Transcript Document

Strongly correlated many-body systems:
from electronic materials to ultracold atoms
Eugene Demler
Harvard University
“Conventional” solid state materials
Bloch theorem for non-interacting
electrons in a periodic potential
Consequences of the Bloch theorem
B
VH
d
Metals
I
EF
EF
Insulators
and
Semiconductors
First semiconductor transistor
“Conventional” solid state materials
Electron-phonon and electron-electron interactions
are irrelevant at low temperatures
ky
kx
Landau Fermi liquid theory: when frequency and
temperature are smaller than EF electron systems
are equivalent to systems of non-interacting fermions
kF
Ag
Ag
Ag
Strongly correlated electron systems
Quantum Hall systems
kinetic energy suppressed by magnetic field
UCu3.5Pd1.5
CeCu2Si2
Heavy fermion materials
many puzzling non-Fermi
liquid properties
High temperature superconductors
Unusual “normal” state,
Controversial mechanism of superconductivity,
Several competing orders
What is the connection between
strongly correlated electron systems
and
ultracold atoms?
Bose-Einstein condensation of
weakly interacting atoms
Density
Typical distance between atoms
Typical scattering length
1013 cm-1
300 nm
10 nm
Scattering length is much smaller than characteristic interparticle distances.
Interactions are weak
New Era in Cold Atoms Research
Focus on Systems with Strong Interactions
• Feshbach resonances
• Rotating systems
• Low dimensional systems
• Atoms in optical lattices
• Systems with long range dipolar interactions
Feshbach resonance and fermionic condensates
Greiner et al., Nature (2003); Ketterle et al., (2003)
Ketterle et al.,
Nature 435, 1047-1051 (2005)
One dimensional systems
1D confinement in optical potential
Weiss et al., Science (05);
Bloch et al.,
Esslinger et al.,
One dimensional systems in microtraps.
Thywissen et al., Eur. J. Phys. D. (99);
Hansel et al., Nature (01);
Folman et al., Adv. At. Mol. Opt. Phys. (02)
Strongly interacting
regime can be reached
for low densities
Atoms in optical lattices
Theory: Jaksch et al. PRL (1998)
Experiment: Kasevich et al., Science (2001);
Greiner et al., Nature (2001);
Phillips et al., J. Physics B (2002)
Esslinger et al., PRL (2004);
and many more …
Strongly correlated systems
Electrons in Solids
Atoms in optical lattices
Simple metals
Perturbation theory in Coulomb interaction applies.
Band structure methods work
Strongly Correlated Electron Systems
Band structure methods fail.
Novel phenomena in strongly correlated electron systems:
Quantum magnetism, phase separation, unconventional superconductivity,
high temperature superconductivity, fractionalization of electrons …
By studying strongly interacting systems of cold atoms we
expect to get insights into the mysterious properties of
novel quantum materials: Quantum Simulators
BUT
Strongly interacting systems of ultracold atoms :
are NOT direct analogues of condensed matter systems
These are independent physical systems with their own
“personalities”, physical properties, and theoretical challenges
Strongly correlated systems of ultracold atoms should
also be useful for applications in quantum information,
high precision spectroscopy, metrology
New Phenomena in quantum
many-body systems of ultracold atoms
New detection methods
Interference, higher order correlations
Nonequilibrium dynamics
Long intrinsic time scales
- Interaction energy and bandwidth ~ 1kHz
- System parameters can be changed over this time scale
Decoupling from external environment
- Long coherence times
Can achieve highly non equilibrium quantum many-body states
Dynamics of many-body quantum systems
Big Bang and Inflation
Cosmic microwave
background radiation.
Manifestation of quantum
fluctuations during inflation
Heavy Ion collisions at RHIC
Signatures of quark-gluon plasma?
Paradigms for equilibrium states of many-body systems
• Broken symmetry phases (magnetism, pairing, etc.)
