Abstract
Quantum entanglement plays an essential role in quantum information processing and quantum networks. One of the commonlyused methods to generate multiple entangled fields is to employ polarizing beam splitters. However, nonclassical input light fields are required and the generated entangled fields are always degenerate in such case. Here, we present a proofofprinciple demonstration of an efficient and convenient way to entangle multiple light fields via electromagnetically induced transparency (EIT) in an atomic ensemble. The atomic spin wave, produced through EIT in the Λtype atomic system, can be described by a Bose operator and can act as an entangler. With such an entangler, any desired number of nondegenerate narrowband continuousvariable entangled fields, in principle, can be generated through stimulated Raman scattering processes. This scheme holds great promise for applications in scalable quantum communication and quantum networks. PACS: 42.50.Gy, 03.67.Bg, 42.50.Dv, 42.65.Lm.
Introduction
Quantum state exchange between light and matter is a basic component for quantum interface in quantum information processing. As is well known, light is the best longdistance quantum information carrier and the atomic ensembles can provide the promising tools for quantum information manipulation and storage. In facilitating quantum information processing and quantum networks, generations of lightlight, atomatom, and atomlight multipartite entanglements play essential roles in the implementations of quantum information protocols^{1,2,3}. So far, the majority studies on entanglement have dealt with the generations of multiple entangled light fields. Apart from the conventional way of generating multipartite entanglement by mixing squeezed fields created through parametric downconversion processes in nonlinear optical crystals with linear optical elements, i.e., polarizing beam splitters (PBS)^{4,5,6,7} as entanglers, the atomic ensembles provide an alternative avenue to the generation of multientangled fields due to the virtue of narrow bandwidth, nondegenerate frequencies, and long correlation time^{8,9,10,11,12,13,14}. Based on the seminal proposal of Duan et al.^{8}, the electromagnetically induced transparency^{15,16,17} (EIT)based doubleΛtype atomic system has been actively implemented for efficiently creating nondegenerate entangled twin fields through either nondegenerate fourwave mixing (FWM) or Raman scattering processes^{9,10,11,12}. The multicolor multipartite continuousvariable (CV) entanglement has also been achieved by using the multiorder coherent Raman scattering^{18} or multiple nondegenerate FWM processes^{13,14}. Moreover, generations of entanglements between an atomic ensemble and light fields, as well as between two atomic ensembles, have also been realized^{8,19,20,21,22}, which are vital for storage and processing of quantum information.
In this study, we propose an efficient and convenient scheme for quantum entangler via EIT in an atomic ensemble. The atomic spin wave, which can be described by a Bose operator and acts as the entangler, is produced through EIT in the Λtype atomic configuration. Through stimulated Raman scattering processes, nondegenerate narrowband multientangled fields up to any desired number can, in principle, be achieved via such an entangler. This proposed entangler is quite distinct as compared to the conventional PBS entangler^{23,24,25}, since only coherent input light fields are needed for generating multipartite entanglement in the present scheme, whereas nonclassical input light fields are required when using the PBS entangler. Moreover, the present proposal is different from the previously proposed ones as given in Refs. 13, 14, 18. Here, the entanglement features between the atomic ensemble and the generated fields are investigated, and it is shown that under the EIT condition the generated atomic spin wave can serve as an entangler. Also, this proposal is different from the one proposed in Ref. 8, since, in principle, any desired number of entangled fields can be produced through this entangler.
Results
Mechanism of excitations
The considered model is shown in Fig. 1a, where energy levels 1〉, 2〉, and 3〉, forming the threelevel Λtype system, correspond, respectively, to the groundstate hyperfine levels 5S_{1/2} (F = 2), 5S_{1/2} (F = 3), and the excited state 5P_{1/2} in D_{1} line of the ^{85}Rb atom with the groundstate hyperfine splitting of 3.036 GHz. A strong coupling field E_{c} (with frequency ω_{c} and Rabi frequency Ω_{c}) and a relatively weak probe field E_{p} (with frequency ω_{p} and Rabi frequency Ω_{p}) are tuned to be on resonance with the transitions 2〉–3〉 and 1〉–3〉, respectively. By applying a third mixing field E_{m1} (with frequency ω_{m}_{1} and Rabi frequency Ω_{m}_{1}), offresonantly coupled to levels 1〉 and 3〉, a Stokes field E_{1} can be created through the nondegenerate FWM process^{13,14}. In fact, the produced Stokes field can be equivalently considered as a result of scattering the mixing field E_{m1} off the atomic spin wave (S) preestablished by the coupling and probe fields in the Λtype EIT configuration formed by levels 1〉, 2〉, and 3〉, as shown in Fig. 1b, where the induced atomic spin wave acts as a frequency converter with frequency equal to the separation between the two lower states^{26,27}. Subsequently, if more laser fields E_{m2}, E_{m3}, … E_{mN} (with N being a positive integer) are applied to offresonantly couple levels 1〉 and 3〉, more Stokes fields E_{2}, E_{3}, … E_{N} can be produced through scattering the mixing fields off the same atomic spin wave. In what follows, we will investigate the entanglement features between the preestablished atomic spin wave and the generated Stokes fields.
