Download Lecture 2 - Harvard Condensed Matter Theory group

Strongly correlated many-body systems: from electronic
materials to ultracold atoms to photons
• Introduction. Systems of ultracold atoms.
• Bogoliubov theory. Spinor condensates.
• Cold atoms in optical lattices. Band structure and
semiclasical dynamics.
• Bose Hubbard model and its extensions
• Bose mixtures in optical lattices
Quantum magnetism of ultracold atoms.
Current experiments: observation of superexchange
• Detection of many-body phases using noise correlations
• Fermions in optical lattices
Magnetism and pairing in systems with repulsive interactions.
Current experiments: Mott state
• Experiments with low dimensional systems
Interference experiments. Analysis of high order correlations
• Non-equilibrium dynamics
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
 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
U
Away from level crossings Mott states have
a gap. Hence they should be stable to small tunneling.
Bose Hubbard Model. Phase diagram
 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
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
 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)
Shell structure in optical lattice
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.
 U
n=3
Mott
n 1
2
n=2
Mott
1
n=1
Mott
0
Jaksch et al.,
PRL 81:3108 (1998)
Superfluid
Shell structure in optical lattice
S. Foelling et al., PRL 97:060403 (2006)
Observation of spatial
distribution of lattice sites
using spatially selective
microwave transitions
and spin changing collisions
n=1
n=2
superfluid regime
Mott regime
Related work
Campbell, Ketterle, et al.
Science 313:649 (2006)
arXive:0904.1532
Extended Hubbard model
Charge Density Wave
and Supersolid phases
Extended Hubbard Model
- on site repulsion
- nearest neighbor repulsion
Checkerboard phase:
Crystal phase of bosons.
Breaks translational symmetry
van Otterlo et al., PRB 52:16176 (1995)
Variational approach
Extension of the Gutzwiller wavefunction
Supersolid – superfluid phase with broken translational symmetry
Quantum Monte-Carlo
analysis
arXiv:0906.2009
Difficulty of identifying supersolid phases
in systems with parabolic potential
Bose Hubbard model away from equilibrium.
Dynamical Instability of strongly interacting
bosons in optical lattices
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)
j
phase
j
phase
phase
Beyond semiclassical equations. Current decay by tunneling
Current carrying states are metastable.
They can decay by thermal or quantum tunneling
Thermal activation
Quantum tunneling
j
Current decay by thermal phase slips
Theory: Polkovnikov et al., PRA (2005)
Experiments: De Marco et al., Nature (2008)
Current decay by quantum phase slips
Theory: Polkovnikov et al., Phys. Rev. A (2005)
Experiment: Ketterle et al., PRL (2007)
Dramatic enhancement
of quantum fluctuations
in interacting 1d systems
d=1
dynamical
instability.
GP regime
d=1
dynamical
instability.
Strongly interacting
regime
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
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.
Two component Bose Hubbard model
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 Hubbard model
Quantum magnetism of bosons in optical lattices
Duan et al., PRL (2003)
• Ferromagnetic
• Antiferromagnetic
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
From two spins to a spin chain
Spin oscillations
?
Data courtesy of
S. Trotzky
(group of I. Bloch)
1D: XXZ dynamics starting from the classical Neel state
P. Barmettler et al, PRL 2009
Y(t=0
)
=
Ising-Order
Quasi-LRO
Equilibrium phase diagram:
1
• DMRG
• Bethe ansatz
• XZ model: exact solution
Time, Jt
D
XXZ dynamics starting from the classical Neel state
D<1, XY easy plane anisotropy
Oscillations of staggered moment,
Exponential decay of envelope
Except at solvable xx point where:
D>1, Z axis anisotropy
Exponential decay of
staggered moment
Behavior of the relaxation time with anisotropy
See also: Sengupta,
Powell & Sachdev (2004)
- Moment always decays to zero. Even for high easy axis anisotropy
- Minimum of relaxation time at the QCP. Opposite of classical critical slowing.
Magnetism in optical lattices
Higher spins and higher symmetries
F=1 spinor condensates
Spin symmetric interaction of F=1 atoms
Ferromagnetic Interactions for
Antiferromagnetic Interactions for
Antiferromagnetic spin F=1 atoms in optical lattices
Hubbard Hamiltonian
Demler, Zhou, PRL (2003)
Symmetry constraints
Nematic Mott Insulator
Spin Singlet Mott Insulator
Nematic insulating phase for N=1
Effective S=1 spin model
When
Imambekov et al., PRA (2003)
the ground state is nematic in d=2,3.
One dimensional systems are dimerized: Rizzi et al., PRL (2005)
Nematic insulating phase for N=1.
Two site problem
2
1
1
0
-2
4
Singlet state is favored when
One can not have singlets on neighboring bonds.
