Ramsey interference as a probe ofRamsey interference as a probe of “synthetic” condensed matter systems

Takuya Kitagawa (Harvard)Di Ab i (H d)Dima Abanin (Harvard)Mikhael Knap (TU Graz/Harvard)E D l (H d)Eugene Demler (Harvard)

Supported by NSF, DARPA OLE, AFOSR MURI, ARO MURI

Ramsey interferencep/2l

p/2lpulse pulse

1

t0

Ramsey interference in atomic clocks

Atomic clocks and Ramsey interference:Working with N atoms improves the precision by         .

y

Using interactions to improve the clock precisionKitagawa, Ueda, PRA 47:5138 (1993)

Classical BECSingle modeSingle mode approximation

Ramsey interference as a probe of 1d dynamics

Interaction induced collapse of Ramsey f i se

y fri

nge

ityfringes. timeR

ams

visi

bili

Only weak spin echo was observed in experiments

Experiments:Widera et al. PRL (2008)PRL (2008)

Decoherence of Ramsey fringes due to many-body dynamics of low dimensional BECsof low dimensional BECsNew probe of dynamics of 1d many-body systems

This talk:This talk:Exploring “synthetic” condensed matter systemswith Ramsey interferencewith Ramsey interference

1. Measuring topological properties of band structures.  Berry and Zak phases

2. Orthogonality catastrophe

Measuring topological propertiesMeasuring topological properties of band structures. Berry and Zak phasesBerry and Zak phases

Topological aspects of band structures

All evidence is indirect:(transport, edge states)Can there be a direct measurement ofBerry phase of Bloch states ?

Example of band structure with Berry phase.H l ( h ) l iHexagonal (graphene) lattice

Tight binding model on a hexagonal lattice

Dirac  fermions in optical lattices, Tarruell et al., Nature 2012

Berry phase in hexagonal lattice

• Eigenvectors lie in the XY plane• Aro nd each Dirac point eigen ector• Around each Dirac point  eigenvector makes 2p rotation

• Integral of the Berry phase is p

How to measure Berry phase in hexagonal latticeNaïve approach:Naïve approach:

Move atom on a closed trajectoryaround Dirac pointMeasure accumulated phase

Problems with this approach: Need to move atom on a complicated curved trajectoryNeed to separate dynamical phase

Integral of the Berry phase is only well defined on a closed trajectory

From Berry phase to Zak phaseIntegral of the Berry phase is only well defined on a closed trajectory

gauge invariantintegral of

Brillouin zone is a torus There are two types of closed trajectories

is not gauge invariantC Berry curvature

Brillouin zone is a torus. There are two types of closed trajectories

How to measure Zak phase usingR i fRamsey interference sequence

Two hyperfine spin states experience the same optical potentialyp p p p p

AdvantagesRequires only straight  trajectoryDynamical phase cancels between two spin states

How to measure Zak phase usingRamsey interference sequenceRamsey interference sequence

Dynamically induced topological phases in a hexagonal lattice T Kitagawa et alin a hexagonal lattice T. Kitagawa et al.,

Phys. Rev. B 82, 235114 (2010)

Dynamically induced topological phases in a hexagonal latticein a hexagonal latticeFloquet spectrum on a strip

Edge states indicate theappearance of topologicallynon-trivial phases

Equivalent to Haldane model

Detection of topological band structure

Detection of topological band structure

Zak phase as a probe of band topologyd i1d version

Su‐Schrieffer‐Heeger model of polyacetylene

Analogous to bichromatic optical lattice potential 

LMU/MPQ

Dimerized model

A B A AB

Topology of the band shows up in the winding of the eigenstateZak phase is equal to p

Characterizing SSH model using Zak phase T h fi i i h i l i lTwo hyperfine spin states experience the same optical potential

a

p/2a-p/2a 0

Zak phase is equal to p

Orthogonality catastrophe

Single impurity problems in condensed matter physics

Rich many-body physics

a e p ys cs

-Edge singularities in the

X-ray absorption spectra-Kondo effect: entangled

state of impurity spin and y p p(exact solution of non-equilibrium

many-body problem)

state of impurity spin and

fermions

Influential area, both for methods (renormalization group) and for strongly correlated materials

