Int J Theor Phys (2012) 51:2307–2311DOI 10.1007/s10773-012-1110-1

Nearly Deterministic Generation of Atomic EntangledState with Weak Cross-Kerr Nonlinearities

Jian Zhou · Zheng-Yuan Xue

Received: 28 September 2011 / Accepted: 23 February 2012 / Published online: 10 March 2012© Springer Science+Business Media, LLC 2012

Abstract Scheme for generating atomic entangled state is revisited. With the help of quan-tum nondemolition measurements, the present generation scheme can be succeeded in anearly deterministic way. This improvement makes the present scheme more efficient thanthe schemes using nonunitary projective measurements, which are of probabilistic nature.Discussions show that our scheme is feasible within the current experimental technology.

Keywords Quantum optics · Generation · Cross-Kerr nonlinearity · Nearly deterministic

Entanglement was firstly introduced by Einstein, Podolsky, and Rosen and then was con-firmed by Bell type experiment: the quantum correlations between entangled states violatingBell inequalities. Entanglement plays an central role in quantum information and quantumcomputation because of its intrinsic nonlocality [1]. The entanglement properties of quan-tum systems are presently attracting great attention in quantum information theory, such asquantum teleportation [2], quantum key distribution [3], quantum dense coding [4], etc. Butquantum entanglement is generally sensitive to practical noise and technical imperfections,which will lead to the infamous decoherence effect. So many schemes with inherent robust-ness to diverse sources of noise have been proposed, such as the well known schemes forentangling atoms [5, 6], macroscopic atomic ensembles [7, 8], as well as for extensive ex-perimental study [9, 10]. However, many schemes to generate entanglement have utilizedvarious tools, such as coherent control, feedback and imperfect measurements, which makethe generation to be probabilistic.

Recently, cross-Kerr nonlinearity was used to implement two-qubit CNOT gate [11] andBell-state analysis (BSA) [12, 13]. The idea of using weak cross-Kerr nonlinearities com-bined with strong coherent fields has been developed by several different authors and applied

J. ZhouAnhui Xinhua University, Hefei, 230088, China

J. Zhou · Z.-Y. Xue (�)Laboratory of Quantum Information Technology, and School of Physics and TelecommunicationEngineering, South China Normal University, Guangzhou, 510006, Chinae-mail: xuezhengyuan@yahoo.com.cn

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Fig. 1 Level configuration of theatoms used in our scheme. Theground state |g〉 can be excited tothe unstable excited state |e〉 byabsorbing a single photon or laserpulse, then it will decay to thedegenerate states (|m−〉 or |m+〉)with a scattered polarized photon(|H 〉 or |V 〉) respectively

in various ways [14–17]. Recently, schemes for implementing gates, entanglement purifi-cation and concentration [18–22] and continuous-variable quantum teleportation [23] viaquantum nondemolition (QND) measurements were also proposed. In this paper, we pro-posed a nearly deterministic quantum generation scheme for atomic state with QND tech-nology. The cross-Kerr nonlinearity based QND measurements, act as an entangler as wellas the BSA, make one free of the CNOT gate and BSA in realizing a generation protocol. Wecan get a maximally entangled atomic state corresponding to every possible measurementresult, that is, the scheme now can be succeeded in a nearly deterministic way, instead ofthe previous probabilistic way. These distinct features make our scheme more efficient andfeasible, and the physical realization of our scheme is also discussed.

