Many-Body Physics with Quantum Gases

Transcription

1 Many-Body Physics with Quantum Gases Christophe Salomon Okinawa Summer school on quantum dynamics September 26-October 6, 2017 Ecole Normale Supérieure, Paris

2 Summary of lectures Quantum simulation with ultracold gases Bose-Einstein condensation and degenerate Fermi gases Experimental techniques Tuning interaction between atoms Ultracold fermions: the crossover between Bose and Fermi superfluids Thermodynamics of quantum gases Bose-Fermi superfluid mixtures

3 The problem Understand many-body quantum systems Examples: high energy physics, condensed-matter, neutron stars, quantum chemistry, Nature is an assembly of interacting particles! Equilibrium properties and dynamics For instance, phase diagrams and phase transitions, time evolution The difficulty: exponential growth of the Hilbert space and consequently of the system s density matrix Example: 25 spin ½, without any spatial degree of freedom has dimension 2 25 = configuration space Approximate solutions: very often uncontrolled

4 Analog Simulators classical vs quantum R.P. Feynman Quantum simulation, 1981 Classical analog simulator: Strasbourg astronomical clock 1574 C. Herlin, and P. Dasypodius, mathematicians.. nature isn t classical, dammit, and if you want to make a simulation of nature, you d better make it quantum mechanical, and by golly it s a wonderful problem, because it doesn t look so easy.

5 The vision Simulating Physics with Computers Richard P. Feynman Received May 7, 1981 Can we simulate quantum Physics with computers? Exponential growth of the Hilbert space when increasing the number of interacting particles: untractable, in particular for fermions 1) Universal Quantum Computer Ongoing research, but extremely challenging 2) Quantum simulator Write an Hamiltonian to describe a physical system Find a well controlled system to simulate this Hamiltonian Measure the system s properties like ground state energy, excitation spectrum, collective modes, non universal Diversity of platforms to realize quantum simulators

6 The goals of quantum simulation Obtain results on the system that cannot be reached by standard methods or numerical simulations Explore novel geometries, parameters, or configurations that are not available in the initial system Invent novel systems or devices based on the acquired knowledge Non-trivial questions: How to verify the simulation results? How to detect and correct errors?

7 Quantum simulators Analog simulator Choose precisely the parameters of the system that we wish to simulate For instance, bosons, fermions, dimension, distance between particles, sign and strength of interaction,... Digital simulator Simulate the time evolution of the state vector by expansion of the evolution operator U(t) () t Uˆ () t (0) Ut ˆ () iht ˆ e / Trotter expansion Ut ˆ() Ut ˆ( / n) Ut ˆ ( / ne e e with )ˆ H Hˆ Hˆ... n iht ˆ / n ihˆ t/ n ihˆ t/ n In practice, useful only when H 1,H 2, involve a few particles See for instance Lanyon et al., Science 2011, Blatt s group, Innsbruck

8 Quantum simulators In these lectures: cold atoms are good quantum simulators

9 Temperature scale of cold gases cold atomic gases Superfluid 3 He liquid 4 He this room liquid N 2 sun surface sun center 1 K 1 mk 1 K 100 K 10 4 K 10 6 K T Dilute, but interacting systems Neutron stars 10 7 K Typical density: Interatomic distance range of interatomic potentials quantum of the motion in the trap thermal energy Equilibrium properties and dynamics are governed by interactions

10 Many-Body Physics with Cold Gases Diluteness: atom-atom interactions described by 2-body (and 3 body) physics. At low energy: a single parameter, the scattering length a Control of the sign and magnitude of interaction Control of trapping parameters: access to time dependent phenomena, out of equilibrium situations, 1D, 2D, 3D n(k) n(k) 1 1 Simplicity of detection Comparison with quantum Many-Body theories: Gross-Pitaevskii, Bose and Fermi Hubbard models, search for exotic phases, dipolar gases disorder effects, Anderson localization, Sherson et al., MPQ 2010 Link with condensed matter (high Tc superconductors, magnetism in lattices), astrophysics (neutron stars) Nuclear physics, high energy physics (quark-gluon plasma), 1 1 k/k k/k F F

11 Quantum statistics Bose-Einstein (1924) Fermi-Dirac (1926)

12 Prix Nobel de physique 1997 S. Chu, C. Cohen Tannoudji, W. Phillips Manipulation d atomes par laser Prix Nobel de physique 2001 E. Cornell, W. Ketterle, C. Wieman Condensation de Bose-Einstein

