Dilute Bose-Einstein condensates and their role in the study of quantum fluids: rotations, vortices, and coherence.

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1 Dilute Bose-Einstein condensates and their role in the study of quantum fluids: rotations, vortices, and coherence. Kirk Madison Vincent Bretin Frédéric Chevy Peter Rosenbuch Wendel Wohlleben Jean Dalibard Laboratoire Kastler Brossel Ecole Normale Supérieure Paris University of British Columbia, April 14, 2003

2 history of quantum fluids characteristic properties of N-body bosonic systems : condensation and superfluidity comparison of liquid He and the new dilute quantum degenerate gases

The birth of experimental quantum fluids occured in the absence of a theory to describe it Heike Kamerlingh-Onnes 1908 Kamerlingh-Onnes liquified He observed aberrant behavior of liquid He - extremum

3 The birth of experimental quantum fluids occured in the absence of a theory to describe it Heike Kamerlingh-Onnes 1908 Kamerlingh-Onnes liquified He observed aberrant behavior of liquid He - extremum of density vs. temperature at 2.2 K - He stops boiling below 2.2 K! discovered superconductivity "Such an extreme could possibly be connected with the quantum theory." Nobel Lecture, Kamerling-Onnes 1913 nascent quantum theory 1901 Max Planck introduces his quantum hypothesis that energy is carried by indistinguishible units, quanta, in his derivation of the black-body radiation law 1905 Einstein explains the photoelectric effect using Planck s quantum hypothesis...is it quantum mechanics?

The theory of quantum condensation was born without a connection to

Bose 1924 Bose explained Planck s radiation law using purely statistical

1 n j = e βej 1 maximizes the ENTROPY - equillibrium distribution - A.

Einstein, Sitzber. Kgl. Preuss. Akand. Wiss.

4 The theory of quantum condensation was born without a connection to experiment S. Bose 1924 Bose explained Planck s radiation law using purely statistical arguments Bose-Einstein distribution - mean occupation of energy level E j 1 n j = e βej 1 maximizes the ENTROPY - equillibrium distribution - A. Einstein 1924 Einstein translated and submitted the paper of Bose S.N. Bose, Z. Phys. 26, 178 (1924) 1924,1925 Einstein postulated quantum condensation A. Einstein, Sitzber. Kgl. Preuss. Akand. Wiss., 261 (1924), 3 (1925) "From a certain temperature on, the molecules condense without attractive forces, that is, they accumulate at zero velocity. The theory is pretty but is there also some truth to it?" - A. Einstein is quantum condensation for real?

The peculiar behavior of liquid He eventually led to its identification as a

discontinuities in the latent and specific heat of He near 2.

discontinuties in He and suggest that this point is a phase transition: He I to He II

5 The peculiar behavior of liquid He eventually led to its identification as a Bose-Einstein condensate 1908 Kamerlingh-Onnes liquifies He 1923 Dana and Onnes measure discontinuities in the latent and specific heat of He near 2.2K 1925 Bose/Einstein postulate quantum condensation 1927 Keesom and Wolfke measure discontinuties in He and suggest that this point is a phase transition: He I to He II 1937 Kapitza and (independently) Allen & Miscner discover vanishing viscosity of He II : superfluidity! 1938 Fritz London postulates that this phase transition is analogous to bosonic condensation Reasoning: low density & low viscosity = gas (non-ideal) 6 fermions = composite boson 4 He = 2p + 2n + 2e p e n e n p F. London

6 history of quantum fluids characteristic properties of N-body bosonic systems : condensation and superfluidity comparison of liquid He and the new dilute quantum degenerate gases Liquid He is a superfluid, but is it a condensate? What s the difference?

7 Bose-Einstein condensation and superfluidity are related but distinct phenomena Bose Einstein Condensation - macroscopic occupation of the ground quantum state - Superfluidity - viscous free flow - occurs because of Bose statistics - evident in the distribution function ideal gas phenomenon - no interactions are needed occurs because interactions modify the behavior of excitations absent for an ideal gas!

