Physics
Bose Einstein Condensate
Bose-Einstein condensate is a state of matter that occurs at extremely low temperatures. It is formed when a group of bosons, which are particles that obey Bose-Einstein statistics, collapse into the same quantum state, resulting in a single entity with unique properties. This phenomenon was first predicted by Satyendra Nath Bose and Albert Einstein in the 1920s.
Written by Perlego with AI-assistance
Related key terms
1 of 5
10 Key excerpts on "Bose Einstein Condensate"
eBook - PDFQuantum Atom Optics
Theory and Applications to Quantum Technology
- Tim Byrnes, Ebubechukwu O. Ilo-Okeke(Authors)
- 2021(Publication Date)
- Cambridge University Press(Publisher)
2 Bose–Einstein Condensation 2.1 Introduction Chapter 1 introduced the formalism for treating many-particle indistinguishable quantum systems. This can describe the wavefunction of any system of bosonic or fermionic particles possibly interacting with each other. In this chapter, we introduce how such a system can undergo Bose–Einstein condensation. The essential feature of a Bose–Einstein condensate (BEC) is the macroscopic occupation of the ground state. The fact that such a state occurs is not completely obvious from the point of view of statistical mechanics – one might naively expect that there is a exponential decay of the probability of occupation following a Boltzmann distribution p n ∝ exp(−E n / k B T ) , where E n is the energy of the state, k B is the Boltzmann constant, and T is the temperature. We first discuss the original argument by Bose and Einstein, showing why such a macroscopic occupation might occur. We then show how macroscopic occupation of the ground state occurs in a grand canonical ensemble and give key results for the condensation temperature and the fraction of atoms in the ground state. We then discuss the effect of interactions on the energy-momentum dispersion relation, which gives rise to the Bogoliubov dispersion relation. This is the key to understanding superfluidity, one of the most astounding features of a quantum fluid. Bose–Einstein condensation is a expansive subject, and the purpose of this chapter is introduce the minimal amount of background such that one can understand the more modern applications of such systems. For a more detailed discussion of the physics of BECs, we refer the reader to excellent texts such as those by Pitaevskii and Stringari [375] and Pethick and Smith [369]. 2.2 Bose and Einstein’s Original Argument To see why macroscopic occupation of the ground state occurs in a system of bosons, we first examine a simple model of N noninteracting two-level particles.
eBook - PDF- Carl E Wieman(Author)
- 2008(Publication Date)
- World Scientific(Publisher)
1, can have only particular 467 on to write down the now familiar Bow-Einstein distribution formula for an ideal gas.’ He noticed, however, that this formula has the peculiar property dlat at very low, but finite temperatures, it predicts that all the atoms will go into the lowest energy level of the container. This is now known as Bose-Einstein condensation and is discussed in every textbook on statistical mechanics. Although this is n o d y discussed in terms of chemical potentials, a more visual way to understand the condition for BEC is to think in tenns of the Debroglie wavelength, I , . As the temperature is reduced, the Dehroglie wavelength of each atom becomes larger. When the sample is so cold that the Debroglie wavelengths are larger than the interparticle spacing, the atoms begin to fidl into the lowest energy state in the container, as illustrated in Fig. lb. Thus the actual condition for BEC is a requirement on the phase space density. The condition usually given in the texts for an infinite homogeneous ideal gas is that (Am )’n > 2.6, where n is the atomic number density. Although this does not apply exactly to our case of a finite inhomogeneous system, it is fairly close, and provides a good indicator for the necessary temperatures (and hence Am) and densities which must be achieved. atoms in a single quantum state. As such, the atoms are indistinguishable in every respect, and hence cannot be considered as separate individual atoms. They have lost their identities as independent atoms, and have now hsed into a sort of “superatom”. Second, the transition to BEC is nonintuitive because, before the transition, the atoms are very fir apart compared to their atomic “size”. The average separation is 10,000 times the Bohr radius, and hence the interactions between them, in the usual sense of electrons pushing up against each other, is extraordinarily small by any measure.
