Degenerate Fermi Gases Alessio Recati

Transcription

1 Degenerate Fermi Gases Alessio Recati CNR-INO BEC Center/ Dip. Fisica, Univ. di Trento (I)

2 BEC Group (28 members): Theory Experiment 4 UNI staff 3 CNR staff 2 CNR staff 7 PostDoc 7 PhD 3 Master 2 Master Trento Algiers

3 Temperature Temperaturescale scale Ultra-cold Gases at/cm3

4 Bosons and Fermions Bosons like staying together, in the same state Fermions hate staying together, in the same state Bose-Einstein statistics (1924) Fermi-Dirac statistics (1926) Dilute gases: 1995, JILA, MIT Dilute gases: 1999, JILA

5 IDEAL FERMI GASES

6 Non-interacting Gas at T=0 The Hamiltonian of a non-interacting gas in second quantization (although we could discuss everything in terms of particle occupation number) Occupation number: for fermions Single-particle energy states Homogeneous case GROUND-STATE: fill up the momentum space till the Fermi momentum simply defined by The Fermi energy is In this case the chemical potential is just the Fermi energy:

7 Non-interacting Gas at T=0 The momentum distribution is But what is the GROUND-STATE structure in real space? Density Pair correlation function

8 Non-interacting Gas at T=0 HARD WALLS: the GROUND-STATE structure in real space Density n(x)/n Friedel Oscillations: N=3, 10, 100 x/l

9 Excitations at Fixed N - particle-hole excitation k+q k ; The excitation energy of the p-h excitation is linear for small q Since the excitation spectrum depends on the actual state of the system its measurement (if possible) is a powerful way to get an insight in the system's state 1p-1h continuum

10 T=0 harmonically trapped Fermi gas Experimentally the atoms are trapped by means of electromegnetic fields properly tailored Typically the confining potential is harmonic-like

11 Cooling & Trapping neutral cold gases 1. Laser light pressure 2. Electric and/or magnetic confinement: harmonic traps 3. Evaporative cooling (a.k.a. cup of coffee cooling) From C. Salomon (ENS)

12 Cooling & Trapping neutral cold gases 1. Laser light pressure 2. Electric and/or magnetic confinement: harmonic traps 3. Evaporative cooling (a.k.a. cup of coffee cooling) From C. Salomon (ENS)

13 T=0 harmonically trapped Fermi gas Experimentally the atoms are trapped by means of electromegnetic fields properly tailored Typically the confining potential is harmonic-like

14 T=0 harmonically trapped Fermi gas Experimentally the atoms are trapped by means of electromegnetic fields properly tailored Typically the confining potential is harmonic-like: = E For large N it looks like a simple function, the Friedel oscillations are almost irrevelant F

15 T=0 harmonically trapped Fermi gas Thomas-Fermi (TF) or Local Density Approximation (LDA) For large particle number N we can assume that the system is locally uniform, i.e., Normalizing to N one finds the (global) chemical potential to be: x

16 T=0 harmonically trapped Fermi gas Thomas-Fermi (TF) or Local Density Approximation (LDA) For large particle number N we can assume that the system is locally uniform, i.e., Normalizing to N one finds the (global) chemical potential to be: GENERALLY: (for many purposes) one studies the equation of state for an homogeneous gas and apply LDA to study the trapped system, where different (homogeneous) phases can coexist

17 Time-Of-Flight (TOF) measurement n(x) Open the trap and observe the cloud after a certiain tof: get the Momentum distribution! n(p) with p=mr/t

18 Fermi see vs. Bose-Einstein condensate 2001 ENS (Paris) 104 7Li atoms, in thermal equilibrium with 104 6Li atoms in a Fermi sea Quantum degeneracy: T= 0.28 mk = 0.2(1) TC= 0.2 TF E F

19 Observing the Fermi Surface Nicely the gas expand like a cube

20 Collective oscillations of a Trapped Gas Collective Oscillations: exp. (MIT '97): ω=1.57ω z teory (HD): ω=1.58ω z ideal gas: ω=2 ω z