• Order parameters
• RG flows and fixed points (e.g. Landau Fermi liquids)
• Topological states
• Effective low energy theories
• Classical and quantum critical points
• Scaling
Do we get any collective (universal?) phenomena in
the case of nonequilibrium dynamics?
Theoretical work on many-body nonequilibrium dynamics of ultracold
atoms: E. Altman, J.S. Caux, A. Cazalilla, K. Collath, A.J. Daley,
T. Giamarchi, V. Gritsev, T.L. Ho, A. Iucci, L. Levitov, M. Lewenstein,
A. Muramatsu, A. Polkovnikov, S. Sachdev, P. Zoller and many more
Strongly correlated many-body
systems of photons
Linear geometrical optics
Strongly correlated systems of photons
Strongly interacting polaritons in
coupled arrays of cavities
M. Hartmann et al., Nature Physics (2006)
Strong optical nonlinearities in
nanoscale surface plasmons
Akimov et al., Nature (2007)
Crystallization (fermionization) of photons
in one dimensional optical waveguides
D. Chang et al., Nature Physics (2008)
Outline of these lectures
• Introduction. Magnetic and optical trapping of ultracold
atoms.
• Cold atoms in optical lattices.
• Bose Hubbard model. Equilibrium and dynamics
• Bose mixtures in optical lattices
Quantum magnetism of ultracold atoms.
• Detection of many-body phases using noise correlations
• Experiments with low dimensional systems
Interference experiments. Analysis of high order correlations
• Fermions in optical lattices
Magnetism and pairing in systems with repulsive interactions.
Current experiments: paramagnetic Mott state, nonequilibrium
dynamics.
• Dynamics near Fesbach resonance. Competition of
Stoner instability and pairing
Emphasis of these lectures: • Detection of many-body phases
• Nonequilibrium dynamics
Ultracold atoms
Ultracold atoms
Most common bosonic atoms: alkali 87Rb and 23Na
Most common fermionic atoms: alkali 40K and 6Li
Other systems:
BEC of
133Cs
(Innsbruck)
BEC of 52Cr (Stuttgart)
BEC of 84Sr (Innsbruck), and
88Sr
(Boulder)
BEC of 168Yb, 170Yb, 172Yb, 174Yb, 176Yb (Kyoto)
Degenerate fermions 171Yb, 173Yb (Kyoto), 87Sr (Boulder)
Magnetic properties of individual alkali atoms
Single valence electron in the s-orbital
and
Nuclear spin
Hyperfine coupling mixes nuclear and electron spins
Total angular momentum (hyperfine spin)
Zero field splitting between
states
and
For 23Na AHFS = 1.8 GHz and for 87Rb AHFS = 6.8 GHz
Magnetic properties of individual alkali atoms
Effect of magnetic field comes from electron spin
gs=2 and mB=1.4 MHz/G
When fields are not too large one can use (assuming field along z)
The last term describes quadratic Zeeman effect q=h 390 Hz/G2
Magnetic trapping of alkali atoms
Magnetic trapping of neutral atoms is due to the Zeeman effect.
The energy of an atomic state depends on the magnetic field.
In an inhomogeneous field an atom experiences a spatially
varying potential.
Example:
Potential:
Magnetic trapping is limited by the requirement that the trapped atoms remain
in weak field seeking states. For 23Na and 87Rb there are three states
Optical trapping of alkali atoms
Based on AC Stark effect
Dipolar moment induced by the electric field
Typically optical frequencies.
- polarizability
Potential:
Far-off-resonant optical trap confines atoms regardless
of their hyperfine state
Ultracold atoms in optical lattices.
Band structure. Semiclasical dynamics.
Optical lattice
The simplest possible periodic optical potential is formed
by overlapping two counter-propagating beams. This results
in a standing wave
Averaging over fast optical oscillations (AC Stark effect) gives
Combining three perpendicular sets of standing waves we
get a simple cubic lattice
This potential allows separation of variables
Optical lattice
For each coordinate we have Matthieu equation
Eigenvalues and eigenfunctions are known
In the regime of deep lattice we get the tight-binding model
- recoil energy
- bandgap
Lowest band
Optical lattice
Effective Hamiltonian for non-interacting
atoms in the lowest Bloch band
nearest
neighbors
Band structure
State dependent optical lattices
Fine structure for 23Na and 87Rb
How to use selection rules for
optical transitions to make
different lattice potentials for
different internal states.