eBook - PDFQuantum Theory
Density, Condensation, and Bonding
- Mihai V. Putz(Author)
- 2013(Publication Date)
- Apple Academic Press(Publisher)
Basics of the Bose–Einstein Condensation (BEC) 107 It can also be generalized to the time-dependent equation -∇ + + = ∂ ∂ 2 2 2 2 m V g i t B ( ) r y y y (5.101) as being viewed like the non-linear Schrödinger equation (NLSE) that reduces to the stationary case under the traditional wave-function ansatz ψ ψ μ ( , ) ( , ) exp r r t t i t = - (5.102) Nonetheless, the present picture is assumed as valid within the next fulfilling criteria: • The existence of an imperfect Bose - gas, that is the dilute gas that is the ideal gas + weak interactions, such that the interactions are described via the single parameter of the length of the two-body scattering process with s-waves: thus the existence of “ a ” ! • The conditions of the ultra-high dilution , acquired for the cases when the in-terparticle distance is far larger than the scattering length: 1/3 v a >> . This is equivalent to saying that no many-body interactions other than two-body inter-actions are present, and this leads for thermodynamic limit ∞ → V N , (see also the Thomas–Fermi approximation in the Chapter 9, Section 9.3) with the con-densate ground state energy proportional with the bosonic interaction term that urns to be E a m N NE B 0 2 4 ~ ~ [ , ] π ρ ψ ψ + , (5.103) with r = N V (5.104) • Fulfilling the healing (or indeterminacy) condition , when scattering wavelength is largely over-passed by the thermal (de Broglie) length, dB a l << , for assuring the losing of bosonic identity in the condensate, since this way, in the low den-sity regime ( 1/3 a r -<< ) it is impossible to localize the particles relative to each other; as a consequence, bosons in the ground state (being characterized by the mean field or the order parameter) are smeared out over large distance (or wave-length, see Figure 5.1) compared with the mean particle distance: 1/3 dB r l -<< .
eBook - ePub- Nicolas Sator, Nicolas Pavloff, Lenaic Couedel(Authors)
- 2023(Publication Date)
- CRC Press(Publisher)
14 Bose–Einstein Condensates of Weakly Interacting Dilute GasesDOI: 10.1201/9781003272427-14Since the middle of the 90's, it has been known how to trap and cool vapours of bosonic atoms (typically rubidium 87 Rb or sodium 23 Na) below the Bose–Einstein condensation temperature (see Section 7.4 ). These systems are very cold (T ~ 100nK) and exhibit strong quantum features. They are also inhomogeneous (because the gas is confined by an external potential) and subject to non-negligible interaction effects: the condensed atoms accumulate at the bottom of the trapping potential well until reaching relatively high densities1 at which the physical properties of the system cannot always be described by the non-interacting model considered in Section 7.4 . The system's low temperature is an asset for the theoretical treatment of the interactions because it is associated with a thermal wavelength that is larger than the typical distance between atoms2 : in this regime, it is legitimate to ignore the details of the interaction and to use a schematic potential, designed only to correctly describe the low energy scattering between two atoms. The simplicity of the resulting model enables to account, in the framework of a mean field approach, for non-trivial interaction effects for which it is possible to obtain precise experimental information.14.1 GROSS-PITAEVSKII EQUATION
The state of a Bose–Einstein condensate at low temperature is described by a complex order parameterϕ (which acts as the “wave function of the condensate”. As in Ginzburg-Landau theory of superconductivity (Section 10.5 ), a complex parameter enables to describe both the local density of the condensate (its modulus) and the current within the condensate [gradient of its phase, see Equation (14.70)]. It is customary to chose a normalisation such thatr →, t )| ϕ (is equal to the local densityr →, t )| 2n (r →, t )
- Hwa-tung Nieh(Author)
- 2003(Publication Date)
- World Scientific(Publisher)
Eur. Phys. J. D 19, 103-109 (2002) DOI: 10.1140/epjd/e20020061 THE EUROPEAN PHYSICAL JOURNAL D EDP Sciences © Societa Italians di Fisica Springer-Verlag 2002 Production of a Bose Einstein Condensate of metastable helium atoms* F. Pereira Dos Santos 1,a , J. Leonard 1 , Junmin Wang 1,b , C.J. Barrelet 1 , F. Perales 2 , E. Rasel 3 , C.S. Unnikrishnan 4 , M. Leduc 1 , and C. Cohen-Tannoudji 1 1 College de France and Laboratoire Kastler Brossel c , Departement de Physique de l'Ecole Normale Superieure, 24 rue Lhomond, 75231 Paris Cedex 05, France 2 Laboratoire de Physique des Lasers d , Universite Paris-Nord, avenue J.B. Clement, 93430 Villetaneuse, France 3 Universitat Hannover, Welfengarten 1, 30167 Hannover, Germany 4 TIFR, Homi Bhabha Road, Mumbai 400005, India Received 15 October 2001 Abstract. We recently observed a Bose-Einstein condensate in a dilute gas of 4 He in the 2 3 Si metastable state. In this article, we describe the successive experimental steps which led to the Bose-Einstein transition at 4.7 fiK: loading of a large number of atoms in a MOT, efficient transfer into a magnetic Ioffe-Pritchard trap, and optimization of the evaporative cooling ramp. Quantitative measurements are also given for the rates of elastic and inelastic collisions, both above and below the transition. PACS. 32.80.Pj Optical cooling of atoms; trapping - 03.75.Fi Phase coherent atomic ensembles; quantum condensation phenomena - 05.30.Jp Boson systems 1 Introduction Recently two groups in France reported the first obser-vation of Bose Einstein condensation (BEC) of helium 4 atoms in the 2 3 Si metastable state [1,2], following the condensation of 87 Rb, 23 Na, and 7 Li in 1995 [3-5] and of atomic hydrogen in 1998 [6]. The method to reach BEC with metastable helium atoms uses similar routes as for alkali atoms, namely laser cooling and trapping followed by a final step of evaporative cooling.
