VARIATIONAL APPROACH FOR THE TWO-DIMENSIONAL TRAPPED BOSE GAS

1
VARIATIONAL APPROACH FOR THE TWO-DIMENSIONAL
TRAPPED BOSE GAS

• L. Pricoupenko Trento, 12-14 June 2003
  
• LABORATOIRE DE PHYSIQUE THEORIQUE DES LIQUIDES
• Université Pierre et Marie Curie (Paris)

2
Motivations

• 2D experiments in the degenerate regime
• Innsbrück (Rudy Grimm)
• Firenze (Massimo Inguscio)
• Villetaneuse (Vincent Lorent)
• MIT (Wolgang Ketterle)
• Why trapped 2D Bose gas interesting ?
• Thermal fluctuations
• Interplay between KT and BEC
• Non trivial interaction induced by the geometry
• Beyond mean-field effects
• 
  

3

• Summary
• Brief review of the actual experimental settings
• Back to the two-body problem
• Contact condition versus Pseudo-potential
• Variational Formulation of Hartree-Fock-Bogolubov
  (HFB)
• Numerical Results

4
The actual experimental settings
Anisotropy parameter

• MIT
• Firenze
• Innsbrück
• Villetaneuse

Reach the 2D regime by decreasing N in an
anisotropic trap
A. Görlitz and al. Phys. Rev. Lett 87, 130402
(2001)
Use a 1D optical lattice ? Slices of 2D
condensates
S. Burger and al. Europhys. Lett., 57, pp. 1-6
(2002)
Evanescent-wave trapping
S. Jochim and al. Phys. Rev. Lett., 90, 173001
(2003)
Evanescent-wave trapping
5
Atoms trapped in a planar wave guide
Two-body problem
Zero range approach
Eigenvalue problem defined by the contact
conditions
The 2D induced scattering length
Maxim Olshanii (private communication)Dima
Petrov and Gora Shlyapnikov, Phys. Rev. A 64,
100503 (2001)
6
The pseudo-potential approach

• Motivation Hamiltonian formulation of the
  problem

Construct a potential which leads to the contact
condition of the 2-body problem
Example the Fermi-Huang potential in 3D
Zero range potential
Regularizing operator
The L-potential in the 2D world
2-body t-matrix at energy
7
Many-body problem for trapped atoms

1. Contact conditions
2. Pseudo-potential

Two possibilities
Constraints on the mean density
Validity of the zero range approach
Validity of the mean-field approach
8
Summary of the zero-range approach
highly anisotropic traps

• Mean inter-particle spacing
• Possible description of a molecular phase
• L freedom
• a2Dgt0 can be tuned via a3D
• (Feshbach resonance)

Observables do not depend on the particular value
of L
Possible study of a highly correlated dilute
system
9
Condensate/Quasi-condensate
T0K Thomas-Fermi
Near TTc
2D character
Actual experiments
Almost BEC Phase in near future experiments
10
The ingredients of HFB

• U(1) symmetry breaking approach
• (Phase of the condensate fixed TltltTF)
• Gaussian Variational ansatz

(Number of atoms fluctuates)
BEC Phase
Use the 2D zero range pseudo-potential

The atomic Bose gas is not the ground state of
the system
A Dangerous game ! ! !

11
HFB Equations

• Generalized Gross-Pitaevskii equation

• Static spectrum

Pairing field (satisfies the contact condition)
Implicit Born approximation
12
The gap spectrum disaster

• Change the phase of f cost no energy
• Anomalous mode solution of the linearized time
  dependent equations (RPA)
• (F,-F) NOT SOLUTION (in general) of the static
  HFB equations

Parameters of the Gaussian ansatzfor the density
operator static spectrum
Eigen-energies of the RPA equations dynamic
spectrum
Spurious energy scale in the thermodynamical
properties
13
Gapless HFB

• Impose that the anomalous mode is solution of the
  static HFB equations

Search L such that
14
Link with the usual regularizing procedure

• Standard approach

UV-div
At the Born level
for the next order
So What !!!
Variational approach
systematic determination of e beyond the LDA
procedure
15
2D Equation Of State (T0)
Popovs EOS
HFB EOS
Schicks EOS
(For Hydrogen )
Possible to probe the EOS using a Feshbach
resonance !
(Example K100)
16
Thomas-Fermi Limit
Trap parameters
Comparison between
LDA Popov EOS .and the
full variational scheme
17
Velocity effects on the coupling constant

• 2-body scattering theory

(Large distance behavior)
Effective coupling constant
with L determined by the mode amplitudes
Expect velocity dependence at the mean field level
18
The anomalous mode of the vortex

• Understanding the tragic fate of a single
• vortex
• The unexpected stabilization of the core at
• finite temperature

D.S. Rokhsar, Phys. Rev. Lett 79, 2164 (1997)
Vortex core
Anomalous mode
T. Isoshima and K. Machida, Phys. Rev. A 59,
2203 (1999)
Usual self-consistent equation
Effective pining potential for the vortex
19
Restoration of the instability

Local Density Appoximation for the t-matrix Full variational approach
function of the local chemical potential depends on the configuration
Calculate the static spectrum without
thermalizing the anomalous mode
20

• Conclusions and perspectives

• lgtgt1 is necessary for observing 2D many-body
  properties
• Closed Formalism from the 2 body problem which
  includes
• velocity effects at the mean-field level ? beyond
  LDA
• Collective modes Time Dependent HFB ? RPA
• ?a possible way to probe the EOS ?
• Variational description of the quasi-condensate
  phase

21
Appendix

• 1) Minimizing the Grand-potential with respect
  to h,D,F

2) The gap equation
3) An equivalent condition for searching L
4) Numerical procedure