Mixed-isotope Bose-Einstein condensates in Rubidium

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2022年自考专业(英语)英语科技文选考试真题及答案37

2022年自考专业(英语)英语科技文选考试真题及答案一、阅读理解题Directions: Read through the following passages. Choose the best answer and put the letter in the bracket. (20%)1、 (A) With the recent award of the Nobel Prize in physics, the spectacular work on Bose-Einstein condensation in a dilute gas of atoms has been honored. In such a Bose-Einstein condensate, close to temperatures of absolute zero, the atoms lose their individuality and a wave-like state of matter is created that can be compared in many ways to laser light. Based on such a Bose-Einstein condensate researchers in Munich together with a colleague from the ETH Zurich have now been able to reach a new state of matter in atomic physics. In order to reach this new phase for ultracold atoms, the scientists store a Bose-Einstein condensate in a three-dimensional lattice of microscopic light traps. By increasing the strength of the lattice, the researchers are able to dramatically alter the properties of the gas of atoms and can induce a quantum phase transition from the superfluid phase of a Bose-Einsteincondensate to a Mott insulator phase. In this new state of matter it should now be possible to investigate fundamental problems of solid-state physics, quantum optics and atomic physics. For a weak optical lattice the atoms form a superfluid phase of a Bose-Einstein condensate. In this phase, each atom is spread out over the entire lattice in a wave-like manner as predicted by quantum mechanics. The gas of atoms may then move freely through the lattice. For a strong optical lattice the researchers observe a transition to an insulating phase, with an exact number of atoms at each lattice site. Now the movement of the atoms through the lattice is blocked due to therepulsive interactions between them. Some physicists have been able to show that it is possible to reversibly cross the phase transition between these two states of matter. The transition is called a quantum phase transition because it is driven by quantum fluctuations and can take place even at temperatures of absolute zero. These quantum fluctuations are a direct consequence of Heisenberg’s uncertainty relation. Normally phase transitions are driven by thermal fluctuations, which are absent at zero temperature. With their experiment, the researchers in Munich have been able to enter a new phase in the physics of ultracold atoms. In the Mott insulator state theatoms can no longer be described by the highly successful theories for Bose-Einstein condensates. Now theories are required that take into account the dominating interactions between the atoms and which are far less understood. Here the Mott insulator state may help in solving fundamental questions of strongly correlated systems, which are the basis for our understanding of superconductivity. Furthermore, the Mott insulator state opens many exciting perspectives for precision matter-wave interferometry and quantum computing.What does the passage mainly discuss?A.Bose-Einstein condensation.B.Quantum phase transitions.C.The Mott insulator state.D.Optical lattices.2、What will the scientists possibly do by reaching the new state of matter in atomic physics?A.Store a Bose-Einstein condensate in three-dimensional lattice of microscopic light traps.B.Increase the strength of the lattice.C.Alter the properties of the gas of atoms.D.Examine fundamental problems of atomic physics.3、Which of the following is NOT mentioned in relation to aweak optical lattice?A.The atoms form a superfluid phase of a Bose-Einstein condensate.B.Each atom is spread out over the entire lattice.C.The gas of atoms may move freely through the lattice.D.The superfluid phase changes into an insulating phase.4、What can be said about the quantum phase transition?A.It can take place at temperatures of absolute zero.B.It cannot take place above the temperatures of absolute zero.C.It is driven by thermal fluctuations.D.It is driven by the repulsive interactions between atoms.5、The author implies all the following about the Mott insulator state EXCEPT that______.A.the theory of Bose-Einstein condensation can’t possibly account for the atoms in the Mott insulator stateB.not much is known about the dominating interactions between the atoms in the Mott insulator stateC.it offers new approaches to exact quantum computingD.it forms a superfluid phase of a Bose-Einstein condensate6、 (B) Gene therapy and gene-based drugs are two ways we would benefit from our growing mastery of genetic science. But therewill be others as well. Here is one of the remarkable therapies on the cutting edge of genetic research that could make their way into mainstream medicine in the c oming years. While it’s true that just about every cell in the body has the instructions to make a complete human, most of those instructions are inactivated, and with good reason: the last thing you want for your brain cells is to start churning out stomach acid or your nose to turn into a kidney. The only time cells truly have the potential to turn into any and all body parts is very early in a pregnancy, when so-called stem cells haven’t begun to specialize. Most diseases involve the death of healthy cells—brain cells in Alzheimer’s, cardiac cells in heart disease, pancreatic cells in diabetes, to name a few; if doctors could isolate stem cells, then direct their growth, they might be able to furnish patients with healthy replacement tissue. It was incredibly difficult, but last fall scientists at the University of Wisconsin managed to isolate stem cells and get them to grow into neural, gut, muscle and bone cells. The process still can’t be controlled, and may have unforeseen limitations; but if efforts to understand and master stem-cell development prove successful, doctors will have a therapeutic tool of incredible power. The same applies to cloning, whichis really just the other side of the coin; true cloning, as first shown, with the sheep Dolly two years ago, involves taking a developed cell and reactivating the genome within, resenting its developmental instructions to a pristine state. Once that happens, the rejuvenated cell can develop into a full-fledged animal, genetically identical to its parent. For agriculture, in which purely physical characteristics like milk production in a cow or low fat in a hog have real market value, biological carbon copies could become routine within a few years. This past year scientists have done for mice and cows what Ian Wilmut did for Dolly, and other creatures are bound to join the cloned menagerie in the coming year. Human cloning, on the other hand, may be technically feasible but legally and emotionally more difficult. Still, one day it will happen. The ability to reset body cells to a pristine, undeveloped state could give doctors exactly the same advantages they would get from stem cells: the potential to make healthy body tissues of all sorts. And thus to cure disease.That could prove to be a true “miracle cu re”.What is the passage mainly about?A.Tomorrow’s tissue factory.B.A terrific boon to medicine.C.Human cloning.D.Genetic research.7、 According to the passage, it can be inferred that which of the following reflects the author’s opinion?A.There will inevitably be human cloning in the coming year.B.The potential to make healthy body tissues is undoubtedly a boon to human beings.C.It is illegal to clone any kind of creatures in the world.D.It is legal to clone any kind of creatures in the world except human.8、Which of the following is NOT true according to the passage?A.Nearly every cell in the human brain has the instructions to make a complete human.B.It is impossible for a cell in your nose to turn into a kidney.C.It is possible to turn out healthy replacement tissues with isolated stem cells.D.There will certainly appear some new kind of cloned animal in the near future.9、All of the following are steps involved in true cloning EXCEPT_______.A.selecting a stem cellB.taking a developed cellC.reactivating the genome within the developed cellD.resetting the developmental instructions in the cell to its original state10、The word “rejuvenated” in para. 5 is closest in meaning to_______.A.rescuedB.reactivatedC.recalledD.regulated参考答案：【一、阅读理解题】1~5CDDAD6~10DBBA。

Bose-Einstein condensation in a one-dimensional interacting system due to power-law trappin

a r X i v :c o n d -m a t /9810197v 1 [c o n d -m a t .s t a t -m e c h ] 16 O c t 1998Accepted to PHYSICAL REVIEW A for publicationBose-Einstein condensation in a one-dimensional interacting system due to power-lawtrapping potentialsM.Bayindir,B.Tanatar,and Z.GedikDepartment of Physics,Bilkent University,Bilkent,06533Ankara,TurkeyWe examine the possibility of Bose-Einstein condensation in one-dimensional interacting Bose gas subjected to conﬁning potentials of the form V ext (x )=V 0(|x |/a )γ,in which γ<2,by solving the Gross-Pitaevskii equation within the semi-classical two-ﬂuid model.The condensate fraction,chemical potential,ground state energy,and speciﬁc heat of the system are calculated for various values of interaction strengths.Our results show that a signiﬁcant fraction of the particles is in the lowest energy state for ﬁnite number of particles at low temperature indicating a phase transition for weakly interacting systems.PACS numbers:03.75.Fi,05.30.Jp,67.40.Kh,64.60.-i,32.80.PjI.INTRODUCTIONThe recent observations of Bose-Einstein condensation (BEC)in trapped atomic gases [1–5]have renewed inter-est in bosonic systems [6,7].BEC is characterized by a macroscopic occupation of the ground state for T <T 0,where T 0depends on the system parameters.The success of experimental manipulation of externally applied trap potentials bring about the possibility of examining two or even one-dimensional Bose-Einstein condensates.Since the transition temperature T 0increases with decreasing system dimension,it was suggested that BEC may be achieved more favorably in low-dimensional systems [8].The possibility of BEC in one -(1D)and two-dimensional (2D)homogeneous Bose gases is ruled out by the Hohen-berg theorem [9].However,due to spatially varying po-tentials which break the translational invariance,BEC can occur in low-dimensional inhomogeneous systems.The existence of BEC is shown in a 1D noninteracting Bose gas in the presence of a gravitational ﬁeld [10],an attractive-δimpurity [11],and power-law trapping po-tentials [12].Recently,many authors have discussed the possibility of BEC in 1D trapped Bose gases relevant to the magnetically trapped ultracold alkali-metal atoms [13–18].Pearson and his co-workers [19]studied the in-teracting Bose gas in 1D power-law potentials employing the path-integral Monte Carlo (PIMC)method.They have found that a macroscopically large number of atoms occupy the lowest single-particle state in a ﬁnite system of hard-core bosons at some critical temperature.It is important to note that the recent BEC experiments are carried out with ﬁnite number of atoms (ranging from several thousands to several millions),therefore the ther-modynamic limit argument in some theoretical studies [15]does not apply here [8].The aim of this paper is to study the two-body interac-tion eﬀects on the BEC in 1D systems under power-law trap potentials.For ideal bosons in harmonic oscillator traps transition to a condensed state is prohibited.It is anticipated that the external potentials more conﬁn-ing than the harmonic oscillator type would be possible experimentally.It was also argued [15]that in the ther-modynamic limit there can be no BEC phase transition for nonideal bosons in 1D.Since the realistic systems are weakly interacting and contain ﬁnite number of particles,we employ the mean-ﬁeld theory [20,21]as applied to a two-ﬂuid model.Such an approach has been shown to capture the essential physics in 3D systems [21].The 2D version [22]is also in qualitative agreement with the results of PIMC simulations on hard-core bosons [23].In the remaining sections we outline the two-ﬂuid model and present our results for an interacting 1D Bose gas in power-law potentials.II.THEORYIn this paper we shall investigate the Bose-Einstein condensation phenomenon for 1D interacting Bose gas conﬁned in a power-law potential:V ext (x )=V 0|x |κF (γ)G (γ)2γ/(2+γ),(2)andN 0/N =1−TF (γ)=1x 1/γ−1dx1−x,(4)and G (γ)=∞x 1/γ−1/2dxNk B T 0=Γ(1/γ+3/2)ζ(1/γ+3/2)T 01/γ+3/2.(6)Figure 1shows the variation of the critical temperature T 0as a function of the exponent γin the trapping po-tential.It should be noted that T 0vanishes for harmonic potential due to the divergence of the function G (γ=2).It appears that the maximum T 0is attained for γ≈0.5,and for a constant trap potential (i.e.V ext (x )=V 0)the BEC disappears consistent with the Hohenberg theorem.0.00.5 1.0 1.5 2.0γ0.00.20.40.6k B T 0 (A r . U n .)FIG.1.The variation of the critical temperature T 0withthe external potential exponent γ.We are interested in how the short-range interactioneﬀects modify the picture presented above.To this end,we employ the mean-ﬁeld formalism and describe the col-lective dynamics of a Bose condensate by its macroscopictime-dependent wave function Υ(x,t )=Ψ(x )exp (−iµt ),where µis the chemical potential.The condensate wavefunction Ψ(x )satisﬁes the Gross-Pitaevskii (GP)equa-tion [24,25]−¯h 2dx 2+V ext (x )+2gn 1(x )+g Ψ2(x )Ψ(x )=µΨ(x ),(7)where g is the repulsive,short-range interaction strength,and n 1(x )is the average noncondensed particle distribu-tion function.We treat the interaction strength g as a phenomenological parameter without going into the de-tails of actually relating it to any microscopic descrip-tion [26].In the semi-classical two-ﬂuid model [27,28]the noncondensed particles can be treated as bosons in an eﬀective potential [21,29]V eﬀ(x )=V ext (x )+2gn 1(x )+2g Ψ2(x ).(8)The density distribution function is given byn 1(x )=dpexp {[p 2/2m +V eﬀ(x )−µ]/k B T }−1,(9)and the total number of particles N ﬁxes the chemical potential through the relationN =N 0+ρ(E )dE2mgθ[µ−V ext (x )−2gn 1(x )],(12)where θ[x ]is the unit step function.More precisely,the Thomas-Fermi approximation [7,20,30]would be valid when the interaction energy ∼gN 0/Λ,far exceeds the kinetic energy ¯h 2/2m Λ2,where Λis the spatial extent of the condensate cloud.For a linear trap potential (i.e.γ=1),a variational estimate for Λis given by Λ= ¯h 2/2m (π/2)1/22a/V 0 1/3.We note that the Thomas-Fermi approximation would breakdown for tem-peratures close to T 0where N 0is expected to become very small.The above set of equations [Eqs.(9)-(12)]need to be solved self-consistently to obtain the various physical quantities such as the chemical potential µ(N,T ),the condensate fraction N 0/N ,and the eﬀective potential V eﬀ.In a 3D system,Minguzzi et al .[21]solved a simi-lar system of equations numerically and also introduced an approximate semi-analytical solution by treating the interaction eﬀects perturbatively.Motivated by the suc-cess [21,22]of the perturbative approach we consider aweakly interacting system in1D.To zero-order in gn1(r), the eﬀective potential becomesV eﬀ(x)= V ext(x)ifµ<V ext(x)2µ−V ext(x)ifµ>V ext(x).(13) Figure2displays the typical form of the eﬀective po-tential within our semi-analytic approximation scheme. The most noteworthy aspect is that the eﬀective poten-tial as seen by the bosons acquire a double-well shape because of the interactions.We can explain this result by a simple argument.Let the number of particles in the left and right wells be N L and N R,respectively,so that N=N L+N R.The nonlinear or interaction term in the GP equation may be approximately regarded as V=N2L+N2R.Therefore,the problem reduces to the minimization of the interaction potential V,which is achieved for N L=N R.FIG.2.Eﬀective potential V eﬀ(x)in the presence of in-teraction(x0=(µ/V0)1/γa).Thick dotted line represents external potential V ext(x).The number of condensed atoms is calculated to beN0=2γa√ze x−1+ 2µ/k B Tµ/k B TH(γ,µ,xk B T)(2µ/k B T−x)1/γ−1/2dxexp[(E−µ)/k B T]−1=κ(k B T)1/γ+1/2J(γ,µ,T),(18) whereJ(γ,µ,T)= ∞2µ/k B T x1/γ+1/2dxze x−1.and Ecis the energy of the particles in the condensateE c=g(1+γ)(2γ+1)gV1/γ.(19)The kinetic energy of the condensed particles is neglected within our Thomas-Fermi approximation to the GP equa-tion.III.RESULTS AND DISCUSSIONUp to now we have based our formulation for arbitrary γ,but in the rest of this work we shall present our re-sults forγ=1.Our calculations show that the results for other values ofγare qualitatively similar.In Figs. 3and4we calculate the condensate fraction as a func-tion of temperature for various values of the interaction strengthη=g/V0a(at constant N=105)and diﬀerent number of particles(at constantη=0.001),respectively. We observe that as the interaction strengthηis increased, the depletion of the condensate becomes more apprecia-ble(Fig.3).As shown in the correspondingﬁgures,a signiﬁcant fraction of the particles occupies the ground state of the system for T<T0.The temperature depen-dence of the chemical potential is plotted in Figs.5and 6for various interaction strengths(constant N=105) and diﬀerent number of particles(constantη=0.001) respectively.0.00.20.40.60.8 1.0T/T 00.00.20.40.60.81.0N 0/NN 0/N=1−(T/T 0)3/2η=10−5η=10−3η=10−1η=10FIG.3.The condensate fraction N 0/N versus temperature T /T 0for N =105and for various interaction strengths η.Eﬀects of interactions on µ(N,T )are seen as large de-viations from the noninteracting behavior for T <T 0.In Fig.7we show the ground state energy of an interacting 1D system of bosons as a function of temperature for dif-ferent interaction strengths.For small η,and T <T 0, E is similar to that in a noninteracting system.As ηincreases,some diﬀerences start to become noticeable,and for η≈1we observe a small bump developing in E .This may indicate the breakdown of our approxi-mate scheme for large enough interaction strengths,as we can ﬁnd no fundamental reason for such behavior.It is also possible that the Thomas-Fermi approximation em-ployed is violated as the transition to a condensed state is approached.0.00.20.40.60.8 1.0T/T 00.00.20.40.60.81.0N 0/NN 0/N=1−(T/T 0)3/2N=108N=105N=103N=101FIG.4.The condensed fraction N 0/N versus temperature T /T 0for η=0.001and for diﬀerent number of particles N .0.00.20.40.60.8 1.0 1.2T/T 0−100100200300400µ/V 0η=1η=0.1η=0.001η=0.00001FIG.5.The temperature dependence of the chemical potential µ(N,T )for various interaction strength and for N =105particles.Although it is conceivable to imagine the full solution of the mean-ﬁeld equations [Eq.(9)-(12)]may remedy the situation for larger values of η,the PIMC simulations [19]also seem to indicate that the condensation is inhibited for strongly interacting systems.The results for the spe-ciﬁc heat calculated from the total energy curves,i.e.C V =d E /dT ,are depicted in Fig.8.The sharp peak at T =T 0tends to be smoothed out with increasing in-teraction strength.It is known that the eﬀects of ﬁnite number of particles are also responsible for such a be-havior [20].In our treatment these two eﬀects are not disentangled.It was pointed out by Ingold and Lam-brecht [14]that the identiﬁcation of the BEC should also be based on the behavior of C V around T ≈T 0.0.00.20.40.60.8 1.0 1.2T/T 0−5050100µ/V 0N=107N=105N=103N=101FIG.6.The temperature dependence of the chemical po-tential µ(N,T )for diﬀerent number of particles N and for η=0.001.0.00.20.40.60.8 1.0 1.2T/T 00.00.20.40.60.8<E >/N k B T 0η=0η=0.001η=0.1η=1Maxwell−BoltzmannFIG.7.The temperature dependence of the total energy of 1D Bose gas for various interaction strengths ηand N =105particles.Our calculations indicate that the peak structure of C V remains even in the presence of weak interactions,thus we are led to conclude that a true transition to a Bose-Einstein condensed state is predicted within the present approach.0.00.20.40.60.81.01.2T/T 00.00.20.40.60.81.0C V /N k Bη=0η=0.001η=0.1Maxwell−BoltzmannFIG.8.The temperature dependence of the speciﬁc heat C V for various interaction strengths ηand N =105particles.IV.CONCLUDING REMARKSIn this work we have applied the mean-ﬁeld,semi-classical two-ﬂuid model to interacting bosons in 1D power-law trap potentials.We have found that for a range of interaction strengths the behavior of the thermo-dynamic quantities resembles to that of non-interactingbosons.Thus,BEC in the sense of macroscopic occu-pation of the ground state,occurs when the short-range interparticle interactions are not too strong.Our results are in qualitative agreement with the recent PIMC sim-ulations [19]of similar systems.Both 2D and 1D sim-ulation results [19,23]indicate a phase transition for a ﬁnite number system,in contrast to the situation in the thermodynamic limit.Since systems of much larger size can be studied within the present approach,our work complements the PIMC calculations.The possibility of studying the tunneling phenomenon of condensed bosons in spatially diﬀerent regions sepa-rated by a barrier has recently attracted some attention [31–34].In particular,Dalfovo et al .[32]have shown that a Josephson-type tunneling current may exist for bosons under the inﬂuence of a double-well trap potential.Za-pata et al .[34]have estimated the Josephson coupling energy in terms of the condensate density.It is inter-esting to speculate on such a possibility in the present case,since the eﬀective potential in our description is of the form of a double-well potential (cf.Fig.2).In our treatment,the interaction eﬀects modify the single-well trap potential into one which exhibits two minima.Thus if we think of this eﬀective potential as the one seen by the condensed bosons and according to the general ar-guments [31–34]based on two weakly connected systems we should have an oscillating ﬂux of particles when the chemical potential in the two wells is diﬀerent.Any con-ﬁguration with N L =N R which is always the case for odd number of bosons will result in an oscillatory mo-tion.It would be interesting to explore these ideas in future work.ACKNOWLEDGMENTSThis work was supported by the Scientiﬁc and Techni-cal Research Council of Turkey (TUBITAK)under Grant No.TBAG-1736and TBAG-1662.We gratefully ac-knowledge useful discussions with Prof.C.Yalabık and E.Demirel.[5]D.J.Han,R.H.Wynar,Ph.Courteille,and D.J.Heinzen,Phys.Rev.A57,R4114(1998).[6]I.F.Silvera,in Bose-Einstein Condensation,Ed.by A.Griﬃn,D.W.Snoke,and S.Stringari(Cambridge Uni-versity Press,Cambridge,1995).[7]F.Dalfovo,S.Giorgini,L.P.Pitaevskii,and S.Stringari,preprint,cond-mat/9806038(to be published in Reviews of Modern Physics);A.S.Parkins and D.F.Walls,Phys.Rep.303,1(1998).[8]W.Ketterle and N.J.van Druten,Phys.Rev.A54,656(1996).[9]P.C.Hohenberg,Phys.Rev.158,383(1967).[10]A.Widom,Phys.Rev.176,254(1968).[11]L.C.Ioriatti,Jr.,S.G.Rosa,Jr.,and O.Hipolito,Am.J.Phys.44,744(1976).[12]V.Bagnato and D.Kleppner,Phys.Rev.A44,7439(1991).[13]T.Haugset and H.Haugerud,Phys.Rev.A57,3809(1998).[14]G.-L.Ingold and mbrecht,Eur.Phys.J.D1,29(1998).[15]W.J.Mullin,J.Low Temp.Phys.110,167(1998)[16]W.Deng and P.M.Hui,Solid State Commun.104,729(1997).[17]H.Monien,M.Linn,and N.Elstner,preprint,cond-mat/9711178.[18]F.Brosens,J.T.Devreese,and L.F.Lemmens,SolidState Commun.100,123(1996).[19]S.Pearson,T.Pang,and C.Chen,Phys.Rev.A58,1485(1998).[20]S.Giorgini,L.P.Pitaevskii,and S.Stringari,Phys.Rev.A,54,4633(1996);J.Low Temp.Phys.109,309(1997).[21]A.Minguzzi,S.Conti,and M.P.Tosi,J.Phys.Cond.Matter9,L33(1997).[22]M.Bayindir and B.Tanatar,Phys.Rev.A58,3134(1998).[23]S.Heinrichs and W.J.Mullin,preprint,cond-mat/9807331.[24]E.P.Gross,Nuovo Cimento20,454(1961).[25]L.P.Pitaevskii,Zh.Eksp.Teor.Fiz.40,646(1961)[Sov.Phys.JETP13,451(1961)].[26]M.Olshanii,Phys.Rev.Lett.81,938(1998).[27]R.J.Dodd,K.Burnett,M.Edwards,and C.W.Clark,Acta Phys.Pol.A93,45(1998).[28]M.Naraschewski and D.M.Stamper-Kurn,Phys.Rev.A58,2423(1998).[29]V.Bagnato,D.E.Pritchard,and D.Kleppner,Phys.Rev.A35,4354(1987).[30]G.Baym and C.J.Pethick,Phys.Rev.Lett.76,6(1996).[31]S.Grossmann and M.Holthaus,Z.Naturforsch.50a,323(1995).[32]F.Dalfovo,L.P.Pitaevskii,and S.Stringari,Phys.Rev.A54,4213(1996).[33]A.Smerzi,S.Fantoni,S.Giovanazzi,and S.R.Shenoy,Phys.Rev.Lett.79,4950(1997).[34]I.Zapata,F.Sols,and A.J.Leggett,Phys.Rev.A57,28(1998).。

