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Entanglement generation by adiabatic navigation in the space of symmetric multi-particle st

Entanglement generation by adiabatic navigation in the space of symmetric multi-particle st
Entanglement generation by adiabatic navigation in the space of symmetric multi-particle st

a r X i v :q u a n t -p h /0205118v 1 19 M a y 2002

Entanglement generation by adiabatic navigation in the space of symmetric

multi-particle states

Razmik G.Unanyan 1?,Michael Fleischhauer 1,Nikolay V.Vitanov 2?,Klaas Bergmann 1

1

Fachbereich Physik der Universit¨a t,67653Kaiserslautern,Germany

2

Theoretical Physics Division,Department of Physics,So?a University,James Boucher 5blvd.,1126So?a,Bulgaria

(February 1,2008)

We propose a technique for robust and e?cient naviga-tion in the Hilbert space of entangled symmetric states of a multiparticle system with externally controllable linear and nonlinear collective interactions.A linearly changing exter-nal ?eld applied along the quantization axis creates a network of well separated level crossings in the energy diagram of the collective states.One or more transverse pulsed ?elds ap-plied at the times of speci?c level crossings induce adiabatic passage between these states.By choosing the timing of the pulsed ?eld appropriately,one can transfer an initial product state of all N spins into (i)any symmetric state with n spin excitations and (ii)the N -particle analog of the Greenberger-Horne-Zeilinger state.This technique,unlike techniques using pulses of speci?c area,does not require precise knowledge of the number of particles and is robust against variations in the interaction parameters.We discuss potential applications in two-component Bose condensates and ion-trap systems.

I.INTRODUCTION

Entanglement is a unique quantum feature which has enjoyed considerable attention in the last few years.It plays a crucial role in many rapidly developing areas of contemporary quantum physics,such as quantum infor-mation [1]and fundamental tests of quantum mechanics [2].Various quantum systems have been suggested for controlled creation of entanglement,e.g.trapped ions [3],spins in magnetic ?eld [4],quantum dots [5],cavity-quantum-electrodynamics systems [6],crystal lattices [7],Josephson junctions [8],and others.In order to entangle N spin-1

?

Permanent address:Institute for Physical Research,Ar-menian National Academy of Sciences,378410Ashtarak,Armenia ?

also at Institute of Solid State Physics,Bulgarian Academy of Sciences,Tsarigradsko chauss′e e 72,1784So?a,Bulgaria

sensitive to variations in the pulse area and resonance

mismatch,which can be caused by temporal and spatial ?uctuations of the external ?eld and may lead to signif-icant errors.Hence an important practical challenge is to design robust and e?cient methods for a controlled navigation in the multi-particle Hilbert space.This is of particular importance in mesoscopic systems,where only limited control over the interaction parameters and the number of particles is possible.

The simplest interaction that can lead to entanglement in a collection of spins involves either pairwise nearest-neighbor interactions or a collective coupling between all particles.An example for the ?rst case is the Ising model [9],while the collective coupling of ions to a phonon mode in an ion trap [3]or the self-interaction in a Bose-Einstein condensate (BEC)[10]is an example for the second.We here analyze the second type of systems and pro-pose a method for controlled,e?cient and robust naviga-tion in the space of symmetric multi-particle entangled states.No precise knowledge of interaction parameters or particle number is needed and only certain adiabaticity criteria have to be ful?lled.The proposed technique is a multi-particle generalization of our earlier proposal [11]for creation of entanglement in a pair of two-state sys-tems by using adiabatic passage induced by a suitably crafted external ?eld.The multi-particle problem adds some new challenges as it involves in general multi-step as well as direct transitions;this opens a variety of paths between any pair of multi-particle states.We use this to advantage and demonstrate that certain paths are in-sensitive to the number of particles N ,a property that is particularly signi?cant for systems with large N ,for which N is not usually known exactly.

