eugene demler harvard university
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Eugene Demler Harvard University
Strongly correlated many-body systems: from electronic materials to ultracold atoms
to photons
“Conventional” solid state materials
Bloch theorem for non-interacting electrons in a periodic potential
B
VH
I
d
First semiconductor transistor
EF
Metals
EF
Insulators andSemiconductors
Consequences of the Bloch theorem
“Conventional” solid state materialsElectron-phonon and electron-electron interactions are irrelevant at low temperatures
kx
ky
kF
Landau Fermi liquid theory: when frequency and temperature are smaller than EF electron systems
are equivalent to systems of non-interacting fermions
Ag Ag
Ag
Strongly correlated electron systems
Quantum Hall systemskinetic energy suppressed by magnetic field
Heavy fermion materialsmany puzzling non-Fermi liquid properties
High temperature superconductorsUnusual “normal” state,Controversial mechanism of superconductivity,Several competing orders
UCu3.5Pd1.5CeCu2Si2
What is the connection between
strongly correlated electron systems
and
ultracold atoms?
Bose-Einstein condensation of weakly interacting atoms
Scattering length is much smaller than characteristic interparticle distances. Interactions are weak
Density 1013 cm-1
Typical distance between atoms 300 nmTypical scattering length 10 nm
New Era in Cold Atoms ResearchFocus on Systems with Strong Interactions
• Atoms in optical lattices
• Feshbach resonances
• Low dimensional systems
• Systems with long range dipolar interactions
• Rotating systems
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
Strongly interacting regime can be reached for low densities
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)
1D confinement in optical potentialWeiss et al., Science (05);Bloch et al., Esslinger et al.,
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 systemsAtoms in optical latticesElectrons in Solids
Simple metalsPerturbation theory in Coulomb interaction applies. Band structure methods wotk
Strongly Correlated Electron SystemsBand structure methods fail.
Novel phenomena in strongly correlated electron systems:
Quantum magnetism, phase separation, unconventional superconductivity,high temperature superconductivity, fractionalization of electrons …
Strongly correlated systems of ultracold atoms shouldalso be useful for applications in quantum information, high precision spectroscopy, metrology
By studying strongly interacting systems of cold atoms we expect to get insights into the mysterious properties of novel quantum materials: Quantum Simulators
BUTStrongly interacting systems of ultracold atoms and photons: are NOT direct analogues of condensed matter systemsThese are independent physical systems with their own “personalities”, physical properties, and theoretical challenges
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
Strong optical nonlinearities innanoscale surface plasmonsAkimov et al., Nature (2007)
Strongly interacting polaritons in coupled arrays of cavitiesM. Hartmann et al., Nature Physics (2006)
Crystallization (fermionization) of photonsin one dimensional optical waveguidesD. Chang et al., Nature Physics (2008)
Strongly correlated systems of photons
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
Most common bosonic atoms: alkali 87Rb and 23Na
Most common fermionic atoms: alkali 40K and 6Li
Ultracold atoms
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.)
Single valence electron in the s-orbital and Nuclear spin
Magnetic properties of individual alkali atoms
Zero field splitting between and states
For 23Na AHFS = 1.8 GHz and for 87Rb AHFS = 6.8 GHz
Total angular momentum (hyperfine spin)
Hyperfine coupling mixes nuclear and electron spins
Magnetic properties of individual alkali atoms
Effect of magnetic field comes from electron spin
gs=2 and B=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
- polarizability
Typically optical frequencies.
Potential:
Dipolar moment induced by the electric field
Far-off-resonant optical trap confines atoms regardlessof their hyperfine state
Bogoliubov theory of weakly interacting BEC. Collective modes
BEC of spinless bosons. Bogoliubov theory
We consider a uniform system first
For non-interacting atoms at T=0 all atoms are in k=0 state.Mean field equations
Minimizing with respect to N0 we find
- boson annihilation operator at momentum p,
- strength of contact s-wave interaction
- volume of the system
- chemical potential
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
- means that p,-p pairs should be counted only once
n0 = N0/V - density and = n0 U0
Mean-field Hamiltonian
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
Short wavelength limit, , free particles
Sound velocity
Probing the dispersion of BECby 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)
ΩΩ
PRL 88:60402 (2002)
Rev. Mod. Phys. 77:187 (2005)
Gross-Pitaevskii equation
Gross-Pitaevskii equationHamiltonian 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-fieldGP 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 afinite densityof both speciesis unstableto 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
Bose mixture: dynamics of phase separationDefine
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
Most unstable mode
Imaginary frequencies indicateexponential growth of fluctuations,i.e. instability. The most unstablemode sets the length for patternformation. Note that when . This is requiredby spin conservation.
