1 INTERNATIONAL SCHOOL OF PHYSICS "ENRICO FERMI" Varenna, July 1st - July 11 th 2008 " QUANTUM COHERENCE IN SOLID STATE SYSTEMS " Introduction to Bose-Einstein condensation 4. STRONGLY INTERACTING ATOMIC FERMI GASES Sandro Stringari University of Trento CNR-INFM
2 Quantum statistics and temperature scales k 1/3 1/ 3 B TC = h! ( 0.83N) kb TF = h! ( 6N)
3 Bosons and Fermions are not separate worlds interaction
4 Interactions can be tuned thanks to availability of Feshbach resonance atoms collide in open channel at small energy same atoms in different hyperfine states form a bound state in closed channel coupling through hyperfine interactions between open and closed channel if open and closed channel have different magnetic moment magnetically tunable ΔE= Δ µ B Resonance when bound-state and continuum become degenerate a = a bg & $ 1+ % B ( ' B 0 #! "
5 Many-body aspects BEC regime unitary limit BCS regime
6 BCS theory (a<0) Theory of superfluidity for a < 0 and k F a << 1 well established (BCS theory). Key results for uniform gases: - Pairing between spin-up and spin-down atoms in momentum space (Cooper pairing) - BCS critical temperature: - BCS theory valid if k F a << 1 T C = 0.28T F e! / 2k (Gorkov, Melik-Barkhudarov, 1961) F a! 4 - Example kfa =! 0.2 " TBCS = 10 TF. BCS critical temperature is too low even smaller than oscillator temperature ( ho "10!2 5 h# T F ) with N!10 ).
7 BEC gas of molecules (a>0): calculation of T C Condition for having dilute molecular gas requires that distance between molecules be much larger than molecular size: d >> a Such a molecular gas behaves like a dilute gas of bosons and exhibits BEC Value of BEC critical temperature for a gas of bosons of mass 2m (molecules) directly related to value of Fermi energy of fermions of mass m. T BEC T F = 1/3 0.94h" hon! = h" ho( 6N! ) 1/3 TBEC = 0. 5T F Critical temperature for superfluidity is much higher in BEC than in BCS side where it is exponentially small. High Tc superfluidity
8 Experimental production of BEC molecules emerging from Fermi sea Two routes have become available (2003): - Cooling of two spin species Fermi gas with a<0. System reaches quantum degeneracy ( T! 0. 1T F ), not enough for BCS. Further cooling is inhibited by Pauli blocking. Scattering length is then tuned through Feshbach resonance until BEC molecules are observed (Jila, Ens)) - Cooling of two spin species Fermi gas with a>0. Molecules are formed, further cooling yields BEC of molecules (Innsbruck, Mit). JILA 2003: 40 K 2 Innsbruck 2003: 6 Li 2 :
9 Some questions concerning the BEC-BCS crossover - search for many-body theory describing the transition - interaction between molecules on BEC side - what happens at unitarity a >>1? (neither molecules, nor Cooper pairs) k F - can we probe superfluidity at the BEC-BCS crossover? - effects of spin polarization on superfluids?
10 Many body theory of the BEC-BCS crossover First theoretical approach developed by Leggett (1980), Nozieres and Schmitt-Rink (1985) generalizes gap equation of BCS theory to the whole resonance region (BCS mean field) Theory predicts (Randeria 1993): - critical temperature and equation of state as a function of dimensionless parameter k F a - formation of molecules with energy h 2 / ma 2 on the BEC side k F a << 1 - molecules on the BEC side interact with scattering length a M = 2a T C / T F 0.4 0.2 0 BEC -1 0 0 1-1/k F a BCS Important effort in recent few years to provide improved many-body schemes (Holland, Griffin, Timmermans, Strinati, Stoof.), including numerical quantum Monte Carlo approaches (Carlson, Giorgini, Prokofeev, Svistunov, Troyer)
11 Equation of state along the BEC-BCS crossover BCS mean field ideal Fermi gas Nnnn Monte Carlo (Astrakharchick et al., 2004) µ < 0 BEC BCS µ > 0 Energy is always smaller than ideal Fermi gas value. Attractive role of interaction along BCS-BEC crossover
12 Critical temperature along the BEC-BCS crossover BEC (Randeria et al. 1993 extension of BCS mf theory to finite T) BCS QMC simulation (Burovskii, Prokofeev, Svistunov and Troyer 2006) QMC result for critical temperature at unitarity recently confirmed by MIT experiment with polarized samples!
