QUANTUM MANY-BODY
SYSTEMS OF ULTRACOLD ATOMS
Eugene Demler
Harvard University
Grad students: A. Imambekov (->Rice), Takuya Kitagawa
Postdocs: E. Altman (->Weizmann), A. Polkovnikov (->U. Boston)
A.M. Rey (->U. Colorado), V. Gritsev (-> U. Fribourg),
D. Pekker (-> Caltech), R. Sensarma (-> JQI Maryland)
Collaborations with experimental groups of I. Bloch (MPQ),
T. Esslinger (ETH), J.Schmiedmayer (Vienna)
Supported by NSF, DARPA OLE, AFOSR MURI, ARO MURI
How cold are ultracold atoms?
feV
pK
peV
nK
current
experiments
10-11 - 10-10 K
neV
µK
µeV
mK
first BEC
of alkali atoms
meV
eV
keV
MeV
GeV
TeV
K
He N
room
temperature
LHC
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
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 …
Quantum simulations with ultracold atoms
Atoms in optical lattice
Antiferromagnetic and
superconducting Tc
of the order of 100 K
Antiferromagnetism and
pairing at nano Kelvin
temperatures
Same microscopic model
Strongly correated 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
First lecture:
experiments with ultracold bosons
Cold atoms in optical lattices
Bose Hubbard model. Superfluid to Mott transition
Looking for Higgs particle in the Bose Hubbard model
Quantum magnetism with ultracold atoms in optical lattices
Low dimensional condensates
Observing quasi-long range order in interference experiments
Observation of prethermolization
Second lecture:
Ultracold fermions
Fermions in optical lattices. Fermi Hubbard model.
Current state of experiments
Lattice modulation experiments
Doublon lifetimes
Strongly interacting fermions in continuum.
Stoner instability
Ultracold Bose 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
n 1
Mott
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.
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.
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.
 U
n=3
Mott
n 1
2
n=2
Mott
1
n=1
Mott
0
Jaksch et al.,
PRL 81:3108 (1998)
Superfluid
Quantum gas microscope
density
Bakr et al., Science 2010
y
x
Nature 2010
The Higgs (amplitude) mode in a
trapped 2D superfluid on a lattice
Elementary Particles (CMS @ LHC)
Cold Atoms (Munich)
Sherson et. al. Nature 2010
Theory: David Pekker, Eugene Demler
Experiments: Manuel Endres, Takeshi Fukuhara, Marc Cheneau, Peter
Schauss, Christian Gross, Immanuel Bloch, Stefan Kuhr
Collective modes of strongly interacting
superfluid bosons
Order parameter
Breaks U(1) symmetry
Figure from Bissbort et al. (2010)
Phase (Goldstone) mode = gapless Bogoliubov mode
Gapped amplitude mode (Higgs mode)
Excitations of the Bose Hubbard model
 U
n=3
Mott
n 1
Superfluid
2
n=2
2
Mott
1
n=1
Mott
Mott
Superfluid
0
Softening of the amplitude mode is the defining characteristic
of the second order Quantum Phase Transition
Is there a Higgs mode in 2D ?
neutron scattering
• Danger from scattering on phase modes
f
Higgs
Higgs
f
• In 2D: infrared divergence
• Different susceptibility has no divergence
lattice modulation
spectroscopy
S. Sachdev, Phys. Rev. B 59, 14054 (1999)
W. Zwerger, Phys. Rev. Lett. 92, 027203 (2004)
N. Lindner and A. Auerbach, Phys. Rev. B 81, 54512 (2010)
Podolsky, Auerbach, Arovas, Phys. Rev. B 84, 174522 (2011)
Why it is difficult to observe the amplitude mode
Bissbort et al., PRL(2010)
Stoferle et al., PRL(2004)
Peak at U dominates and does not
change as the system goes through
the SF/Mott transition
Exciting the amplitude mode
Absorbed energy
Exciting the amplitude mode
Manuel Endres, Immanuel Bloch and MPQ team
n=1
Mott
n=1
Mott
n=1
Mott
Experiments: full spectrum
Manuel Endres, Immanuel Bloch and MPQ team
Time dependent mean-field: Gutzwiller
Similar to Landau-Lifshitz equations in magnetism
Keep two
states per site
only
Threshold for absorption
is captured very well
Plaquette Mean Field
“Better Gutzwiller”
• Variational wave functions better captures local physics
– better describes interactions between quasi-particles
• Equivalent to MFT on plaquettes
Time dependent cluster mean-field
Lattice height 9.5 Er: (1x1 vs 2x2)
single amplitude
mode excited
breathing mode
single amplitude
breathing mode mode excited
2x2 captures width of spectral feature
multiple modes
excited?
