Download Single component and binary mixtures of BECs in double

Single component and
binary mixtures of BECs
in a double-well
Bruno Juliá-Díaz
Departament d’Estructura i Constituents
de la Matèria
Universitat de Barcelona (Spain)
In collaboration with:
D. Dagnino, M. Guilleumas, M. Lewenstein,
J. Martorell, M. Melé-Messeguer, A. Polls
Summary
1.
2.
3.
4.
5.
Introduction
Single component case
Static properties of the
two-site Bose-Hubbard
hamiltonian
Mean-field vs exact
dynamics
(brief) Binary mixtures
of BECs
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
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
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
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
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
Experimentally…
Albiez et al. (2005)
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
A simple, but many-body H
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
Milburn et al (1997)
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
One body density matrix
The one body density matrix reads,

1  aL aL
   
N  aR aL
aL aR 


aR aR 
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
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
semiclassics
The semiclassics is governed by the well known:
z:
:
2J:
population imbalance, (NL-NR)/N
phase difference, R-L
Rabi time (the time it takes for the atoms to go
from left to right and back in absence of atomatom interactions)
Smerzi et al. (1997) (Assuming a two mode ansatz for the Gross Pitaevskii equation)
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
Ground and highest excited state
With the usual base:
|ck|2
Black, ground state
Red, highest excited
|NL,NR>={|N-k,k>}=
{ |N,0>,|N-1,1>,…,|0,N>}
Cat-like state
The hamiltonian can be written
as an N+1 square matrix (here
50+1)
Any N particle vector can be
written as,
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
Properties of the whole spectrum
=0
GS: binomial
=4
In the plot,
x-axis: k index
y-axis: eigenvector index
1, ground,
N+1 highest excited
Color, proportional to |ck|2
GS: Cat-like
=8
GS: Trapped
=12
GS: Trapped
N=50, bias=J/10^10
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
Population imbalance
Ground state: imbalance
NU/J
Blue dashed: Semiclassical prediction: sqrt(1-4/^2)
Red solid: quantum result for the imbalance
Band: dispersion of the imbalance
Julia-Diaz, Dagnino, Lewenstein, Martorell, Polls, PRA (2010)
N=50, bias=J/10^10
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
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)
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
Dispersion of z versus N and 
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
Occupations of the orbitals
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
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
Experimentally observed…
Beautiful experimental exploration ..but just of the mean field
properties. Internal Josephson. (repulsive interactions) (N=500)
T. Zibold et al. (Oberthaler’s group), arXiv 1008.3057
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
G.S. evolution with Lambda
Most occupied eigenstate of ,
normalized to its eigenvalue
dotted: Semiclassical
Red/Blue: quantum
results
N=50, epsilon=J/10^10
Less occupied eigenstate of ,
normalized to its eigenvalue
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
For fixed N and
starting from a ‘meanfield’ like state:
•The smaller the
interaction, the
better the meanfield describes the
exact result.
•Fragmentation
builds up with time
Population imbalance
And orbital ocupations
Time evolution of |N,0
Blue solid: Semiclassical
Black solid: quantum for the imbalance
Red dashed: n1, black dotted, n2
N=50, epsilon=J/10^10
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
Time evolution of |N,0
NU/J=1.5
NU/J=5
Population imbalance
And orbital ocupations
NU/J=1
Mele-Messeguer, M.Thesis. (2010)
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
Time evolution of |N,0
Population imbalance
And orbitalocupations
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/tRabi
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
Time evolution of |N,0
Population imbalance
And orbitalocupations
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/tRabi
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
Binary mixtures
(just a taste)
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
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.
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
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”).
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
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)
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
Beyond two-mode
Mele-Messeguer, et al. arXiv: 1005.5272
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
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
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
A measure of the “spread”
Blue dashed: Semiclassical prediction: sqrt(1-4/^2)
Red solid: quantum result for S=
N=50, epsilon=J/10^10
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010