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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 << 1 Random distribution of atoms superfluidity U/t >> 1 Confined atoms Mott insulator 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 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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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 separation in two component Boson with filling factor = 1 ( a =1/2, b =1/2)

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Phase separated MI

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Two Species Bose-Hubbard Model Exploration of New Quantum Phase Transitions: Present work : t a = t b =1, U a = U b = U Physics of the system is determined by Δ = U ab / U and the densities of the two species ρ a = N a /L and ρ b = N b /L

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Theoretical Approach We calculate the Gap: And the onsite density: For ‘a’ and ‘b’ type bosons, E L (N a,N b ) is the ground state energy and | Ψ 0LNaNb > is the ground state wave function for a system of length L with N a (N b ) 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 ρ a = ρ b = ½, ρ a = 1, ρ b = ½ and ρ a = ρ b = 1 G L = [E L (N a +1,N b ) - E L (N a,N b )] – [E L (N a,N b ) - E L (N a -1,N b )] =

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Result For Δ = 0.95 and for all densities there is a transition from 2SF-MI at some critical value U c. For Δ = 1.05 and ρ a = ρ 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 Δ = 1.05 and ρ a = 1 and ρ b = 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 ρ a = ρ 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 Uc=3.4 for Δ = 0.95 Plots of and 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 LG L and U which showes the SF-MI transition and the lower one between O PS and U. O PS = 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 density When U ab > U we observe phase separation For ρ a = ρ b = ½ we observe PS-SF sandwiched between 2SF and PS-MI For ρ a = 1 and ρ b = ½ there is a transition from 2SF to PS- SF For ρ a = ρ 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).

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