Bose-Einstein condensation of the triplet states in the magnetic insulator TlCuCl3

Bose-Einstein condensation denotes the formation of a collective quantum ground state of identical particles with integer spin or intrinsic angular momentum. In magnetic insulators, the magnetic properties are due to the unpaired shell electrons that have half-integer spin. However, in some such compounds (KCuCl3 and TlCuCl3), two Cu2+ ions are antiferromagnetically coupled to form a dimer in a crystalline network: the dimer ground state is a spin singlet (total spin zero), separated by an energy gap from the excited triplet state (total spin one). In these dimer compounds, Bose-Einstein condensation becomes theoretically possible. At a critical external magnetic field, the energy of one of the Zeeman split triplet components (a type of boson) intersects the ground-state singlet, resulting in long-range magnetic order; this transition represents a quantum critical point at which Bose-Einstein condensation occurs. Here we report an experimental investigation of the excitation spectrum in such a field-induced magnetically ordered state, using inelastic neutron scattering measurements of TlCuCl3 single crystals. We verify unambiguously the theoretically predicted gapless Goldstone mode characteristic of the Bose-Einstein condensation of the triplet states.     

Bose-Einstein condensation of photons in an optical microcavity

Bose-Einstein condensation (BEC)-the macroscopic ground-state accumulation of particles with integer spin (bosons) at low temperature and high density-has been observed in several physical systems, including cold atomic gases and solid-state quasiparticles. However, the most omnipresent Bose gas, blackbody radiation (radiation in thermal equilibrium with the cavity walls) does not show this phase transition. In such systems photons have a vanishing chemical potential, meaning that their number is not conserved when the temperature of the photon gas is varied; at low temperatures, photons disappear in the cavity walls instead of occupying the cavity ground state. Theoretical works have considered thermalization processes that conserve photon number (a prerequisite for BEC), involving Compton scattering with a gas of thermal electrons or photon-photon scattering in a nonlinear resonator configuration. Number-conserving thermalization was experimentally observed for a two-dimensional photon gas in a dye-filled optical microcavity, which acts as a 'white-wall' box. Here we report the observation of a Bose-Einstein condensate of photons in this system. The cavity mirrors provide both a confining potential and a non-vanishing effective photon mass, making the system formally equivalent to a two-dimensional gas of trapped, massive bosons. The photons thermalize to the temperature of the dye solution (room temperature) by multiple scattering with the dye molecules. Upon increasing the photon density, we observe the following BEC signatures: the photon energies have a Bose-Einstein distribution with a massively populated ground-state mode on top of a broad thermal wing; the phase transition occurs at the expected photon density and exhibits the predicted dependence on cavity geometry; and the ground-state mode emerges even for a spatially displaced pump spot. The prospects of the observed effects include studies of extremely weakly interacting low-dimensional Bose gases and new coherent ultraviolet sources.     

Spontaneous coherence in a cold exciton gas

If bosonic particles are cooled down below the temperature of quantum degeneracy, they can spontaneously form a coherent state in which individual matter waves synchronize and combine. Spontaneous coherence of matter waves forms the basis of a number of fundamental phenomena in physics, including superconductivity, superfluidity and Bose-Einstein condensation. Spontaneous coherence is the key characteristic of condensation in momentum space. Excitons--bound pairs of electrons and holes--form a model system to explore the quantum physics of cold bosons in solids. Cold exciton gases can be realized in a system of indirect excitons, which can cool down below the temperature of quantum degeneracy owing to their long lifetimes. Here we report measurements of spontaneous coherence in a gas of indirect excitons. We found that spontaneous coherence of excitons emerges in the region of the macroscopically ordered exciton state and in the region of vortices of linear polarization. The coherence length in these regions is much larger than in a classical gas, indicating a coherent state with a much narrower than classical exciton distribution in momentum space, characteristic of a condensate. A pattern of extended spontaneous coherence is correlated with a pattern of spontaneous polarization, revealing the properties of a multicomponent coherent state. We also observed phase singularities in the coherent exciton gas. All these phenomena emerge when the exciton gas is cooled below a few kelvin.     