天文学常用术语英汉对译及解释LaRivenbsp

天文学常用术语英汉对译及解释LaRive nbsp天文学常用术语英汉对译及解释0000术语解释aberration光行差由于地球的运动所导致的天体的视位置与真实位置之间的差异。

absolute magnitude绝对星等恒星的真正亮度。

定义为恒星在距离我们 10 秒差距 (32.6光年) 时的视星等。

absolute zero绝对零度理论上的最低温度，等于0开尔文(-459.67° F or -273.15° C)。

absorption lines吸收线光谱里的暗线。

来自天体的光，被原子或分子选择性的吸收，导致那部分的光从星光中被消去，留下一条条的暗线。

accretion disk吸积盘指白矮星、中子星或黑洞等致密天体周围，由于物质受到引力作用向中心天体落下所形成的盘状结构。

achromatic lens消色差透镜由两种不同材质的透镜组合而成，消色差透镜的用途是把两种不同颜色的光聚焦到同一点，或称为修正色像差。

active galactic nuclei活动星系核某些星系中的特别明亮的核，被认为是由于物质落向质量极大的黑洞而引起的。

adaptive optics自适应光学计算机控制的望远镜镜面，能做区域性变形，以补偿大气扰动所产生的散焦效应。

albedo反照率行星或卫星反射光能力的标示值，定义为所反射的光和入射光的比值。

反照率的值介于 0 (完美的黑体) 到 1 (完全反射)之间。

月球的反照率为 0.07，而金星为 0.6。

altazimuth mount地平装置一种望远镜支撑方式，使镜筒能在平行和垂直水平的方向自由移动。

altitude地平纬度、高度 1.在海平面以上的高度2..天体在天球上距离地平线的角度 anaglyph立体照片用两台相机拍摄出的一种照片。

将右边拍摄的影像（通常是红色）和左边拍摄的影像（通常是蓝色）叠加起来，通过特殊的色彩滤镜，就能看到三维的效果。

19 Bose-Einstein Condensation

Chapter19Bose-Einstein CondensationAbstract Bose-Einstein condensation(BEC)refers to a prediction of quantum sta-tistical mechanics(Bose[1],Einstein[2])where an ideal gas of identical bosons undergoes a phase transition when the thermal de Broglie wavelength exceeds the mean spacing between the particles.Under these conditions,bosons are stimulated by the presence of other bosons in the lowest energy state to occupy that state as well,resulting in a macroscopic occupation of a single quantum state.The con-densate that forms constitutes a macroscopic quantum-mechanical object.BEC was ﬁrst observed in1995,seventy years after the initial predictions,and resulted in the award of2001Nobel Prize in Physics to Cornell,Ketterle and Weiman.The exper-imental observation of BEC was achieved in a dilute gas of alkali atoms in a mag-netic trap.Theﬁrst experiments used87Rb atoms[3],23Na[4],7Li[5],and H[6] more recently metastable He has been condensed[7].The list of BEC atoms now includes molecular systems such as Rb2[8],Li2[9]and Cs2[10].In order to cool the atoms to the required temperature(∼200nK)and densities(1013–1014cm−3) for the observation of BEC a combination of optical cooling and evaporative cooling were employed.Early experiments used magnetic traps but now optical dipole traps are also common.Condensates containing up to5×109atoms have been achieved for atoms with a positive scattering length(repulsive interaction),but small con-densates have also been achieved with only a few hundred atoms.In recent years Fermi degenerate gases have been produced[11],but we will not discuss these in this chapter.BECs are now routinely produced in dozens of laboratories around the world. They have provided a wonderful test bed for condensed matter physics with stunning experimental demonstrations of,among other things,interference between conden-sates,superﬂuidity and vortices.More recently they have been used to create opti-cally nonlinear media to demonstrate electromagnetically induced transparency and neutral atom arrays in an optical lattice via a Mott insulator transition.Many experiments on BECs are well described by a semiclassical theory dis-cussed below.Typically these involve condensates with a large number of atoms, and in some ways are analogous to describing a laser in terms of a semiclassi-cal meanﬁeld.More recent experiments however have begun to probe quantum39739819Bose-Einstein Condensation properties of the condensate,and are related to the fundamental discreteness of the ﬁeld and nonlinear quantum dynamics.In this chapter,we discuss some of these quantum properties of the condensate.We shall make use of“few mode”approxi-mations which treat only essential condensate modes and ignore all noncondensate modes.This enables us to use techniques developed for treating quantum optical systems described in earlier chapters of this book.19.1Hamiltonian:Binary Collision ModelThe effects of interparticle interactions are of fundamental importance in the study of dilute–gas Bose–Einstein condensates.Although the actual interaction potential between atoms is typically very complex,the regime of operation of current exper-iments is such that interactions can in fact be treated very accurately with a much–simpliﬁed model.In particular,at very low temperature the de Broglie wavelengths of the atoms are very large compared to the range of the interatomic potential.This, together with the fact that the density and energy of the atoms are so low that they rarely approach each other very closely,means that atom–atom interactions are ef-fectively weak and dominated by(elastic)s–wave scattering.It follows also that to a good approximation one need only consider binary collisions(i.e.,three–body processes can be neglected)in the theoretical model.The s–wave scattering is characterised by the s–wave scattering length,a,the sign of which depends sensitively on the precise details of the interatomic potential [a>0(a<0)for repulsive(attractive)interactions].Given the conditions described above,the interaction potential can be approximated byU(r−r )=U0δ(r−r ),(19.1) (i.e.,a hard sphere potential)with U0the interaction“strength,”given byU0=4π¯h2am,(19.2)and the Hamiltonian for the system of weakly interacting bosons in an external potential,V trap(r),can be written in the second quantised form asˆH=d3rˆΨ†(r)−¯h22m∇2+V trap(r)ˆΨ(r)+12d3rd3r ˆΨ†(r)ˆΨ†(r )U(r−r )ˆΨ(r )ˆΨ(r)(19.3)whereˆΨ(r)andˆΨ†(r)are the bosonﬁeld operators that annihilate or create a par-ticle at the position r,respectively.19.2Mean–Field Theory —Gross-Pitaevskii Equation 399To put a quantitative estimate on the applicability of the model,if ρis the density of bosons,then a necessary condition is that a 3ρ 1(for a >0).This condition is indeed satisﬁed in the alkali gas BEC experiments [3,4],where achieved densities of the order of 1012−1013cm −3correspond to a 3ρ 10−5−10−6.19.2Mean–Field Theory —Gross-Pitaevskii EquationThe Heisenberg equation of motion for ˆΨ(r )is derived as i¯h ∂ˆΨ(r ,t )∂t = −¯h 22m ∇2+V trap (r ) ˆΨ(r ,t )+U 0ˆΨ†(r ,t )ˆΨ(r ,t )ˆΨ(r ,t ),(19.4)which cannot in general be solved.In the mean–ﬁeld approach,however,the expec-tation value of (19.4)is taken and the ﬁeld operator decomposed asˆΨ(r ,t )=Ψ(r ,t )+˜Ψ(r ,t ),(19.5)where Ψ(r ,t )= ˆΨ(r ,t ) is the “condensate wave function”and ˜Ψ(r )describes quantum and thermal ﬂuctuations around this mean value.The quantity Ψ(r ,t )is in fact a classical ﬁeld possessing a well–deﬁned phase,reﬂecting a broken gauge sym-metry associated with the condensation process.The expectation value of ˜Ψ(r ,t )is zero and,in the mean–ﬁeld theory,its effects are assumed to be small,amounting to the assumption of the thermodynamic limit,where the number of particles tends to inﬁnity while the density is held ﬁxed.For the effects of ˜Ψ(r )to be negligibly small in the equation for Ψ(r )also amounts to an assumption of zero temperature (i.e.,pure condensate).Given that this is so,and using the normalisationd 3r |Ψ(r ,t )|2=1,(19.6)one is lead to the nonlinear Schr¨o dinger equation,or “Gross–Pitaevskii equation”(GP equation),for the condensate wave function Ψ(r ,t )[13],i¯h ∂Ψ(r ,t )∂t = −¯h 22m ∇2+V trap (r )+NU 0|Ψ(r ,t )|2 Ψ(r ,t ),(19.7)where N is the mean number of particles in the condensate.The nonlinear interaction term (or mean–ﬁeld pseudo–potential)is proportional to the number of atoms in the condensate and to the s –wave scattering length through the parameter U 0.A stationary solution forthe condensate wavefunction may be found by substi-tuting ψ(r ,t )=exp −i μt ¯h ψ(r )into (19.7)(where μis the chemical potential of the condensate).This yields the time independent equation,40019Bose-Einstein Condensation−¯h2 2m ∇2+V trap(r)+NU0|ψ(r)|2ψ(r)=μψ(r).(19.8)The GP equation has proved most successful in describing many of the meanﬁeld properties of the condensate.The reader is referred to the review articles listed in further reading for a comprehensive list of references.In this chapter we shall focus on the quantum properties of the condensate and to facilitate our investigations we shall go to a single mode model.19.3Single Mode ApproximationThe study of the quantum statistical properties of the condensate(at T=0)can be reduced to a relatively simple model by using a mode expansion and subsequent truncation to just a single mode(the“condensate mode”).In particular,one writes the Heisenberg atomicﬁeld annihilation operator as a mode expansion over single–particle states,ˆΨ(r,t)=∑αaα(t)ψα(r)exp−iμαt/¯h=a0(t)ψ0(r)exp−iμ0t/¯h+˜Ψ(r,t),(19.9) where[aα(t),a†β(t)]=δαβand{ψα(r)}are a complete orthonormal basis set and {μα}the corresponding eigenvalues.Theﬁrst term in the second line of(19.9)acts only on the condensate state vector,withψ0(r)chosen as a solution of the station-ary GP equation(19.8)(with chemical potentialμ0and mean number of condensate atoms N).The second term,˜Ψ(r,t),accounts for non–condensate atoms.Substitut-ing this mode expansion into the HamiltonianˆH=d3rˆΨ†(r)−¯h22m∇2+V trap(r)ˆΨ(r)+(U0/2)d3rˆΨ†(r)ˆΨ†(r)ˆΨ(r)ˆΨ(r),(19.10)and retaining only condensate terms,one arrives at the single–mode effective Hamil-tonianˆH=¯h˜ω0a †a0+¯hκa†0a†0a0a0,(19.11)where¯h˜ω0=d3rψ∗0(r)−¯h22m∇2+V trap(r)ψ0(r),(19.12)and¯hκ=U02d3r|ψ0(r)|4.(19.13)19.5Quantum Phase Diffusion:Collapses and Revivals of the Condensate Phase401 We have assumed that the state is prepared slowly,with damping and pumping rates vanishingly small compared to the trap frequencies and collision rates.This means that the condensate remains in thermodynamic equilibrium throughout its prepara-tion.Finally,the atom number distribution is assumed to be sufﬁciently narrow that the parameters˜ω0andκ,which of course depend on the atom number,can be re-garded as constants(evaluated at the mean atom number).In practice,this proves to be a very good approximation.19.4Quantum State of the CondensateA Bose-Einstein condensate(BEC)is often viewed as a coherent state of the atomic ﬁeld with a deﬁnite phase.The Hamiltonian for the atomicﬁeld is independent of the condensate phase(see Exercise19.1)so it is often convenient to invoke a symmetry breaking Bogoliubovﬁeld to select a particular phase.In addition,a coherent state implies a superposition of number states,whereas in a single trap experiment there is aﬁxed number of atoms in the trap(even if we are ignorant of that number)and the state of a simple trapped condensate must be a number state(or,more precisely, a mixture of number states as we do not know the number in the trap from one preparation to the next).These problems may be bypassed by considering a system of two condensates for which the total number of atoms N isﬁxed.Then,a general state of the system is a superposition of number difference states of the form,|ψ =N∑k=0c k|k,N−k (19.14)As we have a well deﬁned superposition state,we can legitimately consider the relative phase of the two condensates which is a Hermitian observable.We describe in Sect.19.6how a particular relative phase is established due to the measurement process.The identiﬁcation of the condensate state as a coherent state must be modiﬁed in the presence of collisions except in the case of very strong damping.19.5Quantum Phase Diffusion:Collapsesand Revivals of the Condensate PhaseThe macroscopic wavefunction for the condensate for a relatively strong number of atoms will exhibit collapses and revivals arising from the quantum evolution of an initial state with a spread in atom number[21].The initial collapse has been described as quantum phase diffusion[20].The origins of the collapses and revivals may be seen straightforwardly from the single–mode model.From the Hamiltonian40219Bose-Einstein CondensationˆH =¯h ˜ω0a †0a 0+¯h κa †0a †0a 0a 0,(19.15)the Heisenberg equation of motion for the condensate mode operator follows as˙a 0(t )=−i ¯h [a 0,H ]=−i ˜ω0a 0+2κa †0a 0a 0 ,(19.16)for which a solution can be written in the form a 0(t )=exp −i ˜ω0+2κa †0a 0 t a 0(0).(19.17)Writing the initial state of the condensate,|i ,as a superposition of number states,|i =∑n c n |n ,(19.18)the expectation value i |a 0(t )|i is given byi |a 0(t )|i =∑n c ∗n −1c n √n exp {−i [˜ω0+2κ(n −1)]t }=∑nc ∗n −1c n √n exp −i μt ¯h exp {−2i κ(n −N )t },(19.19)where the relationship μ=¯h ˜ω0+2¯h κ(N −1),(19.20)has been used [this expression for μuses the approximation n 2 =N 2+(Δn )2≈N 2].The factor exp (−i μt /¯h )describes the deterministic motion of the condensate mode in phase space and can be removed by transforming to a rotating frame of reference,allowing one to writei |a 0(t )|i =∑nc ∗n −1c n √n {cos [2κ(n −N )t ]−isin [2κ(n −N )t ]}.(19.21)This expression consists of a weighted sum of trigonometric functions with different frequencies.With time,these functions alternately “dephase”and “rephase,”giving rise to collapses and revivals,respectively,in analogy with the behaviour of the Jaynes–Cummings Model of the interaction of a two–level atom with a single elec-tromagnetic ﬁeld mode described in Sect.10.2.The period of the revivals follows di-rectly from (19.21)as T =π/κ.The collapse time can be derived by considering the spread of frequencies for particle numbers between n =N +(Δn )and n =N −(Δn ),which yields (ΔΩ)=2κ(Δn );from this one estimates t coll 2π/(ΔΩ)=T /(Δn ),as before.From the expression t coll T /(Δn ),it follows that the time taken for collapse depends on the statistics of the condensate;in particular,on the “width”of the initial distribution.This dependence is illustrated in Fig.19.1,where the real part of a 0(t )19.5Quantum Phase Diffusion:Collapses and Revivals of the Condensate Phase403Fig.19.1The real part ofthe condensate amplitudeversus time,Re { a 0(t ) }for an amplitude–squeezed state,(a )and a coherent state (b )with the same mean numberof atoms,N =250.20.40.60.81-11234560b a is plotted as a function of time for two different initial states:(a)an amplitude–squeezed state,(b)a coherent state.The mean number of atoms is chosen in each case to be N =25.The timescales of the collapses show clear differences;the more strongly number–squeezed the state is,the longer its collapse time.The revival times,how-ever,are independent of the degree of number squeezing and depend only on the interaction parameter,κ.For example,a condensate of Rb 2,000atoms with the ω/2π=60Hz,has revival time of approximately 8s,which lies within the typical lifetime of the experimental condensate (10–20s).One can examine this phenomenon in the context of the interference between a pair of condensates and indeed one ﬁnds that the visibility of the interference pat-tern also exhibits collapses and revivals,offering an alternative means of detecting this effect.To see this,consider,as above,that atoms are released from two conden-sates with momenta k 1and k 2respectively.Collisions within each condensate are described by the Hamiltonian (neglecting cross–collisions)ˆH =¯h κ a †1a 1 2+ a †2a 22 ,(19.22)from which the intensity at the detector follows asI (x ,t )=I 0 [a †1(t )exp i k 1x +a †2(t )expi k 2x ][a 1(t )exp −i k 1x +a 2(t )exp −i k 2x ] =I 0 a †1a 1 + a †2a 2+ a †1exp 2i a †1a 1−a †2a 2 κt a 2 exp −i φ(x )+h .c . ,(19.23)where φ(x )=(k 2−k 1)x .If one assumes that each condensate is initially in a coherent state of amplitude |α|,with a relative phase φbetween the two condensates,i.e.,assuming that|ϕ(t =0) =|α |αe −i φ ,(19.24)40419Bose-Einstein Condensation then one obtains for the intensityI(x,t)=I0|α|221+exp2|α|2(cos(2κt)−1)cos[φ(x)−φ].(19.25)From this expression,it is clear that the visibility of the interference pattern under-goes collapses and revivals with a period equal toπ/κ.For short times t 1/2κ, this can be written asI(x,t)=I0|α|221+exp−|α|2κ2t2,(19.26)from which the collapse time can be identiﬁed as t coll=1/κ|α|.An experimental demonstration of the collapse and revival of a condensate was done by the group of Bloch in2002[12].In the experiment coherent states of87Rb atoms were prepared in a three dimensional optical lattice where the tunneling is larger than the on-site repulsion.The condensates in each well were phase coherent with constant relative phases between the sites,and the number distribution in each well is close to Poisonnian.As the optical dipole potential is increased the depth of the potential wells increases and the inter-well tunneling decreases producing a sub-Poisson number distribution in each well due to the repulsive interaction between the atoms.After preparing the states in each well,the well depth is rapidly increased to create isolated potential wells.The nonlinear interaction of(19.15)then determines the dynamics in each well.After some time interval,the hold time,the condensate is released from the trap and the resulting interference pattern is imaged.As the meanﬁeld amplitude in each well undergoes a collapse the resulting interference pattern visibility decreases.However as the meanﬁeld revives,the visibility of the interference pattern also revives.The experimental results are shown in Fig.19.2.Fig.19.2The interference pattern imaged from the released condensate after different hold times. In(d)the interference fringes have entirely vanished indicating a complete collapse of the am-plitude of the condensate.In(g),the wait time is now close to the complete revival time for the coherent amplitude and the fringe pattern is restored.From Fig.2of[12]19.6Interference of Two Bose–Einstein Condensates and Measurement–Induced Phase405 19.6Interference of Two Bose–Einstein Condensatesand Measurement–Induced PhaseThe standard approach to a Bose–Einstein condensate assumes that it exhibits a well–deﬁned amplitude,which unavoidably introduces the condensate phase.Is this phase just a formal construct,not relevant to any real measurement,or can one ac-tually observe something in an experiment?Since one needs a phase reference to observe a phase,two options are available for investigation of the above question. One could compare the condensate phase to itself at a different time,thereby ex-amining the condensate phase dynamics,or one could compare the phases of two distinct condensates.This second option has been studied by a number of groups, pioneered by the work of Javanainen and Yoo[23]who consider a pair of statisti-cally independent,physically–separated condensates allowed to drop and,by virtue of their horizontal motion,overlap as they reach the surface of an atomic detec-tor.The essential result of the analysis is that,even though no phase information is initially present(the initial condensates may,for example,be in number states),an interference pattern may be formed and a relative phase established as a result of the measurement.This result may be regarded as a constructive example of sponta-neous symmetry breaking.Every particular measurement produces a certain relative phase between the condensates;however,this phase is random,so that the symme-try of the system,being broken in a single measurement,is restored if an ensemble of measurements is considered.The physical conﬁguration we have just described and the predicted interference between two overlapping condensates was realised in a beautiful experiment per-formed by Andrews et al.[18]at MIT.The observed fringe pattern is shown in Fig.19.8.19.6.1Interference of Two Condensates Initially in Number States To outline this effect,we follow the working of Javanainen and Yoo[23]and consider two condensates made to overlap at the surface of an atom detector.The condensates each contain N/2(noninteracting)atoms of momenta k1and k2,respec-tively,and in the detection region the appropriateﬁeld operator isˆψ(x)=1√2a1+a2exp iφ(x),(19.27)whereφ(x)=(k2−k1)x and a1and a2are the atom annihilation operators for the ﬁrst and second condensate,respectively.For simplicity,the momenta are set to±π, so thatφ(x)=2πx.The initial state vector is represented simply by|ϕ(0) =|N/2,N/2 .(19.28)40619Bose-Einstein Condensation Assuming destructive measurement of atomic position,whereby none of the atoms interacts with the detector twice,a direct analogy can be drawn with the theory of absorptive photodetection and the joint counting rate R m for m atomic detections at positions {x 1,···,x m }and times {t 1,···,t m }can be deﬁned as the normally–ordered averageR m (x 1,t 1,...,x m ,t m )=K m ˆψ†(x 1,t 1)···ˆψ†(x m ,t m )ˆψ(x m ,t m )···ˆψ(x 1,t 1) .(19.29)Here,K m is a constant that incorporates the sensitivity of the detectors,and R m =0if m >N ,i.e.,no more than N detections can occur.Further assuming that all atoms are in fact detected,the joint probability density for detecting m atoms at positions {x 1,···,x m }follows asp m (x 1,···,x m )=(N −m )!N ! ˆψ†(x 1)···ˆψ†(x m )ˆψ(x m )···ˆψ(x 1) (19.30)The conditional probability density ,which gives the probability of detecting an atom at the position x m given m −1previous detections at positions {x 1,···,x m −1},is deﬁned as p (x m |x 1,···,x m −1)=p m (x 1,···,x m )p m −1(x 1,···,x m −1),(19.31)and offers a straightforward means of directly simulating a sequence of atom detections [23,24].This follows from the fact that,by virtue of the form for p m (x 1,···,x m ),the conditional probabilities can all be expressed in the simple formp (x m |x 1,···,x m −1)=1+βcos (2πx m +ϕ),(19.32)where βand ϕare parameters that depend on {x 1,···,x m −1}.The origin of this form can be seen from the action of each measurement on the previous result,ϕm |ˆψ†(x )ˆψ(x )|ϕm =(N −m )+2A cos [θ−φ(x )],(19.33)with A exp −i θ= ϕm |a †1a 2|ϕm .So,to simulate an experiment,one begins with the distribution p 1(x )=1,i.e.,one chooses the ﬁrst random number (the position of the ﬁrst atom detection),x 1,from a uniform distribution in the interval [0,1](obviously,before any measurements are made,there is no information about the phase or visibility of the interference).After this “measurement,”the state of the system is|ϕ1 =ˆψ(x 1)|ϕ0 = N /2 |(N /2)−1,N /2 +|N /2,(N /2)−1 expi φ(x 1) .(19.34)That is,one now has an entangled state containing phase information due to the fact that one does not know from which condensate the detected atom came.The corre-sponding conditional probability density for the second detection can be derived as19.6Interference of Two Bose–Einstein Condensates and Measurement–Induced Phase 407n u m b e r o f a t o m s n u m b e r o f a t o m s 8position Fig.19.3(a )Numerical simulation of 5,000atomic detections for N =10,000(circles).The solid curve is a least-squares ﬁt using the function 1+βcos (2πx +ϕ).The free parameters are the visibility βand the phase ϕ.The detection positions are sorted into 50equally spaced bins.(b )Collisions included (κ=2γgiving a visibility of about one-half of the no collision case.From Wong et al.[24]40819Bose-Einstein Condensationp (x |x 1)=p 2(x 1,x )p 1(x 1)=1N −1 ˆψ†(x 1)ˆψ†(x )ˆψ(x )ˆψ(x 1) ˆψ†(x 1)ˆψ(x 1) (19.35)=12 1+N 2(N −1)cos [φ(x )−φ(x 1)] .(19.36)Hence,after just one measurement the visibility (for large N )is already close to 1/2,with the phase of the interference pattern dependent on the ﬁrst measurement x 1.The second position,x 2,is chosen from the distribution (19.36).The conditional proba-bility p (x |x 1)has,of course,the form (19.32),with βand ϕtaking simple analytic forms.However,expressions for βand ϕbecome more complicated with increasing m ,and in practice the approach one takes is to simply calculate p (x |x 1,···,x m −1)numerically for two values of x [using the form (19.30)for p m (x 1,...,x m −1,x ),and noting that p m −1(x 1,...,x m −1)is simply a number already determined by the simu-lation]and then,using these values,solve for βand ϕ.This then deﬁnes exactly the distribution from which to choose x m .The results of simulations making use of the above procedure are shown in Figs 19.3–19.4.Figure 19.3shows a histogram of 5,000atom detections from condensates initially containing N /2=5,000atoms each with and without colli-sions.From a ﬁt of the data to a function of the form 1+βcos (2πx +ϕ),the visibil-ity of the interference pattern,β,is calculated to be 1.The conditional probability distributions calculated before each detection contain what one can deﬁne as a con-000.10.20.30.40.50.60.70.80.91102030405060number of atoms decided 708090100x=0x=1x=2x=4x=6Fig.19.4Averaged conditional visibility as a function of the number of detected atoms.From Wong et al.[13]19.7Quantum Tunneling of a Two Component Condensate40900.51 1.520.500.5Θz ο00.51 1.520.500.5Θx ο(b)1,234elliptic saddle Fig.19.5Fixed point bifurcation diagram of the two mode semiclassical BEC dynamics.(a )z ∗,(b )x ∗.Solid line is stable while dashed line is unstable.ditional visibility .Following the value of this conditional visibility gives a quantita-tive measure of the buildup of the interference pattern as a function of the number of detections.The conditional visibility,averaged over many simulations,is shown as a function of the number of detections in Fig.19.4for N =200.One clearly sees the sudden increase to a value of approximately 0.5after the ﬁrst detection,followed by a steady rise towards the value 1.0(in the absence of collisions)as each further detection provides more information about the phase of the interference pattern.One can also follow the evolution of the conditional phase contained within the conditional probability distribution.The ﬁnal phase produced by each individual simulation is,of course,random but the trajectories are seen to stabilise about a particular value after approximately 50detections (for N =200).19.7Quantum Tunneling of a Two Component CondensateA two component condensate in a double well potential is a non trivial nonlinear dynamical model.Suppose the trapping potential in (19.3)is given byV (r )=b (x 2−q 20)2+12m ω2t (y 2+z 2)(19.37)where ωt is the trap frequency in the y –z plane.The potential has elliptic ﬁxed points at r 1=+q 0x ,r 2=−q 0x near which the linearised motion is harmonic withfrequency ω0=q o (8b /m )1/2.For simplicity we set ωt =ω0and scale the length in units of r 0= ¯h /2m ω0,which is the position uncertainty in the harmonic oscillatorground state.The barrier height is B =(¯h ω/8)(q 0/r 0)2.We can justify a two mode expansion of the condensate ﬁeld by assuming the potential parameters are chosen so that the two lowest single particle energy eigenstates are below the barrier,with41019Bose-Einstein Condensation the next highest energy eigenstate separated from the ground state doublet by a large gap.We will further assume that the interaction term is sufﬁciently weak that, near zero temperature,the condensate wave functions are well approximated by the single particle wave functions.The potential may be expanded around the two stableﬁxed points to quadratic orderV(r)=˜V(2)(r−r j)+...(19.38) where j=1,2and˜V(2)(r)=4bq2|r|2(19.39) We can now use as the local mode functions the single particle wave functions for harmonic oscillators ground states,with energy E0,localised in each well,u j(r)=−(−1)j(2πr20)3/4exp−14((x−q0)2+y2+z2)/r20(19.40)These states are almost orthogonal,with the deviation from orthogonality given by the overlap under the barrier,d3r u∗j(r)u k(r)=δj,k+(1−δj,k)ε(19.41) withε=e−12q20/r20.The localised states in(19.40)may be used to approximate the single particle energy(and parity)eigenstates asu±≈1√2[u1(r)±u2(r)](19.42)corresponding to the energy eigenvalues E±=E0±R withR=d3r u∗1(r)[V(r)−˜V(r−r1)]u2(r)(19.43)A localised state is thus an even or odd superposition of the two lowest energy eigenstates.Under time evolution the relative phase of the superposition can change sign after a time T=2π/Ω,the tunneling time,where the tunneling frequency is given byΩ=2R¯h=38ω0q20r2e−q20/2r20(19.44)We now make the two-mode approximation by expanding theﬁeld operator asˆψ(r,t)=c1(t)u1(r)+c2(t)u2(r)(19.45) where。

