127–137 (1998). [CrossRef]

, 11

11. H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in Context,” Science 298, 1372–1377 (2002). [CrossRef] [PubMed]

] offers one of the most promising and qualified candidates for quantum state engineering and quantum information processing [12

12. J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. 78, 3221–3224 (1997). [CrossRef]

18

18. S. Kang, Y. Choi, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, “Continuous control of the coupling constant in an atom-cavity system by using elliptic polarization and magnetic sublevels,” Opt. Express. 18, 9286–9302 (2010). [CrossRef] [PubMed]

], particularly, for applications in quantum networking [19

19. H. J. Kimble, “The quantum internet,” Nature (London) 453, 1023–1030 (2008). [CrossRef]

], quantum communication, and distributed quantum computation, since atoms trapped in optical cavities are the natural candidates for quantum nodes, and these nodes can be connected by quantum channels such as optical fibers [12

12. J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. 78, 3221–3224 (1997). [CrossRef]

, 19

19. H. J. Kimble, “The quantum internet,” Nature (London) 453, 1023–1030 (2008). [CrossRef]

]. Quantum information is generated, processed and stored locally in each node, which is connected by optical fibers, and is transferred between different nodes via photons through the fibers. Recently, with the development of experimental realization of strong coupling between cold atoms and fiber-based cavity [20

20. Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature (London) 450, 272–276 (2007). [CrossRef]

22

22. D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Hansch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” New J. Phys. 12065038 (2010). [CrossRef]

], the coupled cavity-fiber system has been extensively investigated [23

23. T. Pellizzari, “Quantum networking with optical fibres,” Phys. Rev. Lett. 79, 5242–5245 (1997). [CrossRef]

29

29. X.-Y. Lu, P.-J. Song, J.-B. Liu, and X. Yang, “N-qubit W state of spatially separated single molecule magnets,” Opt. Express 17, 14298–14311 (2010). [CrossRef]

]. Since entangled distant atomic clouds are the building blocks for quantum network and quantum communication, it is desirable to develop a one-step scheme for generating multiparticle entangled states between two cold atomic clouds in distant cavities coupled by an optical fiber.

In this work, we describe a method to construct multiparticle entangled states of the form |ψs=12[eiϕ0|0000001|1111112+eiϕ1|1111111|0000002] or |ψa=12[eiϕ0|0000001|0000002+eiϕ1|1111111|1111112] in a coupled cavity-fiber system, where |000 ··· 000〉j and |111···111〉j (N terms) are product states describing N atoms in the jth cavity which are all in the (same) internal state |0〉 or |1〉. This method can be implemented in one step and in a deterministic fashion. Through suitably choosing the intensities and detunings of the fields and precisely controlling the dynamics of the system, the target entangled states can be engineered, which are immune to the spontaneous emission of the atoms and losses of the fiber, and independent of the states of the cavities. As an application, we also discuss how to use the produced entangled atomic state |ψs〉 to generate the so called NOON state of the cavity modes [30

30. K. T. Kapale and J. P. Dowling, “Bootstrapping approach for generating maximally path-entangled photon states,” Phys. Rev. Lett. 99, 053602 (2007). [CrossRef] [PubMed]

], i.e., |N00N=(|N1|02+|01|N2)/2, where |nj(j = 1, 2) is Fock state for the respective cavity mode. This state is important for quantum lithography and Heisenberg-limited interferometry with photons. We should emphasize that the NOON state, which is mode-entangled, is different from the multiparticle atomic entangled states proposed in this scheme. The generated NOON state in itself is a two-mode entangled state, in which the entanglement is between two different cavity modes. However, the generated atomic entangled states are particle-entangled. The experimental feasibility and technical demands of this scheme are analyzed based on recent experimental advances in the strong coupling between cold 87Rb clouds and fiber-based cavity. Implementing the proposal in experiment would be an important step toward quantum communication and networking with atomic clouds in distant cavities connected by optical fibers.

2. Generation of the entangled states

Fig. 1 (a) Experimental setup. Two distant cold atomic gases are trapped in separate cavities connected by an optical fiber. The two cavity modes are coupled to the fiber mode with the coupling strength ν. (b) Atomic level structure with couplings to the cavity mode and driving laser fields.