Measurement of the spatial coherence of a trapped Bose gas at the phase transition

The experimental realization of Bose-Einstein condensates of dilute gases has allowed investigations of fundamental concepts in quantum mechanics at ultra-low temperatures, such as wave-like behaviour and interference phenomena. The formation of an interference pattern depends fundamentally on the phase coherence of a system; the latter may be quantified by the spatial correlation function. Phase coherence over a long range is the essential factor underlying Bose-Einstein condensation and related macroscopic quantum phenomena, such as superconductivity and superfluidity. Here we report a direct measurement of the phase coherence properties of a weakly interacting Bose gas of rubidium atoms. Effectively, we create a double slit for magnetically trapped atoms using a radio wave field with two frequency components. The correlation function of the system is determined by evaluating the interference pattern of two matter waves originating from the spatially separated 'slit' regions of the trapped gas. Above the critical temperature for Bose-Einstein condensation, the correlation function shows a rapid gaussian decay, as expected for a thermal gas. Below the critical temperature, the correlation function has a different shape: a slow decay towards a plateau is observed, indicating the long-range phase coherence of the condensate fraction.     

二维囚禁广义玻色气体的玻色-爱因斯坦凝聚

应用广义玻色-爱因斯坦分布函数研究在幂函数外势中二维广义玻色气体的玻色-爱因斯坦凝聚(BEC),导出二维广义玻色气体的临界温度、基态粒子占据率和热容量等物理量的解析表达式,讨论了非广延参数q对玻色系统热统计性质的影响.     

Berezinskii-Kosterlitz-Thouless crossover in a trapped atomic gas

Any state of matter is classified according to its order, and the type of order that a physical system can possess is profoundly affected by its dimensionality. Conventional long-range order, as in a ferromagnet or a crystal, is common in three-dimensional systems at low temperature. However, in two-dimensional systems with a continuous symmetry, true long-range order is destroyed by thermal fluctuations at any finite temperature. Consequently, for the case of identical bosons, a uniform two-dimensional fluid cannot undergo Bose-Einstein condensation, in contrast to the three-dimensional case. However, the two-dimensional system can form a 'quasi-condensate' and become superfluid below a finite critical temperature. The Berezinskii-Kosterlitz-Thouless (BKT) theory associates this phase transition with the emergence of a topological order, resulting from the pairing of vortices with opposite circulation. Above the critical temperature, proliferation of unbound vortices is expected. Here we report the observation of a BKT-type crossover in a trapped quantum degenerate gas of rubidium atoms. Using a matter wave heterodyning technique, we observe both the long-wavelength fluctuations of the quasi-condensate phase and the free vortices. At low temperatures, the gas is quasi-coherent on the length scale set by the system size. As the temperature is increased, the loss of long-range coherence coincides with the onset of proliferation of free vortices. Our results provide direct experimental evidence for the microscopic mechanism underlying the BKT theory, and raise new questions regarding coherence and superfluidity in mesoscopic systems.     

简谐势阱中弱相互作用玻色气体的临界温度及基态粒子占据率

应用局域密度近似法（LDA）对简谐势约束下的非理想玻色气体发生玻色－爱因斯坦凝聚（BEC）时的临界温度作出计算，进而讨论了弱相互作用对临界温度及基态粒子占据率的影响。这些研究将有助于对BEC这种新物态性质的进一步了解。     

