Macroscopic Quantum Systems JILA. I. Mazets, J. Schmiedmayer

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1 Macroscopic Quantum Systems JILA I. Mazets, J. Schmiedmayer JILA June 95 Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: /56 Course Outline Degenerate Quantum Gases: basic physics. Quantum statistics: ti ti. Ideal Gas: BEC in 3d, degenerate Fermions.3 Ideal gas in a Trap Experimental Techniques to achieve BEC. Laser cooling. Conservative atom traps.3 Evaporative cooling 3 The interacting Degenerate Quantum Gas 3. Scattering at low energies 3 3. Interacting gas, Gross-Piteavskii equation 3.3 Attractive interactions 3.4 Repulsive interaction 4 Probiyng BEC Properties 4. Imaging techniques 4 4. probing equilibrium i properties 4.3 probing dynamic properties 5 Dynamics of a Quantum Gas 5. Time dependent GP equation 5. Linear response, collective excitations 5.3 Microscopic description, Bogoliubov expansion 5.4 Solitons 6 Rotating Quantum Gas 6. Superfluidity and quantized rotation 6 6. Quantized vorties 6.3 Critical rotation 7 Tunable intreractions 7 7. Feshbach hresonances 7. Molecule formation 7.3 Efimov states 8 Degenerate Fermi Gas 8. Fermi statistics 8. Cooling a Fermi gas 8.3 Signatures of a degenerate Fermi gas 8.4 BEC BCS cross over 9 Low dimensional Quantum Gases 9 9. The ideal Bose gas in low dimensions 9. The trapped interacting Bose gas in D 9.3 The trapped interacting Bose gas in D Coherence on BEC. Interference: Coherent vs. independent BEC. BEC interferometry t in double wells.3 Bragg interferometry.4 Superradiant Rayleigh scattering Quantum Coherence and Quantum Tunneling. Atom lasers. Tunelling in a double well.3 The bosonic Josephson junction Optical Lattices. Lattice basics. D optical lattices.3 Bloch bands.4 Superfluid Mott transition.5 Quantum computation in lattices Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: /56

2 Macroscopic Quantum Systems Introduction to Bose- Einstein condensation Experimental techniques for achieving BEC Interacting quantum gases Basic Theory Guest lectures on BEC Nick Proukakis Univ. Newcastle 6.4. tentative Dynamics of a Quantumgas Rotating Quantumgas Superfluidity and vortices tentative Tunable Interactions ti Feshbach resonance Molecule formation Degenerate Fermi gases BEC-BCS crossover. 6. tentative Low dimensional i quantum gases Coherence properties of BEC tentative Quantum coherence, quantum tunnelling Optical Lattices There will be an extended coffee break in mid afternoon Jörg Schmiedmayer Igor Mazets (experiment) (theory) Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 3/56 Course Modalities Seminars: one seminar session for each lecture block ( in total) short 5+5 min. talks by small teams (up to 3 people) specific topics treated in more detail than in the lecture add experimental details practice for the exam (participation mandatory) Dates announced for each session First seminar session: Wednesday 3 th March 4 h (tentative) Exam: short term-paper (4 pages max.) on a recent scientific publication individual topics defined, given out at last lecture block 7 h home exam, all tools (apart other people) allowed 5 min. discussion when handing in the paper Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 4/56

3 Lecture I: Degenerate Quantum Gases: basic physics. Quantum statistics: Wave function of indistinquishable particles (Bosons Fermions) Statistics and indistinguishability: Counting particle Density of states, Bose- Fermi and Bolzman Statistics (reminder). Ideal Gas: BEC in 3d, degenerate Fermions Citi Critical temperature t for Bosons Condensate fraction.3 Ideal gas in a Trap Density of states in a trap BEC in a Trap Fermions in a Trap Lower dimensions.4 Making a BEC Laser Cooling Magnetic Trapping Evaporative Cooling observing a BEC Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 5/56 BEC basic introduction What is BEC? What is its underlying Physics? What the fundamental concept? Colloquial: all particles are in the same state Broken Gauge Symmetry, Off-diagonal long range order (ODLRO) Long range phase coherence Macroscopic wave function of the condensate These concepts were first introduced in studying superconductivity and superfluidity What is the signature? Delta function of the occupation o number of particles with zero momentum associated with long range phase coherence Bose narrowing (decrease in average energy as density gets higher). For fermions it is the opposite. Process of stimulated scattering: The scattering rate contains a factor (+N f) where N f is the occupation number of the final state Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 6/56

