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:• 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 q
2
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
j ijijiit GmhU ]||2/[ 222
A simple exact solution:
• When all the G’s are the same, a solution exist for ,
• For this case, the composition of the mixture is 1:1.
)sin(1 kxc )cos(2 kxc
Coupled Gross-Pitaevskii equation
• U: interaction potential; G, interaction parameters:
iiit cGmhU ]||2/[ 222
More Generally, in terms of elliptic functions
•
•
• 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.
),()exp( 111 pkxsntic ),()exp( 222 pkxcntic
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:
• Insight:
• The energy becomes :
• Total density normal mode stable.
• The density difference is unstable when
G n G n G n n11 1
2
22 22
12 1 22 G G11 22
2/))((2/))(( 2211211
2211211 nnGGnnGG
1211 GG
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)+Bje-ip(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
• 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
Asymmetric
Symmetric
A
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 • 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
[( ) / ] //G G G122
22 111 2 2
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
• We describe next calculations for different A/B and different densities.
V r An Bn r( ) ( ) 12
22 2
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 =1
• In the Thomas Fermi approximation the amplification factor is proportional to 1/( -1).
2211 /GG
Boundaries of the droplet for 3% offset
• Nearly complete separation.
• Results from Thomas-Fermi approximation.
-0.002 -0.001 0.000 0.001 0.002
-0.002
-0.001
0.000
0.001
0.002
1
2
N1=N2
r (cm)
z (c
m)
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
• Big displacement but not yet separated.
• Results from Thomas-Fermi approximation.
-0.002 -0.001 0.000 0.001 0.002
-0.002
-0.001
0.000
0.001
0.002
N1=N2
1
2
r (cm)
z (c
m)
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%
