The Effective Bosonic Hamiltonian for Excitons Reconsidered

医学考博英语特训教材阅读1试题English:The paragraph discusses the concept of immunological memory, which is a fundamental aspect of the adaptive immune response. Immunological memory refers to the ability of the immune system to remember previously encountered pathogens and mount a faster and more robust response upon re-exposure. This phenomenon is primarily mediated by memory T and B cells, which are generated during the initial infection or vaccination. Memory T cells are capable of recognizing specific antigens presented by infected cells, while memory B cells produce antibodies that can bind to and neutralize pathogens. The paragraph highlights the importance of immunological memory in providing long-lasting protection against infectious diseases and the basis for vaccination strategies. It also emphasizes the role of memory cells in secondary immune responses, which are quicker and more effective due to the presence of pre-existing memory cells. Overall, immunological memory is crucial for the effectiveness of the adaptive immune system in combating recurring infections and is a cornerstone of immunology research and vaccine development.中文翻译:该段讨论了免疫记忆的概念，这是适应性免疫应答的基本方面。

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. 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Figure 11.2 2. The labor supply curve is upward sloping, while the labor demand curve is the marginal
product of labor when the effort level is determined by the efficiency wage 3. The difference between labor supply and labor demand is the amount of unemployment 4. The fact that there’s unemployment puts no downward pressure on the real wage, since firms
226 Abel/Bernanke/Croushore • Macroeconomics, Sixth Edition
b. Another possibility is that a firm may want to pay high wages to get a stable labor force and avoid turnover costs—costs of hiring and training new workers
Chapter 11
Keynesianism: The Macroeconomics of Wage and Price Rigidity
Learning Objectives
I. Goals of Chapter 11 A) Present the central ideas of Keynesian macroeconomics 1. Wages and prices don’t adjust quickly to restore general equilibrium 2. The economy may be in disequilibrium for long periods of time 3. The government should act to stabilize the economy B) Discuss the potential causes of wage and price rigidity

2022年考研考博-考博英语-湖南师范大学考试全真模拟易错、难点剖析AB卷（带答案）一.综合题(共15题)1.单选题Professor Hawking is（）as one of the world’s greatest living physicists.问题1选项A.dignifiedB.clarifiedC.acknowledgedD.illustrated【答案】C【解析】dignify使高贵，授予荣誉；clarify澄清，阐明；acknowledge承认，认可；illustrate阐明，举例。

句意：霍金教授被公认是当今世界上最伟大的物理学家之一。

选项C符合语境。

2.翻译题参与并取胜，这就是奥林匹克精神。

他表现于弱者勇于向强者挑战，也表现于强者力争取得更好的成绩。

胜而又胜，优而更优，这种理想一直鼓舞着运动员奋力前进。

有人说竞技者终宄会是失败者，即使是最佳运动员也终将被更强者淘汰。

成千上万个失败者才涌现一个胜利者，这个胜利者最终仍将被取代，挤出光荣榜——这就是竞技运动的规律。

然而运动员却从不为这种不可避免的失败结局所沮丧，仍然力争最佳发挥。

【答案】To participate and to win—that is the Olympic spirit. It finds expression in the weak daring to defy the strong, and the strong striving for ever better performance. Ever better—the ideal always luring a sportsman forward. It is said that none of the competitors can avoid being defeated—even the best is bound to be surpassed by someone still stronger. Thousands of losers set off one victor who in turn will eventually be replaced by someone on the honor list. This is the rule of sports. However, undaunted by the inevitable failure, he is always striving to do the best he can.3.单选题He gave us an（）consent to take the apples, for he smiled when he saw us do it问题1选项A.exaggeratedB.implicitC.impartialD.extravagant【答案】B【解析】exaggerated夸张的；implicit含蓄的, 暗示的；impartial公正的；extravagant奢侈的, 浪费的。