ULTRACOLD FERMI ALKALI GASES: BOSE CONDENSATION MEETS COOPER PAIRING

Transcription

1 AIP 0 ULTRACOLD FERMI ALKALI GASES: BOSE CODESATIO MEETS COOPER PAIRIG Anthony J. Leggett Department of Physcs Unversty of Illnos at Urbana-Champagn Urbana, IL

2 AIP 1 fracton of condensed partcles ~1 ~ T c /T F á 1 ~ T F ~ T F

3 A UIFYIG COCEPT: ODLRO (Penrose-Onsager, Yang) AIP Consder a general system of ndstngushable partcles (bosons or fermons) occupyng -partcle statesψ n( r1σ 1, rσ... rσ) wth probablty p n. Defne: spn may be absent (0) (a) Sngle-partcle reduced densty matrx (RDM) ρ (rσ, r σ ) dr... dr n σ... σ p Ψ ( rσ, rσ... r σ ) Ψ ( rσ, r σ... r σ ) * n n 1 1 n 1 1 Can dagonalze: ρ ( rσ, rσ ) = n χ ( rσ ) χ ( r σ ) * For bosons, can have n0 ~ 0 (condensate) (b) -partcle RDM: ρ ( rσ, rσ : r σ, r σ ) dr... dr n σ... σ = For bosons or fermons, can have n0 ~ 0 3 p Ψ ( rσ, rσ, rσ... r σ ) Ψ ( r σ, r σ, rσ... r σ ) * n n n n χ σ σ χ σ σ * ( r1 1, r ) ( r1 1, r )

4 AIP 3

5 AIP 4 (very cold!) atoms n dfferent nternal (hyperfne) states possblty of relatve s-wave

6 AIP 5

7 AIP 6 ε ( δ ) F c

8 AIP 7 The problem: fermons, equal nos. and, 3 m tot = k 3π F Ĥ = ( / ) subject to b.c. Ψ ~ const. (1-a s /r j ) for antparallel-spn partcles, j (n dlute lmt, parallel-spn partcles nonnteractng) All (equlbrum) props. must be functons only of ζ = 1/k F a S aïve Ansatz (Eagles 1969, AJL 1980, Randera et al. 1985, Stajc et al ): { ϕ (r1 r : σσ 1 ) ϕ (r3 r 4 : σσ 3 4)... ϕ (r 1 r : σ 1σ )} Ψ = A : : : Ψ H ˆ Ψ = : 1. Parng terms fully taken nto account. Fock terms vansh n dlute lmt 3. Hartree terms?? equvalently: each term of Ψ (naïve) satsfes b.c. for pared partcles only, e.g. 1 st term satsfes t for 1, but not (e.g.) for 1, 3. Output of naïve ansatz: μζ ( ), Δ( ζ) Hence also (E/)( ζ ). (calc n analytc except for D numercal ntegrals)

9 AIP ~Δ/E F

10 AIP 9

11 AIP 10 Some Experments on the BEC-BCS Crossover (mostly 6 L : some 40 K) (s-wave, unpolarzed) EXPERIMET n-stu magng magng after expanson SHOWS/MEASURES (fermonc statstcs) Lfetmes of atoms + molecules Collectve exctatons n trap Sound velocty Specfc heat MR (ESR) Feld sweep (fermonc statstcs) crossover thermodynamcs 3-flud model energy gap (on both sdes of untarty) parng on BCS sde Persstence of vortcty under BEC BCS BEC sweep parng on BCS sde Optcal absorpton onzero closed-channel compt. (on both sdes of untarty) All these experments appear qualtatvely consstent wth naïve ansatz.

12 AIP 11 A SIMPLIFYIG COSIDERATIO I UDERSTADIG (SOME OF) THE EXPERIMETS: DECOUPLIG OF - PARTICLE AD MAY-BODY EFFECTS* Consder a general quantty of the form 1 Ω ΣS( r rj : σ σ j) j wth the range of Sr () r (Exx: potental energy, closed-channel fracton, 1 st 0. moment of ESR spectrum). Intutvely, Ω should depend only on the prob. of fndng two atoms wthn < r 0 of one another. Formally: * ρ ( rσ rσ : r σ rσ ) =Σnχ ( rσ rσ ) χ ( r σ rσ ) then Ω =Σn Σ drdr S( r r : σ σ ) χ ( rr, σ σ ) σσ However, n the lmt kr 0 1 the functonal form of χ F ( rr 1 σσ 1 ) at dstances r1 r < r0 s smply that of the -partcle (free-space) wave functon, and the only dependence on s through the normalzaton. So, wrtng χ ( rr 1σσ 1 r r < r ) C χfs( r1 r : σσ 1 ) approprately 1 0 we can wrte Ω = h( ξ, τ) ϕ 1 ϕ Σ dr S( r : σ σ ) χ ( r : σ σ ) Ω σσ 1 fs 1 h( ζτ, ) Σn( ζτ, ) C ( ζτ, ) Ω many-body effects normalzed -p w.f. *S. Zhang and A. J. Leggett, Phys. Rev. A 79, (009): cf. S. Tan, Ann. Phys. (Y) 33, 95 (008) ncorpo rates -body quantty ALL