4 BEC basic introduction old table from 993: BEC is a common phenomenon occurring in physics on all scales Condensed matter atomic physics nuclear and elementary particle physics astrophysics Bosonic degrees of freedom are composite, they originate from Fermionic degrees of freedom (in most cases). Fundamental Bosons: gauge Bosons : Photon, W,Z Fundamental lfermions: p,n,e. Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 7/56 BEC dilute gas from Eric Cornell s Nobel Prize Lecture Why interesting? Strongly interacting vs. weakly interacting Bose gas Liquid Helium is dominated by interactions. The BEC fraction is in the order of %. Many phenomena are masked by the strong interactions A weakly interacting ti gas (Atoms, Excitons): theoretic ti description is easier using Feshbach resonances the interactions can be tuned by the experimenter: weakly interacting -> strongly correlated Condensation in free space vs. trapped condensates Free space one gets the classic formulas for BEC and its thermodynamic properties. Trapped gases: one has to look at the density of states in the trap. o isotropy of trap potential o dimensionality: 3d, (quasi) d, d o disordered potentials lattices: probe (simulate) solid state problems small number of particles vs. continuum in thermodynamics o what is the minimal size of a system we still can call a Bose condensate? Fermions Pauli principle, FD statistics At low temperatures: BEC vs. BCS o BEC: particle correlation length is very short compared to particle spacing. Molecules o BCS: particle correlation length is larger than the inter particle spacing, Cooper pairs Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 8/56

5 3 Counting Particles classical distinguishable particles 3 particles, total energy = 3 % probability for triple occupancy 3 # Teilchen Wahrscheinl. wie oft rot 3 3 % 6 % 9 3% 3 3 % probability 4% 4 for double occupancy Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 9/ Counting Particles indistinguishable particles 3 particles, total energy = 3 3 Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: /56

6 3 Counting Particles Bosons 3 particles, total energy = 3 33 % probability for triple occupancy 3 Bosons are gregarious! 3 % 4 3 % 44% 33 % probability for double occupancy 33% Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: / fermions % 33% 33% 33% Counting Fermions bosons % % 44% 33% 3 particles, total energy = 3 Fermions are loners! % probability for single occupancy classical % % 3% 4% Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: /56

7 Quantenstatistik 3 verschieden Statistiken ik (Verteilungen) Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 3/56 Quantum Statistics: Two indistinguishable ish Particles Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 4/56

8 Two indistinguishable ngu sha Particles II Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 5/56 N - Particles The above discussion generalizes readily to the case of N particles. Suppose we have N particles with quantum numbers n, n,..., n N. If the particles are bosons, they occupy a totally symmetric state, which is symmetric under the exchange of any two particle labels: Here, the sum is taken over all possible permutations p acting on N elements. The square root on the right hand side is a normalizing constant. The quantity N j stands for the number of ftimes each of fthe single-particle l states t appears in the N-particle l state. t In the same vein, fermions occupy totally antisymmetric states Here, sgn(p) is the signature of each permutation (i.e. + if p is composed of an even number of transpositions, and if odd.) Note that we have omitted the Π j N j term, because each single-particle state can appear only once in a fermionic state. These states have been normalized so that Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 6/56

9 Fermione ionen Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 7/56 Boso onen Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 8/56

10 Verteilung von Teilchen Klassische und Quantensatistik ik Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 9/56 Zustandsdichte s Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: /56

11 Derivation of the density of states for three dimensions Boundary condition: box Quantization of momentum Each point represents sphere with ħ 3 as volume Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: /56 Quanten Statistik Comparison of the Distributions E k B T Ae ( E) k e BT P(E) ( E) k T e B ( E ) k T e B E classical particles Maxwell-Boltzmann statistics F Bosons Bose-Einstein statistics Fermions Fermi-Dirac statistics Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: /56

12 Homogeneous Bose Gas Bose-Einstein i Condensation Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 3/56 Bose-Einstein Condensation II set x=e/kt Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 4/56

13 Bose-Einstein Condensation III /3 Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 5/56 Bose-Einstein Condensation IV Bose Einstein Distribution Number of particles in the condensate k B T e kt e / Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 6/56

14 Bose Distribution other examples Planck radiation law Superfluid He Paired Fermions Superconductivity Superfluid He-3 Fermions form a Pair (=Boson) BCS - pairing Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 7/56 Fermi-Distribution Dstr uton Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 8/56

15 Ferm Ele mi-dis strib butio on ectrons in a solid Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 9/56 BEC in external Potential V. Bagnato et al. Phys. Rev. 35, p4354 (987) free space formulas for power law potentials potential total energy and heat capacity Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 3/56

