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HARVARD UNIVERSITY  MIT CENTER FOR ULTRACOLD ATOMS

Dive into the complexities of measuring entanglement entropy, from microscopic descriptions to simulating quantum matter. Discover the exponential growth of Hilbert space and the challenges of computing entanglement in interacting spin models. Explore quantum simulations using trapped ions, superconducting circuits, and more, to study quantum phase transitions and novel states of matter characterized by entanglement. Unravel the mysteries of entanglement entropy and its significance in understanding the quantum world.

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HARVARD UNIVERSITY  MIT CENTER FOR ULTRACOLD ATOMS

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  1. 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

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

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

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

  5. 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) Defects in diamonds Physics Today 67(10), 38(2014) Superconducting circuits Nature Physics 8, 292–299 (2012) 

  6. Quantum Simulation : Trapped Ions ‘Bottom-up’ approach 1 – 5 μm • Tunable interactions – quantum Ising, XY, 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).

  7. 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, Science340, 583 (2013).

  8. 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)

  9. 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: Shersonet al., Nature 467, 68 (2010)

  10. Single site parity Imaging

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

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

  13. 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)

  14. 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 1 10-1 10-2 10-3 Laguerre-Gauss profile

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

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

  17. 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. GreinerScience 347,1229(2015)

  18. 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?

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

  20. Entanglement Entropy TRACE A B Reduced density matrix: Entangled state  Mixed state Product state  Purestate Quantum purity = RenyiEntanglement Entropy

  21. Entanglement Entropy TRACE A B Reduced density matrix: Entangled state  Mixed state Product state  Purestate Quantum purity = 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)

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

  23. Beam splitter operation: Rabi flopping in a double well P (R) L R -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 +i

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

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

  26. 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)

  27. Quantum interference of bosonic many body systems

  28. Making two copies of a many-body state

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

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

  31. Entanglement in the ground state of a Bose-Hubbard system Mixed H Beam splitter Renyi entropy Purity Purity = Parity 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. Fukuharaet al, PRL 115, 035302 (2015)

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

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

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

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

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

  37. Non equilibrium: Quench dynamics H Beam splitter

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

  39. 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

  40. A ‘quantum gas microscope’ for ions

  41. 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) SimchaKorenblit (now at Hebrew University) Crystal Senko (now at Harvard) Jake Smith Aaron Lee Chris Monroe Guin-Dar Lin (Michigan) LumingDuan (Michigan) Joseph C.-C. Wang (Georgetown) Jim Freericks (Georgetown) Changsuk Noh (Auckland) Howard Carmichael (Auckland) Andrew Daley (strathclyde) Peter Zoller (Innsbruck) Eugene Demler (Harvard) Thank you!!

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