Before presenting our generation protocol, we first review the process of QND measure-ments for the photon number state using cross-Kerr nonlinearity. The cross-Kerr nonlinear-ities can be described with the Hamiltonian

H = �χns np, (1)

where ns(np) denotes the number operator for the mode signal (probe), and χ is the couplingstrength of the nonlinearity, which is determined by the property of the used material. If thesignal state is |ψ〉 = c0|0〉s + c1|1〉s + c2|2〉s while the probe beam initially in a coherentstate of |α〉p , then the cross-Kerr interaction drives the combined signal-probe system toevolve as

U |ψ〉s |α〉p = eiHt/�(c0|0〉 + c1|1〉 + c2|2〉)|α〉p

= c0|0〉|α〉p + c1|1〉∣∣αeiθ⟩p

+ c2|2〉∣∣αei2θ⟩p, (2)

where the Fock state |n〉s is unaffected by the interaction while the coherent state |α〉p picksup a phase shift θ = χt conditioned by the |n〉s states. Generally, this phase factor is pro-portional to the number of photons n in the |n〉s state, i.e., for n photons in the signal mode,the probe beam evolves to |αeiϕ〉p with ϕ = nθ . Because the large nonlinearity θ = π istechnically difficult [24, 25], so we here utilize the weak nonlinearity of the Kerr mediumonly.

Now, let us pay attention to the generation process of atomic entangled state. The levelconfiguration of the alkali atoms utilized in our scheme is depicted in Fig. 1, where |m+〉 and|m−〉 are two degenerate metastable states of the atom and |e〉 (|g〉) is the excited (ground)state.

The atom is initially prepared in the state |g〉, which can be excited to the unstableexcited state |e〉 by absorbing a single photon, then it will decay to the degenerate states(|m−〉 and |m+〉) with a scattered polarized photon ( denoted by |H 〉 or |V 〉 here). With the

Int J Theor Phys (2012) 51:2307–2311 2309

Fig. 2 Schematic diagram of thegeneration. After the interactionbetween the photon and atom, theQND measurements willdistinguish superpositions andmixtures of the states |HH 〉 and|V V 〉 from |HV 〉 and |V H 〉using two cross-Kerrnonlinearities and a coherentlaser probe beam |α〉. Themeasurement box meanshomodyne detection or numberresolving detection

above system incorporated into a high-finesse optical cavity, the process can be determinis-tically expressed as [6, 26]

a+|0〉|g〉 −→ a|H 〉|m−〉 + b|V 〉|m+〉, (3)

where a and b are normalization coefficients with |a|2 +|b|2 = 1. Without loss of generality,we can assume that a and b are both real numbers.

The main setup of our generation scheme is shown in the Fig. 2. Being drived by singlephotons, the atoms in the ground state will be exited to the unstable excited state and thendecay to the degenerate states with scattering polarized photons. The polarized photonicqubits will be split separately by polarizing beam splitters (PBS) into spatial modes, thenthe |H 〉1 and the |H 〉2 will interact with the cross-Kerr nonlinearity medium, as depicted inFig. 2.

At this stage, the state of the combined system can be expressed as

|ψ〉total = (a1|H 〉1|m−〉1 + b1|V 〉1|m+〉1

)

× (a2|H 〉2|m−〉2 + b2|V 〉2|m+〉2

)|α〉p= [

a1a2|HHm−m−〉 + b1b2|V V m+m+〉]1212

|α〉p+ [

a1b2|HV m−m+〉eiθ + b1a2|V Hm+m−〉e−iθ]

1212|α〉p. (4)

We notice that the odd parity terms (|HV 〉 and |V H 〉) will pick up opposite sign phaseshift θ , while no phase is picked for the even parity terms (|HH 〉 and |V V 〉). This enablesus to distinguish the two different parity states by X homodyne measurement without de-stroying them, that is, the even parity terms can be split nearly deterministically from oddparity terms,

|ψ〉1212 ∼ a1a2|HHm−m−〉 + b1b2|V V m+m+〉, (5a)

|ψ〉′1212 ∼ a1b2|HV m−m+〉 + b1a2|V Hm+m−〉. (5b)

We have used the approximate symbol (∼) in the above equations as there is a small but finiteprobability of error (less than 10−5 for αθ2 � 9) to distinguish the states in (5a) and (5b) fromeach other [11]. Generally speaking, natural Kerr media have extremely small nonlinearitieswith a typical dimensionless magnitude of θ < 10−18 [24, 25]. Recent researchers havemake nonlinearities of magnitude up to 10−2 via the systems of cavity QED, EIT and so on[15, 27–32], but a large Kerr nonlinearity at the single-photon level is almost impossible. So

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we could have θ small but would then require α, the amplitude of the probe beam sufficientlylarge, that is, we can still be possible to distinguish (5a) and (5b) from each other in theregime of weak cross-Kerr nonlinearities (θ � π ).

