Non-local generation of entanglement of photons which do not meet each other

合集下载

Being Fascinated by Strings and Membranes Is Kikkawa-Type Physics Possible at Ochanomizu

Abstract On the occasion of the 60th birthday of Professor Keiji Kikkawa, Kikkawa-type physics performed at Ochanomizu was persona metric is discussed with the condensation of the string ﬁelds.
OCHA-PP-77 March 1996
arXiv:hep-th/9603146v1 22 Mar 1996
BEING FASCINATED BY STRINGS AND MEMBRANES: IS KIKKAWA-TYPE PHYSICS POSSIBLE AT OCHANOMIZU?
A. SUGAMOTO Department of Physics, Ochanomizu University, 2-1-1, Otsuka, Bunkyo-ku, Tokyo, 112 Japan
1
Personal Memories with Professor Keiji Kikkawa and the Kikkawa-Type Physics at Ochanomizu
It is my great pleasure to contribute to the proceedings of the workshop held at Osaka to celebrate the 60th birthday of Professor Keiji Kikkawa. I am very much inﬂuenced by his physics, especially by his papers on 1) the light-cone ﬁeld theory of string (Its Japanese version included in Soryushiron Kenkyu was my favorite.), 2) his lecture note on the superstrings given just before the string fever started (I think everybody should begin with this lecture note when he or she wants to do something in strings.), 3) hadronic strings with quarks at the ends, and 4) the path integral formulation of the Nambu-JonaLasinio model. Personally, Professor Kikkawa cited my paper on the dual transformation in gauge theories at the Tokyo conference in 1978, without which I could not have survived in our particle physics community and would deﬁnitely be engaged in another job now. Therefore I am greatly indebted to him for his guidance in physics. It was probably 1979 summer when I went abroad for the ﬁrst time with my late friend Dr. Osamu Sawada, and we stayed at Professor Hirotaka Sugawara’s residence in Honolulu. The topical conference 1

相互纠缠的经典句子

相互纠缠的经典句子In the realm of literature, there exists a phenomenon that captivates readers and writers alike - the entanglement of classic sentences. These sentences, often profound and thought-provoking, have the power to linger in our minds long after we have read them. They intertwine with our emotions, leaving an indelible mark on our souls. This entanglement can be observed from multiple perspectives, such as the impact on readers, the challenges faced by writers, the significance in cultural heritage, the role in shaping personal identity, the exploration of human experiences, and the creation of lasting legacies.For readers, encountering these entangled sentences is like stumbling upon hidden treasures. They evoke deep emotions, resonate with personal experiences, and offer profound insights into the human condition. These sentences have the ability to transport us to different worlds, challenge our perspectives, and ignite our imagination. As we unravel the layers of meaning within these sentences, wefind ourselves entangled in a web of emotions, thoughts, and reflections. The impact of these classic sentences on readers is immeasurable, as they become a part of our literary consciousness, shaping our understanding of the world and ourselves.However, for writers, the creation of entangled sentences is no easy task. It requires a delicate balance between artistry and craftsmanship. Writers must carefully choose their words, crafting sentences that are both captivating and enduring. The challenge lies in finding the perfect combination of words that can evoke the desired emotions and resonate with readers. It is a process oftrial and error, of rewriting and refining, until the sentence reaches its full potential. Writers must also grapple with the weight of their words, knowing that once released into the world, they will become entangled in the minds and hearts of readers.The significance of entangled sentences extends beyond the realm of literature. They are an integral part of cultural heritage, representing the collective wisdom andexperiences of a society. These sentences become symbols of cultural identity, reflecting the values, beliefs, and aspirations of a community. They serve as a bridge between generations, connecting the past with the present and guiding future generations. The preservation and appreciation of these sentences contribute to the richness and diversity of our cultural tapestry, reminding us of our shared humanity and the power of storytelling.On a personal level, entangled sentences play a crucial role in shaping our individual identities. They become touchstones in our lives, guiding our thoughts, beliefs, and actions. These sentences have the power to inspire, motivate, and provide solace in times of uncertainty. They become companions on our journey, offering guidance and wisdom when we need it most. As we encounter these sentences, we find ourselves entangled in a dance of self-discovery, as they help us navigate the complexities oflife and define our own narrative.Beyond their impact on individuals, entangled sentences also serve as a means of exploring the vast spectrum ofhuman experiences. They capture the essence of love, loss, joy, and despair, encapsulating the myriad of emotions that define our existence. These sentences become vesselsthrough which we can delve into the depths of the human psyche, unraveling the intricacies of our thoughts and emotions. They provide a window into the human condition, allowing us to connect with others on a profound andintimate level.Finally, entangled sentences have the power to create lasting legacies. They transcend time and space, transcending the boundaries of language and culture. These sentences become immortalized, passed down from generationto generation, leaving an indelible mark on the literary landscape. They become part of our collective consciousness, shaping the way we perceive the world and ourselves. Through these sentences, writers achieve a form of immortality, as their words continue to resonate with readers long after they are gone.In conclusion, the phenomenon of entangled sentences in literature is a testament to the power of words and theirability to captivate, inspire, and enthrall. From the perspective of readers, these sentences evoke deep emotions and offer profound insights into the human condition. For writers, the creation of these sentences is a challenging yet rewarding endeavor, requiring a delicate balance between artistry and craftsmanship. The significance of entangled sentences extends beyond literature, as they become symbols of cultural heritage, shape personal identities, explore human experiences, and create lasting legacies. These sentences are a testament to the enduring power of literature and its ability to transcend time and space.。

PLA_22660

Thank you for your assistance.
Page 1 of 1
JID:PLA AID:22660 /SCO Doctopic: Quantum physics
Physics Letters A ••• (••••) •••–•••
[m5G; v 1.134; Prn:11/06/2014; 15:18] P.1 (1-5)
Location in article
Q1
Q2
Query / Remark: click on the Q link to go Please insert your reply or correction at the corresponding line in the proof
Please conﬁrm that given names and surnames have been identiﬁed correctly and are presented in the desired order. (p. 1/ line 15)
Please check the address for the corresponding author that has been added here, and correct if necessary. (p. 1/ line 60)
Please check this box if you have no corrections to make to the PDF ﬁle
For correction or revision of any artwork, please consult /artworkinstructions
Any queries or remarks that have arisen during the processing of your manuscript are listed below and highlighted by ﬂags in the proof. Click on the ‘Q’ link to go to the location in the proof.