偶极玻色-爱因斯坦凝聚体中孤子碰撞的理论研究

偶极玻色-爱因斯坦凝聚体中孤子碰撞的理论研究SHI Yu-ren;YANG Xue-ying;TANG Na;LI Xiao-lin;SONG Lin【摘要】对准一维情形下具有偶极相互作用的玻色-爱因斯坦凝聚体(Bose-Einstein condensate,BEC)中孤子的碰撞进行了理论研究.运用虚时演化法数值求解了Gross-Pitaevskii方程的孤子态解,然后构造了实验中可实现的双孤子态以研究其碰撞规律.发现既存在完全弹性碰撞,也存在完全非弹性碰撞.通过调节偶极作用强度,可实现从弹性碰撞到非弹性碰撞的转变.初始时刻孤子的相位差不仅会影响系统的对称性,也会改变孤子的碰撞类型.【期刊名称】《西北师范大学学报（自然科学版）》【年(卷),期】2019(055)004【总页数】8页(P44-51)【关键词】偶极BEC;孤子;碰撞【作者】SHI Yu-ren;YANG Xue-ying;TANG Na;LI Xiao-lin;SONG Lin【作者单位】;;;;【正文语种】中文【中图分类】O145玻色-爱因斯坦凝聚(Bose-Einstein condensate, BEC)是物质的一种新型状态.自从实验上发现BEC中的亮孤子后[1]，冷原子中孤子的行为便受到广泛关注[2-8].在不同的囚禁外势下，孤子的周期、能量变化都有很大不同.研究表明，通过减小轴向频率和径向频率的比值和原子的损耗，可以延长孤子的寿命，产生非常丰富的新奇现象[9-11].混合冷原子中孤子的特性也令研究者产生了极大兴趣[12-13].由于短距离自旋极化费米子之间强烈的泡利阻塞排斥作用，在反射玻色子-费米子相互作用中不可能存在费米亮孤子.Sadhan[14]等证明稳定的费米亮孤子可以在玻色-费米混合气体中形成.此外，当孤子间相互作用不同时，孤子的性质也会发生改变15-19].例如分子类型的相互作用，使得许多孤子可以存在，其中包括串形、环状或规则格子型孤子分子，其动力学行为会发生很大变化[20].近年来，在实验和理论的研究中量子简并气体的远程相互作用受到很多关注.偶极相互作用是长程力，且各向异性，这些特性会影响凝聚体的基态、稳定性及动力学性质.这些性质提供了一种研究多体量子效应的方式，例如,超流晶体的量子相变、超固体，甚至是拓扑序列等.许多学者针对偶极BEC中孤子间相互作用下的动力学性质也做了大量研究[21-24].Pedri[25]数值研究了准二维情形下具有偶极相互作用的亮孤子碰撞，发现亮孤子在碰撞后合并，这是一种典型的完全非弹性碰撞.文中主要研究准一维偶极BEC中孤子的碰撞.采用虚时演化法得到凝聚体的孤子态，在谐振子势下探究初始孤子的相位差对碰撞的影响.1 理论模型考虑束缚在谐振子势阱中的偶极BEC，在平均场理论框架内，体系的动力学行为可以用Gross-Pitaevskii(GP)方程描述[26]其中,m为粒子质量;Ψ为波函数;满足为总粒子数;原子间相互作用强度g=4πћ2as/m,as为s-波散射长度;外势分别为径向和轴向频率;ρ和z为径向和轴向坐标;偶极相互作用项为极化方向单位矢量;对于磁偶极情形，Cdd=μ0μ2,μ0为真空磁导率，对于电偶极情形，为真空介电常数，为玻尔半径.引入可对方程(1)进行无量纲化.当ωρ≫ωz时，方程(1)可化为准一维GP方程[27]其中,变量上面的“～”已略去;“*”表示傅氏卷积;表征变换后的接触相互作用强度;表征变换后偶极作用强度，为偶极作用与接触作用强度的比值，为z方向的特征长度;为余误差函数;波函数满足2 数值结果2.1 准一维单分量BEC的孤子态单分量BEC在实验上容易操控，对其进行深入探究将有利于对BEC的特性更深入的了解.Thierry[28]等在实验上实现了准二维的具有强偶极相互作用的52Cr原子BEC，通过调节外加磁场减弱s-波散射长度，从而使52Cr原子的偶极相互作用强度变得与接触相互作用可比拟，这将导致原子云的长宽比发生变化.下面研究准一维情形下偶极相互作用与接触相互作用对52Cr原子BEC的影响，采用虚时演化法[29]可得到GP方程(2)的孤子态.文献[27]中给出了动能项与偶极项均忽略时孤子态的解析结果.下面用变分法求解当动能项保留而偶极项忽略(即εdd=0)时的孤子解，即GP方程(2).其拉格朗日密度为[30]采用高斯波包作为拟设(假设波包中心位于z=0处)则有效拉格朗日量为变分参数wz的欧拉—拉格朗日方程为(6)一定条件下，方程(6)反映了在给定初始条件下孤立波的振幅及波宽随时间变化的规律.考虑将N=104个52Cr原子束缚在谐振子势阱中. 52Cr原子磁偶极矩为6μB(μB 为玻尔磁子)，原子质量m=8.63×10-26 kg，极化方向取为n=(0,0,1)，谐振子频率(ωρ，ωz)=2π(350,35)Hz,则利用Feshbach共振技术[31]可调节接触相互作用系数β1D，调节外磁场可以改变偶极相互作用强度εdd，这样可保证与Thierry 等实验所用参数[28]一致.图1给出了不同参数情况下的粒子数分布图，其中，NS(Numerical solution)表示用虚时演化法得到的数值结果;AS(Analytical solution)表示用变分法得到的解析结果.可以看出，粒子数分布均呈现出钟状孤立子态.图1(a～b)中所用参数分别为β1D=10,εdd=0.4和β1D=100，εdd=0.6，由此计算得到η≈9.55,λ≈-21.07和η≈63.66,λ≈-316.12.在该参数情形下，均有η>0且λ<0，表明近程作用表现为排斥而长程作用为吸引.从图1(a～b)可看出，解析结果与数值结果在较大范围内基本吻合.但图1a中数值结果的振幅比解析结果的要大，这是因为在变分法计算时忽略了偶极相互作用；而在此参数条件下，偶极作用表现为吸引，这会使得孤子振幅增大.图1b中亦是如此.图1c给出了εdd=0.4时粒子数密度随β1D的变化.可以看出，粒子数密度的峰值(可视为孤子的振幅)随着β1D的增加而增大.图1d给出了β1D=100时粒子数密度随εdd的变化.可以看出，孤子振幅随εdd增加而变大，同时孤子的宽度减小.这是由于偶极作用(此时为吸引)增强的缘故.这些结论与文献[28]中实验结果一致.图1 不同参数情形下粒子数密度分布Fig 1 Particle number density distribution under different parameters为进一步研究粒子数密度与原子间接触相互作用及偶极相互作用的关系，图2给出了孤立波振幅max(|Ψ|2)随β1D及εdd的变化.从图2a可以看出，当εdd较小时，孤立波振幅随β1D的增大而逐渐减小；而当εdd较大时，孤立波振幅则随β1D的增加而增大.图2b给出了不同β1D时孤立波振幅随εdd的变化.可以看出，β1D一定时，孤立波振幅随εdd的增加而单调增加(这一点在图2(a)中也有所体现).这是偶极作用与接触作用相互竞争的结果.可解释如下：在图2所示参数条件下，可见始终有λ<0，表明偶极作用始终表现为吸引.吸引作用将会导致孤立波振幅增加,而排斥作用则相反.当εdd>1时，η<0；而当εdd<1时，η>0.说明通过调节偶极作用强度，也可改变接触相互作用的性质.另外，此时当εdd较小时(接近0)η>0而相对较大，说明接触相互作用表现为排斥且排斥作用强于吸引作用，故孤立波振幅较小.而当εdd较大时，相对较小，说明排斥作用减弱，故会导致孤立波振幅变大.图2 不同参数情形下孤立波振幅随β1D及εdd的变化Fig 2 T he variation of solitary wave amplitude with β1D and εdd under different parameters2.2 准一维单分量BEC孤子态的稳定性孤子的动力学稳定性是一个非常重要的问题.不稳定的孤波结构不能长时间存在，而稳定的孤波具有强的抗干扰能力，可以长时间存在，便于实验上观察和进一步研究.Ueda考虑该系统的平衡态(粒子均匀分布的情形)，通过计算能量得出,无外势情形下，当-0.5≤εdd≤1时,该态呈稳定性,在外势作用下系统将更加稳定[32].我们用数值方法对系统孤子态的稳定性进行研究，发现在所计算的参数范围0<εdd<2内，孤子态均呈现非常强的稳定性.此结果与系统平衡态下的稳定性有很大不同，也符合孤子的特性.数值研究时采用以下做法,用虚时演化法得到GP方程的孤子态Ψ=φ0(z,t)后，当t=0时刻，在该态上加一微小扰动做为初态Ψ(z,0)=φ0(z,0)然后用时间劈裂傅里叶谱方法[33]对GP方程进行长时间动力学演化，便可研究该孤子态的动力学稳定性. 计算时，取A=0.001，W根据孤子的宽度做适当调整.在不同参数情况下的时间步长也需要调整以保证数值稳定性.图3 不同参数下粒子数密度随时间t的变化Fig 3 Variation of particle number density with time t under different parameters图3给出了不同接触作用和偶极作用强度时，粒子数密度随时间t的变化.可以看出，粒子数密度呈钟状孤子态分布，在扰动下并不随时间发生明显的变化，表明该态是动力学稳定的.为保证数值计算精度，图3中空间网格数取为2 048；为保证数值稳定性，图3a计算时时间步长(无量纲化的)取为10-7；图3b中则需取为10-8.这使得计算量急剧增大.比较图3a,b可以看出，随着偶极作用系数的增加，波包明显变得窄而“瘦高”，这是因为偶极作用表现为很强的吸引作用.从图3b可看出，即使在εdd=1.8的情形下，孤子态仍保持稳定.数值计算时，在尝试的参数范围0<β1D≤200,0<ε<εdd内，均没发现不稳定的孤子态.这种较强的稳定性对于孤子在量子信息、非线性光学、原子输运和原子干涉仪等领域内的应用有着重要的理论指导意义.2.3 双孤子碰撞碰撞是孤子重要的动力学性质之一，影响碰撞的因素也有很多.实验中，可以在系统中放置两份制备好的孤子态BEC以观察孤子之间的碰撞现象.理论研究中可采取如下方式来模拟此碰撞过程,首先用虚时演化法得到GP方程(2)的孤立波解，记此态为Ψ=Ψs(z,0).然后通过空间坐标平移，从而得到两份(甚至更多份)孤子态BEC，分别记为Ψ1=Ψs(z-z0,0)和Ψ2=Ψs(z+z0,0)，其中z0为空间平移量.这样制备的双孤子在初始时刻空间位置沿z=0点呈对称分布(理论和实验中均可研究非对称情形，但由于篇幅原因，本文仅研究对称情形).接着取Ψ(z,0)=Ψ1+Ψ2eiΔθ作为初始条件对GP方程(2)进行动力学演化，便可研究双孤子之间的碰撞，这里为初始时刻两孤子态的相位差.进行动力学演化时，采用时间劈裂傅里叶谱方法[33].这种方法精度高，有保持粒子数守恒的优点，被广泛应用于BEC系统的理论模拟[34-37].文献[25]中，Pedri等数值研究了准二维情形下具有偶极相互作用的亮孤子碰撞，发现亮孤子在碰撞后发生合并.这是一种典型的完全非弹性碰撞.但在准一维情形下，我们发现存在两孤立波的完全弹性碰撞.图4a给出了Δθ=0,β1D=50,β1D=50,εdd=0.3时两孤立波的完全弹性碰撞过程.刚开始时两孤立波静止，但在外势和吸引作用下会逐渐加速，相向而行.一段时间后发生碰撞，且在碰撞过程中伴随物质波的干涉现象.碰撞结束后，两孤立波穿过彼此后变为背向而行，并逐渐减速，速度减为0后又重复前述碰撞过程.图4b～e放大给出了孤立波的碰撞过程，从中可以清晰地观察到物质波的干涉现象.Δθ的变化会对干涉条纹产生影响；当Δθ=0时(图4b)，两孤子碰撞时出现五个峰值，且中心位置处也为极大值.当Δθ=π时(图4d)，虽然干涉条纹仍以z=0为中心呈左右对称分布，但中心位置处变为极小值；和时(图4c,e)，碰撞过程中干涉条纹的对称性也不复存在；Δθ取其它值时，也有类似的干涉现象.图4 两孤立波的碰撞(β1D=50,εdd=0.3)Fig 4 Collision of two solitarywaves(β1D=50,εdd=0.3)准一维情形下，也存在类似Pedri等发现的非弹性碰撞.图5给出了β1D=200,εdd=0.6时不同初始相位差情形下双孤子的碰撞.图5a中Δθ=0，初始时刻两孤子均静止.在外势和偶极作用下，它们逐渐加速，相向而行，过段时间后发生碰撞.碰撞后合二为一，然后在z=0附近左右振荡，且振荡幅度随时间逐渐变小.这是一种完全非弹性碰撞,图5b中两孤子在第一次碰撞后穿过彼此，且发生能量转移，对称性也被破坏.这是较典型的非完全弹性碰撞.碰撞后向右运动的孤子振幅变大，且运动相对较短的距离后便向左返回；而向左运动的孤子振幅变小，运动相对较长的距离后便向右运动,然后两孤子再次发生碰撞.二次碰撞后两孤子合并，为完全非弹性碰撞,合并后的孤子在z=0附近左右振荡，但振荡幅度较Δθ=0的情形(参看图5a)大得多.类似的情形在图5c中也存在，和图5b相比仅是在空间上发生了翻转.图5d给出了Δθ=π时双孤子的碰撞.可以看出，两孤子在经过几次非完全弹性碰撞后合二为一，然后在z=0附近作周期性振荡.图5 势场中不同初始相位差时双孤子的碰撞Fig 5 Collisions of double solitons under different initial phase differences in the potential field4 结束语数值研究了准一维偶极玻色-爱因斯坦凝聚体中双孤子的碰撞.运用虚时演化法数值求出GP方程的孤子态解，然后构造了实验中可实现的双孤子态以研究其碰撞规律.发现不仅存在完全弹性碰撞和完全非弹性碰撞，还存在从弹性碰撞到非弹性碰撞的转变.碰撞过程中存在明显的物质波干涉现象.初始时刻孤子的相位差不仅会影响系统的对称性，还会改变孤子的碰撞类型.当初始相位差为0或π时，粒子密度分布具有很好地空间对称性；当初始相位差为其它值时，这种对称性被破坏.这些性质表明BEC在原子运输和量子信息方面具有潜在的应用价值.研究结果可为实验上偶极BEC在实验上的研究提供可能的理论指导.参考文献:【相关文献】[1] STRECKER K E,PARTRIDGE G B,TRUSCOTT A G,et al.Formation and propagation of matter-wave soliton 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[19] SUN J Q,LUO S Y,CAI B G.Interaction of multi-optical solitions in the three-levelgaseous media[J].Acta Phys Sin,2012,61(14):14020.[20] KHAWAJA U A.Interaction of multi-optical solitions in the three-level gaseousmedia[J].Phys Rev E,2012,85(5)：056604.[21] NATH R,PEDRI P,SANTOS L.Soliton-soliton scattering in dipolar Bose-Einstein condensates[J].Phys Rev A,2007,76(1)：013606.[22] ZHANG H,TANG D Y,ZHAO L M,et al．Observation of polarization domain wall solitons in weakly birefringent cavity fiber lasers[J].Phys Rev B,2009,80(5)：052302.[23] LAKOMY K,NATH R,SANTOS L.Soliton molecules in dipolar Bose-Einstein condensates[J].Phys Rev A,2012,86(1)：013610.[24] KÖBERLE P,ZAJEC D,WUNNER G,et al．Creating two-dimensional bright solitons in dipolar Bose-Einstein condensates[J].Phys Rev A,2012,85(2)：023630.[25] PEDRI P,SANTOS L.Two-dimensional bright solitons in dipolar Bose-Einstein condensates[J].Phys Rev Lett,2005,95(2)：200404.[26] OLSON A J,WHITENACK D L,CHEN Y P.Effects of magnetic dipole-dipole interactionsin atomic Bose-Einstein condensates with tunable s-wave interactions[J].Phys RevA,2013,88(4)：043609.[27] CAI Y Y,MATTHIAS R,ZHEN L,et al． Mean-field regime of trapped dipolar Bose-Einstein condensates in one and two dimensions[J].Phys Phys Rev A,2010,82(4)：043623. 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67-理论物理专业硕士研究生培养方案

理论物理专业硕士研究生培养方案
一、培养目标
培养符合国家建设需要, 为祖国和人民服务的, 具有良好道德品质和科学素质的, 具有集体主义精神, 实事求是, 追求真理, 献身科学教育事业的, 具有扎实基础知识和良好科研能力的理论物理专门人才和高等院校师资.
获得本专业硕士学位的研究生应掌握理论物理学科坚实、宽厚的基础知识，较全面和深入的专业知识，熟悉本专业研究方向的发展前沿和热点. 硕士论文选题时，应对国内外研究现状进行较全面的调研和分析，在此基础上，完成具有创造性的研究成果。熟练掌握一门外语, 包括专业阅读和写作，以及能用外语进行简单的学术交流。
预修课程：量子力学及大学物理本科专业相关的课程
主要教材及参考文献：
1、by H.Friedrich．Theortical Atomic Physic［M］．
2、R.N波特，M.卡普路斯．原子与分子［M］．王荣顺等．
预修课程：高等量子力学、场论
主要教材及参考文献：
1、D.F.Wall，G.J.Mibum．Quantum Optics［M］．
2、W.H.Louisell．辐射场的量子统计性质［M］．
3、C.W.Gardiner．Quantum Nois［M］．
4、J.Perina．Quantum statistics of Linear and Nonlinear Optical Phenomena［M］．
周光辉秋 01100720113 孤子微扰理论 60 2 颜家壬秋 01100720114 张量分析与拓扑 60 2 王永久春 01100720115 引力理论基础 60 2 王永久春 01100720116 引力效应 60 2 王永久秋 01100720117 引力场方程严格解 60 2 王永久秋 01100720118 广义相对论 60 2 荆继良秋 01100720119 黑洞物理学 60 2 荆继良春 01100720120 弯曲时空量子场论 60 2 余洪伟春 01100720121 相互作用的统一理论 60 2 余洪伟秋 01100720122 时空大尺度结构与演化 60 2 余洪伟春 01100720123 量子信息理论 60 2 方卯发

物理学中的玻色爱因斯坦凝聚态

物理学中的玻色爱因斯坦凝聚态玻色-爱因斯坦凝聚态（Bose-Einstein Condensate，简称BEC）是20世纪90年代物理学界的一项重大发现。

其意义重大，既推动了基础物理、凝聚态物理等领域的发展，也创造出了一系列的应用，如大功率激光器、量子计算器等等。

本文尝试为大家介绍BEC的相关背景及其物理本质。

1.背景BEC得名自两位物理学家印度的萨提琳德拉·玛萨杜和奥地利的阿尔贝特·爱因斯坦。

经过研究发现，如果把气体冷却到足够低的温度，仅有一个能级能够容纳超过其中一半的原子。

原子的所有空间统计分布现象出现了与此不同的行为，它不再是独立的粒子，而是趋于在相同的能级聚集成一个相干的超原子，也就是玻色-爱因斯坦凝聚态。

2.物理本质在正常的体系中，相互作用的粒子形成了无序的系统，粒子间间距不太相同。

而在低温条件下，粒子间间距小，粒子密度高，由于粒子间相互作用，粒子间的波动也耗费更为复杂、更为巨大的能量。

当温度到达绝对零度以下后，所有粒子全部入同一量子态，并受到同一波动方程的影响，玻色-爱因斯坦凝聚态就形成了。

这个状态的粒子可以被描述成一个巨型波函数，因此它有不同的行为和特性，相对与普通状态的粒子，更易于控制和操纵。

BEC已经成为凝聚态物理中的一个热点，因为这种状态的物理特性与相互作用问题有关，能够在特定材料和设备中进行有效的应用。

3.应用虽然BEC在物理学中得到广泛的应用，但是它同样能够应用于其他领域。

由于BEC可以实现混合物，利用不同的材料来制造化学反应。

而且，BEC在量子计算器方面也是一个无可替代的重要因素之一，提供实现量子算法的最初条件，因此在一项大型科技研究中具有无穷的前景。

总之，BEC是自然界中一个极其神奇和重要的现象，对凝聚态物理学领域以及其他领域具有无限潜力。

BEC的研究已经突破了物理学的范畴，成为了多个重要领域的研究热点，更多的研究还在继续深入。

相信今后，BEC的应用将会越来越广泛。

Exciton-phonon droplets with Bose-Einstein condensate transport and optical properties

2
D. Roubtsov et al.
where ncoh (x, t) ≈ ncore (x − vt) is the ballistic (superﬂuid) part of the packet, ncore (x − vt) ≃ |Ψ0 |2 (x − vt), (4) and ∆n(x, t) is the noncondensed part of it. The following decomposition can be written for the out-of-condensate part: ˆ† δ ψ ˆ(x, t) ≈ δncloud (x, t) + δntail (x, t). ∆n(x, t) = δ ψ The challenging problem is how to describe the spatially inhomogeneous state of the droplet with the exciˆ(x, t), where tonic BEC inside in terms of Ψ0 (x, t) and δ ψ ˆ δ ψ is the “ﬂuctuating” part of the exciton Bose-ﬁeld. For example, within the quasistationary approximation, one has to calculate ncore (x/L0 ) and ∆n(x) ≃ δno, cloud (x/Lch ) + δntail (5) and understand how the diﬀerent characteristic lengthes and coherence properties appear in the theory with the Bose-condensate. Note that if the excitonic packet moves in a crystal (or another semiconductor structure), it interacts with thermal phonons, noncondensed excitons, impurities and other imperfections of the lattice, etc.. Then, the coherent core of the packet can be found in a quasi-stable state, and the ﬂuctuations of φo (x − vt) and, especially, ϕc (x, t) can be of a great importance for possible experimental veriﬁcations of their existence. 2 Exciton-Phonon Condensate To obtain the necessary density of excitons nx in the excitonic cloud and, thus, meet the BEC conditions, the crystals are irradiated by laser pulses with h ¯ ωL ≫ Egap , and the temperature of the crystal is T ≃ 1 ∼ 5 K. If the cross-section area S of an excitation spot on the surface of the crystal can be made large enough, such as S ≈ Ssurf , the hot droplet of paraexcitons can acquire an average momentum during its thermolization process (T ∗ (t) → T ). Indeed, the phonon wind, or the ﬂow of nonequilibrium phonons, blows unidirectionally from the surface into the bulk [5] and transfers the nonzero momentum to the excitonic cloud, Px ≃ Nx ¯ h k0 = 0 and Px ⊥ Ssurf , see Fig. 1. As a result, the packet of moving excitons and nonequilibrium phonons of the phonon wind (Nph ≃ Nx ) is actually the system that undergoes the transition toward developing the Bose-Einstein correlations at T ∗ < Tc . Let us assume that the condensate has been already formed inside the moving excitonic droplet, and the following representation of the exciton Bose-ﬁeld holds: ˆ = Ψ0 + δ ψ ˆ. For the displacement ﬁeld of the crystal ψ ˆ , we introduce a nontrivial coherent part too, i.e., u ˆ= u ˆ , u0 = 0. The important property of the excitonu0 + δ u phonon condensate is a kind of the self-consistency condition, roughly, ∂ u0 (x − vt) ∝ |Ψ0 (x − vt)|2 . (6) In these terms, the moving packet contains both the macroscopically occupied exciton-phonon condensate, or

