Bose-Einstein Condensation and Superfluidity in Nanoscale Liquid and Solid helium Henry R. Glyde Department of Physics & Astronomy University of Delaware Barcelona 23 March 2011

Excitations of liquid 4He in Porous Media Institut Laue Langevin, Grenoble Recent Collaborators: JACQUES BOSSY Institut Néel, CNRS-UJF, Grenoble, France Jonathan Pearce Delaware and National Physical Laboratory, UK Bjorn Fåk - Commissariat à l’Energie Atomique, Grenoble, France Norbert Mulders - University of Delaware Helmut Schober Institut Laue-Langevin, Grenoble, France

BEC and Atomic Momentum Distributions ISIS Faciltiy, Rutherford Appleton Laboratory, UK Recent Collaborators: Richard T. Azuah -NIST Center for Neutron Research, Gaithersburg, USA Souleymane Omar Diallo - Spallation Neutron source, ORNL, Oak Ridge, TN Jonathan Pearce-Delaware and National Physical Laboratory, UK Jon V. Taylor - ISIS, Rutherford Appleton Lab, UK Oleg Kirichek -ISIS, RAL, UK

Phase Diagram of Bulk Helium

Kim and Chan. Science, 305:1941 (2004)

Excitations, BEC, and Superfluidity Organization of Talk 1.Bulk liquid 4He --review Superfluid density, ρ S BEC condensate fraction, n 0 Phonon-roton excitations. 2. Liquid 4He in Porous media Review ρ S, T C Present phonon-roton data. Evidence for localized BEC at temperatures above T C (Bosons in Disorder) 3. Comparisons with Superconductors

B. HELIUM IN POROUS MEDIA B. HELIUM IN POROUS MEDIA AEROGEL*95% porous Open87% porousA 87% porousB - 95 % sample grown by John Beamish at U of A entirely with deuterated materials VYCOR (Corning)30% porous 70Å pore Dia.-- grown with B 11 isotope GELSIL (Geltech, 4F) 50% porous 25 Å pores 44 Å pores 34 Å pores MCM-4130% porous 47 Å pores NANOTUBES (Nanotechnologies Inc.) Inter-tube spacing in bundles 1.4 nm 2.7 gm sample * University of Delaware, University of Alberta

T c in Porous Media

Schematic Phase Diagram of Helium Confined to Nanoscales e.g nm pore diameter

Schematic Phase Diagram He in Nanoporous media Schematic Phase Diagram He in Nanoporous media Bossy et al., PRL 100, (2008)

SUPERFLUIDITY SUPERFLUIDITY 1908 – 4 He first liquified in Leiden by Kamerlingh Onnes 1925 – Specific heat anomaly observed at T λ = 2.17 K by Keesom. Denoted the λ transiton to He II – Superfluidity observed in He II by Kaptiza and by Allen and Misener – Superfluidity interpreted as manifestation of BEC by London v S = grad φ (r)

Kamerlingh Onnes Kamerlingh Onnes

London

BEC and SUPERFLUIDITY BEC and SUPERFLUIDITY 1950’s to today - Remarkable superfluid and other properties of Liquid helium Discovery of superfluidity in liquid 3He onward: Measurement of Bose-Einstein condensate fraction using neutrons – Discovery of BEC in gases of alkali atoms – Superflow in solid helium?

Superfluid Density  s (T) Superfluid Density Bulk Liquid 4 He

BOSE-EINSTEIN CONDENSATION 1924 Bose gas : At T = 0, 100 % in the condensate, p = 0 state is condensate state for uniform fluids. Condensate wave function: ψ(r) = √n 0 e iφ(r)

Bose- Einstein Condensation Bose- Einstein Condensation 1995 BEC in Alkali atom gases Carl Weyman and Eric Cornell

Bose-Einstein Condensation: Atoms in Traps

N. N. Bogoliubov N. N. Bogoliubov

Neutron Scattering: ILL

Bose-Einstein Condensation Glyde, Azuah, and Stirling Phys. Rev., 62, (2000)