We now consider the coupling of the cavity modes to the fiber modes. The number of longitudinal modes of the fiber that significantly interact with the corresponding cavity modes is on the order of ̄/2πc, where l is the length of the fiber and ν̄ is the decay rate of the cavity fields into the continuum of the fiber modes. If we consider the short fiber limit ̄/2πc ≤ 1, then only one resonant mode b of the fiber interacts with the cavity modes [24

24. A. Serafini, S. Mancini, and S. Bose, “Distributed quantum computation via optical fibers,” Phys. Rev. Lett. 96, 010503 (2006). [CrossRef] [PubMed]

]. Therefore, in this case the interaction Hamiltonian describing the coupling between the cavity modes and the fiber mode is
c,f=νb(a^1+eiφa^2)+H.c.,
(4)
where ν is the cavity-fiber coupling strength, and φ is the phase due to propagation of the field through the fiber. Define three normal bosonic modes c, c1, c2 by the canonical transformations c=12(a^1eiφa^2), c1=12(a^1+eiφa^2+2b), c2=12(a^1+eiφa^22b) [24

24. A. Serafini, S. Mancini, and S. Bose, “Distributed quantum computation via optical fibers,” Phys. Rev. Lett. 96, 010503 (2006). [CrossRef] [PubMed]

]. In terms the bosonic modes c, c1 and c2, the interaction Hamiltonian ℋc,f is diagonal. We rewrite this Hamiltonian as 0=2νc1c12νc2c2. So the whole Hamiltonian in the interaction picture is ℋ = ℋ0 + ℋ1.

Now let us perform the unitary transformation ei0t, which leads to
=ei0tei0t=j=1,2[βSj++β*Sj]+Λ*2[ei2νtc1+ei2νtc2+2c]S1eiδt+Λ*2[ei2νtc1+ei2νtc22c]S2eiδt+H.c.
(5)
Then we make another unitary transformation ei𝒱0t, with 𝒱0=j=1,2[βSj++β*Sj]. In the new atomic basis |±nj=(|0nj±|1nj)/2, and under the strong driving limit |β| ≫ |δ|, |Λ|, |ν|, we can bring the effective Hamiltonian (5) to a new form under the rotating-wave approximation [14

14. E. Solano, G. S. Agarwal, and H. Walther, “Strong-driving-assisted multipartite entanglement in cavity QED,” Phys. Rev. Lett. 90, 027903 (2003). [CrossRef] [PubMed]

]
eff={Λ*2[ei2νtc1+ei2νtc2+2c]eiδt+Λ2[ei2νtc1+ei2νtc2+2c]eiδt}×n=1N12(|+n1+||n1|)+{Λ*2[ei2νtc1+ei2νtc22c]eiδt+Λ2[ei2νtc1+ei2νtc22c]eiδt}n=1N12(|+n2+||n2|).
(6)
If we further assume |ν| ≫ {|δ|, |Λ|}, we can take the rotating-wave approximation and safely neglect the nonresonant modes c1, c2. At present, we can obtain the effective Hamiltonian
eff=[θ/2ceiδt+θ*/2ceiδt][Sx1Sx2]=Θ/2[ceiδt+iθ0+H.c.][Sx1Sx2],
(7)
with θ=2Λ2=Θeiθ0, and Sxj=Sj++Sj. It is worth emphasizing that, as the dominant interacting mode c has no contribution from the fiber mode b, the system gets in this instance insensitive to fiber losses. By using the magnus formula, the evolution operator 𝒰 (t) is found as [33

33. A. Sørensen and K. Mølmer, “Entanglement and quantum computation with ions in thermal motion,” Phys. Rev. A 62, 022311 (2000). [CrossRef]

, 34

34. S. L. Zhu, Z. D. Wang, and P. Zanardi, “Geometric quantum computation and multiqubit entanglement with superconducting qubits inside a cavity,” Phys. Rev. Lett. 94, 100502 (2005). [CrossRef] [PubMed]

]
𝒰(t)=eiγ(t)[Sx1Sx2]2e[α(t)cα*(t)c][Sx1Sx2],
(8)
where γ(t) = −(Θ2/4δ2)(δt − sinδt), and α(t) = (Θ/2δ)(1 − eiδt)e0. If the interaction time τ satisfies δτ = 2, the evolution operator for the interaction Hamiltonian (7) can be expressed as
𝒰(τ)=eiλτ[Sx1Sx2]2,
(9)
where λ = −Θ2/4δ. Note that as this operator has no contribution from the cavity modes, thus in this instance the system gets insensitive to the states of the cavity modes, which allows the cavity modes to be in a thermal state.

