Measuring Entanglement Entropy in a Many-body System K. Rajibul Islam MIT-Harvard Center for Ultracold Atoms Caltech Mar 08, 2016 HARVARD UNIVERSITY  MIT CENTER FOR ULTRACOLD ATOMS

Strongly interacting quantum systems Wikipedia.org High Temperature superconductor Quark Gluon ‘plasma’ Microscopic description? Spin network Interacting atoms Macroscopic phenomenon? Simulating quantum matter on computers?

Quantum Superposition Exponential growth of Hilbert space Exponential growth of Hilbert space For N qubits - No. of states = 2N N = 40 240 ~ 1 Tb Entanglement Growth of Entanglement – hard to compute Solving Quantum dynamics of interacting spin models currently limited to about 30 - 40 spins.

Quantum Simulation 2S1/2 = |1,0 Spin states can be initialized and Feynman, International Journal of Theoretical Physics, Vol 21, No. 6/7, 1982 Lloyd, Science, Vol 273, No. 5278, 1996 2S1/2 = |0,0 = |1,0 qubits Spin states can be initialized and Individually detected Long coherence – up to 15 minutes!

Quantum Simulation : Platforms Trapped ions Nature Physics 8, 277–284 (2012)  Polar molecules Nature 501, 521 (2013) Neutral atoms in optical lattices Nature Physics 8,  267–276 (2012)  Photonic networks Nature Physics 8,  285–291 (2012) Superconducting circuits Nature Physics 8, 292–299 (2012)  Defects in diamonds Physics Today 67(10), 38(2014)

? Quantum Simulation : Trapped Ions Ising + ‘Bottom-up’ approach Tunable interactions – quantum Ising, XY, XYZ … ? Frustrated!! Ising + Entanglement in the ground state K. Kim, M.-S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G.-D. Lin, L.-M. Duan, C. Monroe Nature 465, 590 (2010).

Quantum Simulation : Trapped Ions ‘Quantum phase transitions’ 𝐽 𝑖,𝑗 ≈ 𝐽 0 /|𝑖−𝑗| N = 10 R. Islam, E. E. Edwards, K. Kim, S. Korenblit, C. Noh, H. Carmichael, G.-D. Lin, L.-M. Duan, C.-C. Joseph Wang, J. K. Freericks, and C. Monroe Nature Communications 2:377 (2011) R. Islam, C. Senko, W. C. Campbell, S. Korenblit, J. Smith, A. Lee, E. E. Edwards, C.-C. J. Wang, J. K. Freericks, and C. Monroe, Science 340, 583 (2013).

Superconducting circuits Quantum Simulation : Platforms Trapped ions Nature Physics 8, 277–284 (2012)  Neutral atoms in optical lattices Nature Physics 8,  267–276 (2012)  Superconducting circuits Nature Physics 8,  292–299 (2012)  NV defects in diamonds Physics Today 67(10), 38(2014) Photonic networks Nature Physics  8, 285–291 (2012)

Quantum gas microscope Bakr et al., Nature 462, 74 (2009), Bakr et al., Science.1192368 (June 2010) Previous work on single site addressability in lattices: Detecting single atoms in large spacing lattices (D. Weiss) and 1D standing waves (D. Meschede), Electron Microscope (H. Ott), Absorption imaging (J. Steinhauer), single trap (P. Grangier, Weinfurter/Weber), few site resolution (C. Chin), See also: Sherson et al., Nature 467, 68 (2010) 9

Single site parity Imaging

Quantum gas microscope High aperture objective NA=0.8 High resolution imaging 2D quantum gas of Rb-87 in optical lattice Hologram for projecting optical lattice 11

Bose Hubbard Model U/J tunneling J interaction U Superfluid Mott insulator Superfluid U/J Bakr et al., Science. 329, 547 (2010)

Projecting arbitrary potential landscapes hologram Fourier hologram Image: EKB Technologies objective 2D quantum gas of Rb-87 in optical lattice Thesis : P. Zupancic (LMU/Harvard, 2014) 13

Arbitrary beam shaping Weitenberg et al., Nature 471, 319-324 (2011) Zupancic, P., Master’s Thesis, LMU Munich/Harvard 2013 Cizmar, T et al., Nature Photonics 4, 6 (2010) High-order Laguerre Modes Laguerre-Gauss profile 1 10-1 10-2 10-3

A bottom-up system for neutral atoms (Single shot image)

Single-Particle Bloch oscillations F P. M. Preiss, R. Ma, M. E. Tai, A. Lukin, M. Rispoli, P. Zupancic, Y. Lahini, R. Islam, M. Greiner Science 347, 1229 (2015)

Single-Particle Bloch oscillations F Temporal period , spatial width Delocalized over ~14 sites = 10μm. Revival probability 96(3)% P. M. Preiss, R. Ma, M. E. Tai, A. Lukin, M. Rispoli, P. Zupancic, Y. Lahini, R. Islam, M. Greiner Science 347, 1229 (2015)

Entanglement in Many-body Systems Novel states of matter: Order beyond simple broken symmetry Example - Topological order, spin liquid, fractional quantum Hall - characterized by quantum entanglement ! Quantum criticality Quantum dynamics … Challenge: Entanglement not detected in traditional CM experiments Entanglement in ultra-cold atom synthetic quantum matter?