1-s2.0-S0030401810002415-main

Robust scheme to generate N -atom W state in two distant cavitiesWen-An LiSchool of Physics and Engineering,Sun Yat-Sen University,Guangzhou 510275,People's Republic of Chinaa b s t r a c ta r t i c l e i n f o Article history:Received 23December 2009Received in revised form 26February 2010Accepted 10March 2010Keywords:Two distant cavities Entangled stateWe propose a scheme to realize W states for N -atoms trapped in two distant cavities connected by an optical ﬁber.In the scheme,the cavity modes and ﬁber mode are not excited during the process.The quantum information is encoded in two degenerate ground states,so the atom's spontaneous emission can be omitted approximately.Moreover,the operation speed increases with the number of the atoms without a limitation and thus the scheme is extremely robust against decoherence.©2010Elsevier B.V.All rights reserved.Quantum entanglement is a fundamental aspect of quantum mechanics that causes parts of a composite system to show corre-lations stronger than can be explained classically [1,2].Leaving aside its theoretical importance we can point out promising practical applications in diverse areas such as quantum computing [3–5],teleportation [6],cryptography [4,7],precision measurements [8]and quantum information processing [9,10].In general,the more particles that can be entangled,the more clearly nonclassical effects are exhibited and the more useful the states are for quantum applications.For example,GHZ states can not only provide much stronger refutations of local realism and offer possibilities to test quantum mechanics against local hidden theory without inequality [11],but also are useful in quantum information processing [10].Recently,a large number of theoretical [12–15]and experimental [16–18]schemes have been proposed to generate multipartite GHZ states.In addition,Dür et al.[19]have shown that there are two inequivalent classes of tripartite entanglement states,the GHZ class and the W class,under stochastic local operation and classical communication.W states have many interesting features:their entanglement is not only maximally persistent and robust under particle loss,but also immune against global dephasing,and rather robust against bit ﬂip noise.Apart from its intrinsic interest,W state is of practical importance in quantum information processing.Recently,it has been applied to quantum teleportation [20],distilling entanglement [21],quantum key distribution [22]and demonstrating quantum nonlo-cality [23]etc.Considering the importance of W states,the generation of W states has attracted great attention in the last decade [24–30].On the other hand,realizing entanglement and deterministic quantum gates between separate subsystems is of great importance for distributed quantum computation.A lot of schemes of the entanglement and controlled-phase gates between two distantatoms have been proposed [31–35].In these schemes,the entangle-ment between separate subsystems is realized by the detection of leaking photons.So the success probability depends on ef ﬁciency of the linear optics elements.Additionally,Pellizzari [36]proposed another kind of model to realize the reliable transfer of quantum information between two atoms in distant cavities connected by an optical ﬁber.Then a great many of schemes [37–46]based on this cavity –ﬁber –cavity system have been proposed.For example,Peng and Li [39]employed the cavity –ﬁber –cavity model to entangle two atoms in spatially separated cavities through both photon emission and absorption processes.Song et al.[43]proposed a scheme for generating W-type entangled states.However,realizing entangled states with high ﬁdelity in these schemes [38–43]critically relies on the condition that the coupling strength between atoms and the cavity is much smaller than the cavity –ﬁber coupling.This condition means that it may require weak coupling in atom –cavity system because it is not easy to achieve large cavity –ﬁber coupling.Within weak coupling,it is very likely that the atom will suffer an incoherent spontaneous emission,resulting in a photon leaving the cavity undetected to the sides before the electronic degree of freedom has been mapped onto the photonic degree of freedom,which severely damages the quantum entanglement that one intends to create.In this paper,we suggest a robust scheme to generate N -atom W state in the cavity –ﬁber –cavity pared with the previous schemes [41–43],our protocol has the following favorable features:(i)the scheme works without being under the condition that atom –cavity coupling strength is much smaller than cavity –ﬁber coupling.It is of great importance because it can effectively avoid the decoherence arised from the weak coupling in the atom –cavity system.(ii)In the whole procedure,the cavity modes and ﬁber mode are only virtually excited and the system evolves in decoherence-free subspace.In contrast,in ref.[41–43]the quantum state is mediated by photons carried by the optical ﬁber and thus the ﬁdelity is signi ﬁcantly affected by the loss of photons.(iii)The required interaction time decreases asOptics Communications 283(2010)2978–2981E-mail address:liwenan@.0030-4018/$–see front matter ©2010Elsevier B.V.All rights reserved.doi:10.1016/j.optcom.2010.03.020Contents lists available at ScienceDirectOptics Communicationsj o u r n a l h o m e p a g e :w ww.e l s evi e r.c o m /l o c a t e /o p tc o mthe number of the qubits increases,which means that the scheme is extremely robust against decoherence.We consider N identical atoms trapped in two distant single-mode optical cavities,which are connected by an optical ﬁber (see Fig.1).The atom 1is located in cavity 1,while the others are trapped in cavity 2.The atomic level con ﬁguration is shown in Fig.2.There are two degenerate ground states |g 〉,|g ′〉and three degenerate excited states |e 〉,|e ′〉,|e ″〉.The transition |e j 〉↔|g j 〉(j =1,2,3,…,N )is coupled to the corresponding cavity with coupling constant g and detuning Δ.A classical ﬁeld is applied to drive the transition from |g ′〉to |e 〉with coupling coef ﬁcient Ωand detuning Δ+δin each cavity.In addition,atom 1is subjected to two classical ﬁelds with the identical coupling constant Ω′and opposite detuning Δ′as shown in Fig.3.In the interaction picture,the Hamiltonian of the atom –cavity system reads H a −c =ga 1j e 1〉〈g 1j e i Δt +Ωj e 1〉〈g ′1j e i Δ+δðÞt +Ω′j e ′1〉〈g ′1j e i Δ′t +j e ″1〉〈g 1j e −i Δ′t+ga 2∑Nj =2j e j 〉〈g j j e i Δt +Ω∑Nj =2j e j 〉〈g ′j j ei Δ+δðÞt+h :c :;ð1Þwhere a j (j =1,2)is the annihilation operator for photons in the modeof cavity j .In the short ﬁber limit (2Lv )/(2πc )≤1,where L is the length of the ﬁber,c is the velocity of light and v is the decay rate of the cavity ﬁelds into a continuum of ﬁber modes,only one mode of the ﬁber interact with the cavity modes [36,38].In this case,the coupling between the cavity modes and ﬁber is given by the interaction Hamiltonian H c −f=vb a †1+a †2 +h :c :;ð2Þwhere b is the annihilation operator for the ﬁber mode and v is thecoupling strength between the ﬁber mode and the cavity mode.De ﬁne three new bosonic modes as c 0=a 1−a 2ðÞ=ﬃﬃﬃ2p ,c 1=a 1+a 2+ﬃﬃﬃ2p b =2,and c 2=a 1+a 2−ﬃﬃﬃ2p b=2.Then we can rewrite the total Hamiltonian H =H a −c +H c −f in the interaction picture as H =H 0+H I ;ð3Þwhere H 0=ﬃﬃﬃ2p vc †1c 1−ﬃﬃﬃ2p vc †2c 2;ð4Þand H I =12g c 1+c 2+ﬃﬃﬃ2p c 0 j e 1〉〈g 1j e i Δt+Ωj e 1〉〈g ′1j e i Δ+δðÞt +Ω′j e ′1〉〈g ′1j e i Δ′t +j e ″1〉〈g 1j e −i Δ′t+12g c 1+c 2−ﬃﬃﬃ2p c 0 ×∑Nj =2j e j 〉〈g j j ei Δt+Ω∑Nj =2j e j 〉〈g ′j j ei Δ+δðÞt+h :c ::ð5ÞTake the unitary transformation e iH 0t and obtainH ′I =12g c 1e i Δ−ﬃﬃ2p v ðÞt +c 2e i Δ+ﬃﬃ2p v ðÞt+ﬃﬃﬃ2p c 0e i Δt h i j e 1〉〈g 1j +Ωj e 1〉〈g ′1j e i Δ+δðÞt +Ω′j e ′1〉〈g ′1j e i Δ′t +j e ″1〉〈g 1j e −i Δ′t +12g c 1e i Δ−ﬃﬃ2p v ðÞt +c 2e i Δ+ﬃﬃ2p v ðÞt−ﬃﬃﬃ2p c 0ei Δt h i ∑N j =2j e j 〉〈g j j +Ω∑Nj =2j e j 〉〈g ′j j ei Δ+δðÞt+h :c ::ð6ÞHere we assume that Δ;Δ−ﬃﬃﬃ2p v ;Δ+δ≫g ;Ωand Δ′≫Ω′,then the atoms do not exchange energy with the cavity modes,ﬁber mode and classical ﬁelds due to the large detuning.The quantum information is encoded in the two ground states |g 〉and |g ′〉.Since the two atoms are initially prepared in the ground states,they cannot exchange excitation with each other via virtual excitation of cavity modes and they remain in the ground states.However,the three modes c 0,c 1,c 2and classical ﬁeld can be coupled to each other through virtual excitation of the atoms.So the effective Hamiltonian [47,48]isH ′I =−λ1c †1c 1−λ2c †2c 2−λ0c †0c 0 ∑N j =1j g j 〉〈g j j −α∑Nj =1j g ′j 〉〈g ′j j −κ0j g ′1〉〈g ′1j −j g 1〉〈g 1j ðÞ+½ð−β1e −i ﬃﬃ2p vt c †0c 1−β2e i ﬃﬃ2p vt c †0c 2−β3e −2i ﬃﬃ2p vt c †2c 1Þj g 1〉〈g 1j +ðβ1e −i ﬃﬃ2p vt c †0c 1+β2e i ﬃﬃ2p vt c †0c 2−β3e −2i ﬃﬃ2p vt c †2c 1Þ∑N j =2j g j 〉〈g j j +ð−γ0e −i δt c 0−γ1e −i δ+ﬃﬃ2p v ðÞtc 1−γ2e −i δ−ﬃﬃ2p v ðÞt c 2Þj g ′1〉〈g 1j +ðγ0e −i δt c 0−γ1e −i δ+ﬃﬃ2p v ðÞtc 1−γ2e −i δ−ﬃﬃ2p v ðÞt c 2Þ∑Nj =2j g ′j 〉〈g j j +h :c : ;ð7ÞFig.1.Experimental setup.The atom 1is trapped in cavity 1,while others in cavity 2.The two cavities are connected through an optical ﬁber.Fig.2.The atomic level con ﬁguration and transitions for the N identical atoms.Thequantum information is encoded in ground states |g 〉and |g ′〉.The transition |g 〉→|e 〉is coupled to cavity mode with coupling constant g and detuning Δ,and the transition j g ′〉→j e 〉is driven by a classical laser ﬁeld with detuning Δ+δ.Fig.3.The atomic level con ﬁguration and transitions for the atom 1.Apart from the transition depicted in Fig.2,the atom 1is subjected to two extra classical laser ﬁelds with identical coupling coef ﬁcient Ω′and opposite detuning Δ′.2979W.