The scheme proposed in the present paper allows to create robustly and e?ciently maximum entanglement starting from a product state.In particular,it can be used to create various special entangled states,such as of the Greenberger-Horne-Zeilinger (GHZ)type or the so-called W -states.Moreover arbitrary transitions among the collective states can be realized.After introducing the concept of the method we discuss two speci?c imple-mentations:an ion-trap scheme similar to that of M?lmer and S?renson [12]and a coupled two-component BEC in the two-mode approximation [13].

II.NA VIGATION IN THE HILBERT SPACE OF

THE COLLECTIVE STATES

A.Controlled collective spin interaction

We consider a collection of N identical spin-

1

2N

will interact.Thus the 2N -dimensional

Hilbert space reduces to the (N +1)-dimensional sub-space of symmetric multiparticle states.Because each of these states is characterized by a de?nite number of exci-tations n =0,1,...,N (which corresponds to an angular momentum projection m =?J +n ∈{?J,J }),we shall use n to label the states.They are given by

|0 =|↓↓...↓ ,

(2a)

|1 =

N

1

?12

N ?1

i =1N

j =i +1

?σ+i ?σ+

j |0 ,

(2c)

...

|N =|↑↑...↑ ,

(2d)

where ?σ+

i is the Pauli spin-?ip operator,which inverts the spin of the i th particle.In particular,the single-excitation

symmetric state |

1

is an N -particle analog of the W -state [15]:

|1 =

1N

|↑↓↓...↓ +|↓↑↓...↓ +···|↓↓↓...↑ .This state is maximally robust against disposal of any of its qubits [15].Obviously,state |N ?1 ,which has N ?1spins up and one spin down,is also of the W -type.

We assume,similarly to [11],that the particle-?eld cou-pling consists of a linearly changing component along the z -direction (chosen as the quantization axis)and a pulsed

component in the x -direction.Then the Hamiltonian (1)takes the form

?H (t )=ξ?J 2z ?At ?J

z +?(t )?J x ,(3)

where A is assumed real and positive.Because the opera-tor ?J

x connects only adjacent states,the linkage pattern is chainwise.The direct coupling between each pair of adjacent states |n and |n +1 is given by

?n,n +1(t )=

A

.(6)

These crossings are equidistant and separated by a time interval τ=ξ/A ,which does not depend on the number of particles N ,but only on the interaction parameters.Pairs of states with the same total number of excita-tions n +k cross at the same time.However,due to the presence of the nonlinear interaction term m 2ξ,all level crossings are well separated in energy.

The positions of the crossings may or may not depend on N .For example,the crossing between states |0 and |1 is situated at t 01=?(N ?1)ξ/A .Hence this time will be known only if N is known exactly;we will see be-low that this restricts the possible scenarios of adiabatic transfer.In contrast,the crossing between the two prod-uct states |0 and |N does not depend on N ,because it is situated at t 0N =0,and the time t =0is determined by the zero value of the longitudinal ?eld,B z (0)=0.The crossing between states |N and |1 ,which is situated at t 1N =ξ/A ,does not depend on N either.

B.Principles of navigation in Hilbert space

1.Navigation routes

Once a network of level crossings is created,one can design in principle any navigation route in the Hilbert space by choosing properly the timing of the pulsed ex-ternal ?eld.It is most natural to assume that the multi-particle system is prepared initially in one of the product

pulse is applied at the crossings (a)t 01,(b)t 02,(c)t 03,(d)t 04.As a result,four navigation routes are created connecting the initial state |0 to states |1 ,|2 ,|3 ,and |4 ,respectively.

states,for example in |0 .If the system evolves along this state and the designed route requires that it must make a transition to the entangled state |n ,one should apply a su?ciently strong (adiabatic)external pulse at the time of diabatic crossing t 0n between |0 and |n .The interac-tion will open an avoided crossing between the energies of the corresponding adiabatic states and will force the sys-tem to make an adiabatic transition from |0 to |n [16].On the contrary,if the route requires the system to re-main in the same collective state,one should ensure that there is negligible interaction between this state and the other collective states.With the leeway in choosing the time dependence and the intensity of the pulsed external ?eld,one can link any initial state to any ?nal state by using one or more suitably timed pulses.