21 Collective oscillations of a Trapped Gas Both the ground state and the dynamics of the gas (Bose or Fermi or Boltzmann) can be obtained via variational principles on a proper action. ENERGY FUNCTIONAL: Kinetic en. density LDA: GROUND STATE: Ext ptential en. density Number of particle fixed

22 Collective oscillations of a Trapped Gas Both the ground state and the dynamics of the gas (Bose or Fermi or Boltzmann) can be obtained via variational principles on a proper action. Action: Multi-parameter wf describing the mode we are interested in, e.g., breathing, dipole... Dynamics the sloshing mode ideal gas: the brething mode ideal gas: due to the possibility of elastic deformation of the Fermi sphere (i.e., not the relation with the density one has at equilibrium) If there is interaction the equation of state changes and so do the mode frequencies. Indeed the collective mode frequency measurements are in cold trapped gases a precise and sensitive tool to investigate the equation of state of the system

23 INTERACTING FERMI GASES

24 Interaction: s-wave scattering length At low density and temperature the 2- body interaction is conveniently described by an effective contact potential which reproduces the low-energy behaviour of the microscopic potential V(x-x') s-wave scattering length i) a>0 : positive scattering & a Bound State (D=3) ii) a<0 : negative scattering & NO Bound State (D=3) Due to Pauli principle only fermions in different internal states can at this level- interact

25 Interaction: s-wave scattering length [MIT, Nature 392, 151 (1998)] [Innsbruck, PRL 93, (2004

26 Interaction: s-wave scattering length Cold Atoms: possibility of tuning the scattering length [MIT, Nature 392, 151 (1998)] [Innsbruck, PRL 93, (2004)]

27 Interaction: the Hamiltonian Free gases V(q) Depending on the parameters interaction can change completely the possible states of the system, leading to an extremely rich physics. To deal with an interacting Fermi system a huge amount of analitical and numerical methods have been developed. In the next a flavour of (very recentely realized/studied in the context of cold-gases): - Polaron physics (in the spirit Landau quasi-particles) - BCS-BEC crossover - Polarized Fermi gas - Itinerant Ferromagnetism

28 Interaction: the easiest case of N+1 A single blu-down atom (impurity) interacts with the many red-up ones (non-interacting). The energy to first order in perturbation theory for an impurity at rest: Clearly if it moves we have to add its kinetic energy, BUT being in a bath its inertia is modified. One has to calculate the system's energy at finite momentum p and expand to lowest order in p2 : Its coefficient can be identified with an effective mass

29 Interaction: the easiest case of N+1 A single blu-down atom (impurity) interacts with the many red-up ones (non-interacting). The energy to second order in perturbation theory where the small parameter is and p It behaves like a particle with a certain chemical potential (or binding energy ) and an effective mass: a quasi-particle, aka a FERMI-POLARON. A few of them would form a non-interacting Fermi gas of Polaron (Fermi energy and so on...)

30 Interaction: the easiest case of N+1 One can better, e.g., summing a proper class of diagrams with green function formalism or rely on Monte-Carlo calculations. Get more insight using a variational approach based on the simple 1p-1h excitations: p Variational parameter +Σ q k p+q-k The ground state is found by minimizing the energy with respect to the variational parameters.

31 Interaction: the easiest case of N+1 It behaves like a particle with a certain chemical potential (or binding energy ) and an effective mass: a quasi-particle, aka a FERMI-POLARON. A few of them would form a non-interacting Fermi gas of Polaron (Fermi energy and so on...) ATTRACTIVE CASE Notice than nothing special happens at UNITARITY, i.e., Infinity

32 Interaction: the easiest case of N+1 A single blu-down atom (impurity) interacts with the many red-up ones (non-interacting). It behaves like a particle with a certain chemical potential (or binding energy ) and an effective mass: a quasi-particle, aka a POLARON. A few of them would form a non-interacting Fermi gas of Polaron (Fermi energy and so on...) BUT for a>0 there is a bound state, whose energy goes like -1/a2 and size like a, thus at some point our impurities should couple with a red-up atom...and become BOSONS! Polaron Molecule Due to the Fermi see the molecular state appears when its size is smaller than the interparticle distance:

