Insulating Phases of a BEC

in

Novel Optical Lattices

LENS European Laboratory for Nonlinear SpectroscopyDipartimento di Fisica – Università di Firenze

Massimo Inguscio

Quantum Noise in Correlated Systems, January 9, 2008 Weizmann Institute

Noise interferometry

“Ideal” BEC

Ultracold atoms in optical lattices

Adding disorder

Ultracold atoms in disordered potentials

Localization effects

- Bose glasses, spin glasses (strongly interacting systems)

- Anderson localization (weakly interacting systems)

Why disorder?

- Disorder is a key ingredient of the microscopic (and macroscopic) world- Fundamental element for the physics of conduction- Superfluid-insulator transition in condensed-matter systems

�

�

- Ultracold atoms are a versatile tool to study disorder-related phenomena- Precise control on the kind and amount of disorder in the system

- Quantum simulation

Why cold atoms?�

hopping energy interaction energy

J U

At zero temperature the state of the system is determined by the competition between twoenergy scales: the hopping energy J and the on-site interaction energy U

Bose-Hubbard model for interacting bosons in a lattice:

Interacting bosons in a lattice

hopping energy interaction energy

J U

At zero temperature the state of the system is determined by the competition between twoenergy scales: the hopping energy J and the on-site interaction energy U

Bose-Hubbard model for interacting bosons in a lattice:

SUPERFLUID

� Long-range phase coherence� High number fluctuation� Gapless excitation spectrum

� Compressible

MOTT INSULATOR

� No phase coherence� Zero number fluctuation (Fock states)� Gap in the excitation spectrum

� Not compressible

Interacting bosons in a lattice

Experimental scheme

3D optical lattice

strongly interacting regime:

Mott insulator phase first realized in

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

Superfluid to Mott Insulator transition at LENS

increasing the lattice height U/J increases

momentum distribution of the atomic sample after expansiontest of phase coherence

Fallani, Fort , Guarrera, Lye, M.I. (2005)

Measuring the excitation spectrum

U 2U

Mott Insulator spectrum

Measuring the excitation spectrum

U 2U

Mott Insulator spectrum

Noise interferometry

“Ideal” BEC

Ultracold atoms in optical lattices

Adding disorder

Speckle potential

The random potential is produced by shining an off-resonant laser beam onto a diffusive plateand imaging the resulting speckle pattern on the BEC.

optical dipole potentialstationary in time

randomly varying in space

J. Lye et al., PRL 95, 070401 (2005)

LENS, 2004. Now: Orsay, Hannover, Rice, Illinois...

BEC expansion in a disordered waveguide

In-situ images of the BEC expanding in the disordered waveguide:

experiments also in Orsay (Aspect)

Fort et al. PRL 95, 170410 (2005)

issue #1: producing “dense” disorder

bichromatic latticespeckle pattern

issue #2: role of interactions!

The bichromatic lattice

bichromatic lattice

The bichromatic lattice

Energy minima of the lattice

potential along y direction

Non-periodic modulation of the energy minima

with length scale

sites

Localization in a quasi-periodic lattice

extended states

localized states

λ1 = 830 nm s1 = 10 / λ2 = 1076 nm

localization transition in 1D(Aubry-André)Ann.Israel Phys.Soc. 3 133(1980)

square modulus of the ground state wavefunction

Localization in a quasi-periodic lattice + harmonic trap (no interactions)

s2=1

quasi-periodic

potential

s1=10 s2=0.1

s2=0.25

randomsite to site energy

J. Lye et al.,

Phys.Rev.A75,

061603 R (2007)

Localization in a quasi-periodic lattice + harmonic trap adding interactions

Increasing interactions the “size” of the ground state progressively increases going from

a localized state to an extended state…characterization through transport properties at

low atom number

s1=10

s2=0.25 N=0

N=10

N=15000

s2=0.5

s2=1

s2=2

s1=10

N=15000

AUBRY-ANDRE

Thanks to Nir Davidson

Colloquium on Group Theoretical Methods in PhysicsKiriat Anavim (Israel) 25-30 Mars 1979

Analyticity Breaking and Anderson Localization in Incommensurate Lattices

Disordered systems: Role of interactions

hopping energy interaction energy

J U

disorder

∆∆∆∆

In the presence of disorder an additional energy scale ∆ enters the description of the system. The interplay between these energy terms may induce new quantum phase transitions