8 Superfluidity is viscous free flow : dissipation is (excitations are) not allowed before The Landau criterion for dissipation v after energy + momentum conservation p, E 1 mv 2 = 1 mv 2 + E 2 2 v mv = mv + p

9 Superfluidity is viscous free flow : dissipation is (excitations are) not allowed before The Landau criterion for dissipation v after energy + momentum conservation p, E 1 mv 2 = 1 mv 2 + E 2 2 v mv = mv + p 1 2 mv 2 = 1 2 m(v 2-2p v + p 2 /m) + E p v = E + p 2 /2m since p v < pv v > E / p

10 Superfluidity is viscous free flow : dissipation is (excitations are) not allowed The Landau criterion for dissipation v > E / p v p > E energy & momentum conservation Energy excitation spectrum with interactions vp no interactions free particle excitations p

11 Superfluidity is viscous free flow : dissipation is (excitations are) not allowed this picture is incomplete dissipation may involve excitations which are not independent quasi-particles!

Superfluidity is a collective effect which involves more than just a modification of the excitation spectrum Landau s view of superfluidity interactions modify the excitation

with interactions vp no interactions free particle excitations p Modern view the collective flow is metastable (perturbations raise the total energy) thermal fluctuations are

12 Superfluidity is a collective effect which involves more than just a modification of the excitation spectrum Landau s view of superfluidity interactions modify the excitation spectrum energy and momentum conservation preclude excitation generation (dissipation) by a perturbation moving slower than a critical velocity Energy excitation spectrum with interactions vp no interactions free particle excitations p Modern view the collective flow is metastable (perturbations raise the total energy) thermal fluctuations are too small for system to overcome the barrrier quantum tunneling is suppressed because of decoherence Energy Ebarrier >> kbt metastable state ground state collective flow velocity

13 So, what s a BEC?

14 Bose-Einstein condensation involves a macroscopic occupation of the ground state consider an N-body quantum system described by... N-body wavefunction φ N (r 1,...,r N,t) N-particle state N particles vacuum φ 1 N = d 3 r 1...d 3 r N φ N (r 1,...,r N,t) Ψ (r N )...Ψ (r 1 ) 0 N creation operators commutation relations [Ψ(r),Ψ (r )] = δ(r-r ) [Ψ(r),Ψ(r )] = 0 r

15 Bose-Einstein condensation involves a macroscopic occupation of the ground state The idea of condensation is... the ground state wave function of the N-body system can be approximated as a product of single particle wave functions φ (0) (r 1,...,r N,t) N ϕ 0 (rk) ϕ 0 (r) k=1 is the condensate wavefunction Hartree wavefunction

16 The N-body evolution is governed by an N-body Hamiltonian with N-body interactions i h d φ N dt = Η φ N The N-body Hamiltonian couples Hilbert subspaces Η = Η1 + Η2 + Η single particle kinetic energy external potential energy pairs collisions dipole-dipole interactions... triplets collisions Η3 = 1 3 Η 2 = Η1 = d 3 r Ψ (r) ( p 2 + V(r) ) Ψ(r) 2M 1 d 3 r d 3 r Ψ (r)ψ (r ) V 2 (r,r ) Ψ(r )Ψ(r) 2 d 3 r d 3 r d 3 r Ψ (r)ψ (r )Ψ (r ) V3(r,r,r ) Ψ(r )Ψ(r )Ψ(r)... in general the series cannot be truncated!

17 In the Hartree mean field approximation, the only interaction term kept in the Hamiltonian is the one arising from binary collisions Hartree mean field Hamiltonian - effective one-body problem Η Hartree = Η1 + Η2 single particle kinetic energy external potential energy mean field a density dependent potential the mean field term modifies the condensate wavefunction self consistent solution of the Gross-Pitaevskii Eqn. this approximation only works if binary collisions are much more frequent than 3,4,..or N-body collisions i.e. the system is collisionally dilute! and even so it s not always valid because H2 does not, in general, preserve factorizability - QUANTUM DEPLETION -

$collisional volume fraction occupied$ by the particles λ db } a :

18 Diluteness is determined by the collisional volume fraction occupied by the particles λ db } a : interatomic scattering length ri = n-1/3 (collisional size) collisional volume fraction occupied by a particle : na3 probability for a k-body collision ~ (na 3 ) k na 3 << 1 dilute na 3 ~ 1 dense