16 BEC in external Potential II V. Bagnato et al. Phys. Rev. 35, p4354 (987) Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 3/56 General Scaling Density of states in a general setting: (E)=C ) E - using x=e/kt c N ex C ( kt c ) C x dx e ( ) ( ) ( kt ) and for the critical temperature and kt c C N ex N N N N / ( ) ( ) T T c T T c x c / Rieman s zeta-function ( ) n n system infinite -dim 3/.6 box d harm. osz. 5/ d harm. osz. 7/ Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 3/56

17 Low Dimensions Because of the infrared divergence of fthe integral N ( ) d exp( ) / k T there is no BEC for D or D in free space the situation changes dramatically dir trapped quantum gas DOS is different use a finite sum B 3D DOS free space / constant -/ D N T k T c B c - N/N ml ln( N) DOS harmonic trap const D with a finite ground state energy T c N/N k T B c N ln(n) N T ~ N z T c Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 33/56 D-Trap D-Trap Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 34/56

18 Fermions in a Trap Fermi-Dirac Distribution anisotropic harmonic oscillator with and the density of states of a harmonic oscillators one obtains the Fermi energy and Fermi temperature with we find for cigar shaped traps: At T= and equilibrium we require and using the relation between Fermi momentum and density we find the density profile Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 35/56 Making a BEC Nobelprize Nobelprize 997 Nobelprize 997 Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 36/56

19 BEC what we need extremely cold atoms Ways to cool and accumulate large number of atoms Laser cooling Laser trapping (MOT) Evaporative cooling conservative trap to hold the atoms Ways to hold large number of ultra cold atoms without heating Magnetic trap Optical trap good collision i properties High collision rate to achieve thermalization (good collisions) Low inelastic rates (bad collisions) Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 37/56 Cold Atoms for BEC basics Laser Cooling Neutral atoms can be cooled by interacting i with monochromatic light (~thermal equilibrium with the light) Temperature p mk K Velocity.5m/s mm/s debroglie wavelength nm 5nm Typical samples 8 atoms/cm 3 Magnetic Trapping Neutral atoms can be magnetically trapped U=-B Gauss ~ 67 K for a magnetic moment = B Evaporative cooling to BEC Cooling in a magnetic trap by removing the hottest atoms and thermal equilibration (evaporative cooling) Typical samples > 5 4 atoms/cm 3 Temperature <K debroglie wavelength >m Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 38/56

20 Cold Atoms basic relations Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 39/56 Laser cooling MOT.5 cm Laser cooling requires low density to avoid light absorption Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 4/56

21 Recomended Literature Laser Cooling The Quantum Theory of Light; R. Laudon: Oxford Science Publications Laser Cooling and Trapping H. Metcalf, P. van der Straaten (Springer) Nobel prize lectures 997: The manipulation of neutral particles SCh S. Chu; Rev. Mod. Phys (998) Manipulating atoms with photons C. Cohen-Tannoudji: Rev. Mod. Phys (998) Laser cooling and trapping of neutral atoms W. Phillips: Rev. Mod. Phys. 7 7 (998) Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 4/56 Cold Atoms mechanical effects of light Scattering of a photon by an atom photon momentum: k atom momentum: k after excitation of atom: atom momentum: k after spontaneous decay of atom: momentum: k Mean force on atom: F dp dt p t k k mean atom I I I I ( L a kv ) typical forces on the atom can lead to accelerations of 4-6 m/s Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 4/56

22 Cold Atoms laser cooling Atom in counter propagating laser field: optical molasses Total Right Laser Left Laser Close to velocity zero: force is linear in velocity For a detuning F=-v laser atom (red from resonance) and the force is a damping force Heating due to randomness of the photon scattering typical temperature: t k B T=/ (Doppler limit) it) 4 K for =5 MHz Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 43/56 Magneto Optic Trap E. Raab et al. PRL 59 p63 (987) 3d magnetic field realization: Quadrupole Atoms are pushed to the point with B= Typical parameters: Density: > atoms/cm 3 Up to > atoms Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 44/56

23 How to collect ct cold atoms Problem: capture range of light forces is small (m/s) compared with the velocity of thermal atoms (5m/s) Slow atoms from thermal velocity Zeeman slower (tune the atom transitions with external magnetic fields) Collect the slow atoms out of the low energy tail of the Maxwell distribution Vapour cell MOT the 4 solid angle captured by the MOT compensates for the small number of slow atoms. Flux ~ v 3 for mv << kt Maxwell Distr: f(v) ~ v exp(-mv /kt) Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 45/56 Magnetic Trapping Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 46/56