In this EITbased configuration, we assume that the Rabi frequency of the scattering field is far smaller than its frequency detuning, and the established atomic spin wave due to the onresonant EIT is strong enough to ensure that different scattering fields have little influence on it. We also assume that the probe field is relatively weak as compared to the strong coupling field. Under these conditions, the collective atomic spin field S ( with N_{a} being the total number of atoms in the interaction volume) can be viewed as approximately satisfying the bosonic commutation relation , so the produced atomic spin wave S can be treated as a bosonic field and the quantum state of the atomic ensemble can be transferred to that of the generated Stokes fields^{27}.
Generation of bipartite entanglement
We first investigate the bipartite entanglement between the atomic spin wave S and the generated Stokes field E_{1}. As shown in Fig. 1b, we assume that the generated Stokes field is very weak as compared to the scattering field, thus, the scattering field is treated classically, whereas the Stokes field E_{1} and the atomic spin field S are treated quantum mechanically. After adiabatic elimination of the upper excited state, the effective Hamiltonian of this system in the interaction picture can be written as^{27} where a_{1} is the bosonic annihilation operator for the Stokes field E_{1}, and with Δ_{1} = ω_{m1} − ω_{31} being the detuning of the mixing field E_{m1} from the resonant transition 1〉–3〉, and g_{23} being the coupling coefficient between the Stokes field and the respective atomic states. According to the realistic experimental parameters^{13,14}, k_{1} is in the order of several cm^{−1}; in such case, the estimated timescale for k_{1}t = 1 corresponds to about tens of picoseconds. By solving the Heisenberg equations of motion for the operators S(t) and a_{1}(t), one can get the solutions for the two operators as functions of their initial values: where a = cos^{2}θ − sin^{2}θ with tgθ = Ω_{p}/Ω_{c}. As seen in Fig. 1b, the collective atomic state is initially in a coherent superposition state, and the Stokes field is initially in vacuum, so the initial state of the atomfield system can be written as , where 1〉0_{S}〉 (2〉0_{s}〉) represents an atom in state 1〉 (2〉) and the Stokes field in vacuum. We use the criterion V = (Δu)^{2} + (Δν)^{2} < 4 proposed by Duan et al.^{28} to verify the bipartite entanglement between the atomic spin wave S and the generated Stokes field E_{1}, where u = x_{1} − x_{S} and ν = p_{1} + p_{S} with and . Smaller the correlation V is, stronger the degree of the bipartite entanglement becomes.
Figure 2 displays the interaction time evolutions of V under different ratios of Ω_{p}/Ω_{c} with k_{1} = 1. Obviously, it can be seen in Figs. 2a–2d that bipartite entanglement can be created as soon as when the interaction becomes nonzero; under different ratios of Ω_{p}/Ω_{c}, V, with the initial value of about 4, evolves with interaction time and becomes less than 4, which sufficiently indicates the generation of genuine bipartite entanglement between the Stokes field E_{1} and the atomic spinwave field S. In addition, there exists a range of interaction time that V nearly approaches to zero, which means that the generated Stokes field is perfectly quantum correlated with the atomic spin wave. With an increase of the ratio Ω_{p}/Ω_{c} in a moderate range, the interaction time range for nearzero V values would decrease and the degree of bipartite entanglement becomes weakened.
The stimulated Raman scattering process described above for generating bipartite entanglement between the Stokes field E_{1} and the atomic spinwave S has certain similar features as the parametric downconversion process in a nonlinear crystal. The generated Stokes field E_{1} can be viewed as a result of frequency downconversion process through mixing the scattering field E_{m1} with the atomic spin wave S preestablished by the coupling and probe fields. As the generation of a Stokes photon is always accompanied by an atomic spinwave excitation, the downconverted frequency component (i.e., Stokes field) is strongly quantum correlated with the atomic spin wave; as a result, strong bipartite entanglement between the Stokes field and the atomic spin wave can be established. This idea can be generalized to the case with more laser fields E_{m2}, E_{m3}, … E_{mN} tuned to the vicinity of the transition 1〉–3〉 to mix with the same atomic spin wave, then all of the generated Stokes fields E_{1}, E_{2}, … E_{N} would be entangled with the atomic spin wave, and therefore entangled with each other. In such situation any desired number of entangled light fields, in principle, can be created with the atomic spin wave acting as an entangler. This proposed scheme can be further tested by applying another laser field E_{m2} scattering off the atomic spin wave to generate the tripartite entanglement.
