Download 8. Superfluid to Mott-insulator transition

8. Superfluid to Mott-insulator transition
Overview
Optical lattice potentials
Solution of the Schrödinger equation for periodic potentials
Band structure
Bloch oscillation of bosonic and fermionic atoms in optical lattices
Wannier functions
Bose-Hubbard Hamiltonian
Superfluid to Mott-insulator transition
Optical dipole traps
ϕ
1
1
Γ
Potential: U Dip (r ) = d ⋅ E t =
Re(α ) I ( r ) ∝ I (r )
2
2ε 0c
Δ
2
P
1
⎛Γ⎞
Loss rate: γ sc (r ) = abs =
Im(α ) I ( r ) ∝ ⎜ ⎟ I (r )
=ω =ε 0 c
⎝Δ⎠
d: induced dipole moment,
α: polarizability
abs
ω0
ω
Γ: Dipole matrix element between
ground and excited states,
Δ=ω−ω0: detuning
Δ <0 : red detuned
Δ >0 : blue detuned
attractive potential
repulsive potential
Gaussian beam:
I (r ) = I 0 e
−
2r2
w2
Optical lattices
Optical standing wave fields produce periodic
potentials with lattice constants half of the
laser’s wavelenght (λ ~ 800 nm – 10 μm).
1-D
2-D
⎛ 2π
⎞
I ( x ) = I 0 sin 2 ⎜⎜
x + ϕ ⎟⎟
⎝ λlaser
⎠
a=
λlaser
2
3-D
Band model
Solutions of the Schrödinger equation in periodic potentials
SE:
Ansatz:
Comparing coefficients yields conditional equations for parameters C(k)
For a specific k ∈ [-k0/2, k0/2] the wavefunction ψk contains also wavevectors k+nk0 .
In order to find energies and eigenstates for a given k, the equation system (*) has to
be solved for all k ∈ [k+nk0] .
Note: |k> is not an eigenstate of the Hamiltonian.
Band model – energies and eigenstates
system
Band model – example
Dynamics in a lattice potential
Phase difference between
neighboring lattice sites
Bloch-Oscillation (1D lattice)
ϕ j − ϕi =
(E
j
− Ei ) ⋅ t
=
Nonlinear dynamics leads to
dephasing if gradient is left on
for longer times ! (2D lattice)
Δφ = 0
Δφ = π
M. Greiner et al. PRL 87, 160405 (2001)
Bloch oscillation of fermionic atoms
Large contrast for mor than
100 oscillation periodes
Comparison: fermions vs. bosons
(a) Momentum distribution of
fermions in the lattice: 1 ms
(continuous line) and 252 ms
(dashed line).
(b) Momentum distribution of
bosons at 0.6 ms (continuous line)
and 3.8 ms (dashed line).
The much faster broadening for
bosons is due to the presence of
interactions.
TB=1.2 s
G. Roati et al., Phys. Rev. Lett. 92, 230402 (2004)
Fermi surfaces
Bose gas in a 3D lattice potential
Resulting potential consists of a simple cubic lattice where the BEC coherently
populates about 100,000 lattice sites.
M. Greiner et al. PRL 87, 160405 (2001)
Interference of a superfluide bose gas from a 3D lattice
M. Greiner et al. PRL 87, 160405 (2001)
Wave function of single particles in a periodic potential
Bloch states:
Ψ k (r) = u k (r)eikr
Plane waves modulated by a lattice
periodic function
Fourier transform
1 BZ ikR j
Wannier states: w(r − R j ) =
e Ψ k (r)
∑
L k
Localized wave functions. This pictures
allows tunneling as well as localized states.
Inverse Fourier transform
1 lattice sites − ikR j
e
w(r − R j )
Ψ k (r) =
∑
L
j
The Hamiltonian can be written in Wannier basis (TONS). Second quantization results
in the Bose-Hubbard Hamiltonian.
(Necessary condition: only the lowest band is populated ↔ excitation energies are larger than the energy gap.)
Bose-Hubbard Hamiltonian
Expanding the field operator in the Wannier basis of localized wave functions on
each lattice site, yields :
Bosonic operators
ψˆ ( x ) = ∑ aˆi w( x − xi )
i
annihilation
â i ..., N i ,... = N i ..., N i − 1,...
creation
â i+ ..., N i ,... = N i + 1 ..., N i + 1,...