The right circularly polarized light
couples
to two
excited levels P1/2 and P3/2. AC Stark effects have opposite signs and cancel
each other for the appropriate frequency. At this frequency AC Stark effect for
the state
comes only from
polarized light and gives
the potential .
Analogously state
gives the potential
Decomposing
hyperfine
states we find
will only be affected by
.
, which
State dependent lattice
PRL 91:10407 (2003)
Atoms in optical lattices.
Bose Hubbard model
Bose Hubbard model
U
t
tunneling of atoms between neighboring wells
repulsion of atoms sitting in the same well
In the presence of confining potential we also need to include
Typically
Bose Hubbard model. Phase diagram
m U
n=3 Mott
M.P.A. Fisher et al.,
PRB (1989)
n 1
2
n=2 Mott
Superfluid
1
n=1
Mott
0
Weak lattice
Strong lattice
Superfluid phase
Mott insulator phase
Bose Hubbard model
Set .
Hamiltonian eigenstates are Fock states
0
1
mU
Away from level crossings Mott states have
a gap. Hence they should be stable to small tunneling.
Bose Hubbard Model. Phase diagram
m U
n=3 Mott
n 1
2
n=2
Mott
Superfluid
1
n=1
Mott
0
Mott insulator phase
Particle-hole
excitation
Tips of the Mott lobes
z- number of nearest neighbors, n – filling factor
Gutzwiller variational wavefunction
Normalization
Kinetic energy
z – number of nearest neighbors
Interaction energy favors a fixed number of atoms per well.
Kinetic energy favors a superposition of the number states.
Gutzwiller variational wavefunction
Example: stability of the Mott state with n atoms per site
Expand to order
Take the middle of the Mott plateau
Transition takes place when coefficient before
negative. For large n this corresponds to
becomes
Bose Hubbard Model. Phase diagram
m U
n=3
Mott
n 1
2
n=2
Mott
Superfluid
1
n=1
Mott
0
Note that the Mott state only exists for integer filling factors.
For
even when
atoms are localized,
make a superfluid state.
Bose Hubbard model
Experiments with atoms in optical lattices
Theory: Jaksch et al. PRL (1998)
Experiment: Kasevich et al., Science (2001);
Greiner et al., Nature (2001);
Phillips et al., J. Physics B (2002)
Esslinger et al., PRL (2004);
many more …
Nature 415:39 (2002)
Optical lattice and parabolic potential
Parabolic potential acts as a “cut” through
the phase diagram. Hence in a parabolic
potential we find a “wedding cake” structure.
m U
n=3
Mott
n 1
2
n=2
Mott
1
n=1
Mott
0
Jaksch et al.,
PRL 81:3108 (1998)
Superfluid
Nature 2009
arXiv:1006.3799
Nonequilibrium dynamics of
Bose Hubbard model
Dynamics and local resolution in
systems of ultracold atoms
Bakr et al.,
Science 2010
Single site imaging
from SF to Mott states
Dynamics of on-site
number statistics for
a rapid SF to Mott ramp
Moving condensate in an optical lattice. Dynamical instability
Theory: Niu et al. PRA (01), Smerzi et al. PRL (02)
Experiment: Fallani et al. PRL (04)
v
Dynamical instability
Classical limit of the Hubbard model.
Discreet Gross-Pitaevskii equation
Current carrying states
Linear stability analysis: States with p>p/2 are unstable
unstable
unstable
Amplification of
density fluctuations
r
Dynamical instability. Gutzwiller approximation
Wavefunction
Time evolution
We look for stability against small fluctuations
0.5
unstable
0.4
d=3
Phase diagram. Integer filling
d=2
Altman et al., PRL 95:20402 (2005)
p/
0.3
d=1
0.2
stable
0.1
0.0
0.0
0.2
0.4
U/Uc
0.6
0.8
1.0
Center of Mass Momentum
Optical lattice and parabolic trap.
