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. Talk Outline. Brief remarks on quantum phase transitions in a single species ultracold bosonic atoms. - PowerPoint PPT PresentationTRANSCRIPT
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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
hopping onsite interaction
Fisher et al, PRB(1989) U/t << 1 : Superfluid
U/t >> 1 : Mott insulator
Bose-Hubbard Model :
Jaksch et al, PRL(1998)
(for optical lattice)Integer density => SF-MI transition
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SF-MI Transition In Optical Lattice
U/t << 1Random distribution of atoms superfluidity
U/t >> 1Confined atomsMott insulator
Greiner et al, Nature(2002) : 3DStoeferle et al, PRL (2004) : 1D
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SF-MI transition in One component Boson with Filling factor = 1
Mott Insulator
Superfluid
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SF-MI transition in One component Boson with Filling factor = 1
Mott Insulator
Superfluid
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Mott Insulator
Superfluid
SF-MI transition in One component Boson with Filling factor = 1
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Mott Insulator
Superfluid
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 (a=1/2, b=1/2)
Superfluid
Mott Insulator
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Superfluid
Mott Insulator
SF-MI transition in two component Boson with Filling factor = 1 (a=1/2, b=1/2)
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SF-MI transition in two component Boson with Filling factor = 1 (a=1/2, b=1/2)
Superfluid
Mott Insulator
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Phase separation in two component Boson with filling factor = 1 (a=1/2, b=1/2)
Phase separated SF
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Phase separated SF
Phase separation in two component Boson with filling factor = 1 (a=1/2, b=1/2)
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Phase separation in two component Boson with filling factor = 1 (a=1/2, b=1/2)
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 Δ = Uab / U
and the densities of the two species ρa = Na/L and ρb = 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 | Ψ0LNaNb> 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 Δ =0.95 and Δ =1.05 . We have considered three different cases of densities i.e ρa = ρb = ½ , ρa = 1, ρb = ½ and ρa = ρb = 1
GL = [EL(Na+1,Nb) - EL(Na,Nb)] – [EL(Na,Nb) - EL(Na-1,Nb)]
<niα> = <Ψ0LNaNb| ni
α| Ψ0LNaNb>
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Result• For Δ = 0.95 and for all densities there is a transition from
2SF-MI at some critical value Uc .
• For Δ = 1.05 and ρa = ρb = 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 Δ = 1.05 and ρa = 1 and ρb = ½ there is a transition from 2SF to PS-SF. The PS-MI phase does not appear in this case.
• Finally for Δ = 1.05 and ρa = ρb = 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 Uc=3.4 for Δ = 0.95
Plots of <nia> and <ni
b> versus L for U = 1 and U = 4 . These plots are for Δ = 1.05 and L=50.
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The upper plot is between LGL and U which showes the SF-MI transition and the lower one between OPS and U.
OPS = i |<nai> - <nb
i>|
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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 densityWhen Uab > U we observe phase separationFor ρa = ρb = ½ we observe PS-SF sandwiched between 2SF and PS-MI
• For ρa = 1 and ρb = ½ there is a transition from 2SF to PS-SF
• For ρa = ρb = 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).