Strong dipolar effects in a quantum ferrofluid

Symmetry-breaking interactions have a crucial role in many areas of physics, ranging from classical ferrofluids to superfluid (3)He and d-wave superconductivity. For superfluid quantum gases, a variety of new physical phenomena arising from the symmetry-breaking interaction between electric or magnetic dipoles are expected. Novel quantum phases in optical lattices, such as chequerboard or supersolid phases, are predicted for dipolar bosons. Dipolar interactions can also enrich considerably the physics of quantum gases with internal degrees of freedom. Arrays of dipolar particles could be used for efficient quantum information processing. Here we report the realization of a chromium Bose-Einstein condensate with strong dipolar interactions. By using a Feshbach resonance, we reduce the usual isotropic contact interaction, such that the anisotropic magnetic dipole-dipole interaction between 52Cr atoms becomes comparable in strength. This induces a change of the aspect ratio of the atom cloud; for strong dipolar interactions, the inversion of ellipticity during expansion (the usual 'smoking gun' evidence for a Bose-Einstein condensate) can be suppressed. These effects are accounted for by taking into account the dipolar interaction in the superfluid hydrodynamic equations governing the dynamics of the gas, in the same way as classical ferrofluids can be described by including dipolar terms in the classical hydrodynamic equations. Our results are a first step in the exploration of the unique properties of quantum ferrofluids.     

Coherent zero-state and pi-state in an exciton-polariton condensate array

The effect of quantum statistics in quantum gases and liquids results in observable collective properties among many-particle systems. One prime example is Bose-Einstein condensation, whose onset in a quantum liquid leads to phenomena such as superfluidity and superconductivity. A Bose-Einstein condensate is generally defined as a macroscopic occupation of a single-particle quantum state, a phenomenon technically referred to as off-diagonal long-range order due to non-vanishing off-diagonal components of the single-particle density matrix. The wavefunction of the condensate is an order parameter whose phase is essential in characterizing the coherence and superfluid phenomena. The long-range spatial coherence leads to the existence of phase-locked multiple condensates in an array of superfluid helium, superconducting Josephson junctions or atomic Bose-Einstein condensates. Under certain circumstances, a quantum phase difference of pi is predicted to develop among weakly coupled Josephson junctions. Such a meta-stable pi-state was discovered in a weak link of superfluid 3He, which is characterized by a 'p-wave' order parameter. The possible existence of such a pi-state in weakly coupled atomic Bose-Einstein condensates has also been proposed, but remains undiscovered. Here we report the observation of spontaneous build-up of in-phase ('zero-state') and antiphase ('pi-state') 'superfluid' states in a solid-state system; an array of exciton-polariton condensates connected by weak periodic potential barriers within a semiconductor microcavity. These in-phase and antiphase states reflect the band structure of the one-dimensional polariton array and the dynamic characteristics of metastable exciton-polariton condensates.     

Creation of ultracold molecules from a Fermi gas of atoms

Following the realization of Bose-Einstein condensates in atomic gases, an experimental challenge is the production of molecular gases in the quantum regime. A promising approach is to create the molecular gas directly from an ultracold atomic gas; for example, bosonic atoms in a Bose-Einstein condensate have been coupled to electronic ground-state molecules through photoassociation or a magnetic field Feshbach resonance. The availability of atomic Fermi gases offers the prospect of coupling fermionic atoms to bosonic molecules, thus altering the quantum statistics of the system. Such a coupling would be closely related to the pairing mechanism in a fermionic superfluid, predicted to occur near a Feshbach resonance. Here we report the creation and quantitative characterization of ultracold 40K2 molecules. Starting with a quantum degenerate Fermi gas of atoms at a temperature of less than 150 nK, we scan the system over a Feshbach resonance to create adiabatically more than 250,000 trapped molecules; these can be converted back to atoms by reversing the scan. The small binding energy of the molecules is controlled by detuning the magnetic field away from the Feshbach resonance, and can be varied over a wide range. We directly detect these weakly bound molecules through their radio-frequency photodissociation spectra; these probe the molecular wavefunction, and yield binding energies that are consistent with theory.     

激光对磁振子压缩态的作用

压缩态的量子起伏低于相干态相应的量子起伏．同声子和光子比较，自旋波的元激发——磁振子，也属于玻色子；在粒子数表像中，磁振子的哈密顿量在形式上与光子和声子的哈密顿量相似，因而磁振子的压缩态是可能存在的．通过测量在激光场下自旋分量的量子涨落，可以获得材料的性能参数．     