Bose-Einstein condensation

Bose-Einstein condensationShihao LiBJTU ID#:13276013;UW ID#:20548261School of Science,Beijing Jiaotong University,Beijing,100044,ChinaJune1,20151What is BEC?To answer this question,it has to begin with the fermions and bosons.As is known,matters consist of atoms,atoms are made of protons,neutrons and electrons, and protons and neutrons are made of quarks.Also,there are photons and gluons that works for transferring interaction.All of these particles are microscopic particles and can be classified to two families,the fermion and the boson.A fermion is any particle characterized by Fermi–Dirac statistics.Particles with half-integer spin are fermions,including all quarks,leptons and electrons,as well as any composite particle made of an odd number of these,such as all baryons and many atoms and nuclei.As a consequence of the Pauli exclusion principle,two or more identical fermions cannot occupy the same quantum state at any given time.Differing from fermions,bosons obey Bose-Einstein statistics.Particles with integer spin are bosons,such as photons,gluons,W and Z bosons,the Higgs boson, and the still-theoretical graviton of quantum gravity.It also includes the composite particle made of even number of fermions,such as the nuclei with even number ofnucleons.An important characteristic of bosons is that their statistics do not restrict the number of them that occupy the same quantum state.For a single particle,when the temperature is at the absolute zero,0K,the particle is in the state of lowest energy,the ground state.Supposing that there are many particle,if they are fermions,there will be exactly one of them in the ground state;if they are bosons,most of them will be in the ground state,where these bosons share the same quantum states,and this state is called Bose-Einstein condensate (BEC).Bose–Einstein condensation(BEC)—the macroscopic groundstate accumulation of particles of a dilute gas with integer spin(bosons)at high density and low temperature very close to absolute zero.According to the knowledge of quantum mechanics,all microscopic particles have the wave-particle duality.For an atom in space,it can be expressed as well as a wave function.As is shown in the figure1.1,it tells the distribution but never exact position of atoms.Each distribution corresponds to the de Broglie wavelength of each atom.Lower the temperature is,lower the kinetic energy is,and longer the de Broglie wavelength is.p=mv=h/λ(Eq.1.1)When the distance of atoms is less than the de Broglie wavelength,the distribution of atoms are overlapped,like figure1.2.For the atoms of the same category,the overlapped distribution leads to a integral quantum state.If those atoms are bosons,each member will tend to a particular quantum state,and the whole atomsystem will become the BEC.In BEC,the physical property of all atoms is totally identical,and they are indistinguishable and like one independent atom.Figure1.1Wave functionsFigure1.2Overlapped wave functionWhat should be stressed is that the Bose–Einstein condensate is based on bosons, and BEC is a macroscopic quantum state.The first time people obtained BEC of gaseous rubidium atoms at170nK in lab was1995.Up to now,physicists have found the BEC of eight elements,most of which are alkali metals,calcium,and helium-4 atom.Always,the BEC of atom has some amazing properties which plays a vital role in the application of chip technology,precision measurement,and nano technology. What’s more,as a macroscopic quantum state,Bose–Einstein condensate gives a brand new research approach and field.2Bose and Einstein's papers were published in1924.Why does it take so long before it can be observed experimentally in atoms in1995?The condition of obtaining the BEC is daunting in1924.On the one hand,the temperature has to approach the absolute zero indefinitely;on the other hand,the aimed sample atoms should have relatively high density with few interactions but still keep in gaseous state.However,most categories of atom will easily tend to combine with others and form gaseous molecules or liquid.At first,people focused on the BEC of hydrogen atom,but failed to in the end. Fortunately,after the research,the alkali metal atoms with one electron in the outer shell and odd number of nuclei spin,which can be seen as bosons,were found suitable to obtain BEC in1980s.This is the first reason why it takes so long before BEC can be observed.Then,here’s a problem of cooling atom.Cooling atom make the kinetic energy of atom less.The breakthrough appeared in1960s when the laser was invented.In1975, the idea of laser cooling was advanced by Hänsch and Shallow.Here’s a chart of the development of laser cooling:Year Technique Limit Temperature Contributors 1980～Laser cooling of the atomic beam～mK Phillips,etc. 19853-D Laser cooling～240μK S.Chu,etc. 1989Sisyphus cooling～0.1~1μK Dalibard,etc. 1995Evaporative cooling～100nK S.Chu,etc. 1995The first realization of BEC～20nK JILA group Until1995,people didn’t have the cooling technique which was not perfect enough,so that’s the other answer.By the way,the Nobel Prize in Physics1997wasawarded to Stephen Chu,Claude Cohen－Tannoudji,and William D.Phillips for the contribution on laser cooling and trapping of atoms.3Anything you can add to the BEC phenomena(recent developments,etc.)from your own Reading.Bose–Einstein condensation of photons in an optical microcavity BEC is the state of bosons at extremely low temperature.According to the traditional view,photon does not have static mass,which means lower the temperature is,less the number of photons will be.It's very difficult for scientists to get Bose Einstein condensation of photons.Several German scientists said they obtained the BEC of photon successfully in the journal Nature published on November24th,2011.Their experiment confines photons in a curved-mirror optical microresonator filled with a dye solution,in which photons are repeatedly absorbed and re-emitted by the dye molecules.Those photons could‘heat’the dye molecules and be gradually cooled.The small distance of3.5 optical wavelengths between the mirrors causes a large frequency spacing between adjacent longitudinal modes.By pumping the dye with an external laser we add to a reservoir of electronic excitations that exchanges particles with the photon gas,in the sense of a grand-canonical ensemble.The pumping is maintained throughout the measurement to compensate for losses due to coupling into unconfined optical modes, finite quantum efficiency and mirror losses until they reach a steady state and become a super photons.(Klaers,J.,Schmitt,J.,Vewinger, F.,&Weitz,M.(2010).Bose-einstein condensation of photons in an optical microcavity.Nature,468(7323), 545-548.)With the BEC of photons,a brand new light source is created,which gives a possible to generate laser with extremely short wavelength,such as UV laser and X-ray laser.What’s more,it shows the future of powerful computer chip.Figure3.1Scheme of the experimental setup.4ConclusionA Bose-Einstein condensation(BEC)is a state of matter of a dilute gas of bosons cooled to temperatures very close to absolute zero.Under such conditions,a large fraction of bosons occupy the lowest quantum state,at which point macroscopic quantum phenomena become apparent.This state was first predicted,generally,in1924-25by Satyendra Nath Bose and Albert Einstein.And after70years,the Nobel Prize in Physics2001was awarded jointly to Eric A.Cornell,Wolfgang Ketterle and Carl E.Wieman"for theachievement of Bose-Einstein condensation in dilute gases of alkali atoms,and for early fundamental studies of the properties of the condensates".This achievement is not only related to the BEC theory but also the revolution of atom-cooling technique.5References[1]Pethick,C.,&Smith,H.(2001).Bose-einstein condensation in dilute gases.Bose-Einstein Condensation in Dilute Gases,56(6),414.[2]Klaers J,Schmitt J,Vewinger F,et al.Bose-Einstein condensation of photons in anoptical microcavity[J].Nature,2010,468(7323):545-548.[3]陈徐宗,&陈帅.(2002).物质的新状态——玻色-爱因斯坦凝聚——2001年诺贝尔物理奖介绍.物理,31(3),141-145.[4]Boson(n.d.)In Wikipedia.Retrieved from:</wiki/Boson>[5]Fermion(n.d.)In Wikipedia.Retrieved from:</wiki/Fermion>[6]Bose-einstein condensate(n.d.)In Wikipedia.Retrieved from:</wiki/Bose%E2%80%93Einstein_condensate>[7]玻色-爱因斯坦凝聚态(n.d.)In Baidubaike.Retrieved from:</link?url=5NzWN5riyBWC-qgPhvZ1QBcD2rdd4Tenkcw EyoEcOBhjh7-ofFra6uydj2ChtL-JvkPK78twjkfIC2gG2m_ZdK>。

Bose-Einstein condensation in dense quark matter

arXiv:0809.2892v1 [hep-ph] 17 Sep 2008
Bose-Einstein condensation in dense quark matter
Jens O. Andersen
Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway
Preprint submitted to Elsevier 17 September 2008
In this talk, I would like to present some results for Bose condensation of charged pions using the linear sigma model as a low-energy eﬀective for two-ﬂavor QCD. The talk is based on Refs. [8,9] and part of the work is done in collaboration with Tomas Brauner. 2. Interacting Bose gas and 2PI 1/N -expansion The Euclidean Lagrangian for a Bose gas with N species of massive charged scalars is
2 2 2 ωn + m2 4 + ΠLO . The term m4 + ΠLO is thus a local mass term, which we denote by M . The exact propagator D satisﬁes the equation δ Γ/δD = 0. In the present case, it reduces to a local gap equation for M 2 :

High-temperature Bose-Einstein condensation of polaritons realization under the intracavity

High-temperature Bose-Einstein condensation of polaritons: realization under the intracavity laserpumping of matter conditionV.A. Averchenko1), A.P. Alodjants2*), S.M. Arakelian2), S.N. Bagayev3), E.A. Vinogradov4),V.S. Egorov1), A.I. Stolyarov1), I.A. Chekhonin1)1) St. Petersburg State University, Ul'yanovskaya ul. 1, 198504 St. Petersburg, Staryi Peterhof, Russia2) Vladimir State University, ul. Gor'kogo 87, 600000 Vladimir, Russia3) Insitute of Laser Physics, Russian Academy of Sciences, prosp. akad. Lavrent'eva 13/3, 630090Novosibirsk, Russia4) Institute of Spectroscopy, Russian Academy of Sciences, 142190 Troitsk, Moscow region, RussiaAbstract.A quantum model of Bose-Einstein condensation based on processes involvingpolaritons excited in an intracavity absorbing cell with resonance atoms, which is manifested inthe spectral characteristics of the system, is considered. It is shown that the spectral'condensation' appears which is directly related to the degeneracy of a weakly interacting gas ofpolaritons resulting in quasi-condensation at room temperature. The possibility of obtainingpolariton condensation as a new phase state by using the confinement of polaritons in an atomicoptical harmonic trap is discussed.Keywords: polaritons, quasi-condensation, Bose-Einstein condensation, polariton laser.1. IntroductionExperiments on the Bose-Einstein condensation (BEC) of macroscopic numbers of atoms (N ≥ 106 ) is one of the most spectacular recent advances, which have made a great influence on the development of various directions in modern quantum and laser physics and newest technologies (see, for example, [1]). In the case of BEC, when under conditions of the temperature phase transition a macroscopic number of atoms are in the ground (lower) quantum level, a new coherent state of matter is formed. This is manifested in the fact that, for example, at the limiting temperature T=0 an ensemble of condensate atoms, as each individual atom, is described by the common wave function corresponding to a coherent state. In this aspect, the BEC phenomenon is similar to lasing, for example, when strict phase locking of laser modes occurs in laser cavities [2, 3]. In addition, in the case of BEC, we can say about the realisation of a Bose laser (boser) emitting coherent ensembles of atoms [3, 4]. A remarkable feature of such macroscopic quantum states of matter is the possibility to use them for the development of new physical principles of quantum information processing and communication [5, 6].However, despite spectacular achievements in this direction, there exist a number of practical difficulties imposing principal restrictions on the possibility of real applications of the atomic BEC for these purposes. Thus, one of the basic problems is the necessity of maintaining extremely low temperatures (tens of nK) to realise such devices. In this connection the problem of obtaining macroscopic coherent (quantum) states of matter at high (room) temperatures becomes very important.One of the most attractive approaches to the solution of this problem is the preparation of a quasi-condensate of the two-dimensional Bose gas of weakly interacting polaritons (in atomic physics [7]) and _______________________________________________________*) Email: alodjants@vpti.vladimir.ruexcitons (in solid state physics [8-10] *)). Such collective states of the medium (quasi-particles) represent a superposition of photons and spin waves in the atomic medium and can be obtained, for example, within the framework of the Dicke model used to describe superradiance [12]. Although these states cannot be treated as a condensate in a strict thermodynamic sense due to the nonequilibrium state of the system as a whole, under certain conditions imposed on the type of atomic optical interactions in the system, polaritons do form a condensate, their distribution being described by the Bose-Einstein distribution function for an ideal gas of bosons [13].In this paper, we considered the interaction of a system of two-level atoms with an electromagnetic field in the cavity in the case of the so-called strong coupling, when the inequality1/220212c coh d n πωωτ⎛⎞=⎜⎟⎝⎠, (1) is fulfilled, where c ω is the cooperative frequency determining the collective interaction of atoms with the field; 0ω is the atomic transition frequency; d is the transition dipole moment; coh τ is the characteristic coherence time of the atomic medium; n is the atomic gas density; and is Planck's constant. In this case, the field itself is weak (in the number of photons).The so-called condensation of the spectrum occurs when inequality (1) is fulfilled [14, 15]. This effect consists in the fact that under some threshold conditions imposed on the concentration of absorbing atoms and pump intensity, radiation of a broadband laser with a narrowband absorbing intracavity cell is concentrated ('condensed') near the strongest absorption lines of matter.This phenomenon was observed experimentally and interpreted within the framework of a clear classical model of parametric excitation of two coupled oscillators (electromagnetic field and atoms of matter) upon coherent energy transfer between them. However, this model is one-dimensional and is not quantum one, which obviously restricts the field of its applications.In this paper, we propose a detailed quantum model of spectral condensation realised for polaritons excited in an intracavity absorbing cell [16]. We show that spectral condensation can be directly related to the condensation (quasi-condensation) of polaritons in the cavity if a strong coupling between the electromagnetic field and medium is provided. The latter statement is in itself of interest, and in this paper we substantiate for the first time the possibility of obtaining the true BEC in the polariton system at high (room) temperatures in the case of spectral condensation. In this respect, of interest are the experimental data [8, 9] obtained in semiconductor microcavities, which confirm the above assumption.2. Basic relationsConsider the interaction of two-level atoms (with levels a and b) with a quantum electromagnetic field, which is described by the photon annihilation (creation) operators ()k k f f + for the k-th mode. Within the framework of dipole approximation, such a system can be described by the Hamiltonian [13]12()()()()ph k k atk k k k k k k k k kH k E k f f E b b a a g f a b b a f ++++++=+−+−∑∑∑, (2) level a, and the inequality_______________________________________________________*) Here, we are dealing with the so-called Kosterlitz-Thouless phase transition to the superfluid state of two-dimensional Bose systems in which the true Bose-Einstein condensation (in the absence of confinement of gas particles in a trap) is impossible [11].k k k k b b a a ++ (3)is fulfilled. In this approximation, Hamiltonian (2) can be diagonalised by using the unitary transformation1,,k k k k k k Фf a b µν+=− 2,,k k k k k k Фf a b νµ+=+ (4)where the introduced annihilation operators ,j k Ф (j=1,2) characterise quasi-particles (polaritons) in the atomicmedium, corresponding to two types of elementary perturbations, which in approximation (3) satisfy the boson commutation relations,,;i k j k ij ФФδ+⎡⎤=⎣⎦, ,1,2i j =. (5)The transformation parameters k µand k v in expression (4) are real Hopfield coefficients satisfying the condition 221k k v µ+=, which determine the contributions of the photon and atomic (excited) components to a polariton, respectively:22221/2221/242(4)(4)k k k k g g g µδδδ=⎡⎤+++⎣⎦, 221/22221/2(4)2(4)k k k k g g δδνδ++=+ , (6a,b)where ()k at ph E E k δ=−is the phase mismatch determining the contributions of the photon and atomic components to expression (4) for polaritons. In particular, in the limiting case, when 2k g δ− , we have 21k µ→ (20k v →), which corresponds to the negligible contribution of the photon part to the polariton 2,k Ф. Inopposite limit, when 2k g δ , we have 20k µ→ (21kv →), which means that the photon contribution to the coherence of polaritons of this type increases. Expression (6) shows that the polariton is a half-matter and half-photon (221/2µν==) quasi-particle under the resonance condition 0k δ=.Taking expressions (4) and (6) into account, Hamiltonian (2) takes the form11,1,22,2,()()()k k k k k kH k E k ФФE k ФФ++=+∑∑, (7)where 1,2()E k determine the dispersion dependence of polaritons: {}1/2221,21()()()42at ph at ph E k E E k E E k g ⎧⎫⎡⎤=+±−+⎨⎬⎣⎦⎩⎭. (8) Figure la presents dispersion dependences 1,2()E k (8) of polaritons for the interaction of atoms with the quantum field in free space. One can see that the two allowed energy states, polaritons of the upper 1[()]E k and lower 2[()]E k branches, correspond to each value of the wave vector k .When the medium is placed into the resonator, the wave-vector component k ⊥ orthogonal to the mirrorsurface is quantised. At the same time, a continuum of modes exists in the direction parallel to the mirror surface due to the absence of boundary conditions. This means that in the single-mode (single-frequency for each value of k ⊥) regime, the dispersion of polaritons is determined only by the wave-vector component