In Fig.2we show the numerically calculated popu-lations of the ?ve symmetric collective states |n in a four-particle system plotted against the center T 0of a Gaussian coupling pulse.Depending on the timing of this pulse,the population is transferred from the initial state |0 to di?erent collective states.In agreement with the above discussion,the maximum transfer e?ciency for each collective state |n is achieved when T 0is near the crossing t 0n between |0 and |n .One also recognizes that

in ξThe present technique allows also to transfer entan-glement.Indeed,if the many-particle system is initially in the entangled state |n ,it can be transferred adia-batically to another entangled state |k by applying an adiabatic pulse at the crossing time t nk of these states.An entangled state |n can also be transferred into one of the unentangled states |0 or |N ;this can be used for measurement of entanglement.

2.Conditions

For such a state engineering to be successful,the level crossings must be well separated,i.e.the width T of the external pulsed ?eld must be small in comparison with the time separation between the crossings:T ?ξ/A .On the other hand,the condition for Landau-Zener population transfer around the crossing at t nk can be shown to lead to the following conditions

?nk (t nk )T ?

|n ?k |

?1.

|k is very small and cannot induce adiabatic evolution. Thus we have to consider the case?>~Nξ,which is however accessible only numerically.

C.Choice of navigation path

As the energy diagram in Fig.1suggests,there are multiple paths linking each pair of collective states.Each of these paths has certain advantages and disadvantages, depending on the particular experimental situation.

1.Transition between the product states

The two product states|0 and|N can be linked in several di?erent ways.First,one can apply a single adi-abatic pulse at their level crossing at time t0N=0,as shown in Fig.3(upper frame).This approach is indepen-dent of the number of particles N because the crossing time t0N=0is well de?ned by the zero of the linearly increasing z-?eld.In other words,even if we do not know

exact N,we can?nd the exact location of the cross-between the completely unexcited state|0 and the excited state|N .However,using this cross-between|0 and|N requires much stronger?eld be-

these states are not coupled directly,but only via -quanta transition.

one can use a train of pulses centered crossing,so that the population will?ow from through all intermediate states to reach|N at the

|0 →|1 →|2 →...→|N .Because this route(the lowest solid curve in Fig.3) only through crossings of directly coupled adja-levels,much less?eld intensity is needed to satisfy adiabatic condition.However,the?rst crossing at =?(N?1)ξ/A depends on the number of particles

i.e.this approach is only applicable if N is known

third possibility,applicable only to the transition →|N ,is to apply a su?ciently wide single pulse, all crossings,as shown in Fig.3(lower frame).

case,only an approximate knowledge of N is re-

because there are no stringent restrictions on the

width and timing.

2.Creation of W-states

faces similar choices for the transitions from one

product states to any entangled state|n .For ex-

the N-particle W-state can be created using the |0 →|1 by applying a single adiabatic pulse

t01=?(N?1)ξ/A,as shown in Fig.4(upper

Because this is a transition between adjacent col-lective states,only a moderately strong?eld is required. However,this approach is only applicable if the num-ber of particles is known precisely because the crossing time t01depends on N:an error in N,even as small as?N=1,is inadmissible since then the pulse will be applied at a wrong crossing.

Alternatively,state|1 can be populated using the transition|N →|1 ,as shown in Fig.4(lower plot). For this path,the crossing time t N1=ξ/A does not de-pend on N.However,because this is an(N?1)-quanta transition,the coupled states di?er by a large number of spin excitations.Then a much stronger?eld may be needed to widen the much narrower avoided crossing and induce adiabatic evolution.

Similar conclusions apply to the other W-state|N?1 and to any other entangled state|n .

3.Feasibility of the N-invariant scenario

The N-invariance of the latter approach,which uses the(multi-quanta)crossings near t=0rather than the

ing the precise number of particles N.The only problem

is that for multi-quanta transitions a stronger external

?eld is needed to induce adiabatic passage.Therefore we have performed numerical simulations to estimate the

minimal pulse area A min needed to achieve90%transfer e?ciency for the transition|N →|1 from the product

state|N to the W-state|1 .This area is plotted in Fig.5

as a function of the particle number N.

