VARIATIONAL APPROACH FOR THE TWO-DIMENSIONAL TRAPPED BOSE GAS L. Pricoupenko Trento, 12-14 June 2003 LABORATOIRE DE PHYSIQUE THEORIQUE DES LIQUIDES Université.

VARIATIONAL APPROACH FOR THE VARIATIONAL APPROACH FOR THE

TWO-DIMENSIONAL TWO-DIMENSIONAL

TRAPPED BOSE GASTRAPPED BOSE GAS

L. Pricoupenko Trento, 12-14 June 2003

LABORATOIRE DE PHYSIQUE THEORIQUE DES LIQUIDES

Université Pierre et Marie Curie (Paris)

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

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

The actual experimental settings

MIT

Firenze

Innsbrück

Villetaneuse

22

2222

2)(

2)( z

myx

mV zext yx

r

Hz 790 2 Hz 10 2 Hz 30 2yx

z

410345 N

Reach the 2D regime by decreasing N in an anisotropic trap

Hz 2,10 2 Hz 50 2yx

kz

A. Görlitz and al. Phys. Rev. Lett 87, 130402 (2001)

Hz 20 2 Hz 3.16 2 Hz 187 2yx

kz

600107 N

Use a 1D optical lattice Slices of 2D condensates

S. Burger and al. Europhys. Lett., 57, pp. 1-6 (2002)

Evanescent-wave trapping

510200 N S. Jochim and al. Phys. Rev. Lett., 90, 173001 (2003)

Hz 30 2 Hz 30 2yx

kz ?1000 NEvanescent-wave trapping

zAnisotropy parameter

Atoms trapped in a planar wave guide

),()()(),;,( 2120102211 rrzzzrzr

z

22

2)( z

mzV z

ext

ωTk zB zz

N

E

2

Two-body problem:

)(),( 212

2121 ln rrO

aArr

D

rr

zarr 21

21212121

2

),(),()(2

rrforrrErrm

Zero range approach:

Eigenvalue problem defined by the contact conditions :

D

zzD a

aaa

32 2

exp092.2

The “2D induced” scattering length

Maxim Olshanii (private communication)Dima Petrov and Gora Shlyapnikov, Phys. Rev. A 64, 100503 (2001)

The pseudo-potential approach

Motivation : Hamiltonian formulation of the problem

rrrgrr rr

lim)(,0

2121

FermiV

Example : the Fermi-Huang potential in 3D

.ln10

. lim rrrqr

R

Construct a potential which leads to the contact condition of the 2-body problem

Zero range potential Regularizing operator

The « potential » in the 2D world RV )(, 2121 rrgrr

Daqmg

2

2

ln

2

2

)exp(q

2-body t-matrix at energy m

22

Many-body problem for trapped atoms

)(),...,...,...,(

)(2

2221

0

11

22

0

ln jiD

jrirjrir

NN

NN

i

N

iext

N

ii

rrOa

Arjr

irrr

jiEψψH

rVm

H

ji

ji rrHH )(0

V

1) Contact conditions

2) Pseudo-potential jiEψHψ NN ,

1

122

2

D

z

na

na Validity of the zero range approach

Validity of the mean-field approach

Two possibilities

Constraints on the mean density

Summary of the zero-range approach

Mean inter-particle spacing

Possible description of a molecular phase

freedom

a2D>0 can be tuned via a3D (Feshbach resonance)

1 z

highly anisotropic traps

D

D

qa

qar

K

rBound

2

20

)(

zal

Observables do not depend on the

particular value of

Possible study of a highly correlated dilute system

fixed

02

22

z

D

na

na

Condensate/Quasi-condensate

2

6

N

Tk CB

CCTF T

T

T

T

R

Rexp

TTTeTR /

2)(N

TkB

Name DTF

TF 24

ln22

12

22/1

T=0K + Thomas-Fermi

Near T=Tc

Almost BEC Phase in near future experiments

z

1000max

z

2D character

Actual experiments

Hz 10 2

Hz 10 2

kz

The ingredients of HFB

U(1) symmetry breaking approach (Phase of the condensate fixed : T<<T)

Gaussian Variational ansatz

ˆˆ

)(

)(

),(*

),(*

),(),()()(21

1

1

2121

2121

22 ˆ

ˆˆˆ

2

1ˆr

r

rrrr

rrrrrr

h

hrdrdK

)ˆexp(1ˆ

var KZ

D

(Number of atoms fluctuates)

Use the 2D zero range pseudo-potential

A Dangerous game ! ! !