k parallel to the mirror surface. Then, under the condition k k ⊥ which corresponds physically to the paraxialapproximation in optics (see, for example, [17]), the dispersion relation for photons in the resonator has the form23221/22()()2ph k k E k c k k c k O k k ⊥⊥⊥⊥⎡⎤⎛⎞=+=++⎢⎥⎜⎟⎜⎟⎢⎥⎝⎠⎣⎦. (9)a bFigure 1. Dispersion dependences 1()E k (upper branch) and 2()E k (lower branch) of polaritons on the wave vector k in free space (a) and resonator (b). The wave vector is plotted on the abscissa in the units of the resonance wave vector k ⊥ on the ordinate the energy is plotted inthe units of the coupling coefficient g .Here /cav k m L π⊥=is the quantised component of the wave vector parallel to the resonator axis, which corre-sponds to the periodic boundary conditions in the standard field quantisation procedure; L cav is the effective resonator length; and the number m corresponds to the selected mode (frequency). In the case of strong coupling(1), the dispersion curves of a polariton are pushed apart, resulting in the appearance of the upper and lower polariton branches in the resonator (Fig. lb). The principal feature of these curves is the presence of the 'potential' well (for 0k =). The width of the lower polariton well can be found from the condition 2220E k ∂∂= . This condition determines the angular parameters of a polariton beam in the resonator. It is important to note that these effects, which are related to the transverse component of the wave vector of a polariton (k in our case), will not be suppressed due to light diffraction if the angular dimensions of the polariton beam exceed the diffraction-limited divergence ϕ of the light beam, which can be estimated from the expression cav d L ϕ≈ [d and cav L are the beam diameter and resonator (or absorbing cell) length, respectively].3. Spectral 'condensation' and condensation of polaritonsWithin the framework of our approach, the narrowing ('condensation') of the polariton spectrum, which was observed in experiments [14, 15], can be simply explained by BEC. In this connection, taking into account paraxial approximation (9), we represent Hamiltonian (7) in the formlong tr H H H =+, (10a)where''11,1,22,2,()()long k k k k k k H E k ФФE k ФФ⊥⊥⊥⊥⊥⊥++⊥⊥=+∑∑, (10b) ''1,1,1,2,2,2,()()tr tr k k tr k k k k H E k ФФE k ФФ++=+∑∑ (10c)The expression for long H describes polaritons formed along the resonator axis, ''1,21,20()()k E k E k ⊥=≡ determines their dispersion dependence [see (8)] for 0k = . The expression for tr H characterises polaritons produced in the two-dimensional plane perpendicular to the resonator axis. The dispersion of these polaritons is described by the expression '22(1,2)11,2,2/tr pol E k m = . Here,()(1,2)1/222214phpol m m g =∆∆+∓(11)is the mass of polaritons of the upper and lower branches; 20ph m k c E c ⊥=≈ is the effective photon mass inthe medium and 0E ck ⊥∆=− is the detuning of the resonator mode (frequency) from the atomic transitionfrequency.Thus, the BEC of polaritons in the resonator is related to the second term in the expression for the Hamiltonian H in (10a). This term leads in fact to the renormalisation of the photon mass in the medium [see(11)]. Quasi-particles (polaritons) appearing in this case can be treated as an ideal two-dimensional gas [see also (10b)]. Indeed, the possibility of BEC assumes the presence of a stable state with the minimal energy - a 'potential' well (at the point 0k = ), which, as shown in section 2, takes place for polaritons in the resonator (the2well expressed in energy units is of the order of the coupling coefficient222effk g m ∆≈ . (12)In this case, it is possible to introduce formally the effective temperature eff T of the two-dimensional Bose gas of polaritons, which is also of the order of the coupling coefficient within the polariton well [13], i.e., B eff K T g ≈, where B K is the Boltzmann constant.The approach discussed above determines the condensation (more exactly, quasi-condensation) of the two-dimensional gas by assuming that polaritons with large k efficiently relax to the bottom of the dispersion-curve well. In our case, unlike the case of semiconductor microcavities considered in [8, 9], the two-dimensional property of the polariton gas can be provided by the fact that an optically dense medium is excited, as a rule, by the wave packet of synchronised electromagnetic modes, which corresponds to the quasi-monochromatic interaction of the field with medium.The efficient relaxation of polaritons to the bottom of the 'dispersion' well can be related to the intense polariton-polariton interaction discussed in a number of papers (mainly concerning the problems with semiconductor micro-cavities [10, 18]).Consider now in more detail the quasi-condensation of a two-dimensional Bose gas of polaritons described by the last term in (10c). The chemical potential of such a gas is described by the expression [19]22ln 1exp()ln 1exp B T B d T K T n K T T µλ⎡⎤⎛⎞⎡⎤=−−≡−−⎢⎥⎜⎟⎣⎦⎢⎥⎝⎠⎣⎦, (13)where 222/()d eff B T n m K π= is the gas degeneracy temperature; 2n is the two-dimensional density of polaritonsin the plane perpendicular to the resonator axis; and 1/2/(2)T eff B m K T λ= is the thermal wavelength (de Broglie wavelength). The temperature T d in (13) is determined by the condition when the thermal wavelength T λ is of the order of the average distance 1/3V between particles (V is the system volume). Due to the interaction between polaritons (nonideal gas), the additional parameter a scat appears, which is the scattering length depending on the interaction potential. This parameter affects the energy spectrum, which becomes a phonon spectrum [9].It follows from (13) that, strictly speaking, the condensation of the two-dimensional polariton gas (0µ=) occurs at 0T →. At the same time, it is known (see, for example, [9]) that already at the temperature224sd KT eff B n T T m K π= (14)the Kosterlitz-Thouless phase transition to the superfluid state occurs in the two-dimensional weakly interacting Bose gas, when isolated condensate droplets with uncorrelated phases are formed on the two-dimensional surface [n s in (14) is the superfluid liquid density on the two-dimensional surface].For polaritons with the effective mass 33510eff m g −=×and density in a three-dimensional resonator 1133 3.510n cm −=×, the gas degeneracy can appear already at room temperature (T d = 300 K). Indeed, in this case the minimal two-dimensional density of the polariton gas estimated from (13) for d T T = gives the value 82230.310T n n cm λ−≈× [19] for the de Broglie wavelength 41.8410T cm λ−≈×. It is for this value of the atomic concentration 3n that spectral condensation was observed near the yellow doublet of sodium in experiments [14] (Fig. 2). For the upper spectrum (Fig. 2a), 103310n cm −<, and for the lower spectrum (Fig. 2c), 1133 3.510n cm −=×. The similar results obtained in [14] for the neon spectrum also demonstrated the spectralcondensation for polaritons.a bcFigure 2. Spectral condensation near the yellow doublet of sodium (see text) at the atomic concentration 10310n ≤(a), 1110 (b), and 1133.510cm −× (c).Let us find now the conditions under which the true (in thermodynamic sense) condensation of polaritons excited in the resonator can be obtained. It is known [19, 20] that, to obtain such condensation in a two-dimensional weakly interacting (ideal) gas, gas particles should be confined in a trap. For example, for a trap with the trapping potential described by the expression (harmonic potential)222020()2eff eff m r U r U r r Ω== (15)(eff Ω is the trapping (oscillation) frequency of particles, 0r is the transverse size of the trapping region, and r isthe transverse coordinate), the critical BEC temperature for a two-dimensional gas is (cf. [20])2221.645c eff Bn T m K π== , (16)where N is the total number of particles. We also took into account in the right-hand side of (16) that the number N 2 of particles trapped by the potential U(r) on the surface is described by the expression [20]2222B eff eff N n K T m π=Ω. In the absence of a trap (0eff Ω=), as should be, BEC does not occur: 0c T = in (16).(measurements of the first- and second-order coherence degree) is one of the main tools for diagnostics of polariton condensation (see [8, 9]). Because a polariton is a linear superposition of a photon and atomic excitation [see (4)], its coherent properties are caused by the coherence of the light field itself and of an ensemble of atoms with which the field interacts, as well as by their possible quantum interference caused by the condensation process. Within the framework of these experiments, when the condition of the exact resonance0∆= is fulfilled, we have 0k δ≈ and obtain 221/2k k v µ== from expressions (6) and (11), which means thatoptical and atomic parts make identical contributions to a polariton. In this case, the coherent properties of the polariton state can be simply caused by a high coherence of the optical field at the input to the atomic medium irrespective of BEC. However, the problem of measuring the coherence of atomic exitations caused by the interaction and of the intrinsic coherence of the polariton condensate (if it is produced in the system) remains open. In our opinion, this problem can be solved, in particular, by producing polariton BEC based on three-level atoms under conditions of electromagnetic induced transparency (EIT) (see below).Here we consider another possibility based on a small variation of the detuning ∆ [and, therefore, k δ, see (6)] in experiments as the parameter governing the contributions of photon and atomic parts to the resulting coherence of resonator polaritons. In this case, the effective mass of polaritons [see (11)] and, hence, the critical temperatures of degeneracy, condensation, and quasi-condensation in (13), (14), and (16) change. This specific property of a polariton gas means in fact that the formation of a Bose-Einstein condensate can be controlled in experiments.Note, however, that we do not consider in this paper the questions concerning the BEC of a photonic gas in the resonator or, more exactly, the condensation of polaritons of the upper branch of the dispersion curve (see Fig. lb) characterised by the first term in expression (10c). This problem is undoubtedly very important for the scope of questions considered in our paper although it was discussed only in connection with the quantum properties of light in media with cubic nonlinearity (see [3]).In addition, the formation of a photon condensate (or a condensate of polaritons of the upper branch), which is directly connected with lasing in the resonator (cf. [2]), upon varying the parameter ∆ also has an interesting feature. Indeed, for 0∆=, it follows from (11) that polaritons of both dispersion branches have equalmasses, i.e., (1)(2)pol pol eff m m m ==, which corresponds to the equal temperatures of their quasi-condensation [see(14)1. However, in the case of 0∆≠, we have from (11) that (1)(2)pol pol m m ≠ which means physically that thephase-transition temperatures (14) for polaritons of the upper (1)()KT T and lower (2)()KT T dispersion branches aredifferent. Thus, by introducing asymmetry with the help of a small change in the detuning ∆, it is possible to produce a very narrow temperature (energy) gap within which the coherent properties of polaritons of both branches should substantially change. These properties can be observed, for example, by measuring the function of their cross correlation or by using probe radiation under resonance conditions.Therefore, the study of this effect will give the answer to the principal question about the properties ofthe coherence of light, atomic system, and polaritons themselves in the case of BEC.4. ConclusionsWe have developed in the paper the quantum approach for solving the problems of formation of quasi-condensation and realisation of the true (in the thermodynamic sense) Bose - Einstein condensation of a two-dimensional gas of polaritons at room temperature. This approach has allowed us to explain some features of spectral condensation of broadband lasing near strong absorption lines in the laser resonator, which were observed in experiments (in particular, the so-called spectral condensation upon non-resonance pumping). In this aspect, BEC reduces the threshold pump power of parametric excitation of cooperative effects. Consider briefly some phenomena that are directly related to the problem studied in the paper.First, this is the condensation of polaritons, which is of interest in the presence of the EIT effect when alight pulse propagates in a resonance atomic medium without changing its shape in the absence of absorption (see, for example, [5, 21, 22]. A remarkable feature of this effect is the appearance of atomic coherence both for hot [21] and ultracold atoms [5, 22]. The EIT effect can be also explained in terms of bright and dark polaritons, which in the adiabatic approximation corresponding to condition (3) in our case, represent the coherent superposition of atoms in the two states of the hyperfine Zeeman structure and the external probe field maintained with the help of the external probe field at the optical frequency through the third (auxiliary) level (the so-called Λ- scheme [5, 6, 21]).Therefore, upon placing an atomic medium into the resonator to produce the BEC of polaritons, the EITeffect would become a tool for obtaining such a quantum state. In this case, the condensation process could be controlled more precisely by coupling directly two atomic levels with an external weak field, which would provide the ejection of 'hot' polaritons from a trap, as, for example, occurs for condensation of alkali atoms in a magneto-optical trap [1]. On the other hand, upon spectral condensation in the case of BEC, a 'bleaching' of the atomic medium in the resonator caused by a change in its refractive properties can be expected. In this case, the group velocity of a light pulse directed into an atomic medium after switching on probe radiation with the delay time del coh ττ< can decrease, in particular, due to polariton condensation. Indeed, it follows from expressions (10c) and (11) that the group velocities of such quasi-particles in the plane perpendicular to the resonator axis are determined by the expression1,2(1,2)(1,2)()tr gr polE k k m υ∂==∂ In the case of the exact atomic optical resonance (for 0∆=), we have from this that(1)(2)/2gr gr k c k υυ⊥= Therefore, in the paraxial approximation, when k k ⊥ the group velocity of condensedpolaritons is estimated as (1,2)gr c υ , which means in fact that the 'slow' light regime is observed for polaritons inthe resonator.Second, the high-temperature BEC of polaritons is of interest for quantum information, for example, forthe development of new physical principles of quantum memory and data storage. Indeed, as we have shown in[6], such macroscopic polariton states can be used in problems of cloning and quantum information storage.Acknowledgements. This work was partially supported by the Russian Foundation for Basic Research (Grants Nos 04-02-17359 and 05-02-16576) and the Ministry of Education and Science of the Russian Federation. A.P. Alodjants thanks the non-profit Dynasty Foundation for support.AppendixLet us discuss the problem of confinement of the BEC of intracavity polaritons in a trap. Consider a special atomic optical trap whose operation is based on the fact that polaritons represent a coherent superposition of a photon and atomic perturbation. Photons can be confined in the region of atomic-optical interaction in such a trap, where polaritons are produced, by focusing a light beam with a special gradient (cylindrical) lens (or inhomogeneous waveguide) with the refractive index varying along the transverse coordinate as2220()(1)n r n n r ′=−, (A.1)where 'n is the required gradient addition to the refractive index of the lens. The potential for trapping (focusing) photons of the light beam produced by such an optical system can be written in the form [17]222020()'()22opt n r n n r U r n −==,which exactly corresponds to the harmonic-trap potential (15) with the inhomogeneity parameter 2'eff eff n m =Ω.In addition, to trap atoms in the plane perpendicular to the resonator axis, we can use a two-dimensional magnetic trap with the oscillation frequency at Ω, which is widely applied in experiments with 'usual' atomic condensates [1].Thus, to confine polaritons in a trap, it is necessary to confine atoms by a standard method and focus simulta-neously the light beam into the region of atomic-optical interaction by selecting the appropriate parameters at Ωand 'n . This determines the value of eff Ω required in the experiment.References1. Ketterle V. Usp. Fiz. Nauk, 173, 1339 (2003).2. Oraevsky A.N. Kvantovaya Elektron., 24, 1127 (1997) [Quantum Electron., 27, 1094 (1997)].3. Chiao R., Boyce J. Phys. Rev. A, 60, 4114 (1999).4. Imamoglu A., Ram R.J., Pau S., Yamamoto Y. Phys. Rev. A, 53, 4250 (1996).5. Liu C, Dutton Z., Behroozi C.H., Hau L.N. Nature, 409, 490 (2001).6. Alodjants A.P, Arakelian S.M. Int. J. Mod. Phys. B, 20, 1593 (2006).7. Averchenko V.A., Bagayev S.N., et al. Abstract in Technical Digest o/ICONO'05 Conf. (Sankt-Petersburg, Russia, 2005).8. Deng H., Weihs G., Santori C, Bloch J., Yamamoto Y. Science, 298, 199 (2002).9. Kavokin A., Malpuech G., Laussy F.P. Phys. Lett. A, 306, 187 (2003); Richard M., Kasprzak J., Andre R., et al. Phys. Rev. B, 72, 201301(R) (2005).10. Gippius N.A., Tikhodeev S.G., Keldysh L.V., Kulakovskii V.D., Usp. Fiz. Nauk, 175, 327 (2005); Kulakovskii V.D., KrzhizhanovskiiD.N., et al. Usp. Fiz. Nauk, 175, 334 (2005).11. Kosterlitz J.M., Thouless D.J. J. Phys. B: Sol. State Phys., 6, 1181 (1973).12. Dicke R.H. Phys.Rev., 93, 99 (1954).13. Eastham P.R., Littlewood P.B. Phys. Rev. B, 64, 235101 (2001).14. Vasil'ev V.V., Egorov V.S., Fedorov A.N., Chekhonon LA. Opt. Spektr., 76, 146 (1994).15. Bagayev S.N., Egorov V.S., Moroshkin P.V., Fedorov A.N., Chekhonon LA. Opt. Spektr., 86, 912 (1999).16. Kocharovskii V.V., Kocharovskii Vl.V. Kvantovaya Elektron., 14, 2246 (1987) [Sov. J. Quantum Electron., 17, 1430 (1987)].17. Marte M.A., Stenholm S. Phys. Rev. A, 56, 2940 (1997).18. Savvidis P.G., Baumberg J.J., Stevenson P.M., et al. Phys. Rev. Lett., 84, 1547 (2000).19. Petrov D.S., Gangardt G.M., Shlyapnikov G.V. J. Phys. IV France, 116, 3 (2004).20. Bagnato V., Kleppner D.K. Phys. Rev. A, 44, 7439 (1991).21. Lukin M.D. Rev. Mod. Phys., 75, 457 (2003).22. Prokhorov A.V., Alodjants A.P., Arakelyan S.M. Pis'ma Zh. Eksp. Tear. Fiz., 80, 870 (2004).。