-A.Li /Optics Communications 283(2010)2978–2981where λ1=g 24Δ−ﬃﬃﬃ2p v ;λ2=g 24Δ+ﬃﬃﬃ2p v ;λ0=g 2;α=Ω2Δ+δ;κ0=Ω′2Δ′;β1=ﬃﬃﬃ2p g 21Δ−ﬃﬃﬃ2p v+1Δ;β2=ﬃﬃﬃ2p g 21Δ+ﬃﬃﬃ2p v+1Δ;β3=g 281Δ−ﬃﬃﬃ2p v +1Δ+ﬃﬃﬃ2p v;γ0=ﬃﬃﬃ2p g Ω1Δ+δ+1Δ;γ1=g Ω1Δ+δ+1Δ−ﬃﬃﬃ2p v;γ2=g Ω1Δ+δ+1Δ+ﬃﬃﬃ2p v;Furthermore if the conditions δ;ﬃﬃﬃ2p v ;δ−ﬃﬃﬃ2p v ≫β1;β2;β3;γ0;γ1;γ2is satis ﬁed,three bosonic modes c 0,c 1and c 2cannot exchange energy with each other and with classical ﬁelds.The nonresonant couplings between the bosonic modes and the classical ﬁelds lead to energy shifts depending on the excitation number of the modes and the number of atoms in the ground state |g 〉or |g ′〉.Meanwhile,the nonresonant couplings induce the interaction between atom 1and the other atoms and the interaction between atoms in the cavity 2.Therefore the effective Hamiltonian can be written asH ′I =−λ1c †1c 1−λ2c †2c 2−λ0c †0c 0 ∑N j =1j g j 〉〈g j j −α∑Nj =1j g ′j 〉〈g ′j j−κ0j g ′1〉〈g ′1j −j g 1〉〈g 1j ðÞ+½β21ﬃﬃﬃ2p v c †1c 1−c †0c 0 +β22ﬃﬃﬃ2p vc †0c 0−c †2c 2+β232ﬃﬃﬃ2p v c †1c 1−c †2c 2∑Nj =1j g j 〉〈g j j +ðγ20δc †0c 0+γ21δ+ﬃﬃﬃ2p v c †1c 1+γ22δ−ﬃﬃﬃ2p v c †2c 2Þ∑N j =1j g j 〉〈g j j −ðγ20δc 0c †0+γ21δ+ﬃﬃﬃ2p v c 1c †1+γ22δ−ﬃﬃﬃ2p v c 2c †2Þ∑N j =1j g ′j 〉〈g ′j j −ﬃﬃﬃ2p v ½β21c †1c 1−c †0c 0+β22c †0c 0−c †2c 2j g 1〉〈g 1j ðÞ∑N j =2j g j 〉〈g j j+β232ﬃﬃﬃ2p v c †1c 1−c †2c 2 ∑Nj ;k =1;j ≠kj g j 〉〈g j j j g k 〉〈g k j ðÞ+1ﬃﬃﬃ2p v β21c †1c 1−c †0c 0 +β22c †0c 0−c †2c 2h i ∑Nj ;k =2;j ≠k j g j 〉〈g j j j g k 〉〈g k j ðÞ+κ1σ−1∑Nj =2σþj +h :c : !−κ2∑Nj ;k =2;j ≠kσ−j σþk ;ð8Þwhere κ1=γ20δ−γ21δ+ﬃﬃﬃ2p v −γ22δ−ﬃﬃﬃ2p v;ð9Þκ2=γ20δ+γ21δ+ﬃﬃﬃ2p v +γ22δ−ﬃﬃﬃ2p v:ð10ÞHere we de ﬁne σj +≡|g j ′〉〈g j |,σj −≡|g j 〉〈g j ′|(j =1,2,3,…,N ).Obviously,[c j †c j ,H I ′]=0(j =0,1,2),the quantum number of modes c 0,c 1and c 2is conserved during the interaction.Suppose that two cavity modes and ﬁber mode are all initially in the vacuum state,then the three bosonic modes c 0,c 1and c 2remain in the vacuum state during evolution.Now the Hamiltonian reduced to H ′i =−α+κ2ðÞ∑Nj =1j g ′j 〉〈g ′j j −κ0j g ′1〉〈g ′1j −j g 1〉〈g 1j ðÞ+κ1σ−1∑N j =2σþj +h :c :!−κ2∑Nj ;k =2;j n neqkσ−j σþk :ð11ÞThe above Hamiltonian can be rewritten asH ′I =H 1+H 2;ð12ÞwhereH 1=−α+κ2ðÞ∑Nj =1j g ′j 〉〈g ′j j ;ð13ÞH 2=−κ0j g ′1〉〈g ′1j −j g 1〉〈g 1j ðÞ+κ1σ−1∑Nj =2σþj+h :c :!−κ2∑Nj ;k =2;j ≠kσ−j σþk :ð14ÞSince [H 1,H 2]=0,the operator e −iH i ′t can be expressed as e −iH 1t e −iH 2t .We assume that the N -atoms are initially in state |g 1′g 2g 3g 4⋯g N 〉,so after the interaction time t the state of the system isj ψðt Þ〉=ei α+κ2ðÞt ei N −2ðÞκ2t =2½cos At −iN −2ðÞκ2−2κ0sin Atj g ′1g 2g 3g 4⋯g N 〉−iκ1sin At j g 1g ′2g 3g 4⋯g N 〉+⋯+j g 1g 2g 3g 4⋅⋅⋅g ′N 〉ðÞ;ð15Þwhere A ≡ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃN −1ðÞκ21+N −2ðÞκ2−2κ0½ 2=4q .With the choice (N −2)κ2=2κ0,we obtainj ψt ðÞ〉=ei α+κ2+2κ0ðÞt½cos ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃN −1p j κ1j t j g ′1g 2g 3g 4⋯g N 〉−i 1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃN −1p sin ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃN −1p j κ1j tj g 1g ′2g 3g 4⋯g N 〉+⋯+j g 1g 2g 3g 4⋅⋅⋅g ′N 〉ðÞ;ð16Þwhere ≡κ1/|κ1|.In this case the atom 1exchange periodically information with the other atoms in cavity 2.Let cos ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃN −1p j κ1j t=1=ﬃﬃﬃﬃN p ,we have the W state for the N -atoms j ψt ðÞ〉=1ﬃﬃﬃﬃNp e i α+κ2+2κ0ðÞt½j g ′1g 2g 3g 4⋯g N 〉−i j g 1g ′2g 3g 4⋯g N 〉+⋯+j g 1g 2g 3g 4⋯g ′N〉ðÞ :ð17ÞOn the other hand,with the choice ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃN −1p j κ1j t =π=2,the quantum information is completely transferred to the atoms trapped in cavity 2and we have a fully symmetric W state for N −1atoms in cavity 2,j ψt ðÞ〉=−i ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃN −1p e i α+κ2+2κ0ðÞtj g ′2g 3g 4⋯g N 〉+⋯+j g 2g 3g 4⋅⋅⋅g ′N 〉ðÞ;ð18Þwith the atom 1left in the ground state |g 〉.We now brie ﬂy address the experimental feasibility of the proposed scheme.The required atomic level con ﬁguration can be achieved in Cs atoms.The hyper ﬁne levels |F =3,m =−1〉and |F =3,m =1〉of 62S 1/2can act as the states |g 〉and |g ′〉respectively,while the hyper ﬁne levels |F ′=3,m ′=−1〉,|F ′=3,m ′=0〉and |F ′=3,m ′=1〉of 62P 3/2can act as |e ″〉,|e 〉and |e ′〉respectively.In recent cavity QED experiments [49,50],Cs atoms were trapped in an optical cavity with the corresponding coupling strength g =2π×34MHz.The decay rates for the atomic excited state and the cavity mode are Γ=2π×2.6MHz and κ=2π×4.1MHz,respectively.Set Δ=30g ,δ=15g ,ﬃﬃﬃ2p v =10g and Ω=g ,then γ0≃2×10−2g ,γ1≃1.8×10−2g ,γ2≃1.2×10−2g ,κ1≃−1.51×10−5g and κ2≃6.85×10−5g .If there are about 104identical atoms trapped in cavity 2,the time needed to entangle the atoms is about 4.9μs.During the whole procedure both the atomic system and cavity mode are virtually excited.The effective decoherence rates due to the atomic spontaneous emission and cavity decay [51,52]are Γ′=Γg 2/Δ2=2π×2.89kHz and κ′=κγ02/δ2=2π×7.29Hz.2980W.-A.Li /Optics Communications 283(2010)2978–2981The inﬁdelity induced by the decoherence is on the order of(Γ′+κ′) t≃8.9×10−2for the case of N=104.Note that this value will decrease with N.This conﬁrms our claim that the robustness against decoherence becomes stronger with the increase of the number of particles involved.In conclusion,we have suggested a method to realize W states for the N-atoms trapped in two spatially separated cavities.During the evolution,the cavity modes andﬁber mode are only virtually excited, the whole system evolves in decoherence-free subspace.Further-more,the operation speed increases with the number of the atoms without a limitation.So the scheme is robust against decoherence. There is no need to require that the atom–cavity coupling be much smaller than the cavity–ﬁber coupling.This idea can also be extended to realize phase-covariant cloning and anti-cloning between two distant cavities connected by an opticalﬁber.The scheme opens promising perspectives for realizing remote entangled states and quantum cloning.AcknowledgmentsThe author is grateful to Professor Qiong-Gui Lin for discussions, suggestions and encouragement.We also thank an anonymous referee for helpful comments and suggestions.This work was supported by the National Natural Science Foundation of the People's Republic of China(Grant number10675174).References[1]S.Bell,Physics(Long Island City,N.Y.)1(1964)195.[2]J.F.Clauser,A.Shimony,Rep.Prog.Phys.41(1978)1881.[3]H.-K.Lo,S.Popescu,T.Spiller(Eds.),Introduction to Quantum Computation andInformation,World Scientiﬁc,Singapore,1997.[4]M.A.Nielsen,I.L.Chuang,Quantum Computation and Quantum Information,Cambridge University Press,Cambridge,2000.[5] C.H.Bennett,D.P.DiVincenzo,Nature404(2000)247.[6] C.H.Bennett,G.Brassard,C.Crepeau,R.Jozsa,A.Peres,W.K.Wootters,Phys.Rev.Lett.70(1993)1895;L.Davidovich,N.Zagury,M.Brune,J.M.Raimond,S.Haroch,Phys.Rev.A50(1994) R895;J.I.Cirac,A.S.Parkins,ibid50(1994)R4441;G.Pires,N.G.de Almeida,A.T.Avelar,B.Baseia,ibid70(2004)025803;M.Riebe,H.Haffner,C.F.Roos,W.Hansel,J.Benheln,ncaster,T.W.Korber,C.Becher,F.Schmidt-Kaler,D.F.V.James,R.Blatt,Nature429(2004)734;M.D.Barrett,J.Chiaverini,T.Schaetz,J.Britton,W.M.Itano,J.D.Jost,E.Knill,C.Langer,D.Leibfried,R.Ozeri,D.J.Wineland,ibid429(2004)737.[7] A.K.Ekert,Phys.Rev.Lett.67(1991)661.[8]J.Bollinger,W.M.Itano,D.J.Wineland,D.Heinzen,Phys.Rev.A54(1996)R4649.[9]P.Zoller,et al.,Eur,Phys.J.D36(2005)203.[10]M.Hillery,V.Buzek,A.Berthiaume,Phys.Rev.A59(1999)1829.[11] D.M.Greenberger,M.A.Horne,A.Shimony,A.Zeilinger,Am.J.Phys58(1990)1131.[12]X.B.Zou,K.Pahlke,W.Mathis,Phys.Rev.A68(2003)024302;K.Pahlke,X.B.Zou,W.Mathis,J.Opt.B Quantum Semiclass.Opt.6(2004)S142.[13] C.Yu,X.X.Yi,H.Song,D.Mei,Phys.Rev.A75(2007)044301.[14] D.Gonta,S.Fritzsche,T.Radtke,Phys.Rev.A77(2008)062312.[15]S.B.Zheng,Phys.Rev.Lett.87(2001)230404;Phys.Rev.A65(2002)R051804;S.B.Zheng,G.C.Guo,Phys.Rev.A63(2001)044302.[16]J.-W.Pan,D.Bouwmeester,M.Daniell,H.Weinfurter,A.Zeilinger,Nature403(2000)515.[17]R.J.Nelson,D.G.Cory,S.Lloyd,Phys.Rev.A61(2000)022106.[18] D.Leibfried,et al.,Nature438(2000)639.[19]W.Dür,G.Vidal,J.I.Cirac,Phys.Rev.A62(2000)062314.[20] B.-S.Shi,A.Tomita,Phys.Lett.A296(2002)161.[21] B.Fortescue,H.K.Lo,Phys.Rev.Lett.98(2007)260501.[22]J.Joo,J.Lee,J.Jang,Y.J.Park,(2002)Preprint quant-ph/0204003.[23] C.F.Wu,J.L.Chen,L.C.Kwek,C.H.Oh,Phys.Rev.A73(2006)012310.[24]T.Yamamoto,K.Tamaki,M.Koashi,N.Imoto,Phys.Rev.A66(2002)064301.[25]M.Eibl,N.Kiesel,M.Bourennane,C.Kurtsiefer,H.Weinfurter,Phys.Rev.Lett.92(2004)077901.[26]H.Mikami,Y.Li,T.Kobayashi,Phys.Rev.A70(2004)052308.[27]G.P.Guo,C.F.Li,J.Li,G.-C.Guo,Phys.Rev.A65(2002)042102.[28]G.-C.Guo,Y.-S.Zhang,Phys.Rev.A65(2002)054302.[29]S.B.Zheng,Phys.Rev.A70(2004)045804J.Opt.B:Quantum Semiclass.Opt.7(2005)10.[30] E.M.Becerra-Castro,W.B.Cardoso,A.T.Avelar,B.Baseia,J.Phys.B At.Mol.Opt.Phys.41(2008)215505.[31] C.Cabrillo,J.I.Cirac,P.García-Fernández,P.Zoller,Phys.Rev.A59(1999)1025.[32]X.Feng,Z.M.Zhang,X.D.Li,Z.Z.Xu,Phys.Rev.Lett.90(2003)217902.[33]L.M.Duan,H.J.Kimble,Phys.Rev.Lett.90(2003)253601.[34] D.E.Browne,M.B.Plenio,S.F.Huelga,Phys.Rev.Lett.91(2003)067901.[35]J.Cho,H.W.Lee,Phys.Rev.Lett.95(2005)160501.[36]T.Pellizzari,Phys.Rev.Lett.79(1997)5242.[37]Y.L.Zhou,Y.M.Wang,L.M.Liang,C.Z.Li,Phys.Rev.A79(2009)044304.[38] A.Seraﬁni,S.Mancini,S.Bose,Phys.Rev.Lett.96(2006)010503.[39]P.Peng,F.-L.Li,Phys.Rev.A75(2007)062320.[40]Z.Yin,F.Li,Phys.Rev.A75(2007)012324.[41]L.-B.Chen,M.-Y.Ye,G.-W.Lin,Q.-H.Du,X.-M.Lin,Phys.Rev.A76(2007)062304.[42]X.-Y.Lü,P.-J.Song,J.-B.Liu,X.Yang,Opt.Express17(2009)14298.[43]J.Song,Y.Xia,H.S.Song,J.Phys.B At.Mol.Opt.Phys.40(2007)4503.[44]S.-Y.Ye,Z.-R.Zhong,S.-B.Zheng,Phys.Rev.A77(2008)014303.[45]S.B.Zheng,Appl.Phys.Lett.94(2009)154101.[46]X.-Y.Lü,J.-B.Liu,C.-L.Ding,J.-H.Li,Phys.Rev.A78(2008)032305;X.-Y.Lü,L.-G.Si,M.Wang,S.-Z.Zhang,X.Yang,J.Phys.B At.Mol.Opt.Phys.41 (2008)235502.[47] D.F.V.James,Fortschr.Phys.48(2000)823.[48] D.F.V.James,J.Jerke,Can.J.Phys.85(2007)625.[49]J.Mckeever,J.R.Buck,A.D.Boozer,H.J.Kimble,Phys.Rev.Lett.93(2004)143601.[50] A.Boca,ler,K.M.Birnbaum,A.D.Boozer,J.Mckeever,H.J.Kimble,Phys.Rev.Lett.93(2004)233603.[51]S.-B.Zheng,Phys.Rev.A77(2008)045802.[52]S.-B.Zheng,G.-C.Guo,Phys.Rev.Lett.85(2000)2392.2981W.-A.Li/Optics Communications283(2010)2978–2981。