( aˆ i )
*
Bose-Hubbard Hamiltonian
number
= aˆ i+
⎡⎣aˆ i ,aˆ +j ⎤⎦ = δij
aˆ +j aˆ j = n j
1
H = − J ∑ aˆi† aˆ j + ∑ ε i nˆi + U ∑ nˆi (nˆi − 1)
2 i
i, j
i
Single particle energy in the trapping potential
Tunnel matrix element (hopping element)
Onsite interaction matrix element
⎛ =2 2
⎞
J = − ∫ d x w( x − xi ) ⎜ −
∇ + Vlat ( x )⎟ w( x − x j )
⎝ 2m
⎠
4π = 2 a 3
4
U=
d
x
w
(
x
)
m ∫
3
M.P.A. Fisher et al, PRB 40, 546 (1989); D. Jaksch et al., PRL 81, 3108 (1998)
Superfluid Limit
The kinetic energy dominates (weakly interacting bosonic system).
1
H = − J ∑ aˆi† aˆ j + U ∑ nˆi (nˆi − 1)
2 i
i, j
Atoms are delocalized over the entire lattice, macroscopic wave function describes
this state.
Ψ SF
⎛ M †⎞
∝ ⎜ ∑ aˆi ⎟
⎝ i =1 ⎠
N
0
ai ≠ 0
Poissonian atom number distribution per lattice site
n=1
n=2
Mott-insulator
Interaction energy dominates (strongly correlated bosonic system).
1
H = − J ∑ aˆi† aˆ j + U ∑ nˆi (nˆi − 1)
2 i
i, j
Atoms are completely localized to lattice sites.
M
( )
Ψ Mott ∝ ∏ a i
†
i =1
n
0
ai = 0
Fock states with a vanishing atom number fluctuation are formed, e.g. n=1.
n=1
Momentum distribution for different potential depths (Erecoil)
0 Erecoil
22 Erecoil
M. Greiner et al., Nature 415, 39 (2002)
Restoring coherence
a) Ramp up the potential for
generating the Mott-insuator
state and subsequent ramp
down.
Measurement on a phase
incoherent cloud.
b) Width of the zero
momentum central
peak
(Dephasing was applied before
reaching the Mott-insulator state)
The coherence is
restored within the
tunneling time to the
neigboring lattice site.
Measurement on a
phase incoherent cloud.
Before ramping
down the potential
M. Greiner et al., Nature 415, 39 (2002)
Quantum phase transition from a
superfluid to a Mott-insulator
At the critical point gc the
system will undergo a
phase transition from a
superfluid to an insulator.
This phase transition occurs
even at T=0 and is driven by
quantum fluctuations !
Characteristic of the quantum phase transition
• Excitation spectrum is dramatically modified at the critical point.
• U/J < gc (Superfluid regime)
Excitation spectrum is gapless
• U/J > gc (Mott-Insulator regime)
Excitation spectrum is gapped
Critical ratio: U/J = z * 5.8
number of next neighbors (for a cubic lattice 6)
Creating excitations in the MI phase
Mott-insulator with ni=1 atom per lattice site
Without gradient potential
With gradient potential
Special case: ΔEij = U
Energy Scales:
=ωv ≈ 20⋅U
U ≈10 − 300⋅ J
Measuring the excitation gap in the MI phase
(probability vs. gradient)
10 Erecoil tperturb = 2 ms
13 Erecoil tperturb = 4 ms
16 Erecoil tperturb = 9 ms
20 Erecoil tperturb = 20 ms
M. Greiner et al., Nature 415, 39 (2002)
Literature
Bose-Einstein condensates in 1D- and 2D optical lattices
M. Greiner et al. Appl. Phys. B. 73, 769 (2001)
Exporing the phase coherence in a 2D lattice of Bose-Einstein condensates
M. Greiner et al. PRL 87, 160405 (2001)
Fermionic atoms in a three dimensional optical lattice: observing Fermi surfaces, dynamics
and interactions, M. Köhl et al., PRL94, 080403 (2005)
Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms
M. Greiner et al., Nature 415, 39 (2002)