At this stage, we illustrate how to produce the entangled state of the cavity modes |N00N〉. It is known that these entangled states are important for quantum lithography and Heisenberg-limited interferometry with photons. Using the generated atomic entangled state |ψs〉, we wish to produce the state |N00N〉. To this aim, we employ the stimulated Raman transitions between the atomic ground states |0〉 and |1〉. After preparing the two atomic clouds in the target entangled state |ψs〉, we switch off the driving laser fields of frequencies ω1, ω2 and the couplings of the cavity modes to the fiber modes. Then we have two entangled atomic clouds trapped in two separated cavities, where the couplings of the atoms to the driving laser field and cavity modes are the type. Starting from the state |ψs|01|02=12[eiπ/4|0000001|1111112+eiπ/4|1111111|0000002]|01|02, we are able to steer the evolved state towards the target state 12[eiπ/4|N1|02+eiπ/4|01|N2]|1111111|1111112 through stimulated Raman transitions |000···000〉|0〉j → |111···111〉|Nj.

3. Technical considerations

In the discussions above, we have assumed that a cloud of cold atoms can be trapped in an optical cavity and prepared in the ground states, and the atom-field coupling strengths are uniform through the atomic cloud. We now analyze these assumptions are reasonable with the state-of-the-art technology in experiment. The preparation of the initial atomic states can be accomplished through the well-developed optical pumping and adiabatic population transfer techniques. For a cloud of cold 87Rb atoms cooled in the |F = 1〉 ground state and trapped inside an optical cavity, this cloud can be prepared in the |F = 2〉 ground state employing either the optical pumping or adiabatic population transfer techniques [36

36. K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. 70, 1003–1025 (1997). [CrossRef]

]. We note that recent experimental advances are achieved with a BEC or cold cloud of 87Rb atoms positioned deterministically anywhere within the cavity and localized entirely within a single antinode of the standing-wave cavity field [20

20. Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature (London) 450, 272–276 (2007). [CrossRef]

]. In the experiment each atom is identically and strongly coupled to the cavity mode, and a controlled tunable coupling rate has been realized. For a certain lattice site, a well-defined and maximal atom-field coupling could be achieved. Define the average atom-field coupling strength g¯=ρ(r)N|g(r)|2dr, where ρ(r) is the atomic density distribution, g(r)=g0cos(kcz)exp[r2/w2] is the position-dependent single-atom coupling strength (here z and r are respectively the longitudinal and transverse atomic coordinates, and w and kc are respectively the mode radius and wave vector). For a Gaussian cloud centered on a single lattice site with N < 103, in which the distribution can be considered point-like, an average atom-field coupling strength /2π ≃ 200 MHz can be obtained [20

20. Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature (London) 450, 272–276 (2007). [CrossRef]

]. The homogeneous interaction condition requires that the variation of the coupling strength in a single lattice must be very small, i.e., δg/ ≪ 1, or kcδz ≪ 1 and δrw. Therefore, under these conditions all the atoms in the cavity have the nearly same coupling strength and then collectively interact with the cavity field. On the other hand, to avoid the direct interaction between the atoms being in the ground state, the mean atom-atom distance should be larger than the radius of the atom in the ground state. For a combined trap with the shape of flat disk employed in the experiment, the mean atom-atom distance may be estimated as d=π(δr)2/N. Consider an alkali-metal atom with the valence electron in ns state. The orbit radius of the valence electron is approximately as rgn2a0, where a0 is the Bohr radius. This imposes a condition on the atomic number, i.e., N<π(δr)2/n4a02. For n = 5, δr ≃ 2μm, when N < 103, we have d ≫ rg, which implies the single-particle approximation or no-direct interaction condition is valid.