Bose-Einstein Condensation

Phase Diagram of Bulk Helium Kim and Chan. Science, 305:1941 (2004)

Bose-Einstein Condensation Glyde, Diallo, Azuah, Kirichek and Taylor, PRB (2011)

Bose-Einstein Condensation Glyde, Diallo, Azuah, Kirichek and Taylor, PRB (2011)

Bose-Einstein Condensate Fraction Liquid Helium versus Pressure

Supersolid Helium Superfluid Fraction (NCRI)

Bulk Solid Helium Diallo et al. PRB 80, (2009)

B. HELIUM IN POROUS MEDIA B. HELIUM IN POROUS MEDIA AEROGEL*95% porous Open87% porousA 87% porousB - 95 % sample grown by John Beamish at U of A entirely with deuterated materials VYCOR (Corning)30% porous 70Å pore Dia.-- grown with B 11 isotope GELSIL (Geltech, 4F) 50% porous 25 Å pores 44 Å pores 34 Å pores MCM-4130% porous 47 Å pores NANOTUBES (Nanotechnologies Inc.) Inter-tube spacing in bundles 1.4 nm 2.7 gm sample * University of Delaware, University of Alberta

Bulk Solid Helium Condensate fraction

Bose-Einstein Condensate Fraction Liquid and Solid Helium

PHONON – ROTON MODE Landau (1941, 1947) – proposed that superfluid 4 He supports well defined phonon-roton excitations. ---and no other low energy excitation to which superfluid can decay. Superfluidity follows from the nature of the excitations : - have a critical velocity and an energy gap (roton gap  ). - no reference to BEC. -Introduced curl v s = 0 into transport equations.

Landau

Phonon-Roton Dispersion Curve  Donnelly et al., J. Low Temp. Phys. (1981)  Glyde et al., Euro Phys. Lett. (1998) ← Δ

Phonons in bcc solid 4He

Phonon - Roton Mode vs Pressure

Maxon in Bulk Liquid 4 He Talbot et al., PRB, 38, (1988)

Roton in Bulk Liquid 4 He Talbot et al., PRB, 38, (1988)

Roton in Liquid 4 He Zigmond et al. Physica B 388, 43 (2007)

Beyond the Roton in Liquid 4 He Data: Pearce et al. J Phys Conds Matter 13, 4421 (2001) Theory: Sakhel and Glyde PRB 70, (2004)

Excitations, BEC, and Superfluidity Bulk Liquid 4 He -Bose-Einstein Condensation, -well-defined phonon-roton modes at Q > 0.8 Å -1 -superfluidity all coincide/co-exist in same p and T range. e.g., all have same “critical” temperature, T λ = 2.17 K SVP T λ = 1.76 K 25 bar

Phase Diagram of Bulk Helium

B. HELIUM IN POROUS MEDIA B. HELIUM IN POROUS MEDIA AEROGEL*95% porous Open87% porousA 87% porousB - 95 % sample grown by John Beamish at U of A entirely with deuterated materials VYCOR (Corning)30% porous 70Å pore Dia.-- grown with B 11 isotope GELSIL (Geltech, 4F) 50% porous 25 Å pores 44 Å pores 34 Å pores MCM-4130% porous 47 Å pores NANOTUBES (Nanotechnologies Inc.) Inter-tube spacing in bundles 1.4 nm 2.7 gm sample * University of Delaware, University of Alberta

Bosons in Disorder Liquid 4 He in aerogel, Vycor, gelsil Bose gases in traps with disordered potentials Josephson Junction Arrays Granular Metal Films Cooper Pairs in High T c Superconductors Flux Lines in High T c Superconductors Specific Present Goals: Impact of disorder on excitations and Bose-Einstein condensation. Localization of Bose-Einstein Condensation by disorder

T c in Porous Media

Geltech (25 Å pores) Superfluid Density in Porous Media Chan et al. (1988) Miyamoto and Takeno (1996)

- Yamamoto et al, Phys. Rev. Lett. 93, (2004) Phase Diagram of gelsil: 25 A pore diameter