bdg哈密顿量的正则变换

bdg哈密顿量的正则变换摘要：1.哈密顿量的定义与性质2.正则变换的概念与作用3.bdg 哈密顿量的正则变换过程4.bdg 哈密顿量正则变换的意义正文：一、哈密顿量的定义与性质哈密顿量（Hamiltonian）是量子力学中描述系统能量的一个算符，通常由系统哈密顿方程（Schrdinger equation）演化而来。

哈密顿量具有以下性质：首先，它是一个厄米算符（Hermitian operator），即满足H = H（H 的复共轭）；其次，哈密顿量在时间演化过程中保持不变，即满足时间演化规律。

二、正则变换的概念与作用正则变换（Canonical transformation）是一种在量子力学中广泛应用的变换方法，其目的是将一个难以处理的哈密顿量变换为另一个易于处理的哈密顿量，从而简化问题的求解。

正则变换的过程需要引入一个待定参数，通过变换后的哈密顿量可以求解原系统的物理性质。

正则变换在量子力学中具有重要的意义，它可以提高问题求解的效率，使得原本复杂的问题变得简单。

三、bdg 哈密顿量的正则变换过程bdg 哈密顿量（Bose-Einstein condensate Hamiltonian）是描述玻色- 爱因斯坦凝聚（Bose-Einstein condensate, BEC）系统的一个有效哈密顿量。

bdg 哈密顿量的正则变换过程主要包括以下几个步骤：1.引入一个待定参数，如变换矩阵U。

2.对原哈密顿量进行正则变换，得到变换后的哈密顿量H"。

3.通过变换后的哈密顿量H"，求解原系统的物理性质。

四、bdg 哈密顿量正则变换的意义对bdg 哈密顿量进行正则变换具有重要的意义，它可以使得原本复杂的哈密顿量变得简单，从而降低问题求解的难度。

此外，正则变换还可以帮助我们更好地理解玻色- 爱因斯坦凝聚现象，为实验研究提供理论支持。

总之，哈密顿量的正则变换是一种有效的量子力学方法，可以简化问题的求解过程。

UFO最高机密

UFO最高机密:Bob Lazar访谈2015-10-0201:02要常来整理Bob Lazar1959年1月26日出生于弗罗里达州科的柯若盖布尔斯（Coral Gables）。

其声称1988年-1989年间他在S-4区域（第四区）曾以物理专家的身份工作过。

具体地点位于内华达州的格入母湖畔（Groom Lake），毗邻51区。

Bob Lazar根据勒萨（Lazar）的证词，S-4区域是一个秘密的军事基地，目的是为了研究复制外星飞碟的可能性。

勒萨说他看到一共9架飞碟，并且提供了详细的飞碟动力模型。

然而在外界发现无法找到他自称的入学历史，和他自称参加过的科学团体均没有记录时，他的可信度遭到普遍质疑。

Lazar对此回应是此乃政府机构所为，政府无需杀他灭口，将他变成一个骗子要比杀他容易的多。

Bob Lazar提供的飞碟动力模型1989年11月，在拉斯维加斯KLAS电台Lazar和记者乔治·耐普（George Knapp）进行了一次电视采访。

Lazar声称一开始他以为那些飞碟是美国造的秘密武器，那些飞碟目击报告都是军方试飞时被看到了。

然而，当他读完一些简短报告并对飞碟进行一番查看后，他才明白这些飞碟完全来自外星科技。

在这个电视访谈中，Lazar清晰的描述了，进入飞碟后他是何等的震惊。

记者乔治·耐普（George Knapp）对于飞船推进动力来源，Lazar称关键的燃料是元素115，一种重金属。

元素115又叫Ununpentium（UUP），在质子轰击下产生反物质，进而提供能量和创造出反重力效果。

由于115元素的核子力场被放大，扭曲周围的重力场，这样飞船就可以缩短空间的距离。

Lazar同时也声称给他看的简短报告中，有关于外星人呆在地球已超过10000年方面的信息。

报告称这些外星人早年来自Zeta Reticulia1和2星系，通常被称为Zeta 人，也被称为小灰人（Greys）。

51区工作人员与外星人BOB LAZAR的重力产生机Bob Lazar上世纪80年代在51区参加过一个开始于1979年的“复制项目”。

冷知识(很少有人知道的知识)

1. 大家都知道国外认为星期五不吉利，早在18世纪，英国海军为了打破星期五不吉利的传言，特地命名了一艘船叫“H.M.S.星期五”号，在星期五挑选了船员，船长名为James Friday，于一个星期五起航，然后。

这艘船，它神秘的失踪了= =于是本为了打破谣言而生的船，反而成为壮大谣言的一部分~~~其实我爱星期五啊！！管它吉利不吉利！2.皱纹：每皱眉20W次，额头就会出现一条皱纹——大家谨慎皱眉吖~这东西是有配额的3. 靠山：“靠山”的意思神马的不用说了，需要注意的是，这典故出自于唐代的官员张洎拒绝依附安禄山，从而流传下来。

所以那些拍汉代宫廷的古装片导演麻烦你们不要找靠山了好不好！你们到底是有多期待那个一千年后才出生的安禄山啊！4.“如来”和“佛”是一样东西，所以其实没有“如来佛”的称呼。

就像可以叫“爸爸”也可以“爹”，可你见过有人管自己父亲叫“爸爸爹”的吗？如来还有十个名号，分别是：应供、正遍知、明行足、善逝、世间解、无上士、调御丈夫、天人师、佛、世尊。

顺便再说一下，“南无阿弥陀佛”的意思就是：皈依一位名字叫做阿弥陀的如来。

——没错，阿弥陀佛是一尊佛的名字。

是很基础很基础的知识，可是仍然有很多人不知道5.介个是专门扫盲来的~“床笫之欢”不是“床第之欢”。

“第”是大宅子和大门的意思，如“府第”“书香门第”等等。

如果是“床第之欢”的话，意思就是把床搬到大门口去OOXX....以后不要写错哦6.其实唐朝真的没有大家以为的那么好。

唐朝最强盛的“开元之治”时期，全国有户820万，是唐朝的最高值；而隋朝“开皇之治”时期，全国就有户890万。

终唐一世，各项经济指标都没有恢复到隋朝水平。

“主流”专家们诟病为“军事软弱”的宋朝，对外战争（交战规模万人以上，不包括统一战争和国内战争）的胜率超过了70%；而被认为是军事最强盛的唐朝却在对外战争中胜少负多~7.我希望来世可以做一只狮子~不仅仅是因为狮王雄风更重要的是狮子每天平均睡——22个小时= = 并且可以采取各种姿势在各种地点入睡，趴着睡躺着睡，还可以抱着树睡~~~~~~~~~~~~~~··8.古人云~精分神马的最讨厌了~（双子座的我果然又分裂了=。

Bose-Einstein condensates in the presence of a magnetic trap and optical lattice two-mode a

University of New Mexico Albuquerque, NM 87131
P. G. Kevrekidis † Department of Mathematics and Statistics
University of Massachusetts Amherst, MA 01003-4515
4. Numerical results
11
4.1. Veriﬁcation of the two-mode analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.2. Connection to the large nonlinearity solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Bose-Einstein condensates in the presence of a magnetic trap and optical lattice: two-mode approximation
Todd Kapitula ∗ Department of Mathematics and Statistics
form
iqt
+
1 2 qxx
+
ωq
+
δ|q|2q
=
( 1 Ω2x2 2
+
p(x + ξ))q
(1.1)
[18]. Here δ ∈ {−1, +1}, ξ ∈ R, and p : R → R is L-periodic. The term Ω2x2/2 represents the eﬀect of the magnetic trap, with the parameter Ω being the reduced frequency of the trap [18], while the term p(x + ξ) represents the eﬀect of the optical lattice. The parameter ξ corresponds to a shift of the optical lattice relative to the magnetic trap. As it will be seen in the following sections, this shift plays an important role in the stability of the steady-states.

