Detection of topological order with quantum simulators EU - - PowerPoint PPT Presentation

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Detection of topological order with quantum simulators EU - - PowerPoint PPT Presentation

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AvH Feodor Lynen EU IP SIQS Advanced ERC Grant: Advanced ERC Grant: MPG QUAGATUA OSYRIS MPI Garching SGR 874 CERCA/Program Detection of topological order with quantum simulators EU FET-Proactive QUIC a FNP i n o f Polish

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AvH Feodor Lynen EU IP SIQS EU STREP EQuaM Advanced ERC Grant: OSYRIS John Templeton Foundation FOQUS and FISICATEAMO EU FET-Proactive QUIC SGR 874 Advanced ERC Grant: QUAGATUA

FNP Polish Science Foundation NCN Narodowe Centrum Nauki

Detection of topological order with quantum simulators

CERCA/Program MPG MPI Garching

S y m f

  • n

i a

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ICFO – Quantum Optics Theory Teoretyczna Optyka Kwantowa

PhD ICFO:

David Raventos (gauge fields) Emanuele Tirrito (TN) Angelo Piga (TN) Nils-Eric Gnther (many body) Christos Charampoulos (open sys) Zahra Khanian (QI) Albert Aloy (QI) Gorka Muñoz (Brownian) Daniel González (many body) Sergi Julià (QI, many body) Jessica Almeida (quantum optics) Korninian Kottman (many body, ML) Mohit Bera (QI, many body)

Postdocs ICFO:

Miguel Angel García March (all) Emilio Pisanty (atto) Manab Bera (QI) Alexandre Dauphin (many body, atto) Irénée Frerot (many body) Giulia de Rosi (many body) Maria Maffei (QOT) Debraj Rakshit (few/many body) Ex-members and collaborators: Aditi Sen De, Ujjwal Sed (HRI, Alahabad), François Dubin (CNRS), G. John Lapeyre (CSIC), Luca Tagliacozzo (UB), Alessio Celi (IQOQI/UAB), Matthieu Alloing (Paris), Tomek Sowiński (IFPAN), Phillip Hauke (Heidelberg), Omjyoti Dutta (GMV), Christian Trefzger (EC), Kuba Zakrzewski (UJ, Cracow), Mariusz Gajda (IF PAN), Boris Malomed (Haifa), Ulrich Ebling (Kyoto), Bruno Julia Díaz (UB), Christine Muschik (IQOQI), Marek Kuś, Remigiusz Augusiak (CFT), Julia Stasińska (IFPAN), Alexander Streltsov (FUB), Ravindra Chhajlany (UAM), Fernando Cucchietti (MareNostrum), Anna Sanpera (UAB), Veronica Ahufinger (UAB), Tobias Grass (JQI,UMD/NIST), Jordi Tura (MPQ), Alexis Chacón (Los Alamos), Marcelo Ciappina (Prague), Arnau Riera (BCN), Przemek Grzybowski (UAM), Swapan Rana (UW, Warsaw),

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RoY J. GLAUBer and shoucheng zhang in memoriam

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Detection of topological order with quantum simulators

  • 1. Detection of topological order in 1D chiral systems
  • 1.1 Chiral Mean Displacement
  • 1.2 Topological Anderson Insulator
  • 1.3 Photonic random walk in 1D
  • 0. Introduction
  • 0.1 Topology
  • 0.2 Quantum simulators
  • 0.3 Topology in quantum simulators
  • 3. Detection of topological order in 1D interacting systems
  • 3.1 Bosonic Peierls mechanism – minimal instance of lattice dynamics
  • 3.2 Phase diagram of Z2 Bose-Hubbard model
  • 3.3 Correlated symmetry-protected topological states
  • 3.3 Self-adjusted pumping
  • 2. Detection of Chern number in 2D systems
  • 2.1 Measuring Chern numbers in Hofstadter strips
  • 2.2 Probing topology by “heating”
  • 2.3 Loading ultracold gases in Topological Floquet Bands
  • 2.4 Photonic random walk in 2D
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  • 0. Introduction
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Properties that remain unchanged under continuous and smooth deformations Genus: number of holes of a closed surface

Topology

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  • Characterised by global topological invariants
  • Classified on the basis of their symmetries and dimensionality
  • Topologically protected edge-states in systems with boundaries
  • Bulk-edge correspondence: the number of states on each edge is given

by the invariant

  • Beyond the periodic table: interacting/ Anderson / Floquet TIs, …

Chiu, T eo, Schnyder & Ryu, Rev. Mod. Phys. (2016)

