Bose-Fermi mixed condensates of atomic gas with Boson-Fermion quasi-bound state

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。

关于原子物理认识的英语作文The journey of understanding the fundamental building blocks of our universe has been a captivating and ever-evolving pursuit for scientists and thinkers alike. At the heart of this exploration lies the intriguing field of atomic physics, which delves into the intricate workings of the smallest known particles that make up the matter around us. As we delve deeper into the realm of atomic structure and behavior, we uncover a world of incredible complexity and wonder, shedding light on the very essence of our physical reality.One of the most significant milestones in the understanding of atomic physics was the groundbreaking work of Ernest Rutherford, a New Zealand-born physicist who is often referred to as the father of nuclear physics. In the early 20th century, Rutherford and his colleagues conducted a series of experiments that challenged the prevailing understanding of the atom, leading to the development of the Rutherford model of the atom. This model, which depicted the atom as a dense, positively charged nucleus surrounded by orbiting electrons, was a significant departure from the earlier plum pudding model proposed by J.J. Thomson.Rutherford's experiments involved bombarding thin sheets of gold foil with alpha particles, which are positively charged helium nuclei. The vast majority of the alpha particles passed through the foil undeflected, as expected, but a small percentage were unexpectedly deflected at large angles. This observation led Rutherford to conclude that the atom was not a solid, uniform sphere, as previously believed, but rather a dense, concentrated nucleus with a significant amount of empty space surrounding it. This groundbreaking discovery paved the way for a deeper understanding of the structure and behavior of atoms.Building upon Rutherford's work, the Danish physicist Niels Bohr further refined the understanding of atomic structure by proposing a model that incorporated the concept of quantized energy levels. Bohr's model suggested that electrons within an atom could only occupy specific, discrete energy levels, and that they could only transition between these levels by emitting or absorbing a specific amount of energy in the form of a photon. This model, known as the Bohr model of the atom, provided a more accurate description of the behavior of electrons within an atom and laid the foundation for the development of quantum mechanics.The advent of quantum mechanics, pioneered by physicists such as Max Planck, Werner Heisenberg, and Erwin Schrödinger, marked apivotal shift in our understanding of atomic physics. Quantum mechanics introduced the concept of the wave-particle duality, which posits that particles, including electrons, can exhibit both particle-like and wave-like properties. This revelation challenged the classical, deterministic view of the physical world and led to the development of probabilistic interpretations of atomic and subatomic phenomena.One of the most intriguing aspects of quantum mechanics is the principle of uncertainty, as formulated by Heisenberg. This principle states that there is a fundamental limit to the precision with which certain pairs of physical properties, such as position and momentum or energy and time, can be simultaneously measured. This limitation has profound implications for our understanding of the behavior of atoms and subatomic particles, as it suggests that the act of measurement can itself influence the outcome of an experiment.Another key concept in atomic physics is the wave function, which was in troduced by Schrödinger. The wave function is a mathematical representation of the state of a particle, and its square modulus is interpreted as the probability density of finding the particle in a particular location. The wave function evolves over time according to the Schrödinger equation, which describes the dynamics of quantum systems. The wave function's ability to represent the superposition of multiple possible states, known as quantum superposition, is acornerstone of quantum mechanics and has led to the development of groundbreaking technologies, such as quantum computing.As our understanding of atomic physics has progressed, we have also gained insights into the fundamental forces that govern the interactions between particles at the atomic and subatomic scales. These forces, known as the four fundamental forces of nature, include the strong nuclear force, the weak nuclear force, the electromagnetic force, and the gravitational force. The study of these forces and their interplay has led to the development of theories such as quantum electrodynamics (QED) and quantum chromodynamics (QCD), which provide a comprehensive description of the behavior of particles and the interactions between them.One of the most significant developments in atomic physics in recent decades has been the exploration of the behavior of atoms and molecules at extremely low temperatures, known as the field of atomic, molecular, and optical (AMO) physics. In this realm, researchers have been able to observe and manipulate the behavior of individual atoms and molecules, leading to groundbreaking discoveries and the development of technologies such as atomic clocks, Bose-Einstein condensates, and quantum sensors.The ongoing exploration of atomic physics has not only deepened our understanding of the fundamental nature of matter and energybut has also paved the way for numerous technological advancements that have transformed our world. From the development of nuclear power and medical imaging techniques to the emergence of quantum computing and nanotechnology, the insights gained from the study of atomic physics have had a profound impact on our lives and continue to shape the future of scientific and technological progress.As we continue to delve into the mysteries of the atomic world, we are reminded of the enduring power of human curiosity and the relentless pursuit of knowledge. The journey of understanding atomic physics is a testament to the human spirit, as we strive to unravel the intricacies of the universe, one particle at a time. With each new discovery and every breakthrough, we inch closer to a more comprehensive understanding of the fundamental building blocks of our reality, unlocking the potential to transform our world in ways we can scarcely imagine.。