Bose-Einstein condensation of exciton polaritons

Phase transitions to quantum condensed phases--such as Bose-Einstein condensation (BEC), superfluidity, and superconductivity--have long fascinated scientists, as they bring pure quantum effects to a macroscopic scale. BEC has, for example, famously been demonstrated in dilute atom gas of rubidium atoms at temperatures below 200 nanokelvin. Much effort has been devoted to finding a solid-state system in which BEC can take place. Promising candidate systems are semiconductor microcavities, in which photons are confined and strongly coupled to electronic excitations, leading to the creation of exciton polaritons. These bosonic quasi-particles are 10(9) times lighter than rubidium atoms, thus theoretically permitting BEC to occur at standard cryogenic temperatures. Here we detail a comprehensive set of experiments giving compelling evidence for BEC of polaritons. Above a critical density, we observe massive occupation of the ground state developing from a polariton gas at thermal equilibrium at 19 K, an increase of temporal coherence, and the build-up of long-range spatial coherence and linear polarization, all of which indicate the spontaneous onset of a macroscopic quantum phase.     

囚禁势中玻色-爱因斯坦凝聚体的半轴宽度和粒子数密度

文章计算了准一维、准二维、球对称、轴对称系统的玻色-爱因斯坦凝聚体的半轴宽度和粒子数密度,采用求基态能量极小值和令势能等于相互作用能的办法,讨论的问题包括凝聚体半轴宽度的解析表达式、粒子数密度的理论计算值与实验数据的对比分析和吸引相互作用系统的塌缩粒子数。     

谐振势阱中有限粒子玻色-爱因斯坦凝聚的数值计算

从统计力学原理出发,用数值方法研究了三维等方谐振势阱中有限粒子数玻色子系统的化学势及其导数随温度的变化．结果表明,粒子数有限的系统没有一级相变,但在有限温度发生玻色-爱因斯坦凝聚;利用化学势二阶导数的极小值定义的玻色-爱因斯坦凝聚临界温度很好地符合实验结果．     

Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms.

For a system at a temperature of absolute zero, all thermal fluctuations are frozen out, while quantum fluctuations prevail. These microscopic quantum fluctuations can induce a macroscopic phase transition in the ground state of a many-body system when the relative strength of two competing energy terms is varied across a critical value. Here we observe such a quantum phase transition in a Bose-Einstein condensate with repulsive interactions, held in a three-dimensional optical lattice potential. As the potential depth of the lattice is increased, a transition is observed from a superfluid to a Mott insulator phase. In the superfluid phase, each atom is spread out over the entire lattice, with long-range phase coherence. But in the insulating phase, exact numbers of atoms are localized at individual lattice sites, with no phase coherence across the lattice; this phase is characterized by a gap in the excitation spectrum. We can induce reversible changes between the two ground states of the system.     

一维光晶格中玻色-爱因斯坦凝聚的粒子数分布

对捕陷在三维轴对称谐振势阱叠加一维光晶格的组合势中的玻色凝聚气体,基于平均场Gross-Pitae-vskii方程理论,并运用G-P能量泛函和变分方法,得出了非线性薛定谔方程的一维形式,运用数值计算的方法,研究了组合势中子凝聚体的粒子数分布与光晶格深度之间的关系,同时分析了磁势阱对子凝聚体粒子数分布的影响。     

Magnetic and non-magnetic phases of a quantum spin liquid

A quantum spin-liquid phase is an intriguing possibility for a system of strongly interacting magnetic units in which the usual magnetically ordered ground state is avoided owing to strong quantum fluctuations. It was first predicted theoretically for a triangular-lattice model with antiferromagnetically coupled S = 1/2 spins. Recently, materials have become available showing persuasive experimental evidence for such a state. Although many studies show that the ideal triangular lattice of S = 1/2 Heisenberg spins actually orders magnetically into a three-sublattice, non-collinear 120° arrangement, quantum fluctuations significantly reduce the size of the ordered moment. This residual ordering can be completely suppressed when higher-order ring-exchange magnetic interactions are significant, as found in nearly metallic Mott insulators. The layered molecular system κ-(BEDT-TTF)(2)Cu(2)(CN)(3) is a Mott insulator with an almost isotropic, triangular magnetic lattice of spin-1/2 BEDT-TTF dimers that provides a prime example of a spin liquid formed in this way. Despite a high-temperature exchange coupling, J, of 250 K (ref. 6), no obvious signature of conventional magnetic ordering is seen down to 20 mK (refs 7, 8). Here we show, using muon spin rotation, that applying a small magnetic field to this system produces a quantum phase transition between the spin-liquid phase and an antiferromagnetic phase with a strongly suppressed moment. This can be described as Bose-Einstein condensation of spin excitations with an extremely small spin gap. At higher fields, a second transition is found that suggests a threshold for deconfinement of the spin excitations. Our studies reveal the low-temperature magnetic phase diagram and enable us to measure characteristic critical properties. We compare our results closely with current theoretical models, and this gives some further insight into the nature of the spin-liquid phase.     