We consider the traveling-plane-wave driving laser fields with electric fields E⃗t = ℰ⃗iei[k⃗ir⃗ωit](i = 0, 1, 2, 3). Then the actual coupling coefficients between the atoms and the laser fields have the spatial phases eik⃗ir⃗n for the individual atom with the position coordinate r⃗n. Therefore, the effective Hamiltonian (2) should be in a more general form as
1=j=1,2{n=1N[Ω1Ω2Δ1ei(k1k2)rnj|1nj0nj|+Ω0g0*Δ0eik3rnja^j|1nj0nj|eiδt]}+H.c.
(14)
After performing a time-independent unitary transformation U=exp{12j=1,2n=1N[i(k1k2)rnj(|1nj1nj||0nj0nj)]} [37

37. I. E. Linington and N. V. Vitanov, “Decoherence-free preparation of Dicke states of trapped ions by collective stimulated Raman adiabatic passage,” Phys. Rev. A 77, 062327 (2008). [CrossRef]

], we can bring the Hamiltonian (14) to the form
1=j=1,2{n=1N[Ω1Ω2*Δ1|1nj0nj|+Ω0g0*Δ0ei(k3k1+k2)rnja^j|1nj0nj|eiδt]}+H.c.
(15)
If the relative orientation of the three driving laser fields are adjusted in such a way that k⃗3k⃗1 + k⃗2 = 0, then the spatial phase terms will not appear in the effective Hamiltonian. Thus our discussion in the above section still holds. The only difference is that the state vector for each atom should be redefined as |0nei/2(k1k2)rn|0n and |1nei/2(k1k2)rn|1n [37

37. I. E. Linington and N. V. Vitanov, “Decoherence-free preparation of Dicke states of trapped ions by collective stimulated Raman adiabatic passage,” Phys. Rev. A 77, 062327 (2008). [CrossRef]

]. On the other hand, if the atoms in the trap move around a mean position, i.e., r⃗n + δrn, and the deviation from the mean position is much smaller than the wavelength of the laser—Lamb-Dicke regime, i.e., δ rnλL, then the spatial phase terms are global and can be absorbed into the complex Rabi frequencies.

Now let us verify the model and study the performance of this protocol under realistic circumstances through numerical simulations. In a realistic experiment, the effect of the spontaneous emission of the atoms and cavity and fiber losses on the produced entangled atomic states should be taken into account. We choose the parameters as Δ0 = 100|g0|, Δ1 = − 100|g0|, |Ω0| = |g0|, |Ω1| = 10|g0|, |Ω2| = 10|g0|, ν = 0.1|g0|, δ = 0.01|g0|. The probability for the Raman transition |0〉 ↔ |1〉 induced by the classical field and the normal modes c1, c2 is on the order of 𝒫f ∼ |Ω0g0|2/(Δ0ν)2. For fiber loss at a rate κf, we get the effective loss rate Γf = κf0g0|2/(Δ0ν)2 ∼ 0.01 κf. The occupation of the excited state |e〉 can be estimated to be 𝒫e ∼ |Ω1|2/|Δ1|2. Spontaneous emission from the excited state at a rate γe thus leads to the effective decay rate Γe = γe1|2/|Δ1|2 ∼ 0.01γe. For this proposal, the probability for cavity excitation can be estimated as 𝒫c|θ|2/4δ2|Ω0g0|2/(8δ2Δ02). For cavity decay of photons at a rate κc, this leads to an effective decay rate Γc=κc|Ω0g0|2/(8δ2Δ02)0.1κc. Therefore, The main decoherence effects in our scheme is due to cavity decay. Coherent interaction thus requires the preparing time τ{Γc1,Γe1,Γf1}. In the following, we present some numerical results to show how the cavity decay affects the performance of this scheme. The evolution of the system is governed by the following master equation
dρdt=i[,ρ]+κcj=12[a^j]ρ
(16)
where the superoperator ℒ[ô] = 2ôρôôôρρôô. To solve the master equation numerically, we have used the Monte Carlo wave function formalism from the quantum trajectory method [38

38. R. Schack and T. A. Brun, “A C++ library using quantum trajectories to solve quantum master equations,” Comput. Phys. Commun. 102, 210–228 (1997). [CrossRef]

]. All the simulations are performed under one trajectory with the atomic number amounting to ten. (To perform Monte Carlo simulations for the case of much larger number of atoms is time-consuming, which is even unable to be accomplished under present calculating conditions.) In Fig. 2 we display the time evolution of the population and coherences of joint atomic ground states |000···000〉 and |111···111〉, as well as the fidelity F = 〉ψf|ρa|ψf〉, where ρa is the final reduced density matrix of the atoms, under different values for the parameter κc. The atomic system starts from the joint ground state |000···000〉 in each cavity. Figures 2(a) and (b) show the calculated populations, coherences, and fidelity for the case of two atoms trapped in each cavity. We can see that (Fig. 2(a)) at the time τ = 2π/δ the state (13) is obtained with a fidelity higher than 99% in the relatively strong coupling regime. However, from Fig. 2(c) and (d), we find that, for the larger number of atoms even the same cavity decay rate leads to more pronounced degradation of the generated state. It seems that for large number of atoms the produced entangled states are much easier to be spoiled by losses, i.e., off-diagonal elements of the atomic density matrix may decay more rapidly. Therefore, for a cloud containing a few hundred of atoms, to generate the target entangled states with a high fidelity may demand more stringent conditions.