Probing impurity physics in solids is limited -Many unknowns;Simple models hard to test( l d b d k(complicated band structure, unknown

impurity parameters, coupling to phonons)

Probing impurity physics in solids is limited -Many unknowns;Simple models hard to test( l d b d k(complicated band structure, unknown

impurity parameters, coupling to phonons)

X-ray absorption in Na

Probing impurity physics in solids is limited -Many unknowns;Simple models hard to test( l d b d k(complicated band structure, unknown

impurity parameters, coupling to phonons)

-Limited probesLimited probes(usually only absorption spectra)

-Dynamics beyond linear response

out of reach (relevant time scales GHz-THz,

i t ll diffi lt)X-ray absorption in Na

experimentally difficult)

Cold atoms: new opportunities for studying impurity physics

-Parameters known

pu y p ys cs

allow precise tests of theoryexact solutions available

-Tunable by the Feshbach resonanceresonance

f (k) a1 ika

-Fast control of microscopic parameters(compared to many-body scales)

-Rich toolbox for probing many-body states

Cold atoms: new opportunities for studying impurity physics

-Parameters known

pu y p ys cs

allow precise tests of theoryexact solutions available

New challenges for theory:

Describe dynamics

-Tunable by the Feshbach resonance

-Describe dynamics-Characterize complicated transient many-body states

resonancef (k) a

1 ika-Explore new many-body phenomena

-Fast control of microscopic parameters(compared to many-body scales)

-Rich toolbox for probing many-body states

Introduction to Anderson orthogonalitycatastrophe (OC)ca as op e (OC)

-Overlap S FS | FS '

- as system size , “orthogonality catastrophe”S 0 L

-Infinitely many low-energy electron-hole pairs produced

Fundamental property of the Fermi gas

Orthogonality catastrophe in X-ray absorption spectraabso p o spec a

Without impurity

With impurity

-Relevant overlap: S(t) FS | eiH0teiH f t | FS t2 / 2

-- scattering phase shift at Fermi energy, tan kFa

-Manifests in a power-law singularity in the absorption spectrum

1A() exp(it)S(t)dt 112 / 2

Orthogonality catastrophe in X-ray absorption spectraabso p o spec a

Without impurity

With impurity

-Relevant overlap: S(t) FS | eiH0teiH f t | FS t2 / 2

-- scattering phase shift at Fermi energy, tan kFa

-Manifests in a power-law singularity in the absorption spectrum

1

OBSERVATION IN SOLIDS CHALLENGING

A() exp(it)S(t)dt 112 / 2

Orthogonality catastrophe with cold atoms:Setup

-Fermi gas+single impurity

Se up

-Two pseudospin states of impurity and impurity, and

- -state scatters fermions

state scatte s e o s-state does not

-Scattering length

a-Fermion Hamiltonian for pseudospin -- , H0, H f

Earlier theoretical work on Kondo in cold atoms: Zwerger, Lamacraft, …

Ramsey fringes – new manifestation of OC-Utilize control over spin -Access coherent coupled dynamics of spin and Fermi gas-Ramsey interferometry

1) p/2 pulse FS 1 FS 1 FS1) p/2 pulse

2) Evolution

FS 2 FS

2 FS

1 eiH0t FS 1 eiH f t FS2) Evolution

3) Use p/2 pulse to measure )](Re[ tSSx 2 e FS

2 e FS

S(t) FS | eiH0teiH f t | FS

Direct measurement of OC in the time domain

Ramsey fringes as a probe of OCFirst principle calculationsFirst principle calculations

