Single component and binary mixtures of BECs in a double-we - PowerPoint Presentation

Slide1

Single component and binary mixtures of BECs in a double-well

Bruno Juliá-DíazDepartament d’Estructura i Constituents de la MatèriaUniversitat de Barcelona (Spain)

In

collaboration with:

D.

Dagnino

, M.

Guilleumas

, M.

Lewenstein

, J.

Martorell

,

M.

Melé-Messeguer

, A. PollsSlide2

Introduction

Single component case

Static properties of the

two-site

Bose-Hubbard

hamiltonian

Mean-field

vs

exact dynamics

(brief) Binary mixtures of BECs

SummarySlide3

Josephson (GP)

1150 atoms 87Rb, trap conditions as in Albiez/OberthalerThe initial phase is set to ZERO for all z

Definitions (standard):Z(t) = (

Nleft(t)-N

right(t))/N

total

Phase difference=

=

right

-

leftSlide4

Self trapping (GP)

1150 atoms, trap conditions as beforeIf the initial imbalance is large enough, no Josephson oscillation occurs. Instead a self trapping regime appears Smerzi et al. (1997)

This

is a

genuine effect of having atom-atom contact interactionsSlide5

Experimentally…

Albiez

et al. (2005)Slide6

Lets consider the following

two-site Bose-Hubbard model:

J:

hopping

parameter >

0

U:

atom-atom

interaction >0

(proportional to

g

)(

attractive)

Epsilon:

Bias>0

, promotes the left well

The bias is here taken very small, Epsilon<<J

It is customary to define,

=NU/J

A simple, but many-body H

Milburn et al (1997)Slide7

The one body density matrix reads,

Eigenvalues

, n

1

+n

2

=1

If the system is

fully condensed

,

then the

eigenvalues are 1 and 0. The eigenvector corresponding to 1 is,

Departure from 0,1 indicates the system is

fragmented

One body density matrixSlide8

The

semiclassics

is governed by the well known

:

z: population imbalance, (N

L

-N

R

)/N

:

phase difference, R-

L

2J:

Rabi time

(the time it takes for the atoms to go from left to right and back in absence of atom- atom interactions)

semiclassics

Smerzi et al. (1997

)

(

A

ssuming

a two

mode

ansatz

for the Gross

Pitaevskii

equation

)Slide9

Black, ground state

Red, highest excited

Ground and highest excited state

Cat-like state

With the usual base:

|N

L

,N

R

>={|N-

k,k

>}= { |N,0>,|N-1,1>,…,|0,N>}

The

hamiltonian

can be written as an N+1 square

matrix (here 50+1)

Any

N particle

vector can be written as,

|

c

k

|

2Slide10

GS: binomial

In the plot,

x-axis: k index

y-axis: eigenvector index

1, ground,

N+1 highest excited

Color, proportional to |

c

k

|

2

N=50, bias=J/10^10

Properties of the whole spectrum

=0

=4

=8

=12

GS: Cat-like

GS: Trapped

GS: TrappedSlide11

Blue dashed:

Semiclassical

prediction:

sqrt

(1-4/



^2)

Red solid: quantum result for the imbalance

Band: dispersion of the imbalance

N=50, bias=J/10^10

Ground state: imbalance

NU/J

Julia-Diaz,

Dagnino

,

Lewenstein

,

Martorell

, Polls, PRA (2010)

Population imbalanceSlide12

Variation with N

The

semiclassical

behavior is the same in all cases (the bias is taken the same)

The size of the highly disperse region decreases abruptly as N is increased

For which value of



does

the ‘quantum hop’ take place?

It turns out to be an interesting interplay between N, U, J and the bias:

Julia-Diaz,

Martorell

, Polls (2010)Slide13

Dispersion of z versus N and

Slide14

Blue dashed

:

Semiclassical

prediction



1,0

Red solid:

quantum result for the

eigenvalues

of the one body density matrix

N=50, epsilon=J/10^10

Occupations of the

orbitalsSlide15

T.

