Download Lecture 1 - Harvard Condensed Matter Theory group

Strongly correlated many-body systems:
from electronic materials to ultracold atoms
to photons
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 wotk
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 and photons:
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
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
New detection methods
Interference, higher order correlations
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. Systems of ultracold atoms.
Bogoliubov theory. Spinor condensates.
Cold atoms in optical lattices.
Bose Hubbard model and extensions
Bose mixtures in optical lattices
Quantum magnetism of ultracold atoms.
Current experiments: observation of superexchange
• Fermions in optical lattices
Magnetism and pairing in systems with repulsive interactions.
Current experiments: Mott state
• Detection of many-body phases using noise correlations
• Experiments with low dimensional systems
Interference experiments. Analysis of high order correlations
• Non-equilibrium dynamics
Emphasis of these lectures:
• Detection of many-body phases
• 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
(e.g. Grimm et al.)
BEC of 52Cr (Pfau et al.)
BEC of 84Sr (e.g. Grimm et al.),
87Sr
and
88Sr
(e.g. Ye et al.)
BEC of 168Yb, 170Yb, 172Yb, 174Yb, 176Yb
Quantum degenerate fermions 171Yb 173Yb (Takahashi et al.)
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
Bogoliubov theory of
weakly interacting BEC.
Collective modes
BEC of spinless bosons. Bogoliubov theory
We consider a uniform system first
- boson annihilation operator at momentum p,
- strength of contact s-wave interaction
- volume of the system
- chemical potential
For non-interacting atoms at T=0 all atoms are in k=0 state.
Mean field equations
Minimizing with respect to N0 we find
BEC of spinless bosons. Bogoliubov theory
We now expand around the nointeracting solution for small U0 .
From the definition of Bose operators
When N0 >>1 we can treat b0 as a c-number and replace
by
Mean-field Hamiltonian
- means that p,-p pairs should be counted only once
n0 = N0/V - density and m = n0 U0
Bogoliubov transformation
Bosonic commutation relations
are preserved when
Bogoliubov transformation
Mean-field Hamiltonian becomes
Bogoliubov transformation
Cancellation of non-diagonal terms requires
To satisfy
take
Solution of these equations
Mean-field
Hamiltonian
Bogoliubov modes
Dispersion of collective modes
Define healing length from
Long wavelength limit,
, sound dispersion
Sound velocity
Short wavelength limit,
, free particles
Probing the dispersion of BEC
by off-resonant light scattering
For details see
cond-mat/0005001
Treat optical field as classical
Excitation rate out of the ground state |g>
Dynamical structure factor
For small q we find
PRL 83:2876 (1999)
PRL 88:60402 (2002)
Ω
Rev. Mod. Phys. 77:187 (2005)
Gross-Pitaevskii equation
Gross-Pitaevskii equation
Hamiltonian of interacting bosons
Commutation relations
Equations of motion
This is operator equation. We can take classical limit by assuming
that all atoms condense into the same state
.
The last equation becomes an equation on the wavefunction
Gross-Pitaevskii equation
Analysis of fluctuations on top of the mean-field
GP equations leads to the Bogoliubov modes
Two-component mixtures
Two-component Bose mixture
Consider mean-field (all particles in the condensate)
Repulsive interactions. Miscible and immiscible regimes
System with a
finite density
of both species
is unstable
to phase
separation
Bose mixture: dynamics of phase separation
What happens if we prepare a mixture of two
condensates in the immiscible regime?
Two component GP equation
Assume equal densities
Analyze fluctuations
Bose mixture: dynamics of phase separation
Define
density fluctuation
phase fluctuation
Equations of motion: charge conservation and Josephson relation
Equation on collective modes
Bose mixture: dynamics of phase separation
Here
When
we get imaginary frequencies
Imaginary frequencies indicate
exponential growth of fluctuations,
i.e. instability. The most unstable
mode sets the length for pattern
formation. Note that
when
. This is required
by spin conservation.
Most unstable mode
PRL 82:2228 (1999)
Spinor condensates
F=1
Spinor condensates. F=1
Three component order parameter: mF=-1,0,+1
Contact interaction depends on relative spin orientation
When g2>0 interaction is antiferromagnetic. Example 23Na
When g2<0 interaction is antiferromagnetic. Example 87Rb
F=1 spinor condensates. Hamiltonian
- spin operators for F=1
Total Fz is conserved so linear Zeeman term should (usually)
be understood as Lagrange multiplier that controls Fz.
Quadratic Zeeman effect causes the energy of mF=0
state to be lower than the energy of mF=-1,+1 states.
