Quantum Simulations with Yb + crystal ~5  m Trapped Atomic Ions

Raman beatnotes:  HF   † upper sidebands frequency  HF +  carrier lower sidebands  HF  global spin-dependent oscillating force

† † phonons interaction between qubits (entangling gates etc..) evolution operator

“Adiabatically eliminate” phonons: |  k | >>    “SLOW MOLMER” General effective Hamiltonian theory: D. F. James, Canadian J. Phys. 85, 625 (2007) upper sidebands frequency carrier lower sidebands  kk sideband linewidth =

Raman beatnote:   HF   upper sidebands frequency  HF +  carrier lower sidebands  HF   HF  (  )  HF control normal mode eigenvectors (ion i mode m ) Ising Model global spin-dependent oscillating force

Quantum Simulation: What is it?  Describes N interacting systems, each system having D degrees of freedom D N coupled differential equations

Physical System  Trial H Physical System  Choose H Two approaches (1) (2)

Quantum simulations with trapped ions Porras and Cirac, PRL 92, (2004) Deng, Porras, Cirac, PRA 72, (2005) Taylor and Calarco, PRA 78, (2008) A. Friedenauer, et al., Nature Phys. 4, 757 (2008) K. Kim et al., Phys. Rev. Lett. 102, (2009) K. Kim et al., Nature 465, 590 (2010) E. Edwards et al., Phys. Rev. B 82, (2010) J. Barreiro et al., Nature 470, (2011) R. Islam, et al., Nature Comm. 2, 377 (2011) B. Lanyon et al., Science 334, 57 (2011) J. Britton et al., Nature 484, 489 (2012) A. Khromova et al., PRL 108, (2012) R. Islam, et al., Science 340, 583 (2013) P. Richerme, et al., ArXiv (2013) P. Richerme, et al., ArXiv (2013)

Frustration and Entanglement AFM Spin Liquids

1936: Giauque and Stout, “The Entropy of Water and the Third Law of Thermodynamics. Heat Capacity of Ice from 15 to 273°K” Zero-point entropy in 'spin ice’, A. P. Ramirez, A. Hayashi, R. J. Cava, R. Siddharthan and B. S. Shastry, Nature 399, 333 (1999) (pyrochloric “spin ice” Dy 2 Ti 2 O 7 ) 1945: L. Pauling The Nature of the Chemical Bond (Cornell Univ. Press), pp Ice

Control Range of Interaction! Theory Power Law Exponent  upper sidebands frequency  HF carrier lower sidebands   COM tune laser here tune laser here

Initialization Cooling Optical Pumping Spins along y (or against y) Detection Measure each spin along x time Adiabatic Quantum Simulation

A. Friedenauer, et al., Nature Phys. 4, 757 (2008) N=2

B/J rms  (  s) B/J rms  (  s) Exact Ground State Measured Populations J 12 =J 13 =J 23 < 0 Initialization Cooling Optical Pumping Spins along y Detection Measure each spin along x time P ↓↓↓ P ↓↓↑ P ↓↑↓ P ↓↑↑ P ↑↓↓ P ↑↓↑ P ↑↑↓ P ↑↑↑ P ↓↓↓, P ↑↑↑ E. Edwards, et al., Phys. Rev. B 82, (2010) N=3

B/J rms  (  s) B/J rms  (  s) J 12 =J 13 =J 23 > P ↓↓↓ P ↓↓↑ P ↓↑↓ P ↓↑↑ P ↑↓↓ P ↑↓↑ P ↑↑↓ P ↑↑↑ P ↓↓↓, P ↑↑↑ Initialization Cooling Optical Pumping Spins along y Detection Measure each spin along x time 0.2 E. Edwards, et al., Phys. Rev. B 82, (2010) Exact Ground State Measured Populations N=3

FM Ferromagnetic couplings FM J 12 =J 13 =J 23 < 0 K. Kim, et al., Nature 465, 590 (2010) |  = |  +|  ground state is entangled P 0 P 1 P 2 P 3 B x =0 |    = |  no entanglement |    = |  no entanglement Bx0Bx0 P 0 P 1 P 2 P 3 symmetry breaking field B x N=3

Competing AFM: spin frustration AFM J 12 =J 13 =J 23 > 0 ? K. Kim, et al., Nature 465, 590 (2010) |  = |  +|  +|  +|  +|  +|  ground state is entangled B x =0 P 0 P 1 P 2 P 3 |    = |  +|  +|  still entangled! symmetry breaking field B x |    = |  +|  +|  still entangled! Bx0Bx0 P 0 P 1 P 2 P 3 Frustration  Entanglement N=3

Emergence of ferromagnetism vs. # spins N (all FM couplings: J ij <0) |mx||mx| R. Islam et al., Nature Communications 2, 377 (2011) t(ms) B/|J| N

Ion index, j Time/τ B Long Range Antiferromagnetism (N=10) pair correlation G ,j = J i,j  1 |i  j| 1.1

Frustration and energy gaps Short range: exponent 1.5 Long range: exponent 0.5 Ground state Neel ordered: Abandoning adiabaticity probes frustration Low-lying energy states in antiferromagnetic model

Structure Function Spatial frequency k (2  ) Short range Long range R. Islam et al., Science 340, 583 (2013) Frustration of Magnetic Order (N=10)

Antiferromagnetic Néel order of N=10 spins All in state  2600 runs,  =1.12 AFM ground state order 222 events 441 events out of 2600 = 17% Prob of any state at random =2 x (1/2 10 ) = 0.2% 219 events R. Islam et al., Science 340, 583 (2013) All in state 