ACTION AND PASSION AT A DISTANCE An Essay in Honor of Professor Abner Shimony

TAUP2334-96 ACTION AND PASSION AT A DISTANCEAn Essay in Honor of Professor Abner Shimony∗Sandu PopescuDepartment of Physics,Boston University,Boston,MA02215,U.S.A.Daniel RohrlichSchool of Physics and Astronomy,Tel-Aviv University,Ramat-Aviv69978Tel-Aviv,Israel(May7,1996)AbstractQuantum mechanics permits nonlocality—both nonlocal correlations andnonlocal equations of motion—while respecting relativistic causality.Is quan-tum mechanics the unique theory that reconciles nonlocality and causality?We consider two models,going beyond quantum mechanics,of nonlocality—“superquantum”correlations,and nonlocal“jamming”of correlations—andderive new results for the jamming model.In one space dimension,jammingallows reversal of the sequence of cause and eﬀect;in higher dimensions,how-ever,eﬀect never precedes cause.∗To appear in Quantum Potentiality,Entanglement,and Passion-at-a-Distance:Essays for Ab-ner Shimony,R.S.Cohen,M.A.Horne and J.Stachel,eds.(Dordrecht,Netherlands:Kluwer Academic Publishers),in press.1I.INTRODUCTIONWhy is quantum mechanics what it is?Many a student has asked this question.Some physicists have continued to ask it.Few have done so with the passion of Abner Shimony.“Why is quantum mechanics what it is?”we,too,ask ourselves,and of course we haven’t got an answer.But we are working on an answer,and we are honored to dedicate this work to you,Abner,on your birthday.What is the problem?Quantum mechanics has an axiomatic structure,exposed by von Neumann,Dirac and others.The axioms of quantum mechanics tell us that every state of a system corresponds to a vector in a complex Hilbert space,every physical observable corre-sponds to a linear hermitian operator acting on that Hilbert space,etc.We see the problem in comparison with the special theory of relativity.Special relativity can be deduced in its entirety from two axioms:the equivalence of inertial reference frames,and the constancy of the speed of light.Both axioms have clear physical meaning.By contrast,the numerous axioms of quantum mechanics have no clear physical meaning.Despite many attempts, starting with von Neumann,to derive the Hilbert space structure of quantum mechanics from a“quantum logic”,the new axioms are hardly more natural than the old.Abner Shimony oﬀers hope,and a diﬀerent approach.His point of departure is a remark-able property of quantum mechanics:nonlocality.Quantum correlations display a subtle nonlocality.On the one hand,as Bell[1]showed,quantum correlations could not arise in any theory in which all variables obey relativistic causality[2].On the other hand,quantum correlations themselves obey relativistic causality—we cannot exploit quantum correlations to transmit signals at superluminal speeds[3](or at any speed).That quantum mechanics combines nonlocality and causality is wondrous.Nonlocality and causality seem prima facie incompatible.Einstein’s causality contradicts Newton’s action at a distance.Yet quan-tum correlations do not permit action at a distance,and Shimony[4]has aptly called the nonlocality manifest in quantum correlations“passion at a distance”.Shimony has raised the question whether nonlocality and causality can peacefully coexist in any other theory2besides quantum mechanics[4,5].Quantum mechanics also implies nonlocal equations of motion,as Yakir Aharonov[6,7] has pointed out.In one version of the Aharonov-Bohm eﬀect[8],a solenoid carrying an isolated magneticﬂux,inserted between two slits,shifts the interference pattern of electrons passing through the slits.The electrons therefore obey a nonlocal equation of motion:they never pass through theﬂux yet theﬂux aﬀects their positions when they reach the screen[9]. Aharonov has shown that the solenoid and the electrons exchange a physical quantity,the modular momentum,nonlocally.In general,modular momentum is measurable and obeys a nonlocal equation of motion.But when theﬂux is constrained to lie between the slits, its modular momentum is completely uncertain,and this uncertainty is just suﬃcient to keep us from seeing a violation of causality.Nonlocal equations of motion imply action at a distance,but quantum mechanics manages to respect relativistic causality.Still,nonlocal equations of motion seem so contrary to relativistic causality that Aharonov[7]has asked whether quantum mechanics is the unique theory combining them.The parallel questions raised by Shimony and Aharonov lead us to consider models for theories,going beyond quantum mechanics,that reconcile nonlocality and causality. Is quantum mechanics the only such theory?If so,nonlocality and relativistic causality together imply quantum theory,just as the special theory of relativity can be deduced in its entirety from two axioms[7].In this paper,we will discuss model theories[10–12] manifesting nonlocality while respecting causality.Theﬁrst model manifests nonlocality in the sense of Shimony:nonlocal correlations.The second model manifests nonlocality in the sense of Aharonov:nonlocal dynamics.Weﬁnd that quantum mechanics is not the only theory that reconciles nonlocality and relativistic causality.These models raise new theoretical and experimental possibilities.They imply that quantum mechanics is only one of a class of theories combining nonlocality and causality;in some sense,it is not even the most nonlocal of such theories.Our models raise a question:What is the minimal set of physical principles—“nonlocality plus no signalling plus something else simple and fundamental”as Shimony put it[13]—from which we may derive quantum mechanics?3II.NONLOCALITY I:NONLOCAL CORRELATIONSThe Clauser,Horne,Shimony,and Holt [14]form of Bell’s inequality holds in any classical theory (that is,any theory of local hidden variables).It states that a certain combination of correlations lies between -2and 2:−2≤E (A,B )+E (A,B )+E (A ,B )−E (A ,B )≤2.(1)Besides 2,two other numbers,2√2and 4,are important bounds on the CHSH sum ofcorrelations.If the four correlations in Eq.(1)were independent,the absolute value of the sum could be as much as 4.For quantum correlations,however,the CHSH sum ofcorrelations is bounded [15]in absolute value by 2√2.Where does this bound come from?Rather than asking why quantum correlations violate the CHSH inequality,we might ask why they do not violate it more .Suppose that quantum nonlocality implies that quantum correlations violate the CHSH inequality at least sometimes.We might then guess that relativistic causality is the reason that quantum correlations do not violate it maximally.Could relativistic causality restrict the violation to 2√2instead of 4?If so,then nonlocalityand causality would together determine the quantum violation of the CHSH inequality,and we would be closer to a proof that they determine all of quantum mechanics.If not,then quantum mechanics cannot be the unique theory combining nonlocality and causality.To answer the question,we ask what restrictions relativistic causality imposes on joint probabilities.Relativistic causality forbids sending messages faster than light.Thus,if one observer measures the observable A ,the probabilities for the outcomes A =1and A =−1must be independent of whether the other observer chooses to measure B or B .However,it can be shown [10,16]that this constraint does not limit the CHSH sum ofquantum correlations to 2√2.For example,imagine a “superquantum”correlation functionE for spin measurements along given axes.Assume E depends only on the relative angle θbetween axes.For any pair of axes,the outcomes |↑↑ and |↓↓ are equally likely,and similarly for |↑↓ and |↓↑ .These four probabilities sum to 1,so the probabilities for |↑↓4and|↓↓ sum to1/2.In any direction,the probability of|↑ or|↓ is1/2irrespective of a measurement on the other particle.Measurements on one particle yield no information about measurements on the other,so relativistic causality holds.The correlation function then satisﬁes E(π−θ)=−E(θ).Now let E(θ)have the form(i)E(θ)=1for0≤θ≤π/4;(ii)E(θ)decreases monotonically and smoothly from1to-1asθincreases fromπ/4to 3π/4;(iii)E(θ)=−1for3π/4≤θ≤π.Consider four measurements along axes deﬁned by unit vectorsˆa ,ˆb,ˆa,andˆb separated by successive angles ofπ/4and lying in a plane.If we now apply the CHSH inequality Eq.(1)to these directions,weﬁnd that the sum of correlationsE(ˆa,ˆb)+E(ˆa ,ˆb)+E(ˆa,ˆb )−E(ˆa ,ˆb )=3E(π/4)−E(3π/4)=4(2)violates the CHSH inequality with the maximal value4.Thus,a correlation function could satisfy relativistic causality and still violate the CHSH inequality with the maximal value4.III.NONLOCALITY II:NONLOCAL EQUATIONS OF MOTIONAlthough quantum mechanics is not the unique theory combining causality and nonlocal correlations,could it be the unique theory combining causality and nonlocal equations of motion?Perhaps the nonlocality in quantum dynamics has deeper physical signﬁcance.Here we consider a model that in a sense combines the two forms of nonlocality:nonlocal equations of motion where one of the physical variables is a nonlocal correlation.Jamming,discussed by Grunhaus,Popescu and Rohrlich[11]is such a model.The jamming paradigm involves three experimenters.Two experimenters,call them Alice and Bob,make measurements on systems that have locally interacted in the past.Alice’s measurements are spacelike separated from Bob’s.A third experimenter,Jim(the jammer),presses a button on a black box.This event is spacelike separated from Alice’s measurements and from Bob’s.The5black box acts at a distance on the correlations between the two sets of systems.For the sake of deﬁniteness,let us assume that the systems are pairs of spin-1/2particles entangled in a singlet state,and that the measurements of Alice and Bob yield violations of the CHSH inequality,in the absence of jamming;but when there is jamming,their measurements yield classical correlations(no violations of the CHSH inequality).Indeed,Shimony[4]considered such a paradigm in the context of the experiment of Aspect,Dalibard,and Roger[17].To probe the implications of certain hidden-variable the-ories[18],he wrote,“Suppose that in the interval after the commutators of that experiment have been actuated,but before the polarization analysis of the photons has been completed, a strong burst of laser light is propagated transverse to but intersecting the paths of the propagating photons....Because of the nonlinearity of the fundamental material medium which has been postulated[in these models],this burst would be expected to generate exci-tations,which could conceivably interfere with the nonlocal propagation that is responsible for polarization correlations.”Thus,Shimony asked whether certain hidden-variable theories would predict classical correlations after such a burst.(Quantum mechanics,of course,does not.)Here,our concern is not with hidden-variable theories or with a mechanism for jamming; rather,we ask whether such a nonlocal equation of motion(or one,say,allowing the third experimenter nonlocally to create,rather than jam,nonlocal correlations)could respect causality.The jamming model[11]addresses this question.In general,jamming would allow Jim to send superluminal signals.But remarkably,some forms of jamming would not; Jim could tamper with nonlocal correlations without violating causality.Jamming preserves causality if it satisﬁes two constraints,the unary condition and the binary condition.The unary condition states that Jim cannot use jamming to send a superluminal signal that Alice (or Bob),by examining her(or his)results alone,could read.To satisfy this condition,let us assume that Alice and Bob each measure zero average spin along any axis,with or without jamming.In order to preserve causality,jamming must aﬀect correlations only,not average measured values for one spin component.The binary condition states that Jim cannot use6jamming to send a signal that Alice and Bob together could read by comparing their results, if they could do so in less time than would be required for a light signal to reach the place where they meet and compare results.This condition restricts spacetime conﬁgurations for jamming.Let a,b and j denote the three events generated by Alice,Bob,and Jim, respectively:a denotes Alice’s measurements,b denotes Bob’s,and j denotes Jim’s pressing of the button.To satisfy the binary condition,the overlap of the forward light cones of a and b must lie entirely within the forward light cone of j.The reason is that Alice and Bob can compare their results only in the overlap of their forward light cones.If this overlap is entirely contained in the forward light cone of j,then a light signal from j can reach any point in spacetime where Alice and Bob can compare their results.This restriction on jamming conﬁgurations also rules out another violation of the unary condition.If Jim could obtain the results of Alice’s measurements prior to deciding whether to press the button,he could send a superluminal signal to Bob by selectively jamming[11].IV.AN EFFECT CAN PRECEDE ITS CAUSE!If jamming satisﬁes the unary and binary conditions,it preserves causality.These con-ditions restrict but do not preclude jamming.There are conﬁgurations with spacelike sep-arated a,b and j that satisfy the unary and binary conditions.We conclude that quantum mechanics is not the only theory combining nonlocal equations of motion with causality.In this section we consider another remarkable aspect of jamming,which concerns the time sequence of the events a,b and j deﬁned above.The unary and binary conditions are man-ifestly Lorentz invariant,but the time sequence of the events a,b and j is not.A time sequence a,j,b in one Lorentz frame may transform into b,j,a in another Lorentz frame. Furthermore,the jamming model presents us with reversals of the sequence of cause and eﬀect:while j may precede both a and b in one Lorentz frame,in another frame both a and b may precede j.To see how jamming can reverse the sequence of cause and eﬀect,we specialize to the7case of one space dimension.Since a and b are spacelike separated,there is a Lorentz frame in which they are simultaneous.Choosing this frame and the pair(x,t)as coordinates for space and time,respectively,we assign a to the point(-1,0)and b to the point(1,0). What are possible points at which j can cause jamming?The answer is given by the binary condition.It is particularly easy to apply the binary condition in1+1dimensions,since in 1+1dimensions the overlap of two light cones is itself a light cone.The overlap of the two forward light cones of a and b is the forward light cone issuing from(0,1),so the jammer, Jim,may act as late as∆t=1after Alice and Bob have completed their measurements and still jam their results.More generally,the binary condition allows us to place j anywhere in the backward light cone of(0,1)that is also in the forward light cone of(0,-1),but not on the boundaries of this region,since we assume that a,b and j are mutually spacelike separated.