Lecture 1. 2D quantum gases: the static case. Low dimension quantum physics. Physics in Flatland. The 2D Bose gas:

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1 Lecture 1 2D quantum gases: the static case Low dimension quantum physics Quantum wells and MOS structures Jean Dalibard, Laboratoire Kastler Brossel*, ENS Paris * Research unit of CNRS, ENS, and UPMC Review article: I. Bloch, J. Dalibard, W. Zwerger, Many-Body Physics with Ultracold Gases arxiv: See also a very recent preprint: L. Giorgetti, I. Carusotto, Y. Castin arxiv: High T c superconductivity also Quantum Hall effect, films of superfluid helium, Physics in Flatland Peierls 1935, Mermin Wagner - Hohenberg 1966 in 2D: No long range order at finite temperature whereas this tends to a finite value in 3D no real crystalline order in 2D at finite T 1. The 2D Bose gas: Ideal vs. interacting Homogeneous vs. trapped The kind of order a physical system can possess is profoundly affected by its dimensionality.

2 The ideal Bose gas The uniform case at the thermodynamic limit The ideal Bose in a harmonic potential In 3D: when the phase space density reaches, BEC occurs In 2D: for any phase space density, the occupation of the various states in non singular, and no BEC is epected. In 3D, BEC occurs when In 2D, BEC occurs when Thermodynamic limit: Bagnato Kleppner (1991) Does harmonic trapping make 2D and 3D equivalent? What about interaction? The density in the trap (semi-classical approach) 3D 2D Adding interactions At BEC: At BEC: Note:

3 Interactions in «true 2D»: the square well problem How a 3D pseudopotential is transformed in 2D Petrov-Holmann-Shlyapnikov is the 3D scattering length associated to the 3D pseudopotential The scattering amplitude is a dimension-less quantity that characteries the scattering process Scattering cross-section (length): when with is always positive: there is always a bound state in the 2D problem, irrespective of the sign of the 3D scattering length The relevant amplitude for eperimentaly trapped 2D atomic gases no Born approimation for low energy scattering in 2D How a 3D pseudopotential is transformed in 2D (II)? The 3D interaction energy: The role of (weak) interactions on BEC in a 3D trap The simple shift due to mean field: Repulsive interactions slightly decrease the central density, for given N and T The result can be recovered simply by taking the 3D For an ideal gas, the central density at the condensation point is (semi-classical) energy with trial wave functions For a gas with repulsive interactions, one needs to put a bit more atoms to obtain the proper Mean-field analysis: start from with Note: In addition there is a shift of Tc due to comple non-mean field effects

4 The role of (weak) interactions on BEC in a 2D trap Problem with 2D BEC in a harmonic trap (continued) The procedure similar to the 3D mean field analysis fails. Indeed, Treat the interactions at the mean field level: where the mean field density is obtained from the self-consistent equation where Two remarkable results One cannot make a perturbative treatment of interactions around this point! One can accommodate an arbitrarily large atom number. The effective frequency deduced from tends to ero when Badhuri et al Holmann et al. Repulsive interactions tend to reduce the density, which cannot be infinite anymore at the center of the trap Similar to a 2D gas in a flat potential However there is more than just (no) BEC in a 2D interacting gas Eistence of a phase transition in a 2D Bose fluid Bereinski and Kosterlit Thouless Nelson Kosterlit 1977 T c superfluid normal T 2. algebraic decay of g 1 eponential decay of g 1 The superfluidity in a 2D Bose fluid and the BKT mechanism Bound vorteantivorte pairs Unbinding of vorte pairs Proliferation of free vortices

5 The KT mechanism for pedestrians BKT transition for pedestrians (2) Probability for a vorte to appear as a thermal ecitation? One has to calculate the vorte free energy E-TS and compare it with kt Free energy of a vorte: ξ R Thermodynamic limit in 2D: T Energy: no free vorte proliferation of free vortice Entropy: Free energy of a vorte: Superfluidity in 2 dimensions A 2D film of helium becomes superfluid at sufficiently low temperature (Bishop and Reppy, 1978) When does the Kosterlit-Thouless transition occurs? The result is only a partial answer, because is a quantity that depends on temperature and it is usually unknown. Torsion pendulum adsorbed He film shift of the oscillation period (ns) T (K) Analytical calculation by Fisher & Hohenberg, Monte-Carlo by Prokof ev et al dimensionless interaction strength universal jump to ero of superfluid density at T = T c For our setup with Rb cold atoms: Also Safonov et al, 1998: variation of the recombination rate in a H film This is not universal by contrast to the 3D result

