Bose-Einstein CondensateFundaments, Excitation and

Turbulence

Vanderlei Salvador Bagnato

Instituto de Física de São Carlos – Universidade de São Paulo

USHUAIA -2012

Lectures:

1) Basic concepts for BEC

2) Excitations – collective modes

Thermodynamics – Global variables

3) Vortices and Quantum turbulence

Future directions

BEC is a superfluid

Can one make an analysis of tem Thermodynamics of an heterogeneous

system using a single marcoscopic variable?

VOLUME PRESSURE

(,,) = Extensive x Intensive

)()(

43

3

zgkT

kTP

3

1

V

)(

)(

3

4

zg

zgNkTPV

rd)r(U)r(n3

2P 3

30

Alternatively, we can use the famous Gibbs-Duhem relation:

The Gibbs-Duhem Relation

dV

NdT

V

SdP

),( TPP P is a fundamental quantity when expressed

in terms of T and - contains all the thermodynamics information of the system.

Taking T constant, N/VConsider an atomic gas of bosons of mass m in a trap

The Gibbs-Duhem Relation

221 ).()( rmrVext

The volume parameter is321

1

V

22

322

222

133

2

1)(

3

2zyxmrrdP

Where )(r is the density profile measured.

rdrrVrV

P ext3)()(.

3

1

To obtain P:

1.T Tc – Thermal Cloud2.T < Tc – Condesate fraction3.Extrapolation to T0

Measurements in three situations

TOFafter )(r

Measurement of situin )(r

Processing to obtain

Based on Y. Castin, and R. Dum: Phys. Rev. Lett. 77, 5315(1996).

rdrrVrV

P ext3)()(.

3

1

Double component velocity distribution

Thomas-Fermi profile for the center

Experimental data

Total fit

Gaussian fit for the wings

Overrall view across the transitionVarying the trapped number of atomssince is constant:variation of N variation of density

The transition line P vs T – Phase DiagramIt occurs from the discontinuity of the derivative of Pc vs Tc

0,0 0,1 0,2 0,3 0,4 0,5 0,6-2,00E-020

0,00E+000

2,00E-020

4,00E-020

6,00E-020

8,00E-020

1,00E-019

1,20E-019

1,40E-019

1,60E-019

1,80E-019

Pc

Pc

Tc

BEC + normal

normal

The relation Pc vs Tc is not like cc TP

Extrapolation T0

For zero temperature:

m

aUzyxm

UrTF

2

022

322

222

10

4where

2

11

density) (harmonic~ where

~157

1 yieldswhich

3

5/75/220

N

amP

Extrapolation T0

Taking the points after the transition we could extrapolate the curves to zero:

Dependence of P(T0) with Number

Finally we plot the zero temperature harmonic pressure as a function of N7/5:

Pressure T0

4,1 and

1,5 found wee wher

~say can ely weAlternativ

theory

0,30,2-

0

P

1-By knowing the equation of state one can measure the heat capacity at constant harmonic volume:

This measurement requires the possibility to adiabatically change the harmonic volume, i.e., change the frequencies of the harmonic trap.

This can be done in an optical trap since:

0, Pzr

Perspectives

SNNVV T

V

T

PTC

,,

Perspectives

2-Another relevant quantity that could be measured is the isothermal compressibility:

-both must diverge at the critical temperature.

Displacement,

Rotation and

Deformation of the potential

ADDITION OF “SHAKING” COILS

EXCITATION BY OSCILLATION OF THE POTENTIAL

Atomic washing machine

Regular BEC

5ms 5,5ms 6ms 6,5ms

7ms 7,5ms 8ms 8,5ms

9ms 9,5ms 10ms 10,5ms

11ms 11,5ms 12ms

QUADRUPOLE AND DIPOLE EXCITATIONS AND ……..

FORMATION OF VORTICES BY OSCILLATORY EXCITATION

Fluctuations at the surface of the BEC

BEC and thermal cloud counter flow

Phys. Rev. A 79, 043618 (2009)

How to form the vortices?