Comparison of experiments
and Gutzwiller theories
Experiment
2x2 Clusters
Key experimental facts:
• “gap” disappears at QCP
• wide band
• band spreads out deep in SF
Single site Gutzwiller
Captures gap
Does not capture width
Plaquette Gutzwiller
Captures gap
Captures most of the width
Beyond Gutzwiller: Scaling at low frequencies
signature of Higgs/Goldstone mode coupling
Higgs
w
2 Goldstones
vacuum
External drive couples vacuum to Higgs
Higgs can be excited only virtually
Higgs decays into a pair of Goldstone modes with conservation of energy
Matrix element w2/w=w
Density of states w
Fermi’s golden rule: w2xw = w3
Open question: observing discreet modes
disappearing amplitude mode
Breathing mode
details at the QCP
spectrum remains
gapped due to trap
Higgs Drum Modes
1x1 calculation, 20 oscillations
Eabs rescaled so peak heights
coincide
Quantum magnetism with ultracold
atoms in optical lattices
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 et al., 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
Probing low dimensional
condensates with interference
experiments
Quasi long range order
Prethermalization
Interference of independent condensates
Experiments: Andrews et al., Science 275:637 (1997)
Theory: Javanainen, Yoo, PRL 76:161 (1996)
Cirac, Zoller, et al. PRA 54:R3714 (1996)
Castin, Dalibard, PRA 55:4330 (1997)
and many more
Experiments with 2D Bose gas
Hadzibabic, Dalibard et al., Nature 2006
z
Time of
flight
x
Experiments with 1D Bose gas Hofferberth et al. Nat. Physics 2008
Interference of two independent condensates
r’
r
Assuming ballistic expansion
1
r+d
d
2
Phase difference between clouds 1 and 2
is not well defined
Individual measurements show interference patterns
They disappear after averaging over many shots
Interference of fluctuating condensates
d
Polkovnikov et al., PNAS (2006); Gritsev et al., Nature Physics (2006)
Amplitude of interference fringes,
x1
x2
For independent condensates Afr is finite
but Df is random
For identical
condensates
Instantaneous correlation function
FDF of phase and contrast
• Matter-wave interferometry
phase, contrast
FDF of phase and contrast
• Matter-wave interferometry
phase, contrast
• Plot as circular statistics
contrast
phase
FDF of phase and contrast
• Matter-wave interferometry: repeat
many times
phase, contrast
accumulate statistics
contrasti
• Plot
i>100
phase
Calculate average contrast
Fluctuations in 1d BEC
Thermal fluctuations
Thermally energy of the superflow velocity
Quantum fluctuations
Interference between Luttinger liquids
Luttinger liquid at T=0
K – Luttinger parameter
For non-interacting bosons
For impenetrable bosons
Finite
temperature
Experiments: Hofferberth,
Schumm, Schmiedmayer
and
and
Distribution function of fringe amplitudes
for interference of fluctuating condensates
Gritsev, Altman, Demler, Polkovnikov, Nature Physics 2006
Imambekov, Gritsev, Demler, PRA (2007)
is a quantum operator. The measured value of
will fluctuate from shot to shot.
L
Higher moments reflect higher order correlation functions
We need the full distribution function of
Distribution function of interference fringe contrast
Hofferberth et al., Nature Physics 2009
Quantum fluctuations dominate:
asymetric Gumbel distribution
(low temp. T or short length L)
Thermal fluctuations dominate:
broad Poissonian distribution
(high temp. T or long length L)
Intermediate regime:
double peak structure
Comparison of theory and experiments: no free parameters
Higher order correlation functions can be obtained
Interference between interacting 1d Bose liquids.
Distribution function of the interference amplitude
Distribution function of
Quantum impurity problem: interacting one dimensional
electrons scattered on an impurity
Conformal field theories with negative
central charges: 2D quantum gravity,
non-intersecting loop model, growth of
random fractal stochastic interface,
high energy limit of multicolor QCD, …
2D quantum gravity,
non-intersecting loops
Yang-Lee singularity
Fringe visibility and statistics of random surfaces
Distribution function of
Mapping between fringe
visibility and the problem
of surface roughness for
fluctuating random
surfaces.
Relation to 1/f Noise and
Extreme Value Statistics
h ( )
2
Roughness
  h( ) d
Interference of two dimensional condensates
Experiments: Hadzibabic et al. Nature (2006)
Gati et al., PRL (2006)
Ly
Lx
Lx
Probe beam parallel to the plane of the condensates
Interference of two dimensional condensates.