Entanglement in Many-body Systems A B Many-body system: Bipartite entanglement Product state: e.g. Mott insulator Entangled state: e.g. Superfluid

Entanglement Entropy TRACE A B Tr( 𝜌 𝐴 2 ) =1 <1 Reduced density matrix: Product state  Pure state Entangled state  Mixed state Tr( 𝜌 𝐴 2 ) Quantum purity = =1 <1 𝑆 2 𝜌 𝐴 =− log Tr( 𝜌 𝐴 2 ) Renyi Entanglement Entropy

Entanglement Entropy TRACE A B Tr( 𝜌 𝐴 2 ) =1 <1 Reduced density matrix: Product state  Pure state Entangled state  Mixed state Tr( 𝜌 𝐴 2 ) Quantum purity = =1 <1 Many-body Hong-Ou-Mandel interferometry Alves and Jaksch, PRL 93, 110501 (2004) Mintert et al., PRL 95, 260502 (2005) Daley et al., PRL 109, 020505 (2012)

No coincidence detection Hong-Ou-Mandel interference No coincidence detection for identical photons Hong C. K., Ou Z. Y., and Mandel L. Phys. Rev. Lett. 59 2044 (1987)

Beam splitter operation: Rabi flopping in a double well P (R) L R -i +i Also see: Kaufman A M et al., Science 345, 306 (2014) Without single atom detection: Trotzky et al., PRL 105, 265303 (2010) also Esslinger group

Two bosons on a beam splitter Hong-Ou-Mandel interference Beam splitter

limited by interaction measured fidelity: Beam splitter Beam splitter P(1,1) 4(4)% 96(4)% limited by interaction measured fidelity: Time in double well (ms) Also see : Kaufman A M et al., Science 345, 306 (2014), R. Lopes et al, Nature 520, 7545 (2015)

Quantum interference of bosonic many body systems ? How “identical” are the particles? vs. How “identical” are the states? If , deterministic number parity after beam splitter Alves and Jaksch, PRL 93 (2004) Daley et al., PRL 109 (2012)

Quantum interference of bosonic many body systems

Making two copies of a many-body state

Measuring many-body entanglement Mott Insulator Locally pure Product State Globally pure Superfluid Locally mixed Entangled Globally pure

Measuring many-body entanglement Mott Insulator always even  locally pure HOM even even  globally pure Superfluid odd or even  locally mixed HOM  Entangled! even even  globally pure Ref: Alves C M, Jaksch D, PRL 93, 110501 (2004), Daley A J et al, PRL 109, 020505 (2012)

Entanglement in the ground state of a Bose-Hubbard system Mixed H Purity = Parity Renyi entropy Purity Beam splitter complete 2-site 1-site Pure Rajibul Islam et al, Nature 528, 77 (2015) U/J Mott insulator Superfluid Entanglement in optical lattice systems: M. Cramer et al, Nature Comm, 4 (2013), T. Fukuhara et al, PRL 115, 035302 (2015)

Entanglement in the ground state of a Bose-Hubbard system Mixed H Purity = Parity Renyi entropy Purity Beam splitter complete 2-site 1-site Pure Rajibul Islam et al, Nature 528, 77 (2015) U/J Mott insulator Superfluid

Entanglement in the ground state of a Bose-Hubbard system Mixed H Purity = Parity Renyi entropy Beam splitter complete 2-site 1-site Pure Rajibul Islam et al, Nature 528, 77 (2015) U/J Mott insulator Superfluid

Entanglement in the ground state of a Bose-Hubbard system Mixed H Purity = Parity Renyi entropy Purity Beam splitter complete 2-site 1-site Pure Rajibul Islam et al, Nature 528, 77 (2015) U/J Mott insulator Superfluid

Entanglement in the ground state of a Bose-Hubbard system Renyi entropy complete 2-site 1-site U/J Mott insulator Superfluid

Mutual Information IAB IAB = S2(A) + S2(B) - S2(AB) Renyi entropy IAB Mutual Information Boundary Area U/J Mott insulator Superfluid

Non equilibrium: Quench dynamics Beam splitter

Non equilibrium- Quench dynamics Outlook: Non equilibrium- Quench dynamics Greiner group - unpublished

Scaling of entanglement entropy and mutual information – probe critical points, violation of area law etc. Dynamical phenomena with entanglement – MBL phase. Overlap of two wave functions Sensitivity to perturbation signaling quantum phase transitions. Higher order Renyi entropies by interfering more than two copies. Ψ1 Ψ2

A ‘quantum gas microscope’ for ions

Thank you!! Theory Experiments Harvard Philipp Preiss Ruichao Ma Eric Tai Matthew Rispoli Alex Lukin Markus Greiner Maryland Kihwan Kim (Now at Tshinhua, China) Ming-Shien Chang (now at Academia Sinica, Taiwan) Emily Edwards Wes Campbell (now at UCLA) Simcha Korenblit (now at Hebrew University) Crystal Senko (now at Harvard) Jake Smith Aaron Lee Chris Monroe Guin-Dar Lin (Michigan) Luming Duan (Michigan) Joseph C.-C. Wang (Georgetown) Jim Freericks (Georgetown) Changsuk Noh (Auckland) Howard Carmichael (Auckland) Andrew Daley (strathclyde) Peter Zoller (Innsbruck) Eugene Demler (Harvard)