MOMENT OF INERTIA AND SUPERFLUIDITY

Transcription

1 1 Chaire Européenne du College de France (004/005) Sandro Stringari Lecture 6 1 Mar 05 MOMENT OF INERTIA AND SUPERFLUIDITY Previous lecture: BEC in low diensions - Theores on long range order. Algebraic deca in low D. - Mean field and beond ean field. - Collective oscillations in 1D gas. This lecture. Irrotational vs rotatational flow. Moent of inertia and scissors ode. Epansion of rotating BEC.

2 Rotating Bose-Einstein condensates (low angular velocities) Superfluids rotate ver differentl fro classical fluids In classical fluid, due to viscosit, the velocit field of stead rotation is given b the rigid value v ( r) Ω r and is characterized b unifor vorticit v(r) Ω Superfluids are characterized b irrotationalit constraint (lecture ) consequence of the phase of order paraeter ( Ψ n0 ep is ), ielding irrotational value v( r) ( h / ) S v( r) vorticit is hence vanishing ( ) ecept along lines of singularit (vorticites, see lecture 7). 0 Difference in velocit field shows up in observable quantities

3 3 Moent of inertia When the container (bucket in heliu, agnetic or optical trap In atoic gases) is put in rotation with angular velocit Ω the sste acquires angular oentu L z ( Ω) drn( r) r v Moent of inertia is defined b relationship If the sste rotates classicall ( v Ω r ) one finds rigid value of oent of inertia Θ ΩΘ Superfluid ehibits different behaviour: - at low angular velocit has L z 0 - at higher Ω ehibits jups in angular oentu due to vortices L z rig N rig < r Θ > Lz Ω rigid value Angular oentu vs angular velocit easured in heliu (Hess and Fairbank, 1967)

4 4 Moent of inertia of a trapped gas Questions: - can we evaluate Θ in a trapped gas? - what is the role of the roughness of the trap (needed to transfer angular oentu) -can weeasure Θ and probe superfluidit? Soe answers: - calculation of Θ in - ideal gas (role of statistics, teperature) - hdrodnaic theor of superfluids at T0 - eperiental inforation fro - scissors ode, - epansion of a rotating gas,

5 5 Moent of inertia of haronicall trapped quantu gas Moent of inertia can be regarded as linear response to static perturbation H. pert ΩL z Perturbation theor then provides result Lz n L E z Θ e β Q e n n, Ω Q En E with n, E n eigenstates and eigenvalues of unperturbed Hailtonian H - Response function can be calculated if H is not ai-setric ( [ H, L z ] 0 ). - Deforation of haronic trap provides natural setr breaking ( esosocpic roughening ) : V ho ( + + z ) z β E n [ H, Lz ] i( ) i i i 0

6 6 Moent of inertia is obtained b solution of L z X H ], [ ielding ( ) L n n X X n n L e Q E E L n e Q z z n E n z n E 1,, Θ β β copleteness relation ], [ X L z Θ ho sp V p H + Algebraic solution for X available with Hailtonian [ ] p p i X i i i i i / ) ( ) ( + + h ielding ( ) ) ( ) ( rig Θ Θ holds for Bose and Feri statistics. An T Result for oent of inertia adits well defined liit when

7 7 Moent of inertia of ideal Bose gas For a Bose gas above critical teperature or for Feri gas one can use seiclassical result for square radii: /, 1/ 1 Moent of inertia takes rigid value Θ Θ rig For a Bose gas at T0 one instead has 1/, 1/ Moent of inertia takes irrotational value Θ δ where Θ rig δ + is deforation of the condensate. Vanishes for ai-setric configuration (superfluidit) Calculation is easil etended to finite T (Stringari, 1996) condensate theral coponent Θ Θ rig δ r r 0 0 N N r r T T N N T T

8 8 Moent of inertia in therodnaic liit For fied value of T /T c one has as N (see Lecture 4) < r < r > > 0 T T T C 1 N 1/3 0 Hence Θ Θ rig also below critical teperature When N, oent of inertia of ideal gas is deterined b theral coponent even if fraction of atos in the condensate is finite Interactions stabilize the ratio < r > T < r > 0 T ( n(0) a 1/ 6 and hence provide finite reduction of oent of inertia for an T below critical teperature C T 3 ) N 5 10 N ideal gas therodnaic liit with interaction