Generation of tripartite entanglement
In the following, we consider the case of generating the Stokes fields E_{1} and E_{2} through scattering the laser fields E_{m1} and E_{m2} off the same atomic spin wave, respectively, as shown in Fig. 1b. The effective Hamiltonian of the system in the interaction picture has the form^{27} where with Δ_{2} = ω_{m2} − ω_{31} being the frequency detuning of the mixing field E_{m2} from the transition 1〉–3〉. In the similar way as calculating the bipartite entanglement generation, we solve the Heisenberg equations of motion for the operators S(t), a_{1}(t), and a_{2}(t), and the solutions for the three operators as functions of their initial values can be expressed as: where . We demonstrate the tripartite entanglements of the generated Stokes fields E_{1}, E_{2}, and atomic spin wave S by employing the criterion proposed by van LockFurusawa (VLF)^{6,7} with inequalities as follows: where and g_{i} (i = 1,2,3) are arbitrary real numbers. In a similar way as that used in Ref. 29, we set , , and . Satisfying any pair of these three inequalities is a sufficient condition of realizing the tripartite entanglement, and also smaller the correlations V_{12}, V_{1s}, and V_{2s} are, higher the degree of tripartite entanglement will be.
Figures 3a–3d depict the evolutions of the VLF correlations as a function of interaction time with k_{1} = 1 and Ω_{p}/Ω_{c} = 1/20 under different k_{2} values. It can be seen that when k_{2} is far smaller than k_{1} (see Fig. 3a), there only exists a very limited range of the interaction time within which the inequalities for V_{12} and V_{2s} are fulfilled, whereas V_{1s} with the minimal value nearly to zero is smaller than 4 in a wide range of the interaction time; this means that strong entanglement between the generated Stokes field E_{1} and the atomic spin wave is obtained, whereas the generated Stokes field E_{2} is weakly entangled with both the Stokes field E_{1} and atomic spin wave S. With an increase of k_{2}, the minimal values of V_{12} and V_{2s} would decrease and the ranges of interaction time with both V_{12} and V_{2s} being smaller than 4 would increase, whereas V_{1s} displays an opposite behavior, that is, the degree of bipartite entanglement between the generated Stokes field E_{1} and atomic spin wave is weakened and that between the Stokes field E_{2} and Stokes field E_{1} (or atomic spin wave) is strengthened. When k_{1} = k_{2} = 1 (see Fig. 3c), V_{12}, approaching to 2 after certain interaction time, is smaller than 4 over the entire interaction time range, and there exists a broad range of interaction time within which the inequalities for V_{12}, V_{1s}, and V_{2s} are all satisfied. In this case, the Stokes fields E_{1} and E_{2} and the atomic spin wave S are CV entangled with each other, and the strongest tripartite entanglement is obtained. Further increasing k_{2} would enhance the degree of bipartite entanglement between the Stokes field E_{2} and atomic spin wave, and degrade the entanglement between the Stokes field E_{1} and Stokes field E_{2} (or atomic spin wave). In principle, this scheme can be generalized to generate any desired number of maximallyentangled fields by using the atomic spin wave as an entangler through applying more laser fields E_{m3}, E_{m4}, … E_{mN} tuned to the vicinity of the transition 1〉–3〉 under the condition of k_{1} = k_{2} = k_{3} … = k_{N}. Detailed calculations show that the VLF correlations V_{12}, V_{1s}, and V_{2s} exhibit weak dependence on the variation of ratio Ω_{p}/Ω_{c}, as long as the probe field is relatively weak as compared to the coupling field.
Discussion
In the above EITbased configuration, the Rabi frequencies of the scattering fields should be small as compared to their frequency detunings so that coupling between different scattering fields can be neglected, and the atomic spin wave should be strong enough to ensure that different scattering fields have little influence on it, which can be realized by using substantially strong probe and coupling fields as compared to the scattering fields. Moreover, the probe field should be relatively weak as compared to the coupling field, so that the produced atomic spin wave S can be treated as a bosonic field. In an atomic medium, there will be realistic imperfections, such as finite coherence times, atomic diffusion, and Doppler broadening. In the above analysis, we did not take into account these effects, so that the main physics in such system can be resolved and better understood. The influences to entanglement due to those realistic imperfections will be analyzed in the future work by using a HeisenbergLangevin approach.
