Theory of interacting Bose and Fermi gases in traps

1. Crete, July 2007 Summer School on Bose-Einstein Condensation Theory of interacting Bose and Fermi gases in traps Sandro Stringari 1st lecture Role of the order parameter University of Trento CNR-INFM

2. Quantum statistics and temperature scales

3. Bosons • When T tends to 0 a macroscopic fraction of bosons • occupies a single particle state (BEC) • Wave function of macroscopically occupied single particle state • defines order parameter • Actual form of order parameter depends on two-body interaction • (Gross-Pitaevskii equation) Fermions • - In the absence of interactions the physics of fermions deeply • differs from the one of bosons (consequence of Pauli principle) • Interactions can change the scenario in a drastic way: • - pairs of atoms can form a bound state (molecule) and give rise to BEC • - pairing can affect the many-body physics also in the absence • of two-body molecular formation (many-body or Cooper pairing) • giving rise toBCS superfluidity

4. First lecture Theory of order parameter for both Bose and Fermi gases. Microscopic nature of order of parameter (and corresponding equations) very different in the two cases Second lecture Unifying approach to dynamics of interacting Bose and Fermi gases in the superfluid regime. Structure of equations of superfluid dynamics (irrotational hydrodynamics) in the macroscopic regime is the same for fermions and bosons

5. Bosons 1-body density matrix and long-range order (Bose field operators) Relevant observables related to 1-body density: - Density: - Momentum distribution: In uniform systems

6. Bosons Long range order and eigenvalues of density matrix BEC occurs when . It is then convenient to rewrite density matrix by separating contribution arising from condensate: For large N the sum can be replaced by integral which tends to zero at large distances. Viceversa contribution from condensate remains finite up to distances fixed by size of BEC and long range order: consequence of macroscopic occupation of a single-partice state ( ) . Procedure holds also in non uniform and in strongly interacting systems .

7. Bosons In bulk matter Off-diagonal long range order (Landau, Lifschitz, Penrose, Onsager) or Example of calculation of density matrix in strongly correlated superfluid: liquid He4 (Ceperley, Pollock 1987)

8. Bosons ORDER PARAMETER Diagonalization of 1-body density matrix permits to identify single particle wave functions . In terms of these functions one can write field operator in the form: If (BEC) one can use Bolgoliubov approximation (non commutativity unimportant for most physical properties within 1/N approximation). Order parameter (gauge symmetry breaking) Quantum and thermal fluctuations

9. Bosons Dilute Bose gas at T=0 Basic assumption: Almost all the particles occupy a single particle state (no quantum depletion; no thermal depletion) Field operator can be safely replaced by classical field Density coincides with condensate density Many-body Hamiltonian Zero range potential a =s-wave scattering length

10. Bosons Energy can be written in the form Variational procedure yields equation for order parameter (Gross-Pitaevskii, 1961) Conditions for applicability of Gross-Pitaevskii equation • diluteness: (quantum fluctuations negligible) • low temperature (thermal fluctuations negligble)

11. Bosons • Gross-Pitaevskii (GP) equation for order parameter plays role • analogous to Maxwell equations in classical electrodynamics. • Condensate wave function represents classical limit of • de Broglie wave (corpuscolar nature of matter no longer important) Important difference with respect to Maxwell equations: GP contains Planck constant explicitly. Follows from different dispersion law of photons and atoms: from particles to waves: photons atoms particle (energy) wave (frequency) GP eq. is non linear (analogy with non linear optics) GP equation often called non linear “Schroedinger equation” Equation for order parameteris not equation for wave function

12. Bosons BEC in harmonic trap Non interacting ground state depends on Gaussian with width Role of interactions Using and as units of lengths and energy, and GP equation becomes normalized to 1 dimensionless Thomas-Fermi parameter If Non interacting ground state If Thomas-Fermi limit (a>0)

13. Bosons In Thomas Fermi limit kinetic energy can be ignored and density profile takes the form (for n>0) Does not depend on Thomas-Fermi radius R is fixed by condition of vanishing density with fixed by normalization. One finds Thomas-Fermi condition implies

14. Bosons Some conclusions concerning equilibrium profiles a >0 non interacting Thomas-Fermi parameter drives the transition from non interacting to Thomas-Fermi limit wave function column density non interacting Huge effects due to interaction at equilibrium; good agreement with experiments GP exp: Hau et al, 1998

15. Bosons Thomas-Fermi regime is compatible with diluteness condition Gas parameter in the center of the trap Thomas-Fermi Diluteness example: Gross-Pitaevskii theory is not perturbative even if gas is dilute (role of BEC)!

16. Fermions Microscopic approach to superfluid phase is much more difficult in Fermi than in Bose gas (role of the interaction and of single particle excitations is crucial to derive equation for the order parameter) Order parameter is proportional to (pairing !!) rather than to Fermi field operator Equation for order parameter follows from proper diagonalization of many body Hamiltonian. • - Interaction at short distances is active only in the presence of two spin • species (consequence of Pauli principle) • ( ) regularized potential • (Huang and Yang 1957) • (needed to cure ultraviolet divergencies, arising from 2-body problem)

17. Fermions Many-body Hamiltonian can be diagonalized if one treats pairing correlations at the mean field level. Order parameter (Bogoliubov - de Gennes Eqs.) • Mean field Hamiltonian is bi-linear • in the field operators • can be diagonalized by Bogoliubov • transformation which transforms • particle into quasi particle operators