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。

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>。

a r X i v :c o n d -m a t /0607711v 2 [c o n d -m a t .s t r -e l ] 1 S e p 2006Magnon Bose condensation in symmetry breaking magnetic field S.V.Maleyev,V.P.Plakhty,S.V.Grigoriev,A.I.Okorokov and A.V.Syromyatnikov Petersburg Nuclear Physics Institute,Gatchina,Leningrad District 188300,Russia E-mail:maleyev@.spb Abstract.Magnon Bose condensation (BC)in the symmetry breaking magnetic field is a result of unusual form of the Zeeman energy that has terms linear in the spin-wave operators and terms mixing excitations which momenta differ in the wave-vector of the magnetic structure.The following examples are considered:simple easy-plane tetragonal antiferromagnets (AFs),frustrated AF family R 2CuO 4,where R=Pr,Nd etc.,and cubic magnets with the Dzyaloshinskii-Moriya interaction (MnSi etc.).In all cases the BC is important when the magnetic field is comparable with the spin-wave gap.The theory is illustrated by existing experimental results.1.Introduction Magnon Bose condensation (BC)in magnetic field was intensively studied in spin singlet materials (see for example [1]and references therein).In this case magnons condens in the field just above the triplet gap.In this paper we consider magnon BC that appears in the symmetry breaking magnetic field.The theoretical discussion is illustrated by experimental observation of this BC in frustrated antiferromagnet (AF)Pr 2CuO 4and cubic helimagnets MnSi and FeGe.To clarify our idea we begin with consideration of conventional AFs.In textbooks two limiting cases are considered.First,the magnetic field is directed along the sublattices.In this case the system remains stable up to the critical field H C =∆,where ∆is the spin-wave gap.Then the first order transition occurs to the state in which the field is perpendicular to sublattices (spin-floptransition).Second,the field is perpendicular to initial staggered magnetization.The system remains stable but the spins are canted toward the field by the angle determined by sin ϑ=−H/(2SJ 0),where J 0=Jz ;J and z are the exchange interaction and the number of nearest neighbors,respectively.At H +2SJ 0the spin-flip transition occurs to the ferromagnetic state.To the best of our knowledge the first consideration of the symmetry breaking field was performed theoretically in [2]in connection with experimental study of the magnetic structure of the frustrated AF R 2CuO 4,where R=Pr,Nd,Sm and Eu [3,4].In these papers the non-collinear structure was observed using the neutron scattering in the field directed at angle of δ=450to the sublattices.It was found in[2]that in inclinedfield the Zeeman energy has unusual form with terms which are linear in the spin-wave operators and term mixing magnons which momenta differ in the AF vector k0.As a result the BC arises of the spin-waves with momenta equal to zero and±k0.Similar situation exists in cubic helimagnets MnSi etc.[5].If thefield is directed along the helix wave-vector k the plain helix transforms into conical structure and then the ferromagnetic spin state occurs at criticalfield H C.But if H⊥k the magnon condense with momenta zero,±k,±2k etc.This leads to the following observable phenomena:i)a transition to the state with k directed along thefield at H⊥∼H C1=∆√S/2(a l−a+l),l=1,2 and a l(a+l)are Bose operators.As a result the Zeeman energy has unusual formH Z=H a+q+k0a q+iϑN,i.e. these operators has to be considered as classical variables as in the Bogoliubov theory of the BC in dilute Bose gas.Due to thefirst term in(2)we must consider the operatorsa±k0and a+∓kas classical variables too.Minimizing the full Hamiltonian with respectto these variables we obtainE=(∆2sin22ϕ)/(16J0)−S2J0ϑ2−(H H⊥)2/[4J0(∆2(ϕ,H)],(3)where thefirst term is the energy of the square anisotropy.In cuprates with S=1/2ithas quantum origin and arises due to pseudodipolar in-plane interaction[9].The secondterm is the energy of the spin canting in perpendicularfield.The last term is the BCenergy and∆2(ϕ,H)=∆2cos4ϕ+H2⊥−H2 is the spin-wave gap in thefield[2].This contribution becomes important at H∼∆.The spin configuration is determined byd E/dϕ=0and equilibrium condition d2E/dϕ2≥0.This theory was verified by neutrons scattering[10,11].In diagonalfield H (1,1,0)the spin configuration in frustrated Pr2CuO4is governed by Eq.