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

Single component and binary mixtures of BECs in a double-

wellBruno 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. Polls

B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010

1. Introduction2. Single component case3. Static properties of the

two-site Bose-Hubbard hamiltonian

4. Mean-field vs exact dynamics

5. (brief) Binary mixtures of BECs

Summary


Josephson (GP) 1150 atoms 87Rb, trap conditions

as in Albiez/Oberthaler

The initial phase is set to ZERO for all z

Definitions (standard): Z(t) = (Nleft(t)-Nright(t))/Ntotal Phase difference=

=right-left


Self trapping (GP)

1150 atoms, trap conditions as before

If 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 interactions


Experimentally…

Albiez et al. (2005)


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 wellThe bias is here taken very small, Epsilon<<JIt is customary to define, =NU/J

A simple, but many-body H

Milburn et al (1997)


The one body density matrix reads,

RRLR

RLLL

aaaaaaaa

N1

Eigenvalues, n1+n2=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 matrix


The semiclassics is governed by the well known:

z: population imbalance, (NL-NR)/N : phase difference, R-L2J: 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) (Assuming a two mode ansatz for the Gross Pitaevskii equation)


Black, ground stateRed, highest excited

Ground and highest excited state

Cat-like state

With the usual base: |NL,NR>={|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,

|ck|2


GS: binomial

In the plot, x-axis: k indexy-axis: eigenvector index

1, ground, N+1 highest excited

Color, proportional to |ck|2

N=50, bias=J/10^10

Properties of the whole spectrum=0

=4

=8

=12

GS: Cat-like

GS: Trapped

GS: Trapped


Blue dashed: Semiclassical prediction: sqrt(1-4/^2)Red solid: quantum result for the imbalanceBand: dispersion of the imbalanceN=50, bias=J/10^10

Ground state: imbalance

NU/J

Julia-Diaz, Dagnino, Lewenstein, Martorell, Polls, PRA (2010)

Popu

latio

n im

bala

nce


Variation with NThe 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)


Dispersion of z versus N and


Blue dashed: Semiclassical prediction 1,0Red solid: quantum result for the eigenvalues of the one body density matrixN=50, epsilon=J/10^10

Occupations of the orbitals


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)


dotted: Semiclassical Red/Blue: quantum resultsN=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 eigenvalue


Blue solid: Semiclassical Black solid: quantum for the imbalanceRed dashed: n1, black dotted, n2N=50, epsilon=J/10^10

Time evolution of |N,0For 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 Po

pula

tion

imba

lanc

eA

nd o

rbita

l ocu

patio

ns


Time evolution of |N,0Po

pula

tion

imba

lanc

eA

nd o

rbita

l ocu

patio

ns

NU/J=1 NU/J=1.5 NU/J=5

Mele-Messeguer, M.Thesis. (2010)


Time evolution of |N,0When 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/tRabi

Popu

latio

n im

bala

nce

And

orb

italo

cupa

tions


Time evolution of |N,0When 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/tRabi

Popu

latio

n im

bala

nce

And

orb

italo

cupa

tions


Binary mixtures(just a taste)


Binary Mixture 1150 atoms, trap conditions as the

Heidelberg experiment.

We consider now the other limit: (50%,0%,50%) configuration

The initial phases are all ZERO Initial 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.


Binary Mixture (II) 1150 atoms,

We consider a (98%,0%,2%) configuration

The initial phases are all ZERO Initial population imbalances

are: z1 (0) = - z-1(0)

The most populated component follows the usual Josephson oscillation

The less populated one follows the most populated one (“anti-Josephson”).


A novel way of extracting a0 and a2

We proposed an experimental way of accessing the spin independent and spin dependent scattering lenghts: a0 and a2

The key points are:

a) 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~gab

b) Perform two measurements:1) highly polarized, Na>>Nb

2) Na~Nb and za(0)=-zb(0)

c) Extract from za(t) and zb(t) the scattering lengths.Julia-Diaz, Guilleumas, Lewenstein, Polls, Sanpera (PRA 2009)


Beyond two-mode

Mele-Messeguer, et al. arXiv: 1005.5272


Summary Single component case

Static properties of the Bose-Hubbard hamiltonian with small bias

Existence of strongly correlated ‘cat-like’ ground states Cat-like states appear in the spectrum when the mean

field predicts a bifurcation Can 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: 1005.5272


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”

Documents

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