Fig. 2 Time evolution of the population and coherences of joint atomic ground states |000···000〉 and |111···111〉, as well as the fidelity. The first full curve (counted from above at δτ < 2) is the population of the joint ground state 000 |···000〉, the second one is the fidelity, the third one is population of the joint ground state |111···111〉, the last two curves are the imaginary and real part of the off-diagonal elements of the atomic density matrix, respectively. Results are displayed for different atomic numbers and cavity decay rates: (a) N = 2, κc = 0.1g0; (b) N = 2, κc = 0.5g0; (c) N = 5, κc = 0.1g0; (d) N = 5,κc = 0.5g0.

For experimental implementation of this scheme with the fiber-connected coupled cavity system, the recent experimental setup of integrated fiber-based cavity system [20

20. Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature (London) 450, 272–276 (2007). [CrossRef]

22

22. D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Hansch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” New J. Phys. 12065038 (2010). [CrossRef]

] is particularly suitable. The core of this kind of cavity design is a concave, ultralow-roughness mirror surface fabricated directly on the end face of an optical fiber, or two closely spaced fiber tips placed face to face. Light couples directly in and out of the resonator through the optical fiber, which can be either single mode or multi-mode. Atoms or other emitters can be transported into the cavity using optical lattices or other traps. For realization of this protocol, the best candidate for atomic system is 87Rb, with the ground states |0〉 and |1〉 corresponding to the |5S1/2〉 hyperfine levels, and the excited state |e〉 corresponding to the |5P1/2〉 sub-states. It is noted that the strong coupling of cold 87Rb clouds with the fiber-based cavity has been realized in recent experiments [20

20. Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature (London) 450, 272–276 (2007). [CrossRef]

22

22. D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Hansch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” New J. Phys. 12065038 (2010). [CrossRef]

]. The coupling strength between cavity mode and cold atoms ranges from |g0|/2π = 10.6MHZ to |g0|/2π = 215MHz. We choose the cavity QED parameters as those in Ref. [20

20. Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature (London) 450, 272–276 (2007). [CrossRef]

], |g0|/2π = 215MHz, κ/2π = 53MHz, γ/2π = 3MHz. Other experimental parameters can be chosen as Δ0/2π = 20GHz, Δ1/2π = −20GHz,Δ3/2π = 40GHz, |Ω0|/2π = 200MHz,|Ω1|/2π = 2GHz, |Ω2|/2π = 2GHz, |Ω3|/2π = 3GHz,ν/2π = 20MHz, δ/2π = 2MHz. The fiber length can be chosen as L ≲ 1m in most realistic experimental situations. With the chosen parameters, for a few hundred of cold atoms trapped in the cavity, the time to prepare the entangled state τ ∼ 0.5μs.

4. Conclusions

To conclude, we have introduced an efficient scheme for producing multiparticle entangled states between distant atomic clouds in two cavities coupled by an optical fiber. This proposal can be implemented in just one step, and is robust against the atomic spontaneous emission and fiber losses. We have discussed the experimental feasibility of this proposal based on recent experimental advances in the strong coupling between cold 87Rb clouds and fiber-based cavity. We have also discussed how to use the generated multiatom states to produce the NOON state for the two cavities. Experimental implementation of this proposal may offer a promising platform for implementing long-distance quantum communications with atomic clouds trapped in separated cavities connected by optical fibers.

Acknowledgments

This work is supported by the National Key Project of Basic Research Development under Grant No. 2010CB923102, and National Nature Science Foundation of China under Grant No. 60778021. P.-B. L. acknowledges the support from the New Staff Research Support Plan of Xi’an Jiaotong University under No.08141015 and the useful discussions with H.-Y. Li.

References and links

1.