- Nick P. Proukakis, David W. Snoke, Peter B. Littlewood(Authors)
- 2017(Publication Date)
- Cambridge University Press(Publisher)
Despite a common misconception, it is not a rule of BEC that the particles condense into the p = 0 state. The rule instead is that they condense into the lowest energy available state, as defined earlier. Only in empty space, and only if the total linear momentum and the total angular momentum of the particles are zero, is the lowest energy state a state of zero momentum. It should be obvious that the particles do not condense in the p = 0 state if they are moving or rotating. Nonetheless, Bose-Einstein condensation occurs. 31.5 Observational Implications For a long time, it was thought that axions and the other proposed forms of cold dark matter behave in the same way on astronomical scales and are therefore indis- tinguishable by observation. Axion BEC changed that. On time scales longer than their thermalization time scale τ , axions almost all go to the lowest energy state available to them. The other dark matter candidates, such as weakly interacting masssive particles (WIMPs) and sterile neutrinos, do not do this. It was shown in Ref. [16] that, on all scales of observational interest, density perturbations in axion BEC behave in exactly the same way as those in ordinary cold dark matter provided the density perturbations are within the horizon and in the linear regime. On the other hand, when density perturbations enter the horizon, or in second order of perturbation theory, axions generally behave differently from ordinary cold dark matter because the axions rethermalize so that the state most axions are in tracks the lowest energy available state. A distinction between axions and the other forms of cold dark matter arises in second order of perturbation theory, in the context of the tidal torquing of galactic 618 N. Banik, P. Sikivie halos. Tidal torquing is the mechanism by which galaxies acquire angular momen- tum.
- Hossein R Sadeghpour, David E Pritchard, Eric J Heller(Authors)
- 2003(Publication Date)
- World Scientific(Publisher)
New BECs This page intentionally left blank This page intentionally left blank ICAP 2002 31 All-Optical Atomic Bose-Einstein Condensates M. D. Barrett,. M.-S. Chang, C. Hamley, K. Fortier, J. A. Sauer, and M. S. Chapman School of Physics, Georgia Institute of Technology,Atlanta, GA We have created an atomic Bose-Einstein condensate (BEC) using all-optical methods, realizing a long-term objective in the field. Our method is simpler and faster than traditional BEC experiments and offers unique capabilities for atoms and molecules not amenable to magnetic trapping. 1 Introduction Given the tremendous impact of BEC research in last 7 years and the continued growth of the field, it is important to explore different methods for reaching BEC, particularly methods that offer new capabilities, simplicity, or speed. We have recently demonstrated such a method by creating a Bose condensate of Rb atoms directly in a crossed-beam optical dipole force trap using tightly focused COz gas laser beams [ 11. In the broader scope of research with ultracold degenerate gases, our system stands out for several reasons. First, all-optical BEC provides the first new path to achieving BEC since the first pioneering demonstrations [2-4], and it is surprising simple and an order of magnitude faster than standard BEC experiments. Also, optical trapping potentials are essentially spin-independent and hence are well suited for studying the formation and dynamics of spinor condensates. Finally, we can engineer a rich variety of spatial confinements, including large period one- and three- dimensional lattices that offer the possibility of optically resolving individual lattice sites. All-optical methods of reaching the BEC phase transition have been pursued since the early days of laser cooling.