(In particular,j cannot be at(0,1)itself.)Such reversals may boggle the mind,but they do not lead to any inconsistency as long as they do not generate self-contradictory causal loops[19,20].Consistency and causality are intimately related.We have used the term relativistic causality for the constraint that others call no signalling.What is causal about this constraint?Suppose that an event(a“cause”) could inﬂuence another event(an“eﬀect”)at a spacelike separation.In one Lorentz frame the cause precedes the eﬀect,but in some other Lorentz frame the eﬀect precedes the cause; and if an eﬀect can precede its cause,the eﬀect could react back on the cause,at a still earlier time,in such a way as to prevent it.A self-contradictory causal loop could arise.A man could kill his parents before they met.Relativistic causality prevents such causal contradictions[19].Jamming allows an event to precede its cause,but does not allow self-contradictory causal loops.It is not hard to show[11]that if jamming satisﬁes the unary and binary conditions,it does not lead to self-contradictory causal loops,regardless of the number of jammers.Thus,the reversal of the sequence of cause and eﬀect in jamming is consistent.It is,however,suﬃciently remarkable to warrant further comment below,and we also show that the sequence of cause and eﬀect in jamming depends on the space dimension in a surprising way.8The unary and binary conditions restrict the possible jamming conﬁgurations;however, they do not require that jamming be allowed for all conﬁgurations satisfying the two con-ditions.Nevertheless,we have made the natural assumption that jamming is allowed for all such conﬁgurations.This assumption is manifestly Lorentz invariant.It allows a and b to both precede j.In a sense,it means that Jim acts along the backward light cone of j; whenever a and b are outside the backward light cone of j and fulﬁll the unary and binary conditions,jamming occurs.V.AN EFFECT CAN PRECEDE ITS CAUSE??That Jim may act after Alice and Bob have completed their measurements(in the given Lorentz frame)is what may boggle the mind.How can Jim change his own past?We may also put the question in a diﬀerent way.Once Alice and Bob have completed their measurements,there can after all be no doubt about whether or not their correlations have been jammed;Alice and Bob cannot compare their results andﬁnd out until after Jim has already acted,but whether or not jamming has taken place is already an immutable fact. This fact apparently contradicts the assumption that Jim is a free agent,i.e.that he can freely choose whether or not to jam.If Alice and Bob have completed their measurements, Jim is not a free agent:he must push the button,or not push it,in accordance with the results of Alice and Bob’s measurements.We may be uncomfortable even if Jim acts before Alice and Bob have both completed their measurements,because the time sequence of the events a,b and j is not Lorentz invariant;a,j,b in one Lorentz frame may transform to b,j,a in another.The reversal in the time sequences does not lead to a contradiction because the eﬀect cannot be isolated to a single spacetime event:there is no observable eﬀect at either a or b,only correlations between a and b are changed.All the same,if we assume that Jim acts on either Alice or Bob—whoever measures later—we conclude he could not have acted on either of them, because both come earlier in some Lorentz frame.9What,then,do we make of cause and eﬀect in the jamming model?We oﬀer two points of view on this question.One point of view is that we don’t have to worry;jamming does not lead to any causal paradoxes,and that is all that matters.Of course,experience teaches that causes precede their eﬀects.Yet experience also teaches that causes and eﬀects are locally related.In jamming,causes and eﬀects are nonlocally related.So we cannot assume that causes must precede their eﬀects;it is contrary to the spirit of special relativity to impose such a demand.Indeed,it is contrary to the spirit of general relativity to assign absolute meaning to any sequence of three mutually spacelike separated events,even when such a sequence has a Lorentz-invariant meaning in special relativity[20].We only demand that no sequence of causes and eﬀects close upon itself,for a closed causal loop—a time-travel paradox—would be self-contradictory.If an eﬀect can precede its cause and both are spacetime events,then a closed causal loop can arise.But in jamming,the cause is a spacetime event and the eﬀect involves two spacelike separated events;no closed causal loop can arise[11].This point of view interprets cause and eﬀect in jamming as Lorentz invariant;observers in all Lorentz frames agree that jamming is the eﬀect and Jim’s action is the cause.A second point of view asks whether the jamming model could have any other interpretation. In a world with jamming,might observers in diﬀerent Lorentz frames give diﬀerent accounts of jamming?Could a sequence a,j,b have a covariant interpretation,with two observers coming to diﬀerent conclusions about which measurements were aﬀected by Jim?(No ex-periment could ever prove one of them wrong and the other right[21].)Likewise,perhaps observers in a Lorentz frame where both a and b precede j would interpret jamming as a form of telesthesia:Jim knows whether the correlations measured by Alice and Bob are nonlocal before he could have received both sets of results.We must assume,however,that observers in such a world would notice that jamming always turns out to beneﬁt Jim;they would not interpret jamming as mere telesthesia,so the jamming model could not have this covariant interpretation.Finally,we note that a question of interpreting cause and eﬀect arises in quantum me-10chanics,as well.Consider the measurements of Alice and Bob in the absence of jamming. Their measured results do not indicate any relation of cause and eﬀect between Alice and Bob;Alice can do nothing to aﬀect Bob’s results,and vice versa.According to the con-ventional interpretation of quantum mechanics,however,theﬁrst measurement on a pair of particles entangled in a singlet state causes collapse of the state.The question whether Alice or Bob caused the collapse of the singlet state has no Lorentz-invariant answer[11,22].VI.JAMMING IN MORE THAN ONE SPACE DIMENSIONAfter arguing that jamming is consistent even if it allows reversals of the sequence of cause and eﬀect,we open this section with a surprise:such reversals arise only in one space dimension!In higher dimensions,the binary condition itself eliminates such conﬁgurations; jamming is not possible if both a and b precede j.To prove this result,weﬁrst consider the case of2+1dimensions.We choose coordinates(x,y,t)and,as before,place a and b on the x-axis,at(-1,0,0)and(1,0,0),respectively.Let A,B and J denote the forward light cones of a,b and j,respectively.The surfaces of A and B intersect in a hyperbola in the yt-plane.To satisfy the binary condition,the intersection of A and B must lie entirely within J.Suppose that this condition is fulﬁlled,and now we move j so that the intersection of A and B ceases to lie within J.The intersection of A and B ceases to lie within J when its surface touches the surface of J.Either a point on the hyperbola,or a point on the surface of either A of B alone,may touch the surface of J.However,the surfaces of A and J can touch only along a null line(and likewise for B and J);that is,only if j is not spacelike separated from either a or b,contrary to our assumption.Therefore the only new constraint on j is that the hyperbola formed by the intersection of the surfaces of A and B not touch the surface of J.If we place j on the t-axis,at(0,0,t),the latest time t for which this condition is fulﬁlled is when the asymptotes of the hyperbola lie along the surface of J.They lie along the surface of J when j is the point(0,0,0).If j is the point(0,0,0),moving j in either the x-or y-direction will cause the hyperbola to intersect the surface of J.We conclude that11there is no point j,consistent with the binary condition,with t-coordinate greater than0. Thus,j cannot succeed both a and b in any Lorentz frame(although it could succeed one of them).For n>2space dimensions,the proof is similar.The only constraint on j arises from the intersection of the surfaces of A and B.At a given time t,the surfaces of A and B are (n−1)-spheres of radius t centered,respectively,at x=−1and x=1on the x-axis;these (n−1)-spheres intersect in an(n−2)-sphere of radius(t2−1)1/2centered at the origin. This(n−2)sphere lies entirely within an(n−1)-sphere of radius t centered at the origin, and approaches it asymptotically for t→∞.The(n−1)-spheres centered at the origin are sections of the forward light cone of the origin.Thus,j cannot occur later than a and b.Weﬁnd this result both amusing and odd.We argued above that allowing j to succeed both a and b does not entail any inconsistency and that it is contrary to the spirit of the general theory of relativity to exclude such conﬁgurations for jamming.Nonetheless,weﬁnd that they are automatically excluded for n≥2.VII.CONCLUSIONSTwo related questions of Shimony[4,5]and Aharonov[7]inspire this essay.Nonlocality and relativistic causality seem almost irreconcilable.The emphasis is on almost,because quantum mechanics does reconcile them,and does so in two diﬀerent ways.But is quantum mechanics the unique theory that does so?Our answer is that it is not:model theories going beyond quantum mechanics,but respecting causality,allow nonlocality both ways.We qualify our answer by noting that nonlocality is not completely deﬁned.Relativistic causality is well deﬁned,but nonlocality in quantum mechanics includes both nonlocal correlations and nonlocal equations of motion,and we do not know exactly what kind of nonlocality we are seeking.Alternatively,we may ask what additional physical principles can we impose that will single out quantum mechanics as the unique theory.Our“superquantum”and “jamming”models open new experimental and theoretical possibilities.The superquantum12model predicts violations of the CHSH inequality exceeding quantum violations,consistent with causality.The jamming model predicts new eﬀects on quantum correlations from some mechanism such as the burst of laser light suggested by Shimony[4].Most interesting are the theoretical possibilities.They oﬀer hope that we may rediscover quantum mechanics as the unique theory satisfying a small number of fundamental principles:causality plus nonlocality“plus something else simple and fundamental”[13].ACKNOWLEDGMENTSD.R.acknowledges support from the State of Israel,Ministry of Immigrant Absorption, Center for Absorption in Science.13REFERENCES[1]J.S.Bell,Physics1,195(1964).[2]The term relativistic causality denotes the constraint that information cannot be trans-ferred at speeds exceeding the speed of light.This constraint is also called no signalling.[3]G.C.Ghirardi,A.Rimini and T.Weber,Lett.Nuovo Cim.27(1980)263.[4]A.Shimony,in Foundations of Quantum Mechanics in Light of the New Technology,S.Kamefuchi et al.,eds.(Tokyo,Japan Physical Society,1983),p.225.[5]A.Shimony,in Quantum Concepts in Space and Time,R.Penrose and C.Isham,eds.(Oxford,Claredon Press,1986),p.182.[6]Y.Aharonov,H.Pendleton,and A.Petersen,Int.J.Theo.Phys.2(1969)213;3(1970)443;Y.Aharonov,in Proc.Int.Symp.Foundations of Quantum Mechanics,Tokyo, 1983,p.10.[7]Y.Aharonov,unpublished lecture notes.[8]Y.Aharonov and D.Bohm,Phys.Rev.115(1959)485,reprinted in F.Wilczek(ed.)Fractional Statistics and Anyon Superconductivity,Singapore:World-Scientiﬁc,1990;[9]It is true that the electron interacts locally with a vector potential.However,the vectorpotential is not a physical quantity;all physical quantities are gauge invariant.[10]S.Popescu and D.Rohrlich,Found.Phys.24,379(1994).[11]J.Grunhaus,S.Popescu and D.Rohrlich,Tel Aviv University preprint TAUP-2263-95(1995),to appear in Phys.Rev.A.[12]D.Rohrlich and S.Popescu,to appear in the Proceedings of60Years of E.P.R.(Work-shop on the Foundations of Quantum Mechanics,in honor of Nathan Rosen),Technion, Israel,1995.14[13]A.Shimony,private communication.[14]J.F.Clauser,M.A.Horne,A.Shimony and R.A.Holt,Phys.Rev.Lett.23,880(1969).[15]B.S.Tsirelson(Cirel’son),Lett.Math.Phys.4(1980)93;ndau,Phys.Lett.A120(1987)52.[16]For the maximal violation of the CHSH inequality consistent with relativity see also L.Khalﬁn and B.Tsirelson,in Symposium on the Foundations of Modern Physics’85,P.Lahti et al.,eds.(World-Scientiﬁc,Singapore,1985),p.441;P.Rastall,Found.Phys.15,963(1985);S.Summers and R.Werner,J.Math.Phys.28,2440(1987);G.Krenn and K.Svozil,preprint(1994)quant-ph/9503010.[17]A.Aspect,J.Dalibard and G.Roger,Phys.Rev.Lett.49,1804(1982).[18]D.Bohm,Wholeness and the Implicate Order(Routledge and Kegan Paul,London,1980);D.Bohm and B.Hiley,Found.Phys.5,93(1975);J.-P.Vigier,Astr.Nachr.303,55(1982);N.Cufaro-Petroni and J.-P.Vigier,Phys.Lett.A81,12(1981);P.Droz-Vincent,Phys.Rev.D19,702(1979);A.Garuccio,V.A.Rapisarda and J.-P.Vigier,Lett.Nuovo Cim.32,451(1981).[19]See e.g.D.Bohm,The Special Theory of Relativity,W.A.Benjamin Inc.,New York(1965)156-158.[20]We thank Y.Aharonov for a discussion on this point.[21]They need not be incompatible.An event in one Lorentz frame often is another eventin another frame.For example,absorption of a virtual photon in one Lorentz frame corresponds to emission of a virtual photon in another.In jamming,Jim might not only send instructions but also receive information,in both cases unconsciously.(Jim is conscious only of whether or not he jams.)Suppose that the time reverse of“sending instructions”corresponds to“receiving information”.Then each observer interprets the sequence of events correctly for his Lorentz frame.15。