As the?gure demonstrates,A min increases nearly

quadratically for small N and approaches a linear depen-

dence for large N.This(slow)linear increase implies that the conditions on the required resourses for application

of this technique to many-particle systems are not very strong.The linear behavior can be understood qualita-

tively by noting that,as the numerical simulations show,

the coupling term?J x in Eq.(3)(which is∝?0∝A min) is of the same order of magnitude as the vertical energy

splitting(which is∝Nξ).Hence,the transition proba-

bility must scale with the parameter?0/(Nξ).

interesting application of the present scheme,

a slight modi?cation,is the creation of the so-called

state[18],

|GHZ =12 |0 +|N .(8)

from the product state|0 one has to create an equal superposition of this state and the W-state for example,by a singleπ/2-pulse.Then the present can be used to transfer the population of|1 to other product state|N by applying a single adiabatic

at the crossing t1N,

|0 π/2?→12(|0 +|1 )adiab?→12(|0 +|N ).

III.IMPLEMENTATIONS

In the following we brie?y discuss two implementations of the technique discussed above.The?rst one involves two-photon transitions in trapped ions,the second a co-herently coupled two-component Bose-Einstein conden-sate.

A.Ion-trap system

Recently S?renson and M?lmer suggested a realization of a nonlinear Hamiltonian in a two-level ion trap sys-tem[12],displayed in Fig.6.In their scheme,two laser ?elds(in the Lamb-Dicke limit of light coupling)are ap-plied with frequenciesω1andω2that are symmetrically detuned from the single-photon resonance by a detuning δ.Then there is a two-photon resonance between states

eg n+eg n eg n?ge n+ge n ge n?11

FIG.6.Ion-trap system for realization of an e?ective non-linear spin Hamiltonian.The application of a symmetrically detuned bichromatic ?eld to ions with ground state |g and excited state |e leads to two-photon resonance between the collective states |gg and |ee independent of the trap oscilla-tion quantum number n .

with two ions in the ground level |gg and two ions in the

excited level |ee .If the detuning is larger than the Rabi frequencies of the two lasers (δ??1=?2≡?)there is no single-photon excitation.

In a coarse-grained time-averaged picture an e?ec-

tive

Hamiltonian

proportional to

ξ

?J 2z emerges [12],with

ξ=2η2?2ν/(ν2?δ2

),ηbeing the Lamb-Dicke param-eter,and νthe phonon frequency.If,in addition to the detuned bichromatic ?elds,a resonant laser coupling the z component and one coupling the x component are ap-plied,the Hamiltonian (3)is obtained.Since ξis pro-portional to the Lamb-Dicke parameter η~1/

√2

a ?2a 2+

U bb

2

a ?a

b ?b +?(t )[ab ?e ?iφ(t )+ba ?e iφ(t )].(9c)

Here a and b are the annihilation operators for the bosons

in the two states |a and |b ,and E a and E b are their

energies.U aa ,U bb and U ab characterize the particle-particle interaction when the particles are in states |a and |b .The time-dependent functions ?(t )and φ(t )are the amplitude and the phase of the coherent cou-pling between the two components.If the elastic s -wave scattering lengths of the two components are equal (U aa =U bb =U ),as in the sodium experiment [19],the nonlinear spin coupling ξwill be independent of the number of particles N .

In terms of the Schwinger bosonic represenation of the angular momentum,?J

x =1

2i (ab

?

?ba ?),?J

z =1

2U ab .

After a unitary

transformation |Ψ(t ) =exp[?iφ(t )?J

x ]|Φ(t ) we obtain a Hamiltonian equivalent to (9)

?H (t )=ξ?J 2z +[α?˙φ

(t )]?J z +2?(t )?J x .IV.SUMMARY

In the present paper we have proposed and analyzed a robust adiabatic scheme for generating symmetric en-tangled states of a many-particle system.A constant

nonlinear interaction in the collective spin projection ?J 2z combined with a time-dependent linear interaction in ?J

z creates a web of level crossings between the collective multiparticle states.Application of pulsed ?elds along the x component of the collective spin at the times of appropriate level crossings induces single or multi-step transitions between the states.In this way a controlled adiabatic navigation in the (N +1)-dimensional subspace of the symmetric collective states is possible allowing,for example,generation of N -particle W and GHZ states.The suggested method is robust against parameter vari-ations and does not request an exact knowledge of the particle number,but only demands proper timing of the pulsed ?eld.It should be noted that the nonlinear inter-action ?J 2z

guarantees the separation of the level crossings and is hence necessary for the navigation in the Hilbert space.

