Atomic Bose-Einstein Condensates Mixtures

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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

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Collaborators

• P. Ao
• Hong Chui
• Wu-Ming Liu
• V. Ryzhov
• Hulain Shi
• B. Tanatar
• E. Tereyeva
• Yu Yue
• Wei-Mou Zheng

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Introduction to BEC

• Optical, and Magnetic traps
• Evaporative Cooling
• http//jilawww.colorado.edu/bec/

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Formation of BEC
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Slow expansion after 6 msec at TltTc, TTc and
TgtgtTc
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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

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Classical phase separation spinodial
decomposition

• At intermediate times a state with a periodic
  density modualtion forms.
• Domains grow and merge at later times.

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Physics of the spinodial decomposition

• ?2lt0 for small q.
• From Goldstones theorem, ?q20 when q0.
• For large enough q, ?q2 gt0

?2
q
qsd
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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.

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Densities at different times

• D. Hall et al.,
• PRL 81, 1539 (1998).
• Right 1gt
• Middle2gt
• Left total

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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.

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Metastability

• Sometimes the state with the periodic density
  exists for a long time

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H-J Miesner at al. (PRL 82, 2228 1999)
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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

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Coupled Gross-Pitaevskii equation

• U interaction potential Gij, interaction
  parameters

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A simple exact solution

• When all the Gs are the same, a solution exist
  for ,
  
• For this case, the composition of the mixture is
  11.

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Coupled Gross-Pitaevskii equation

• U interaction potential G, interaction
  parameters

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More Generally, in terms of elliptic functions

• 
  
• N1/N2(G12-G22)/(G11-G12) for G11gtG22gtG22 (
  correspons to Rb)
• N1/N21 for G11G22G12. This can be related to
  Na (G11G12gtG22) by perturbation theory.

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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?

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Density of component 1 Numerical Results

• Na, 1D
• MIT parameters
• 11

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Total density

• Na
• MIT parameters
• 11
• Gij are close to each other

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Phase Separation Instability

• Interaction energy
• Insight
• The energy becomes
• Total density normal mode stable.
• The density difference is unstable when

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Results from Linear Instability Analysis

• Period is inversely proportional to the square
  root of the dimensionless coupling constant.
• Time is proportional to period squared.

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Hypothesis of stability

• System is stable only for compositions close to
  11.

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Quantum nonlinear dynamics a very rich area

• Rb
• 41
• Periodic state no longer stable
• Very intricate pattern develops.

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Self assembled quantum devices

• For applications such as atomic intereferometer
  it is important to put equal number of BEC in
  each potential well.

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Self-assembled quantum devices

• Phase separation in a periodic potential.
• Two length scales the quantum spinodial
  wavelength ?qs and the potential period l2(ab).

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Density distribution of component 1 as a function
of time

• Density is uniform at time t0.
• As time goes on, the system evolves into a state
  so that each component goes into separate wells.

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How to pick the righ parameters

• Linear stability analysis can be performed with
  the transfer matrix method.
• In each well we have ? ?jAjeip(x-nl)Bje-ip(x-nl
  )ei? t
• Get cos(kl)cos2qa cos2pb-(p2q2)sin2qa
  sin2pb/2pq.

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How to pick the right parameters?

• kk1ik2 real wavevector k1 l (solid line) and
  imaginary wavevector k2 l (dashed line) vs ?2.
• Fastest mode occurs when k1 l¼ ?

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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.

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A schematic illustraion

• Top initial homogeneous state.
• Middle separated symmetric state.
• Bottom separated asymmetric state.

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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
A
Symmetric
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Different Giis 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.

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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

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Some three dimensional example
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Broken symmetry state

• Density at z0 as a function of x and y for the
  TOPS trap.
• Right density difference.
• Left total density of 1 and 2.

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Broken symmetry state

• Right density of component 1.
• Left density of component 2.

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Symmetric state

• Right density difference of 1 and 2
• Left sum of the density of 1 and 2

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Smaller droplets Back to symmetric state
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Different confining potentials

• The TOP magnetic trap provides for a confing
  potential
• We describe next calculations for different A/B
  and different densities.

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A/B2, Back to symmetric State
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A/B1.5, back to symmetric state
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When the final phase is more symmetric

• Na
• 21
• Now G11gtG22
• Before G22gtG11

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Symmetric final State Domain growth

• G11G22
• 21

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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
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Expet. Result

• Hall et al.

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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

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Physics Close to the critical point of change of
symmetry

• Asymmetric solution favored by domain wall energy
• for G11 gtG22, 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).

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Boundaries of the droplet for 3 offset

• Nearly complete separation.
• Results from Thomas-Fermi approximation.

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Density of components 1 and 2

• Trap offset is only 3 per cent of the radius of
  the droplet.
• y0
• Results from Monte Carlo simulation.

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Boundaries for 0.3 potential offset

• Big displacement but not yet separated.
• Results from Thomas-Fermi approximation.

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Density of components 1 and 2

• Trap offset is 0.3 per cent the radius of the
  droplet.

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Density of component 2

• Trap offset 0.3

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Density of component 1

• Trap offset 0.3