Strongly correlated many-body systems: from electronic materials

to ultracold atoms to photons

Eugene Demler Harvard University

Thanks to: E. Altman, I. Bloch, A. Burkov, H.P. Büchler, I. Cirac, D. Chang, R. Cherng, V. Gritsev, B. Halperin, W. Hofstetter, A. Imambekov, M. Lukin, G. Morigi, A. Polkovnikov, G. Pupillo, G. Refael, A.M. Rey, O. Romero-Isart , J. Schmiedmayer, R. Sensarma, D.W. Wang, P. Zoller, and many others

“Conventional” solid state materials

Bloch theorem for non-interacting electrons in a periodic potential

“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

One dimensional electron systemsnon-perturbative effects of interactions in 1d

High temperature superconductors,Heavy fermion materials,Organic conductors and superconductorsmany puzzling non-Fermi liquid properties

Bose-Einstein condensation of weakly interacting atoms

Scattering length is much smaller than characteristic interparticle distances. Interactions are weak

Strongly correlated systems of cold atoms

• Low dimensional systems

• Optical lattices

• Feshbach resonances

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., arXive:0712.1817

Strongly correlated systems of photons

We understand well: many-body systems of non-interactingor weakly interacting particles. For example, electron systems in semiconductors and simple metals,When the interaction energy is smaller than the kinetic energy,perturbation theory works well

We do not understand: many-body systems with stronginteractions and correlations. For example, electron systems in novel materials such as high temperature superconductors. When the interaction energy is comparable or larger than the kinetic energy, perturbation theory breaks down. Many surprising new phenomena occur, including unconventional superconductivity, magnetism, fractionalization of excitations

Ultracold atoms have energy scales of 10-6K, compared to 104 K for electron systems

We will also get new systems useful for applications in quantum information and communications, high precision spectroscopy, metrology

By engineering and studying strongly interacting systems of cold atoms we should get insights into the mysterious properties of novel quantum materials

Strongly interacting systems of ultracold atoms and photons: NOT the analogue simulatorsThese are independent physical systems with their own “personalities”, physical properties, and theoretical challenges

Focus of these lectures: challenges of new strongly correlated systems

Part IDetection and characterization of many body statesQuantum noise analysis and interference experiments

Part IINew challenges in quantum many-body theory: non-equilibrium coherent dynamics

Quantum noise studies of ultracold atoms

Part I

Introduction. Historical review

Quantum noise analysis of the timeof flight experiments with utlracold atoms (HBT correlations and beyond)

Quantum noise in interference experiments with independent condensates

Quantum noise analysis of spin systems

Outline of part I

Quantum noiseClassical 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

Measuring spin of a particle in the left detectorinstantaneously determines its value in the right detector

Einstein-Podolsky-Rosen experiment

Aspect’s experiments:tests of Bell’s inequalities

SCorrelation function

Classical theories with hidden variable require

Quantum mechanics predicts B=2.7 for the appropriate choice of ‘s and the state

Experimentally measured value B=2.697. Phys. Rev. Let. 49:92 (1982)

+

-

+

-1 21 2

S

Hanburry-Brown-Twiss experimentsClassical theory of the second order coherence

Measurements of the angular diameter of SiriusProc. Roy. Soc. (London), A, 248, pp. 222-237

Hanbury Brown and Twiss, Proc. Roy. Soc. (London), A, 242, pp. 300-324

Shot noise in electron transport

e- e-

When shot noise dominates over thermal noise

Spectral density of the current noise

Proposed by Schottky to measure the electron charge in 1918

Related to variance of transmitted charge

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 the time of flight experiments with utlracold atoms (HBT correlations and beyond)

Theory: Altman, Demler, Lukin, PRA 70:13603 (2004)

Experiments: Folling et al., Nature 434:481 (2005); Greiner et al., PRL 94:110401 (2005); Tom et al. Nature 444:733 (2006);

see also Hadzibabic et al., PRL 93:180403 (2004) Spielman et al., PRL 98:80404 (2007); Guarrera et al., preprint (2007)

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); Ketterle et al., PRL (2006)

Bose Hubbard model

tunneling of atoms between neighboring wells

repulsion of atoms sitting in the same well

U

t

Superfluid to insulator transition in an optical lattice

M. Greiner et al., Nature 415 (2002)

U

1n

t/U

SuperfluidMott insulator

Why study ultracold atoms in optical lattices?

t

U

t

Fermionic atoms in optical lattices

Experiments with fermions in optical lattice, Kohl et al., PRL 2005

Antiferromagnetic and superconducting Tc of the order of 100 K

Atoms in optical lattice

Antiferromagnetism and pairing at sub-micro Kelvin temperatures

Same microscopic model

Positive U Hubbard model

Possible phase diagram. Scalapino, Phys. Rep. 250:329 (1995)

Antiferromagnetic insulator

D-wave superconductor

Atoms in optical lattice

Same microscopic model

Quantum simulations of strongly correlated electron systems using ultracold atoms

Detection?

