Transcript Seminar C

Atomic Bose-Einstein
Condensates Mixtures
• Introduction to BEC
• Dynamics: (i) Quantum spinodial
decomposition, (ii) Straiton, (iii) Quantum
nonlinear dynamics.
• Self-assembled quantum devices.
• Statics: (a) Broken symmetry ? (b)
Amplification of trap displacement
Collaborators:
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P. Ao
Hong Chui
Wu-Ming Liu
V. Ryzhov
Hulain Shi
B. Tanatar
E. Tereyeva
Yu Yue
Wei-Mou Zheng
Introduction to BEC
• Optical, and Magnetic traps
• Evaporative Cooling
• http://jilawww.colorado.edu/bec/
Formation of BEC
Slow expansion after 6 msec at
T<Tc, T~Tc and T>>Tc
Mixtures:
Different spin states of Rb (JILA) and
Na (MIT).
Dynamics of phase separation: From
an initially homogeneous state to a
separated state.
Static density distribution
Classical phase separation:
spinodial decomposition
• At intermediate times a state with a periodic
density modualtion forms.
• Domains grow and merge at later times.
Physics of the spinodial
decomposition
• 2<0 for small q.
• From Goldstone’s
theorem, q2=0 when
q=0.
• For large enough q,
q2 >0
2
q
qsd
Dynamics: Quantum spinodial
state
In classical phase separation, for
example in AlNiCo, there is a
structure with a periodic density
modulation called the spinodial
decomposition. Now the laws are
given by the Josephson
relationship. But a periodic density
modulation still exists.
Densities at
different times
• D. Hall et al.,
• PRL 81, 1539
(1998).
• Right: |1>
• Middle:|2>
• Left: total
Intermediate time periodic state:
• Just like the classical case, the fastest
decaying mode from a uniform phase
occurs at a finite wavevector.
• This is confirmed by a linear instability
analysis by Ao and Chui.
Metastability:
• Sometimes the state with the periodic
density exists for a long time
H-J Miesner at al. (PRL 82, 2228
1999)
Metastability:
• Solitons are metastable because they are
exact solutions of the NONLINEAR
equation of motion
• Solitons are localized in space. Is there an
analog with an EXTENDED spatial
structure?---the ``Straiton’’
Coupled Gross-Pitaevskii
equation
• U: interaction potential; Gij, interaction
parameters
 t i  [U  i  h  / 2m   j Gij |  j | ] i
2
2
2
A simple exact solution:
• When all the G’s are the same, a solution
exist for 1  c sin(kx ) ,  2  c cos(kx )
• For this case, the composition of the
mixture is 1:1.
Coupled Gross-Pitaevskii
equation
• U: interaction potential; G, interaction
parameters:
 t i  [U  i  h  / 2m  G | c | ] i
2
2
2
More Generally, in terms of
elliptic functions
1  c1 exp( i1t )sn(kx, p)
•
2  c2 exp( i2t )cn(kx, p)
•
• N1/N2=(G12-G22)/(G11-G12) for
G11>G22>G22 ( correspons to Rb)
• N1/N2=1 for G11=G22=G12. This can be
related to Na (G11=G12>G22) by
perturbation theory.
Domains of metastability
• Exact solutions can be found for the one
dimensional two component Gross-Pitaevskii
equation that exhibits the periodic density
modulation for given interaction parameters
only for certain compositions.
• Exact solutions imply metastability: that the
nonlinear interaction will not destroy the state.
• Not all periodic intermediate states are
metastable?
Density of component 1:
Numerical Results
• Na, 1D
• MIT
parameters
• 1:1
Total density
• Na
• MIT
parameters
• 1:1
• Gij are close to
each other
Phase Separation Instability:
• Interaction energy: G11n1  G22n22  2G12n1 n2
• Insight: G11  G22
• The energy becomes :
2
(G11  G12 )( n1  n2 )2 / 2  (G11  G12 )( n1  n2 )2 / 2
• Total density normal mode stable.
• The density difference is unstable when
G11  G12
Results from Linear Instability
Analysis
• Period is inversely proportional to the
square root of the dimensionless coupling
constant.
• Time is proportional to period squared.
Hypothesis of stability:
• System is stable only for compositions close
to 1:1.
Quantum nonlinear dynamics: a
very rich area
• Rb
• 4:1
• Periodic
state no
longer
stable
• Very
intricate
pattern
develops.
Self assembled quantum devices
• For applications such as atomic
intereferometer it is important to put equal
number of BEC in each potential well.
Self-assembled quantum devices
• Phase separation in a
periodic potential.
• Two length scales: the
quantum spinodial