J. S. Bell, “On the einstein-podolsky-rosen paradox,” Phys. 1, 195–200 (1964).

2.

D. M. Greenberger, M.A. Horne, A. Shimony, and A. Zeilinger, “Bell’s theorem without inequalities,” Am. J. Phys. 58, 1131–1143 (1990). [CrossRef]

3.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge2000).

4.

W. Dur, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62, 062314 (2000). [CrossRef]

5.

H. J. Briegel and R. Raussendorf, “Persistent entanglement in arrays of interacting particles,” Phys. Rev. Lett. 86, 910–913 (2001). [CrossRef] [PubMed]

6.

For a review see, K. Hammerer, A. S. Sorensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041–1093 (2010), and references therein. [CrossRef]

7.

For a review see, L.-M. Duan and C. Monroe, “Colloquium: Quantum networks with trapped ions,” Rev. Mod. Phys. 82, 1209–1224 (2010), and references therein. [CrossRef]

8.

R. Blatt and D. Wineland, “Entangled states of trapped atomic ions,” Nature (London)453, 1008–1015 (2008). [CrossRef]

9.

For a review see, D. Jaksch and P. Zoller, “The cold atom Hubbard toolbox,” Ann. Phys. 315, 52–79 (2005), and references therein. [CrossRef]

10.

H. J. Kimble, “Strong interactions of single atoms and photons in cavity QED,” Phys. Scr. T76, 127–137 (1998). [CrossRef]

11.

H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in Context,” Science 298, 1372–1377 (2002). [CrossRef] [PubMed]

12.

J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. 78, 3221–3224 (1997). [CrossRef]

13.

A. D. Boozer, A. Boca, R. Miller, T. E. Northup, and H. J. Kimble, “Reversible state transfer between light and a single trapped atom,” Phys. Rev. Lett. 98, 193601 (2007).

14.

E. Solano, G. S. Agarwal, and H. Walther, “Strong-driving-assisted multipartite entanglement in cavity QED,” Phys. Rev. Lett. 90, 027903 (2003). [CrossRef] [PubMed]

15.

P.-B. Li, Y. Gu, Q.-H. Gong, and G.-C. Guo, “Quantum-information transfer in a coupled resonator waveguide,” Phys. Rev. A 79, 042339 (2009). [CrossRef]

16.

F. Mei, M. Feng, Y.-F. Yu, and Z.-M. Zhang, “Scalable quantum information processing with atomic ensembles and flying photons,” Phys. Rev. A 80, 042319 (2009). [CrossRef]

17.

P.-B. Li, Y. Gu, Q.-H. Gong, and G.-C. Guo, “Generation of two-mode entanglement between separated cavities,” J. Opt. Soc. Am. B 26, 189–193 (2009). [CrossRef]

18.

S. Kang, Y. Choi, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, “Continuous control of the coupling constant in an atom-cavity system by using elliptic polarization and magnetic sublevels,” Opt. Express. 18, 9286–9302 (2010). [CrossRef] [PubMed]

19.

H. J. Kimble, “The quantum internet,” Nature (London) 453, 1023–1030 (2008). [CrossRef]

20.

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature (London) 450, 272–276 (2007). [CrossRef]

21.

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, “Microfabricated high-finesse optical cavity with open access and small volume,” Appl. Phys. Lett. 87, 211106 (2005). [CrossRef]

22.

D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Hansch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” New J. Phys. 12065038 (2010). [CrossRef]

23.

T. Pellizzari, “Quantum networking with optical fibres,” Phys. Rev. Lett. 79, 5242–5245 (1997). [CrossRef]

24.

A. Serafini, S. Mancini, and S. Bose, “Distributed quantum computation via optical fibers,” Phys. Rev. Lett. 96, 010503 (2006). [CrossRef] [PubMed]

25.

Z. Q. Yin and F. L. Li, “Multiatom and resonant interaction scheme for quantum state transfer and logical gates between two remote cavities via an optical fiber,” Phys. Rev. A 75, 012324 (2007). [CrossRef]

26.

P. Peng and F. L. Li, “Entangling two atoms in spatially separated cavities through both photon emission and absorption processes,” Phys. Rev. A 75, 062320 (2007). [CrossRef]

27.

Y. L. Zhou, Y. M. Wang, L. M. Liang, and C. Z. Li, “Quantum state transfer between distant nodes of a quantum network via adiabatic passage,” Phys. Rev. A 79,044304 (2009). [CrossRef]

28.

