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THE STRUCTURE’ VORTEX OF CONDENSED GAS IN TRAPPED CLOUD

Bose-Einstein condensation, which has a very low particle density and a highly complex quantum structure, has been experimentally proven to date for rubidium, sodium, lithium, hydrogen, metastable helium, cesium and chromium atoms. Due to the differences between the properties and binary interactions of these atoms, experimental studies on condensation have found many impressive results. The structure of a vortex in a trapped environment and the factors affecting it were investigated in this study. Angular Momentum was calculated using Thomas-Fermi Approach.


INTRODUCTION
The Bose-Einstein concentration, first proposed by Einstein in 1925 for the first time, was observed experimentally in 1995 in the rubidium, sodium and lithium alkali atoms. Bose-Einstein estimates that condensation is for particles that do not interact under certain temperatures. A phase transition will occur with each other by condensing the macroscopic distribution of the gas.
At the center of the atomic cloud of the Bose-Einstein condensate, the particle density is in the order of 10 13 -10 15 cm -3 . This can be interpreted as the dilution of the Bose-Einstein condensation when compared to the density of air molecules at room temperature and atmospheric pressure, 10 19 cm -3 . In systems with such low density, the temperature must be in the order of 10 -5 K for the quantum phenomenon to be examined.
Bose-Einstein condensate has been experimentally obtained to date for rubidium, sodium, lithium, hydrogen, metastable helium, cesium and chromium atoms. Due to the differences between the properties of these atoms and their binary interactions, many impressive results have been obtained from experimental studies on condensation. When nuclear and electronic spin rates are included in the system, the system content is richer [1].

FORMALISM
Bose-Einstein gas properties that do not interact with each other in the trap can be determined by statistical mechanics. The Bose distribution function is given by for the thermal balanced uninterrupted bosons, where the average number of settlements of the singleparticle state is the single-particle state energy for a given trapped potential. For a free particle in a given 3-dimensional internal state, there is an average state in each volume (2h) 3 of the phase space. The momentum space region, which is less than the p momentum, has a cubic volume ( ) with a p radius, since it is the energy of the particle with momentum is given by , the total number of states with lower energies than the energy is given by where V is the volume in this system.
The number of states between and energetic levels is usually given by the state density: The transition temperature can also be defined as the maximum temperature at which the macroscopic settlement occurs in the lowest-energy state. The number of particles in excited levels is given by (4) When , it takes its maximum value. The transition temperature is when examined for the case where the total number of particles is in the evoked levels: The number density ( ) defined as the number of particles per unit volume for uniform Bose gas in the V volume three-dimensional box can be expressed by the Equation (6).
(6) the number of particles in the excited levels below the transition temperature, as in Equation (4)  ( ): For particles ( ) in the three-dimensional box, the number of excited particles in the unit volume is given by Equation (7): Utilizing the thermodynamic properties of the ideal Bose gas, energy, entropy and other properties of condensed phase, can be determined. Since the number of basal augmentations and the interactions between particles are low in the temperatures below, the chemical potential is lifted off and the internal energy is given by The specific heat is therefore given by; The intrinsic heat is also found Equation (11), because it can be expressed by using the term entropy S by :

(11)
It should be noted here that under temperature, energy, entropy and specific heat are not dependent on the total number of particles.
When these results are examined in the classical limit, the result that the Bose-Einstein distribution transforms into the Boltzmann distribution at high temperatures is reached. The total number of particles and energy is Equation (12) and Equation (13) in this case: In the case of full condensation, all bosons are at the same single-particle level, and for this reason the wave function is given by (14) for the system consisting of N particles. If the single-particle wave function is normalized and written energy, then (15) is found. By equating the variation to zero according to , the time-independent Gross-Pitaevski equation in the form of (16) is reached [2].

DYNAMICS OF CONDENSATION
The continuity equation is derived to understand the nature of the velocity of the condensate. When the time-dependent Gross-Pitaevski equilibrium is multiplied by , the complex conjugate of the equation obtained is subtracted and (17) result is reached. This is the same as that obtained from the linear Schrödinger equation because the nonlinear potential in the Gross-Pitaevski equation is the reel.