9 9 Irrotational hdrodnaics and T0 value of oent of inertia Role of interactions in a superfluid at T0 can be investigated using equations of irrotational hdrodnaics (Lecture ) It is convenient to write HD equations in rotating frae where rotating trap is at rest and one can look for stead solutions. Equations are derived using Hailtonian H ΩLz H Ω r v t n + [ n( v t v S + ( S 1 Ω r)] v S 0 + µ ( n) + V et v S Ω r) 0 where is superfluid velocit (irrotational) in lab frae v S S h

10 10 For haronic trapping the hdrodnaic equations adit stationar solutions of the for v S α α δω Equation of continuit ields relationship where δ + is deforation of the condensate. Angular oentu is given b L Ω ( ) drn( r) r δ ΘrigΩ z v S Irrotationalit of oent of inertia follows fro irrotationalit of superfluid otion

11 11 Scissors ode Direct easureent of oent of inertia is difficult because iages of atoic cloud probe densit distribution (not velocit distribution) In defored traps rotation is however coupled to densit oscillations. Eact relation, holding also in the presence of -bod forces: [ H, L ] i( ) z i i i angular oentu quadrupole operator Response to transverse probe easurable thorugh densit response function!! Eaple of coupling is provided b scissor ode. If confining (defored) trap is suddenl rotated b angle θ the gas is no longer in equilibriu. Behaviour of resulting oscillation depends cruciall on value of oent of inertia (irrotational vs rigid)

12 1 Qualitative estiate of scissors frequenc (role of oent of inertia) K Θ deforation of haronic trap Restoring force K ε is proportional to (no energ cost for setric trap) Mass paraeter Θ is given b Θ rig δ Θ rig δ ε - irrotational value in superfluid phase ( ) - rigid value in non superfluid phase ε 0 As scissors frequenc - approaches finite value in superlfuid - vanishes in non superfluid phase

13 13 Scissors frequencies (Guer-Odelin and Stringari, 1999) Superfluid (T0) With the irrotational ansatz v S α ( t ) one finds eact solution of HD equations. If trap is defored ( ) the solution corresponds to rotation of the gas around the principal ais in, plane + Result is independent of equation of state (surface ode) Noral gas (above T C ). Gas is dilute and interactions can be ignored (collisionless regie). Ecitations are provided b ideal gas Hailtonian. Two frequencies: Differentl fro superfluid the noral gas ehibits low frequenc ode ε (crucial to ensure rigid value of oent of inertia) ± ±

14 14 Scissors easured at Oford (Marago et al, PRL 84, 056 (000)) T C Above (noral) odes: ± ± T C Below (superfluid) : single ode: +

15 15 Scissors and superfluidit Is the easureent of the scissors ode at hdrodnaic frequenc a proof of superfluidit? + - If noral gas is in collisional regie its dnaics is governed b sae hdrodnaic equations as in the superfluid. - In this case stud of scissors ode does not perit to distinguish between superfluid and non superfluid regies. - Question relevant in Feri gases near Feshbach resonance (Lecture 8) In general to eploit superfluidit one should stud collective oscillations in the presence of rotating trap! In this case superfluid and noral gas behave differentl even if noral gas is in collisional regie.

16 16 Irrotational vs rotational flow In the presence of rotating trap the stationar velocit field behaves differentl depending on whether the sste is superfluid or noral. In a superfluid the velocit field is subject to the constraint of Irrotationalit : v α Rotating Anisotropic Trap V M M ( + + z ) [ ( 1+ ε) + ( ε) + z ] et z 1 Rigid rotation v0 Ω r : z ( ) : Irrotational flow v0 α A noral gas, in stead configuration, instead rotates in rigid wa v Ω r

17 17 Tie needed to achieve rigid rotation in non superflid phase - In the absence of viscosit the rotating trap will never be able to transfer angular oentu to the noral gas and to generate rigid rotation. - Tie needed to spin up the noral gas can be calculated b solving Boltzann equations. - In collisional regie τ < 1 one finds (Guer-Odelin, 000) τ τ up 1/( ε τ ) where is average collisional tie. τ up - Tie becoes large if - deforation of trap is sall - sste is too deepl in hdrodnaic regie ( τ <<1 )

18 18 Ecitation of scissors ode with rotating trap What happens to the cloud if we suddenl stop the rotation Ω of the confining trap? Sste will be no longer in equilibriu and will start oscillate (scissors ode). If the gas is superfluid we ecite the scissors ode according to the predictions irrotational hdrodnaics + If the gas is noral and collisional, the scissors ode will be described b the equations of rotational hdrodnaics (in laborator frae) 1 P v + ( v + Vet ) + v v t n ter absent in superfluid HD

19 19 Scissors ode (after stopping rotation of the trap) Superfluid (T0) + Noral (collisional) beating between + ± Ω Noral (collisionless) beating between ± Ω 0., ε 0.

