Lecture 9 : superfluid-Mott insulator transition
mardi 20 mai 14 Reminder on band structure
1D sinusoidal potential :
Energy scale :
Bloch waves : unk : Bloch function periodic with period d k : quasi-momentum in 1st BZ n: band index
V =0E V =0.5E V =2E 0 R 0 R 0 R V =5E 10 10 10 10 0 R
8 8 8 8
6 6 6 6
4 4 4 4 Energy (Er) Energy (Er) Energy (Er) Energy (Er)
2 2 2 2
0 0 0 0 −0.5 0 0.5 −0.5 0 0.5 −0.5 0 0.5 −0.5 0 0.5 quasi−momentum (2 /d) quasi−momentum (2 /d) quasi−momentum (2 /d) quasi−momentum (2 /d)
Dashed : harmonic oscillator approximation
mardi 20 mai 14 Reminder on band structure
V =4E 0 R
4
3
Wannier functions : 2
1 Wannier functions 0
−2 −1 0 1 2 position x (d) Dashed : harmonic oscillator approximation
In the Bloch basis, we can express the hamiltonian as
In the Wannier basis, we can express the hamiltonian as
Single band + tight binding approximations (valid for V0>>ER):
mardi 20 mai 14 Three-dimensional optical lattices
Cut of the potential Beam geometry in the x-y plane
Bravais lattice :
Reciprocal lattice :
Bloch waves indexed by
Wannier functions :
Energy bands :
mardi 20 mai 14 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 Fraction of atoms in Calculation of Tc : the lowest band @Tc
full band structure tight-binding approximation
mardi 20 mai 14 How to prepare quantum gases in optical lattices ?
In standard traps, one achieves quantum degeneracy using evaporative cooling Evaporation rate proportional to the population of atoms in the wings of the thermal distribution, near an energy ~V0
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.
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).
mardi 20 mai 14 Isentropic loading (thermodynamical)
Increase of the lattice depth from zero to 10 ER at constant entropy
Isentropic path goes from the blue curve to the red horizontally
Red dots mark the location of Tc for each case
A: adiabatic cooling path
B: adiabatic heating path
P B Blakie and J. V. Porto. PRA 69,13603 (2004) mardi 20 mai 14 Isentropic loading (thermodynamical)
Increase of the lattice depth from zero at constant entropy
Red dots mark the location of Tc for each case
C: path where the gas adiabatically uncondenses
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 :
: width of the many-body ground state distribution in Fock space
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 mardi 20 mai 14 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
average filling factor
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 ? mardi 20 mai 14 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]
The ground state many-body wavefunction µ/U corresponds to an array of Fock states 3
n0=3 First excited state corresponds to removing a 2 particle or adding one, which requires an energy n0=2 ~Un0 (interaction gap). 1
n0=1 When µ/U is an integer : n0 and n0-1 are degenerate
n0=0 mardi 20 mai 14 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 :
average filling factor
Commensurate filling : average filling =n0 = integer
Minimized when
mardi 20 mai 14 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
mardi 20 mai 14 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 mardi 20 mai 14 Mean-field phase diagram
Uncommensurate filling : average filling not integer
For J=0, degeneracy between the Fock states n0 and n0-1 Atoms can always tunnel between sites, the system remains superfluid µ/U
3
2
1
mardi 20 mai 14 Mean-field phase diagram
Uncommensurate filling : average filling not integer
For J=0, degeneracy between the Fock states n0 and n0-1 Atoms can always tunnel between sites, the system remains superfluid
Generalizing the theory for integer filling 3 one finds lobe-like domains where a superfluid solution is stable:
2
1 Outside of these domains, the system enters a Mott insulator phase.
mardi 20 mai 14 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
mardi 20 mai 14 T.o.f. interference pattern across the Mott insulator transition
0 ER 12ER 20 ER
Lattice depth V0
Ramping back down 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)
mardi 20 mai 14 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 :
mardi 20 mai 14 Single-site imaging of the Mott shells
Sherson et al., Nature 2009 Bakr et al., Nature 2008/2009 mardi 20 mai 14