Massive creation of entangled exciton states in semiconductor quantum dots

with heﬀ = ho + hop − iΓ accounting for: ho , the Coulomb-renormalized few-particle states x; hop , the light-coupling described within the usual rotating-wave and dipole approximations;19 iΓ, dephasing and relaxation due to environment interactions; ﬁnally, J accounts for in-scatterings which guarantee that the trace of ρ is preserved at all times.20,21 In this paper we shall consider low temperatures throughout, and thus take spontaneous photon emissions as the only source of dephasing and relaxation.8,22 As regarding the time evolution of the light pulse, we assume the geometry depicted in Fig. 1a of a laser
PACS numbers: 42.50.Md, 03.65.Ud, 78.6one of the most intriguing consequences of quantum mechanics which completely lacks a classical counterpart. In particular within the context of the emerging ﬁelds of quantum computation1,2 and quantum communication3,4 it has become clear that entanglement provides the utmost viable element for such future technology, and numerous recent work has been devoted to the preparation and measurement of entangled states in real physical systems. However, the interaction of a quantum system with its environment unavoidably introduces an uncontrollable element to the system’s time dynamics, thus spoiling the direct exploitation of entanglement. Noteworthy, such environment losses hitherto seem to be only controllable in a few atomic and photonic systems, but become prohibitively large in the technologically more interesting solid state. Consequently, the identiﬁcation of long-lived and suﬃciently well protected solid-state excitations has now turned into the most pertinent issue within this area of research. In this respect, semiconductor quantum dots,5,6,7 or artiﬁcial atoms as they are sometimes called because of their atomic-like carrier states, provide a promising new class of material, which resembles many of the atomic properties whilst oﬀering at the same time all the ﬂexibility of semiconductor nanostructures. Quantum dots consist of a small island of lower-bandgap material embedded in a solid-state matrix of higher-bandgap material. Proper choice of the material and dot parameters thus allows the conﬁnement of a few carrier states within this lower-bandgap region, resulting in discrete spectra and strongly enhanced lifetimes: indeed, remarkably long dephasing times have been recently reported for optical excitations (excitons) which were solely governed by radiative decay.8 Another property that has attracted enormous interest is the possibility to create several electronhole pairs (multi-excitons) within a single quantum dot, where, because of Coulomb renormalziations, the spectra exhibit a surprisingly rich ﬁne structure.9,10 We emphasize that it is precisely this Coulomb correlation eﬀect which is at the heart of quantum-dot based single-photon sources11,12 and which recently allowed for an optically induced exciton entanglement.13,14 In this paper it is shown that a strong laser pulse propagating in a macroscopic sample of inhomogeneously broadened quantum dots massively creates en-

Tavis Cummings模型中原子的纠缠度

Journ al of Sha ngha i Univ er sity (En glish Edition ),2006,10(3):215-218Article ID :1007 6417(2006)03 0215 04Entanglement of atoms in Tavis Cummings modelDONG Chuan hua (董传华), ZHANG Ya li (张亚利)Depar tm ent of Phy sics ,College of Scien ces ,Shan ghai Un iver sity ,Shan gha i 200444,P .R .ChinaReceived Sep.24,2004; Revised Ja n.7,2005DO NG Chuan hua,Prof.,E mail:chdong@ma Abstract Entanglem ent is used to measure correlation between separated subsystems.von Neumann entropy is used to study evolu tions of entanglement of atoms in processes of interaction between atoms with the field prepared in coherent state.The effects of field intensity and detuning on entanglement a re investiga ted.It is shown that the entanglement exhibited osc illations in its evolutions,their amplitudes and mean values decrease with inc reasing field intensity.Oscillation frequenc ies increase with detuning,but the maximum values a re almost independent of detuning.Key words entanglem ent,entangled state,Jaynes Cummings model.PACS 2001 42.50. P1 IntroductionSuperposition principle is one of the fundamentals in quantum m echanics.A two level atom may be in the ground state |0 ,the excited state |1 o r the superpo sition states C 0|0 +C 1|1 .The coefficients C 0and C 1are complex probability amplitudes of |0 and |1 respectively.For a system consisting of two atoms,the state of total system may be in the direct p roduct states |0,1 or |1,0 .Acco rding to the superposition principle,the total system may also be in the superpo sition states 1 2(C 0|0,1 +C 1|1,0 ).This kind of superposition states of multi particle is called entan gled state.Recently,people have devoted much attention to the applications of entangled states in quantum information,such as quantum teleporting [1,2],quantumcopying and cloning [3],quantum cryptography andquantum secret sharing [4],and quantum computation [5].Various schemes of preparing entangledstates have been proposed [6].The concept of entan gling has been generalized to the states of multi parti cle system,i .e .,G reenberger Ho rne Zeilinger states.Various methods of preparing G H Z states have alsobeen proposed [7,8].Entangling can be realized with non local correlation and entanglement is defined to measure the degree ofnon local correlation.Some definitions of entanglement have been given [9,10].Zuo and Xia have studied the evolution properties of three body entanglement of two identical two level atoms in the presence of fieldin vacuum [11].Rendell and Rajagopal have studied the entanglement of initially entangled mixed state in damped Jaynes Cummings model and found the timeevolutions of the concurrence lower bound [12].In this paper,we will use the partial entropy to study evolu tions of entanglement for two atoms in the presence of the field in coherent state with Tavis Cummings model.Effects of light intensity and the detunning on evolutions of entanglement are investigated.2 Definition of entanglementConsidering a system consisting of two atoms A and B,its entanglement (partial entropy entanglement)E is defined with von Neumann entropy,i .e .E =S ( A )=S ( B ),(1)where A and B are reduced density operators of at oms A and B.von Neumann entropy is a generalization of Shannon entropy in classic system to quantum sys tem,which describes the quantum co rrelation betweenthe sub systems [13].von Neumann entropy is defined asS ( )=-tr( log 2 ).(2)With Schmidt decomposition,the density operato rs of atoms A and B can be written in the fo rmA=tr B( A B)=mj=12j|j A A!j|,B=tr A( A B)=mj=1 2j|j B B!j|,(3)where m∀min{dim H A,dim H B},H A and H B are the Hilbert subspaces of atoms A and B,respectively. It follows that A and B have m non zero eigenvalues j(j=1-m).The relation between E and j isE=-mj=1j log2 j.(4)For two atoms in the entangled states| AB=C1|1,1 +C2|1,0 +C3|0,1+C4|0,0 ,(5) the eigenvalues of A(or B)arej=121-(-1)j1-4|C1C4-C2C3|2,j=1or2.(6)The partial entropy entanglement can be calculated as follows.E=-( 1log2 1+ 2log2 2).(7) 3 Entanglement of two atoms in TavisCummings modelConsidering a system consisting of two level atoms A and B,the ground state and excited state are denot ed|0 and|1 respectively.The light field is initially prepared in the coherent state|! =nf n|n ,where f n=exp(- n 2)!n n!and n is mean photon num ber.Two atoms are in the entangled state|(0) ato m=1 2(|0,1 +|1,0 ).(8) This entangled state is one of Bell bases,which is maximally entangled state with entanglement E=1. The state evolves from the initial state to|(t) at time t>0|(t) =C1(t)|1,1;n-1 +C2(t)|1,0;n+C3(t)|0,1;n +C4(t)|0,0;n+1 ,(9) where the coefficients C j(t)(j=1-4)are to be de termined.In the interaction picture,the Hamiltonian of the system consisting of two atoms and a field in Tavis Cummings model is,H I=g(a+S-e i∀t+S+a e-i∀t),(10)where∀=#-#0,#0is the transiting frequency of at oms and#is the f requency of field,S+=S(A)++S(B)+, S-=S(A)-+S(B)-,and S(j)#(j=A or B)are the transit ing operato rs of atom A(or B).Substituting the inter action Hamiltonian Eq.(10)and the state vector Eq.(9)into the Schr dinger equation and solving it using a standard p rocedure with the initial condition Eq.(8), the coefficients C j(t),(j=1∃4)in Eq.(9)can be obtained as follows.C1(t)=-2n gm1∃1-∀[ei(∃1-∀)t-1]+m2∃2-∀[ei(∃2-∀)t-1]+m3∃3-∀[ei(∃3-∀)t-1],C2(t)=C3(t)=m1e i∃1t+m2e i∃2t+m3e i∃3t,C4(t)=-2n+1gm1∃1+∀[ei(∃1+∀)t-1]+m2∃2+∀[ei(∃2+∀)t-1]+m3∃3+∀[ei(∃3+∀)t-1],(11)wherem1=f n2∃2∃3+2(2n+1)g2(∃1-∃2)(∃1-∃3),m2=f n2∃1∃3+2(2n+1)g2(∃2-∃1)(∃2-∃3),m3=f n2∃1∃2+2(2n+1)g2(∃3-∃1)(∃3-∃2)(12)and∃1=2r1 3cos%,∃2=2r1 3cos(%+2& 3),∃3=2r1 3cos(%+4& 3),(13) r=[2(2n+1)g2+∀2]3 27,%=1 3arccos(-∀g2 r).(14) In the case of∀=0,C j(t)can be reduced toC1(t)=-i2nf n g sin(∋n t) ∋n,C2(t)=C3(t)=2-1 2f n cos(∋n t),C4(t)=-i2(n+1)f n g sin(∋n t) ∋n,(15) where Rabi frequency is2∋n and.(16)216J our nal of Shan gha i Univer sityThe reduced density operator of atom A can beobtainedA = n(|C 1|2+|C 2|2)|1 !1|+(|C 3|2+|C 4|2)|0 !0|+(C *2C 4+C *1C 3)|0 !1| +(C 2C *4+C 1C *3)|1 !0|,(17)its eigenvalues are =0.5(1#1-(),(18)where(=4[ n (|C 1|2+|C 2|2) n(|C 3|2+|C 4|2)- n(C 1C *3+C 2C *4)2].(19)Using Eq.(7),the evolutions of entanglement can be obtained,which are shown in Fig.1for the case of resonance.The atoms A and B are prepared initially in Bell base |1,0 +|0,1 ,so the initial entanglement is 1.In the processes of interaction between the atoms and the field,the state of the system consisting of at oms A and B will deviate from this Bell base.In case of resonance,the states will evolve to the state | (t ) at a time t >0,which can be rewritten as | (t ) =1 2 nf n {cos (∋n t )(|1,0;n +|0,1;n ) -i sin (∋n t )[1-1 (2n +1)|1,1n -1+1+1 (2n +1)|0,0;n +1 ]}=1 2 n f n {cos (∋n t )(|1,0;n +|0,1;n ) -i sin (∋n t )[(|1,1;n -1 +|0,0;n +1 )-1 (4n +2)(|1,1;n -1 -|0,0;n +1 )+%]}.(20)It can be seen from Eq.(20)that the other two Bellbases |1,1 #|0,0 appear in atom field interaction.These Bell bases evolve according to cosine and sine with frequency ∋n that is different for various pho ton numbers n .Superposing these Bell bases form the os cillations of entanglement in evolution. In a weaker field ( n <4),the valley values of oscil lations rise obviously with increasing field intensity so that the mean values of entanglements in the oscil lations increase (see Fig.1(a)-(c)).In a stronger field ( n >4),the mean values of entanglements de crease rapidly (see Fig.1(d)and (e)),and the am plitudes shrink sharply with enhancing the field.When n =15,entanglement is maintained roughly constant inevolution.Thus,enhancing the field makes the atomsdetangle.This can be explained by the fact that thequantum co rrelation is weakened due to enhancing the interaction of two atoms with the field respectively in a stronger field.Fig.1 Entanglement of two atoms in the case of resonanc e((a) n =0.2,(b) n =1.0,(c) n =4.0,(d) n =9.0,(e) n =15.0)Evolution of entanglement in off resonance are shown in Fig.2(for weaker field, n =0.2)and Fig.3(for stronger field, n =9).From these figures,it can be seen that oscillation frequencies increase and the amplitudes shrink with increasing detuning in a weaker field.Except these oscillations,there are slow varia tions with large amplitudes in evolution for a stronger field and large detuning.The minimum values rise in a weaker field (see Fig.2)and drop in a stronger field (see Fig.3)with increasing the detuning.The maxi mum values of entanglement in the evolution are inde pendent of detuning almost,while dependent only on the field intensity.In summary,in interactions between atoms and the field,entanglements of two atoms will oscillate due to the Rabi oscillations.The field intensity and the detun ing affect evolutions of atom entanglement.Entanglements of two atoms depend on the field in tensity.The mean entanglem ents increase slightly in a weak field and decrease obviously in a strong field217Vol.10 No.3 Jun.2006DONG C H,et al .: Entanglem ent of atom s in Tavis Cummings modelFig.2 Entanglement of two atoms in the case of off resonanc efor n =0.2((a)∀=0.5g ,(b)∀=2.0g ,(c)∀=4.0g ,(d)∀=10.0g ,(e)∀=20.5g)Fig.3 Entanglement of two atoms in the case of off resonanc efor n =9.0((a)∀=0,(b)∀=0.5g ,(c )∀=1.0g ,(d)∀=4.0g ,(e)∀=10g )with increasing field intensity.It follows that a strong field makes the atoms detangle.The amplitudes of oscillations of entanglements shrink sharply with in creasing field intensity.The maximum values of entan glement are independent of detuning but dependent on the field intensity.The oscillation frequencies increase and the amplitudes shrink in a weak field with the increasing detuning.References[1] Bennett C H,Wiesner S munication v ia one andtwo particle operators on Einstein Podolsky Rosen states [J].Phys .Rev .Lett .,1992,69(20):2881-2884.[2] Bennett C H,Brassard G,Cr peau C,et al .Teleportingan unknown quantum state via dual classical and Ein stein Podolsky Rosen c hannels [J].Phy s .Rev .Lett .,1993,70(13):1895-1898.[3] Buzek V,H illery M.Quantum copying:Beyond the nocloning theorem[J].Phys .Rev .,1996,A54(3):1844-1852.[4] Hillery M,Buzek V,Berthiaume A.Quantum secret sharing[J].Phys .Rev .,1999,A59(3):1829-1834.[5] Anders S rensen,Klaus M lme r.Entanglement and quantum computation with ions in the therm al motion [J ].Phys .Rev .,2000,A 62(2):022311 1-022311 11.[6] Cirace J I,Zoller P.Preparation of macroscopic superposition in many atom systems [J ].Phy s .Rev .,1994,A50(4):R2799-R2802.[7] Yao C M,Guo G C.Generation of spin type G HZ statesof the cavity field in squeezed coherent states[J].Acta Phy sica Sinica ,2001,50(1):59-62(in Chinese).[8] Liu X,Li H C.Prepa ration of multi atom GHZ states v iathe Raman interaction of V type three level atoms and one cavity field[J].Acta Phy sica Sin ica ,2001,50(9):1689-1692(in Chinese).[9] Vedral V,Plenio M B,Rippin M A,et al .Quantifyingentanglement[J].Phys .Rev .Lett .,1997,78(12):2275-2279.[10] Vedral V,Plenio M B.Entanglement measures and purification procedures[J].Phy s .Rev .,1998,A57:1619-1633.[11] Zhou Z C,Xia Y J.The evolution property of three bodyentanglement measure in Tavis Cumm ings model [J ].Acta Physica Sinica ,2003,52(11):2687-2693(in Chinese).[12] Rendell R W,Rajagopal A K.Revivals and entanglementfrom initially entangled mixed sta tes of a damped Jaynes Cumm ings model [J].Phys .Rev .,2003,A67(6):062110 1-062110 11.[13] von Neumann.Mathem atical F oundat ion s of Qua ntumMechan ics [M].Princeton University Press,Princeton,NJ,1955.(Editor YAO Yu e yuan )218J our nal of Shan gha i Univer sity。