An initial product state is connected to symmetric en-tangled states via various pathways involving avoided crossings induced by single and/or multiparticle inter-actions.If the exact number of particles is known,it is in general possible to ?nd a pathway that runs only through single-excitation crossings;then only a moder-ately strong ?eld is needed to induce adiabatic passage.If N is not known,a pathway can be chosen that involves multi-photon avoided crossings.Our numerical studies

indicate that in this case the required?eld scales nearly linearly with N.

We have given two explicit examples for the applica-tion of our scheme,ion traps and coherently coupled two-component Bose condensates.

This work has been supported in part by the EU-network HPRN-CT-1999-00129(COCOMO)and by NATO grant1507-826991.The work of RGU was sup-ported by the Deutsche Forschungsgemeinschaft,grant Fl210/10.RGU and NVV acknowledge support by the Alexander von Humboldt Foundation.

IELTS作文——Generation gap

Generation gap There always has been generation gap since the dawn of civilization. Old people act like a frog in the well. They are fully convinced with their ideas as ultimate and ideal. They ignore certain vital factors that are no longer valid in modern days. There is a great hue and cry raised by the elders and the young both that the communication between them has broken down and they accuse each other for and bemoan this state of affairs. They attribute it to generation gap. Majority in the society carries along with this moroseness and never pauses to think the why and how of the problem. 为什么?? The elders are more critical of the younger generation with a big inventory of complaints against the young and the young mostly tend to ignore the grumbling, mumbling and occasional loud protestations of the older generation. But now and then they do protest. They resent the petting attitude of the elders. It is generally observed that the old behave like a frog in the well. They are fully convinced that the ideas they have had throughout their lives are the ultimate and ideal. They ignore certain vital factors that are no longer valid in the case of the modernity. There always has been generation gap since the dawn of civilization. The young have always deviated from the older standards and it was well that they did or there wouldn’t have been any progress today. The younger generation naturally differs in dress, food, habits etc, as all these were not available to the older generation. These changes are indicative of progress. With the passage of time when the society, right from the basic unit of the family through society and the entire humanity, is changing, how can the new generation be asked to stagnate or move bac kward to the ancestors’ way of life? In the olden days, in the Indian society the arranged marriage took place first and love between husband and the wife were thought to be a natural development. The moral standards differed in consonance with the contemporary situation, as human society is essentially a utilitarian one. The elders couldn’t have visualized what was to come in their lifetime i.e. live-in relationship, one night stand, car key pooling and fishing and things that are yet to come out in the open. It will be generally agreed that most fathers want their offspring to adopt the same profession that they themselves followed without giving due weight to the aptitude of the offspring. This causes tension between the elders and the children. The younger generation is more intellectually evolved than the preceding one. But the ego of the older people does not permit them to accept

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2、人有两件宝 人有两件宝,双手和大脑(nǎo)。双手会做工,大脑(nǎo)会思考。用手不用脑(nǎo),事情做不好。用脑(nǎo)不用手,啥(shá)也做不好。用手又用脑,才能有创造。一切创造靠(kào)劳动,劳动要用手和脑。 (一)这是一首儿歌,一共有()句话。 (二)填空: (1)人有两件宝是指( )和( )。 做工靠( ),思考靠( )。 (2)做事情要用( )又用( )。这样才能 ( )。 (三)词语搭配: (1)认真地劳动(2)一双手指 辛勤地双手一根手表 勤劳的头脑一只小手 聪明的思考一块手套 3、夏天 初(chū)夏,石榴(liú)花开了。远看,那红色的花朵像一簇(cù)火焰(yàn)。近看,一朵朵石榴(liú)花像一个个小喇叭(lǎba)。

