Download Anderson localization of ultra

LENS European Laboratory for Nonlinear Spectroscopy,
Dipartimento di Fisica
Università di Firenze
INFM-CNR
Massimo Inguscio
Anderson localization of ultra-cold atoms
NANO OPTICS and ATOMICS: TRANSPORT of LIGHT and MATTER WAVES
Varenna, 23 June – 3 July
LENS European Laboratory for Nonlinear Spectroscopy,
Dipartimento di Fisica
Università di Firenze
INFM-CNR
Anderson localization of ultra-cold atoms
Speckles and bichromatic optical lattices
Bloch oscillations and control of interactions
Interplay disorder-interactions
Glassy phases
Two-species
Mott, highly correlated phases, new diagnostics
Leonardo Fallani
Temperature scales
Light/matter interaction: mechanical effects
Fundamental process: momentum transfer between photons and atoms
Quasi-resonant interaction:
Non-resonant interaction:
absorption (+ spontaneous emission)
dispersive interaction
dissipative force (e.g.
conservative force
Laser cooling
(MOT, optical molasses, ...)
)
trapping
(optical traps, optical lattices, ...)
Optical trapping
Far off resonance light induces
an electric dipole
r
r
p = αE
The atomic induced electric
dipole then interacts with the e.m.
wave
r
r r r
U(r) = − p ⋅ E(r)
↓
Optical trapping
Optical traps
Red detuning
Blue detuning
Optical trapping
L.Fallani, C.Fort, M.Inguscio
Bose Einstein Condensates in Optical Potentials
Riv.Nuovo Cimento 28, serie 4 n.2 (2005)
Quantum simulations with crystals of light
M. Greiner and S. Fölling, Nature 453, 736 (2008)
Disordered interacting bosons
Subtle interplay between disorder and interactions
(granular superconductors, superfluid He in porous media, high-Tc, …)
Phase diagram of disordered interacting bosons
Anderson
Bose glass...
Superfluid
Mott
interaction
T. Giamarchi and H. J. Schultz,
Anderson localization and interactions in one-dimensional metals
Phys. Rev. B 37, 325 - 340 (1988)
Superfluid to Mott Insulator transition
momentum distribution of the 87Rb atomic sample after expansion (LENS, 2006)
first experimental demonstration in M. Greiner et al., Nature 415, 39 (2002)
Superfluid to Mott Insulator transition
momentum distribution of the 87Rb atomic sample after expansion (LENS, 2006)
P. Pedri et al., Phys. Rev. Lett. 87, 220401 (2001)
first experimental demonstration in M. Greiner et al., Nature 415, 39 (2002)
Dipole oscillations and Josephson effect
The oscillation frequency ω*
depends on the effective mass m*
ω* =
m
ω
m*
depending on the tunnelling energy J:
2h 2
m* = 2 m
λ J
F. S. Cataliotti et al.,
Josephson junctions arrays with BECs,
Science 293, 843 (2001)
Ultracold atoms in disordered potentials
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
Why cold atoms?
- 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
Localization effects
- Bose glasses, spin glasses (strongly interacting systems)
- Anderson localization (weakly interacting systems)
Bose-Einstein condensates in disordered potentials
Beginning of the story: BEC in a disordered potential
(Florence, Orsay, Hannover, Rice, Illinois...)
How to produce disorder
speckle pattern
bichromatic lattice
L.Fallani, C.Fort, M.Inguscio
Bose-Einstein condensates in disordered potentials
Advances Atomic, Molecular and Optical Physics vol 56, pp 119-160
edited by E.Arimondo, P.Berman, C.Lin (Academic Press 2008)
BICHROMATIC OPTICAL LATTICE – QUASI CRYSTALS
Adding a weak incommensurate optical lattice...
Anderson localization
Single particle tight binding model with random
on-site energies (electrons in a crystal lattice)
tunneling (J)
amplitude of disorder (∆)
No diffusion for V < Vc～ W
Extended wave function
Localized wave
function
ψ
2
W
Anderson localization
one electron in a periodic lattice
DIFFUSION
Anderson localization
introducing disorder in the lattice
Anderson localization
one electron in a disordered lattice
LOCALIZATION
How to realize the Anderson model with cold atoms!
A deep optical lattice realizes a tight binding lattice model...
Atoms trapped in the sites with a hopping probability
How to realize the Anderson model with cold atoms!
An inhomogeneous external potential breaks the lattice translational invariance
Aubry-André model with cold atoms!
Adding a weak incommensurate optical lattice...
The second lattice controls the site energies
Aubry-André model with cold atoms!
Adding a weak incommensurate optical lattice...
The second lattice controls the site energies
Localization models
Localization depends on the kind of disorder and dimensionality!
1D Anderson model
1D Aubry-André model
pure random
quasiperiodic
localization for any ∆
localization transition at finite ∆ = 2J
The Aubry-André model
S. Aubry and G. André,
Analyticity breaking and Anderson localization in incommensurate lattices
Ann. Israel Phys. Soc. 3, 133 (1980).
irrational number
➤ Anderson transition from extended to exponentially localized eigenstates
∆c = 2 J
➤ Dual model in momentum space: ∆ → 4 J2 /∆
The Aubry-André model
S. AubryLocalization
and G. André,
transition for β approximating an
Analyticity
breaking
andthrough
Anderson
in incommensurate lattices
irrational
number
thelocalization
Fibonacci sequence
Ann. Israel Phys. Soc. 3, 133 (1980).
