By Carlo F. Barenghi

*The target of this primer is to hide the fundamental theoretical info, speedy and concisely, in an effort to let senior undergraduate and starting graduate scholars to take on initiatives in topical study components of quantum fluids, for instance, solitons, vortices and collective modes.*

*The collection of the fabric, either in regards to the content material and point of presentation, attracts at the authors research of the luck of appropriate examine initiatives with novices to the sphere, in addition to of the scholars suggestions from many taught and self-study classes at the topic matter.*

*Starting with a quick old evaluation, this article covers particle information, weakly interacting condensates and their dynamics and eventually *superfluid helium* and quantum turbulence.*

*At the tip of every bankruptcy (apart from the 1st) there'll be a few workouts. exact options might be made to be had to teachers upon request to the authors.*

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**Example text**

A. Cornell, Phys. Rev. Lett. 77, 4984 (1996) 11. A. S. Rokhsar, Phys. Rev. A 55, 4346 (1997) 12. F. London, Nature 141, 643 (1938) Chapter 3 Gross-Pitaevskii Model of the Condensate Abstract The Gross-Pitaevskii equation (GPE) is a successful and well-established model for describing an atomic Bose-Einstein condensate. Here we introduce this model, along with its assumptions. Throughout the rest of this chapter we explore its properties and key time-independent solutions. 1 The Gross-Pitaevskii Equation We assume that the gas is at zero temperature, such that the thermal gas and thermallydriven excitations of the condensate are non-existent.

13) The continuity equation expresses conservation of the number of atoms (or, when written in terms of ρ(r, t), conservation of mass). By integrating the equation over a given volume, we see that, if the number of atoms changes in that volume, it is because fluid has moved in or out of it. The second equation is, ∂v = −∇ m ∂t √ 2 1 2 ∇2 n mv + V + gn − √ 2 2m n . 14) √ √ The ∇ 2 n/ n term is termed the quantum pressure term (see below). 15) where P and P are respectively the pressure and the quantum pressure, P= gn 2 , 2 P =− 2 4m n∇ 2 (ln n).

23) V (r) = ωr2 r 2 , 2 where r 2 = x 2 + y 2 + z 2 . The characteristic length scale of this potential is the harmonic oscillator length, /mωr . 24) r = There is no general analytic solution for the ground state (lowest energy) solution of the BEC in a harmonic trap; usually the ground state is found by numerically solving Eq. 9). However, there exist useful analytic results for certain regimes which we describe below. It is useful to work in terms of the interaction parameter,2 N as / r . Below we distinguish the following cases: no interactions, strong repulsive 1) and weak interactions (|N as / r | 1).