量子相变中的量子纠缠度量研究-英文版

RG flow
• c-theorem:
cUV cIR
fixed point
(Zamolodchikov, 1986)
UV
fixed point
IR
• Massive theory (off critical) Block entropy saturation
fixed point
RG flow
UV
Irreversibility of RG trajectories
Loss of entanglement
Open Systems & Quantum Information Milano, 10 Marzo 2006
• Local Entropy: when the subsystem A is a single site.
• The concurrence measures the entanglement between two sites after having traced out the remaining sites. • The transition is signaled by the first derivative of the concurrence, which diverges logarithmically (specific heat).

h 1 lnh 1
S1 has the same singularity as
Local (single site) entropy: 1 ρ1 = I + σ z σ z S1 = Tr ρ1lnρ1 2
σz

Local measures of entanglement based on the 2-site density matrix depend on 2-point functions

新托福TPO3阅读原文(三)The Long-Term Stability of Ecosystems及译文

新托福TPO3阅读原文(三):The Long-Term Stability of Ecosystems TPO-3-3：The Long-Term Stability of EcosystemsPlant communities assemble themselves flexibly, and their particular structure depends on the specific history of the area. Ecologists use the term “succession”to refer to the changes that happen in plant communities and ecosystems over time. The first community in a succession is called a pioneer community, while the long-lived community at the end of succession is called a climax community. Pioneer and successional plant communities are said to change over periods from 1 to 500 years. These changes—in plant numbers and the mix of species—are cumulative. Climax communities themselves change but over periods of time greater than about 500 years.An ecologist who studies a pond today may well find it relatively unchanged in a year’s time. Individual fish may be replaced, but the number of fish will tend to be the same from one year to the next. We can say that the properties of an ecosystem are more stable than the individual organisms that compose the ecosystem.At one time, ecologists believed that species diversity made ecosystems stable. They believed that the greater the diversity the more stable the ecosystem. Support for this idea came from the observation that long-lasting climax communities usually have more complex food webs and more species diversity than pioneer communities. Ecologists concluded that the apparent stability of climax ecosystems depended on their complexity. To take an extreme example, farmlands dominated by a single crop are so unstable that one year of bad weather or the invasion of a single pest can destroy the entire crop. In contrast, a complex climax community, such as a temperate forest, will tolerate considerable damage from weather to pests.The question of ecosystem stability is complicated, however. The first problem is that ecologists do not all agree what “stability”means. Stability can be defined as simply lack of change. In that case, the climax community would be considered the most stable, since, by definition, it changes the least over time. Alternatively, stability can be defined as the speed with which an ecosystem returns to a particular form following a major disturbance, such as a fire. This kind of stability is also called resilience. In that case, climax communities would be the most fragile and the least stable, since they can require hundreds of years to return to the climax state.Even the kind of stability defined as simple lack of change is not always associated with maximum diversity. At least in temperate zones, maximum diversity is often found in mid-successional stages, not in the climax community. Once a redwood forest matures, for example, the kinds of species and the number of individuals growing on the forest floor are reduced. In general, diversity,by itself, does not ensure stability. Mathematical models of ecosystems likewise suggest that diversity does not guarantee ecosystem stability—just the opposite, in fact. A more complicated system is, in general, more likely than a simple system to break down. A fifteen-speed racing bicycle is more likely to break down than a child’s tricycle.Ecologists are especially interested to know what factors contribute to the resilience of communities because climax communities all over the world are being severely damaged or destroyed by human activities. The destruction caused by the volcanic explosion of Mount St. Helens, in the northwestern United States, for example, pales in comparison to the destruction caused by humans. We need to know what aspects of a community are most important to the community’s resistance to destruction, as well as its recovery.Many ecologists now think that the relative long-term stability of climax communities comes n ot from diversity but from the “patchiness”of the environment, an environment that varies from place to place supports more kinds of organisms than an environment that is uniform. A local population that goes extinct is quickly replaced by immigrants from an adjacent community. Even if the new population is of a different species, it can approximately fill the niche vacated by the extinct population and keep the food web intact.译文：TPO-3-3 生态系统的长期稳定植物群体可以自由地聚集，它们特殊的结构取决于聚集区域的具体历史。