21种提高思维能力的方法

21种提高思维能力的方法 大脑就是一台三磅重的超级计算机。它是身体运行的命令和控制中心。它几乎涉及你所做的每一件事。你的大脑决定你如何思考,如何感觉,如何行动,以及如何与他人相处。你的大脑甚至决定你是哪一类的人。它决定了你有多善解人意;你有多友善或是有多粗鲁。它决定了你思维有多敏捷,这还涉及到你工作完成的如何以及你的家庭。你的大脑还影响你的情感活动,以及你如何对待异性。 大脑比我们可以想象到的任何计算机都要复杂。你的大脑里一千亿个神经细胞,每一个细胞都与其他许多细胞有联系,你知道吗?事实上,大脑内部的联系比宇宙中的星星还要多!无论是在工作,休息还是恋爱中,要做到最好的自己的本质上就是要优化你的大脑。 显然,你做的所有事,你所有的感觉和思想,你与人相处的每一处细微差别,其中心就是你的大脑。它既是一个带动你复杂生命的超级计算机,也是一个为你的灵魂提供住所的温柔器官。而当你跑步、举重或者做瑜伽以保持良好身体状态时,你忽略了你的大脑以及相信它给它做它的工作的机会。 无论你的什么年纪,精神锻炼都带给大脑普遍积极的影响。所以,这儿有22条方法提升你的脑力。 1.驱动你的大脑细胞 研究表明得到足够运动的人,其大脑也更好。加州拉由拉市的萨克生物研究学院的科学家发现,与整天坐那儿在网络聊天室里讨论指环王的老鼠相比,只要觉得想要跑步就在转轮上跑动的成年老鼠的海马得到的新细胞是他们的两倍,海马是大脑控制学习和记忆的部分。研究者们也不能确定为什么更活跃的啮齿动物的大脑会有这种反应,但可以知道的是这种自愿的运动可以减压,因此而更有益。这意味着找到了享受运动,而不是强迫自己去运动的方法,会让你更聪明,也更有幸福感。 所以,做点运动,选择一个训练项目比如马拉松,三项全能或者“趣味赛跑”,或者找个伴儿一起让运动变得有趣。 2.锻炼你的思维 让你的大脑细胞活跃起来并不只是物理运动。就像那些出租车司机和玩儿钢琴的人,你可以通过使用大脑里各种各样的区域来建立它们。杜克大学神经生物学教授罗伦斯·C·凯兹,也是《让你大脑New一下》(Keep Your Brain Alive)的合著者,找到使用大脑退化部分的简单途径不仅有助于维护神经细胞,也有助于细胞上的树突和轴突接收传递信息。凯兹说就像一次新的举重练习帮助建立未充分使用的肌肉,用新奇的方式思考和观察世界有助于改善大脑不活跃部分的功能。 体验新的味觉和嗅觉;用你平时不用的那只手做事;找些新的路开车上班;到新的地方去旅行;进行艺术创作;读读陀思妥耶夫斯基的小说;为泰德·肯尼迪和拉什·林宝写一篇兄弟喜剧——基本上,做一切可以让你的大脑跳出常轨的事。 3.多问为什么 我们的大脑与好奇心联系在一起。随着我们长大变“成熟”,许多人开始抑制或否认自己天生的好奇心。让你充满好奇心吧!问问自己为什么会发生这些事情。问问知道的人们。锻炼好奇心最好的方式就是问“为什么?”让一天至少问十个“为什么”成为你的习惯吧。你的大脑会变的愉快,你也会惊讶地发现你的生活工作中有那么多的机遇和解决方案。 4.笑 科学家告诉我们笑有益健康;它可以促进体内释放内啡肽及其它有积极力量的化学物质。事实上我并不需要科学家来告诉我们笑会带来良好感觉。笑可以帮助我们减压,还可以打破旧有模式。因此笑可以看做是对大脑的快速充电。多多大笑吧!

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