Lecture 9 : superfluid-Mott insulator transition

mardi 20 mai 14

Reminder on band structure

Bloch waves : unk : Bloch function periodic with period dk : quasi-momentum in 1st BZn: band index

1D sinusoidal potential :

Energy scale :

−0.5 0 0.50

2

4

6

8

10

quasi−momentum (2//d)

Ener

gy (E

r)

V0=0.5ER

−0.5 0 0.50

2

4

6

8

10

quasi−momentum (2//d)

Ener

gy (E

r)

V0=0ER

−0.5 0 0.50

2

4

6

8

10

quasi−momentum (2//d)

Ener

gy (E

r)

V0=2ER

−0.5 0 0.50

2

4

6

8

10

quasi−momentum (2//d)

Ener

gy (E

r)

V0=5ER

Dashed : harmonic oscillator approximation

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Reminder on band structure

Single band + tight binding approximations (valid for V0>>ER):

Wannier functions :

−2 −1 0 1 2

0

1

2

3

4

position x (d)

Wan

nier

func

tions

V0=4ER

In the Wannier basis, we can express the hamiltonian as

In the Bloch basis, we can express the hamiltonian as

Dashed : harmonic oscillator approximation

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Three-dimensional optical lattices

Beam geometryCut of the potential

in the x-y plane

Bravais lattice :

Reciprocal lattice :

Wannier functions :

Energy bands :

Bloch waves indexed by

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Ideal Bose gas in a 3D lattice

Grand canonical ensemble :

BEC occurs when this sum saturates, i.e. when (in 3D)

Application to a gas of bosons in a 3D lattice :

We fix the filling factor (average number of atoms per lattice site) and the temperature

Calculation of Tc :

tight-binding approximation

full band structure

Fraction of atoms in the lowest band @Tc

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Evaporation rate proportional to the population of atoms in the wings of the thermal distribution, near an energy ~V0

How to prepare quantum gases in optical lattices ?

In a lattice, atoms tend to accumulate in the lowest bands.

Population of band n :

Temperature scale for atoms mostly in the lowest band

Then exponentially small

Evaporation stops.

In standard traps, one achieves quantum degeneracy using evaporative cooling

To achieve quantum gases, one first prepare a gas using evaporation in a regular trap, leading to some temperature. Then one ramps up adiabatically the lattice potential to transfer the cloud (eventually removing as well the initial harmonic trap).

The best one can do is to do this without increasing entropy (isentropic transfer).

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Isentropic loading (thermodynamical)

Red dots mark the location of Tc for each case

A: adiabatic cooling path

B: adiabatic heating path

Increase of the lattice depth from zero to 10 ER at constant entropy

P B Blakie and J. V. Porto. PRA 69,13603 (2004)

Isentropic path goes from the blue curve to the red horizontally

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Isentropic loading (thermodynamical)

Red dots mark the location of Tc for each case

C: path where the gas adiabatically uncondenses

Increase of the lattice depth from zero at constant entropy

P B Blakie and J. V. Porto. PRA 69,13603 (2004)mardi 20 mai 14

Bose-Hubbard model for BECs in double well potentials

Basis of localized states for the low-energy subspace:

Boe-Hubbard model :

In the limit of many atoms per well, the ground state for U=0 (ideal gas) shows a binomial distribution in Fock space.

With increasing interaction strength U, the distribution progressively narrows down until J >>U/N, where the ground state approaches the symmetric Fock states with N/2 atoms in each well.

The reduction of number fluctuations («number squeezing») is accompanied by an increase in phase fluctuations (reduction of phase coherence) detectable in t.o.f. images

: width of the many-body ground state distribution in Fock space

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Bose-Hubbard model in 3D optical lattices

In the Wannier basis, a derivation essentially identical to the one used for the case of two wells leads to the Bose-Hubbard model

U: on-site interaction energy between two bosons

J: tunneling matrix element, quantifies the kinetic energy

Weakly-interacting bosons behave qualitatively as in the double-well case.

What about strong interactions ?

average filling factor

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Disconnected wells, or «atomic limit»

We set J=0 and compute the free energy for a given well in the GC ensemble :

ni=0,1,2,...

The ground state energy corresponds to a particular integer filling n0 that changes when the chemical potential increases:

n0 = Int[µ/U]

When µ/U is an integer : n0 and n0-1 are degenerate

µ/U

n0=1

n0=2

n0=3

1

2

3

n0=0

The ground state many-body wavefunction corresponds to an array of Fock states

First excited state corresponds to removing a particle or adding one, which requires an energy ~Un0 (interaction gap).

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Gutzwiller variational wavefunction

Variational wavefunction that works in both extreme cases (U=0 and J=0):

On-site wavefunction:

Truncation to the three most important states (n0 = closest integer to the average filling) :

4 variational parameters :

Commensurate filling : average filling =n0 = integer

average filling factor

Minimized when

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Bose-Hubbard model

Quantum phase transition from a BEC (=superfluid in 3D) to a Mott insulator state

Limitations of the model , e.g. number fluctuations do not vanish at the transition

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Quantum phase transitions

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 for a QPT

Excitation spectrum is dramatically modified at the critical point.

U/J < gc (Superfluid regime) Excitation spectrum is gapless (phonon modes at low energies)

U/J > gc (Mott-Insulator regime) Excitation spectrum is gapped (particle-hole modes)

Critical ratio for U/J = 36 for a cubic lattice

see Subir Sachdev, Quantum Phase Transitions, Cambridge University Press

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Mean-field phase diagram

Uncommensurate filling : average filling not integer

For J=0, degeneracy between the Fock states n0 and n0-1Atoms can always tunnel between sites, the system remains superfluid

1

2

3

µ/U

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Mean-field phase diagram

Uncommensurate filling : average filling not integer

For J=0, degeneracy between the Fock states n0 and n0-1Atoms can always tunnel between sites, the system remains superfluid

Generalizing the theory for integer filling one finds lobe-like domains where a superfluid solution is stable:

Outside of these domains, the system enters a Mott insulator phase.

1

2

3

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Time of flight

T.o.f. pattern results from the interference of many matter waves emitted from each lattice site considered as a point source : same interference pattern as a square grating

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T.o.f. interference pattern across the Mott insulator transition

M. Greiner et al., Nature 415, 39 (2002)

see also :C. Orzel et al., Science 291, 2386 (2001) Z. Hadzibabic et al., PRL 93, 180403 (2004)

0 ER 12ER 20 ER

Lattice depth V0

Ramping back down

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Shell structure in a trap

Within a Mott lobe, changing the chemical potential does not change the density: Incompressibility

Consequence of the gap for producing particle/hole excitations, which vanishes at the phase boundaries.

Simple picture in 1D :

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Single-site imaging of the Mott shells

Sherson et al., Nature 2009Bakr et al., Nature 2008/2009

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