Nonequilibrium quantum dynamics of many‐body systems of ultracold atomsmany body systems of ultracold atoms

Eugene Demler

Harvard University

$$ NSF, AFOSR MURI, DARPA,  AROHarvard-MIT

Antiferromagnetic and

Atoms in optical lattice

Antiferromagnetism andAntiferromagnetic and superconducting Tc of the order of 100 K

Antiferromagnetism and pairing at sub-micro Kelvin temperatures

Same microscopic model

New Phenomena in quantum e e o e a qua umany-body systems of ultracold atomsLong intrinsic time scalesLong 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

Other theoretical work on many-body nonequilibrium dynamics of ultracold atoms: E Altman A Cazalilla K Collath A J Daleyultracold atoms: E. Altman, A. Cazalilla, K. Collath, A.J. Daley, T. Giamarchi, V. Gritsev, L. Levitov, A. Muramatsu, A. Polkovnikov, A. Yazbashian, P. Zoller, and many more

Universaility in dynamics of nonlinear classical systems

Universality in quantum many-body systems in equilibrium

Solitons in nonlinear wave propagation Broken symmetries

B d ll i th f T di t F i li id t tBernard cells in the presence of T gradient Fermi liquid state

Do we have universality in nonequilibriumDo we have universality in nonequilibrium dynamics of many-body quantum systems?

OutlineOutline

Dynamical splitting of 1d condensates and prethermalizationp

Resonant scattering of fermions in reduced dimensions

Dynamical splittingDynamical splittingof 1d condensates

and prethermalization

Takuya Kitagawa, Adilet Imambekov, Vladimir Gritsev Joerg SchmiedmayerVladimir Gritsev, Joerg Schmiedmayer, Eugene Demler

Phys. Rev. Lett. 104:255302 (2010)arXiv:1104.5631

Experiments: D. Simith et al.

Interference of macroscopic condensates

Experiments: Andrews et al., Science 275:637 (1997)

zExperiments with 2D Bose gas

Hadzibabic, Dalibard et al., Nature 2006

Time of

fli h xflight

Experiments with 1D Bose gas Hofferberth et al., Nat. Physics 2008

Interference experiments with condensates

12rd

Assuming ballistic expansionr+d

d

Interference of fluctuating condensates

d

Polkovnikov et al. (2006)

Interference of fluctuating condensatesx1

Amplitude of interference fringesx2

C CC

Interference of fluctuating condensatesP lk ik Alt D l PNAS 103 6125(2006)d

Amplitude of interference fringes

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

x1For independent condensates |C | is finite

C

but is random x2

For identical condensates

Instantaneous correlation function

Experiments with 2D Bose gas

xfi t ti

Hadzibabic et al., Nature 441:1118 (2006)

z

Contrast afterintegration

0.4

integration

over x axis z

integration

i 0 2middle T

low T

integration

over x axisz

0.2

high Tg

over x axisz integration distance Dx

00 10 20 30

Dx(pixels)

Experiments with 2D Bose gasH d ib bi t l N t 441 1118 (2006)

fit by:

trast 0.4

l T

22

12 1~),0(1~

D

Ddxxg

DC

x

Hadzibabic et al., Nature 441:1118 (2006)

rate

d co

nt

0.2low T

middle T

Exponent

xx DD

integration distance D

Inte

gr

00 10 20 30

high T

0.5integration distance Dx

if g1(r) decays exponentially ith

0.4

0.3 high T low Twith :

if g1(r) decays algebraically or central contrast

0 0.1 0.2 0.3

g ow

if g1(r) decays algebraically or exponentially with a large :

central contrast

“Sudden” jump!?

Distribution function of fringe amplitudes for interference of fluctuating condensates

Polkovnikov et al. (2006), Gritsev et al. (2006), Imambekov et al. (2007)

is a quantum operator. 

The measured value of  CC

will fluctuate from shot to shotx1 C< C n >

Higher moments reflect x2

C< C  >

ghigher order correlation functions

Experiments analyzeC Experiments analyze 

distribution  function of C

C

Measurements of distribution functions of interference amplitudefunctions of interference amplitude

rAccumulate statistics Distribution function of contrast

obability

C|2Pro |C|2

Interference of independent 1d condensatesSh t t h t fl t ti f f i t t

Theory: Altman, Imambekov, Polkovnikov, Gritsev, Demler

Shot to shot fluctuations of fringe contrast as a probe of many-body correlations

y , , , ,Experiments: Hofferberth et al., Nature Physics (2008)

Thermalion

Quantum Thermalnoise

Quantumnoise

on f

unct Quantum

and ThermalD

istri

buti

Observation of quantum fluctuations in 1d Bose gas

contrast normalized by the average value

q g

Probe of higher order correlation functionsfringe contrast

Measurements of dynamics of split condensate

Theoretical analysis of dephasingL ttinger liq id modelLuttinger liquid model

Luttinger liquid model of phase dynamics

Luttinger liquid model of phase dynamics

For each k‐mode we have simple harmonic oscillatorsFast splitting prepares states with small fluctuations of relative phase

Time evolutionsmall fluctuations of relative phase

Phase relaxation Bistrizer, Altman (2007)

Initial state Phase dynamics

x1

x2x2

Coarse‐graining on the scale of x is impliedShort wavelength modes have faster dynamicshave faster dynamics

Experimental observablesAmplitude of interference fringesp g

Result of averaging interference x1

x2

amplitudes over many shots

2

Contrast of individual shots primarily depends on

C

Reduction of average amplitude comes from modes at all wavevectors (uncertainty in both the amplitude and the overall phase)both the amplitude and the overall phase).Reduction of contrast comes only from modeswith l > l

Phase diffusion vs Contrast Decay

h h h d ff

Phase diffusion vs Contrast ecay

When                              we have phase diffusion

When                            we have decay of contrast

At long times the difference between the two regime occurs for

Length dependent phase dynamics

“Short” = phase diffusion “Long” = contrast decay

Energy distribution

Initially the system is in a squeezed state with large number fluctuations

Energy stored in each mode initially

Equipartition of energyEquipartition of energyThe 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?