Gutzwiller approximation
0.00
0.17
0.34
0.52
0.69
0.86
N=1.5
N=3
0.2
0.1
The first instability
develops near the edges,
where N=1
0.0
-0.1
U=0.01 t
J=1/4
-0.2
0
100
200
300
Time
400
500
Gutzwiller ansatz simulations (2D)
PRL (2007)
Beyond semiclassical equations. Current decay by tunneling
phase
phase
phase
Polkovnikov et al., Phys. Rev. A (2005)
site index j
site index j
site index j
Current carrying states are metastable.
They can decay by thermal or quantum tunneling
Thermal activation
Thermal phase slips observed
by DeMarco et al., Nature (2008)
Quantum tunneling
Quantum phase slips observed
by Ketterle et al., PRL (2007)
Engineering magnetic systems
using cold atoms in an optical lattice
Two component Bose mixture in optical lattice
Example:
t
. Mandel et al., Nature (2003)
t
Two component Bose Hubbard model
We consider two component Bose mixture in the n=1
Mott state with equal number of and atoms.
We need to find spin arrangement in the ground state.
Quantum magnetism of bosons in optical lattices
Duan, Demler, Lukin, PRL (2003)
• Ferromagnetic
• Antiferromagnetic
Two component Bose Hubbard model
In the regime of deep optical lattice we can treat tunneling
as perturbation. We consider processes of the second order in t
We can combine these processes into
anisotropic Heisenberg model
Two component Bose mixture in optical lattice.
Mean field theory + Quantum fluctuations
Altman et al., NJP (2003)
Hysteresis
1st order
Two component Bose Hubbard model
+ infinitely large Uaa and Ubb
New feature:
coexistence of
checkerboard phase
and superfluidity
Exchange Interactions in Solids
antibonding
bonding
Kinetic energy dominates: antiferromagnetic state
Coulomb energy dominates: ferromagnetic state
Realization of spin liquid
using cold atoms in an optical lattice
Theory: Duan, Demler, Lukin PRL (03)
Kitaev model
Annals of Physics (2006)
H = - Jx S six sjx - Jy S siy sjy - Jz S siz sjz
Questions:
Detection of topological order
Creation and manipulation of spin liquid states
Detection of fractionalization, Abelian and non-Abelian anyons
Melting spin liquids. Nature of the superfluid state
Superexchange interaction
in experiments with double wells
Theory: A.M. Rey et al., PRL 2008
Experiments: S. Trotzky et al., Science 2008
Observation of superexchange in a double well potential
Theory: A.M. Rey et al., PRL 2008
J
J
Use magnetic field gradient to prepare a state
Observe oscillations between
and
states
Experiments:
S. Trotzky et al.
Science 2008
Preparation and detection of Mott states
of atoms in a double well potential
Reversing the sign of exchange interaction
Comparison to the Hubbard model
Beyond the basic Hubbard model
Basic Hubbard model includes
only local interaction
Extended Hubbard model
takes into account non-local
interaction
Beyond the basic Hubbard model
Summary of lecture I
•
•
•
•
Introduction. Systems of ultracold atoms.
Cold atoms in optical lattices.
Bose Hubbard model. Equilibrium and dynamics
Bose mixtures in optical lattices.
Quantum magnetism of ultracold atoms.
Outline of future lectures
•
•
•
•
Introduction. Systems of ultracold atoms.
Cold atoms in optical lattices.
Bose Hubbard model. Equilibrium and dynamics
Bose mixtures in optical lattices.
Quantum magnetism of ultracold atoms.
• Detection of many-body phases using noise correlations
• Experiments with low dimensional systems
Interference experiments. Analysis of high order correlations
• Fermions in optical lattices
• Dynamics near Fesbach resonance. Competition of
Stoner instability and pairing