33 Interaction: the easiest case of N+1 RadioFrequency Wave Binding energy obtained via RF-frequency measurement in Zwierlein's group (MIT, 2009) Molecular side Unitarity

34 Oscillation of a Polaron in the trapped case Write the polaron Energy as: Supposing the bath (red-up Fermi atoms) is not essentially modified by the interaction with the impurity and that it is well described by LDA, one can write an effective single-particle Hamiltonian for the impurity: Which gives a renormalized dipole oscillation frequency: Spin-dipole mode E.g., AT UNITARITY

35 Momentum relaxation time Untill now we assumed that the (quasi-particle) polaron is well defined for each value of its momentum. This is generally not the case! At finite momentum it will have a finite lifetime due to the collisions with the red-up particles We consider the momentum relaxation of an homogeneous highly polarized Fermi gas. The minority component have a mean momentum k with respect to the majority one: total momentum per unit volume

36 Decaying time of the collective modes { Collisionless regime: possible to see the dipole mode Hydrodynamic regime: the dipole mode overdamped : MIT regime :

37 INTERACTING FERMI GASES: Superfluidity

38 Attractive Interaction For a>0 there is a bound state and at some point our impurities couple with a red-up atom...and become BOSONS BUT in the balanced case (equal number of red and blue atoms) a similar effect is present for any value of the interaction also for small negative scattering length: COOPER PAIR FORMATION and BCS-superfluidity In 3D in vacuum an attractive potential has to be attractive enough to bound a pair of (red-blue) atoms. In 2D and 1D any attractive potential make it due to the different density of states. Well, if we consider (Cooper problem) 2 atoms on top a frozen Fermi sea also in 3D any attractive potential allows for a state with an energy smaller than the Fermi Energy itself!

39 Attractive Interaction Cooper problem: Schroedinger Equation continuum Since from one also gets calculate as: Non-perturbative result no matter how weak is the interaction the size of the pair can be easily Much larger than the interparticle distance

40 BCS vs Bose-Einstein Condensation Note on finite T: Except for very weak coupling (BCS) pairs/molecules form and condense at different temperature, T* and Tc

41 BCS vs Bose-Einstein Condensation Molecular Bose-Einstein condensation from a fermionic gas [JILA, Innsbruck, MIT, ENS, RICE, 2003] Vortex lattice on the BCS-BEC crossover [MIT, 2005] Court. M. Zwierlein

42 BCS vs Bose-Einstein Condensation Landau critical velocity for a system to give rise to energy dissipation Due to pair breaking Due to phonon excitation (as in a BEC (Pitaevskii lecture))

43 Superfluid fermions at unitarity The only scales at unitarity are the Fermi energy and the temperature. The thermodynamic properties have an universal form. In particular at T=0 energy density, pressure, chemical potential are proportional to the ones of an ideal Fermi gas with a density equal to the superfluid one. The universal parameter (via Montecarlo & Experiments) S

44 Collective Oscillation of a SF Fermi-gas Both the ground state and the dynamics of the gas can be obtained via variational principles on a proper action. Action: Multi-parameter wf describing the mode we are interested in, e.g., breathing, dipole... Dynamics the sloshing mode SF gas: The (axial) brething mode (elongated) SF gas: due to coherence which assures local equilibrium (i.e., the relation with the density one has at equilibrium) If there is interaction the equation of state changes and so do the mode frequencies. Indeed the collective mode frequency measurements are in cold trapped gases a precise and sensitive tool to investigate the equation of state of the system

45 Fermi gas at Unitarity : expansion Hydrodynamic equation for a superfluid or a perfect (collisional) fluid At Unitarity one finds same expansion for T<Tc<<T F and T close to TF, but different from a wealy interacting Fermi gas Strongly Interacting 6Li gas T = 10-7 K [Duke, Science (2002) ]