Bose-Hubbard model with bounded disorder in the external potential

Adding disorder

Phase diagrams

Qualitative phase-diagram for a system of interacting bosons in a disordered lattice

for ultracold atoms see B. Damski et al., PRL 91, 080403 (2003); R. Roth et al., PRA 68, 023604 (2003).

from M. P. A. Fisher et al., PRB 40, 546 (1989)

Phase diagrams

Qualitative phase-diagram for a system of interacting bosons in a disordered lattice

for ultracold atoms see B. Damski et al., PRL 91, 080403 (2003); R. Roth et al., PRA 68, 023604 (2003).

from M. P. A. Fisher et al., PRB 40, 546 (1989)

BOSE-GLASS

� No long-range phase coherence� Gapless excitation spectrum� Finite compressibility

(T. Giamarchi and H. J. Schulz, PRB 37, 325 (1988))

Broadening the MI spectrum

Starting from a Mott Insulator and adding disorder, the energy required for the

hopping of a boson from a site to a neighboring one becomes a function of position

When ∆j = U the excitation energy goes to zero and the gap disappears

---

Experimental geometry

λ1 = 830 nm s1 = 40

λ1 = 830 nm s1 < 20

λ2 = 1076 nm s2 < 3

x

z

y

1D atomic systems + 1D disorder

Excitation spectraL. Fallani et al., PRL 98, 130404 (2007)

Excitation spectrum for s1=16 and increasing disorder strength from s2=0 to s2=2.5:

MI spectral brodening

Excitation maximum at U

as a function of disorder strength:

∆=U

No agreement for strong

disorder ∆>U when the gap goes to zero

Good agreement with the

MI broadening for weak

disorder ∆<U

L. Fallani et al., PRL 98, 130404 (2007)

Phase coherence

Visibility of the interference pattern after time-of-flight:

L. Fallani et al., PRL 98, 130404 (2007)

Insulating state (no long-range phase coherence)

with broad excitation spectrum

Bose-Glass

Noise interferometry

“Ideal” BEC

Ultracold atoms in optical lattices

Adding disorder

HB&T noise interferometry in quantum gases

in a single image we have approx. 30000 detectors!

absorption image of a Mott Insulator state

Noise correlations

Nature 434, 481 (2005)

first proposal by E. Altman et al., Phys. Rev. A 70, 013603 (2004).

noise correlations

Experiments: Mainz (Nature 2005)

JILA (PRL 2005) – pairs from molecules

NIST- Maryland (PRL 2007)

LENS (2007) – Breaking of Mott order

Noise correlations (Mott phase) V.Guarrera, N.Fabbri, L.Fallani, C.Fort,

K.M.R.van der Stam, M.I. (2007)

k1

uniform filling of regularly-spaced lattice

Breaking the MI order

1 optical lattice

uniform filling of regularly-spaced lattice

Breaking the MI order

1 optical lattice2 optical lattices

controlled creation of particle / holes

inhomogeneous filling of (almost) regularly-spaced lattice

Breaking the MI order

1st and 2nd order correlations

noise correlations reveal the inhomogeneous filling of the lattice

when first-order coherence does not provide any spatial information

1st: Harmonic potential

site filling

site energy

site filling

site energy

2nd: Harmonic potential + incommensurate lattice

s2=0 s2=0.5 s2=1 s2=3 s2=5

Destroying the MI domains

the disordering lattice destroys the “wedding cake”

Calculating noise correlations

calculation of the ground state for J=0

the ground state is a disordered Mott insulator with

the site filling minimizing the total energy

Controlled breaking of Mott insulator order detected via noise correlation interferometry

theory (disordered MI)experiment

V.Guarrera, N.Fabbri, L.Fallani, C.Fort, K.M.R.van der Stam, M.I. (2007)

k1k2

k1 - k2

Noise correlations in the bichromatic lattice

Observable:

the ratio between the heights of the k2 peak and the k1 peak along the horizontal median line

Quantitative analysis of the k2 peak growth

Disordered systems: Role of interactions

CONTROLLED BREAKING of

MOTT DOMAINS

Noise interferometry

“Ideal” BEC

Ultracold atoms in optical lattices

Adding disorder

87Rb

Science 2001Modugno et al

41K boson

87Rb

40K fermion

PRL 2002Roati et al

2007Roati et al

...the last stable alkali isotope!

