Exploring “Big” questions on “Small” scales with ultracold atoms

$$ NSF, AFOSR MURI, DARPA OLE,  MURI ATOMTRONICS

Harvard-MIT

Theory: Takuya Kitagawa (Harvard), Mehrtash Babadi(Harvard/Caltech), Fabian Grusdt (U. Kaiserslautern) Dima Abanin(Harvard/Perimeter Institute),  David Pekker (Caltech/U Pittsburgh),Eugene Demler (Harvard)

Experiments: I. Bloch’s group (MPQ/LMU) J. Schmiedmayer’ group (TU Vienna)

Scales of ultracold

keV MeV GeV TeVfeV peV µeV meV eV

pK nK µK mK K

neV

roomtemperature

LHC

He Ncurrent

experiments10-11 - 10-10 K

first BECof alkali atoms

Higgs mode in ultracold atoms, 2012Higgs mode of the standard model, 2012

Higgs excitationgeneral feature of systems withspontaneous symmetry breaking

by Steven Weinberg

Outline

Observation of Higgs mode in superfluid phaseof ultracold atoms in 2dM. Enders et al., Nature 487:454 (2012)

Zak/Berry phase measurements of topological order parameterM. Atala et al., Nature Physics 9:795 (2013)D. Abanin et al., Phys. Rev. Lett. 110:165304 (2013)

Demonstration of Prethermalizationin split 1D Bose condensates

D. Gring et al., Science 337:1318 (2012)M. Kuhnert et al., Phys. Rev. Lett. 110:090405 (2013)D. Smith et al., New Journal of Physics 15:075011 (2013)

Observation of Higgs mode in superfluidphase of ultracold atoms in 2d

M. Enders et al., Nature 487:454 (2012)

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

U

t

tunneling of atoms between neighboring wells

repulsion of atoms sitting in the same well

M.P.A. Fisher et al., PRB (1989)D. Jaksch et al. PRL (1998)

4

Bose Hubbard model

1n

U

02

0

M.P.A. Fisher et al.,PRB40:546 (1989)

MottN=1

N=2

N=3

Superfluid

Superfluid phase

Mott insulator phase

Weak interactions

Strong interactions

Mott

Mott

Parabolic trapping potentialBakr et al., Science 2010

density

21n

U

1

0

Mottn=1

n=2

n=3 Mott

Mott Superfluidphase

Order parameter 

Phase (Goldstone) mode = gapless Bogoliubov mode

Breaks U(1) symmetry

Gapped amplitude mode (Higgs mode)

Collective modes of strongly interactingsuperfluid bosons

Excitations of the Bose Hubbard model

2

Mott Superfluid

21n

U

1

0

Mottn=1

n=2

n=3

Superfluid

Mott

Mott

Softening of the amplitude mode is the defining characteristicof the second order Quantum Phase Transition

Is there a Higgs resonance 2d?D. Podolsky et al., (2012)

Earlier work: S. Sachdev (1999), W. Zwerger (2004)

Exciting the amplitude mode

Absorbed energy

Mottn=1

Exciting the amplitude modeM. Endres et al., Nature (2012)

Mottn=1 Mottn=1

Experiments: full spectrum

Higgs Drum Modes

Similar to Higgs mode in compactified dimensions

Experimental demonstration of prethermalization in split 1d condensates

D. Gring et al., Science 337:1318 (2012)M. Kuhnert et al., Phys. Rev. Lett. 110:090405 (2013)D. Smith et al., New Journal of Physics 15:075011 (2013)

Heavy ions collisionsQCD

Where does statistical mechanics begins?How do isolated quantum systems relax towards the equilibrium state?

Pump and probe experimentsin condensed matter

Expanding universe

Relaxation to equilibrium in quantum systems 

Equilibriumquantum systemsare well understood

Non‐equilibriumquantum systems

are notwell understood

time

Most simple picture: one single timescale

Thermalization: an isolated interacting system approaches thermal equilibrium at  microscopic timescales. All memory about the initial conditions except energy is lost.