19 The condensate is the single particle state with the largest occupation describe the N-body state by a single particle density N-body wavefunction ρ(r,r ) = dr 2,...,dr N φ * 0 (r,r2,...,r N ) φ 0 (r,r 2,...,r N ) ρ(r,r ) = Σ n ρ n ϕ n *(r ) ϕ n (r) single particle wavefunctions the largest value of ρ n divided by N is the condensate fraction the rest is the quantum depletion

21 history of quantum fluids characteristic properties of N-body quantum systems : condensation and superfluidity comparison of liquid He and the new dilute quantum degenerate gases

22 Traditional quantum degenerate liquids are dense while the new atomic gas systems are dilute Traditional QDL s The new dilute systems (since 1908) (since 1995) particle density critical temperature 3 He, 4 He Bosons (electrons) n 0 ~ cm -3 n 0 ~ cm -3 T c ~ 2.17 K Fermions H 4 He * 6 Li, 7 Li 23 Na 40 K, 41 K 85 Rb, 87 Rb 133 Cs T c ~ 500 nk why are they all alkali metals? collisional size dense a ~ 3-50 A a ~ 50 A na 3 ~ 1/ na 3 ~ 10-6 dilute

23 Traditional quantum degenerate liquids are dense while the new atomic gas systems are dilute Traditional QDL s The new dilute systems (since 1908) (since 1995) 3 He, 4 He Bosons Fermions H 4 He * 6 Li, 7 Li 23 Na 40 K, 41 K 85 Rb, 87 Rb 133 Cs why are they all alkali metals? dense Theory difficult (impossible?) Theory tractable dilute (a condensate is a good description)

24 Atomic gas condensates have so far been used for quantitative tests of mean field theories but may eventually be used to realize strongly correlated systems too Tests of the Hartree approximation / Gross-Pitaevskii eqn. macroscopic interference measurement collective oscillations : frequencies / spectrum Despite their diluteness, do these atomic condensates behave like Liquid He II? classic superfluidity (Landau - like) Do vortices form??? Beyond Mean-Field : but not too far beyond T = 0 (thermal fluctuations) condensation physics, dissipation, relaxation Future : highly correlated systems??

25 What are vortices and what do they have to do with superfluidity?

26 What are the consequences of condensation? Macroscopic coherence All of the particles are in the same single-particle state and contribute in the same way to a given single-particle measurement : an intereference pattern. quantum depletion can reduce fringe visibility Matter wave interference source : a Bose condensate theory experiment M.R. Andrews et al., Science 275, 637 (1997) Potential flow field given then ϕ 0 (r) = V = h m ρ e is S probability current h j = ρv = ρ m S

27 What are the consequences of condensation? Macroscopic coherence All of the particles are in the same single-particle state and contribute in the same way to a given single-particle measurement : an intereference pattern. quantum depletion can reduce fringe visibility Potential flow field given then ϕ 0 (r) = V = h m ρ e is S the flow is irrotational! Ω x v = 0 rigid body rotation x v = 2Ω can a condensate rotate?

28 A condensate cannot rotate, but it can circulate The circulation is... A V = h m S Γ c = v dl dl quantized Γ c = h m 2π n n = 0,1,2... because the wavefunction is single-valued the phase is allowed to wind by n 2π associated with a phase singularity : a vortex ϕ 0 must vanish so that A is not simply connected vortex singularity dl A Γ c = v dl = Stoke s Theorem ( x v) da = 0! a vortex is the only way a condensate can circulate

29 Some properties of vortices circulation current v = nh mr velocity field analogous to the magnetic field around a line of current θ ρ vortex core is empty Vorticity is localized on singular lines where the density vanishes over a distance ξ ξ healing length (interaction energy) ξ = h 2 / mµ vortices interact vortices with the same sense of circulation repel phase winding : n ϕ 0 (r) = ρ e inθ