46 Fermi gas at Unitarity : expansion At Unitarity one finds same expansion for T<Tc<<T F and T close to TF, but different from a wealy interacting Fermi gas Moment of Inertia Red normal fluid Blue superfluid y2 x2

47 Fermi gas at Unitarity : a perfect fluid? He near λ λαµ βδ αpoint QGP simulations String theory limit

48 Unbalanced Fermi gases: From Polaron to Superfluid Physics P=1 Normal Liquid P=0 Superfluid Phase Transition [Phase Transition to a normal phaase for large magnetic field B. S. Chandrasekhar (1962), A. M. Clogston (1962)]

49 Recent Experiments on imbalanced Fermi gases at unitarity 1: Collective mode Measurement of the (axial) breathing mode of the majority component in an elongated (cigar-shaped) trap, reducing the amount of minority atoms P

50 Recent Experiments on imbalanced Fermi gases at unitarity 2: Vortices MIT, Science 311, 492 (2006)

51 Recent Experiments on imbalanced Fermi gases at unitarity 3: Condensate Fraction BEC Unitarity BCS [MIT, Phys. Rev. Lett. 97, (2006)]

52 Normal phase of polarized Fermi gas at unitarity Assumption: at high polarization homogeneous phase, NORMAL FERMI LIQUID: consider a very dilute mixture of spin- atoms immersed in non-interacting gas of spin- atoms Energy expansion for small concentration... Non interacting gas single-particle energy quantum pressure of a Fermi gas of quasi-particles with an effective mass

53 Superfluid-Normal phase coexistence at unitarity Quasi-particles interaction Values using FN-QMC A = 0.99(2) m*/m = 1.09(3) B = 0.14 x=1: EN=1.12(2) [S. Pilati and S. Giorgini, Phys. Rev. Lett. 100, (2008)] Critical concentration xc: PSF = PN SF N with xc=0.44 Phase Separation xc=0.44 Coexistence line x=1: ES=0.84(2)

54 Exploring Phase diagram in the Trap: LDA LDA : Decreasing outward Constant also inside the trap x By the total number of atoms SuperFluid N Fully Polarized By the imbalance Critical imbalance

55 Exploring Phase diagram in the Trap: LDA LDA : Decreasing outward Constant also inside the trap x N SF FP FP N SF

56 Normal phase of polarized Fermi gas at unitarity: TRAP Density profiles 3D density Density Jump Superfluid Cooper pair condensate Normal liquid: POLARON GAS

57 Normal phase of polarized Fermi gas at unitarity: TRAP 1) Critical Polarization (IN TRAP): PC = 0.77

58 Outside the unitarity regime Creator:GPL Ghostscript SVN PRE-RELEASE CreationDate:2007/11/07 17:00:40 LanguageLevel:2 SF N SF [S. Pilati and S. Giorgini, Phys. Rev. Lett. 100, (2008)]

59 INTERACTING FERMI GASES: Ferromagnetism

60 Repulsive Interaction Let us consider the energy to the first order in the interaction Which P minimizes the energy? a) Without interaction: P=0, i.e., better to have same number of up and down atoms: paramagnetic phase) b) With interaction: { P=0; i.e., better to have more up (+) or down (-) atoms: ferromagnetic phase The system prefers to have separates in 2 islands when the majority of atoms are either the blue or the red ones a.k.a: STONER INSTABILITY

61 Repulsive Interaction Let us consider the energy to the first order in the interaction Paramagnet Which P minimizes the energy? a) Without interaction: P=0, i.e., better to have same number of up and down atoms: paramagnetic phase) b) With interaction: { P=0; Ferromagnet i.e., better to have more up (+) or down (-) atoms: ferromagnetic phase The system prefers to have separates in 2 islands when the majority of atoms are either the blue or the red ones a.k.a: STONER INSTABILITY

62 Susceptibility An important quantity in a spin-like system is its susceptibility, which tell us how easy is to polarise a the system. For the repulsive interaction close to P=0 one has: Thus: which diverges at the critical value