Relaxation to equilibrium in quantum systems 

Equilibriumquantum systemsare well understood

Non‐equilibriumquantum systems

are notwell understood

time

Prethermalization:two timescales

(decay to non‐thermalquasi‐steady state, partial

loss of information)

Prethermalization

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)

Heavy ions collisionsQCD

New frontier in quantum many‐body physics: nonequilibrium dynamics

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

Experimental demonstration of prethermalizationProbing thermolization using local resolution and complete characterization of quantum noise

Analysis of full distribution functions of fringe contrast can be used to demonstrate Prethermalization atall lengthscales and In all correlation functions

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

x

z

Time of

flight

Experiments with 2D Bose gasHadzibabic, Krüger, Dalibard, et al., Nature 2006

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

Interference of two independent condensates

1

2

r

r+d

d

r’

Phase difference between clouds 1 and 2is not well defined

Assuming ballistic expansion

Individual measurements show interference patternsThey disappear after averaging over many shots

x1

dAmplitude 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 (2006)

Distribution function of phase and contrast

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

Higher moments reflect higher order correlation functions

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

Experiments analyze distribution  function of C

C

r1

r2

Fluctuations in 1d BECThermal fluctuations

Thermally energy of the superflow velocity

Quantum fluctuations

Weakly interactingatoms

Distribution function of interference fringe contrastHofferberth et al., Nature Physics 4:489 (2008)

Comparison of theory and experiments: no free parameters

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

Measurements of dynamics of split condensate

• Matter-wave interferometry

phase, contrast

FDF of phase and contrast

• Matter-wave interferometry

contrast

phase, contrast

FDF of phase and contrast

phase

• Plot as circular statistics

• Matter-wave interferometry: repeat many times

• Plot

i>100

contrasti

phase

phase, contrast

FDF of phase and contrast

accumulate statistics

• Matter-wave interferometry: repeat many times

• Plot as

This is the full distribution function                of phase & contrast

i>100

contrasti

phasei

phase, contrast

FDF of phase and contrast

accumulate statistics

Experimental demonstration of prethermalizationProbing thermolization using local resolution and complete characterization of quantum noise. M. Gring et al., Science (2012) 

Initial T=120 nK (blue line). After  27.5 ms identical to thermal system at T= 15 nKAt all lengthscalesIn all correlation functions

time

Integration

length

Luttinger liquid model of phase dynamicsBistrizer, Altman, PNAS (2007)Burkov, Demler, PRL (2007)

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 evolution

Energy distribution

Equipartition of energy For 2d quasi‐conendsates pointed out by Mathey, Polkovnikov (2010) 

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

The system should look thermal like after different modes dephase.Effective temperature is not related to the physical temperature

Energy stored in each mode initially

Similar discussion for general CFT in Calabrese and Cardy PRL (2006) 

PrethermalizationTheory: T. Kitagawa, A. Imambekov, et al., PRL(2010), NJP (2011)Expt: M. Gring et al., Science 2012

From prethermalization to thermalization

From prethermalization to thermalizationT. Langen, et al., unpublished

8% imbalamce

3% imbalamce

Measurement of topological order parameter

Order beyond symmetry breaking

Topological order is the “quantum protectorate” of this precise quantization

Current along x, measure voltage along y.On a plateau

In 1980 the first ordered phase beyond symmetry breaking was discovered

Integer Quantum Hall Effect: 2D electron gas in strong magnetic field shows plateaus in Hall conductance

with an accuracy of 10-9

Magnetization - order parameter in ferromagnets

Order parameters

Berry/Zak phase in 1d

Vanderbilt, King-SmithPRB 1993

How to measure topological order parameter?

Measure the Berry/Zak phase itself, not its consequence

Su-Schrieffer-Heeger Model

B A B BA

When dz(k)=0, states with t>0 and t<0 are topologically distinct.