均匀相互作用玻色气体的凝聚温度

最近人们已经在实验中实现了相互作用玻色气体的玻色－爱因斯坦凝聚,相互作用玻色气体的凝聚温度是玻色－爱因斯坦凝聚问题中一个重要的热力学量,运用鹰势方法和巨正则分布,作者对盒子中相互作用玻色气全的凝聚温度给预了详细考察,研究表明凝聚温度的移动为δTc/Tc=-2.9an^1/3,此处a和n分别为两粒子的S波散射长度和粒子精密度。     

Towards Bose-Einstein condensation of excitons in potential traps

An exciton is an electron-hole bound pair in a semiconductor. In the low-density limit, it is a composite Bose quasi-particle, akin to the hydrogen atom. Just as in dilute atomic gases, reducing the temperature or increasing the exciton density increases the occupation numbers of the low-energy states leading to quantum degeneracy and eventually to Bose-Einstein condensation (BEC). Because the exciton mass is small--even smaller than the free electron mass--exciton BEC should occur at temperatures of about 1 K, many orders of magnitude higher than for atoms. However, it is in practice difficult to reach BEC conditions, as the temperature of excitons can considerably exceed that of the semiconductor lattice. The search for exciton BEC has concentrated on long-lived excitons: the exciton lifetime against electron-hole recombination therefore should exceed the characteristic timescale for the cooling of initially hot photo-generated excitons. Until now, all experiments on atom condensation were performed on atomic gases confined in the potential traps. Inspired by these experiments, and using specially designed semiconductor nanostructures, we have collected quasi-two-dimensional excitons in an in-plane potential trap. Our photoluminescence measurements show that the quasi-two-dimensional excitons indeed condense at the bottom of the traps, giving rise to a statistically degenerate Bose gas.     

Using photoemission spectroscopy to probe a strongly interacting Fermi gas

Ultracold atomic gases provide model systems in which to study many-body quantum physics. Recent experiments using Fermi gases have demonstrated a phase transition to a superfluid state with strong interparticle interactions. This system provides a realization of the 'BCS-BEC crossover' connecting the physics of Bardeen-Cooper-Schrieffer (BCS) superconductivity with that of Bose-Einstein condensates (BECs). Although many aspects of this system have been investigated, it has not yet been possible to measure the single-particle excitation spectrum (a fundamental property directly predicted by many-body theories). Here we use photoemission spectroscopy to directly probe the elementary excitations and energy dispersion in a strongly interacting Fermi gas of (40)K atoms. In the experiments, a radio-frequency photon ejects an atom from the strongly interacting system by means of a spin-flip transition to a weakly interacting state. We measure the occupied density of single-particle states at the cusp of the BCS-BEC crossover and on the BEC side of the crossover, and compare these results to that for a nearly ideal Fermi gas. We show that, near the critical temperature, the single-particle spectral function is dramatically altered in a way that is consistent with a large pairing gap. Our results probe the many-body physics in a way that could be compared to data for the high-transition-temperature superconductors. As in photoemission spectroscopy for electronic materials, our measurement technique for ultracold atomic gases directly probes low-energy excitations and thus can reveal excitation gaps and/or pseudogaps. Furthermore, this technique can provide an analogue of angle-resolved photoemission spectroscopy for probing anisotropic systems, such as atoms in optical lattice potentials.