Then we can measure the photons 1, 2 in the basis:

|+〉 = 1√2

(|H 〉 + |V 〉), (6a)

|−〉 = 1√2

(|H 〉 − |V 〉). (6b)

For (5a), if the measurement results are | + +〉 or | − −〉, the ionic state will be left ina1a2|m−m−〉12 + b1b2|m+m+〉12. If the measurement results are | + −〉 or | − +〉, the ionicstate will be left in a1a2|m−m−〉12 − b1b2|m+m+〉12. Similar to the form above, for (5b), ifthe measurement results are | + +〉 or | − −〉, the ionic state will be left in a1b2|m−m+〉12 +b1a2|m+m−〉12. If the measurement results are | + −〉 or | − +〉, the ionic state will be leftin a1b2|m−m+〉12 − b1a2|m+m−〉12.

Generally a1 = b1, a2 = b2, that is, the probability of every atom decay from |e〉 to |g〉is equal. Therefore, we can get bipartite maximally entangled state from our scheme, anddiffract final states can be changed to each other only by local single-qubit operations. Soit is possible for our scheme to create two-qubit maximal atomic entangled states almostdeterministically via weak cross-Kerr nonlinearities.

In our scheme, the ideal single photon resources, photon-atom interaction, Kerr nonlin-earity and the QND measurement are essential. In the real situation, we can get the idealsingle photon resource by using the parameter down conversion (PDC) process. One of thetwin photons generated from the PDC process can be used as the single photon used in ourcurrent scheme, and the redundant photons can be thrown away by the corresponding mea-surement. However, take the intensity and directionality into account, there is really difficultin experiment driving by single photon. The idea interaction of photon-atom can be realizedby trapping atoms in the cavities and driving adiabatically through a classical laser pulsewith the corresponding Rabi frequency denoted by �(t) [6, 33]. Based on this, our genera-tion scheme can be realized near deterministically and far more efficient than the previous’in experiment via the QND measurement. Generally, it was thought that strong nonlinear-ities are required to do this near deterministically, however, our scheme here is using onlyweak nonlinearities (θ = 3×10−2) provided α can be made sufficiently large (α ∼ 106). Butthe strong probe coherent field with a large amplitude will make the fidelity of the schemedecrease simultaneously because of the decoherence (photon loss) during nonlinear inter-action and transmission in optical fibre. Then an arbitrary strong coherent state associatedwith a displacement D(−α) was required to performed on the coherent state and the QNDphoton-number-resolving detection [34, 35], which can make the decoherence arbitrarilysmall. In addition, the entanglement of the degraded entangled states after transmission canbe enhanced by entanglement purification [21, 22, 36]. Alternatively, we also can distributethe entanglement resources in a fault-tolerant way [37]. So our generation scheme based onthe weak cross-Kerr nonlinearity is robust to the decoherence.

In conclusion, we proposed a nearly deterministic generation protocol in atomic systemusing weak cross-Kerr nonlinearities. In particular, we realized the generation only by linearoptics and photon number QND detectors, which are far fewer physical resources than theprevious schemes. These distinct features make our scheme more efficient and feasible.

Acknowledgements We acknowledge fruitful discussions and hints with Zhuo-liang Cao and Ming Yang.This work is supported by the NSFC (Nos. 11004065 and 10905024), the Program of the Education Depart-ment of Anhui Province (No. KJ2012B075) and the project of Anhui Xinhua University (No. 2011zr005).

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