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Experiments with Fermi e Bose
atomic gases in optical lattices
Giovanni Modugno
LENS, Università di Firenze, and INFM
XXVII Convegno di Fisica Teorica, Cortona, May 2005
Introduction
Motivations
Ultracold atomic gases in optical lattices are potentially a powerful
model system to study condensed-matter problems (almost
everything can be easily tuned)
Interesting applications beyond condensed matter are arising
Outline of the talk
Production and properties of atomic quantum gases; optical lattices
Experiments with Bose-Einstein condensates: superfluid transport,
instabilities and localization driven by interactions
Experiments with Fermi gases: fundamental transport phenomena and
applications
Future directions
Production methods
Laser cooling in magneto-optical traps:
T =10µK
Evaporative/sympathetic cooling in magnetic traps:
T =10nK
40K
87Rb
fermions
bosons
145 nK
110 nK
80 nK
Typical parameters:
N = 105-107
n =1012 -1014 cm-3
l = 10-1000 µm
Tmin=0.1 TF, 0.1 Tc
Detection of momentum distribution
by absorption imaging with resonant
light
Roati et al. Phys.Rev. Lett. 89, 150403
(2002).
Ultracold collisions
Molecular interaction between neutral atoms: contact interaction
σ=
-
8π
k2
2
∑ (2l + 1) sin δ
l even
l odd
l
Even waves for identical bosons, odd waves for identical fermions
All waves with l≠0 are thermally suppressed as E2l
σ s = 4πa 2
No interactions between identical
fermions below 100µK
De Marco and Jin, Phys. Rev. Lett. 1999
Fano-Feshbach resonances
Magnetically tunable resonances
 tunable interaction in s, p, and other waves
 observed or expected for all alkali species
(both homo- and hetero-nuclear)
Molecules formation and Cooper pairing in Fermi gases
Bose-Einstein condensation of molecules
Molecules formation at FanoFeshbach resonances
JILA, Innsbruck, ENS,
MIT, Rice University
Condensation of Cooper pairs
F. Chevy and C. Salomon, Physics World, March 2005
Optical lattices
Optical dipole potential:
r 3πc 2 Γ r
U (r ) =
I (r )
3
2ω Δ
1D optical lattice:
U ( z , r ) = U 0 (1 + cos(2π
z / λ )) × exp(−r 2 / w2 )
z
Natural energy and
momentum scales:
h2
ER =
2mλ2
h
qB =
λ
EER
R
00
}}εε
}}22δδ
-q
-qBB
λ= 1µm, qB= 5 mm s-1, ER = 100 nK, U = 1-100 ER
q
xxq
+q
+qBB
Cubic lattices with various dimensionalities 1D, 2D, 3D, other geometries,
lattices with large spacing 1-10 µm, …
Bose gases in optical lattices
Superfluidity and interactions in periodic potentials
 macroscopic transport at low interaction strengths
 insulating phases due to interactions
Transport
of di
a superfluid
Gas
Bose in reticoli ottici: trasporto superfluido
Collective dipole oscillations
Horizontal Position (pix)
___________________________________________________________________
44
42
40
38
36
0
50
100
Thermal cloud
150
200
250
300
350
Time (ms)
BEC
0 ms
20 ms
40 ms
60 ms
80 ms
F. Cataliotti, et al. Science 293, 843 (2001).
Band spectroscopy and dynamical instabilities
2.0
s=0
s = 1.3
s = 3.8
B )
velocity (v/v
1.5
1.0
0.5
0.0
0.0
0.5
1.0
1.5
quasimomentum (q/q
)
B
2.0
Optical lattices can be put in motion:
Spectroscopy of the lattice band dispersion with a BEC
What is the role of atomic interactions?
cos()2RisEkxgtmψψψ∂=−∇++∂hh
2222
L. Fallani, et al. Phys. Rev. Lett. 93, 140406 (2004).
Band spectroscopy and dynamical instabilities
growth of excitations (lattice on)
0.08
-1
loss rate [ms]
0.40qB
0.06
0.04
10
200
500
800
1000
1200
2
5
10
20
30
35
1500 ms
0.02
0.55qB
0.00
0.0
0.5
1.0
1.5
quasimomentum (q/q
)
B
55
What is the role of atomic interactions?
cos()2RisEkxgtmψψψ∂=−∇++∂hh
2222
L. Fallani, et al. Phys. Rev. Lett. 93, 140406 (2004).
ms
Localization in a Mott insulator
(M. Greiner et al., Nature 415, 39 (2002))
1ˆˆˆˆˆ (1)2ijiiiiijiiHJaanUnnε=−++−∑∑∑
†,
Bose-Hubbard Hamiltonian
SUPERFLUID PHASE
MOTT INSULATOR PHASE
1. Long-range phase coherence
2. High number fluctuations
3. No gap in the excitation spectrum
1.
2.
3.
4.
5.