13 Fermions and Bosons Fermions Particles with half- integer spin Examples : electron, proton, neutron 2 identical fermions cannot occupy the same quantum state (Pauli exclusion principle) Bosons Particles with integer spin Examples : photon, atoms, molecules, Bose statistics: tendency to occupy the same quantum state Composite systems: atoms An atom is a boson if it contains an even number of fermions (Ex : H, He 4, Li 7, Na 23, Rb 87 ), or a fermion if it contains an odd number of fermions (Ex : D, He 3, Li 6, Sr 88 )

14 Ideal Gas H h h h h N identical particles without interaction in a box or a trap Hamiltonian h N Basis of eigenvectors of one body- hamiltonian aa et N H a a N, N, N,... Basis of eigen states in Fock space: ' '' where N are occupation numbers of an individual quantum state 0 or 1 for fermions, positive integer for bosons Total number of particles N N

15 Quantum statistics Fermions: N 1 ( ) e 1 : chemical potential: Energy to add a particle to the system 1/ kt B Boltmann gas positive and large compared to kt: degenerate Fermi gas N Bosons: ( ) e 1 can take all values from - min 1 to min When tends toward, N 0 tends to infinity: Saturation of excited states N 0 kt B min

16 Bose-Einstein condensation of an ideal Bose gas N identical bosons in a trap, at thermal equilibrium at temperature T DB 2 mk T When the temperature T is lowered, the de Broglie wavelength increases. When T < T c, a macroscopic number of bosons N 0 condenses in the trap ground state. The critical temperature T c corresponds to a situation where the de Broglie wavelength becomes on the order of the average distance between particles. The waves associated to different atoms overlap and interfere. B

17 DB 2 mk T B Pictorial image when temperature is lowered

18 Boson accumulation in the trap ground state T T C δe BEC is not a trivial thermal effect that occurs when thermal excitation energy kt B C is smaller than the trap energy level spacing E between levels

19 Bose-Einstein condensation: order of magnitude Dilute gas of atoms at temperature T confined in harmonic potential : V( r) m r 2 N Condensation threshold: k B T 3 kbt 3 n n 0 : central density h mk T 2 B Liquid helium : atoms/m 3 n -1/3 0 = 10 Å T ~1 K Gaseous condensate at/m 3 n -1/3 0 = 0.5 m T ~1 K

20 The ideal Fermi gas: a reminder Zero temperature Fermi sea: E(p) E kf k 2m 2 2 F F kt B F (6 n) ~ (particle distance) 2 1/3 1 E F p Fermi pressure: 1 2m /2 E 5/2 F Approximation valid as long as T<<T F

21 Electrons vs cold atoms Electrons in metal Ultra cold atoms Density /m /m 3 Mass kg kg Fermi temperature 10 4 K 1µK T/T F <10-5 ~ 10-2 Lifetime Infinite ~10s Size 1cm 10 µm Particle number

22 Preparation of a quantum gas Create an atomic beam of atoms or a vapor Laser cooling to ~100 K n Magnetic trapping or optical trapping Evaporative cooling to ~1 K n 3 =2.612

23 Loading a magnetic or optical trap Laser slowing and laser cooling 10 9 atoms, 1 cm K n Photo: Bell Labs optical molasses Radiation pressure of the laser L 0 Atom

24 Hänsch, Schawlow Wineland, Dehmelt Doppler Cooling Doppler effect 0 L L 0 L 0 v Laboratory frame L kv L kv Atom frame Absorption of the photon L +kv, followed by a spontaneous emission equiprobable in all directions of the space. Act as: F = - v (friction force) =mdv/dt

25 Optical Molasses S. Chu, Scientific American, 174, 1992 Na molasses

26 Sisyphus cooling J. Dalibard, C. Cohen-Tannoudji, S. Chu Optical pumping Light-shifted sublevels kt B U0 /4 Limit Temperature: about 10 times recoil energy: K range

27 Magneto-optical trap F = - v k r 3D Molasses Doppler effect Trapping Zeeman effect b' = 10 Gauss / cm I = a few mw per beam

28 Evaporative cooling The main method to reach quantum degeneracy Laser cooling to BEC, Sr : Stellmer, R. Grimm, F. Schreck (2013) - Remove hot atoms - Elastic collisions ensure re-thermalisation / 150 elastic inelastic