18. Fermions Diagonalization is analytic in uniform matter. Hamiltonian takes the form of Hamiltonian of a gas of independent quasi-particles with energy spectrum Coupled equations for and are obtained by imposing self-consistency condition for pairing field F(s) and value of density: T=0 + extensions to finite T: Eagles (1969) Leggett (1980) Nozieres and Schmitt-Rink (1985) Randeira (1993) BCS mean field equations

19. Fermions What is BCS mean field theory useful for ? Provides prediction for equation of state and hence for compressibility - Predicts gapped quasi-particle excitation spectrum - According to Landau’s criterion for critical velocity occurrence of gap implies superfluidity (absence of viscosity and existence of persistent currents) Key role plaid by order parameter !! Results for uniform matter can be used in trapped gases using LDA

20. Fermions When expressed in units of Fermi energy Equation of state, order parameter and excitation spectrum depend on dimensionless combination This feature is not restricted to BCS mean field, but holds in general for broad resonances where the scattering length is the only interaction parameter determining the macroscopic properties of the gas Holds if scattering length is much larger than effective range of the potential Scattering length a iskey interaction parameter of the theory: Determined by solution of Schrodinger equation for the two-body problem

21. Fermions In the presence of Feshbach resonance the value of a can be tuned by adjusting the external magnetic field At resonance a becomes infinite When scattering length is positive weakly bound molecules of size a and binding energy are formed If size of molecules is much smaller than average distance between molecules the gas is a BEC gas of molecules In opposite regime of small small and negative values of a size of pairs is larger than interparticle distance (Cooper pairs, BCS regime)

22. Fermions Some key predictions : • BEC regime ( ) • - Chemical potential (gas of independent molecules) • Single particle gap (energy needed to break a molecule) • BCS regime ( ) • - Chemical potential ( weakly interacting Fermi gas) • Single particle gap • (Gap coincides with order parameter • and is exponentially small)

23. Fermions Many-body aspects (BEC-BCS crossover) BEC regime BCS regime unitary limit

24. 2003: Molecular Condensates Fermions JILA: 40K2 MIT 6Li2 6Li2:Innsbruck ENS 6Li2 6Li2 7Li Also Rice 6Li2

25. Fermions • - Basic many body features well accounted for by BCS mean field theory. • - However BCS mean field is approximate and misses important features • For example: on BEC side of resonance this theory correctly describes gas • of molecules with binding energy . • However these molecules interact with wrong scattering length • correct value is Petrov et al, 2004)) Equation of state predicted by BCS mean field is approximate. - Exact many-body calculatons of equation of state are now available along the whole BEC-BCS crossover using Quantum Monte Carlo techniques (Carlson et. al; Giorgini et al 2003-2004)) - QMC calculations gives also access to gap parameter.

26. Fermions Equation of state along the BEC-BCS crossover BCS mean field ideal Fermi gas Nnnn Monte Carlo (Astrakharchick et al., 2004) BEC BCS Energy is always smaller than ideal Fermi gas value. Attractive role of interaction along BCS-BEC crossover

27. Fermions • - Behavior of equation of state is much richer than in dilute Bose gases where • (Bogoliubov equation of state) • Possibility of exploring both positive and negative values of scattering • length including unitary regime where scattering length takes infinite value

28. Fermions Behaviour at resonance (unitarity) - At resonance the system is strongly correlated but its properties do not depend on value of scattering length a (independent even of sign of a). UNIVERSALITY. - UNIVERSALITY requires (dilute, but strongly interacting system) All lengths disappear from the calculation of thermodynamic functions (similar regime in neutron stars) Example: T=0 equation of state of uniform gas should exhibit same density dependence as ideal Fermi gas (argument of dimensionality rules out different dependence): Atomic chemical potential for ideal Fermi gas Values of beta: Mean field -0.4 Monte Carlo: -0.6 dimensionless interaction parameter characterizing unitary regime

29. Fermions Equation of state can be used to calculate density profiles using Local density approximation: For example at unitarity • - From measurement of density profiles one can determine value • of interaction parameter • Value of measurable also from release energy (ENS 2004) • and sound velocity (Duke 2006) (see next lecture)

30. Fermions Measurement of in situ column density: role of interactions (Innsbruck, Bartenstein et al. 2004) non interacting Fermi gas BEC BCS More accurate test of equation of state and of superfluidity available from study of collective oscillations (next lecture)

31. Summary: role of order parameter in superfluids Key parameter of theory (Gross-Pitaevskii eqs. for BEC ) (Bogoliubov de Gennes eqs. for Fermi superfluids ) Directly related to basic features of superfluids: - density profiles in dilute BEC gases (easily measured) - gap parameter in Fermi superfluids (relevant for Landau’s criterion of superfluidity, measurable with rf transitions ?) In both Bose and Fermi superfluids order parameter is a complex quantity. (modulus + phase). This lecture mainly concerned with equilibrum configurations where order parameter is real Phase of order parameter plays crucial role in the theory of superfluids: - accounts for coherence phenomena (interference) - determines superfluid velocity field: important for quantized vortices, solitons and dynamic equations (next lecture)

32. General reviews on BEC and Fermi superfluidity • - Theory of Bose-Einstein Condensation in trapped gases • F. Dalfovo et al., Rev. Mod. Phys. 71, 463 (1999) • Bose-Einstein Condensation in Dilute Gases • C. Pethick and H. Smith (Cambridge 2001) • - A. Leggett, Rev. Mod. Phys. 73, 333 (2001) • Bose-Einstein Condensation • L. Pitaevskii and S. Stringari (Oxford 2003 • Ultracold Fermi gases • Proccedings of 2006 Varenna Summer School • W. Ketterle, M. Inguscio and Ch. Salomon (in press) • - Theory of Ultracold Fermi gases • S. Giorgini et al. cond-mat/0706.3360