THE WEAKLY INTERACTING BEC

THE WEAKLY INTERACTING BEC

4. Superfluid hydrodynamics

Summary of the previous lectures

• The properties of dilute Bose-Einstein condensates can be described within the mean-field approximation.

• All particles possess the same wave-function F solution of the Gross-Pitaevskii Equation (GPE)

• In the Thomas-Fermi approximation, Na/aho>>1, the GPE reduces to the LDA.

22 2( ) | |

2ti U r Ng

m

Low-lying excitations

Bose-Einstein condensate close to equilibrium

0 /( , ) ( ) ( , )

i tt e t

r r r

For a weak perturbation: dF<<

Expansion of the time-dependent EGP to first order:

GP* *ti

with 2 2

0

GP *2 2

0

2 | |

2 | |

h gN gN

gN h gN

22

0 0( )2

h Um

r

! LGP is not a hermitian operator!

The Bogoliubov spectrum

Find the spectrum of LGP for a uniform system: n0=N||2; µ0=gn0.

GP

u u

v v

( )

( )

ik

k

uu e

vv V

k.rr

rand

2 2

0 0

GP 2 2

0 0

/ 2

/ 2

k m gn gn

gn k m gn

w is eigenvalue of

Bogoliubov spectrum: 2 2 2 2

0/ 2 ( / 2 2 )k m k m gn

cosh( )

sinh( )

k k

k k

u

v

2 2

tanh(2 )/ 2

k

gn

k m gn

Bogoliubov Spectrum (II)

g<0: collapse instablity

g>0:Bogoliubov spectrum

k

w

w2

k

Acoustic branch

ck

Sound velocity 2

0 mc

Crossover for 2 2

02

k

m

Measurement of the Bogoliubov spectrum

Bragg spectroscopy of elementary excitations

1 1,k 2 2,k

Energy momentum conservation

Resonant transfer of photons for

Bogo 1 2 1 2( ) | | k k

w

k

w1,2

w2,1

Steinhauer et al. Phys. Phys. Lett. 88, 120407, (2002).

Landau’s criterion for superfluidity

Energy-momentum conservation: the motion of the impurity is damped by emission of elementary excitations if

min kV ck

Raman et al. PRL 83, 2502 (1999).

M

V V’

k,w

Superfluid Hydrodynamics

Alternative approach to the dynamical properties of a Bose-Einstein condensate

Madelung’s Transform: ( , )( , )

( , ) i tn tt e

N

rrr

Using Gross-Pitaevskii’s Equation

( ) 0tn n v

22 2/ 2

2tm v gn U mv n

m n

Long wave-length/Thomas-Fermi/large cloud limit

Hydrodynamic equations for a viscousless fluid.

m v Potential flow

Conservation equation

Euler’s Equation

Simple solutions of the hydrodynamic equations

Stationnary solution:

0

( ) ( ) cte

v

gn U

r r Thomas-Fermi profile!

Homogeneous system close to equilibrium:

( )

1

( )

0 1

i t

i t

e

n n n e

k.r

k.rck with

In a harmonic trap: scaling solutions

0 /c gn m

1 2

Cylindrical trap: 2 2 2 4 4 2 21

3 4 9 16 162

z z z

THE WEAKLY INTERACTING BEC

5. Strongly interacting gases

The Bogoliubov Hamiltonian

Second quantized Hamiltonian:

3 † † †

0ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) ( ) ( )

2

gH d h r r r r r r r

ˆ ˆ( ) ( )r N r

Expand H with dY

†

3

0 † †

ˆ ˆ1...

ˆ ˆ2GPH E d

r

The Bogoliubov Hamiltonian: Homogeneous system

Expand the field operators in the plan-wave basis ˆ ˆie

aV

k.r

k

k

† 2 2

0 0

0 † †2 2

0 0

ˆ ˆ/ 21

ˆ ˆ2 / 2

k k

k k k

a ak m gn gnH E

a agn k m gn

Bogoliubov transform:

†

† †

ˆ ˆ ˆ

ˆ ˆ ˆ

k k k k k

k k k k k

b u a v a

b u a v a

†

0ˆ ˆˆ

k k k

k

H E b b

For uk2-vk

2=1, the bk‘s are bosonic annihilation operators.

Quantum depletion and Lee-Huang Yang corrections

The ground state corresponds to the vacuum of the Bogoliubov excitations.

Even at T=0, states of finite momentum are populated

† 2

0

0

k k k k

k k

N a a v

Quantum depletion of the ground state:

38

3

Nna

N

Energy of the ground state (Lee-Huang-Yang formula)

3

Mean Field

1281

15E E na

Experimental demonstration of the Lee-Huang-Yang corrections

Averaging images of 7Li between 1440 and 2150 a0

We find : 4.5(7)

Navon et. al., PRL 107,135301 (2011)

Pm

a5/

2

And beyond Lee-Huang-Yang?

When getting closer to the Feshbach resonance the lifetime of the could decreases!

Three-body recombination:

The binding energy is large compared to the potential height: the three atoms are lost.

Phenomenological law : 2

3

dNL n N

dt

Three-body losses in a Bose gas close to a Feshbach resonance

Gross et. al., PRL 2009 (also Pollack et. al., Science 2010 ), also Innsbruck, Florence, Rice…)

Dimensionally: 4

3

aL

m

7Li

Quasi-equilibrium for µ/>>L3n2 3 1na

Hard to enter to strongly correlated regime with free bosons!