ROTATING CONDENSATION
One of the properties of superfluidity is the response to rotation. In addition, for charged superfluids, the response to the magnetic field is also a evidence. One of the important properties of superfluids is the result of the forced movements that occur due to the rate of coagulation being proportional to the gradient of the phase of the wave function. The speed of a condensate is the gradient of the scalar: This equation is one of the consequences of the condensation's possible actions, affecting many features of the condensation.
In general, the condensate wave function is monovalent. The change in in the wave function phase around a closed contour must be a multiple of . Or it must have conditions, an integer l. Thus, circulation around a closed contour: Equation (20) shows that the circulation is quantized with h/m. The value of the quantum circulation is about m 2 s -1 , which A is the mass number. As a simple example of such a flow, a completely azimuthal flow in the invariant trap under rotation around the axis can be considered. In order to provide a single valued condition, the condensation wave function must be changed to with the azimuthal angle. In order to satisfy the condition of being a mono-valued condition, the condensate wave function must be changed by . Here is azimuth angle.
 is the distance from the axis of the trap, velocity equation; is reached. The angular momentum per particle is not quantized, but the circulation of the other levels is quantized. The generalized state is where is for the level having the vortex extending along the z-axis. Where is the twodimensional Dirac delta function in the plane, and and are the unit vector in the z direction. When there are a large number of vortices, the right side of the above equation is the sum of the two-dimensional delta functions on the perpendicular surfaces in the direction of the vortex line. The intensity of the delta function is a vector oriented along the vortex line, and the value is equal to the value of the circulation associated with the vortex [3]. If the wave function for the trap with axial symmetry is , the wave function of the condensation is (23).
in the spherical polar coordinates. is a real number. Equation (15) uses for the energy value: The difference between the energy obtained from Equation (24)

VORTEX IN UNIFORM MEDIUM
When the infinite mean of the uniform potential is taken into account, the wave function in the base state is not dependent on z, and therefore the derivatives related to z are off. For distances farther distally, the radial derivative and the centrifugal barrier in the form of lose their significance and therefore the value of the choke wave function becomes . The derivative and centrifugal superposition dominates near the axis and the appropriate solution on the axis behaves like in a free particle with unit angular momentum in two dimensions. The terms in the Gross-Pitaevski equation in Equation 25 show that, in the course of the distance from the axis to a certain distance, the transition between the two states occurs. For this reason, it is possible to scale the lengths taking advantage of a certain length of [3]: Where is the intensity in the distant regions of the vortex. is the amplitude of the vortex wave function in distant regions from the vortices, and . is the energy density when is transformed: is found [4].
Another magnitude that must be taken into account for calculating the energies of the annulus is the extra energy that represents the energy of the particle and the enthalpy of the same number in uniform level. The energy in the unit length of the vortex is: The number of particles per unit length is Thus, the uniform system has a unit-length energy (31) The energy at unit length pertaining to the vortex is the difference between Equations 28 and 30: The above expression is found in in the unit length.

VORTEX IN A TRAPED CLOUD
The vortex energy in the Bose-Einstein condensate cloud in the trap is important for predicting the lowest angular velocity that is appropriate for energy in order to generate vortex in the cloud. For this purpose, considering the rotationally invariant harmonic trap potential around the z-axis and the number of atoms is sufficiently high for the Thomas-Fermi approach to be good, the localized vortex core radius on the z-axis of the trap determined by the consistent length is short compared to the cloud size.
Considering the two-dimensional problem by neglecting the trap force in the z-direction, the cloud is cylindrical to have the radius 2  and the energy of unit length The angular momentum L per unit length is obtained by multiplying the total number of particles  per unit length. The second case for   is found using the Thomas-Fermi approach as follows: When the three-dimensional problem is considered, z-axis is formed in the Z-direction and the size of the cloud in this direction is greater than the consistent length. In this case, the energy of the cloud can be asserted as the sum of the horizontal cloud slice energies, so that kinetic energy from the vertical gradient of the condensation wave function is neglected. The total kinetic energy and is integrated over the vertical axis of the cloud and found:

CONCLUSION
The vortex energy in the Bose-Einstein condensate cloud in the trap is important for predicting the lowest angular velocity that is appropriate for energy in order to generate vortex in the cloud. Angular momentum for a vortex in a uniform environment is a result of rotational symmetry. The angle of the angular momentum for the cloud that is not in the input trap axis depends on the position of the input. In addition, the angular momentum is not maintained for an invariant trap by rotating about the axis of rotation, and therefore has no particular value. The expression of total angular momentum is simply a velocity-dependent magnitude.