BOSE-EINSTEIN CONDENSATION AND SUPERFLUIDTY

College de France, June 1st, 2016BOSE-EINSTEIN CONDENSATIONAND SUPERFLUIDTYSandro StringariUniversità di TrentoCNR-INO

Quantum gases and fluids:the ‘magic’ idity

This Lecture:Bose-Einstein Condensation with superfluidity(liquid He4, 3D Bose and Fermi gases)Superfluidity without Bose-Einstein condensation(2D superfluids)Bose-Einstein Condensation without superfluidity ?(spin-orbit coupled BECs)

Definition of Bose-Einstein condensationBose-Einstein condensation emerges from long rangebehavior of off-diagonal 1-body density matrix:n(1) (r , r ' )ˆ (r ) ˆ (r ' )r r' 2 0- It implies macroscopic occupation of single particle state eiSdescribed by complex order parameter- In uniform systems one finds n( p) N0 ( p) n ( p) withN 0 / N condensate fraction. In general momentumdistribution exhibits bimodal structure- BEC implies coherence phenomena associated with phaseof the order parameter (interference of matter waves)

Bose-Einstein condensation can begeneralized also to fermionic systemsLong range behavior of off-diagonal 2-body density matrixdefines the pairing field Flim rˆ (r r ) ˆ (r r ) ˆ (r ) ˆ (r )2112and the condensate fraction of pairsncond1ds F ( s ) 2n/2 F (r1 , r2 ) 2

Definition of superfluid densityNormal (non superfluid ) density is defined by static responseto transverse current operatorn1Q lim q0m,neEm m J xT (q) n 2E n Em(qq)iqykTJ(q)pewhere xis transverse current operator.k k ,x[Equivalent definition for S based on phase twist method]At T 0 normal density vanishes in Galilean invariantsuperfluids (liquid Helium, usual BEC and Fermi gases) andsystem is fully superfluid.(new surprises in spin-orbit coupled BECs, see later)Key difference withe respect to BEC fraction which is alwaysdifferent from 1 at T 0 in interacting systems, because ofquantum depletion (huge effect in liquid He4)

Bose-Einstein Condensation with superfluidity(liquid He4, 3D Bose and Fermi gases)

Measurement of BEC in ultracold atomic gases1996 MitN0 / N0coherence wave naturene i1995(Jila Mit)Macroscopicoccupationof sp state)N 0 (T ) / N1999Phase transition(Jila 1996)(Bloch et al)Long rangeorder

Bose-Einstein Condensation can be measured alsoin interacting Fermi gases with pairing

Fermi Superfluidity: the BEC-BCS Crossover(Eagles, Leggett, Nozieres, Schmitt.Rink, Randeria)Tuning the scattering length througha Feshbach resonanceBEC regime(molecules)BCS regime(Cooper pairs)unitary limitDilute Bose gas(size of molecules muchsmaller thaninterparticle distanceAt unitarity scattering lenghtis much larger thaninterparticle distance: stronglyinteracting superfluid

In a Fermi gas, Bose-Einstein condensation of pairs wasmeasured by ramping the scattering length to the BEC sideand detecting the resulting bimodal distributionKu et al. 2012Astrakharchicket al, 20051/a 0TT 0-1/kfaCondensation of pairs measured at unitarity (1/a 0) as afunction of temperature. Only 50% of pairs are Bose-Einsteincondensed at zero temperature (strongly interacting gas).Result agrees with predictions of MC simulations

What about Bose-Einstein condensationin superfluid He 4 ?

What about Bose-Einstein condensationin superfluid He 4 ?In He4 the condensate fraction is determined from the measurement ofmomentum distribution via neutron scattering at high momentum andenergy transfer.Impulse approximation permits to write the dynamic structure factor interms of the so called scaling function J(Y)S IA (q, )d p n( p ) (( p q)22mp2 mm)J (Y ) with Y(2m qqIn the presence of BEC the scaling functionshould exhibit a delta peak at Y 0.In practice final state interactions and finiteresolution effects smooth the behavior of J(Y).Sokol et al. were able to extract thecondensate fraction from such measurements.q2)2m