39K boson

0

Density d

istr

ibutio

n

Horizontal position (µm)

0 200-200 0 200-200 0 200-200

3 s 3.2 s 3.5 s

0 1 2 30,0

0,5

1,0

39K BEC

Lase

r p

ow

er (

arb

. u

nit

s)

B=395.2 G

aKRb

= 28 a0

Time (s)

aK = -33 a

0

aKRb

= 150 a0

aK = 150 a

0

B = 316 G

T=800 nKT=150 nK

K-K

K-Rb

280 300 320 340 360 380 400 420 440-200

-100

0

100

200

aB

B (a

0)

B (G)

(1) (2)

NK = 7*104

Tc~ 100 nK

B0 = 317.9 G

B0 = 402.4 G

BEC of 39K Roati et al PRL 99, 010403 (2007)

Feshbach assisted sympatetic cooling in a mixture with 87Rb

39K BEC with tunable interactionsRoati et al Phys.Rev.Lett. 99, 010403 (2007)

Collisional physics: see also D’Errico et al N.J.P. 9, 223 (2007)

Interference with degenerate trapped atoms___________________________________________________________________

• Confined-atom interferometers are promising in terms of

compactness and portability

• They offer the possibility of extending interrogation times beyond the

typical 0.5 s achievable in atomic fountains

• They offer high spatial resolution for study of atom surface

interaction, Casimir Polder forces, deviations from Newtonian

gravitational law at small distances

• Main problems: interaction induced shift, decoherence induced by

the trapping potential

CCD

Resonant laser (imaging)

Ultracold atoms

Non resonantlattice

MirrorHigh vacuum cell

Interferometry with a vertical optical lattice

A multiple well interferometer

2

λFE =∆

Wannier-Stark states

-2 -1 0 1 2Momentum

Interference of Wannier-Stark states:

Bloch oscillations

h2/λω FB =

Combination of high brightness and ultralow momentum spread: in principle ideal sources for

matter wave interferometry …

222 2cos ( )

2R

i sE kx gt m

ψψ ψ

∂= − ∇ + +

∂

hh

… however

g

Bloch oscillations of 40K fermions trapped in a

vertical optical lattice in presence of gravity

G.Roati, E. De Mirandes, F.Ferlaino, H.Ott, G.Modugno, M.I. Phys.Rev.Lett. 92, 230402 (2004)

Bloch oscillations with fermions

Atom interferometry with 39K BEC Fattori et al Phys.Rev.Lett. In press___________________________________________________________________

Lattice parameters: λ = 1032 nm, slattice= 6, νradial=40 Hz 100 a0 vs 1 a0

0 ms 0.4 ms 0.8 ms 1.2 ms 1.6 ms 2 ms 2.4 ms 2.8 ms 3.2 ms 3.6 ms 4 ms

100 a0

1 a0

-100 -50 0 50 100

100

200

300

400

500

600

700

800

900

Ato

mic

density (

a.u

.)

z axis (px)

-100 -50 0 50 100

100

200

300

400

500

600

700

800

900

Ato

mic

density (

a.u

.)

z axis (px)

T =0 ms T =4 ms

Widths of central peak!

0 100 1300 1310

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

y

or

sig

ma

y (

pix

)

Time (ms)

Towards a high sensitivity local sensing of forces

Sensitivity:

6×10-5g per shot (10-6g achievable)

∆g/g = [ ∆x/(xbragg) / Nosc ]

also Innsbruck with cesium

Conclusions and outlook

Atom-atom interaction in 39K Bose condensate can be accurately

tuned over a large range and and controlled around zero.

This is a very interesting system to investigate phenomena that can

be destroyed by atom-atom interactions:

BEC-based interferometry – forces at short distance

disorder (Anderson localization, tuning U versus J)

Vortices, with IDEAL BEC (J.Dalibard talk)

2D physics (Z. Hadzibabic talk)

R.Van der Stam M.Jona-Lasinio M.Modugno G.Modugno F.Minardi G.Roati M.Fattori J.Catani

L.Fallani G.Barontini C.Fort M.I. N.Fabbri V.Guarrera C.D’Errico M.Zaccanti L.DeSarlo