The atomic Bose gas is not the ground state of the system

BEC Phase

HFB Equations

• Generalized Gross-Pitaevskii equation

)()()()(2

)()()()(2

12*11

22

12112

2

rvrurvrVm

rurvrurVm

nnn*

next

nnnnext

)()(2ˆ2ˆ)()(2)()(2

*222

rr/rR/rRgrrnrgrVm excext

R

• “Static spectrum”

2ˆ2ˆ12 /rR/rRgR

R RRgR

ˆˆ211

Implicit Born approximationPairing field (satisfies the contact condition)

The gap spectrum “disaster”

Change the phase of cost no energy

Anomalous mode solution of the linearized time dependent equations (RPA)

(*NOT SOLUTION (in general) of the static HFB equations

ii

ii

)exp(

)exp(* *

*v

u

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

Gapless HFB

02ˆ2ˆ*

/rR/rR

RSearch such that

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

D

excext

aqmg

rrrnrgrVm

2*

2

222

ln

2andequationsmodal

)()()(2)()(2

*

*

Link with the usual regularizing procedure

Standard approach :

)(2 Bbare tgAt the Born level

)()( )2( rgrV bare

mk

kd

tg kkBbare

222

2

2

1

2)(

11max

2/(ˆ)2/(ˆ

2

0

212

2

11

rRrRg

ng

rbare

excbare

lim

UV-div

…for the next order

systematic determination of beyond the LDA procedure

Variational approach

2D Equation Of State (T=0)

2

212

2

2

4ln

4)(

Dame

mn

exc

D

exc

ng

gm

aqmg

mgn

2

ln

124

2

*

2

*

2*22

*

2

*

2

2*

22

2

ln

4

Dnam

HFB EOS

Popov’s EOS

Schick’s EOS

322 102 Dna(For Hydrogen : )

Possible to probe the EOS using a

Feshbach resonance !

212

max

2

22 4

exp4

K

N

e

am DTF

KTF max (Example: K=100)

Thomas-Fermi Limit

atoms Cesium and Hz 10 2 Hz 10 2 kz

22

2)( R

mRnlocal

)(ln2

2ˆ2ˆ *12 rrqRm

/rR/rR Ο

Trap parameters:

Comparison between … LDA +Popov EOS

….and the full variational scheme

03 250

10000

aa

N

D

03 2700

4000

aa

N

D

Velocity effects on the coupling constant

2-body scattering theory

k

kfk

2

)()(

2

2ln

2)(

2

ikqa

kf

D

*2

*

ifm

g

with* determined by the mode amplitudes

4exp

8

)().exp()(

krikr

kfrkir

(Large distance behavior)

Effective coupling constant

Expect velocity dependence at the mean field level

The anomalous mode of the vortex

Understanding the tragic fate of a single

vortex

The unexpected stabilization of the core at

finite temperature

0

00

z

f

trapz L

tt

L

t

D.S. Rokhsar, Phys. Rev. Lett 79, 2164 (1997)

T. Isoshima and K. Machida, Phys. Rev. A 59, 2203 (1999)

corecorelcore xx 11

)()()(2)()(2

222

rrrgnrgrVm excext

Usual self-consistent equation

Effective “pining potential” for the vortex

Anomalous mode

Vortex core

Restoration of the instability

)(2 Bt

R R

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

Conclusions and perspectives

>>1 is necessary for observing 2D many-body properties

Closed Formalism from the 2 body problem which includesvelocity effects at the mean-field level beyond LDA

Collective modes : Time Dependent HFB RPAa possible way to probe the EOS ?

Variational description of the quasi-condensate phase

Appendix1) Minimizing the Grand-potential with respect to h,

)()()()()(2

)(

)()(2

*1211

22

212)(),(

2112)()(

22

),(

21

2121

2121212121

rrrrngrVm

Vm

h

excext

rrrrrr

rrrrrrrrrr ext

)(ln2

2ˆ2ˆ *12 rrqRm

/rR/rR Ο

3) An equivalent condition for searching

rmNg

qm

kdRvRug/rR/rR

kEE

*nnn

kn

r

2

2

*12

122

2

ln2

2)2(212ˆ2ˆ

0

4) Numerical procedure

mm

kr

rkikdrRrRg 2222

).exp(

)2(2/(ˆ)2/(ˆ

2

2

120

212

lim

2) The “gap equation