Condensate fraction in superfluid He4.Only 10% of atoms are condensed at T 0

Measurement of superfluidity(selection of superfluid phenomena)- Quantized vortices and solitons- Quenching of moment of inertia- Absence of viscosity(Landau critical velocity and supercurrents)- Lambda transition in specific heat(He4 and Fermi superfluids)- Hydrodynamic behavior (irrotationality of superfluid flow,collective oscillations, first and second sound)

Quantized VorticesBEC gasLiquid He4(Chevy et al. 2000)(Yarmchuck and Packard,1982)

Quantization of angular momentumand circulation (quantized vortices)BEC gas(Chevy et al. 2000)Liquid He4(Vinen, 1967)

Absence of viscosityLandau’s critical velocityin Fermi gas at unitarityFountain effect in LiquidHe4(MIT, 2007)Kapitza, Allen,Miesener 1937

Specific heat and Lambda transition: identification of TcTC0.167 TFFermi gas at unitarity(High Tc superfluid)(MIT, 2007)TC2.17 0 KSuperfluid transition inliquid He4

Can one measure the superfluid densityand its temperature dependence ?In liquid helium superfluid density isdetermined through the measurement ofthe moment of inertian(bucket experiment)rigand the velocityof second soundc221mTs 2nCPSTheoretical predictions, based on PIMCcalculations provide excellent agreementExp: from Dash and Taylor, 1957Theory: from Ceperley, 1995

What about ultra cold atomic gases ?In dilute 3D Bose gases superfluid density coincides in practice withcondensate fractionSituation is more interesting in 2D Bose gases where BEC is absent(see later). T-dependence of superfluid density not yet measured.In 3D interacting Fermi gases (at unitarity) temperature dependenceof superfluid density has been recently measured through the velocityof second sound (Innsbruck-Trento collaboration)

First and second sound in the unitary Fermi gasFirst soundpropagates also beyond theboundary between thesuperfluid and the normal partsSecond soundpropagates only within theregion of co-existence of thesuper and normal fluids.Second sound is basically anisobaric wave, but signal isvisbile because of small, butfinite thermal expansion.

From measurement of second sound velocity one extractstemperature dependence of superfluid density(first measurement in a Fermi superfluid)Sidorenkov et al., Nature 2013Superfluidhelium1 (T / TC )3 / 2ideal Bose gas

Superfluidity without Bose-Einstein condensation(2D superfluids)

- In two dimensions Hohenberg-Mermin-Wagner theorem rulesout long range order (and hence Bose-Einstein condensation)because of thermal fluctuations of the order parameter- System exhibits algebraic long range order below the criticaltemperature (Berezinskii-Kosterlitz-Thouless phase transition)- Algebraic long range order is enough to ensure coherence andsuperfluid phenomenaImportant relationship between BKT temperature and superfluiddensity at the transition (Nelson and Kosterlitz, 1977)2TBKT2k B2Sm2

- Bimodal velocity ditribution and coherence phenomena inuniform 2D Bose gases recently measured at College de France(Chomaz et al. Nature Comm. 2015)Bose-Einstein condensation isabsent, but velocitydistribution still exhibitsbimodal distributionCoherence of two overlappingcoplanar 2D Bose gase shows upin interference fringes

- Thermodynamic functions of 2D Bose gas were measured atENS (Yefsah et al., 2011). Excellent agreeement with theory(Prokfeev and Svistunov, 2002)- Reduced pressure2T 2P /Tvs dimensionless parameter/ k BT- In 2D thermodynamic fuctions (including specific heat) do notreveal any specific feature at the BKT temperature- Transport properties are requested in order to measurethe superfluid density

Critical velocity acrossthe BKT transitionDesbuquois et al.Nature Physics 8, 645 (2012)While in the normal phase theLandau’s critical velocity ispractically zero, below a criticaltemperature it exhibits a suddenjump to a finite valuerevealing the occurrence of aphase transition associated witha jump of the superfluiddensity

Measurement of temperature dependence of superfluiddensity, incuding jump at the BKT transition, could beprovided by measurement of second sound velocityKey features in 2DSuperfluid density andsecond sound velocityhave discontinuityat the BKT transition2nd soundOzawa and S.S. PRL 2014

Bose-Einstein Condensation without superfluidity ?The case of spin-orbit coupled BEC’s