20 0 Epansion of a rotating superfluid gas : consequences of irrotationalit In the absence of rotation the epansion of a cigar condensate is faster in the radial direction (Lecture ). After tie t c such that R t c Z the shape of the sste becoes spherical (aspect ratio 1) For longer ties the densit profile takes a pancake for. What happens if the gas is rotating? At t0 the gas carries irrotational angular oentu L z δ ΘrigΩ A superfluid cannot appraoch spherical shape during the epansion because the oent of inertia would vanish and angular oentu would not be conserved. The gas starts rotating fast when t approaches t c, but deforation reains finite (aspect ratio 1 ).

21 1 Skater increases angular velocit b reducing radial size Superfluid gas increases angular velocit during the epansion. It cannot reach setric configuration (aspect ratio 1) because of angular oentu conservation. Theor: Edwards et al., 00 Ep: Hechenblaickner et al. 00 Ω / π 8Hz Ω / π 0Hz Ω / π 0

22 What happens at higher angular velocities? Ω B increasing vorte lines becoe energeticall favourable (see lecture 7). Sste has then two ain possibilities: A) Sste keeps irrotationalit and is still described b HD (etastabilit, angular velocit should be increased adiabaticall) B) Lines of singular vorticit are created if sste is allowed to jup into lowest energ configuration (lecture 7) Hpothesis A) can be eplored b finding stationar solutions of irrotational HD equations as a function of angular velocit Ω t n + [ n( v t v S S 1 + ( v Ω r)] 0 S + µ ( n) + V et v S Ω r) 0 equations in rotating frae

23 3 For a dilute Bose gas ( ) one finds stationar solutions with irrotational velocit and parabolic densit profile v α + + ) ~ ~ ( ~ 1 ) ( z g r n z µ Ω δ α gn µ The new distribution is characterized b the renoralized oscillator frequencies: Ω + + Ω + α α α α ~ ~ 0 ) ( 3 + Ω Ω + ε α Hdrodnaic equations ield cubic equation α (Recati et al. 001)

24 4 Isotropic trapping ( ε 0 ) - If Ω > / one finds 3 solutions for α δω - solutions with α 0 have lowest energ. (for α 0 quadrupole oscillation becoes energeticall unstable ( δe Ω Ω < 0 ). Spontaneous breaking of rotational setr (siilar to bifurcation phenoena in rotating classical fluids) ε 0 1/ Ω /

25 5 Role of trap deforation Even sall trap deforations can produce sizable effects For ε 0 one identifies two different branches: Main branch starting fro Ω 0. This branch can be followed adiabaticall b increasing slowl the angular velocit up to soe critical angular velocit where the sste ehibits dnaic instabilit (Sinha, Castin, 001). α ideal gas Second branch etends up to angular velocities larger than trapping oscillator frequenc (overcritical branch). ain branch overcritical dnaic instabilit

26 6 Rotating configurations can be realized eperientall b raping up adiabaticall the angular velcocit (black circles). For larger angular velocities the sste nucleates vortices due to dnaic instabilit of collective frequencies Overcritical branch can be also followed eperientall (white circles). Madison et al., 001

27 7 Energetic vs dnaic instabilit Sstes out of equilibriu can develop instabilities - Energetic instabilit. Corresponds to occurrence of oscillations with negative ecitation energ. Energetic instabilit is effective onl in the presence of theralization effects or echanical activation of the unstable ode. - Dnaic instabilit. Corresponds to oscillations with cople frequenc. In this case the perturbation growth up spontaneousl, also without theral activation. BEC gases in rotating haronic traps eploit both cases of instabilit.

28 8 This Lecture. Moent of inertia and superfluidit Irrotational vs rotatational flow. Moent of inertia and scissors ode. Epansion of rotating BEC. Net Lecture. Quantized vortices. Quantization of circulation. Nucleation of vortices. Measureent of angular oentu. Vorte lattice. Collective oscillations.