PhD thesis in English

Chapter 3Thermodynamics of a rotating ideal BECThe behavior of a BEC under rotation is essential for understanding many fundamentalphenomena [68, 7, 19]. The response of a quantum fluid to rotation representsone of the seminal hallmarks of superfluidity characterized by a nucleationof vortices with a quantized circulation. In the past, quantum vortices were studiedexperimentally in the superfluid helium and in type-II superconductors. An importantadvantage of the cold atomic gases for studying vortices is that the typical sizeof a vortex core is 3 orders of magnitude larger than in the superfluid Helium, and,even more importantly, it is large enough to be observed optically [69, 7]. Earlyexperiments with rotating cold gases explored different regimes: for slow rotation asmall number of vortices was observed [70, 69], while the triangular Abrikosov lattice[71] composed of about 100 vortices was detected in the case of a higher rotationfrequency [72]. Eventually, in the case of a very rapid rotation it is expected thatbosons would go from the superfluid phase into a strongly correlated phase that isrelated to the quantum Hall physics [73].The equivalence of the rotation and the motion of a charged particle in a magneticfield can be easily understood by considering the Hamiltonian of a single particle ina harmonic trap in a rotating reference frame with the rotation frequency ⃗ Ω = Ω⃗e z[68]:Ĥ rot = Ĥ − Ω ⃗ · ˆ⃗ L= 12M (ˆp2 x + ˆp2 y + ˆp2 z ) + 1 2 Mω2 (ˆx 2 + ŷ 2 + λ 2 zẑ2 ) − Ω⃗e z · (ˆ⃗r × ˆ⃗p)= 12M (ˆp x + MΩŷ) 2 + 12M (ˆp y − MΩˆx) 2+ 1 2 M(ω2 − Ω 2 )(ˆx 2 + ŷ 2 ) + 12M ˆp2 z + 1 2 Mλ2 z ω2 ẑ 2 , (3.1)where ˆ⃗ L is the angular momentum and all the variables in the last expression aregiven in the rotating reference frame. From Eq. (3.1), we see that in the limit Ω → ω,the Hamiltonian becomes formally equivalent to the one describing particles in the57