6 How to make a cold 2D gas? Take a 3D condensate and slice it 3. The critical point in an atomic 2D Bose gas Z. Hadibabic, P. Krüger B. Battelier, M. Cheneau, P. Rath, S. Stock superposition of a harmonic magnetic potential + periodic potential of a laser standing wave 2 independent planes with 1 5 Rb atoms in each Phys. Rev. Lett. 95, 1943 (25) Nature 441, 1118 (26) cond-mat/732 other 2D eperiments at MIT, Innsbruck, Oford, Florence, Heidelberg, NIST, etc. Why 2 planes? The crucial properties of a 2D Bose gas sit in its coherence function accessible in an interference eperiment The trapping configuration: lattice + harmonic potential Getting an interferogram for 2D gases CCD camera Time of Waistofthelasers along: 12 μm y flight imaging laser We vary the temperature between 5 and 1 nk (2 to «a few» planes) Tunnelling between planes is negligible. However adjacent sites may echange particles from the high energy tail of the distribution. This equalies T and μ 2D character of the gas: at center

7 Overlap between epanding planar gases Transition in a planar atomic gas 5 1, Integrated optical density along 8 6 We maintain a constant temperature and vary the atom number Apparition of a bi-modal distribution above a critical atom number Interference fringes Central bright component Halo in the periphery Characteristic of each gas individually 2 Position along Bimodal fit: Gauss + Thomas-Fermi give info on numbers, sies, temperature TF profile true BEC Atom number calibrated using 3D BEC Number of atoms in the central component ( 1 3 ) Temperature etracted from a gaussian fit of the pedestal N c = 85 T = 92 (6) nk Total number of atoms ( 1 3 ) Interference between two planar gases Bimodality and interferences y Time of flight The interfering part coincides with the central part of the bimodal distribution Number of atoms in the central component ( 1 3 ) atoms in TF part of distribution interference amplitude T = 92 (6) nk Total number of atoms ( 1 3 ) Interference amplitude (arb.units) Within our accuracy, onset of bimodality and interference agree

8 Variation of the critical atom number with T Can it be the Kosterlit-Thouless transition point? 15 For a uniform system with our interaction strength, the KT transition is epected to occur for Critical atom number ( 1 3 ) 1 5 Fit using ideal gas BEC in Local density approimation + gaussian density profile (as seen eperimentally) T (nanokelvin) 5.3 larger than the ideal gas prediction This factor 4.9 has to be compared with the eperimental factor 5.3: not bad The superfluid phase investigated using matter-wave interferometry cold hot 4. Quasi-long range order in a superfluid 2D Bose gas Time of Z. Hadibabic, P. Krüger Phys. Rev. Lett. 95, 1943 (25) B. Battelier, M. Cheneau, S. Stock Nature 441, 1118 (26) y flight sometimes: dislocations! Theory: Shlyapnikov-Gangardt-Petrov, Holtman et al., Kagan et al., Stoof et al., Mullin et al., Simula-Blackie, Hutchinson et al. Polkovnikov-Altman-Demler

9 Dislocation = evidence for free vortices Investigation of the long range order: π The interference signal between and gives Similar results at NIST uniform phase fraction of images showing a dislocation 4% 3% 2% 1% temperature control (arb.units) = full contrast at center 1 = no contrast at center Investigation of the long range order (2) Integrated contrast: average over many images: integration over ais Contrast after integration.4 Polkovnikov, Altman, Demler.2 Integrated contrast D (micrometers) Investigation of the long range order (3) Polkovnikov, Altman, Demler low T large T fit by integration over ais.2 low T large T Just below the transition, decays algebraically like.5 D integration over ais integration distance D (micrometers) Just above the transition, decays eponentially temperature control (arb.un.)

