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
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
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)
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
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
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
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)
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
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
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
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
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
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
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
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)
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
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
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
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)
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
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
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
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
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
Time evolution of |N,0Po
pula
tion
imba
lanc
eA
nd o
rbita
l ocu
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NU/J=1 NU/J=1.5 NU/J=5
Mele-Messeguer, M.Thesis. (2010)
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 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
B. Juliá-Díaz, Trobades Científiques de la Mediterrània, Menorca, 2010
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
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
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”