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 << 1Random distribution of atoms superfluidity

U/t >> 1Confined atomsMott insulator

Greiner et al, Nature(2002) : 3DStoeferle 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)

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 separated SF

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

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 : 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

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>

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

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.

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>|

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 > U we observe phase separation

For 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.

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).