Comparison of mean-field theories for vortices in trapped Bose-Einstein condensates

a r X i v :c o n d -m a t /0108462v 1 [c o n d -m a t .s o f t ] 28 A u g 2001Comparison of mean-ﬁeld theories for vortices in trapped Bose-Einstein condensatesS M M Virtanen,T P Simula and M M SalomaaMaterials Physics Laboratory,Helsinki University of Technology P.O.Box 2200(Technical Physics),FIN-02015HUT,Finland(February 1,2008)We compute structures of vortex conﬁgurations in a har-monically trapped Bose-Einstein condensed atom gas within three diﬀerent gapless self-consistent mean-ﬁeld theories.Outside the vortex core region,the density proﬁles for the condensate and the thermal gas are found to diﬀer only by a few percent between the Hartree-Fock-Bogoliubov-Popov the-ory and two of its recently proposed gapless extensions.In the core region,however,the diﬀerences in the density pro-ﬁles are substantial.The structural diﬀerences are reﬂected in the energies of the quasiparticle states localized near the vortex core.Especially,the predictions for the energy of the lowest quasiparticle excitation diﬀer considerably between the theoretical models investigated.PACS number(s):03.75.Fi,05.30.Jp,67.40.DbThe landmark experiments to realize Bose-Einstein condensation in dilute atomic gases [1]have sparked vig-orous investigation on the physical properties of these novel quantum ﬂuids.Due to the weak interactions,such systems are rare examples of interacting quantum ﬂu-ids amenable to quantitative microscopic analysis,and thus provide unique possibilities to test the fundamental principles and theories of many-body quantum physics.Theoretical approaches yield several quantities,such as density proﬁles for the condensate and the thermal gas component,stability estimates,speciﬁc heats,and prop-erties of various propagating sound modes,to be com-pared with experiments.Experiments also yield detailed information on the energies of the individual excitation modes of these systems [2].Such information provides the most direct and stringent tests for the accuracy of dif-ferent theoretical approaches,as compared to the above-mentioned “collective”quantities which depend on the excitation spectrum as a whole.The Bogoliubov equations [3]are a widely used start-ing point to compute the excitation spectra for dilute Bose-Einstein condensates (BECs).They can be seen as eigenmode equations for the condensate described by the Gross-Pitaevskii equation [4],neglecting eﬀects of the thermal,noncondensed gas component in the sys-tem.The Hartree-Fock-Bogoliubov (HFB)theory [5]takes self-consistently into account the condensate and the thermal gas densities,as well as the lowest order anomalous average of the boson ﬁeld.However,it is plagued by an unphysical gap in the excitation spec-trum,which violates Goldstone’s theorem and invalidatesits value in predicting the lowest collective mode excita-tion frequencies.Goldstone’s theorem can be restored by neglecting the anomalous average mean ﬁeld in the HFB formalism.This yields the gapless Popov version of the HFB theory [6].At low temperatures,predic-tions of the Popov approximation (PA)for the lowest excitation frequencies of irrotational condensates are in good agreement with experimental results,but at tem-peratures T >∼T bec /2(T bec denotes the critical temper-ature of condensation)the deviations become apparent [7].The main inadequacies of the PA are that it neglects the eﬀects of the background gas on atomic collisions and the dynamics of the thermal gas component.As an improvement to overcome the ﬁrst limitation within a computationally manageable formalism,the so-called G1and G2approximations have been suggested [8,9].They are gapless mean-ﬁeld theories which take into account eﬀects of the medium on atomic collisions by allowing the interaction couplings to depend on the correlation mean ﬁelds in a self-consistent manner.The two versions are based on diﬀerent approximations for the momentum de-pendence of the full many-body T -matrix in the homo-geneous gas limit,and their precision for inhomogeneous systems remains to be investigated.To assess the accuracy of the above-mentioned gapless HFB-type approximations,their predictions for the exci-tation frequencies of irrotational,harmonically trapped atomic BECs have been computed and compared with experiments [7,8,10–12].For temperatures T <∼T bec /2,the predictions of the PA,G1and G2for the lowest ex-citation frequencies diﬀer only a few percent [8,9].For higher temperatures,the diﬀerences are larger and ex-ceed the experimental uncertainty estimates for measure-ments,but none of the theories agrees satisfyingly with experiments [8].However,in this temperature range the dynamics of the thermal gas component,which these ap-proximations do not take into account,is expected to have an increasingly important inﬂuence on the excita-tion eigenmodes.Consequently,results for irrotational condensates remain somewhat inconclusive in determin-ing the validity of these theories.Recently,vortex states in dilute atomic BECs have been experimentally realized [13].Furthermore,by ob-serving the precession of vortices,the energy of the low-est excitation,the so-called lowest core localized state (LCLS),has been measured [14].Interestingly enough,the experimental results for this energy agree well with the Bogoliubov approximation [15],while they deﬁnitely1disagree with the picture given by the self-consistent mean-ﬁeld theories:the latter predict the energy of the precession mode to be positive w.r.t.the condensate state [16,17],but experiments imply negative energies.We suggest that this puzzling fact could be due to incomplete thermalization of the(moving)vortex and/or the limita-tions of the quasi-stationary,i.e.,adiabatic HFB formal-ism in describing time-dependent phenomena.The adi-abatic approximation essentially fails if the kinetic rates of the system exceed frequency separations of the tely,we have shown that the requirement of adiabaticity leads to a criterion for the velocity of the moving vortex,which is violated in the precession obser-vations so far[18].However,if the precession radii—and thus the velocities—of the vortices could be reduced,or the physical parameter values appropriately adjusted in order for the system to better fulﬁll the criteria for adia-baticity and thermalization,one should be able to mean-ingfully compare experimental data with the predictions of the self-consistent equilibrium theories for the vortex states.In this paper,we present results of computations for the structures and excitation frequencies of vortex states within the G1and the G2approximations,and compare them with the previously computed predictions of the Popov approximation[16,17].Outside the vortex core region,the density proﬁles for the condensate and the thermal gas component are found to diﬀer by only a few percent between the PA,G1and G2.However,in the core region the diﬀerences are considerably larger.This is reﬂected in substantial diﬀerences in the energy of the LCLS,which is localized in the core region.The gapless HFB-type theories considered in this paper can be expressed in the form of the generalized Gross-Pitaevskii(GP)equation[5,9][H0(r)+U c(r)|φ(r)|2+2U e(r)ρ(r)]φ(r)=µφ(r)(1)for the condensate wavefunctionφ(r),and the eigenvalue equationsL(r)u q(r)+U c(r)φ2(r)v q(r)=E q u q(r),(2a) L(r)v q(r)+U c(r)φ∗2(r)u q(r)=−E q v q(r)(2b)for the quasiparticle amplitudes u q(r),v q(r),and eigenenergies E q.Above,H0(r)=−¯h2∇2/2m+V trap(r) is the bare single-particle Hamiltonian for atoms of mass m conﬁned by a harmonic trapping potential V trap(r)= 1in reference[11].This method allows one to use rather small values for E c with excellent accuracy,thus essen-tially improving the computational eﬃciency.In order to stabilize the iteration,we use underrelaxation in updat-ing the mean-ﬁeld potentials.To facilitate comparison with previously presented re-sults for the Popov approximation,the physical parame-ter values for the gas and the trap were chosen to be the same as those in reference[16].We modelled a sodium gas with the atomic mass m=3.81×10−26kg and the scattering length a=2.75nm in a trap with the radial frequencyνr=ωr/2π=200Hz.The density of the gas was determined by treating N=2×105atoms per length L=10µm in the axial direction.Altogether,these val-ues yield the condensation temperature T bec≈0.8µK. Figures1and2present results of our computations for axisymmetric single-quantum vortex states.The density proﬁles for the condensate,the noncondensate and the anomalous average are displayed inﬁgure1at temper-atures T=50nK and400nK.Outside the vortex core region,the diﬀerences in the density proﬁles between the PA,G1and G2are at most a few percent at temperatures T<∼T bec/2.In the core region,however,the diﬀerences are considerably larger.The total density of the gas is approximately20%larger on the vortex axis in the G2 than within the PA.This squeezing behavior is associated to the“softening”of the repulsive eﬀective interaction in the core region due to many-body eﬀects[9].The diﬀerences in the core densities between the var-ious approximations also suggest diﬀerences in the ener-gies of the quasiparticle excitations localized in the core region.Figure2displays the energies of three such states, the lowest excitations with angular momentum quantum numbers qθ=−1,0,and1,as functions of tempera-ture.For the G1and the G2,the increased core densities are compensated by smaller eﬀective couplings,and the shifts in the excitation energies are generally only a few percent for temperatures T<∼T bec/2;at higher tempera-tures the softening eﬀect of the interaction becomes more pronounced[9,12],also increasing the shifts in the exci-tations.However,the lowest Kelvin mode(consisting of the lowest qθ=−1excitations)state,the LCLS,is espe-cially sensitive to the structure of the core region.The diﬀerences in the energies of the LCLS between the ap-proximations are25–40%even at temperatures for which the predictions of the PA for the excitation frequencies of irrotational condensates diﬀer by less than5%from the experimental data.In addition,the temperature de-pendence of the LCLS is found to be much stronger than for the other states.In fact,the lowest excitation en-ergy vanishes in the zero-temperature limit for all the approximations[17];the remainder of the spectrum is essentially temperature-independent,except in the vicin-ity of T bec.The state displayed inﬁgure2with qθ=1 is the Kohn mode,which should have the exact energy E=¯hωr according to Kohn’s theorem for parabolic con-ﬁnement[21].Kohn’s theorem is satisﬁed to an accuracy of1–6%for all the approximations,suggesting that dy-namical eﬀects of the thermal gas component are small in the temperature range studied.In conclusion,we argue that future measurements of the lowest excitation frequencies of the vortex states could provide stringent tests for the validity of the mean-ﬁeld theories considered.Especially,they could be used to estimate the degree to which the approximations for the many-body T-matrices based on the homogeneous limit remain valid for highly inhomogeneous systems. We thank the Center for Scientiﬁc Computing for com-puter resources,and the Academy of Finland and the Graduate School in Technical Physics for support.[14]Anderson B P,Haljan P C,Wieman C E and Cornell EA2000Phys.Rev.Lett.852857[15]Svidzinsky A A and Fetter A L2000Phys.Rev.Lett.845919[16]Isoshima T and Machida K1999Phys.Rev.A592203[17]Virtanen S M M,Simula T P and Salomaa M M2001Phys.Rev.Lett.862704[18]Virtanen S M M,Simula T P and Salomaa M M2001Preprint cond-mat/0105398[19]Morgan S1999PhD Thesis(Oxford University)[20]Svidzinsky A A and Fetter A L2000Phys.Rev.A62063617Feder D L,Svidzinsky A A,Fetter A L and Clark C W2001Phys.Rev.Lett.86564Fetter A L and SvidzinskyA A2001J.Phys.:Condens.Matter13R135[21]Dobson J1994Phys.Rev.Lett.732244FIG.1.Density proﬁles of the vortex state for the conden-sate(|φ|2),thermal gas component(ρ)and anomalous average (|∆|)in the PA(solid),G1(dashed)and G2(dashed-dotted)at temperatures(a)T=400nK and(b)T=50nK.Axes for the values ofρand|∆|are on the left-hand sides,and for|φ|2on the right-hand sides.Figures(c)and(d)display the vortex core region,where the diﬀerences in the density proﬁles between the approximations are substantially larger than farther from the vortex axis.FIG.2.(a)Energies of the lowest excitation modes with angular momentum quantum numbers qθ=−1,0,and1in the PA(solid),G1(dashed)and G2(dashed-dotted)as func-tions of temperature.The qθ=−1state is the so-called lowest core localized state(LCLS),and the qθ=1excitationis the Kohn mode.(b)Temperature dependence of the energyof the LCLS within the PA,G1and G2.Note the substantial relative diﬀerences between the theories in this energy.4。