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
g2<0
Ferromagnetic
Shaded region: mixture of all three states. There is XY component of the spin.
Shaded region: mixture ofmF=-1,+1 states. This statelowers interaction energy.
Nature 396:345 (1998)
Using magnetic field gradient to explore the phase diagramof F=1 atoms with AF interactions23Na
Nature 443:312 (2006)
Magnetic dipolar interactionsin ultracold atoms
Magnetic dipolar interactions in spinor condensates
Comparison of contact and dipolar interactions.
Typical value a=100aB
For 87Rb =B and =0.007 For 52Cr =6B and =0.16
Bose condensationof 52Cr. T. Pfau et al. (2005)
Review: Menotti et al.,arXiv 0711.3422
Magnetic dipolar interactions in spinor condensates
Interaction of F=1 atoms
Ferromagnetic Interactions for 87Rb
Spin-depenent part of the interaction is small. Dipolar interaction may be important (D. Stamper-Kurn)
a2-a0= -1.07 aB
A. Widera, I. Bloch et al., New J. Phys. 8:152 (2006)
Spontaneously modulated textures in spinor condensates
Fourier spectrum of thefragmented condensate
Vengalattore et al.PRL (2008)
Energy scales: importance of Larmor precession
S-wave Scattering•Spin independent (g0n = kHz)•Spin dependent (gsn = 10 Hz)
Dipolar Interaction•Anisotropic (gdn=10 Hz)•Long-ranged
Magnetic Field•Larmor Precession (100 kHz)•Quadratic Zeeman (0-20 Hz)
BF
Reduced Dimensionality•Quasi-2D geometry spind
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
XY components of the spins can lower the energy using modulation along z.
Z components of thespins can lower the energyusing modulation along x
X
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
Strong instabilities of systems with dipolar interactions after averaging over precession
X
Instabilities of F=1 Rb (ferromagnetic) condensate due to dipolar interactions
Theory: unstable modes in the regimecorresponding to Berkeley experiments.Cherng, Demler, PRL (2009)
Experiments.Vengalattore et al. PRL (2008)
Berkeley Experiments: checkerboard phase
Instabilities of magnetic dipolar interactions: general analysis
– angle between magneticfield and normal to the plane
dn – layer thickness
q measures the strengthof quadratic Zeeman effect
Instabilities of collective modes
Magnetoroton softening
Q measures the strengthof quadratic Zeeman effect
87Rb condensate: magnetic supersolid ?
Phase diagram of 4He
Possible supersolid phase in 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 tosoftening of roton excitations
Resonant period as a function of T
Interlayer coherence in bilayer quantum Hall systems at =1
Hartree-Fock predicts roton softeningand transition into a state with both interlayer coherence and stripe order. Transport experiments suggest first order transition into a compressible state.
L. Brey and H. Fertig (2000)Eisenstein, Boebinger et al. (1994)
Low energy effective model for ferromagnetic condensates
Mass superflow currentcaused 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 87Rbferromagnetic condensates
Experimental constraints:
1. Dihedral symmetry of the magnetic crystal2. Rectangular lattice3. Non-planar spin orientation
Magnetic crystal latticesoptimized within each class
Optimal spin configuration.
Cherng, Demler, unpublished
Spin correlation functionsfor spin componentsparallel and perpendicularto the magnetic field
Magnetic crystal phases in F=1 87Rbferromagnetic condensates
Ultracold atoms in optical lattices.Band structure. Semiclasical dynamics.
Optical lattice
The simplest possible periodic optical potential is formedby overlapping two counter-propagating beams. This resultsin 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 latticeFor each coordinate we have Matthieu equation
Eigenvalues and eigenfunctions are known
In the regime of deep lattice we get the tight-binding model
- bandgap
- recoil energy
Lowest band
Optical lattice
Effective Hamiltonian for non-interacting atoms in the lowest Bloch band
nearestneighbors
Band structure
Semiclassical dynamics in the lattice1. Band index is constant
2.
3.
Bloch oscillations
Consider a uniform and constant force
PRL 76:4508 (1996)
State dependent optical latticesHow to use selection rules for optical transitions to make different lattice potentials for different internal states.
Fine structure for 23Na and 87Rb
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 will only be affected by , which gives the potential .
Decomposing hyperfine states we find
PRL 91:10407 (2003)
State dependent lattice
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