J. Busch and A. Beige, “Generating single-mode behavior in fiber-coupled optical cavities,” arXiv:1009.1011v2 (2010).

29.

X.-Y. Lu, P.-J. Song, J.-B. Liu, and X. Yang, “N-qubit W state of spatially separated single molecule magnets,” Opt. Express 17, 14298–14311 (2010). [CrossRef]

30.

K. T. Kapale and J. P. Dowling, “Bootstrapping approach for generating maximally path-entangled photon states,” Phys. Rev. Lett. 99, 053602 (2007). [CrossRef] [PubMed]

31.

T. Brandes, “Coherent and collective quantum optical effects in mesoscopic systems,” Phys. Rep. 408, 315–474 (2005). [CrossRef]

32.

D. F. V. James, “Quantum computation with hot and cold ions: an assessment of proposed schemes,” Fortschr. Phys. 48, 823–837 (2000). [CrossRef]

33.

A. Sørensen and K. Mølmer, “Entanglement and quantum computation with ions in thermal motion,” Phys. Rev. A 62, 022311 (2000). [CrossRef]

34.

S. L. Zhu, Z. D. Wang, and P. Zanardi, “Geometric quantum computation and multiqubit entanglement with superconducting qubits inside a cavity,” Phys. Rev. Lett. 94, 100502 (2005). [CrossRef] [PubMed]

35.

K. Mølmer and A. Sørensen, “Multiparticle entanglement of hot trapped ions,” Phys. Rev. Lett. 82, 1835–1838 (1999). [CrossRef]

36.

K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. 70, 1003–1025 (1997). [CrossRef]

37.

I. E. Linington and N. V. Vitanov, “Decoherence-free preparation of Dicke states of trapped ions by collective stimulated Raman adiabatic passage,” Phys. Rev. A 77, 062327 (2008). [CrossRef]

38.

R. Schack and T. A. Brun, “A C++ library using quantum trajectories to solve quantum master equations,” Comput. Phys. Commun. 102, 210–228 (1997). [CrossRef]

OCIS Codes
(270.5580) Quantum optics : Quantum electrodynamics
(270.5585) Quantum optics : Quantum information and processing

ToC Category:
Quantum Optics

History
Original Manuscript: October 26, 2010
Revised Manuscript: November 15, 2010
Manuscript Accepted: November 26, 2010
Published: January 11, 2011

Citation
Peng-Bo Li and Fu-Li Li, "Deterministic generation of multiparticle entanglement in a coupled cavity-fiber system," Opt. Express 19, 1207-1216 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-2-1207


