strength (t U0). In the non interacting limit (U0 = 0), the many body ground state is essentially a BEC where all the atoms occupy identical Bloch states with zero momentum (k=0) in the lowest band which is essentially a superfluid phase.
Since the hopping strength is dominant which is responsible for delocalizing the atoms across the lattice sites and thus a quantum correlation in atom numbers is negligible. Under this condition, the system can be described by a giant wave function and subsequently, the many body ground state for N atoms is just the sum over all the Bloch states as [54],
|ΨiU0=0 ∝ X
i
ai†
!N
|0i 1.12
An MI phase on the other hand is observed when the on-site interaction po- tential dominates over the tunneling amplitude (U0 t). In the strong interacting limit, atoms tend to localize across the lattice site and remain so as long as the hopping amplitude remains below the on-site potential and hence a perfect corre- lation in atom numbers exist. Thus in the MI phase, the many body ground state is a product of local Fock state with fixed numbers of atoms (n) per lattice site is given by [54],
|Ψit=0∝ Y
i
ai†
!n
|0i 1.13
0 1 2 3
4
0 0.05 0.1 0.15 0.2
(a)
Eg=U0
MI
SF
ρ=1 ρ=2 ρ=3 ρ=4
µ /U 0
zt/U0
(a)
Eg=U0
MI
SF
ρ=1 ρ=2 ρ=3
ρ=4 (b)
MI
SF (c)
Figure 1.4: The zero temperature phase diagram of a homogeneous BHM in (a).
The lobes indicate MI phase with occupation densities, ρ. The vertical line corre- sponds to the critical tunneling strength, ztc/U0 from the MI to SF phase. Pictorial representations of the occupation densities in the MI lobe (ρ= 1) in (b) and in the SF phase in (c) of bosons on an optical lattice. Figure courtesy of Markus Greiner [12].
The zero temperature phase diagram of a homogeneous BHM in theµ/U0-zt/U0
plane which schematically shows the MI and SF phases is presented in Fig1.4(a).
Here z is the coordination number which is related with the lattice dimension, d as z = 2d. At smaller values of zt/U0, each lobe denotes a Mott insulating phase with integer occupation densities per lattice site, that is ρ = 1,2, . . . due to localization [Fig1.4(b)]. The compressibility, κ =∂ρ/∂µ vanishes inside the MI lobes implying that the MI phase is an incompressible phase. Also, there exists an energy gap (Eg = U0) in the particle-hole excitation spectrum and it is equal to the on-site interaction potential, U0 at zero hopping limit (t = 0). As soon as the tunneling strength is increased, the system will remain in the MI phase below a certain critical value, ztc/U0 since the kinetic energy required for an atom to hop from one site to another is still insufficient to overcome the potential energy cost. Further increase in hopping amplitude above this critical value will push the system to undergo a phase transition to an SF phase which corresponds to a non- integer number of bosons per lattice site that isρ ,1,2, . . . due to delocalization and hence shows finite compressibility Fig1.4 (c). The point in the vicinity of the two MI lobes withρ =n andρ =n+1 is degenerate along the vertical axis (µ) and corresponds to the SF phase. Thus a quantum phase transition from a gapped, incompressible MI phase is accompanied to a gapless, compressible SF phase with the existence of finite values for the off-diagonal long range order. The mean field calculation shows that the tunneling strength, zt/U0 for the MI-SF phase transition now depends on the occupation density of the MI lobes, n and lattice dimension,d as [68]
1 zt =
"
n+1
nU0−µ − n (n−1)U0−µ
#
1.14
1.3.2 Superfluid to Mott insulator transition: QPT
After such theoretical prediction by Jacksh et al., the first experimental sig- nature of the quantum phase transition involving the SF and MI phases was demonstrated by Greiner et al. by loading 87Rb atoms from a BEC into a three- dimensional optical lattice potential using a magneto optical trap (MOT) [2]. They recorded the absorption spectra via a ’Time of flight (TOF)’ experiment, a commonly used method for a system of cold atoms loaded in optical lattices [54]. In TOF, the system is initially prepared in a BEC state and trapped by using a confining potential. Later, all the trapping as well as optical lattice potentials are suddenly turned off to allow the atoms to expand for a certain interval of time,τ. After such expansion, the wave packets associated with the atoms confined at each lattice site grow and start to overlap with each other giving rise to the interference pat- tern in the momentum space. The absorption image in Fig.1.5 corresponding to different optical lattice potential,V0 shows a SF-MI phase transition. In absence of lattice potential, a sharp absorption spectra, which corresponds to a construc-
Figure 1.5: Absorption images of multiple matter wave interference pattern for different potential depths, after a time of flight period of 15 ms. In the superfluid regime for potential depth, V0 = 0, a sharp peak in the interference pattern is observed and such trend is visible up to about V0 = 12Er. For a potential depth, V0≥ 13Er, such interference peaks are gradually diminishing indicating a transition to the Mott insulator phase. Figure courtesy of Markus Greiner [12].
tive interference pattern atk=0, indicates that the system is in the SF phase. In that phase, since atoms are delocalized across the lattice sites, a coherent matter wave prevails during the expansion which results in sharp peaks in the interfer- ence pattern and similar trend is maintained for relatively smaller values of the potential strength. With further increase in the potential height beyond a critical valueV0c '13Er [2], sharp peaks in the absorption spectra disappears indicating a destructive interference pattern, thereby signaling the transition to the MI phase.
In MI phase, the atoms are localized at the lattice sites and hence demonstrates a destructive interference pattern with no definite phase relationship. Surprisingly, this critical value of the potential strength at which the absorption spectra shows a transition is in excellent agreement with the theoretical prediction of the BHM.