30 Does a dilute BEC (just like He II) form vortices in response to a rotating perturbation?

31 Well, yes, but... What did we do? What did we see?

32 a TF = 5 µm x The BEC lives in a magnetic trap y axi-symmetric harmonic potential Utrap = m ωr 2 (x 2 +y 2 ) + m ωz 2 z Typical frequencies ω r ~ 2π 250 Hz atomic cloud ω z ~ 2π 10 Hz Length scales cigar shaped atomic cloud Typical parameters N 0 = 300,000 atoms T = 100 nk 100 µm Typical size z harmonic oscillator size a HO 2 = h mω ar ~ 1 µm az ~ 5 µm Thomas-Fermi size a TF 2 = 2µ mω 2

33 An optical potential is used as a spoon to stir the condensate magnetic + optical potential still harmonic U = m ωr 2 (ε X X 2 +ε Y Y 2 ) + m ωz 2 z z elliptical intensity profile in X-Y plane can rotate Y Ωt X not axi-symmetric ε X = ε Y optical potential can rotate at a frequency Ω

34 Optical absorption detection follows a free expansion free fall expansion τ = 30 ms 5 µm Radial expansion factor: 1+(ω r τ) 2 ~ 40 resonant laser beam shadow imaged onto a CCD camera 170 µm

35 Nucleation of a single vortex - identified by its core - was observed for Ω 0.64 ωr Ω = 2π 120 Hz Ω = 2π 114 Hz vortex core size consistent with µm after TOF x expansion Position (mm) 0.22 µm in situ ξ ~ 0.2 µm Position (mm) K.W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, PRL 84, 806 (2000)

36 What about superfluidity? does the supercurrent flow forever? do the vortices live forever?

37 The vortex state is a persistent current 1.0 Survival probability atoms T < 80 nk time (ms) vortex metastability - persistent current vortex lifetime ~ 1 s trap oscillation period ~ 5 ms 1 / interaction energy ~ 0.7 ms K.W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, PRL 84, 806 (2000)

38 What about nucleation? do the vortices nucleate as they do in superfluid Helium?

Nucleation of multiple vortices was the system s response to faster rotations

diagram one vortex complicated many structures vortices turbulent? unperturbed?

nucleation threshold frequency was 91 Hz. What happened?

39 Nucleation of multiple vortices was the system s response to faster rotations Abrikosov lattices form because the vortices repel no vortices vortex phase diagram one vortex complicated many structures vortices turbulent? unperturbed? Ω/2π 0 Hz 147 Hz 159 Hz 172 Hz 189 Hz 210 Hz ω r Two questions: 1) The expected nucleation threshold frequency was 91 Hz. What happened? 2) What is going on above 172 Hz? K.W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, J. Mod. Opt. 47, 2715

The first experimental detection of quantized

rotates, the frequency of the two lowest vibration

40 The first experimental detection of quantized circulation used the modes of a vibrating wire Vinen (1961) a vortex core pinned to a vibrating wire rotating superfluid helium When the fluid rotates, the frequency of the two lowest vibration modes of the wire are no longer degenerate two peaks excitation spectrum

41 Vortices can be detected from the precession of the quadrupolar oscillation axes with no vortices the oscillation axes are fixed t = 1 ms t = 3 ms t = 5 ms long axis of ellipse toggles between two orthogonal axes θ with a single vortex present the axes precess long axis of ellipse toggles between two orthogonal 180 axes which slowly precess in the lab frame θ 90 t and the precession frequency yields the average angular momentum per particle in the condensate Lz = 2mR 2 θ 0 θ = 5.9 degrees/ms Zambelli & Stringari: valid for a superfluid with strong enough interactions t

42 Condensate angular momentum (per particle) one vortex no vortices 3.0 Lz 2.0 h multiple vortices unperturbed turbulent ωr / 2π = 170 Hz stirring frequency (Hz) Discontinuity in Lz with entry of the first vortex. Continuous variation of Lz with vortex position for a non-centered vortex, Lz < h. a single vortex a pair of vortices 123 Hz 123 Hz 124 Hz 126 Hz F. Chevy, K.W. Madison, and J. Dalibard, PRL 85, 2223 (2000)

43 Vortex nucleation was found to coincide with the quadrupolar resonance 6 5 quadrupolar resonance Ω = ω r / 2 Lz/h ω r / 2π = 177 Hz weak stirring small ε strong stirring large ε strirring frequency (Hz) this nucleation mechanism is extrinsic see film...