Zibold

et al. (

Oberthaler’s

group),

arXiv

1008.3057

Experimentally observed…

Beautiful experimental exploration

..but just of the mean field properties. Internal Josephson. (repulsive interactions) (N=500)Slide16

dotted:

Semiclassical

Red/Blue: quantum results

N=50, epsilon=J/10^10

G.S. evolution with Lambda

Most occupied

eigenstate

of

, normalized to its

eigenvalue

Less occupied

eigenstate

of

, normalized to its

eigenvalueSlide17

Blue solid:

Semiclassical

Black solid: quantum for the imbalance

Red dashed: n

1

, black dotted, n

2

N=50, epsilon=J/10^10

Time evolution of |N,0



For

fixed N

and

starting from a ‘mean-field’ like state

:

The smaller the interaction, the better the mean-field describes the exact result.

Fragmentation builds up with time

Population imbalance

And orbital

ocupationsSlide18

Time evolution of |N,0



Population imbalance

And orbital

ocupations

NU/J=1

NU/J=1.5

NU/J=5

Mele-Messeguer

,

M.Thesis

. (2010)Slide19

Time evolution of |N,0



When starting from a ‘mean-field state’:

For small N (here 10), clear deviations are quickly seen (less than a Rabi time here) between the mean-field/

semiclassics

results and the full quantum behavior

Correspondingly, the cloud is far from condensed as time evolves

t/t

Rabi

Population imbalance

And

orbitalocupationsSlide20

Time evolution of |N,0



When starting from a ‘mean-field state’:

For large N (here 1000), the mean field provides an excellent account of the full dynamics during long times (here almost two Rabi periods)

The cloud, thus, remains condensed for a while.

t/t

Rabi

Population imbalance

And

orbitalocupationsSlide21

Binary mixtures

(just a taste)Slide22

Binary Mixture

1150 atoms, trap conditions as the Heidelberg experiment.We consider now the other limit: (50%,0%,50%) configurationThe initial phases are all ZEROInitial population imbalances are: z1 (0) = - z-1(0)Note there would be no

josephson at all if both components were just one

A longer oscillation is seen.Slide23

Binary Mixture (II)

1150 atoms,We consider a (98%,0%,2%) configurationThe initial phases are all ZEROInitial population imbalances are: z1 (0) = - z-1(0)

The most populated component follows the usual Josephson oscillationThe less populated one follows the most populated one

(“anti-Josephson”).Slide24

A novel way of extracting a

0 and a2We proposed an experimental way of accessing the

spin independent and spin dependent scattering lenghts:

a0 and a

2The key points are:

Consider a

binary mixture

made by populating the (F=1, m=1) and (F=1, m=-1) states of an F=1

spinor

. In this way,

gaa=

gbb~gabPerform two measurements:highly polarized, N

a

>>

N

b

N

a

~N

b

and

z

a

(0)=-

z

b

(0)

Extract from

z

a

(t) and

z

b

(t) the scattering lengths.

Julia-Diaz,

Guilleumas

,

Lewenstein

, Polls,

Sanpera

(PRA 2009)Slide25

Beyond two-mode

Mele-Messeguer, et al. arXiv: 1005.5272Slide26

Summary

Single component caseStatic properties of the Bose-Hubbard hamiltonian with small biasExistence of strongly correlated ‘cat-like’ ground statesCat-like states appear in the spectrum when the mean field predicts a bifurcationCan we see any ‘traces of the cats’?

When does the mean-field description break down?

J-D, Dagnino

, Lewenstein,

Martorell

, Polls, PRA A 81, 023615 (2010)

J-D,

Martorell

, Polls, PRA81, 063625 (2010)

Binary mixtures populating m=+/-1

of an F=1

spinor

Extraction of the scattering lengths

Complete analysis, beyond two-mode, …

J-D,

Guilleumas

,

Lewenstein

, Polls,

Sanpera

, PRA 80, 023616 (2009)

Mele-Messeguer

, J-D,

Guilleumas

, Polls,

Sanpera

,

arXiv

: 1005.5272Slide27

Blue dashed

:

Semiclassical

prediction:

sqrt

(1-4/



^2)

Red solid

: quantum result for S=

N=50, epsilon=J/10^10

A measure of the “spread”