Quantum noise analysis as a probe of many-body states of ultracold

atoms

Time of flight experiments

Quantum noise interferometry of atoms in an optical lattice

Second order coherence

Second order coherence in the insulating state of bosons.Hanburry-Brown-Twiss experiment

Experiment: Folling et al., Nature 434:481 (2005)

Hanburry-Brown-Twiss stellar interferometer

Hanburry-Brown-Twiss interferometer

Quantum theory of HBT experiments

For bosons

For fermions

Glauber,Quantum Optics and Electronics (1965)

HBT experiments with matter

Experiments with 4He, 3HeWestbrook et al., Nature (2007)

Experiments with neutronsIanuzzi et al., Phys Rev Lett (2006)

Experiments with electronsKiesel et al., Nature (2002)

Experiments with ultracold atomsBloch et al., Nature (2005,2006)

Second order coherence in the insulating state of bosons.Hanburry-Brown-Twiss experiment

Experiment: Folling et al., Nature 434:481 (2005)

Second order coherence in the insulating state of bosons

Bosons at quasimomentum expand as plane waves

with wavevectors

First order coherence:

Oscillations in density disappear after summing over

Second order coherence:

Correlation function acquires oscillations at reciprocal lattice vectors

Second order coherence in the insulating state of bosons.Hanburry-Brown-Twiss experiment

Experiment: Folling et al., Nature 434:481 (2005)

0 200 400 600 800 1000 1200

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Interference of an array of independent condensates

Hadzibabic et al., PRL 93:180403 (2004)

Smooth structure is a result of finite experimental resolution (filtering)

0 200 400 600 800 1000 1200-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Quantum theory of HBT experiments

For bosons

For fermions

Second order coherence in the insulating state of fermions.Hanburry-Brown-Twiss experiment

Experiment: Tom et al. Nature 444:733 (2006)

How to detect antiferromagnetism

Probing spin order in optical lattices

Correlation Function Measurements

Extra Braggpeaks appearin the secondorder correlationfunction in theAF phase

How to detect fermion pairing

Quantum noise analysis of TOF images: beyond HBT interference

Second order interference from the BCS superfluid

)'()()',( rrrr nnn

n(r)

n(r’)

n(k)

k

0),( BCSn rr

BCS

BEC

kF

Momentum correlations in paired fermionsGreiner et al., PRL 94:110401 (2005)

Fermion pairing in an optical lattice

Second Order InterferenceIn the TOF images

Normal State

Superfluid State

measures the Cooper pair wavefunction

One can identify unconventional pairing

How to see a “cat” statein the collapse and revival experiments

Quantum noise analysis of TOF images: beyond HBT interference

Collapse and revival experimentswith bosons in an optical lattice

Coherent state in each well at t=0

Increase the heightof the optical latticeabruptly

Time evolution within each well

Initial superfluid state

Individual wells isolated

Collapse and revival experimentswith bosons in an optical lattice

At revival times tr =h/U, 2h/U, … all number states

are in phase again

The collapse occurs due to the loss of coherence betweendifferent number states

collapse

tc=h/NU

revival

tr=h/U

time

coherence

Collapse and revival experimentswith bosons in an optical lattice

Greiner et al., Nature 419:51 (2002)

Dynamical evolution of the interference pattern after jumping the optical lattice potential

Collapse and revival experimentswith bosons in an optical lattice

Greiner et al., Nature 419:51 (2002)

Quantum dynamics of a coherent states

Cat state at t=tr /2

How to see a “cat” statein collapse and revival experiments

pairing-like correlations

no first order coherence

Properties of the “cat” state:

In the time of flight experiments this should lead to correlations between and

Define correlation function(overlap original and flipped images)

Romero-Isart et al., unpublished

How to see a “cat” statein collapse and revival experiments

Exact diagonalization of the 1d lattice system. U/t=3, N=2

Collapse and revival experimentswith bosons in an optical lattice

Greiner et al., Nature 419:51 (2002)

