Measuring Entanglement Entropy in a Manybody System K

Measuring Entanglement Entropy in a Many-body System K. Rajibul Islam MIT-Harvard Center for Ultracold Atoms Caltech Mar 08, 2016 H A R V A R D U N I V E R S I T Y M I T CENTER FOR ULTRACOLD ATOMS

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

Quantum Superposition Exponential growth of Hilbert space For N qubits - No. of states = 2 N 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 • Feynman, International Journal of Theoretical Physics, Vol 21, No. 6/7, 1982 • Lloyd, Science, Vol 273, No. 5278, 1996 = |1, 0 2 S 1/2 = |0, 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) Neutral atoms in optical lattices Nature Physics 8, 267– 276 (2012) Photonic networks Polar molecules Nature Physics 8, 285– 291 (2012) Nature 501, 521 (2013) Superconducting circuits Defects in diamonds Physics Today 67(10), 38(2014) Nature Physics 8, 292– 299 (2012)

Quantum Simulation : Trapped Ions ‘Bottom-up’ approach 1 – 5 μm • Tunable interactions – quantum Ising, XYZ … Ising Frustrated!! + + ? + 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’ 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).

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

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 1 D standing waves (D. Meschede), Electron Microscope (H. Ott), Absorption imaging (J. Steinhauer), single trap (P. CENTER FOR ULTRACOLD ATOMS Grangier, Weinfurter/Weber), few site resolution (C. Chin), See also: Sherson et al. , Nature H A R V A R D U N I V E R S I T Y M I T

Single site parity Imaging

Quantum gas microscope Hologram for projecting optical lattice High resolution imaging High aperture objective NA=0. 8 2 D quantum gas of Rb-87 in optical lattice

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

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

Arbitrary beam shaping • • High-order Laguerre Modes 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) 1 10 -2 10 -3 Laguerre-Gauss profile

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

Entanglement Entropy E C A R T A B Reduced density matrix: Quantum purity = Product state Entangled state Pure state Mixed state Renyi Entanglement Entropy

Entanglement Entropy E C A R T A B Reduced density matrix: Quantum purity = Product state Entangled state Pure state Mixed state 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)

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)

P (R) Beam splitter operation: Rabi flopping in a double well 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

Beam splitter P(1, 1) Beam splitter measured fidelity: 96(4)% Time in double well (ms) 4(4)% limited by interaction 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 Globally pure Entangled

Measuring many-body entanglement Mott Insulator always even locally pure HOM even globally pure Superfluid odd or even locally mixed HOM even Entangled! 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 Purity = Parity Renyi entropy H complete Beam splitter 2 -site 1 -site Pure Rajibul Islam et al, Nature 528, 77 (2015) Mott insulator U/J 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 Purity = Parity Renyi entropy H complete Beam splitter 2 -site 1 -site Pure Rajibul Islam et al, Nature 528, 77 (2015) Mott insulator U/J Superfluid

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

Mutual Information IAB Renyi entropy IAB = S 2(A) + S 2(B) - S 2(AB) Mott insulator Mutual Information IAB U/J Superfluid Boundary Area

Non equilibrium: Quench dynamics H Beam splitter

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. Ψ 1 Ψ 2 • Overlap of two wave functions Sensitivity to perturbation signaling quantum phase transitions. • Higher order Renyi entropies by interfering more than two copies.

A ‘quantum gas microscope’ for ions

Thank you!! Theory Experiments Harvard Philipp Preiss Ruichao Ma Eric Tai Matthew Rispoli Alex Lukin Markus Greiner Maryland 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) 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