Bose-Einstein Condensation Liquid 4 He in Vycor Azuah et al., JLTP (2003) T c (Superfluidity) T c = K

Phonon-Roton Dispersion Curve  Donnelly et al., J. Low Temp. Phys. (1981)  Glyde et al., Euro Phys. Lett. (1998) ← Δ

Phonons, Rotons, and Layer Modes in Vycor and Aerogel

Intensity in Single Excitation vs. T T c = 2.05 K Glyde et al., PRL, 84 (2000) T c = 2.05 K

P-R Mode in Vycor, T = 1.95 K T c = 2.05 K

P- R Mode in Vycor: T = 2.05 K T c = 2.05 K

Fraction, f s (T), of Total Scattering Intensity in Phonon-Roton Mode- Vycor 70 A pores T c = 2.05 K

Fraction, f s (T), of total scattering intensity in Phonon-Roton Mode- gelsil 44 A pore T c = 1.92 K

T c ~ 1.3 K Liquid 4 He in gelsil 25 A pore diameter

Conclusions: Localization of Bose-Einstein Condensation in disorder Observe phonon-roton modes up to T ~ T λ = 2.17 K in porous media, i.e. above T c for superfluidity Well defined phonon-roton modes exist because there is a condensate. Thus have BEC above T c in porous media. At SVP Vycor T c = 2.05 K gelsil (44 Å) T c = 1.92 K gelsil (25 Å) T c = 1.3 K At temperatures above T c - BEC is localized by disorder - No extended phase coherence across the sample - No superflow

Schematic Phase Diagram He in Nanoporous media Schematic Phase Diagram He in Nanoporous media Bossy et al., PRL 100, (2008)

- Yamamoto et al, Phys. Rev. Lett. 93, (2004) Phase Diagram of gelsil: 25 A pore diameter

Temperature dependence of S(Q,ω) at Q = 2.1 Å -1 (roton) at p = 31.2 bars.

Temperature dependence: Roton Energy at pressure 31.2 bar in MCM-41 Bossy et al. (in progress) 2011

He in Nanoporous media: points where P-R modes last observed He in Nanoporous media: points where P-R modes last observed Bossy et al., PRL 100, (2008)

Pressure dependence: 44 Å gelsil phonon (Q = 0.7 Ǻ -1 ) roton (Q=2.1Å -1 )

Pressure dependence: 44 Å gelsil Roton (Q=2.1Å -1 ) Vranjes et al. PRL 2005

Phase diagram MCM - 41 (47 A) and gelsil (25A): Points where P-R modes are last observed Bossy et al., PRL 100, (2008)

S(Q) of Amorphous Solid Helium in MCM-41 S(Q) of Amorphous Solid Helium in MCM-41 Cooling at 37.8 bars. Liquid in MCM-41 remains liquid at 0.4 K. Height of maximum in S(Q) decreases slightly on cooling. MCM-41: 47 A pore diameter

Amorphous Solid Helium MCM A S(Q) on cooling at 48.6 bars

Amorphous Solid Helium MCM A S(Q) on warming at 48.6 bars

Amorphous Solid Helium Difference in S(Q) between Amorphous solid and liquid

Amorphous Solid Helium S(Q), simulations of layer by layer freezing. Rossi, Galli and Reatto, Phys. Rev B 72, (2005)

Schematic Phase Diagram He in Nanoporous media Schematic Phase Diagram He in Nanoporous media Bossy et al., PRL 100, (2008)

Conclusions: Liquid 4 He in Disorder and Boson Localization Extended BEC at temperatures below T c in the superfluid phase. Superfluid - Normal liquid transition in porous media associated with an extended to localized BEC cross over. Above T c have only localized BEC (separated islands of BEC).Above T c have only localized BEC (separated islands of BEC). At higher temperatures and pressures have no BEC at all.At higher temperatures and pressures have no BEC at all.

