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

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

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

SF-MI transition in One component Boson with Filling factor = 1 Mott Insulator Superfluid

SF-MI transition in One component Boson with Filling factor = 1 Mott Insulator Superfluid

Mott Insulator Superfluid SF-MI transition in One component Boson with Filling factor = 1

Mott Insulator Superfluid SF-MI transition in One component Boson with Filling factor = 1

SF-MI transition in two component Boson with Filling factor = 1 (  a =1/2,  b =1/2) Superfluid Mott Insulator

Superfluid Mott Insulator SF-MI transition in two component Boson with Filling factor = 1 (  a =1/2,  b =1/2)

Superfluid Mott Insulator

Phase separation in two component Boson with filling factor = 1 (  a =1/2,  b =1/2) Phase separated SF

Phase separation in two component Boson with filling factor = 1 (  a =1/2,  b =1/2)

Phase separated MI

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

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 )] =

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/

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.

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 | - |

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.

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

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

Experimental verification of SF-MI transition M. Greiner, et al. Nature 415, 39 (2002).