囚禁离子多体W·类纠缠态的简单制备方案

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摘 要 提出一种囚禁离子多体W类纠缠态的简单制备方案.在该方案中，N个离子被一束红边带激光同时照射，从而可以实现N体对称W态或N+1体反对称W态的一步产生.原则上，粒子数N可以任意大.另外，离子数越大，制备目标纠缠态所需要的时间就越短.

关键词 W态；囚禁离子；一步产生

Quantum entanglement is a consequence of quantumstate superposition principle applied to composite quantum systems. It not only provides possibilities to test the foundations of quantum physics， but also lies at the heart of the growing field of quantum information science[1]. Compared with bipartite entangled states， multipartite entangled states would exhibit clearer nonclassical effects and are more useful for quantum applications[23]. Thus characterizing and preparing multipartite entangled states have been attracting much attention. However， preparation of multipartite entangled states is an experimental challenge， because of the technical difficulty of manipulating individual particles. So far， tripartite GreenbergerHorneZeilinger （GHZ） state in cavity QED[4]， 10photon GHZ state[5]， 6photon cluster state[6]， 3photon W state[7]， and 14ion GHZ state[8] have been observed. Wtype entangled state is a special type of multipartite entangled state[9]， and has widespread applications in quantum information processing （see， e.g.， Refs. [1014]）. Thus， generation of multiparticle Wtype states has been paid much attention.

湖南师范大学自然科学学报 第36卷第3期

汪新文等：囚禁离子多体W类纠缠态的简单制备方案 Not long before， the Innsbruck group has experimentally prepared 40Ca+ionic 8qubit W state[15]， with S1/2 （mj=1/2） acting as the lower level and D5/2 （mj=-1/2） acting as the upper level. In the experiment， they individually derived each of the ions with sequential blue sideband pulses， and created a W state step by step with one motional quantum as memory. In this paper， we propose an alternative scheme to generate multipartite entanglement of the W type in the iontrap system， which can be regarded as an improvement on that of Ref.[15]. In the scheme， an Nion W state can be created in one step by driving simultaneously the ions with one red sideband laser beam. Moreover， the larger the number of ions N is， the less time it will take to prepare the desired state. Thus our scheme is much faster and simpler than that of Ref.[15]， which is very important in view of decoherence.

We consider that N identical ions （for example， 40Ca+ ions） are confined in a linear Paul trap. Each of them has the ground state g〉and the excited state e〉.We simultaneously drive the ions using a red sideband laser beam with frequency of ω0-ν， where ω0 is the frequency of electronic transition e〉g〉 and ν is the frequency of the centerofmass mode of the collective motion of the ions. In the rotatingwave approximation， the Hamiltonian for this system can be described by （let = 1）

H=H0+Hi，

H0=νa+a+12ω0∑Nj=1σZj，

Hi=Ω2e-iφ∑Nj=1σ+jei[η（a++a）-（ω0-ν）]+H.c.，

（1）

where a+（a） denotes the creation （annihilation） operator for the centerofmass mode of the motion of the ions， σZj=ej〉〈ej-gj〉〈gj is the conventional Pauli operator， σ+j=ej〉〈gj and σ-j = gj〉〈ej are the spin flip operators， η is the LambDicke parameter， Ω and φ are the Rabi frequency and the phase of the laser field. In the interaction picture， the Hamiltonian reads

HI=Ω2e-iφ-η2/2∑Nj=1σ+j∑∞l=0（iη）2l+1l！（l+1）！（a+）lal+1+H.c.，

（2）

In the LambDicke regime， i.e.， η+11 with being the mean phonon number of the centerofmass mode， the Hamiltonian of Eq. （2） can be approximated by

HI=λ（e-iφaS++eiφa+S-），

（3）

where λ=iΩη/2， and S±=Sx±iSy with Sα=∑Nj=1（σαj+σαj）（α=x，y，z） being the usual three components of the total spin operator of the ions. The Hamiltonian is mathematically identical to that describing the interaction between N twolevel atoms and a singlemode cavity field， with the cavity mode replaced by the vibrational mode [16]. Since [HI，]=0， where =a+a+∑Nj=1σ+jσ-j is the excitation number operator， the dynamics is separable into subspaces having a prescribed eigenvalue M of . We now assume that the vibrational state of the ions is initially in the onephonon state1〉 and the electronic state of each of them is initially in the ground stateg〉， i.e.， the initial state of the system is 1g1g2…gN〉. After an interaction time τ the state of the system is

Ψ（τ）〉=cos（Nλτ）1〉g1g2…gN〉-ieiφsin（Nλτ）0〉WN〉S，

（4）

where WN〉S=（1N）N-1，1〉 with N-1，1〉 denoting all the totally symmetric states including N-1g and 1 e， i.e.， a symmetric W state. In general， the state of Eq. （4） is an asymmetric W state involving both the external and the internal degree of freedom of the ions， as long as τ≠nπ（Nλ） with n being a nonnegative integer. By choosing cos（Nλτ）=0，i.e.，τ=π（2Nλ）， we can obtain the Nion symmetric W state WN〉S involving only internal degree of freedom. This result can be easily understood in physics： the initial one phonon is equally shared by the N ions. Setting τ=π（4Nλ）， we can obtain a special （N+1）partite asymmetric W state

WN+1〉A=12（1〉g1g2…gN〉-ieiφ0〉WN〉S）.

（5）

Such a configuration of asymmetric W state can be utilized to carry out some quantuminformation processing tasks more efficiently than the symmetric W state [1114]. The state WN+1〉A involves both the external and the internal degree of freedom of the ions. If mapping the vibrational phonon state to the electronic state of another ion， one can also obtain an （N+1）ion asymmetric W state that involves only internal degree of freedom of the ions. Note that the time needed for preparing the desired W stated decreases with the increase of the number of ions.

In conclusion， we have demonstrated a very simple scheme for generating multipartite Wtype entangled states with trapped ions. The scheme does not require us to address the ions individually， and create a W state in a single step. The number of ions can be arbitrarily large in principle. Besides， the more ions that are involved， the less time it takes to generate the desired W state. The presented scheme can be regarded as an improvement on that of Ref.[15]. Then one Zeeman level of the S12 ground state of a 40Ca+ ion acts as g〉 and one Zeeman level of the metastable D52 state acts as e〉. However， if we use a pair of the hyperfine S12 ground states of a 9Be+ ion as the two carrying levels e〉 and g〉 through the Raman transition[17]， our scheme is also effective by appropriately redefining the symbols of equation （1）.

References：

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