Schematic Phase Diagram He in Nanoporous media Schematic Phase Diagram He in Nanoporous media

A Brief Introduction to Cuprates AF Mott Insulator Insulator Metal T Doping Level Superconductor Pseudo-gap Metal

Schematic Phase Diagram High Tc Superconductors Schematic Phase Diagram High Tc Superconductors Alvarez et al. PRB (2005)

Patches of Antiferromagnetic and Superconducting regions Patches of Antiferromagnetic and Superconducting regions Alvarez et al. PRB (2005)

Phase Diagram High Tc uperconductor Phase Diagram High Tc uperconductor Yazdani, J. Phys. Condens. Matter (2009)

Patches of Energy gap, T c = 93 K Patches of Energy gap, T c = 93 K Yazdani, J. Phys. Condensed. Matter (2009)

Conclusions Bulk Liquid 4 He -Bose-Einstein Condensation, -well-defined phonon-roton modes at Q > 0.8 Å -1 -superfluidity coincide/co-exist in same p and T range. e.g., all have same “critical” temperature, T λ = 2.17 K SVP T λ = 1.76 K 25 bar

Schematic Phase Diagram He in Nanoporous media Schematic Phase Diagram He in Nanoporous media Bossy et al., PRL 100, (2008)

Conclusions: Liquid 4 He in Disorder and Boson Localization Extended BEC at temperatures below T c in the superfluid phase. Superfluid - Normal liquid transition in porous media associated with an extended to localized BEC cross over. Above T c have only localized BEC (separated islands of BEC).Above T c have only localized BEC (separated islands of BEC). At higher temperatures and pressures have no BEC at all.At higher temperatures and pressures have no BEC at all.

Excitations, BEC, and Superfluidity Landau Theory: Superfluidity follows from existence of well defined phonon-roton modes. The P-R mode is the only mode in superfluid 4He. Bose-Einstein Condensation (BEC): Superfluidity follows from BEC. An extended condensate has a well defined magnitude and phase, = √ n 0 e ιφ ; v s ~ grad φ Bose-Einstein Condensation: Well defined phonon-roton modes follow from BEC.

Net Scattering intensity, gelsil 34 Å and bulk liquid simulation compared. Pearce et al. (in progress) ← 60 bars Bulk liquid

4 He remains liquid in 34 A gelsil up to what pressure? 4 He remains liquid in 34 A gelsil up to what pressure? Δp = p L – p S = 2α / R c p S = 25.3 bars R c = 14 Å (a)α = 0.17 erg/cm 2 -- constant p L = 50 bars (b)α = -increases with pressure (Maris and Caupin, JLTP 131, 145 (2003)) p L = 70 bars Vycor, p L = 45 bars R c = 35 Å

Excitations, BEC, and Superfluidity Collaborators: ILL JACQUES BOSSY Institut Néel, CNRS-UJF Grenoble, France Jonathan Pearce Delaware and National Physical Laboratory, UK Francesco Albergamo ESRF, Grenoble, France Bjorn Fåk - Commissariat à l’Energie Atomique, Grenoble, France Norbert Mulders - University of Delaware Helmut Schober Institut Laue-Langevin Grenoble, France

Excitations, BEC, and Superfluidity Collaborators: ISIS Richard T. Azuah -NIST Center for Neutron Research, Gaithersburg, USA Souleymane Omar Diallo - Spallation Neutron source, ORNL, Oak Ridge, TN Jonathan Pearce-Delaware and National Physical Laboratory, UK Jon V. Taylor - ISIS Facility, UK Oleg Kirichek -ISIS Facility, UK

Excitations, BEC, and Superfluidity Collaborators: Jonathan PearceInstitut Laue-Langevin Grenoble, France Francesco Albergamo -ESRF, Grenoble, France Richard T. Azuah -NIST Center for Neutron Research, Gaithersburg, Maryland, USA Jacques Bossy -Centre de Recherche sur Les Très Basses Temperature CNRS, Grenoble, France Bjorn Fåk -Commissariat à l’Energie Atomique Grenoble, France Helmut Schober Institut Laue-Langevin Grenoble, France