By changing the geometry of the spoon, different surface modes can be resonantly excited

HP Ω QP Condensate profiles Hexapolar mode 6 fold symmetry one Gaussian spot generates

symmetry ω r = 2π 99 Hz two Gaussian spots generate only even orders in potential 20 40 60

44 By changing the geometry of the spoon, different surface modes can be resonantly excited and induce vortex nucleation at new frequencies Spoon potential 5 Lz/h Vortex nucleation Ω HP Ω QP Condensate profiles Hexapolar mode 6 fold symmetry one Gaussian spot generates both odd and even orders in potential Ω HP = ω r / Quadrupolar mode 4 fold symmetry ω r = 2π 99 Hz two Gaussian spots generate only even orders in potential stirring frequency (Hz) K.W. Madison, F. Chevy and J. Dalibard cond-mat/ Ω QP = ω r / 2

classical vorticity: 2Ω expect a uniform surface density: S v h m d v = S v -1/2 = S v = 2Ω h 2mΩ -> 2.

45 Abrikosov lattices were observed to assemble into uniform, triangular arrays Abrikosov,Tkachenko the lowest energy configuration is a triangular lattice triangular lattice square lattice correspondance principle quantum vorticity = classical vorticity: 2Ω expect a uniform surface density: S v h m d v = S v -1/2 = S v = 2Ω h 2mΩ -> 2.17 µm common N = 12 R = 5 µm Ω/2π = 77 Hz rare observed vortex spacing d v = 2.55 µm K.W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, J. Mod. Opt. 47, 2715

46 What about the phase of condensate wavefunction associated to the vortex? inteference measurement

47 Radio frequency coupling between the magnetic sublevels is used to perform evaporation but also provides a handle for spin state manipulation Radio frequency coupling : υ RF evaporation coherent spin state manipulation m = +2 Zeeman energy υ 0 +1 υ RF x -1 J = 2-2

48 Interferometric (homodyne) detection is performed by splitting the condensate between two magnetically trapped states and allowing them to spatially separate p apply 1st π/2 pulse and wait p apply 2nd π/2 pulse and wait p x x m F =2 m F =1 x x 1 x 1 x 1 g and image after TOF release atoms fringe spacing d = ht / ma

49 A vortex phase singularity produces a phase slip or fringe dislocation in the homodyne interferogram Predicted interferograms no vortex a single vortex Measured interferograms no vortex a single vortex fork signature a single vortex several vortices F. Chevy, K.W. Madison, V. Bretin, and J. Dalibard, PRA 64, 31601(2001)

50 Conclusions : vortices do form in a dilute BEC achieved vortex nucleation by stirring and measured the vortex density singularity -> the healing length observed Abrikosov lattice formation and anealing developed a homodyne interferometric technique to measure the vortex phase singularity developed a technique for measuring the condensate angular momentum using surface mode spectroscopy identified an extrinsic nucleation mechanism : a resonantly excited rotating surface mode which can nucleate a vortex state identified and studied a stationary (eigen) state of the GP equation and demonstrated that this mode can decay via dynamic instabilities into a vortex state

51 Recent experimental work with vortices in gaseous BECs Vortex creation by phase imprinting Vortex nucleation in a rotating trap Observation of Abrikosov lattices Measurement of <Lz> of a vortex state Study of vortex precession Observation of vortex lattices Use of surface waves to see vortex tilting Dark soliton decay into vortex rings Routes to vortex nucleation Observation of vortex lattices Interferometric detection of a vortex Phase singularities from vortex shedding Nucleation from a rotating thermal cloud Nucleation from higher order modes (JILA) PRL 83, 2498 (1999) (ENS) PRL 84, 806 (2000) (ENS) J. Mod. Opt. 47, 2715 (ENS) PRL 85, 2223 (2000) (JILA) PRL 85, 2857 (2000) (ENS) cond-mat/ (JILA) PRL 86, 2922 (2001) (JILA) PRL 86, 2926 (2001) (ENS) PRL 86, 4443 (2001) (MIT) Science 292, 476 (ENS) PRA 64, 31601(2001) (MIT) PRL 87, (2001) (JILA) cond-mat/ (ENS) cond-mat/

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