63 Susceptibility An important quantity in a spin-like system is its susceptibility, which tell us how easy is to polarise a the system. Attractive case: In the normal phase the susceptiblity should decrease In the superfluid phase the susceptibility should instead goes to zero (reflecting the gap in the spin excitation spectrum)

64 Relative number (spin) fluctuations in V V At large enough temperature T & volume cell V thermodynamic relation: MIT measured the reduction of the susceptibility in the attractive (superfluid) regime. On the repulsive branch the susceptibility should diverge if there exists a ferromagnetic transition. Interesting the quantum fluctuations (within Fermi-Landau liquid theory) also diverge but much more slowly[4] (logarithmically in ) [4] A.R. and S. Stringari, PRL 106, (2011)

65 (Not) The END...

66 ...more......more... More exotic phases (polarized Superfluid, FFLO, Sarma...) More than 2 species (analogies with color superfluidity?) Bose-Fermi mixtures Include disorder and noise Cold and Dipolar Molecules Low dimensional systems Antiferromagnetic order: Néel transition Quantum Hall effect Cold gases on atom chip Cold gases in Cavity QED Trapped ions Quantum Information/Computation Cold gases in optical lattices and Hubbard models

67 Why Why are are Cold Cold Gases Gases interesting? interesting? Diluteness: atom-atom interactions described by 2-body and 3-body physics At low energy: a single parameter, the scattering length Comparison with theory: Gross-Pitaevskii, Bose and Fermi Hubbard models, search for exotic phases, «Simplicity of detection» New way to address some pending questions in the physics of interacting many-body systems Link with condensed matter (high Tc superconductors, magnetism in lattices), nuclear physics, high energy physics (quark-gluons plasmas), astrophysics Experimental tunability of almost all the parameters which enter in the physics of the system under study!

68 Why Why are are Cold Cold Gases Gases interesting? interesting? Experimental tunability of almost all the parameters which enter in the physics of the system under study! Control of sign of interaction and of trapping parameters: weakly and strongly interacting systems 1D, 2D, 3D geometry & (optical) lattice access to time dependent phenomena & out of equilibrium situations and more Towards quantum simulations with cold atoms «a la Feynman»,i.e., the first idea of quantum computation

69 Number fluctuations in a cell V V At large enough temperature T & volume cell V thermodynamic relation (TR): In general we have deviations from it. In particular at T=0 we have quantum fluctuations which scales differently with N (e.g., [1,2]) EXP: Ideal Fermi gas: ETH (T. Esslinger), MIT (W. Ketterle) Interacting Fermi gas: MIT (W. Ketterle)

70 Decaying time of the collective modes We consider the momentum relaxation of an homogeneous highly polarized Fermi gas. The minority component have a mean momentum k with respect to the majority one: total momentum per unit volume p p' p-q p' + q

71 Decaying time of the collective modes { Collisionless regime: possible to see the dipole mode Hydrodynamic regime: the dipole mode overdamped : MIT regime :

72 Why Why are are Cold Cold Gases Gases interesting? interesting? Diluteness: atom-atom interactions described by 2-body and 3-body physics At low energy: a single parameter, the scattering length Comparison with theory: Gross-Pitaevskii, Bose and Fermi Hubbard models, search for exotic phases, «Simplicity of detection» New way to address some pending questions in the physics of interacting many-body systems Link with condensed matter (high Tc superconductors, magnetism in lattices), nuclear physics, high energy physics (quark-gluons plasmas), astrophysics Experimental tunability of almost all the parameters which enter in the physics of the system under study!

73 Why Why are are Cold Cold Gases Gases interesting? interesting? Experimental tunability of almost all the parameters which enter in the physics of the system under study! Control of sign of interaction and of trapping parameters: weakly and strongly interacting systems 1D, 2D, 3D geometry & (optical) lattice access to time dependent phenomena & out of equilibrium situations and more Towards quantum simulations with cold atoms «a la Feynman»,i.e., the first idea of quantum computation

74 The End