Excitations, BEC, and Superfluidity Collaborators (Con’t): Norbert Mulders -University of Delaware Newark, Delaware USA Oliver Plantevin -ESRF, Grenoble Reinhard Scherm -Physikalisch-Technische Bundesanstalt, Braunschweig John Beamish -University of Alberta Edmonton, Canada Gerrit Coddens - Laboratoire des solides irradiés Ecole Polytechnique Palaiseau, France

Excitations, BEC, and Superfluidity Landau Theory: Superfluidity follows from existence of well defined phonon-roton modes. The P-R mode is the only mode in superfluid 4He. Bose-Einstein Condensation (BEC): Superfluidity follows from BEC. An extended condensate has a well defined magnitude and phase, = √n 0 e ιφ ; v s ~ grad φ Bose-Einstein Condensation: Well defined phonon-roton modes follow from BEC. Single particle and P-R modes have the same energy when there is BEC. No low energy single particle modes.

Phase diagran and excitations of superfluid 4 He in 44 Å gelsil Pearce et al., PRL (2004)

Phase diagran and excitations of superfluid 4 He in 44 Å gelsil Pearce et al., PRL (2004)

Excitations of superfluid 4 He at pressures up to 40 bars

Bose-Einstein Condensation Liquid 4 He in Vycor Azuah et al., JLTP (2003) T c (Superfluidity) = K

Graduate Students Jonathan DuBois Bose-Einstein Condensation of Bosons in Traps, Variational Monte Carlo, Diffusion MC Asaad Sakhel Models of excitations in liquid 4 He BEC in traps Souleymane Omar Diallo Neutron scattering measurements at ISIS, n(p) of solid 4 He, condensate and n(p) in liquid 3 He/ 4 He mixtures Ali Shams Models of localization of BEC in porous media

Focused Research Group: NSF Neutron Scattering Studies of Surface and bulk Disordered Quantum Systems Oscar VilchesUniversity of Washington John LareseUniversity of Tennessee Henry Glyde (PI)University of Delaware From left to right, J. Pearce, J. Larese, H. Glyde and T. Arnold

Excitations and Bose-Einstein Condensation in Quantum Liquids in Disorder Henry R. Glyde, University of Delaware, DMR Figure 1. Top: The Insitiut Laue Langevin (just behind the ESRF synchrotron ring) in Grenoble. Bottom: Left to right, Jacques Bossy, Henry Glyde, Francesco Albergamo and Olivier Plantevin in front of the IN6 neutron spectrometer of ILL.

S. N. Bose S. N. Bose

Liquid helium in porous media supports well defined phonon-roton excitations – up to wave vectors Q ≈ 2.8 Å. Energies and widths (within precision) are the same as in bulk 4 He at all T. Liquid also supports “layer modes” at roton wave vectors. MCM-41: at partial fillings, can also see ripplons on 4 He liquid surfaces. Conclusions: Liquid 4 He in Disorder and Boson Localization

Phonons and Rotons Arise From Bose-Einstein Condensation Bogoliubov (1947) showed: Bose gas with BEC -- quasiparticles have energy: - phonon (sound) Quasiparticle mode coincides with sound mode. Only one excitation when have BEC.

Liquid 4 He in Disorder and Boson Localization - Vycor Well defined p-r excitations (Q > 0.8 Å) exist because there is Bose- Einstein condensation (BEC). Measure superfluid density ρ s (T) and determine the normal to superfluid transition temperature T c in Vycor (same sample). Find: T c = 2.05 K <T λ = 2.17 K (Vycor) (Bulk) - disorder suppresses T c below T λ Find well defined phonon–roton excitations in Vycor at temperatures T > T c, up to T = T λ = 2.17 K Thus BEC in Vycor above T c, at temperatures T c < T < T λ. - localized BEC.

