Nonequilibrium quantum dynamics of synthetic matter:of synthetic matter:

ultracold atoms and photons

Tak a Kitaga a Da id Pekker Rajdeep SensarmaTakuya Kitagawa, David Pekker, Rajdeep Sensarma, Ehud Altman, Erez Berg, Mark Rudner, Liang Fu, Takashi Oka, Eugene Demler

Collaboration with experimental groups of T. Esslinger (ETH) and A. White (U. Queensland)

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

T. Esslinger (ETH) and A. White (U. Queensland)

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 modelSame microscopic model

U

t

U

t

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

Need to understand questions of relaxation and equilibrationq qeven when we are interested in equilibrium phase diagrams

Strongly correlated systems of photons

Strongly interacting polaritons in coupled arrays of cavitiesM H t t l N t Ph i (2006)M. Hartmann et al., Nature Physics (2006)

Strong optical nonlinearities innanoscale surface plasmonsAkimov et al Nature (2007)Akimov et al., Nature (2007)

Crystallization (fermionization) of photonsin one dimensional optical waveguidesD. Chang et al., Nature Physics (2008)

Never in thermal equilibrium. Need to understand coherent many-body dynamics

Nonequilibrium quantum dynamics q q yis much more important for

synthetic quantum many-body systemsy q y y ysuch as

ultracold atoms and nonlinear photonsp

This talk: interesting dynamical questions of familiar modelsThis talk: interesting dynamical questions of familiar models

Outline

Doublon decay in repulsive Hubbard d lmodel

Strohmaier et al., PRL 104:80401 (2010)Expts by T Esslinger’s group at ETHExpts by T. Esslinger s group at ETHTheory by Pekker et al., Harvard

Exploring topological phases with photonsp g p g p pT. Kitagawa et al., PRA 82:33429 (2010)Phys. Rev. B 82, 235114 (2010) Expts by A White’s group QueenslandExpts by A. White s group, QueenslandarXiv:1105.5334

Doublon relaxation inDoublon relaxation in ultracod Fermi gasesultracod Fermi gases

Realization of Hubbard model withultracold atoms in Optical Latticeultracold atoms in Optical Lattice

U

tt

Repulsive Hubbard model at half-filling

currentexperimentsTN experimentsN

Mott statewithout AF order

U

Signatures of incompressible Mott state of fermions in optical latticeof fermions in optical lattice

Suppression of double occupanciesR. Joerdens et al., Nature (2008)

Compressibility measurementsU. Schneider et al., Science (2008)R. Joerdens et al., Nature (2008) U. Schneider et al., Science (2008)

Doublon relaxation inDoublon relaxation in ultracod Fermi gasesultracod Fermi gases

Lifetime of doublons in Hubbard model

Modulate lattice potentialModulate lattice potential

Create doubly occupied sites= doublons

M h l it t k fMeasure how long it takes fordoublons to decay

Fermions in optical lattice.Decay of repulsively bound pairsDecay of repulsively bound pairs

Experiments: N. Strohmaier et. al.

Doublon decay in a compressible stateEnergy U converted to kinetic energy of single atoms. Compressible state: excitationsof single atoms.

Doublon can decay only if it d l

Kinetic energy scale

Compressible state: excitations with energies set by tunneling

if it produces several particle-hole pairs

Perturbation theory to order n=U/w

gyset by bandwidth

Decay probability

Doublon decay in a compressible state

To calculate the rate: considerTo calculate the rate: consider processes which maximize the number of particle-hole excitations

Doublon relaxation in organic gMott insulators ET-F2TCNQ

One dimensional Mott insulator ET-F2TCNQ

t=0.1 eV U=0.7 eV

Photoinduced metallic stateH. Okamoto et al., PRL 98:37401 (2007)S. Wall et al. Nature Physics 7:114 (2011)

Surprisingly long relaxation time 840 fsg y g

h/t = 40 fs

Photoinduced metallic stateH. Okamoto et al., PRL 98:37401 (2007)S. Wall et al. Nature Physics 7:114 (2011)

1400 ft=0.1 eVw=4t=0.4eV

1400 fscomparable to experimentally

d 840U=0.7 eV measured 840 msEstimate suggested by T. Oka

E l i t l i l h ith h tExploring topological phases with photons

T Kitagawa et al PRA 82:33429 (2010)T. Kitagawa et al., PRA 82:33429 (2010)Phys. Rev. B 82, 235114 (2010) Expts by A. White’s group, QueenslandarXiv:1105.5334

Topological states of electron systems

R b t i t di d d t b tiRobust against disorder and perturbationsGeometrical character of ground states

Can dynamics possess topological properties ?Ca dy a cs possess opo og ca p ope es

One can use dynamics to make stroboscopic implementations of static topological Hamiltonians

D i it i t l i lDynamics can possess its own unique topological characterization

Both can be realized experimentally with “synthetic matter”: ultracold atoms and photonsy p

This talk: realization of topological states p gwith quantum walk

Discreet time quantum walk

Definition of 1D discrete Quantum Walk

1D lattice particle1D lattice, particle starts at the origin

Spin rotation

Spin-dependent pTranslation

Analogue of classical random walk.