Topological insulators

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Integer Quantum Hall effect – 1980 2D semiconductor at very low temperature under a strong magnetic field

  • K. von Klitzing,

Nobel lecture

  • M. Konig et al., Science 318: 766 (2007)

Quantum Spin Hall effect – 2007 HgTe quantum well

Enormous progresses in the last ten years

Topology in condensed matter systems

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Ultracold atoms in optical lattices: Simulating quantum many-body physics

  • M. Lewenstein, A. Sanpera, V. Ahufinger, Oxford University Press (2012),

reprint-paperback (2017)

Quantum simulators

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Controllable experimental platforms simulating the dynamics of the system of interest

T . Torres et al., Nat. Phys. 13:883

Many simulations of condensed matter systems with ultracold atoms and photons in the last 10 years

Water bath - 2017 black holes

  • M. Greiner et al., Nature 415:39–44

Ultracold atoms in optical lattice - 2002 Superfluid – Mott insulator transition

Simulators and quantum simulators

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  • T. Kitagawa et al.,
  • Nat. Commun. (2012)
  • F. Cardano et al.,Sci. Adv., (2015)
  • F. Cardano,...M.M....,
  • Nat. Commun. (2017)

A.D’Errico...M.M...., arXiv:(2018)

  • Y. E. Kraus et al.,

Phys.Rev. Lett. (2012) M . Aidelsburger et al., Phy s . Rev. Lett. (2013)

Laser assisted tunneling

T

  • pology in quantum simulators

ynthetic dimension

  • G. Jotzu et al., Nature

(2014)

ime periodic attic e shaking

Mancini et al., Science (2015 M.C.Rechtsman et al., Nature (2013)

  • J. M. Zeuner et al.,
  • Phys. Rev. Lett. (2015)

With photons:

Array of shaped waveguides Quantum walk

With ultracold atoms:

  • M. Aidelsburger et al.,Phys. Rev. Lett

(2013)

Laser assisted t Synthetic dimension

G Jotzu et al., Nature (2014)

Time-periodic lattice shaking

Mancini et al. Science (2015) Stuhl et al. Science (2015)

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  • M. Lohse et al., Nat. Phys.

(2015)

Transverse displacement under external driving

  • M. Aidelsburger et al., Nat.Phys.

(2015) A.D’Errico...M.M...., arXiv:1811.04001 (2018)

Edge states Direct measure

  • f the topological

invariant

M.C.Rechtsman et al., Nature (2013) E.J. Meier et al., Nat. Commun. (2016)

  • M. Atala et al., Nat. Phys. (2013)

Detecting topology in simulators

  • E. J. Meier...M.M...., 14

Science(2018)

  • F. Cardano,...M. M....,
  • Nat. Commun. (2017)
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  • 1. Detection of topological order

in 1D chiral systems

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Detecting topology in 1D chiral systems: the mean chiral displacement

  • Detection of Zak phases and topological invariants in a chiral quantum walk of twisted

photons, F . Cardano, A. D’Errico, A. Dauphin, M.M., B. Piccirillo, C. de Lisio, G. De Filippis,

  • V. Cataudella, E. Santamato, L. Marrucci, M. Lewenstein and P

. Massignan, Nature Communications 8:15516 (2017)

  • Topological characterization of chiral models through their long time dynamics,

M.M., A. Dauphin, F . Cardano, M. Lewenstein and P . Massignan, New Journal of Physics 20 (2018)

  • Observation of the topological Anderson Insulator in disordered atomic wires,
  • E. J. Meier, F

. Alex An, A. Dauphin, M.M., P . Massignan, T . L. Hughes, B. Gadway Science 362:6417 (2018) Maria Maffei: 1 PhD Thesis co-tutelle Università degli studi di Napoli Federico II and IFCO

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The SSH model An Hamiltonian is chiral-symmetric if there exists an Hermitian and unitary operator such that:

with

Chiral-symmetric topological insulators

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Windingnumber

The winding number is the topological invariant characterising the Chiral class in 1D

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with h(k) = a + beik

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In the limit of zero intra-cell hopping localised states arise at the edges of the chain Bulk-edge correspondence: the winding number counts the states on each edge

Bulk-edge correspondence

The edge states are topologically protected against chiral- and gap-preserving perturbations

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Starting from an arbitrary localized state The MCD reads

Winding number and Mean Chiral Displacement (MCD)

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No external elements nor filled bands required for the MCD detection

F . Cardano, A. D’Errico, A. Dauphin, M.M., B. Piccirillo, C. de Lisio, G. De Filippis, V . Cataudella,