Excitations, BEC, and Superfluidity Liquid 4 He in disorder BEC, well-defined excitations are separated from superfluidity in disorder e.g., T c - superfluidity Have phonon-roton excitations and BEC at temperatures T > T c Disorder localizes the condensate, T > Tc New Here Measurements of phonon-roton excitations and BEC in disorder

Willem Keesom

A. Einstein A. Einstein

Superfluid Density  s (T) Superfluid Density Bulk Liquid 4 He

Excitations, BEC, and Superfluidity Landau Theory: Superfluidity follows from existence of well defined phonon-roton modes. The P-R mode is the only mode in superfluid 4He. Bose-Einstein Condensation (BEC): Superfluidity follows from BEC. An extended condensate has a well defined magnitude and phase, = √n 0 e ιφ ; v s ~ grad φ Bose-Einstein Condensation : Well defined phonon-roton modes follow from BEC. Single particle and P-R modes have the same energy when there is BEC. No low energy single particle modes.

Phonons and Rotons arise from Bose-Einstein Condensation Bogoliubov (1947) showed: Bose gas with BEC -- quasiparticles have energy: - phonon (sound) form Quasiparticle mode coincides with sound mode. Only one excitation when have BEC.

Phonon-roton mode of liquid 4 He under pressure (26 bars) in MCM-41 and phonon modes of bulk solid helium around the MCM-41

Superfluidity Landau Theory Superfluidity follows from the nature of the excitations: that there are phonon-roton excitations only and no other low energy excitations to which superfluid can decay have a critical velocity and an energy gap (roton gap  ). Via P-R excitations, superflow arises from BEC. BEC and Phase Coherence, Ø (r) Superfluidity follows directly from BEC, phase conherence.

Topic of Talk: Well defined p-r excitations (Q > 0.8 Å) exist because there is Bose- Einstein condensation (BEC). Measure superfluid density ρ s (T) and determine the normal to superfluid transition temperature T c in Vycor (same sample). Find: T c = 2.05 K <T λ = 2.17 K (Vycor) (Bulk) - disorder suppresses T c below T λ Find well defined phonon–roton excitations in Vycor at temperatures T > T c, up to T = T λ = 2.17 K Thus BEC in Vycor above T c, at temperatures T c < T < T λ. - localized BEC.

- Yamamoto et al, Phys. Rev. Lett. 93, (2004) Quantum Phase Transition in 25 A pore diameter gelsil ?

Phonons and Rotons Arise From Bose-Einstein Condensation Bogoliubov (1947) showed: Bose gas with BEC -- quasiparticles have energy: - phonon (sound) form Quasiparticle mode coincides with sound mode. Only one excitation when have BEC.

Superfluid Properties at Nanoscales Confinement reduces T c below. Confinement modifies (T dependence). Confinement reduces (magnitude). Porous media is a “laboratory” to investigate the relation between superfluidity, excitations, and BEC. Measure corresponding excitations and condensate fraction, n o (T). (new, 1998) Localization of Bose-Einstein Condensation by Disorder

Condensate Fraction: Bose Gas

Bose-Einstein Condensation: Atoms in Traps

BOSE- EINSTEIN CONDENSATION Bose Statistics introduced by S. N. Bose in 1924: N ( ε ) = [ exp (β ε) – 1 ] -1 ε = single Boson energy ε = p 2 /2m - free Bosons ε = ħω - Bosons in harmonic trap Possibility of BEC noted by Einstein also 1924: n 0 = N(0)/N - condensate fraction. Below T c, becomes macroscopic. n 0 = 1 at T =0 for a Bose gas

Liquid 4 He and 3 He Strongly Interacting Liquid: T ~ 1K He- He Pair Potential: v(r) = 4 ε [ (σ/r) 12 – (σ/r) 6 ] σ ~ 2.6 Å ε = 10 K Zero point energy: K ~ 15 K Total Energy: E = K + U ~ - 7 K Dense Liquid: nσ 3 ~ (σ/R) 3 ~ 0.2 i.e. R ~ σ = a, hard core diameter

Pressure dependence: 44 Å gelsil Roton (Q=2.1Å -1 ) Vranjes et al. PRL 2005

Phase Diagram of Bulk Helium

- Yamamoto et al. Superfluid Density in Gelsil (Geltech) – 25 A diameter

BEC, Excitations, and Superfluidity

Inter- atomic Potential in Helium v(r) = 4ε [ (σ/r) 12 – (σ/r) 6 ] (σ = a –hard diameter)

Bose-Einstein Condensation