Simplest realization of (1D) spin-orbit coupling in s 1/2Bose-Einstein condensates (Spielman, Nist, 2009)Two detuned and polarized laserbeams non linear Zeeman fieldprovide Raman transitionsbetween two spin states, givingrise to new s.p. Hamitonianh01[ p x k022z1p2 ]2xi xis canonical momentumk 0 is laser wave vector differenceis strength of Raman couplingpxSpin orbit Hamiltonian istranslationally invariant.However it breaksGalilean invariance(physical momentumPxmvx( p x k0z)does not commute with h0 )

For small values oftwo sp states canhost BEC with canonical momentumk1k0 12/ 4k04Transition between two phases(plane wave and zero-th momentum phase)is second order.It has been observed2at the predicted value2kc0of Raman couplingSpin polarizability divergesat the transition(theory: Martone et al. EPL 2012Exp: Zhang et al. PRL 2012)

Suppression of superfluidity inSOC Bose-Einstein condesed gasesCollaboration with Lev Pitaevskii (Trento)and Shizhong Zhang (Hong Kong)Yi-Cai Zhang et al. arXiv: 1605.02136

To calculate normal density at T 0n1lim qN0n 0 J xT (q) n 2En E0(qq)one needs to know spectrum of elementary excitations

To calculate normal density at T 0n1lim qNJ xT (q)0n 0 J xT (q) n 2En E0( p k , x k0k(qq)iqyk)ek ,zone needs to know spectrum of elementary excitationsSpinor BEC’s exhibit two branchesin the excitation spectrumDue to Raman couplingonly one branch is gaplessand exhibits phononbehavior at small qExp: Si-Cong Ji et al., PRL 2015;Khamehchi et al, PRA 2014Theory: Martone et al., PRA 2012gapped branchphonons

Phonon branch has longitudinal natureand cannot contribute to nContribution from gapped branch can be evaluatedin terms of energy weighted sum rulen1lim q2N1N 2 0 J xT (q) n 2 ( En00 [ J xT (qE0 ) (qq)n0), [ H , J xT (q0)]] 02k022xis q 0 value of energy gap-Only spin component of current J xT (q 0)contributes to energy weighted sum rule(canonical component commutes with H)k( p k , x k0- Values of andx are available in both plane waveand zero-momentum phase (Martone et al, PRA2012)k,z)

Results for normal densityin plane wave and zero-th momentum phasePlane wave phaseZero-momentum phasecn4(k02ck02 22G2 )3 2G2Cn22(k02 2G2 ) ; G22k024G2n( g g ) / 4At the transitionone finds nand superfluid densitySn identically vanishes !!parameters of Rb87

Superfluid density is strongly suppressed near the phasetransition between the plane wave and zero-momentum phaseBEC fraction is instead practically unperturbed (quantumdepletion always remains very small in 3D gas, less than 1%)Superfluid density(present work)Quantum depletion(W. Zheng et al. JPhysB 2013)At the transition:Bose-Einstein condensation without superfluidity !

Another example where suppression of superfluidity is moreimportant than quenching of BEC.This is the case of a weakly interacting BEC in the presenceof weak diroder. Bogoliubov theory predicts(Huang and Meng, PRL 1992; Giorgini et al. PRB 1994)nandn0nm21/ 2R(na)06 3/ 283with343 1/ 2(na )1/ 2Because of factor ¾SR0 m2 U k 2 / Vnn0if a is sufficently small.n

CAN WE MEASUREs?

- f-sum rule analysis. Differently from Galilean invariantsystems, f-sum rule is not exhausted by phonon branch.- Contribution from upper branch is given by Nq 2 n /Contribution from phonon branch is then Nq 2 s /- One then finds expressionsc2- In a Galilean invariant system c 2for superfluidity density1and hences- Compressibility is not modified by SO coupling.Sound velocity instead exhibits strong suppression nearc

- f-sum rule analysis. Differently from Galilean invariantsystems, f-sum rule is not exhausted by phonon branch.- Contribution from upper branch is given by Nq 2 n /Contribution from phonon branch is then Nq 2 s /- One then finds expressionsc2- In a Galilean invariant system c 2for superfluidity density1and hences- Compressibility is not modified by SO coupling.Sound velocity instead exhibits strong suppression nearc