Dynamical evolution of the interference pattern after jumping the optical lattice potential

Perfect correlations “hiding” in the image

Quantum noise in interference experiments with independent condensates

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

Nature 4877:255 (1963)

x

z

Time of

flight

Experiments with 2D Bose gasHadzibabic, Dalibard et al., Nature 441:1118 (2006)

Experiments with 1D Bose gas S. Hofferberth et al. arXiv0710.1575

Interference of two independent condensates

1

2

r

r+d

d

r’

Clouds 1 and 2 do not have a well defined phase difference.However each individual measurement shows an interference pattern

x1

d

Amplitude of interference fringes,

Interference of fluctuating condensates

For identical condensates

Instantaneous correlation function

For independent condensates Afr is finite but is random

x2

Polkovnikov, Altman, Demler, PNAS 103:6125(2006)

Fluctuations in 1d BEC

Thermal fluctuations

Thermally energy of the superflow velocity

Quantum fluctuations

For a review see Shlyapnikov et al., J. Phys. IV France 116, 3-44 (2004)

For impenetrable bosons and

Interference between Luttinger liquids

Luttinger liquid at T=0

K – Luttinger parameter

Finite temperature

Experiments: Hofferberth,Schumm, Schmiedmayer

For non-interacting bosons and

Distribution function of fringe amplitudes for interference of fluctuating condensates

L

is a quantum operator. The measured value of will fluctuate from shot to shot.

Higher moments reflect higher order correlation functions

Gritsev, Altman, Demler, Polkovnikov, Nature Physics 2006Imambekov, Gritsev, Demler, cond-mat/0612011

We need the full distribution function of

Distribution function of interference fringe contrastExperiments: Hofferberth et al., arXiv0710.1575Theory: Imambekov et al. , cond-mat/0612011

Comparison of theory and experiments: no free parametersHigher order correlation functions can be obtained

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

Distribution function of fringe amplitudes for interference of fluctuating condensates

L

is a quantum operator. The measured value of will fluctuate from shot to shot.

Higher moments reflect higher order correlation functions

Gritsev, Altman, Demler, Polkovnikov, Nature Physics 2006Imambekov, Gritsev, Demler, cond-mat/0612011

We need the full distribution function of

L

Change to periodic boundary conditions (long condensates)

Explicit expressions for are available but cumbersome Fendley, Lesage, Saleur, J. Stat. Phys. 79:799 (1995)

Calculating distribution function of interference fringe amplitudes

Method I: mapping to quantum impurity problem

Impurity in a Luttinger liquid

Expansion of the partition function in powers of g

Partition function of the impurity contains correlation functions taken at the same point and at different times. Momentsof interference experiments come from correlations functionstaken at the same time but in different points. Euclidean invarianceensures that the two are the same

Relation between quantum impurity problemand interference of fluctuating condensates

Distribution function of fringe amplitudes

Distribution function can be reconstructed fromusing completeness relations for the Bessel functions

Normalized amplitude of interference fringes

Relation to the impurity partition function

is related to a Schroedinger equation Dorey, Tateo, J.Phys. A. Math. Gen. 32:L419 (1999) Bazhanov, Lukyanov, Zamolodchikov, J. Stat. Phys. 102:567 (2001)

Spectral determinant

Bethe ansatz solution for a quantum impurity can be obtained from the Bethe ansatz followingZamolodchikov, Phys. Lett. B 253:391 (91); Fendley, et al., J. Stat. Phys. 79:799 (95)Making analytic continuation is possible but cumbersome

Interference amplitude and spectral determinant

0 1 2 3 4

Pro

babi

lity

P(x

)

x

K=1 K=1.5 K=3 K=5

Interference of 1d condensates at T=0. Distribution function of the fringe contrast

Narrow distributionfor .Approaches Gumbeldistribution.

Width

Wide Poissoniandistribution for

When K>1, is related to Q operators of CFT with c<0. This includes 2D quantum gravity, non-intersecting loop model on 2D lattice, growth of randomfractal stochastic interface, high energy limit of multicolor QCD, …

Yang-Lee singularity

2D quantum gravity,non-intersecting loops on 2D lattice

correspond to vacuum eigenvalues of Q operators of CFT Bazhanov, Lukyanov, Zamolodchikov, Comm. Math. Phys.1996, 1997, 1999

From interference amplitudes to conformal field theories