1. Superfluidity and Quantum Vortices

2. Outline of the presentation Bose-Einstein Condensation Superfluidity Quantum Vortix

3. Is a rare state (or phase) of matter in which a large percentage of bosons collapse into their lowest quantum state, allowing quantum effects to be observed on a macroscopic scale. The bosons collapse into this state in circumstances of extremely low temperature, near the value of absolute zero. Based on 1. the indistinguishability 2. the wave nature of particles A Bose-Einstein condensate :

4. Fig2: The phase diagram of 4 He. The liquid phase exists over a range of pressure up to about 25 atm even at the absolute zero of temperature; and there are two liquid phases, helium I, which is conventional in its properties, normal fluid, and helium II, which is superfluid. The normal fluid, with density ρ n, flow velocity field v n and conventional viscosity η n, carries all the thermal energy and entropy in the system. The superfluid component, with density ρ s and flow velocity field v s, can flow without friction and carries no thermal energy.

5. Fluids that can exhibit frictionless flow(superfluidity) liquid 4 He below about 2.17 K liquid 3 He below about 2 mK (discovery by Osheroff, Richardson and Lee in 1973) Ultra-cold atomic Bose gases A superfluid, which is a partial Bose Einstein condensate, has many exotic properties including 1.Free flow without friction 2.Entropy is zero. 3. Ballistic instead of diffusive heat transport. 4. Superfluid is also uniquely characterized by the presence of quantized vortices.

6. The assembly of condensed atoms is called the condensate, and the associated wavefunction is called the condensate wave function (CWF). For a more general motion of the condensate we can write We identify this velocity with the velocity of the superfluid component QUANTUM RESTRICTIONS ON SUPERFLUID FLOW

7. This is really a consequence of the quantization of angular momentum, But there can be a finite hydrodynamic circulation, defined as round any circuit that cannot shrink to nothing while remaining in the fluid; for example, a circuit round a solid cylinder passing through the fluid This means that the superfluid circulation must be quantized for the superfluid velocity that This means that there can be no local rotational motion of the superfluid component.

8. QUANTIZED VORTEX LINES IN SUPERFLUID HELIUM A free quantized vortex line in the superfluid component is a quantum of circulation round a tiny cylindrical hole in the helium. Such a line always has one quantum of circulation, and the hole then has a natural size, determined by a balance between the kinetic energy of flow and the surface energy of the hole, that is less than an interatomic spacing. Such vortex lines can exist in superfluid helium, and, as we shall show, they play an important role in its behaviour. Most obviously, perhaps, they allow the superfluid component to rotate if the helium is placed in a rotating vessel; otherwise such rotation would be forbidden. A parallel array of lines, as shown in Fig. 4 gives rise to a flow field that mimics uniform rotation on length scales larger than the line spacing, which is about 0.2 mm.

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