1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。

Non-local generation of entanglementof photons which do not meet each otherJürgen Rösch 1, Xian-Min Jing 2, Juan Yin 2, Tao Yang 2, Jian-Wei Pan 1,21 Physikalisches Institut, Universit¨at Heidelberg, Philosophenweg 12, D-69120 Heidelberg, Germany2 Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics,University of Science and Technology of China, Hefei, Anhui 230026, ChinaWe report for the first time in an ancilla-free process a non-local entanglement between two single photons which do not meet. For our experiment we derive a simple and efficient method to entangle two single photons using post-selection technology. The photons are guided into an interferometer setup without the need for ancilla photons for projection into the Bell-states. After passing the output ports, the photons are analyzed using a bell state analyzer on each side. The experimental data clearly shows a non-local interaction between these photons, surpassing the limit set by the CHSH-inequality with an S-value of 2.54 and 24 standard deviations.In the year 1935 Einstein, Podolsky and Rosen (EPR) published their famous article ”Can Quantum- Mechanical Description of Physical Reality Be Considered Complete?” [1, 2] in which they did show the incompatibility between a local realist point of view and the way quantum mechanics looks upon the world. Originally, EPR wanted to show that quantum mechanics in its current form is incomplete - and needs further description which later led to the concept of hidden variables. Since the introduction of Bell’s in 1964 many experiments (seee.g. [4–6]) were performed indicating that nature behaves like the mathematical model of quantum mechanics predicts. The first Bell tests have been carried out by Freedman and Clauser [9] who investigated polarization correlations for photons emitted in a cascade process from a calcium source. Experiments which came closer to Bell’s’ original idea were performed by Aspect and his colleagues[4, 10, 11] where polarization settings where changed within the flight time of the photons and by Weihs et al. [5] who implemented a much stronger condition on locality with truly independent observers where the decision about measurement settings was based on pure statistics.Creation of entanglement has been demonstrated by various methods which are illustrated in figure 1(a)-1(c). Most of the experiments use a down-conversion source (SPDC) to generate entangled pairs of photons. In this way photons get already entangled at the source where they are created (1a). A technically more challenging method demonstrated by Shih and Alley in [12] is to take orthogonally polarized photons, e.g. one H and one V polarized photon (from preferably independent sources), and overlap them spatially and temporally on a beamsplitter with equal reflectivities (illustration 1b). It is to mention that the processes in figure 1a and 1b require for the two photons (as shown in figure 1a) to interact with each A completely different approach (illustrated in 1c) has been undertaken by Pan et al. in [13] where they use four photons produced by two down-conversion sources prepared in the state()()12341212343412H V V H H V V H ψ=−⊗−Conditional on a projective measurement performed on photon 2 and 3, which are prepared in the antisymmetric |Ψ->-state and which overlap on a beamsplitter, the photons 1 and 4 are projected in that one of the four Bell states that is equal to the state of photons 2 and 3. Using this method to create entanglement of photon 1 and photon 4, two ancilla photons are needed as mediator.In figure 1(d) a simplified drawing is given which shows the basic principle of our experiment. In contrast to previous setups, our experiment is the first of its kind (ancilla-free process) that creates an entangled Bell State using two individual photons which do not meet each other. The essential experimental setup is illustrated in figure 2. The idea is based on a slightly modified version of the theoretical concept of Pati an Zukowski [21]. Two single photons are guided onto an 8-port interferometer consisting of two beamsplitters with transmission/reflection ratio of 50/50 and a pair of polarizing beamsplitters. For each photon 1 and 2 there is a 50 percent chance to finally reach either one of the detectors D1/D2 on Alice’s side and a 50 percent chance to reach either oneof the detectors D3/D4 on Bob’s side. The electronic circuit connected to the detectors registers the number of coincidences between the detectors D1 andD3, D1 and D4, D2 and D3 and D2 and D4, i.e. coincidences are defined as cases where one photon arrives at Bob’s side and one photon arrives at Alice’s side. Given these cases, each photon will travel a different path and the individual geometric paths of the photons do not cross each other. Whenanalyzing the photons after the polarizing beamsplitters we can show that an entanglement takes place even when there is no direct ”contact/encounter” of one photon with the other. It can be said that there is no possibility of local interaction of the photons similar to the entanglement swapping experiment of Pan et al. [13]. In our experiment we go a step further and present a sinusoidal curve which has visibility high enough to conduct a Bell-type measurement [7] with a convincing violation of the CHSH inequality. For the setup we use a 460 mW single mode Argon-Ion UV Laser beam focused (waist 80 µm) into the center of a 2 mm long beta-barium-borate (BBO) crystal for type-II phase matching. Unwanted laser-fluorescence is minimized with a dispersion prism. Extraordinary and ordinary photons have different velocities and travel along different paths inside the crystal due to the birefringence of the BBO. The resulting walk-off effects are compensated by a combination of a half wave plate and an additional BBO of half the length of the downconversion crystal in each arm. Further polarization optics allow to create a Phi state, namely)−Φ=Since we need in our experiment a high quality single photon source, but not an entangled state of photons, we destroy the entanglement. This can be achieved by filtering out either one, either the HH case or the VV case and can be realized using polarization filters. Hence, the input for the interferometer becomes:()12input V V ψ=One key to our experiment is to eliminate any timing information which would make the photons distinguishable. To ensure that both photons enter the interferometer at the same time, we allow path adjustments using an optical trombone prism (prism 2 in figure 2). Also it is important to ensure identical temporal modes inside the interferometer, therefore we introduce two additional optical trombones, prism 1 and prism 3. To check whether the condition of equal path lengths is fulfilled a coincidence measurement at PBS 3 and PBS 4 is performed. As can be seen in figure 3 and 4, we achieve a visibility of 92.3% and 93,4% respectively. Theory would expect a 100% visibility, which is equivalent to a complete breakdown of coincidences for two impinging photons on different sides of the beamsplitter. Practically, this value suffers from small imperfections which are mainly a tribute to slightly imperfect mode-matching. At BS2, Photon 2, which is vertically polarized, has the choice to go to either Alice’s side or Bob’s side, but its state will in any case be converted to H. Analyzing all pathways which could lead to one of the four possible combinations of coincidences between any of the detectors on Alice’s side and one on Bob’s side shows:Therefore photons are projected into a maximally entangled state:)Ψ=where we can arbitrarily change φ with a nanometer stepsize piezo and therewith have full control over the phase. It is important to note, that for all coincidences the two photons do not encounter each other. The argument is independent of the dualism of light and works in the particle framework as well as in the framework of wavepackets. To test for non-local interactions we make use of the inequality first derived by Clauser, Horne, Shimony, and Holt (CHSH) [7, 8] which is more suitable for measurements with polarized photons. This variant of Bell’s inequality can be expressed by:2≤Swhere()()()()1234,',,'S E E E E','αβαβαβα=++−βis the so called Bell-parameter with()()()()()()()(),,,,,,,,xN N N NEN N N Nαβαβαβαβαβαβαβαβ++−−+−−+++−−+−−++−−=+++.In each run of the four subexperiments of the CHSH-test the polarizers take one of two possible settings. The measured E-values are coincidence expectation values for different settings of the half wave plates in front of the polarizing beamsplitters PBS 5 and PBS 6. A comparison of classically expected coincidences with quantum mechanically expected coincidences [7, 8] shows that under classical assumptions an absolute value for the Bell-parameter is expected which is not larger then 2 whereas the laws of quantum mechanics≈.allow for S to reach values up to 2.81We took data for the following settings of the half wave plates: α = 22.5° (⇒ 45°), β = 45° (⇒ 90°), α = 67.5° (⇒ 135°), β' = 0° (⇒ 0°).Parameters for the angles of the half wave plates are chosen such that the violation is supposed to be at its maximum value. Hence, with the data from table 1 we obtain as the result for the Bell parameter S a value of 2.54 (σ = ±0.023). Therefore the classical value of 2 is violated with 24.08 standard deviations. Applying the CHSH-inequality implicates the ’fair sampling’ assumption [17]. For light intensities on the single photon level, highly sensitive photodetectors are needed in the experiment (we used SPCM-AQR13 operated in Geiger-mode). At the current level of technology such detectors have typical quantum efficiencies of 74% at our wavelength (702 nm) and as consequence not all of the created photons can be registered. ’Fair sampling’ assumes that the registered and analyzed photons are a fair sample of all emitted photons and do behave thesame way as the uncollected photons, i.e. we assume that the photons registered by the detectors are a representative sample of all photons created [10].Ultra high stability has been a major challenge for our experiment. The interferometer’s geometry has carefully been designed in a way which minimizes unwanted changes in path length due to disturbances. Laboratory air-conditioning did show a strong effect on the data. A thick acrylic glass cover was constructed protecting the interferometer from unwanted air-flow and temperature drifts. The cover was designed to allow control for all necessary angular settings from the outside for all four subexperiments. After each subexperiment we performed a measurement in the 45° basis to confirm that we are still in the ψ−−state. Compactness of the setup, highly stable pillar posts and the custom built glass cover enabled us to keep the phase stable for several minutes. Detection events were registered by a self-developed constant fraction discriminator and self-developed counting card in combination with NIM-electronics for the logic.In summary, our work supports that creation of entanglement in between two photons is possible, even if there is no direct local interaction between the involved photons. The possible choices of the paths the photons can take can be seen as the underlying cause for the observed non-local entanglement-generation.It can be argued that 50 percent of all produced photons that enter the interferometer meet each other (25 percent at PBS 3 and 25 percent at PBS4 assuming no other losses). In all these cases the photons go to the same side, either both to Alice or both to Bob such that no coincidences occur. However, these events are filtered out (post selection) and are not registered by the electronics. In different words, 50 percent of all incoming photons meet but these are not the cases for which we prove entanglement. As shown by Popescu, Hardy and Zukowski in [20], post selection does not prevent us to perform entanglement verification.It is to mention that the observed two-photon interference is deeply rooted in the fundamentals of quantum mechanics and the question for hidden variables. The practical applications of the EPR-principle extend to important topics like quantum cryptography (see [18, 19]) and will be helpful in optics-based quantum information processing. We would like to thank Mr. Flammia for enlightening discussions on the topic. We acknowledge the support by the Marie Curie Excellence Grant of the EU and the Alexander von Humboldt Foundation, the National Natural Science Foundation of China and the Chinese Academy of Sciences.This version is a rough draft. We put it onto the quantum server to share our measurement results with the scientific community. Discussion is encouraged.References:[1] A. Einstein, B. Podolsky, N. Rosen, Institute for Advanced Study, Princeton, New Jersey ( March 25, 1935).[2] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935).[3] J.S. Bell, Physics (Long Island City, N.Y.) 1, 195 (1964).[4] A. Aspect, J. Dalibard, G. Roger, Phys. Rev. Lett. 49, 1804 (1982)[5] G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, A. Zeilinger Phys. Rev. Lett. 81, 5039 (1998)[6] T. Yang, Q. Zhang, J. Yin, Z-B. Chen, Z. Zhao, M. Zukowski, J-W. Pan, Phys. Rev. Lett. 95 240406 (2005)[7] J.F. Clauser, M.A. Horne, A. Shimony, and R.A. Holt. Phys. Rev. Lett. 23, 880 (1969)[8] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, New York, 2002).[9] J. Freedman, J.F. Clauser, Phys. Rev. Lett. 28 938 (1972)[10] A. Aspect, P. Grangier, G. Roger, Phys. Rev. Lett. 47, 460 (1981)[11] A. Aspect, P. Grangier, G. Roger, Phys. Rev. Lett. 49, 91 (1982)[12] Y.H. Shih, C.O. Alley, Phys. Rev. Lett. 61 2921 (1988)[13] J-W. Pan, D.Bouwmeester, H. Weinfurter, A. Zeilinger, Phys. Rev. Lett. 80, 3891 (1998)[14] C.K. Hong, Z.Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044 (1987).[15] H. de Riedmatten et al. ,Phys. Rev. A 67 22301, (2003)[16] C. Santori, D. Fattal, J. Vuˇckovi´c, G.S. Solomon, and Y. Yamamoto, Nature (London) 419, 594 (2002).[17] A. Garruccio, V.A. Rapisarda, Nuovo Cimento 65A, 269 (1981)[18] D. Aerts, M. Czachor, M. Pavlovski, Phys. Rev. A 73 034303, (2006)[19] D. Aerts, M. Czachor, M. Pavlovski, Phys. Rev. A 73 059901(E), (2006)[20] S. Popescu, L. Hardy, M. Zukowski, Phys. Rev. A, 56, 6, R4353, (1997)[21] A. K. Pati, M. Zukowski. Pramana-J. Phys.,56, 393, (2001)Appendix - tables:ANGLES VALUE SIGMA E(22.5,45) 0.578065 σ = ± 0.011966E(22.5,0) -0.67484 σ = ± 0.010916E(67.5,45) 0.600959 σ = ± 0.011296E(67.5,0) 0.689742 σ = ± 0.010944Table 1: Angular settings of half wave plates and measured values of E x with standard deviation.Appendix -figures:Figure 1 Different methods for creation of entanglement. (a) A high intensity laser beam generates two entangled photons in a down conversion process. (b) Two single-photons generated independently overlap on a 50/50 beamsplitter. (c) Entanglement swapping as demonstrated by Pan et al. using two down-converted photon pairs.Photon 2 and 3 undergo a joint Bell-measurement. In consequence, Photon 1 and 4 are projected into a Bell-state despite they do not locally interact. (d) Creation of entanglement without the need for ancilla photons using our non-local method.Figure 2Simplified schematic of the experimental setup. The two single photon enter the interferometer at the 50/50 Beamsplitters BS1 and BS2 and exit at the polarizing Beamsplitters PBS3 and PBS4. Rotating the HWP in front of PBS5 and PBS6 it is possible (a) to choose in which basis the measurement shall be performed and (b) to select polarization states that maximize the violation of the inequality of Clauser, Horne, Shimony and Holt. Successful events are registered whencoincidences between opposite sites occur.Data of 1-3-dip (Alice) measurement to verify path length inside the interferometer.Figure 4 (a) Measurement of the coherent envelope between detector D2 and detector D3 with visibility of 95.1. The visibilities of the other 3 combinations (D1-D2, D2-D4, D1-D4) are all above the required limit of 0.71. (b) 45-45-Measurement of relative phasechange of all four coincidences dependent on piezo shift.。