13 Some obvous questons: AIP 1 1a. Statcs (T=0): how good s naïve ansatz? In partcular, at untarty have smple problem: (Bertsch) Mn. e. v. of Ĥ = m subject to b.c. Ψ rσ r σ... r σ ~r j whenever r 0 for σ σ. j j On dmensonal grounds, E / = AE = 3 ε FG 5 F Δ= BE FG Chang + Pandharpande: Jastrow-BCS ansantz, Π j { } Ψ= f(r ) Ψ r,σ j BCS accomodates Hartree affect CP ave Expt A ± ± ± ± 0 05 B

14 AIP 13 1b. More questons on statcs: Behavor of Crossover n (ζ, T) Plane dssocaton tetracrtcal pont T F μ = T c Tc (ζ)(onset of ODLRO) T π 1 ~ TFexp k F a s δ $64K queston: how to go beyond the naïve ansatz? (and why does t seem to be qualtatvely correct?) rgorous upper lmt on T c? On (T)/ ρ (T)? o s Other questons: Dynamcs, knetcs...

15 AIP 14 SOME GEERALIZATIOS A. S-wave parng, unequal spn populatons Effect of magnetc feld on parng n neutral superconductor μbh (Clogston, Chandrasekhar, Mak and Tsuzuk...) Effect observed, n real superconductor, by Messner effect (and small polarzablty) Δ Δ / Δ / supercoolng T thermodynamc superheatng Experments on 6 L wth unequal spn populatons (separate detecton of speces) phase separaton nto pure pared regons and normal (nonzero-spn) regons profles sometmes nonmonotonc crtcal polarzaton for parng at untarty 70% Fully polarzed system descrbed by nonnteractng Ferm sea (for k F r 0 á1). What s MBWF for a sngle reversed spn?

16 AIP 15 GEERALIZATIOS (cont.) 0 B. The case 1. Qualtatve dfference from s-wave case: (-body prob). In s-wave case, general E=0 soluton outsde potental s Ψ ( r ) = 1 a s /r and n partcular, at untarty, Ψ ( r ) ~r 1 n manybody cases expect strong 3, body nteracton effects. In 0 case, c ~ r ( ) suggests untary lmt may be (almost) trval n 1/ 3 lm r a n! o. The angular momentum problem: Ψ r In BEC of tghtly bound 0 datonc modules, overwhelmngly plausble that L = ˆ What s stuaton n BCS lmt? Most obvous number-conservng ansatz: ( ) / + + Ψ~ Σca a, c υ /u k k k k k k k wth (e.g.) c~ exp ϕ k k. Ths has L = ˆ just as n BEC lmt, rrespectve of magn. of Δ. Problem: macroscopc dscontnuty at transton to normal state L = 0! ( )

17 ULTRACOLD FERMI ALKALI GASES: SOME APPLICATIOS AIP Smulaton of other systems (nuclear matter, quark-gluon plasma, exctons ) when parameters not adjustable : none of these s n dlute lmt kr F Smulaton of specfc models: case of most nterest s D Hubbard model (beleved by many to descrbe cuprate superconductors) Ths s a lattce model: Hˆ = t a + a + U n n j= nn To smulate, need optcal lattce: j U t U/t tunable va V 0 or va Feshbach resonance. : may not be model of real cuprate. 3. Topologcal quantum computng: requres p-wave parng (Feshbach resonance?) Accordng to standard pcture, a vortex n a (sngle-spncomponent) p-wave Ferm superflud can accommodate Majorana fermons, whch behave as nonabelan anyons and can thus be used for TQC. : s standard pcture correct? V 0

18 AIP 17 SOME QUESTIOS ABOUT THE ESTABLISHED WISDOM 1. ature of MBWF of (p + p) Ferm superflud Recap: standard ansatz s (for say ÆÆ) ( ) / + + Ψ ~ caa vac, c ~expϕ k k k k k k.e. all pars of states n Ferm sea have anyon momentum. Alternatve ansatz: frst shot: Ψ(, ) P h ~Δ / p + + ~ caa k k k, k> kf unchanged from E F Δ: k< k F da a k k k h / vac keeps pp pp and hh hh, but not (e.g.) pp hh. Remedy: Ψ ~ Q Ψ (, ), p, K p k p k Q slowly varyng as f(, ) degenerate wth standard ansatz to 0( 1/ ), but L~( /) ( Δ / E F ) p k IS GS OF (p + p) UIQUE?