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References

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  2. D. M. Greenberger, M. A. Horne, A. Shimony, and A. Zeilinger, “Bell’s theorem without inequalities,” Am. J. Phys. 58, 1131–1143 (1990). [CrossRef]
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  6. For a review see,K. Hammerer, A. S. Sorensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041–1093 (2010) (and references therein). [CrossRef]
  7. For a review see, L.-M. Duan, and C. Monroe, “Colloquium: Quantum networks with trapped ions,” Rev. Mod. Phys. 82, 1209–1224 (2010) (and references therein). [CrossRef]
  8. R. Blatt and D. Wineland, “Entangled states of trapped atomic ions,” Nature 453, 1008–1015 (2008). [CrossRef]
  9. For a review see, D. Jaksch, and P. Zoller, “The cold atom Hubbard toolbox,” Ann. Phys. 315, 52–79 (2005) (and references therein). [CrossRef]
  10. H. J. Kimble, “Strong interactions of single atoms and photons in cavity QED,” Phys. Scr. T 76, 127–137 (1998). [CrossRef]
  11. H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in Context,” Science 298, 1372–1377 (2002). [CrossRef] [PubMed]
  12. J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. 78, 3221–3224 (1997). [CrossRef]
  13. A. D. Boozer, A. Boca, R. Miller, T. E. Northup, and H. J. Kimble, “Reversible state transfer between light and a single trapped atom,” Phys. Rev. Lett. 98, 193601 (2007).
  14. E. Solano, G. S. Agarwal, and H. Walther, “Strong-driving-assisted multipartite entanglement in cavity QED,” Phys. Rev. Lett. 90, 027903 (2003). [CrossRef] [PubMed]
  15. P.-B. Li, Y. Gu, Q.-H. Gong, and G.-C. Guo, “Quantum-information transfer in a coupled resonator waveguide,” Phys. Rev. A 79, 042339 (2009). [CrossRef]
  16. F. Mei, M. Feng, Y.-F. Yu, and Z.-M. Zhang, “Scalable quantum information processing with atomic ensembles and flying photons,” Phys. Rev. A 80, 042319 (2009). [CrossRef]
  17. P.-B. Li, Y. Gu, Q.-H. Gong, and G.-C. Guo, “Generation of two-mode entanglement between separated cavities,” J. Opt. Soc. Am. B 26, 189–193 (2009). [CrossRef]
  18. S. Kang, Y. Choi, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, “Continuous control of the coupling constant in an atom-cavity system by using elliptic polarization and magnetic sublevels,” Opt. Express 18, 9286–9302 (2010). [CrossRef] [PubMed]
  19. H. J. Kimble, “The quantum internet,” Nature 453, 1023–1030 (2008). [CrossRef]
  20. Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature 450, 272–276 (2007). [CrossRef]
  21. M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, “Microfabricated high-finesse optical cavity with open access and small volume,” Appl. Phys. Lett. 87, 211106 (2005). [CrossRef]
  22. D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Hansch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” N. J. Phys. 12, 065038 (2010). [CrossRef]
  23. T. Pellizzari, “Quantum networking with optical fibres,” Phys. Rev. Lett. 79, 5242–5245 (1997). [CrossRef]
  24. A. Serafini, S. Mancini, and S. Bose, “Distributed quantum computation via optical fibers,” Phys. Rev. Lett. 96, 010503 (2006). [CrossRef] [PubMed]
  25. Z. Q. Yin and F. L. Li, “Multiatom and resonant interaction scheme for quantum state transfer and logical gates between two remote cavities via an optical fiber,” Phys. Rev. A 75, 012324 (2007). [CrossRef]
  26. P. Peng and F. L. Li, “Entangling two atoms in spatially separated cavities through both photon emission and absorption processes,” Phys. Rev. A 75, 062320 (2007). [CrossRef]
  27. Y. L. Zhou, Y. M. Wang, L. M. Liang, and C. Z. Li, “Quantum state transfer between distant nodes of a quantum network via adiabatic passage,” Phys. Rev. A 79, 044304 (2009). [CrossRef]
  28. J. Busch and A. Beige, “Generating single-mode behavior in fiber-coupled optical cavities,” arXiv:1009.1011v2 (2010).
  29. X.-Y. Lu, P.-J. Song, J.-B. Liu, and X. Yang, “N-qubit W state of spatially separated single molecule magnets,” Opt. Express 17, 14298–14311 (2010). [CrossRef]
  30. K. T. Kapale and J. P. Dowling, “Bootstrapping approach for generating maximally path-entangled photon states,” Phys. Rev. Lett. 99, 053602 (2007). [CrossRef] [PubMed]
  31. T. Brandes, “Coherent and collective quantum optical effects in mesoscopic systems,” Phys. Rep. 408, 315–474 (2005). [CrossRef]
  32. D. F. V. James, “Quantum computation with hot and cold ions: an assessment of proposed schemes,” Fortschr. Phys. 48, 823–837 (2000). [CrossRef]
  33. A. Sørensen and K. Mølmer, “Entanglement and quantum computation with ions in thermal motion,” Phys. Rev. A 62, 022311 (2000). [CrossRef]
  34. S. L. Zhu, Z. D. Wang, and P. Zanardi, “Geometric quantum computation and multiqubit entanglement with superconducting qubits inside a cavity,” Phys. Rev. Lett. 94, 100502 (2005). [CrossRef] [PubMed]
  35. K. Mølmer and A. Sørensen, “Multiparticle entanglement of hot trapped ions,” Phys. Rev. Lett. 82, 1835–1838 (1999). [CrossRef]
  36. K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. 70, 1003–1025 (1997). [CrossRef]
  37. I. E. Linington and N. V. Vitanov, “Decoherence-free preparation of Dicke states of trapped ions by collective stimulated Raman adiabatic passage,” Phys. Rev. A 77, 062327 (2008). [CrossRef]
  38. R. Schack and T. A. Brun, “A C++ library using quantum trajectories to solve quantum master equations,” Comput. Phys. Commun. 102, 210–228 (1997). [CrossRef]

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