No phase coherence
Zero number fluctuations
Gap in the excitation spectrum
Vanishing superfluid fraction
Vanishing compressibility
Fermi gases in optical lattices
Identical fermions: an ideal gas in a perfect periodic
potential
 transport properties of a perfect crystal of atoms
 applications
Transport
of di
a non
interacting
Fermi ottici:
gas
Gas
Bose
in reticoli
trasporto superfluido
___________________________________________________________________
Collective dipole oscillations
Position
s=7
Fermions remain trapped on the
side of the harmonic potential
0
2
4
Time
6
s=0
Transport of a non-interacting Fermi gas
E
ε
EF
An ideal crystal
is an insulator.
}2δ
x
s=5
Ott, et al. Phys. Rev. Lett. 93, 120407 (2004), Rigol and Muramatsu, Phys.Rev. A 63, (2004),
Hooley and Quintanilla, Phys. Rev. Lett. 93,080404, (2004).
Collision-induced transport
decay time (ms)
10000
Esaki-Tsu model for electrons
in superlattices
ω
1000
BO
100
0
100
200
300
400
500
600
-1
collisional rate (s )
Tuning collisions in a boson-fermion mixture: crossover from an ideal
conductor (that behaves like an insulator) to a real conductor
Pezzè et al., Phys. Rev. Lett. 93, (2004); Ott et al., Phys. Rev. Lett. 92, 160601 (2004).
Spectroscopy of localized states
Atoms in delocalized states can be
selectively removed with a RF knife
E
ε
RF
sweep
}2δ
x
Ott, et al., Phys. Rev. Lett. 93, 120407 (2004).
s=5
Applications
 quantum computing
 atom interferometry for force sensing
Quantum registers: Bose vs Fermi
What is needed:
 Macroscopic array of indidually addressable qubits
 Lowest possible number of defects
 Controllable, coherent interactions to perform operations
Bose:
Bose interaction-induced localization
 One localized particle per lattice site
 Controllable interactions between neighbouring
sites via spin-selective lattices
Fermi:
Fermi potential-induced localization
 Two localized particle per lattice site
 Loading procedure confines defects to the outer shell
 Tunable interactions between two states via F-F resonances
Wannier-Stark states and Bloch oscillations
Wannier-Stark states in a
lattice tilted by gravity:
Δϕ = 0
Δx = 2δ / F
ΔE =
mgλ
2
Their interference oscillates:
Δϕ = π
U 0 = 2 ER
-2
-1
0
1
Momentum (q
)
B
ω B = mgλ / 2h
2
q&= −mg
Semiclassical picture: Bloch oscillations
-qB
q
+qB
Bloch oscillations
+ qB
− qB
Time-resolved Bloch oscillations of trapped, non-interacting fermions
2 ms
2.4 ms
2.8 ms
3.2 ms
3.6 ms
4 ms
4.4 ms
4.8 ms
5.2 ms
5.6 ms
G. Roati, et al., Phys. Rev. Lett. 92, 230402 (2004).
Bloch oscillations
1.0
Momentum B(q)
0.5
0.0
-0.5
-1.0
0
5
10
15
250
255
Time (ms)
TB = 2h / mgλ
TB = 2.32789 (22)ms ⇒ Δg / g = 10 −4
Fermions trapped in lattices: a force sensor with high spatial resolution
( presently 50µm, but no fundamental limitations down to a few lattice
sites)
Force sensing at the micrometer lengthscale
Casimir-Polder potential in proximity
of a dielectric surface
VCP = −
k BTα 0 ε 0 − 1
4z3 ε 0 +1
Features:
 high resolution in presence of gravity
 direct measurement of forces
 low sensitivity to gradients
 high sensitivity (10-7g)
Applications:
 atom-surface interactions out of thermal
equilibrium
 possible deviations from Newton’s
gravitational law at short distances
I. Carusotto, L. Pitaevskii, S. Stringari, G. Modugno, M. Inguscio, cond-mat/0503141.
Search for non newtonian forces
10-7 g
10-10 g
S. Dimopulos and A. A. Geraci, Phys. Rev. D 68, 124021 (2003)
Future directions
Bose and Fermi gases in 1D optical lattices
 phenomenology of the band transport, transport of bosonic and
fermionic superfluids
 fermionic Bloch oscillator: application to high precision study of
fundamental phenomena
Bose, Fermi and Fermi-Bose gases in 2D and 3D optical lattices
 condensed matter physics: Mott insulators, high Tc superfluidity, …
 low dimensionality systems: Luttinger liquids, BEC-BCS, …
 applications to quantum computing
Optical lattice and random potentials
 Anderson localization, Bose and Fermi glasses, …
 BEC-BCS in presence of disorder
Fermi surface in
a 2D lattice
The quantum gas team at LENS
Estefania De Mirandes, Leonardo Fallani, Francesca Ferlaino, Vera Guarrera, Iacopo Catani,
Luigi De Sarlo, Jessica Lye, Giacomo Roati, Herwig Ott
Chiara Fort, Francesco Minardi, Michele Modugno
Giovanni Modugno, Massimo Inguscio