First Excited States (Pop. ~ 2% each)

Second Excited States (Pop. ~ 1% each)

Distribution of all 2 10 = 1024 states Probability Nominal AFM state B << J Probability Initial paramagnetic state B >> J 0 R. Islam et al., Science 340, 583 (2013)

Distribution of states ordered by energy (N=10) Energy/J 0 R. Islam et al., Science 340, 583 (2013)  = 1.12  = 0.86 Thermalization?? Cumulitive Prob

AFM order of N=14 spins (16,384 configurations)

= = At B y = 0: AFM Ising with AXIAL field

AFM Ising with AXIAL field

AFM Ground States 2-Bright Ground State 1-Bright Ground States 0-Bright Ground State P. Richerme, et al., ArXiv (2013) AFM Ising with AXIAL field

0-Bright Ground State 1-Bright Ground States 2-Bright Ground States 3-Bright Ground States 4-Bright Ground States 5-Bright (AFM) Ground States System exhibits a complete devil's staircase for N → ∞ P. Bak and R. Bruinsma, PRL 49, 249 (1982) P. Richerme, et al., ArXiv (2013) AFM Ising with AXIAL field

Modulate transverse B field to drive transitions between ground and excited states time ByBy J x i,j amplitude Dynamics: many-body spectroscopy C. Senko et. al., in preparation

time ByBy J x i,j amplitude Start from Drive to N = 6 Dynamics: many-body spectroscopy C. Senko et. al., in preparation

Start from Drive to N = 5 time ByBy J x i,j amplitude Dynamics: many-body spectroscopy C. Senko et. al., in preparation

Start from Drive to N = 5 Dynamics: many-body spectroscopy C. Senko et. al., in preparation

N = 5 Modulation frequency (kHz) Measurement Theory Spin states in order of energy Dynamics: many-body spectroscopy Complete spectrum of 5 spins C. Senko et. al., in preparation

C. Senko et. al., in preparation Dynamics: many-body spectroscopy N = 12

Modulation frequency (kHz) Measurement Theory C. Senko et. al., in preparation Dynamics: many-body spectroscopy N = 12

FM Population  = Drive system with all frequencies simultaneously (and control relative phases) Create equal superposition of single-spin flip states (W state) Dynamics: quantum engineering (FM: N=4) C. Senko et. al., in preparation

time ByBy J x i,j amplitude ++ +  = entangled  = 340°  = 160° Dynamics: quantum engineering (FM: N=4) C. Senko et. al., in preparation

“Ising Quench” (a)Prepare (↓+↑)  N “kT =  ” (b)Meaure correlations C midpoint, j (t) Dynamics: “light cone” of interaction propagation with long range interactions Theory: Z. Gong and A. Gorshkov (JQI)  =2.5 N=41  =1.5  =0.5 N=11 J 0 =0.5kHz  =0.81 shorter range longer range N=11 J 0 =0.5kHz  =1.3 neutrals (nearest-neighbor interactions): M. Cheneau et al., Nature 481, 484 (2012) E.H. Lieb and D.W. Robinson, “The finite group velocity of quantum spin systems,” Commun. Math. Phys. 28, 251–257 (1972).

Dynamics: L-R bounds with long range interactions Preliminary Data N=11 spins

Formation of localized defects: nonequilibrium dynamics M. Knap, E. Demler, I. Bloch (in preparation) XY model Spin-1: topological excitations Programmable fully connected spin network Up next… N beams, each with N spectral components interactions S. Korenblit, et al., New. J. Phys. 14, (2012)

Example: programming a 2D kagome lattice with a linear chain of 36 ions atom # spectral component Theory J. Garcia-Ripoll et al., Phys. Rev. A 71, (2005) S. Korenblit et al., ArXiv (2012)

GET MORE SPINS!! B/J = 0.01 B/J = 5 16 spin AFM simulation18 spin FM simulation m x = total spin along x Probability

N=16 () N=1 N=0 () Photon count histograms for N=16 ions # photons Global Spin Detection: 16 ions N=8

Quantum simulation with N=16 ions B>>J B ~ J B << J Ferro couplings Quantum Phase Transition Decreasing B/J FM/AFM order paramagnetic polarization Anti-ferro couplings G(1,j) Distance from 1 st ion, j B ~ J B << J N=16 () N=0 () Theoretical photon count histograms # photons N=8 B>>J Ferro couplings

~few 100 Be + ions in a Penning Trap J. Britton et al., Nature 484, 489 (2012) Quantum Hard-drive?

Going Cold: N>50

GaTech Res. Inst. Al/Si/SiO 2 Maryland/LPS GaAs/AlGaAs Sandia Nat’l Lab: Si/SiO 2 NIST-Boulder Au/Quartz

a (C.O.M.) b (stretch) c (Egyptian) d (stretch-2) Mode competition – example: axial modes, N = 4 ions Fluorescence counts Raman Detuning  R (MHz) a b c d a b c d 2a c-a b-a 2b,a+c b+c a+b 2a c-a b-a 2b,a+c b+c a+b carrier axial modes only mode amplitudes cooling beam D. Kielpinski, CM, D. Wineland, Nature 417, 709 (2002)

Large scale vision (10 3 – 10 6 atomic qubits?) New hierarchical and modular quantum computer architecture Different model for circuit optimization Error correction thresholds exist! (R. Raussendorf) C.M., et al., ArXiv (2012) Hz then, ~1 Hz now, ~1 kHz soon

ENIAC (1946)Solid-state transistor (1947)

right idea, wrong platform