Quasi long range order and the BKT transition
Ly
Lx
Above BKT transition
Below BKT transition
Experiments with 2D Bose gas
z
Hadzibabic, Dalibard et al., Nature 441:1118 (2006)
Time of
flight
x
Typical interference patterns
low temperature
higher temperature
Experiments with 2D Bose gas
Hadzibabic et al., Nature 441:1118 (2006)
x
integration
over x axis z
z
Contrast after
integration
0.4
low T
integration
middle T
0.2
over x axis
z
high T
integration
over x axis
Dx
0
z
0
10
20
30
integration distance Dx
(pixels)
Experiments with 2D Bose gas
Integrated contrast
Hadzibabic et al., Nature 441:1118 (2006)
0.4
fit by:
C2 ~
low T
1
Dx
 1 

 Dx 
Dx
2


g
(
0
,
x
)
dx ~ 
 1
middle T
0.2
Exponent a
high T
0
0
10
20
30
integration distance Dx
if g1(r) decays exponentially
with
:
0.5
0.4
0.3
high T
0
if g1(r) decays algebraically or
exponentially with a large
:
0.1
low T
0.2
0.3
central contrast
“Sudden” jump!?
2a
Experiments with 2D Bose gas. Proliferation of
thermal vortices
Hadzibabic et al., Nature (2006)
30%
Fraction of images showing
at least one dislocation
Exponent a
20%
0.5
10%
0.4
low T
high T
0
0
0.1
0.2
0.3
central contrast
The onset of proliferation
coincides with a shifting to 0.5!
0.4
0.3
0
0.1
0.2
central contrast
0.3
Quantum dynamics of split
one dimensional condensates
Prethermalization
Theory: Takuya Kitagawa et al., PRL (2010)
New J. Phys. (2011)
Experiments: D. Smith, J. Schmiedmayer, et al.
arXiv:1112.0013
Relaxation to equilibrium
Thermalization: an isolated interacting systems approaches thermal
equilibrium at long times (typically at microscopic timescales). All
memory about the initial conditions except energy is lost.
Bolzmann equation
U. Schneider et al.,
arXiv:1005.3545
Prethermalization
Heavy ions collisions
QCD
We observe irreversibility and approximate thermalization. At large
time the system approaches stationary solution in the vicinity of, but
not identical to, thermal equilibrium. The ensemble therefore retains
some memory beyond the conserved total energy…This holds for
interacting systems and in the large volume limit.
Prethermalization in ultracold atoms, theory: Eckstein et al. (2009);
Moeckel et al. (2010), L. Mathey et al. (2010), R. Barnett et al.(2010)
Measurements of dynamics of split condensate
Theoretical analysis of dephasing
Luttinger liquid model
Luttinger liquid model of phase dynamics
Luttinger liquid model of phase dynamics
For each k-mode we have simple harmonic oscillators
Phase diffusion vs Contrast Decay
Segment size is smaller than the fluctuation lengthscale
Segment size is longer than the fluctuation lengthscale
At long times the difference between
the two regime occurs for
Length dependent phase dynamics
15 ms 15.5
16
16.5
17
19
21
24
27
32
37
110 µm
61µm
41µm
30 µm 20µm 10µm
“Short segments” = phase diffusion
“Long segments” = contrast
decay
47
62
77
107
137
167
197
Energy distribution
At t=0 system is in a squeezed state with large number fluctuations
Energy stored in each mode initially
Equipartition of energy
For 2d also pointed out by Mathey, Polkovnikov in PRA (2010)
The system should look thermal like after different modes dephase.
Effective temperature is not related to the physical temperature
Comparison of experiments and LL analysis
Do we have thermal-like distributions at longer times
Prethermalization
Interference contrast is described by thermal
distributions but at temperature much lower
than the initial temperature
Testing Prethermalization
First lecture:
experiments with ultracold bosons
Cold atoms in optical lattices
Bose Hubbard model. Superfluid to Mott transition
Looking for Higgs particle in the Bose Hubbard model
Quantum magnetism with ultracold atoms in optical lattices
Low dimensional condensates
Observing quasi-long range order in interference experiments
Observation of prethermolization
Beyond Gutzwiller: Scaling at low frequencies
signature of Higgs/Goldstone mode coupling
Excite virtual Higgs excitation
Virtual Higgs decays into a pair of Goldstone excitations
Matrix element of Higgs to Goldstone coupling scales as w2
Phase space scales as 1/w
Fermi’s golden rule: (w2)2x(1/w) = w3