eBook - PDFBoulevard of Broken Symmetries
Effective Field Theories of Condensed Matter
- Adriaan M J Schakel(Author)
- 2008(Publication Date)
- WSPC(Publisher)
By 132 Bose-Einstein Condensation Eq. (4.189) with the ideal-gas expression (4.190) for n, the result (4.202) leads to the interaction-induced shift in the condensation temperature ΔT BEC T BEC = 8π 3ζ (3/2) a λ T BEC = 8π 3ζ 4/3 (3/2) an 1/3 ≈ 2.33 an 1/3 . (4.203) Although this result is valid only for large N, it is independent of this parameter. The next-to-leading order in 1/N was found to give only a moderate correction to this leading result of order 25% for N = 2. Monte Carlo studies typically agree within a factor of 2 with the estimate (4.203). 4.12 Two-Fluid Model The Bogoliubov theory, describing a weakly interacting Bose gas, does not apply to strongly interacting superfluid 4 He. Both systems share, however, many fea- tures. In this section, a two-fluid description of a weakly interacting Bose gas is given to underscore these similarities. In the context of superfluid 4 He, the two- fluid model is a phenomenological model that successfully describes many of its startling properties. It is based on the assumption that the system can be separated in a condensate and a normal liquid consisting of elementary excitations. Two types of elementary excitations can be identified. The first consists of phonons, the quanta of sound waves, which we already met in the theory of weakly inter- acting Bose gases. Excitations of the second type were dubbed rotons by Landau. They are sometimes pictured as almost free particle excitations surrounded by a cloud of phonons. Strictly speaking, it is impossible to divide the elementary ex- citations into two types as both are part of a single-branch spectrum, consisting of phonons at long wave lengths and rotons at shorter wave lengths. Figure 4.6 shows an experimental curve of the elementary excitation spectrum in superfluid 4 He obtained in recent high-precision inelastic neutron scattering measurements.
- Raymond F Bishop, Klaus A Gernoth, Niels R Walet(Authors)
- 2000(Publication Date)
- World Scientific(Publisher)
32. Y. Castin and R. Dum, Phys. Rev. Lett 77, 5315 (1996). 33. E. W. Hagley et al., Phys. Rev. Lett. 83, 3112 (1999). 34. I. Bloch, T. W. Hansch, and T. Esslinger, 1999, preprint. 35. D. M. Stamper-Kurn et al., Phys. Rev. Lett. 80, 2072 (1998). 36. S. Inouye et al, Nature 392, 151 (1998). 357 QUANTUM KINETIC THEORY FOR A BOSE-EINSTEIN CONDENSED ALKALI GAS M. J. HOLLAND, J. COOPER, AND R. WALSER JILA, University of Colorado, Boulder, CO 80309-0440, USA E-mail: [email protected] The most salient features of the Bose-Binstein condensation of a magnetically con-fined alkali vapor is the diluteness of the gas and the extremely weak effective in-teractions. From a theoretical point of view, the interesting aspect is the potential formulation of the many-body quantum theory for a non-uniform and potentially non-equilibrium system founded entirely on microscopic physics. The crucial pos-tulate is the rapid attenuation of many particle quantum correlations in the dilute system which can be motivated from universal considerations. In principle, it wil be possible to provide direct comparison between theory and experiment over all temperature scales with no phenomenological parameters—a challenge facing the theoretical community in the near future. The dilute gas experiments provide an exciting stage on which to build bridges linking the theory of complex and collec-tive phenomena in superconducting and superfluid systems, with the single particle microscopic physics described in quantum optics and laser physics. 1 Development of a kinetic theory Since the recent demonstration of Bose-Einstein condensation in 1995 in experi-ments probing the physics of ultracold gases, 1 there has been renewed theoretical activity investigating many novel properties of confined and weakly-interacting con-densates.
- Claude Cohen-tannoudji(Author)
- 2004(Publication Date)
- World Scientific(Publisher)
phase. However, the detection processes induce a dispersion on n, so that the relative phase of the two condensates becomes better and better known. Indeed, the first detected boson can come either from mode 1 or from mode 2. After this detection, the state vector becomes: |^> = a|JV 1 -l, J (V 2 >+/3|JV 1 ,JV 2 -l) (93) where a and j3 are coefficients depending on the position of the first detected boson. Similarly, the second detection process changes the state vector into: ip) = XN 1 -2,N 2 )+nN 1 -1, N 2 -1) + uNi,N 2 -2} (94) The off-diagonality of p = ip) (ip in n = n — n 2 increases with the number of detected bosons and a relative phase 9 builds up. Note that 9 is an unpredictable random variable and takes different values from one experimental realization to the other. 462 658 BOSE-EINSTEIN CONDENSATES AND ATOM LASERS Relative phase and interference More precisely, the emergence of a relative phase between two condensates has been studied analytically by Y. Castin and J. Dalibard [21] and numerically by J. Javanainen and S.M. Yoo [22] as well as by the groups of P. Zoller [20] and of D.F. Walls [23]. 4. Beyond the variational approximation In Section 3, we have described the condensate within the variational approximation. This approach gives an approximate expression only for the ground state, but does not yield any information about the excited states or the elementary excitations. Besides, one may wonder if the approximation of the ground state by a product state is sufficient, and what the first corrections to this treatment are. Indeed, the interpretation of some physical effects requires to go beyond the product state description, as we will see in Section 5 for the total intensity of the light scattered by a condensate. In this paragraph, we briefly review the Bogolubov treatment [24] for a homogeneous condensate, which gives analytical results (see also [6], Chapter 19). This approach can be extended to a gas of bosons in a trap [25].
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