(3)and the intensityof the(1/2,1/2,−1)is given by I∼1+sin2ϕ[2].Neglecting the BC term we getsin2ϕ=−(H/H C)2,where H C=∆.As a result at H→H C we obtain I∼H C−H. But very close to H C the BC term becomes important and we have a crossover to I∼(H C−H)1/2.It is clearly seen infigure2.This crossover was observed in[10,11].3.Frustrated AFsIn frustrated R2CuO4AFs there are two copper spins in unit cell belonging to different CuO2planes(see inset infigure1).From symmetry considerations these spins do not interact in the exchange approximation.The orthogonal spin structure is a result of the interplane pseudodipolar interaction(PDI)[2,3]and the ground state energy is given by∆2E=on T we obtain H C≃7.8T andγC≃5.30that is in stronger disagreement wit the experiment.The experimentally obtained anglesαandγat T=18K andδ=9.50are shown infigure4[12].The transition to the collinear state withα∼−450andγC∼200 was observed.Again the non-BC theory can not explain the experimental data.For example it givesγC≃2.50.Explanation of all these experimental data using the BC theory will be given elsewhere.4.BC in helimagnetsIn helimagnets MnSi etc.Dzyaloshinskii-Moriya interaction(DMI)stabilizes the helical structure and the helix wave-vector has the form k=SD[ˆa׈b]/A,where D is the strength of the DMI,A is the spin-wave stiffness at momenta q≫k,ˆa andˆb are unit orthogonal vectors in the plane of the spin rotation.The classical energy depends on thefield component H along the vector k and the cone angle of the spin rotation is given by sinα=−H/H C,where H C=Ak2is the criticalfield of the transition into ferromagnetic state[5].However at H⊥≪H C rotation of the helix axis toward thefield direction and the second harmonic2k of the spin rotation were observed[6-8].Both phenomena are related to the magnon BC in perpendicularfield[5].The linear and mixing terms appear in the Zeeman energy in much the same way as it was discussed above:H Z=(H a−i H b)2the real form of the BC energy is not so simple.It is determined by nonlinear interactions but consideration of this problem is out of the scope of this paper.As a result the perpendicular susceptibility is proportional to1/(∆2−H2⊥/2)and 2k harmonic appears.The last was observed by neutron scattering[6-8].Intensities of corresponding Bragg satellites have the formI±∼[∆2/(∆2−H2⊥/2)]2[1∓(kP)]δ(q∓2k),(7) where P is the neutron polarization.If H⊥→∆√5.ConclusionsWe discuss a few examples of the magnon BC in symmetry breaking magneticfield.BC appears due to unusual terms in Zeeman energy.Obviously this phenomenon is very general and can be observed in other ordered magnetic systems.Effects related to the BC has to be more pronounced in thefield of order of the sin-wave gap.6.AcknowledgmentsThis work is partly supported by RFBR(Grants03-02-17340,06-02-16702and00-15-96814),Russian state programs”Quantum Macrophysics”,”Strongly Correlated electrons in Semiconductors,Metals,Superconductors and Magnetic Materials””Neutron Research of Solids”,Japan-Russian collaboration05-02-19889-JpPhysics-RFBR and Russian Science Support Foundation(A.V.S.).References1Crisan M,Tifrea I,Bodea D and Grosu I2005Phys.Rev.B721644142Petitgrand D,Maleyev S,Bourges P and Ivaniv A1999Phys.Rev.B5910793D.Petitgrand,Moudden A,Galez P and Boutrouille P1990J.Less-Common Metals164-165768 4ChattpodhayaT,Lynn J,Rosov N,Grigereit T,Barilo S and Zigunov D1994Phys.Rev.B49 99445Maleyev S2006Phys.Rev.B731744026Lebech B,Bernard J and Feltfoft1989J.Phys.:Condens.Matter161057Okorokov A,Grigoriev S,Chetverikov Yu,Maleyev S,Georgii R,B¨o ni P,Lamago D,EckerslebeH and Pranzas P2005Physica B3562598Grigoriev S,Maleyev S,Chetverikov Yu,Georgii R,B¨o ni P,Lamago D,Eckerlebe H and Pranzas P2005Phys.Rev B721344029Yildirim T,Harris A,Aharony A and Entin-Wohlman O1995Phys.Rev.B521023910Plakhty V,Maleyev S,Burlet P,Gavrilov S and Smirnov O1998Phys.Lett.A25020111Ivanov A and Petitgrand D2004J.Mag.Mag Materials272-27622012Plakhty V,Maleyev S,Gavrilov S,Bourdarot F,Pouget S and Barilo S2003Europhys.Lett.61 53413Ivanov A,Bourges P and Petitgrand D1999Physica B259-261879FiguresaFigure 1.Spin configuration in the field.Full and dashed arrows correspond to zero and nonzero field,respectively.Addition spin canting in H ⊥is shown by broken arrows.Inset:spin configuration in neighboring planes of frustrated AF.Figure 2.Log-Log plot of the (1/2,1/2,−1)Bragg intensity in diagonal field,h ∼(H C −H ).Figure3.Thefirst order transition in thefield directed along b axis.Calculated intensities for the spinflop configurations when spins are perpendicular to thefield (white arrows).Figure4.Field dependence of anglesαandγatδ=9.50.)10a H mT =)50b H mT =)150c H mT=Figure 5.Bragg reflections in the field along (1,1,0).a)Four strong spots corresponds to ±(1,1,1)and ±(1,1,−1)reflections.Weak spots are the double Bragg scattering.b)The 2k satellites appear.c)The helix vector is directed along the field.。