The antiferromagnetic interaction (g2>0) favors the nematic
(polar) state (mF=0 and its rotations). The ferromagnetic
interaction (g2<0) favors spin polarized state (mF=+1) and
its rotations).
Phase diagram of F=1 spinor condensates
g2>0
Antiferromagnetic
Shaded region: mixture of
mF=-1,+1 states. This state
lowers interaction energy.
g2<0
Ferromagnetic
Shaded region: mixture of all three
states. There is XY component
of the spin.
Nature 396:345 (1998)
Using magnetic field gradient
to explore the phase diagram
of F=1 atoms with AF interactions
23Na
Nature 443:312 (2006)
Magnetic dipolar interactions
in ultracold atoms
Magnetic dipolar interactions in spinor condensates
q
Comparison of contact and dipolar interactions.
Typical value a=100aB
For 87Rb m=mB and e=0.007
Bose condensation
of 52Cr.
T. Pfau et al. (2005)
Review:
Menotti et al.,
arXiv 0711.3422
For 52Cr m=6mB and e=0.16
Magnetic dipolar interactions in spinor condensates
Interaction of F=1 atoms
Ferromagnetic Interactions for 87Rb
a2-a0= -1.07 aB
A. Widera, I. Bloch et al.,
New J. Phys. 8:152 (2006)
Spin-depenent part of the interaction is small.
Dipolar interaction may be important (D. Stamper-Kurn)
Spontaneously modulated textures in spinor condensates
Vengalattore et al.
PRL (2008)
Fourier spectrum of the
fragmented condensate
Energy scales: importance of Larmor precession
Magnetic Field
•Larmor Precession (100 kHz)
•Quadratic Zeeman (0-20 Hz)
B
F
S-wave Scattering
•Spin independent (g0n = kHz)
•Spin dependent (gsn = 10 Hz)
Dipolar Interaction
•Anisotropic (gdn=10 Hz)
•Long-ranged
Reduced Dimensionality
•Quasi-2D geometry
d   spin
Stability of systems with static dipolar interactions
Ferromagnetic configuration is robust against small
perturbations. Any rotation of the spins conflicts with
the “head to tail” arrangement
Large fluctuation required to reach a lower energy configuration
Dipolar interaction averaged after precession
“Head to tail” order of the transverse spin components is violated
by precession. Only need to check whether spins are parallel
XY components of the
spins can lower the energy
using modulation along z.
X
X
Z components of the
spins can lower the energy
using modulation along x
Strong instabilities of systems with dipolar interactions
after averaging over precession
Instabilities of F=1 Rb (ferromagnetic)
condensate due to dipolar interactions
Theory: unstable modes in the regime
corresponding to Berkeley experiments.
Cherng, Demler, PRL (2009)
Experiments.
Vengalattore et al. PRL (2008)
Berkeley Experiments: checkerboard phase
Instabilities of magnetic dipolar interactions:
general analysis
a – angle between magnetic
field and normal to the plane
dn – layer thickness
q measures
the strength
of quadratic
Zeeman effect
Instabilities of collective modes
Magnetoroton softening
Q measures
the strength
of quadratic
Zeeman effect
87Rb
condensate:
magnetic supersolid ?
Possible supersolid phase in 4He
Phase diagram of 4He
A.F. Andreev and I.M. Lifshits (1969):
Melting of vacancies in a crystal due
to strong quantum fluctuations.
Also
G. Chester (1970); A.J. Leggett (1970)
T. Schneider and C.P. Enz (1971).
Formation of the supersolid phase due to
softening of roton excitations
Resonant period as a function of T
Interlayer coherence in bilayer
quantum Hall systems at n=1
Hartree-Fock predicts roton softening
and transition into a state with both
interlayer coherence and stripe order.
Transport experiments suggest first
order transition into a compressible
state.
Eisenstein, Boebinger et al. (1994)
L. Brey and H. Fertig (2000)
Low energy effective model for
ferromagnetic condensates
Mass superflow current
caused by spin texture
Topological spin winding
(Pontryagin index)
Magnetic dipolar interaction.
Favors spin skyrmions
Requires net zero
spin winding
Magnetic crystal phases in F=1 87Rb
ferromagnetic condensates Cherng, Demler, unpublished
Experimental constraints:
1. Dihedral symmetry of the magnetic crystal
2. Rectangular lattice
3. Non-planar spin orientation
Magnetic crystal lattices
optimized within each class
Optimal spin configuration.
Magnetic crystal phases in F=1 87Rb
ferromagnetic condensates
Spin correlation functions
for spin components
parallel and perpendicular
to the magnetic field
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
Semiclassical dynamics in the lattice
1. Band index is constant
2.
3.
Bloch oscillations
Consider a uniform and constant force
PRL 76:4508 (1996)
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)