wavelength qs and the
potential period
l=2(a+b).
Density distribution of component 1
as a function of time
• Density is uniform at
time t=0.
• As time goes on, the
system evolves into a
state so that each
component goes into
separate wells.
How to pick the righ parameters:
• Linear stability analysis can be performed
with the transfer matrix method.
• In each well we have  j=[Ajeip(x-nl)+Bjeip(x-nl)]ei t
• Get cos(kl)=cos2qa cos2pb-(p2+q2)sin2qa
sin2pb/2pq.
How to pick the right parameters?
• k=k1+ik2; real
wavevector k1 l (solid
line) and imaginary
wavevector k2 l
(dashed line) vs 2.
• Fastest mode occurs
when k1 l¼ 
Topics
• Quantum phase segregation: domains of
metastability and exact solutions for the
quantum spinodial phase. The dynamics
depends on the final state.
• What are the final states? Broken
symmetry: A symmetric-asymmetric
transition.
• Amplification of trap offsets due to
proximity to the symmetric-asymmetric
transition point.
A schematic illustraion:
• Top: initial
homogeneous state.
• Middle: separated
symmetric state.
• Bottom: separated
asymmetric state.
Asymmetric states have lower
interface area and energy
Asymmetric
A
Symmetric
• Illustrative example:
equal concentration in
a cube with hard walls
• For the asymmetric
phase, interface area is
A.
• For the asymmetric
phase, it is 3.78A
Different Gii’s favor the
symmetric state:
• The state in the middle has higher density.
The phase with a smaller Gii can stay in the
middle to reduce the net inta-phase
repulsion.
Physics of the interface
• Interface energy is of the order of
2
1/ 2
2
• [(G12  G22G11 ) / ] /   in the weakly
segragated regime
• The total density from the balance between
the terms linear and quadratic in the density,
the gradient term is much smaller smaller
• The density difference is controlled by the
gradient term, however
Some three dimensional example
Broken symmetry state:
• Density at z=0 as a
function of x and y for
the TOPS trap.
• Right: density
difference.
• Left: total density of 1
and 2.
Broken symmetry state
• Right: density of
component 1.
• Left: density of
component 2.
Symmetric state
• Right: density
difference of 1 and 2
• Left: sum of the
density of 1 and 2
Smaller droplets: Back to
symmetric state
Different confining potentials:
• The TOP magnetic trap provides for a
confing potential
V (r )  ( An12  Bn22 )r 2
• We describe next calculations for different
A/B and different densities.
A/B=2, Back to symmetric State
A/B=1.5, back to symmetric state
When the final phase is more
symmetric:
• Na
• 2:1
• Now
G11>G22
• Before
G22>G11
Symmetric final State: Domain
growth
• G11=G22
• 2:1
Amplification of the trapping
potential displacement
• Trapping potential of
the two components:
dz is the displacement
of one of the potential
from the center.
• The displacement of
the two components
are amplified.
dz
Expet.
Result
• Hall et al.
Amplicatifation of the center of
mass difference as a function of
potential offset
• Thomas Fermi
approximation: Ratio
is about 70 for small
offsets. For large
offsets the ratio is
much smaller.
• ``Exact calculation’’:
The trend is smoother
Physics: Close to the critical
point of change of symmetry
• Asymmetric solution favored by domain
wall energy
• for G11 >G22, component 2 is inside where
the density is higher and the self repulsion
can be lowered.
• Critical point occurs when   G11 /G22 =1
• In the Thomas Fermi approximation the
amplification factor is proportional to 1/(
1).
Boundaries of the droplet for 3%
offset
0.002
N1=N2

0.001
z (cm)
• Nearly complete
separation.
• Results from ThomasFermi approximation.
2
0.000
1
-0.001
-0.002
-0.002
-0.001
0.000
r (cm)
0.001
0.002
Density of components 1 and 2
• Trap offset is only 3
per cent of the radius
of the droplet.
• y=0
• Results from Monte
Carlo simulation.
Boundaries for 0.3% potential
offset
0.002
N1=N2

0.001
z (cm)
• Big displacement but
not yet separated.
• Results from ThomasFermi approximation.
2
0.000
1
-0.001
-0.002
-0.002
-0.001
0.000
r (cm)
0.001
0.002
Density of components 1 and 2
• Trap offset is 0.3 per
cent the radius of the
droplet.
Density of component 2
• Trap offset 0.3%
Density of component 1
• Trap offset 0.3%