It should be noted that the present scheme is different from the one proposed by Duan et al.^{8}. In the current scheme, the atomic spin wave is created in advance via EIT, which acts as an entangler and provides a way to generate nondegenerate multiple CVentangled fields to any desired number. In addition, in Ref. 8, the atomic spin wave is produced through spontaneous Raman scattering, so the generated Stokes or antiStokes field would emit in all directions and the photon production efficiency is very low, whereas in the present scheme, the generated Stokes field propagates along a particular direction determined by the phasematching condition for FWM^{13,14}, so the photon production efficiency is much higher, which would bring great facility in realistic quantum information processing protocols. Moreover, the present scheme is different from our previous proposal^{14}. In the present EITbased configuration, we assume that the probe field is relatively weak as compared to the strong coupling field, so the produced atomic spin wave S can be treated as a bosonic field and serve as an entangler, and the quantum state of the atomic ensemble can be transferred to that of the generated Stokes field. Also, the generated antiStokes fields are relatively weak as compared to the Stokes fields, so the entanglement properties between the Stokes and antiStokes fields do not need to be taken into account. However, as discussed in Refs. 13, 14, strong coupling and probe fields with nearly equal Rabi frequencies are employed in the previous scheme in order to enhance the production efficiencies of both the Stokes and antiStokes fields and promote entanglement between them.
Note also that, as compared to the routinelyemployed method to produce multifield entanglement by using PBS entanglers^{4,5,6,7}, where the interferometric stabilization of the optical paths is required, and the entangled multifields are degenerate and suffer from short correlation time (~ps), the present scheme can be employed to generate narrowband multicolor entangled CV fields with a long correlation time (~ms or even ~s^{30,31}) by using a single entangler, where the correlation time is determined by the coherence decay time between the two lower atomic states due to the finite interaction time between atoms and light fields. Furthermore, the proposed entangler is different from and superior to the PBS entangler^{23,24,25} in that the generation of multipartite entanglement using PBS entanglers is a linear process, whereas it is a nonlinear one for the present entangler, so as nonclassical input light fields are required for using PBS entanglers to generate multipartite entanglement, only coherent input light fields are needed for using the present entangler to create multipartite entanglement, which will greatly simplify its practical implementation.
In conclusion, we have proposed an efficient and convenient scheme for realizing an entangler from an EITbased atomic ensemble. The entangler has the virtue of being able to generate, in principle, arbitrary number of nondegenerate and narrowband CVentangled fields with a long correlation time, which can satisfy all the essential requirements for constructing a scalable quantum repeater for the realization of longdistance quantum communication. In addition, only coherent input light fields are required for generating multipartite entanglement. Together with the entanglement generations between two atomic ensembles and entanglement swapping^{8,19,20,21,22}, lightlight, atomatom, and atomlight multipartite entanglements can be achieved, which will find promising applications in quantum information processing and quantum networks.
Methods
Calculation of timedependent operators
According to the Heisenberg equation of motion , one can get the solutions for the field operators as functions of their initial values by substituting the effective Hamiltonian H_{I} into the equations of motion. We assume that the atomic spin wave is strong enough, so different scattering fields will have little influence on it. We also assume that the generated Stokes fields are very weak as compared to the corresponding scattering fields, so the approximations of undepleted coupling and probe fields as well as scattering fields can be employed in the above calculations. Similar hyperbolic solutions for generating tripartite entanglement by applying cascaded or concurrent nonlinearities have been examined by Olsen et al.^{29}.
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Acknowledgements
This work is supported by NBRPC (Nos. 2012CB921804 and 2011CBA00205), the National Natural Science Foundation of China (Nos. 11274225, 10974132, 50932003, and 11021403), and Innovation Program of Shanghai Municipal Education Commission (No. 10YZ10). Yang's email is yangxihua@yahoo.com or yangxih@yahoo.com.cn; M. Xiao's email is mxiao@uark.edu.
Author information
Affiliations
Department of Physics, Shanghai University, Shanghai 200444, China
 Xihua Yang
 & Yuanyuan Zhou
National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China
 Min Xiao
Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701, USA
 Min Xiao
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Contributions
X.Y. and M.X. conceived the idea for this project. X.Y. and Y.Z. performed the calculations and prepared figures 1–3. X.Y. and M.X. wrote the main manuscript text. All authors reviewed the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to Xihua Yang or Min Xiao.
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