量子力学中的超导性与超流性

量子力学中的超导性与超流性量子力学中超导性与超流性量子力学是研究微观物质行为的理论框架，它在解释粒子行为和相互作用方面取得了巨大的成功。

在量子力学的领域中，超导性和超流性是两个重要的现象。

本文将就量子力学中的超导性与超流性进行探讨，以便更好地理解这两个现象的本质和研究意义。

1. 超导性超导性是指在超导体中电阻为零的现象。

在超导态下，电流可以在材料内部无阻力地流动，能够形成超导电流和超导态。

1957年，BCS 理论被提出，解释了超导现象的基本原理。

BCS理论认为，超导的本质是由于电子之间通过库仑相互作用产生电子和声子的相互吸引，从而形成了库仑配对。

这种配对使得电子可以以无阻力的方式在晶格中移动。

超导体的临界温度是指超导态能够在该温度以下形成的最高温度。

对于不同的超导体来说，其临界温度是不同的。

例如，铜氧化物和镁二硼化铜等高温超导体的临界温度远高于液氦的沸点温度。

目前，研究者们正在努力寻找更高临界温度的超导体，以便将超导技术应用于更广泛的领域。

2. 超流性超流性是指某些物质在低温下流体能够表现出无粘性和无黏滞阻力的现象。

特别的是，在低于临界温度之下，超流体可以在闭合容器内无限制地流动，即使出现了爬壁现象也不会有能量的损耗。

最早对超流性的研究是在液体氦上进行的。

超流性现象的解释由London方程和两流体模型提供。

London方程描述了超导态中超流体流动的宏观性质。

两流体模型则将超流体分为正常流体和超流体两部分，分别对应了非约束和无粘性的流动。

这种模型成功地解释了超流性的一些重要行为，例如超流态的纳秒响应时间和扭曲辐射。

超流性在量子力学中有着重要的应用。

例如，在超冷原子气体中，玻色—爱因斯坦凝聚（Bose-Einstein Condensate，BEC）是一种新的物质态，也被称为超流体。

BEC是由由低温、高密度和玻色子组成的物质形成的，在BEC中，原子将以波的形式存在，形成一个超流性态。

BEC的研究不仅帮助我们更好地理解了超流性的本质，也对相干和相干性的研究具有重要意义。

2023-2024学年黑龙江省实验中学高一下学期开学考试英语试题

2023-2024学年黑龙江省实验中学高一下学期开学考试英语试题FAMOUS CHINESE PAINTINGS, ARTISTIC TREASURESWith thousands of years of continuous history, China is one of the most culturally unique nations. Over many centuries, Chinese artists have created paintings that are now in the hearts of more than a billion people.Nymph of the Luo River—Gu KaizhiThe legend has it that Cao Zhi, a prince of the state of Cao Wei, fell in love with the governor’s daughter. However, she married his brother, Cao Pi, and the prince became upset. Later, he composed an emotional poem about the love between the goddess and common people. In the 4th century, Gu Kaizhi, a Chinese artist, was moved by the story and illustrated the poem.Court Ladies Adorning Their Hair with Flowers—Zhou FangDuring the Tang Dynasty, China had a prosperous economy and flourishing culture. In this period, the genre of “beautiful women painting” enjoyed popularity. Coming from a noble background, Zhou Fang, a Chinese artist, created artworks in this genre. In his painting, the ladies stand as though they are fashion models, but one of them is entertaining herself by teasing a cute dog.Along the River During the Qingming Festival—Zhang ZeduanZhang Zeduan depicted the landscape in his work Along the River During the Qingming Festival. However, instead of concentrating on the vastness of nature, he captured the daily life of the people of Bianjing, present-day Kaifeng. His work shows much about life in the Northern Song Dynasty.A Thousand Li of Rivers and Mountains—Wang XimengNot only did officials and scholars enjoy listening to music, but they also found pleasure in depicting nature. One such painter was Wang Ximeng. He was a prodigy. Wang Ximeng painted A Thousand Li of Rivers and Mountains when he was only seventeen years old. He died several years later, but he left one of the largest and most beautiful paintings in Chinese history.1. Where do you think this passage is taken from?A．A novel. B．A travel journal.C．A magazine. D．A book review.2. What can we learn from this passage?A．Wang Ximeng created his masterpiece in his teens.B．Nymph of the Luo River is a poem written by Cao Zhi.C．Zhou Fang painted fashion models in his painting.D．Zhang Zeduan’s painting describes daily life of officials.3. Which is your best choice if you want to enjoy a painting with the beauty of nature?A．Nymph of the Luo River—Gu Kaizhi.B．A Thousand Li of Rivers and Mountains—Wang Ximeng.C．Court Ladies Adorning Their Hair with Flowers—Zhou Fang.D．Along the River During the Qingming Festival—Zhang Zeduan.You run into the grocery store to quickly pick up your item. You grab what you need and head to the front of the store. After quickly sizing up the check-out lines, you choose the one that looks fastest. You chose wrong. People getting in other lines long after you have already checked out and headed to the parking lot. Why does this seem to always happen to you?Well, as it turns out, it's just math that is working against you. A grocery store tries to have enough employees at the checkout lines to get all their customers through with minimum delay. But sometimes, like on a Sunday afternoon, they get super busy. Because most grocery stores don't have the physical space to add more checkout lines, their system becomes overburdened. Some small interruption — a price check, a particularly talkative customer — will have downstream effects, holding up the entire line behind them.If there are three lines at the store, these delays will happen randomly at different registers (收银台). Think about the probability. The chances of your line being that fastest one are only one in three, which means you have a two-thirds chance of not being in the fastest line. So it's not just in your mind: Another line is probably moving faster than yours.Now, mathematicians have come up with a good solution, which they call queuing theory, to this problem: Just make all customers stand in one long snaking line, called a serpentine line, and serve each person at the front with the next available register. With three registers, this method is about three times faster on average than the more traditional approach. This is what they do at most banks, Trader Joe's, and some fast-food places. With a serpentine line, a long delay at one register won't unfairly punish the people who lined up behind it. Instead, it will slow everyone down a little bit. 4. What phenomenon is described in the first paragraph?A．Queuing in a line. B．A shopping experience.C．A rush in the morning. D．Cutting in a line.5. According to the article, what may cause delays in checking out?A．The lack of employees in the grocery store.B．Some unexpected delays of certain customers.C．The increasing items bought by customers.D．A worsening shopping system of the store.6. What is the solution given by mathematicians?A．Employing more workers for checking out.B．Limiting the number of queuing people.C．Making only one line available.D．Always standing in the same line.7. What's the principle behind the queuing theory?A．To pursue the maximum benefit.B．To leave success or failure to luck.C．To avoid the minimum loss.D．To spread the risk equally among everyone.Most glitter(小发光物品), which is made up of tiny pieces of plastic, is a huge danger to the environment. “Everyone talks about the mountain of plastic floating in the ocean. You can grab empty bottles from the water, but with tiny pieces, it’s impossible,” says Victor Alvarez, a chemical engineer who sells an eco-friendly alternative to glitter.In the early 2,000s, Alvarez worked for Mercedes-Benz in Germany, where he became fond of any technology that protected the environment. A few years after leaving Mercedes-Benz, he founded Blue Sun International in Miami, which makes specialty ingredients for the skin and hair care industries.Glitter is a popular ingredient in cosmetics, such as eye shadows and lipsticks. So Alvarez began researching an alternative that didn’t contain plastic to make his products safer for the environment. That’s when he came across Ronald Britton Ltd., a company which had developed a plastic-free, biodegradable product called Bioglitter. It is made from regenerative cellulose(纤维素) sourced from hardwoods, primarily eucalyptus(桉树). Alvarez worked with the company to become the first retailer to sell Bioglitter in America. In 2018, he formed Today Glitter in order to sell the biodegradable glitter directly to consumers through its website.Today Glitter sells two kinds of biodegradable glitter Bioglitter Sparkle and Bioglitter Pure. Both are almost plastic-free and can biodegrade in a short time. Meanwhile, they are as shiny as regular glitter. All these products are third-party tested by TÜV, an international organization that provides testing and certification for compostable (可降解的) and biodegradable products.Despite its benefits, the hardwoods needed to make biodegradable glitter cause it to cost about twice as much as conventional glitter. A small glass container that contains just 6 grams of Bioglitter costs $10, while the same amount of regular glitter could cost at least half that amount. Alvarez expects the price will come down over time. He also expects the company’s sales to cross $1 million next year. But more importantly, Alvarez says, his main goal is to effect a meaningful change.8. While at Mercedes Benz, Alvarez .A．developed a way to grab glitter in the sea B．became interested in the environmentC．attempted to live a plastic-free life D．created a kind of harmless glitter9. What do we know about Bioglitter?A．It is a plastic-free ingredient for eye shadows.B．It will soon be on sale in the American market.C．It is very difficult to break down in the wild.D．It was invented by Blue Sun International.10. Why does the author mention TÜV in the text?A．To show Bioglitter Sparkle and Bioglitter Pure are popular.B．To stress it provides a broad range of testing services.C．To prove Today Glitter’s products are eco-friendly.D．To explain many plastic products are low-quality.11. What is the problem faced by Today Glitter at present?A．The complex process of making glitter. B．How to expand its overseas market.C．How to attract potential investors. D．The high cost of raw materials.Carl Wieman, a Nobel Prize-winning physicist at Stanford University, excelled in the lab, where he created the Bose-Einstein condensate (玻色-爱因斯坦凝聚态). However, his mastery in the lab did not extend to the classroom. For years, he wrestled with what seemed to be a straightforward task: making undergraduates comprehend physics as he did. Laying it out for them—explaining, even demonstrating the core concepts of the discipline—was not working. Despite his clear explanations, his students’ capacity to solve the problems he posed to them rema ined inadequate.It was in an unexpected place that he found the key to the problem: not in his classrooms but among the graduate students (研究生) who came to work in his lab. When his PH. D. candidates entered the lab, Wieman noticed, their habits of thought were no less narrow and rigid than the undergraduates. Within a year or two, however, these same graduate students transformed into the flexible thinkers he was trying so earnestly, and unsuccessfully, to cultivate. “Some kind of intellectual process mu st have been missing from the traditional education,” Wieman recounts.A major factor in the graduate students’ transformation. Wieman concluded, was their experience of intense social engagement around a body of knowledge — the hours they spent advising, debating with, and recounting anecdotes to one another. In 2019, a study published in the Proceedings of the National Academy of Sciences backed this idea. Tracking the intellectual advancement of several hundred graduate students in the sciences over the course of four years, its authors found that the development of crucial skills such as generating hypotheses (假设), designing experiments, and analyzing data was closely related to the students’ engagement with their peers in the lab, rather than the guidance they received from their faculty mentors (导师).Wieman is one of a growing number of Stanford professors who are bringing this “active learning” approach to their courses. His aspiration is to move science education away from the lecture format, toward a model that is more active and more engaged.12. What problem did Carl Wieman have with his undergraduates?A．Making them excel in the lab. B．Demonstrating lab experiments.C．Facilitating their all-round development. D．Enhancing their physics problem-solving.13. Which of the following best describes the graduate students who first joined Wieman’s lab?A．Limited in thinking. B．Resistant to new ideas.C．Flexible and earnest. D．Experienced and cooperative.14. What is crucial for developing st udents’ intelligent thought according to the 2019 study?A．Intense lab work. B．Peer pressure and evaluation.C．Academic interaction with fellows. D．Engagement with external society.15. Which of the following can be a suitable title for the text?A．Transforming Graduates’ Habits B．Carl Wieman’s Nobel Prize JourneyC．The Nobel-Prize Winner’s Struggles D．Carl Wieman’s Education InnovationFour Tips to Discover Your True PassionTrue passion is the emotion, feeling, and desire that arises out of love for something. It’s the force that magnifies (放大) your capabilities for the benefit of excellent performance. This is because it’s a powerful motivation for best performance and everyone wants to get there. 16 Finding what you’re passionate about is a long road that requires effort and the following are some useful tips.Search your childhood.The purity and truth of your being are in the early stages of life. To that end, going back to the things you loved back then is always an excellent indication of marked preferences. Did you want to be a doctor? 17 These are all clues that begin to point the way. While it’s true that children have several preferences in their early stages, it’s actually quite easy to figure out which they prioritize (优先考虑).Experiment and discover.Doing the same activities day after day closes any probabilities to succeed in finding your true passion. 18 This creates a trial-and-error scenario (设想) that promotes a positive outcome in less time.Focus to find your true passion.You must forget about the myth that doing what you’re passionate about will be easy and pleasant. This is because there’ll always be inconveniences in any activity you undertake. Beyond that, don’t lose focus on the fact that these are temporary moments. 19Remember that your age doesn’t matter.20 Encouraging yourself to take the first step can be difficult due to some of society’s obsolete (被淘汰的) barriers. The truth is that you must allow yourself to achieve and show your tastes leaving aside any obstacles, especially those that are self-imposed (自己强加的). It’s never too late!My phone was an extension of myself. I couldn’t go anywhere _________ it. I scrolled through my friends’ _________ into late night and turned my phone on before my eyes were fully open. I enjoyed having it until that day when I realized that what had started as a useful tool had turned into a(n) _________. So, I made a _________ decision to turn off that smartphone and _________ my old dumb phone out of hiding.When I was watching TV shows or movies and there was a boring part, I _________ picking up the phone and checking out “what everyone else was doing.” Now I had to pay attention or find a way to _________ myself with my thoughts instead. When I visited New York City, I had to write out_________ and ask people where an intersection (十字路口) was _________ pulling out my phone.I have welcomed silence into my life with open arms and am enjoying the increased __________ from not constantly having to fight with the __________ to check my phone while I’m trying to get something done. I am allowing my thoughts to speak to me instead of my phone. I am connecting with others in a __________ way that isn’t all about me. Instead of always posting about what’s going on in my __________ perfect life, I am connecting with my friends one-on-one, talking about the reality of our lives and being there for them from a place of __________.As much as having a smartphone was __________, not having one is freeing on a much bigger level.21.A．through B．without C．despite D．upon22.A．updates B．instructions C．assessments D．decisions23.A．symbol B．responsibility C．reality D．addiction24.A．conscious B．shameful C．foolish D．random25.A．sell B．pull C．throw D．mind26.A．avoided B．denied C．missed D．allowed27.A．assist B．improve C．associate D．occupy28.A．requests B．thoughts C．directions D．comments29.A．instead of B．regardless of C．but for D．apart from30.A．creativity B．productivity C．knowledge D．independence31.A．routine B．necessity C．loneliness D．temptation32.A．genuine B．grateful C．direct D．desperate33.A．naturally B．certainly C．seemingly D．gradually34.A．wonder B．imagination C．interest D．authenticity35.A．depressive B．comfortable C．stressful D．boring阅读下面短文，在空白处填入1个适当的单词或括号内单词的正确形式，并将正确答案填写在答题卡上。

2023-2024学年四川省内江市第六中学高三下期入学考试英语试题

2023-2024学年四川省内江市第六中学高三下期入学考试英语试题Not everyone knows that Mogao Caves in China has a “little sister” — Yulin Caves, which is smaller but better preserved than Mogao Caves. This “little sister” has the most brilliant murals (壁画) which are large in scale (规模) and diverse in forms and skills, with much art value. Here is some information to refer to when planning your tour here.Daily Itinerary (行程)DAY 1: Visit Mingshashan in the middle afternoon when it is not very hot and sunburn is low.DAY 2: Visit Yulin Caves. You can visit up to 6 caves. After that, drive about 30 minutes to Suoyangcheng.DAY 3: Visit the newly opened Mogao Caves Digital Exhibition Center. You will watch two short but well-produced educational documentary movies about Mogao Caves. After the movies, take the interzonal bus to Mogao Caves.Highlights of This Tour●In-depth tour of Mogao Caves and Yulin Caves●Visiting Suoyangcheng — an ancient ruined city 2,000 years ago●Private tour package covering airport pick-up, accommodation, sightseeing and private tour guideBasic Information about Price●$380 per person●This price is based on a group of 2 adults sharing one standard double-bed room in 4-star hotels and traveling with our private tour in low seasons.●This price is subject to change according to your traveling season, group size, hotel, class, etc. If you want a lower price, you can get more people to join you, or use economy class hotels.1. When can tourists see the documentary movies?A．On Day 1. B．On Day 2. C．On Day 3. D．Any day.2. What is a most interesting part of the tour?A．An old living city. B．In-depth travel in private.C．Free accommodations. D．Two brilliant tour guides.3. Where can you find the text probably?A．In a history textbook. B．On a travel agency website.C．In an academic journal. D．On an exhibition of murals.Jeremiah Thoronka was born in the fighting of the Sierra Leone civil war and grew up in the gutter (贫民窟) on the suburbs of the capital Freetown, having to burn wood for lighting and heating. Jeremiah saw with his own eyes how, in addition to the smog making breathing problems common, his young contemporaries fell behind in their schoolwork because of a lack of decent lighting.Energy poverty is a major issue in Sierra Leone—with just 26% of the population having access to electricity. In rural parts of the country, only 6% of people have electricity access, most of whom turn to solar lanterns and dry-cell batteries. As a result, it’s led to the destruction of forests as people cut down trees for firewood, which leaves Sierra Leone highly sensitive to extreme events like flooding and landslides. Families’ reliance on firewood also leads to frequent house fires.These life-threatening disadvantages and hardships fuelled Jeremiah’s passion for renewable energy and climate change advocacy. At 17, when studying at the African Leadership University in Rwanda, he launched a start-up called Optim Energy that transforms vibrations (震动) from vehicles and footfall on roads into an electric current. It is different from established renewable energy sources including wind or solar because it generates power without relying on changeable weather.Optim Energy ran a successful pilot program in Jeremiah’s neighbourhoods, Makawo in the northern part of Sierra Leone and Kuntoluh east of Freetown. With just two devices, the start-up provided free electricity to 150 households comprising around 1,500 citizens, as well as 15 schools where more than 9,000 students attend.Jeremiah is currently developing plans to expand into the healthcare field, which needs power to cool medicines and create enough light for treating patients after dark.4. What affected the young fellows’ academic performance?A．The war. B．The smog. C．The poorlighting. D．The breathing problem.5. What can we learn about “energy poverty” from paragraph 2?A．It is caused by solar lanterns and dry-cellbatteries.B．It has a worse impact on the city area.C．It brings about the destruction of forests. D．It increases the risk of forest fires.6. What is special about Optim Energy?A．It draws on vibrations to makeelectricity.B．It upgrades the use of solar energy.C．It runs a pilot program throughout thecity.D．It helps 9,000 students return to school.7. What message does the text want to convey?A．Electricity is the lifeblood of the city.B．Whoever is happy will make others happy too.C．Life is either a daring adventure or nothing at all.D．We are all in the gutter, but some are making it better.Carl Wieman, a Nobel Prize-winning physicist at Stanford University, excelled in the lab, where he created the Bose-Einstein condensate (玻色-爱因斯坦凝聚态). However, his mastery in the lab did not extend to the classroom. For years, he wrestled with what seemed to be a straightforward task: making undergraduates comprehend physics as he did. Laying it out for them—explaining, even demonstrating the core concepts of the discipline—was not working. Despite his clear explanations, his students’ cap acity to solve the problems he posed to them remained inadequate.It was in an unexpected place that he found the key to the problem: not in his classrooms but among the graduate students (研究生) who came to work in his lab. When his PH. D. candidates entered the lab, Wieman noticed, their habits of thought were no less narrow and rigid than the undergraduates. Within a year or two, however, these same graduate students transformed into the flexible thinkers he was trying so earnestly, and unsuccessfully, to cultivate. “Some kind of intellectual process must have been missing from the traditional education,” Wieman recounts.A major factor in the graduate students’ transformation. Wieman concluded, was their experience of intense social engagement around a body of knowledge — the hours they spent advising, debating with, and recounting anecdotes to one another. In 2019, a study published in the Proceedings of the National Academy of Sciences backed this idea. Tracking the intellectual advancement of several hundred graduate students in the sciences over the course of four years, its authors found that the development of crucial skills such as generating hypotheses (假设), designing experiments, and analyzing data was closely related to the students’ engagement w ith their peers in the lab, rather than the guidance they received from their faculty mentors (导师).Wieman is one of a growing number of Stanford professors who are bringing this “active learning” approach to their courses. His aspiration is to move science education away from the lecture format, toward a model that is more active and more engaged.8. What problem did Carl Wieman have with his undergraduates?A．Making them excel in the lab. B．Demonstrating lab experiments.C．Facilitating their all-round development. D．Enhancing their physics problem-solving.9. Which of the following best describes the graduate students who first joined Wieman’s lab?A．Limited in thinking. B．Resistant to new ideas.C．Flexible and earnest. D．Experienced and cooperative.10. What is crucial for developing students’ intelligent thought according to the 2019 study?A．Intense lab work. B．Peer pressure and evaluation.C．Academic interaction with fellows. D．Engagement with external society.11. Which of the following can be a suitable title for the text?A．Transforming Graduates’ Habits B．Carl Wieman’s Nobel Prize JourneyC．The Nobel-Prize Winner’s Struggles D．Carl Wieman’s Education Innovation Since the 1950s, some 9.2 billion tonnes of plastic have been produced globally, of which only about10% has ever been recycled. Yet environmentally conscious companies and consumers continue to look to recycling as a way to ease the plastic problem. Manufacturing giants claim to be committedto making more of their products and packaging from recycled materials. However, this confidence masks (掩饰) a complex web of issues around plastic recycling. Recycling rates remain extremely low and critics argue that we should look at alternative ways to tackle plastic pollution.While many plastics have the potential to be recycled, most are not because the process is costly, complicated and the resulting product of a lower quality than the original. Despite rising demand for recycled plastic, few waste companies turn a profit. Part of this is because virgin plastic-linked to oil prices - is often cheaper than recycled plastic, meaning there is little economic incentive to use it. Worse yet, much of our plastic waste is difficult to recycle. Lightweight food packaging, like a mozzarella packet, contains different plastics, dyes and toxic additives (添加剂). This dirty mix means plastic recycled through mechanical methods- the most common form- can only be melted down and moulded (浇铸,塑造) again a couple of times before it becomes too fragile to be reused. And the nature of the process means plastic recycling has a carbon footprint of its own.Given all of these difficulties, environmental critics say recycling is not the solution-and argue that creating more products from recycled material to attract environmental consciousness merely worsens the problem. “The solution is to use less plastic and to stop misleading the public about the recyclability,” says Enck, president of Beyond Plastics, a US campaign group with a mission to end single-use plastic. “They should stop making false claims about the recyclability of plastics since they know most will either be littered (乱扔) or burned or landfilled (填埋). Using less plastics means shifting to reusable products and relying more on paper, cardboard, glass and metal- -all of which should be made from recycled content.”12. What is an environmentally conscious customer’s attitude towards recycling plastics?A．Suspicious. B．Favorable. C．Indifferent. D．Disapproving. 13. What does the underlined word “incentive” in Paragraph 2 probably mean?A．motive. B．issue. C．crisis. D．policy.14. What is Paragraph 2 mainly about?A．The recycling process of plastics. B．Pollutants contained in recycled plastics.C．Reasons why users dislike recycled plastics. D．Contributing factors to low plastic recycling rates.15. What will the environmental critics be happy to see according to the text?A．Using metal or glass food containers. B．Littering recycled plastics in a landfill.C．Processing plastics in a mechanical way. D．Launching campaigns to promoterecyclability.Science fiction television has done a lot to shape how we view the meals of the future, from an evening dinner in pill form to machines giving us any meal we desire, on demand. 16 However, while these ideas for food creation are more than fifty years old, the way we produce and consume food has not changed very much. Even though there is enough food available today to feed the world, more than 870 million people do not have enough to eat. 17With the global population expected to grow to more than nine billion by 2050, demand for food will only increase. No pills or machines have solved our food problems yet.However, agricultural science has been responsible for saving huge numbers of lives. Science and technology helped us out of starvation during the 1960s and 70s when the world’s population exploded. 18 The result of this panic was the “green revolution”, which saw the intr oduction to farming of high production grains, improved irrigation systems and hybrid seeds, saving over one billion people from starvation19 Having enough food is only the first step. There are complications with natural disasters, conflict, poverty and environmental problems. All these can mean that the food supply is put at risk 20 Many people are concerned about “playing around with nature”, adjusting how food looks and tastes and smells, and other human interferences(干预), all of which are widespread.Anyhow, the world has not been completely saved from starvation, but we are on the way to get there.Have you ever found yourself in a situation where something you enjoy on a daily basis is suddenly not available to you? For several weeks I volunteered on a ________ on the big island of Hawaii. My ________ in doing so: to learn how to grow my own food in a sustainable way, along with experiencing a simpler ________.Being an average twenty-something citizen of the ________ world, I spend a lot of free time using mobile devices. Before coming to Hawaii, I’d n ever made an effort to ________ that. When I arrived at the farm and discovered that the WiFi, ________ mainly by solar panels on the roof of my small room, wasn’t working normally, I knew a ________ to learn about myself had come knocking. Over the first couple days, I frequently thought about ________ my social media before realizing that I couldn’t. I ________ this source of entertainment and felt ________ from the world. I was________ that someone would message me and think I was ________ them. After it fully sank in that I couldn’t connect to my social networks, I felt more ________ in my immediate environment.My attention was less ________ while working on tasks. Though I still periodically felt ________, my anxiety faded and suddenly, I realized how much more time I had for the things that gave me a deeper, more real sense of achievement and ________: for me, this included yoga and mindful movement, reading, and being out in ________.My digital detox was not something I chose to do ________, but I’m thankful that it happened this way. ________ I value my devices just as much as I did before the digital detox, I realize that being ________ about how I use them is key to keeping my phone a positive addition to my life.21.A．farm B．beach C．playground D．river22.A．method B．purpose C．advantage D．suggestion23.A．solution B．custom C．principle D．lifestyle24.A．unique B．modern C．perfect D．complex25.A．change B．blame C．explore D．destroy26.A．challenged B．repaired C．powered D．closed27.A．vacation B．difficulty C．routine D．chance28.A．checking B．inventing C．building D．quitting29.A．respected B．missed C．praised D．accepted30.A．broken up B．fed up C．cut off D．paid off31.A．anxious B．surprised C．embarrassed D．cautious32.A．repeating B．answering C．ignoring D．criticizing33.A．experienced B．absorbed C．concerned D．disappointed34.A．received B．escaped C．caught D．distracted35.A．delighted B．greedy C．lonely D．convinced36.A．peace B．urgency C．ambition D．loss37.A．return B．place C．danger D．nature38.A．nervously B．reasonably C．carelessly D．willingly39.A．If B．After C．While D．Because40.A．bored B．selective C．familiar D．patient阅读下面材料，在空白处填入适当的内容(1个单词)或括号内单词的正确形式。