24 Atoms in Magnetic Field Breit Rabi Formula for F=I+/ E A 4 E 4m I HFS F B ( F I, mf ) mf g K K B x x x g g J B K K B E E Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 47/56 E A( I ) Magnetic Trapping for : B Trappingpotential: potential: U Gauss Magnetic states: t U mag < U mag > U mag mag - B 67K ev high field seeking (attracted to maximum) low field seeking (attracted to minimum) Earnshaw Theorem: No maximum of a static field (combination of fields) in a source free region Magnetic traps are low field seeker traps, Atoms trapped in minimum of field but not in the ground state of potential (this would be a high field seeking state) Avoid Zeros in the field (Majorana transitions) Quadrupole trap has a zero in the field at the centre! Even at non zero weak field, there are Landau-Zener transitions possible. μ Rate for a harmonic minimum: e B ip πω e B ip field at minimum i... trap frequency quadrupole hexapole Time Orbiting Potential trap Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 48/56

25 Configurations with non-zero Field Minimum Baseball Trap Ioffe Pritchard Trap Other Realizations Clover Leaf Trap (MIT) Z-Wire Trap (Innsbruck/HD) Trapping field is created by superposition of the field of a current carrying Z-shaped Wire and a homogeneous bias field Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 49/56 Evaporative cooling Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 5/56

26 Evaporative Cooling basic principles Energy / position-selective removal of hot atoms by RF radiation, inducing spin-flips to untrapped states; E(r) = g F m F µ B B(r) E(m( F F)( )(r)) = ħω RF Rethermalisation to Maxwell- Boltzmann distribution leads to lower mean temperature ħ RF 87 Rb F=, m F T i T i T f T i Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 5/56 Evaporative Cooling H.F. Hess Phys.Rev.B 34 p R3476 (986) KB K.B. Davis, M-O. Mewes, W. Ketterle Appl.Phys.B 6, p55 (995) O.J.Luiten, M. Reynolds, J. Walraven Phys.Rev.A 53 p38 (996) General Scaling Evaporative cooling happens at exponential scale Time scale is given by the thermalisation (collision) time (it takes about 5 collision for a truncated distribution to thermalise) Characteristic quantities are logarithmic derivatives d (lnt ) A key parameter is d (ln N ) Evaporation is controlled by a potential depth kt (temperature decrease per particle loss) The density of states () of trap pot. determines scaling. For power law potentials: A pl square well:, harmonic:, linear: Ioffe-Pritchard trap: A IP ( +U IP ) Exponential scaling for a d-dimensional potential U(r) ~r d Number of atoms N d (lnt ) Temperature T with 3 d (ln N ) Volume V Density n average energy in potential Phase space density D <>= kt Elastic coll. Rate nv Run-away evaporation: Achieve faster and faster thermalsation with cooling Collision rate has to grow: Collisions: good collisions: bad collisions: elastic collisions inelastic coll., trap loss coll., etc limit the trap lifetime How many collisions per tapping time for run away evap.? RF induced evaporation Linear trap: > 5 collisions per lifetime Harmonic trap >5 collisions per lifetime Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 5/56

27 BEC in a Dilute System example Na Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 53/56 Roadmap to BEC with an atomic gas Laser cooling, magnetic trapping, evaporative cooling Nobelprize Nobelprize 997 Nobelprize 997 Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 54/56

28 Observing the BEC Methods of imaging: 3 processes: absorption, emission, shifting the phase 3 methods: absorptive, fluorescence, dispersive imaging Description: complex index of refraction n n i ref with 4 / Transmission T and phase shift ~ ~ ~ ~ D / exp( n n T e ) with D ~ ~ D n Imaging dense clouds (D >): Optimal absorption imaging i is at optical density of Need large detuning but there is refraction at > For diffraction limited imaging we need a phase shift < For optical density we need =(D ) / At this detuning the phase shift (D ) / much too large For < we need =D / Need phase contrast imaging Non destructive imaging: Example of an image: 3x3 pixel with photons each ( 5 photons) This can be non perturbative for large condensates. Important figure of merit: ratio signal / heating Absorption imaging: each photon gives one recoil energy Dispersive imaging: there are more forward scattered photons than absorbed photons for large detuning the gain is D /4! elastic scattered photons contribute not to heating if imaging is done in the trap and light pulse is longer that trap one can make many pictures of the condensate For low density clouds there is no advantage of dispersive imaging Lens setup (3m resolution) Spot diagram Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 55/56 Observing BEC phase transition, expansion Expansion man sieht den Grndzustand und in der Expansion erkennt man die Unschärferelation Kurze Zeiten: sieht x Lange Zeiten: sieht p kleines x -> großes p (schnelle Expansion) g p p phase transition großes x -> kleines p (langsame Expansion) ms 5 ms ms ms 3 ms 45 ms MIT, Sept. 95 Degenerate Quantum gases SS I. Mazets, J. Schmiedmayer Lecture I Folie: 56/56