QUANTUM DEGENERATE BOSE SYSTEMS IN LOW DIMENSIONS

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QUANTUM DEGENERATEBOSE SYSTEMSIN LOW
DIMENSIONS

• G. Astrakharchik
• S. Giorgini
• Istituto Nazionale per la Fisica della Materia
  Research and Development Center onBose-Einstein
  Condensation
• Dipartimento di Fisica Università di Trento

Trento, 14 March 2003
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• Bose Einstein condensates of alkali atoms
• dilute systems na3ltlt1
• 3D mean-field theory works
• low-D role of fluctuations is enhanced
• 2D thermal fluctuations
• 1D quantum fluctuations
• beyond mean-field effects
• many-body correlations
• 
  

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• Summary
• General overview
• Homogeneous systems
• Systems in harmonic traps
• Beyond mean-field effects in 1D
• Future perspectives

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• BEC in low-D homogeneous systems
• Textbook exercise Non-interacting Bose gas in
  a box
• Thermodynamic limit
• Normalization condition

fixed density
momentum distribution
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• D3 if
• D?2 for any T gt0
• If ?0 infrared divergence in nk

? ? 0 chemical potential
D3 converges
D?2 diverges
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• Interacting case
• T?0 Hohenberg theorem (1967)
  Bogoliubov 1/k2 theorem
• per absurdum argumentatio
• If
• Rules out BEC in 2D and 1D at finite temperature
• Thermal fluctuations destroy BEC in 2D and 1D
• quantum fluctuations?

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• T0 Uncertainty principle
  (Stringari-Pitaevskii 1991)
• If
• But

fluctuations of particle operator
fluctuations of density operator
static structure factor
sum rules result
Rules out BEC in 1D systems even at T0 Quantum
fluctuations destroy BEC in 1D (Gavoret
Nozieres 1964 ---- Reatto Chester 1967)
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• Are 2D and 1D Bose systems trivial as they enter
  the quantum degenerate regime ?

Thermal wave-length
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• One-body density matrix
• central quantity to investigate the coherence
  properties of the system

condensate density
long-range order
liquid 4He at equilibrium density
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• 2D
• Something happens at intermediate temperatures

low-T from hydrodynamic theory (Kane
Kadanoff 1967)
high-T classical gas
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• Berezinskii-Kosterlitz-Thouless transition
  temperature TBKT
• (Berezinskii 1971 --- Kosterlitz Thouless 1972)
• Universal jump (Nelson Kosterlitz 1977)
• Dilute gas in 2D Monte Carlo calculation
  (Prokofev et al. 2001)

TltTBKT system is superfluid
TgtTBKT system is normal
Thermally excited vortices destroy
superfluidity Defect-mediated phase transition
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• Torsional oscillator experiment on 2D 4He films
• (Bishop Reppy 1978)

Dynamic theory by Ambegaokar et al. 1980
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• 1D
• From hydrodynamic theory (Reatto Chester 1967)
• T0
• T?0
• 4He adsorbed in carbon nanotubes
• Cylindrical graphitic tubes 1 nm
  diameter 103 nm long
• Yano et al. 1998 superfluid behavior
• Teizer et al. 1999 1D behavior of binding energy

degeneracy temperature in 1D
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• BEC in low-D trapped systems
• a)
• )
• )

anisotropy parameter
motion is frozen along z kinematically the gas is
2D
motion is frozen in the x,y plane kinematically
the gas is 1D
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• Goerlitz et al. 2001

3D ?2D
3D ?1D
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• Finite size of the system
• cut-off for long-range fluctuations
  fluctuations are strongly quenched
• BEC in 2D (Bagnato Kleppner 1991)
• Thermodynamic limit

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• But density of thermal atoms
• Perturbation expansion in terms of g2D n breaks
  down
• Evidence of 2D behavior in Tc
• (Burger et al. 2002)
• BKT-type transition ?
• Crossover from standard BEC to BKT ?

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• 1D systems
• No BEC in the thermodynamic limit N??
• For finite N macroscopic occupation of lowest
  single-particle state
• If

(Ketterle van Druten 1996)
2-step condensation
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• Effects of interaction (Petrov - Holzmann
  Shlyapnikov 2000)
• (Petrov Shlyapnikov Walraven 2000)
• Characteristic radius of phase fluctuations
• 2D
• 1D

true condensate
(quasi-condensate) condensate with fluctuating
phase
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• Dettmer et al. 2001
• Richard et al. 2003

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• Beyond mean-field effects in 1D at T0
• Lieb-Liniger Hamiltonian
• Exactly solvable model with repulsive zero-range
  force
• Girardeau 1960 --- Lieb Liniger 1963 --- Yang
  Yang 1969
• at T0 one parameter na1D

a1D scattering length
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• Equation of state

mean-field
Tonks-Girardeau fermionization
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• One-body density matrix
• Quantum Monte-Carlo (Astrakharchik Giorgini
  2002)

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• Momentum distribution

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• Lieb-Liniger harmonic confinement
• Exactly solvable in the TG regime (Girardeau -
  Wright - Triscari 2001)
• Local density approximation (LDA) (Dunjko -
  Lorent - Olshanii 2001)
• If
• 1D behavior is assumed from the beginning

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• 3D-1D crossover
• Quantum Monte-Carlo (Blume 2002 --- Astrakharchik
  Giorgini 2002)
• Harmonic confinement
• Interatomic potential (a s-wave scattering
  length)

highly anistropic traps
hard-sphere model
soft-sphere model (R5a)
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• Compare DMC results with
• Mean-field Gross-Pitaevskii equation
• 1D Lieb-Liniger
• (Olshanii 1998)

with
with
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• Possible experimental evidences of TG regime
• size of the cloud (Dunjko-Lorent-Olshanii 2001)
• collective compressional mode (Menotti-Stringari
  2002)
• momentum distribution (Bragg scattering TOF)

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• Infrared behavior kltlt1/? --- Finite-size
  cutoff kgtgt1/Rz

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• Future perspectives
• Low-D and optical lattices
• many-body correlations
• ? superfluid Mott insulator quantum phase
  transition
• (in 3D Greiner et al. 2002)
• Thermal and quantum fluctuations
• ? low-D effects
• Investigate coherence and superfluid properties

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• Tight confinement and Feshbach resonances
• (Astrakharchik-Blume-Giorgini)
• Quasi-1D system
• confinement induced resonance (Olshanii 1998 -
  Bergeman et al. 2003)