10 .5 The inhomogeneous density profile in the trap High Temperature y Red: Orange : Region of integration for measuring α The 2D atomic Bose gas at present Several signatures of a Kosterlit-Thouless cross-over have been identified Apparition of a bi-modal structure (superfluid core?) proliferation of vortices loss of long range order temperature control (arb.un.) Yellow : The observed phenomena do not match the ideal Bose gas condensation Good agreement with quantitative predictions of the KT mechanisms Low Temperature One important remaining question: direct test the superfluidity of the core Lecture 2 2D Bose gases: the rotating case Physics in a rotating frame Cylindrically symmetric trap potential in the y plane: Stir at frequency (with a rotating laser beam for eample) Hamiltonian in the rotating frame: Jean Dalibard, Laboratoire Kastler Brossel*, ENS Paris * Research unit of CNRS, ENS, and UPMC Same physics as charged particles in a magnetic field + harmonic confinement

11 Macroscopic quantum objects in rotation Ω Rotating bucket with superfluid liquid helium 1. Neutron stars and pulsars Bose gases in rotation and vorte nucleation Superconductor in a magnetic field Rotating nuclei Classical vs. quantum rotation Vortices in a stirred condensate Rotating classical gas velocity field of a rigid body Cylindrical trap + stirring ENS, Boulder, MIT, Oford Rotating a quantum macroscopic object macroscopic wave function: In a place where, irrotational velocity field: The only possibility to generate a non-trivial rotating motion is to nucleate quantied vortices (points in 2D or lines in 3D) Time of flight analysis (25 ms) ENS: Chevy,Madison,Rosenbusch, Bretin 2 imaging beam Feynman, Onsager, Pitaevskii Ω c Ω ω centrifugal limit

12 Nucleation of vortices: a simulation using Gross-Pitaevskii equation After time-of-flight epansion: The single vorte case C. Lobo, A. Sinatra, Y. Castin, Phys. Rev. Lett. 92, 243 (24) Questions which have been answered: Total angular momentum Nħ (i.e. ħ per particle)? Is the phase pattern varying as e iθ? What is the shape of the vorte line? Can this line be ecited (as a guitar string)? Kelvin mode The intermediate rotation regime The fast rotation regime The number of vortices is notably larger than 1. However one keeps the rotation frequency Ω notably below ω The centrifugal force nearly balances the trapping force. The radius of the gas increases and the density ρ drops. core sie ξ << vorte spacing The vorte core increases to infinity??? Uniform surface density of vortices n v with The vorte spacing decreases MIT Coarse-grain average for the velocity field Are there still vortices? Do they still form a regular lattice? cf. H c2 field for a superconductor?

13 Landau levels for a rotating gas Isotropic harmonic trapping in the y plane with frequency ω Hamiltonian in the rotating frame: 2. The fast rotation regime: Physics in the Lowest Landau Level m=-2 m=-1 Ω= Ω < ω m= m=2 m= m= ω LLL Ω ω When Ω = ω, macroscopically degenerate ground state for the one-body hamiltonian (Rohshar, Ho) Reaching the lowest Landau level The lowest Landau level Ω Characteristic length scale Assume that the direction is froen, with a etension LLL If the chemical potential and the temperature are much smaller than, the physics is restricted to the lowest Landau level General one-particle state in the LLL: This LLL regime corresponds to Typically: N=1 atoms, l =.5 μm, a =5 nm a: scattering length Ω >.99 ω Polynomial or analytic function The sie of the vortices is comparable to their spacing The atom distribution is entirely determined from the vorte position

14 Ground state in the mean-field LLL approimation Minimiation of A typical LLL solution 1 1 Rb atoms Ω=.99 ω ν = 15 H 5 In the LLL: atom distribution -5-1 vorte distribution Radial distribution Minimiation of The distortion of the vorte distribution on the edges is essential for producing a quasi-parabolic atomic density profile only 1 parameter: Aftalion, Blanc, Dalibard, Phys. Rev. A 71, (25) A. Aftalion, X. Blanc, and J. Dalibard, Phys. Rev. A 71, (25) also Watanabe, Baym, Pethick Komineas, Read, Cooper Sheehy, Radihovsky How to reach the LLL eperimentally Evaporative spin-up method (Boulder): first rotate the gas at a moderate frequency and then evaporate of atoms with small angular momentum. Limits of the mean field approach When Ω increases even more than the LLL threshold, the number of vortices N v increases and becomes similar to the atom number N It occurs for Total angular momentum: ( per particle) side view of the rotating BEC mean-field region LLL region 1 V. Schweikhard et al., PRL 92, 444 (24) Note that for a 3D description of the binary collisions