Download Quantum Phase Transitions in Ultracold Bosonic Atoms

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
Bose-Hubbard Model :
hopping
Fisher et al, PRB(1989)
U/t << 1 : Superfluid
U/t >> 1
Jaksch et al, PRL(1998)
(for optical lattice)
onsite interaction
: Mott insulator
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
Superfluid
Mott
Insulator
SF-MI transition in One component Boson with
Filling factor = 1
Superfluid
Mott
Insulator
SF-MI transition in One component Boson with
Filling factor = 1
Superfluid
Mott
Insulator
SF-MI transition in One component Boson with
Filling factor = 1
Superfluid
Mott
Insulator
SF-MI transition in two component Boson with
Filling factor = 1 (a=1/2, b=1/2)
Superfluid
Mott Insulator
SF-MI transition in two component Boson with
Filling factor = 1 (a=1/2, b=1/2)
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 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 : 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:
GL = [EL(Na+1,Nb) - EL(Na,Nb)] – [EL(Na,Nb) - EL(Na-1,Nb)]
And the onsite density:
<niα> = <Ψ0LNaNb| niα| Ψ0LNaNb>
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
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 <nib> versus L for U = 1
and U = 4 . These plots are for Δ = 1.05 and
L=50.
OPS = i |<nai> - <nbi>|
The upper plot is between LGL and U which showes the SF-MI
transition and the lower one between OPS and U.
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 BoseHubbard 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 PSSF
• 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).