irrational number
➤ Anderson transition from extended to exponentially localized eigenstates
∆c = 2 J
➤ Dual model in momentum (courtesy
space: ∆ of
→A.4Minguzzi)
J2 /∆
Extended and localized states
Localization transition in 1D incommensurate bichromatic lattice
Michele Modugno, arXiv:0901.0210 (2009)
localized states:
extended states:
J. Lye et al., PRA 75, 061603(R) (2007)
MICHELE MODUGNO
extended superposition of
many localized states
interactions
Effects of interactions on the localized state
one localized state
potassium - rubidium
39,40,41K
/ 87Rb sympathetic cooling
Symultaneous trapping of a mixture Rb/K
K
Rb
Selective evaporation of Rb
microwave
@ 6.8 GHz
Repulsive potential
Thermalization of the K / Rb mixture
No K losses
BEC of 39K
Roati et al PRL 99, 010403 (2007)
Feshbach assisted sympatetic cooling in a mixture with 87Rb
200
(1)
(2)
0
-100
-200
280 300 320 340 360 380 400 420 440
0,5
-200
39
K BEC
0,0
0
1
Density distribution
3.2 s
0
200
-200
aKRb = 28 a0
T=150 nK
2
Time (s)
B0 = 317.9 G
3s
aK = 150 a0
T=800 nK
B (G)
B0 = 402.4 G
B=395.2 G
B = 316 G
aK = -33 a0
aKRb = 150 a0
1,0
Laser power (arb. units)
aBB (a0)
100
K-K
K-Rb
NK = 7*104
Tc~ 100 nK
3.5 s
0
200 -200
Horizontal position (µm)
3
0
200
2007
PRL
39K boson
...the last stable alkali isotope!
39K
BEC with tunable interactions
G. Roati et al., PRL 99, 010403 (2007)
1
100
∆a = 0.06 a0
0
K3 = 1.3(5)×10-29 cm6s-1
-1
348
350
352
0
0
-400
340
-100
350
360
370
380
390
magnetic field (G)
Interferometry: Fattori et al., PRL 100, 080405 (2008)
.
400
binding energy (MHz)
scattering
cattering length (a0)
400
410
Dipolar effects: Fattori et al., PRL 101, 190405 (2008)
Disordered people
M. Fattori
G. Roati
M. Zaccanti
L. Fallani
C. D’Errico
M. Modugno
G. Modugno
C. Fort
Experimental scheme
G. Roati et al., Nature 453, 895 (2008)
Probing the transport properties
The noninteracting BEC is initially confined in a harmonic trap and then left
free to expand in the bichromatic lattice
Absence of diffusion
G. Roati et al., Nature 453, 895 (2008)
∆=0
Ballistic expansion:
∆/J=1
Ballistic expansion
with reduced velocity
∆/J=7
Absence of diffusion:
Absence of diffusion
G. Roati et al., Nature 453, 895 (2008)
Expansion in the bichromatic lattice
G. Roati et al., Nature 453, 895 (2008)
Size of the condensate after 750 ms expansion in the bichromatic lattice:
Scaling law: onset of localization
only depends on ∆/J!
Localized states
Diffusion stops because the eigenstates are localized!
Periodic: wavefunction is delocalized on the whole system size
Disordered: eigenstates are localized in a finite region of space
exponentially decaying amplitude of wavefunction
Exponential localization
G. Roati et al., Nature 453, 895 (2008)
Fit of the density distribution with a generalized exponential function:
Exponential localization
G. Roati et al., Nature 453, 895 (2008)
Fit of the density distribution with a generalized exponential function:
gaussian
exponential
LIGHT in PHOTONIC
LATTICES
Phys.Rev.Letters in press
Momentum distribution
Momentum distribution
experiment
G. Roati et al., Nature 453, 895 (2008)
theory
Density distribution after
time-of-flight of the initial
stationary state
Width of the central peak
Visibility
Momentum distribution
experiment
G. Roati et al., Nature 453, 895 (2008)
theory
Density distribution after
time-of-flight of the initial
stationary state
Universal behavior with ∆/J!
Width of the central peak
Visibility
AUBRY- ANDRE Hamiltonian
L. Fallani and M. Inguscio
Controlling cold-atom conductivity
Science 322 (5th December 2008)
…Can physics be simulated by a universal computer?
Richard P. Feynman, Int. J. Theor. Phys 21, 467 (1982)
Richard P. Feynman realized that certain phenomena in
Quantum Field Theory are well imitated by certain
Condensed Matter systems…
He thought that there should be a certain class of quantum
mechanical systems which would simulate any other system, a
UNIVERSAL QUANTUM SIMULATOR
that could serve as a quantum laboratory where the validity
of several theoretical models may be tested.
L. Fallani and M. Inguscio
Controlling cold-atom conductivity
Science 322 (5th December 2008)
only the beginning...
work in progress also at
Institute d’Optique, Hannover, Rice, Illinois…
P. W. Anderson, Nobel lecture (1977)
… about the role of interactions
A second reason why I felt discouraged in the early days was that
I couldn’t fathom how to reinsert interactions, and I was afraid
they, too, would delocalize.
The realization that, of course, the Mott insulator localizes without
randomness, because of interactions, was my liberation on this:
one can see easily that Mott and Anderson effects supplement,
not destroy, each other…
The present excitement of the field for me is that a theory of
localization with interactions is beginning to appear…
It is remarkable that in almost all cases interactions play a vital
role, yet many results are not changed too seriously by them.
Disordered interacting bosons
Experiments are in progress to investigate the whole
phase diagram of lattice bosons in presence of disorder
3D systems, exp. in progress
Roati et al, Nature 453, 895 (2008)
Billy et al, Nature 453, 891 (2008)
R.Hulet (Rice)
1D systems, exp. in progress
Fallani et al, Phys. Rev. Lett. 98, 130404 (2007)
Disorder is beautiful