Spin echo: probing non-trivial dynamics of the Fermi gase e gas

-Response of Fermi gas to process in which impurity

switches between different states several times

iH t iH t

-Such responses play central role in analysis of Kondo

S1(t) FS | eiH0teiH f teiH0teiH f t | FSp p y y

problem but could not be studied in solid state systems

-Advantage: insensitive to slowly fluctuating magnetic fields

(unlike Ramsey)

Spin echo response: features

32 / 2

-Power-law decay at long times with an enhanced exponent

S1(t) t32 / 2

-Unlike the usual situation

(spin-echo decays slower

than Ramsey)

Cancels magnetic field-Cancels magnetic field

flutuations

-UniversalUniversal

-Generalize to n pi-pulses

to study even more complex

response functions

RF spectroscopy of impurity atomFree atom

Atom in a Fermi sea – many-body effects completely changey y p y gabsorption function

RF spectra can be calculated exactly

Edge singularities in the RF spectra:Intuitive pictureu ve p c u e

Photon pseudospin flipped+massive production of Photon pseudospin flipped+massive production of

excitations

Exact RF spectra

a<0; no impuritybound state

a>0; bound stateTwo thresholds

Single thresholdin absorption

-bound state filled after absorption

b d t t t-bound state empty

Generalizations: non-equilibrium OC, non-abelian Riemann-Hilbert problemo abe a e a be p ob e

-Impurity coupled to several Fermi seas at different chemical

i lpotentials

-Theoretical works in the context of quantum transportq p

-Mathematically, reduces to non-abelian Riemann-Hilbert problem (challenging)

Muzykantskii et al’03Abanin Levitov ‘05 (challenging)

-Experiments lacking

Abanin, Levitov 05

Generalizations: non-equilibrium OC, non-abelian Riemann-Hilbert problemo abe a e a be p ob e

-Multi-component Fermi gas coupled

to impurityp y

-Imbalance different species

-Mix them by pi/2 pulses

-Realization of non-equilibrium OC problem

-”Simulator” of quantum transport Simulator of quantum transport

and non-abelian Riemann-Hilbert problem

-Charge full counting statistics can be probed

Hyperfine state 1

Hyperfine t t 2state 1 state 2

Other directions

-Multi-component Fermi gas: non-equilibrium orthogonality catastrophe, non-abelian Riemann-Hilbert problemp , p

-Spinful fermions dynamics in the Kondo problem

-Dynamics: many-body effects in Rabi oscillations ofimpurity spin

Very different physics for bosons -Very different physics for bosons

Exploring “synthetic” condensed matter systemsExploring  synthetic  condensed matter systemswith Ramsey  interference

Measuring topological properties of band structures.  Berry and Zak phases

Orthogonality catastropheOrthogonality catastrophe

Interaction induced collapse of Ramsey fringesTwo component BEC. Single mode approximation

Ramsey fringe visibility

time

Experiments in 1d tubes:A. Widera et al. PRL 100:140401 (2008)

Spin echo. Time reversal experiments

Single mode approximation

The Hamiltonian can be reversed by changing a12

Predicts perfect spin echo

Spin echo. Time reversal experiments

Expts: A. Widera, I. Bloch et al.

Experiments done in array of tubes. S fl i i 1d

No revival?Strong fluctuations in 1d systems.Single mode approximation does not apply.Need to analyze the full model

Interaction induced collapse of Ramsey fringesin one dimensional systems

Only q=0 mode shows complete spin echoOnly q=0 mode shows complete spin echoFinite q modes continue decay

The net visibility is a result of competition between q=0 and other modes

Decoherence due to many‐body  dynamics of low dimensional  systems

New probe of dynamicsof 1d many‐body systems

Manifestations of topological character of band structurescharacter of band structures

All evidence is indirect:(transport, edge states)(transport, edge states)Can there be a direct measurement ofBerry phase of Bloch states ?Berry phase of Bloch states ?

Topologically different dimerizations

Dimerized tunneling Dimerized on‐site potential

topological non‐topological

Zak phase is equal to p Zak phase is equal to 0

Dimerized without inversion symmetry

Zak phase can be arbitrary