Quantum Phase Transition in Ultracold bosonic atoms

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Quantum Phase Transition in Ultracold bosonic
atoms

• Bhanu Pratap Das
• Indian Institute of Astrophysics
• Bangalore

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Talk Outline

• Brief remarks on quantum phase transitions in a
  single species ultracold bosonic atoms.
• Quantum phase transitions in a mixture of two
  species ultracold bosonic atoms.
• Special reference to new quantum phases and
  transitions between them.

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SF-MI transition for bosons in a periodic
potential
Bose-Hubbard Model
onsite interaction
hopping
Fisher et al, PRB(1989)
U/t ltlt 1 Superfluid U/t gtgt 1 Mott
insulator
Jaksch et al, PRL(1998) (for optical lattice)
Integer density gt SF-MI transition
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SF-MI Transition In Optical Lattice
Greiner et al, Nature(2002) 3D Stoeferle et
al, PRL (2004) 1D
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SF-MI transition in One component Boson with
Filling factor 1
Superfluid
Mott Insulator
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SF-MI transition in One component Boson with
Filling factor 1
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SF-MI transition in One component Boson with
Filling factor 1
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SF-MI transition in One component Boson with
Filling factor 1
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SF-MI transition in two component Boson with
Filling factor 1 (?a1/2, ?b1/2)
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SF-MI transition in two component Boson with
Filling factor 1 (?a1/2, ?b1/2)
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SF-MI transition in two component Boson with
Filling factor 1 (?a1/2, ?b1/2)
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Phase separation in two component Boson with
filling factor 1 (?a1/2, ?b1/2)
Phase separated SF
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Phase separation in two component Boson with
filling factor 1 (?a1/2, ?b1/2)
Phase separated SF
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Phase separation in two component Boson with
filling factor 1 (?a1/2, ?b1/2)
Phase separated MI
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Two Species Bose-Hubbard Model
Exploration of New Quantum Phase Transitions
Present work ta tb 1 , Ua Ub
U Physics of the system is determined by ? Uab
/ U and the densities of the two species ?a
Na/L and ?b Nb/L
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Theoretical Approach

• We calculate the Gap
• And the onsite density
• For a and b type bosons, EL(Na,Nb) is
  the ground state energy and
• ?0LNaNbgt is the ground state wave
  function for a system of length L with Na (Nb)
  number of a(b) type bosons obtained by DMRG
  method which involves the iterative
  diagonalization of a wave function and the energy
  for a particular state of a many-body system. The
  size of the space is determined by an appropriate
  number of eigen values and eigen vectors of the
  density matrix.
• We study the system for ? 0.95 and ? 1.05 .
• We have considered three different cases of
  densities
• i.e ?a ?b ½ , ?a 1, ?b ½ and ?a
  ?b 1

GL EL(Na1,Nb) - EL(Na,Nb) EL(Na,Nb) -
EL(Na-1,Nb)
ltniagt lt?0LNaNb nia ?0LNaNbgt
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Result

• For ? 0.95 and for all densities there is a
  transition from 2SF-MI at some critical value Uc
  .
• For ? 1.05 and ?a ?b ½ there is a
  transition from 2SF to a new phase known as PS-SF
  at some critical value of U and there is a
  further transition to another new phase known as
  PS-MI for some higher value of U.
• For ? 1.05 and ?a 1 and ?b ½ there is a
  transition from 2SF to PS-SF. The PS-MI phase
  does not appear in this case.
• Finally for ? 1.05 and ?a ?b 1 there is a
  transition from 2SF to PS-MI without an
  intermediate PS-SF phase. This result is very
  intriguing.
• Tapan Mishra, Ramesh. V. Pai, B. P. Das,
  cond-mat/0610121

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Results....
This plots shows the SF-MI transition at the
critical point Uc3.4 for ? 0.95
Plots of ltniagt and ltnibgt versus L for U 1 and
U 4 . These plots are for ? 1.05 and L50.
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OPS ?i ltnaigt - ltnbigt
The upper plot is between LGL and U which showes
the SF-MI transition and the lower one between
OPS and U.
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Conclusion

• For the values of the interaction strengths and
  the density considered here we obtain phases like
  2SF, MI, PS-SF and PS-MI
• The SF-MI transition is similar to the single
  species Bose-Hubbard model with the same total
  density
• When Uab gt U we observe phase separation
• For ?a ?b ½ we observe PS-SF sandwiched
  between 2SF and PS-MI
• For ?a 1 and ?b ½ there is a transition from
  2SF to PS-SF
• For ?a ?b 1 no PS-SF was found and the
  transition is directly from 2SF to MI-PS.

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• Co-Workers
• Tapan Mishra, Indian Institute of Astrophysics,
  Bangalore
• Ramesh Pai, Dept of Physics, University of Goa,
• Goa

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Bragg reflections of condensate at reciprocal
lattice vectors showing the momentum distribution
function of the condensate
M. Greiner, et al. Nature 415, 39 (2002).
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Experimental verification of SF-MI transition
M. Greiner, et al. Nature 415, 39 (2002).