Quasi‐steady state

Prethermalization

Prethermalization

Heavy ions collisionsHeavy ions collisionsQCD

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 retainsnot 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)

Resonant scattering of fermions gin reduced dimensions

Vill Pi till (H d) D id P kkVille Pietilla (Harvard), David Pekker (Caltech), Yusuke Nishida (MIT), Dima Abanin (Princeton) Ivar Zapata (U Madrid)Abanin (Princeton), Ivar Zapata (U. Madrid), Adilet Imambekov (Rice), Eugene Demler

Feshbach resonance

I i i ll kl i i F i h d h lInitially weakly interacting Fermi quenched to the strongly  interacting regime in the vicinity of Feshbach resonance  [Jo et al., Science 325, 1521 (2009)]: 

BCS instabilityStoner instability

Molecule formationand condensation

System unstable to both  molecule formationand Stoner ferromagnetism. Which instability dominates?

A. Turlapov et al.,PRL 105, 030404 (2010)

B. Frohlich et al., PRL 106:105301 (2011)

G. Lamporesi et al.,PRL 104, 153202 (2010)

Experimental observationof mix-dimensional scattering resonances

Low dimensional many-body systemsnear Feshbach resonancesnear Feshbach resonances

Interplay of confinement, resonant interactions, and b d ff tmany-body effects

Low dimensional systems have an extra tunable control

Experiments are always done at finite density. It may be

parameter in addition to and

difficult to disentangle two-body and many-body effects

Example: Fermi mixture. When size of the bound state becomes comparable to interparticle distance,Pauli blocking becomes important

Introduction.Resonant two particle scattering in reducedResonant two particle scattering in reduced dimensions 

Two‐particle scattering 

Separate center of mass and relative motion

Dyson’s equation

T T

Two‐particle scattering 

Microscopic interaction strength depends on high energycut‐off We want to trade it for 3d scattering lengthcut‐off. We want to trade it for 3d scattering length.Direct approach: introduce cut‐offs into GE. Alternative approach by Zwerger et al. (RMP 2008):pp y g ( )

Two‐particle resonant scattering In d=3In d=3

Express T‐matrix in terms of 3d scattering length   

Tubes (Olshanii)

Pancakes  (Petrov, Shlyapnikov)

Many‐body problem: two particle scattering in the presence of the Fermi seascattering in the presence of the Fermi sea

No separation of center of mass and relative motion

T T

Poles of             describe molecules 

Many‐body problem

T T

Express many‐body T‐matrix in terms of the vacuum one

Balanced mixture                          Imaginary part of poles gives rate of molecule formation

Strong spin imbalance              and              . Polarons

Spin balanced mixture.Molecule formation rateMolecule formation rate

th tgrowth rateof molecules

In the limit of tight transverse confinement

maximum of pairing rate is shifted to the BCS side             p g

Many‐body effects on molecular binding energy

In 3d many‐body effects increase molecular binding energy, e.g. Cooper pair binding exists even on the BCS side

In 2d confinement induced resonance is present  even pon the BCS side. So Pauli blocking results in weaker binding of 2d molecules

PRL 106, 105301 (2011) 

Strongly interactingRF

NoninteractingRF

Molecules vs Polarons

Comparison ofpolarons andpolarons and molecules in 3dProkof’ev, Svistunov,

polaron

PRA 2008

moleculemolecule

Polarons

Impurity atom

First order contribution tolf hiself‐energy. This

approximation works wellfor 3d systems.y

Equivalent  to Chevy’s variational wavefunction

Impurity atom spectral function in d=2

RF spectra theoretical

polaron

atom interacting with environment

free atom (initial state)

polaron

free atom (initial state)

First peak in RFmatches k 0 polaronk=0 polaron

molecules

polaronsp

Comparison of polaronsand molecules was also

Location of the first peak inRF spectra of Frolich et al.

given by Zollner et al in Phys. Rev.  A (2011)

SummaryDynamical splitting of 1d condensates and prethermalization

Resonant scattering of fermions in reduced dimgDynamical molecule formation Dynamical formation of polarons in RF

Universaility in dynamics of nonlinear classical systems

Universality in quantum many‐body systems in equilibrium

Solitons in nonlinear wave propagation Broken symmetries

d ll h f d l dBernard cells in the presence of T gradient Fermi liquid state

Do we have universality in nonequilibriumDo we have universality in nonequilibrium dynamics of  many‐body quantum systems?

Relaxation to equilibrium

Thermalization: an isolated interacting systems approaches thermalThermalization: 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., 2010

Absence of relaxationCollapse and revival in Opical Lattice Quantum Newton’s craddle

D. Weiss et al.

Collapse and revival in Opical Lattice Quantum Newton s craddle

I blFinite system Integrable system

Prethermalization

Heavy ions collisionsHeavy ions collisionsQCD

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 retainsnot 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)

First experimental demonstration of prethermalization

Probing thermolization not only in the average value but  in the complete quantum noise

Initial T=120 nK (blue line). After  27.5 ms identical to thermal system at T= 15 nK

Interference experimentsInterference experiments with ultracold bosonic atoms