Nematic state is a compromise. It corresponds
and
to a superposition of
on each bond
SU(N) Magnetism with Ultracold Alkaline-Earth Atoms
A. Gorshkov et al., Nature Physics (2010)
Example: 87Sr (I = 9/2)
nuclear spin decoupled from electrons
SU(N=2I+1) symmetry
SU(N) spin models
Example: Mott state with nA atoms in sublattice A and nB atoms in sublattice B
Phase diagram for
nA + nB = N
There are also extensions to models with additional orbital degeneracy
Learning about order from noise
Quantum noise studies of ultracold atoms
Quantum noise
Classical measurement:
collapse of the wavefunction into eigenstates of x
Histogram of measurements of x
Probabilistic nature of quantum mechanics
Bohr-Einstein debate on spooky action at a distance
Einstein-Podolsky-Rosen experiment
Measuring spin of a particle in the left detector
instantaneously determines its value in the right detector
Aspect’s experiments:
tests of Bell’s inequalities
+
+
1
-
q1
S
q2
2
-
S
Correlation function
Classical theories with hidden variable require
Quantum mechanics predicts B=2.7 for the appropriate choice of q‘s and the state
Experimentally measured value B=2.697. Phys. Rev. Let. 49:92 (1982)
Hanburry-Brown-Twiss experiments
Classical theory of the second order coherence
Hanbury Brown and Twiss,
Proc. Roy. Soc. (London),
A, 242, pp. 300-324
Measurements of the angular diameter of Sirius
Proc. Roy. Soc. (London), A, 248, pp. 222-237
Quantum theory of HBT experiments
Glauber,
Quantum Optics and
Electronics (1965)
HBT experiments with matter
For bosons
Experiments with neutrons
Ianuzzi et al., Phys Rev Lett (2006)
Experiments with electrons
Kiesel et al., Nature (2002)
For fermions
Experiments with 4He, 3He
Westbrook et al., Nature (2007)
Experiments with ultracold atoms
Bloch et al., Nature (2005,2006)
Shot noise in electron transport
Proposed by Schottky to measure the electron charge in 1918
e-
e-
Spectral density of the current noise
Related to variance of transmitted charge
When shot noise dominates over thermal noise
Poisson process of independent transmission of electrons
Shot noise in electron transport
Current noise for tunneling
across a Hall bar on the 1/3
plateau of FQE
Etien et al. PRL 79:2526 (1997)
see also Heiblum et al. Nature (1997)
Quantum noise analysis of time-of-flight
experiments with atoms in optical lattices:
Hanburry-Brown-Twiss experiments
and beyond
Theory: Altman et al., PRA (2004)
Experiment: Folling et al., Nature (2005);
Spielman et al., PRL (2007);
Tom et al. Nature (2006)
Time of flight experiments
Cloud before expansion
Cloud after expansion
Quantum noise interferometry of atoms in an optical lattice
Second order coherence
Quantum noise analysis of time-of-flight
experiments with atoms in optical lattices
Experiment: Folling et al., Nature (2005)
Quantum noise analysis of time-of-flight
experiments with atoms in optical lattices
Cloud before expansion
Cloud after expansion
Second order correlation function
Here
and
are taken after the expansion time t.
Two signs correspond to bosons and fermions.
Quantum noise analysis of time-of-flight
experiments with atoms in optical lattices
Relate operators after the expansion to operators before the
expansion. For long expansion times use steepest descent
method of integration
TOF experiments map momentum
distributions to real space images
Second order real-space correlations after TOF expansion
can be related to second order momentum correlations
inside the trapped system
Quantum noise analysis of time-of-flight
experiments with atoms in optical lattices
Example: Mott state of spinless bosons
Only local correlations present in the Mott state
G - reciprocal
vectors of the
optical lattice
Quantum noise in TOF experiments in optical lattices
We get bunching when
corresponds to one
of the reciprocal vectors of the original lattice.
Boson bunching arises from the Bose enhancement factors.
A single particle state with quasimomentum q is a
supersposition of states with physical momentum q+nG.
When we detect a boson at momentum q we increase
the probability to find another boson at momentum q+nG.
Quantum noise in TOF experiments in optical lattices
Another way of understanding noise correlations comes from
considering interference of two independent condensates
After free
expansion
When condensates 1 and 2 are not correlated
We do not see interference in
.
Oscillations in the second order correlation function
Quantum noise in TOF experiments in optical lattices
Mott state of spinless bosons
Interference of an array of independent condensates
Hadzibabic et al., PRL 93:180403 (2004)
Smooth structure is a result of finite experimental resolution (filtering)
3
1.4
2.5
1.2
2
1
1.5
0.8
1
0.6
0.5
0.4
0
-0.5
0.2
-1
0
-1.5
0
200
400
600
800
1000
1200
-0.2
0
200
400
600
800
1000
1200
Second order correlations. Experimental issues.
Autocorrelation function
Complications we need to consider:
- finite resolution of detectors
- projection from 3D to 2D plane
s – detector resolution
- period of the optical lattice
In Mainz experiments
The signal in
and
is
Second order coherence in the insulating state of bosons.
Experiment: Folling et al., Nature (2005)
Quantum noise analysis of time-of-flight
experiments with atoms in optical lattices
Example: Band insulating state of spinless fermions
Only local correlations present in the band insulator state
Quantum noise analysis of time-of-flight
experiments with atoms in optical lattices
Example: Band insulating state of spinless fermions
We get fermionic antibunching. This can be understood
as Pauli principle. A single particle state with quasimomentum
q is a supersposition of states with physical momentum q+nG.
When we detect a fermion at momentum q we decrease the
probability to find another fermion at momentum q+nG.
Second order coherence in the insulating state of fermions.
Experiment: Tom et al. Nature (2006)
Second order correlations as
Hanburry-Brown-Twiss effect
Bosons/Fermions
Quantum noise analysis of time-of-flight
experiments with atoms in optical lattices
Bosons with spin. Antiferromagnetic order
Second order correlation function
New local correlations
Additional contribution to second order correlation function
- antiferromagnetic wavevector
We expect to get new peaks in the correlation function when
Probing spin order in optical lattices
Correlation function measurements after TOF expansion.
Extra Bragg peaks appear in the second
order correlation function in the AF phase.
This reflects
doubling of the
unit cell by
magnetic order.