29 Evaporative cooling (2) N N / 100 T T / 1000 Phase-space density n 3 multiplied by 10 7 Duration : 1 to 30 seconds, N f =10 5 to 10 7 atoms, T f = 0.2 to 2 K

30 Imaging cold atomic clouds and condensates Absorption imaging 1) In situ measurement: spatial distribution in the trap 2) After Mesure time of flight in situ: expansion: distribution velocity en position distribution ou après temps de vol: distribution en impulsion

31 Bose-Einstein Condensation in Rubidium 87 JILA - Boulder 1000 atoms in ground state of magnetic trap. Remark: Metastable systems The true ground state of Rb at 1 K is a piece of solid Science, 269, 198 (1995) M. Anderson, E. Cornell and C. Wieman + Sodium, Lithium, Hydrogen, Potassium Helium (2s state), Cesium, Ytterbium, Calcium, Strontium, Erbium, Dysprosium

32 Bimodal distributions Condensate Narrow peak corresponding to The velocity width of ground state of harmonic trap non condensed atoms Thermal atoms in excited states: broader distribution

33 Condensate signature La signature d'un condensat A few millions atoms in anisotropic magnetic trap T > T c T < T c 0,5 to 1 K Time of flight 100 m * 5 m Boltzmann Gas mvi kt 2 2 isotropic anisotropic condensate mvi 2 4 without interactions i

34 Pauli Exclusion Principle and evaporative cooling of ultra-cold Fermi gases Collision between two atoms. Effective potential in the l-wave: ( 1) 2mr 2 eff () () 2 V r V r l >0 Interatomic potential (long range~-1/r 6 ) l=0 centrifugal potential At low temperature, atoms cannot cross the centrifugal barrier: only s-wave collisions. Symmetrization for identical particles: even l-wave collisions forbidden for polarized fermions.

35 Suppression of elastic collisions in a spin polarized Fermi gas Spin mixture (s-wave) B. DeMarco, J. L. Bohn, J.P. Burke, Jr., M. Holland, and D.S. Jin, Phys. Rev. Lett. 82, 4208 (1999). Use spin mixtures or several atomic species (eg 6 Li- 7 Li, K-Rb, different spin states )

36 Quantum gases in harmonic traps Bose-Einstein statistics (1924) Fermi-Dirac statistics (1926) Bose-Einstein condensate Fermi sea E F Bose enhancement h T = (0.83 N) 1/3 C k B Dilute gases: 1995, JILA, MIT Pauli Exclusion h T << T = (6N) F k B 1/3 Dilute gases: 1999, JILA

37 Bose-Einstein condensate and Fermi sea Lithium ENS Lithium Li 7 atoms, in thermal equilibrium with 10 4 Li 6 atoms in a Fermi sea. Quantum degeneracy: T= 0.28 K = 0.2(1) T C = 0.2 T F Now: T=0.03 T F

38 Magnetic trap F=1,m=1 F=1,m=0 z E. B B F=1,m=-1 Local minimum of B + Photo: spin polarisation Bell Labs Atoms cannot be magnetically trapped in the lower energy state. Two-body inelastic collisions Example: Ioffe-Pritchard trap Trap depth 1 mk Loaded with laser cooled atoms Or cryo-cooled atoms (Harvard) V= B Maxwell's equations: No max of B in vacuum.

39 Optical Trapping Laser field Induced dipole polarizability e 0 g Interaction energy Dipole potential I: laser intensity Potential depth ~1 mk Dipole force See R. Grimm and Y. Ovchinikov, Adv. At, Mol. and Opt. Physics, 42, 95, 2000

40 Two YAG beams with 5W and waist of 38 m The core of the experiment

41 The non-interacting Fermi gas Gaussian Fit Fermi-Dirac T/T F <0.05 Atom number~10 5

42 Fermions in a box B. Mukherjee et al. PRL 2017, MIT T=0.49 T F T=0.32 T F T=0.16 T F Bosons: A. Gaunt et al., PRL 2013, Cambridge

43 The setup A typical experiment

44 Next lecture: Tuning Atom-atom interactions in a 3D Fermi gases

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46 Coherence of Bose-Einstein condensates Young slit experiment, Munich, 2002 T > T c T < T c E m=1 Radiofrequency 1 Radiofrequency 2 z m=0 High contrast reveals macroscopic occupation of single quantum state n ( z) ( zz ) ( zz ) out out 1 out cos q z ( 12 ) t z qm( z z ) 2 g / 2 1 2

47 Examples of Atom lasers MIT YALE ORSAY