New Lecture 9 : superfluid-Mott insulator transitionchevy/AtomesFroids/Lecture9.pdf · 2014. 5....
Transcript of New Lecture 9 : superfluid-Mott insulator transitionchevy/AtomesFroids/Lecture9.pdf · 2014. 5....
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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