Bose-Einstein condensate space

The two-photon spectroscopy described above had made it possible to study three characteristic features of Bose-Einstein condensation condensation in real space, condensation in momentum space (Fig. 5) and the mapping the phase boundary (Fig. 6) [9]. 

Bose-Einstein condensates are unusual in numerous ways. With careful study physicists will gain basic knowledge about the material and quantum worlds. The atoms in a condensate are indistinguishable. All atoms move at the same speed in the same space. One can ask How can two objects occupy the same place at the same time A condensate is a macroscopic quantum wave packet and a macroscopic example of Heisenberg s uncertainty principle. Condensates hold the promise of bringing new insights to the strange world between the microscopic quantum and the macroscopic classical domains. 

We recall that Bose-Einstein condensation is the macroscopic occupation of the ground state of a system at finite temperature. For a weakly interacting gas, this phase transition occurs when the inter-particle spacing becomes comparable to the thermal de Broglie wavelength A = /2nh /mkBT, where ks is the Boltzmann constant and T is the temperature. A rigorous treatment for the ideal Bose gas yields n > 2.61221 , where n is the density [35]. At a temperature of 50 yuK, for instance, the critical density for hydrogen is 1.8 x 10 cm. ... 

What happens to a gas when cooled to nearly absolute zero More than seventy years ago, Albert Einstein, extending work by the Indian physicist Satyendra Nath Bose, predicted that at extremely low temperatures gaseous atoms of certain elements would "merge" or "condense" to form a single entity and a new form of matter. Unlike ordinary gases, liquids, and solids, this super-cooled substance, which was named the Bose-Einstein condensate (BECj, would contain no individual atoms because the original atoms would overlap one another, leaving no space in between. 

Classically, there are three distinct states of matter gas, liquid, and solid. (The newer two, plasma and Bose-Einstein condensates, are not applicable to our discussion so we omit them.) In the previous section we noted how as temperature increases it is thermodynamically favorable for transitions to occur from a more ordered form to a less ordered one. The atoms or molecules that make up a gas are randomly arranged (E and S are high) and widely separated. A gas will fill all the available space inside a container. The atoms or molecules that make up a liquid are also randomly arranged, but they are closer together than those in a gas and they move relative to one another. The characteristic of a liquid is that it will fill a container... 

One of the major motivating factors in the study of colhsions of cold ground state neutral atoms has been the quest to achieve Bose-Einstein condensation (BEC). This is a phase transition which occurs in a gas of identical bosons when the phase space density becomes large enough, namely, when there is about one particle per cubic thermal de Broglie wavelength. The specific criterionfor condensation is... 

The superfluidity of " He solid is usually described in terms of Bose-Einstein condensation or quantum statistics in energy space. All particles occupy the lowest... 

G(R,Rq) is not known explicitly (or by quadrature) for any but the most simple (and uninteresting problems). But it is clearly related to the solution of a diffusion problem for a particle starting at Rq in a 3N dimension space and subject to absorption probability V(R) + Vq per unit time. We therefore expect to be able to sample points R from G(R,Rq) conditional on Rq. It turns out to be possible by means of a recursive random walk in which each step is drawn from a known Green s function for a simple subdomain of the full space for the wavefunction. References ( 4) and ( ) contain a thorough discussion of this essential technical point, and also of the methods which permit the accurate computation of the energy and other quantum expectations such as the structure function, momentum density, Bose-Einstein condensate fraction, and... 

It has long been known from statistical mechanical theory that a Bose-Einstein ideal gas, which at low temperatures would show condensation of molecules into die ground translational state (a condensation in momentum space rather than in position space), should show a third-order phase transition at the temperature at which this condensation starts. Nonnal helium ( He) is a Bose-Einstein substance, but is far from ideal at low temperatures, and the very real forces between molecules make the >L-transition to He II very different from that predicted for a Bose-Einstein gas. 

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