Domain wall states in SSH ModelAn interface between topologically different states has protected midgap states

Absorption spectra onneutral and doped trans‐(CH)x

Probing band topology with Ramsey/Bloch interference

C. Salomon et al., PRL  (1996) 

Tools of atomic physics:Bloch oscillations 

More than 20,000 Bloch oscillations:Innsbruck, Florence

/2 pulse

Evolution

Tools of atomic physics:Ramsey interference

Used for atomic clocks, gravitometers, accelerometers, magnetic field measurements

/2 pulse + measurement ot Szgives relative phase accumulated by the two spin components

EvolutionEvolution

Zak phase probe of band topology in 1d

One dimensional superlattices Su‐Schrieffer‐Heeger model

Experiments Marcos Atala, Monika Aidelsburger,Julio Barreiro, I. Bloch (LMU/MPQ)

Theory: Takuya Kitagawa (Harvard), Dima Abanin (Harvard/Perimeter),  Immanuel Bloch (MPQ), Eugene Demler (Harvard)

M. Atala et al., Nature Physics 9:795 (2013)

SSH model of polyacetylene

Analogous to bichromatic optical lattice potential 

I. Bloch et al.,LMU/MPQ

B A B BA

Su, Schrieffer, Heeger, 1979

Characterizing SSH model using Zak phase Two hyperfine spin states experience the same optical potential

/2a/2a

a

Zak phase is equal to 0

Problem: experimentally difficult to control Zeeman phase shift

Dynamic phases due todispersion and magnetic field fluctuations cancel.Interference measuresthe difference of Zakphases of the two bands in two dimerizations.Expect phase

Spin echo protocol for measuring Zak phase 

Bloch oscillations measurements in LMU/MPQWith -pulse but no swapping of dimerization

Bloch oscillations measurements in LMU/MPQWith -pulse and with swapping of dimerization

Zak phase measurements in LMU/MPQ

Zak phase measurements can be used to probetopological propertiesof Bloch bands in 2D

D. Abanin et al., Phys. Rev. Lett. 110:165304 (2013)

Berry phase in hexagonal lattice

• Eigenvectors lie in the XY plane• Around each Dirac point  eigenvector makes 2 rotation

• Integral of the Berry phase is 

Berry’s phase of Dirac fermionsShifted positions of Integer quantum Hall plateaus

Measurement of the Berry and Zak phases in 2Dwith Ramsey/Bloch method

Experimental realizationTarruell et al., Nature (2012)

When jumps by Sz changes sign

In Ramsey interference

How to measure the p-Berry phaseof Dirac fermions“Parallel” measurements with a cloud of fermions

Measurement of the Berry and Zak phases in 2DSpin echo protocol

Measurement of the Berry and Zak phases in 2DModified spin echo protocol

Summary

Zak/Berry phase measurements of topological order parameterusing Ramsey/Bloch interference

Observation of Higgsmode in superfluid phaseof ultracold atoms in 2d

Demonstration of prethermalizationin split 1d condensates

M. Enders et al., Nature 487:454 (2012)

D. Gring et al., Science 337:1318 (2012)M. Kuhnert et al., PRL 110:090405 (2013)D. Smith et al., NJP 15:075011 (2013)

M. Atala et al., Nature Physics 9:795 (2013)D. Abanin et al., PRL 110:165304 (2013)

Interacting bosons in 1d at T=0

K – Luttinger parameter

Low energy excitations and long distance correlation functions can be described by the Luttinger Hamiltonian.

Connection to original bosonic particles

Gutzwiller model for the amplitude mode

Bogoliubov mode comes from the phase and charge degrees of freedom:        and 

Amplitude/Higgs mode comes from       and  

Time dependent mean-field: project dynamics to factorizable Gutzwiller wavefunctions. It is equivalent to Landau-Lifshitz eqs. It gives collective modes but not coupling between them.

Threshold for absorption is captured very well

Time dependent cluster mean‐field

2x2 captures width of spectral feature

breathing mode

single amplitude mode excited multiple modes

excited?

breathing mode

single amplitude mode excited

Lattice height 9.5 Er: (1x1 vs 2x2)

Absorption spectra. Theory (1x1 calculations)

Breathing mode

disappearing amplitude mode

details at the QCP

spectrum remains gapped due to trap

Higgs Drum Modes1x1  calculation, 20 oscillationsEabs rescaled so peak heights coincide

Similar to Higgs mode in compactified dimensions

keV MeV GeV TeVfeV peV µeV meV eV

pK nK µK mK K

neV

LHCCold atoms experiments10‐11 ‐ 10‐10 K

Higgs mode in ultracold atoms, 2012Higgs mode of the standard model, 2012

Scales of ultracold