13 Behaviour near unitarity At unitarity ( k F a >>1 ) system is strongly correlated properties do not depend on value (even sign) of scattering length a UNIVERSALITY ( k r <<1 F 0 ) All lengths disappear from the calculation of thermodynamic functions. (Bertsch 2002) 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): 2 2/3 2/ 3 ( 6" ) (1! ) n 2 µ = h + 2m " = 0 - ideal Fermi gas "! 0 - at unitarity Many-body calculation needed to determine value of! ("!0. 6 ).! is negative, reflecting attractive role of interaction. Equation of state can be used to determine density profiles, release energy and collective frequencies in Thomas-Fermi approximation.
14! -dependence of radii at unitarity Equilibrium profile in local density approximation (isotropic trapping for simplicity) 2 h 2m 2 2/3 2/3 1 2 2 ( 6# ) (1 + " ) n + m! r = µ 2 ho Same result for density profile as for ideal gas with replacement " " / 1+!, µ # µ /(1 +! ) ho # ho In ideal gas 1/ 6 R = a (48N), a = h / m! ideal ho ho ho R = R ideal ( 1+! ) 1/ 4 Simple rescaling for Thomas-Fermi radii with respect to ideal gas prediction!!
15 Measurement of in situ column density: role of interactions (Innsbruck, Bartenstein et al. 2004) non interacting Fermi gas # "!0.7 BEC BCS More accurate test of equation of state and of superfluidity available from study of collective oscillations (next lectures)
16 The quest for superfluidity in Fermi gases Experimental probe of superfluidity in trapped Fermi gases - expansion and aspect ratio - collective oscillations - pairing gap - thermodynamic functions - quantized vortices - spin polarization
17 Collective oscillations and hydrodynamic theory In linear regime ( n = neq +! n ) hydrodynamic equations take the form " " t 2 2! n = # ( n 0 " µ #(! n)) " n Solutions of HD equations predict surface and compression modes Surface modes -Surface modes are unaffected by equation of state.! = l l! ho - For isotropic trap one finds where is angular momentum - surface mode is driven by external potential, not by surface tension! = l! ho - Dispersion law differs from ideal gas value (interaction effect)
18 Compression modes - Sensitive to the equation of state - analytic solutions for collective frequencies available for polytropic equation of state! [at unitarity (1/a=0)! = 2 / 3 ] µ " n - Example: radial compression mode in cigar trap - At unitarity one predicts universal value - For a BEC one finds " = 2"! 10 " = "! 3 - First correction on BEC side due to Lee-Huang-Yang (LHY) eq. of state 2 h & 32 3 1/ 2 # µ + = g $ 1+ ( ) +... 2 M n na M 2ma! % 3 ' " 105! " = 2" #[1 + 256 a 3 M n (0)] beyond mean field effect (Pitaevskii, Stringari 1998)
19 Radial breathing mode at Innsbruck (2006) MC equation of state (Astrakharchick et al., 2005) includes LHY effect does not include LHY effects BCS mean field (Hu et al., 2004) 10 / 3 = 1.83 Measurement of collective frequencies provides accurate test of equation of state!!
20 Quantized vortices in Fermi gases observed along the BEC-BCS crossover (MIT, Nature June 2005, Zwierlein et al.) Scattering length is suddenly ramped to small and positive values in order to increase visibility of vortex lines
21 Spin polarized Fermi superfluids N " # N! Differently from BEC s phase separation is not easily observed by imaging density profiles of Fermi gas (bimodal distribution is absent at unitarity as well as in BCS Phase separation can be nevertheless observed in spin polarized samples
22 Occurrence of phase separation in spin polarized Fermi gas observed experimentally at unitarity (see also Rice exp) Density difference n # n "! (phase contrast imaging, MIT 2006) In superfluid phase n " = n! In polarized normal phase n " > n!
23 Phase diagram of uniform matter at T=0 P = N N " " # + N N!! FFLO phase C l o g s t o Chandrasekher-Clogston limit at unitarity Interactions in normal phase play a crucial role in determining critical polarization. Example: neglecting interactions in normal phase yields!1 P C P C = 0.39
24 Density jump at the interface Spin up density practically continuous at the interface Exp: MIT (Shin et al. 2007) Theory: Trento (Lobo et al.1996) Spin down density exhibits jump at the interface Based on MC equations of state for superfluid and polarized normal phase Theory predicts critical polarization in excellent agreement with exps