  • E. Santamato, L. Marrucci, M. Lewenstein and P

. Massignan,

  • Nat. Commun. 8:15516 (2017)
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The MCD detects the winding number in chiral systems with any internal dimension D

Generalization of the method

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m

We write the MCD as the trace over a basis of the internal space

  • M. M. et al., New J. Phys. 20 (2018)
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Generalization of the method

The MCD detects the Winding number in chiral systems with long-range hopping

m

The MCD detects the Winding also at the transition points

  • M. M. et al., New J. Phys. 20 (2018)
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  • In a 1D chiral symmetric system
  • A. Altland et al., Phys. Rev.B, 91: 085429 (2015)
  • I. Mondragon-Shem et al., Phys. Rev. Lett., 113: 046802 (2014)

In 1- and 2D a disorder can induce Anderson localization A strong disorder can drive a system from a trivial to a topological phase

  • In a 2D metallic quantum well
  • J. Li et al., Phys. Rev. Lett., 102:136806 (2009)
  • S. Stutzer
  • In a 2D array of optical

waveguides

et al., Nature 560:461 (2018)

Theory Observation

Topological Anderson Insulator (TAI)

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Simulating the TAI with cold atoms

  • Bose-Einstein condensate (BEC)
  • Interference between a single and a

multi-frequency beam

  • Lattice of discrete momentum states of

the BEC

  • Laser-driven coupling between

momentum states

  • 1D Hamiltonian with tunable hopping

Phase difference between the laser fields

  • E. J. Meier, F

. A. An, A. Dauphin, M. M., P . Massignan, T . L. Hughes, B. Gadaway, Science 362: 6417 (2018)

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Random disorder can be included in the hopping bn an Color map: real space Winding number for a system of 200 cells and 1000 disorder realizations Red line: critical boundary (diverging localization length in the therm. limit)

r (inter/intra cell hopp.)

W

1 1

Simulating the TAI with cold atoms

p/2hk

  • E. J. Meier, F

. A. An, A. Dauphin, M. M., P .Massignan, T . L. Hughes, B. Gadaway, Science 362: 6417 (2018)

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  • E. J. Meier, F

. A. An, A. Dauphin, M. M., P . Massignan, T . L. Hughes, B. Gadaway, Science 362: 6417 (2018)

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Revealing the TAI phase through the MCD

Numerical simulation (exp.times, 200 dis. conf.) W (winding number therm. lim.) Numerical simulation for longer times (250 cells, 200 dis. conf.) Experimental data (20 cells, 50 dis. conf.)

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Simulating a chiral insulator with a photonic 1D quantum walk

  • Statistical moments of quantum-walk dynamics reveal topological quantum

transitions, F . Cardano, M.M., F . Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato and L. Marrucci, Nature Communications 7:11439 (2016).

  • Detection of Zak phases and topological invariants in a chiral quantum walk
  • f twisted photons, F

. Cardano, A. D’Errico, A. Dauphin, M.M., B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, L. Marrucci, M. Lewenstein and P . Massignan, Nature Communications 8:15516 (2017).

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Quantum Walk(QW)

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Quantum Walk’s effective Hamiltonian

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Quantum Walk with light’s OrbitalAngular Momentum

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Quantum Walk with light’s OrbitalAngular Momentum

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Quantum Walk with light’s OrbitalAngular Momentum

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Topological characterization of 1D chiral QuantumWalks

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Detection of the invariants through the MCD

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  • 2. Detection of Chern number in 2D systems
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Sam Mugel: 1 PhD Thesis co-tutelle University of Southhampton and ICFO

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arXiv:1804.06345, Imaging topology of Hofstadter ribbons, Dina Genkina, Lauren M. Aycock, Hsin-I Lu, Alina M. Pineiro, Mingwu Lu, I.B. Spielman

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Conjecture/Open Problem?

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Simulating a Chern insulator with a photonic 2D quantum walk

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Integer Quantum Hall effect on the lattice

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Integer Quantum Hall effect on the lattice

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Quantum Walk with light’s transverse wavevector

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Quantum Walk with light’s transverse wavevector

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Quantum Walk with light’s transverse wavevector

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Measuring Chern numbers through transverse displacements

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Wavepackets dynamics

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Wavepackets dynamics

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  • 3. Detection of topological order in

1D interacting systems

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Motivations

Simon Wall Leticia Tarruell

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Phase diagram

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ICFO-QOT

Progress Status Workshop Barcelona 18-20 March 2015

Conclusions: Quantum Narcissism