Introduced in quantumIntroduced in quantum information:

Q Search, Q computations

Quantum walk with photons

A. A. White’s group in Univ. QueenslandT. Kitagawa et al., arXiv:1105.5334

Rotation is implemented by

g ,

Rotation is implemented by half-wave platesTranslation by birefringent

l it t l th t di lcalcite crystals that displace only horizontally polarized light

Earlier realization of QW with photons: A. Schrieber et al., PRL 104:50502 (2010)

PRL 104 100503 (2010)PRL 104:100503 (2010)

Also Schmitz et alAlso Schmitz et al.,PRL 103:90504 (2009)

From discreet timeFrom discreet timequantum walks to

T l i l H il iTopological Hamiltonians

T. Kitagawa et al., Phys. Rev. A 82, 033429 (2010)

Discrete quantum walkSpin rotation around y axisSpin rotation around y axis

Translation

One stepOne stepEvolution operator

Effective Hamiltonian of Quantum WalkInterpret evolution operator of one step

as resulting from Hamiltonian.

Stroboscopic implementation of p pHeff

Spin-orbit coupling in effective Hamiltonianp p g

From Quantum Walk to Spin-orbit Hamiltonian in 1d

k-dependent“Zeeman” field

Winding Number Z on the plane defines the topology!

Winding number takes integer valuesWinding number takes integer values.Can we have topologically distinct quantum walks?

Split-step DTQW

Split-step DTQWPhase Diagram

Topological Hamiltonians in 1DTopological Hamiltonians in 1D

Schnyder et al PRB (2008)Schnyder et al., PRB (2008)Kitaev (2009)

Detection of Topological phases:localized states at domain boundaries

Phase boundary of distinct topological h h b d t tphases has bound states

Bulks are insulators Topologically distinct, so the “gap” has to close

near the boundarynear the boundary

a localized state is expected

Split-step DTQW with site dependent rotationsApply site-dependent spin

rotation for

Split-step DTQW with site dependent rotations: Boundary Staterotations: Boundary State

Experimental demonstration of topological quantum walk with photonstopological quantum walk with photonsT. Kitagawa et al., arXiv:1105.5334

Rotation is implemented by half-wave platesT l ti b bi f i tTranslation by birefringentcalcite crystals that displace only horizontally polarized light

Quantum Hall like states:Quantum Hall like states:2D topological phase

with non-zero Chern number

Chern NumberThis is the number that characterizes the topologyThis is the number that characterizes the topology

of the Integer Quantum Hall type states

Chern number is quantized to integers

2D triangular lattice, spin 1/2“One step” consists of three unitary and translation operations in three directions

Phase Diagram

Topological Hamiltonians in 2DTopological Hamiltonians in 2D

Schnyder et al PRB (2008)Schnyder et al., PRB (2008)Kitaev (2009)

C bi i diff t d f f d lCombining different degrees of freedom one can also perform quantum walk in d=4,5,…

What we discussed so farWhat we discussed so far

Split time quantum walks provide stroboscopic implementationSplit time quantum walks provide stroboscopic implementationof different types of single particle Hamiltonians

By changing parameters of the quantum walk protocolwe can obtain effective Hamiltonians which correspond to different topological classesto different topological classes

Topological properties unique toTopological properties unique to dynamics

Topological properties of evolution operatorTi d dTime dependent periodic Hamiltonian

Floquet operator Uk(T) gives a map from a circle to the space of

Floquet operator

q p k( ) g p punitary matrices. It is characterized by the topological invariant

This can be understood as energy winding.This is unique to periodic dynamicsThis is unique to periodic dynamics. Energy defined up to 2p/T

Example of topologically non-trivial evolution operator

d l i Th l l i l iand relation to Thouless topological pumpingSpin ½ particle in 1d lattice. S i d i l dSpin down particles do not move. Spin up particles move by one lattice site per period

group velocity

n1 describes average displacement per period.Q ti ti f d ib t l i l i f ti lQuantization of n1 describes topological pumping of particles. This is another way to understand Thouless  quantized pumping

Experimental demonstration of topological quantum walk with photonstopological quantum walk with photons

A. White et al., Univ. Queensland

Boundary withBoundary with Boundary with topologically different evolution operators

Boundary with topologically similar evolution operators

Dynamically induced topological phases y y p g p

Ultracold atoms Photoinduced Hall effect i l t tin electron systems

T Kitagawa et alT. Oka, and H. Aoki, Ph R B 79 081406 (R) (2009)T. Kitagawa et al.,

Phys. Rev. B 82, 235114 (2010) Phys. Rev. B 79, 081406 (R) (2009);J. Inoue, A. Tanaka,Phys. Rev. Lett. 105, 017401 (2010)

SummaryInteresting phenomena in non-equilibrium quantum dynamics of synthetic matter

Doublon decay in repulsiveDoublon decay in repulsive Hubbard modelStrohmaier et al., PRL 104:80401 (2010)

Exploring topological phases with photonsp g p g p pT. Kitagawa et al., PRA 82:33429 (2010)Phys. Rev. B 82, 235114 (2010) arXiv:1105 5334arXiv:1105.5334

OutlineDoublon decay in repulsive Hubbard modelStrohmaier et al., PRL 104:80401 (2010)Expts by T. Esslinger’s group at ETHTheory by Pekker et al., HarvardTheory by Pekker et al., Harvard

Related experiments in ET-F2TCNQOkamoto et al., PRL 98:37401 (2007)

Exploring topological phases with photons

, ( )Wall et al. Nature Physics 7:114 (2011)

p g p g p pT. Kitagawa et al., PRA 82:33429 (2010)Phys. Rev. B 82, 235114 (2010) Expts by A White’s group QueenslandExpts by A. White s group, QueenslandarXiv:1105.5334

Topological properties of evolution operatorD i i th f b dDynamics in the space of m-bandsfor a d-dimensional system

Floquet operator is a mxm matrixwhich depends on d-dimensional k

New topological invariants

Example:d 3d=3

Diagramatic Flavors Comparison of approximations

me

(h/t

)

Doublon Propagator

feti

me

tim

Interacting “Single” ParticlesU/6t

lif“Missing” Diagrams