Back: Brad Blakestad, Till Rosenband, Jonathan Home, Jim Bergquist, Hermann Uys, Michael Biercuk, David Wineland, John Bollinger, Yves Colombe; Middle: Joe Britton, Luca Lorini, John Jost, Sarah Bickman, James Chou, David Hanneke, Wayne Itano, Bob Drullinger; Front: Manny Knill, Dietrich Leibfried, Aaron VanDevender, Jason Amini, Christian Ospelkaus, Kenton Brown; Photo taken October 3, 2008.

Quantum Simulation Possibilities with 2-d Ion Arrays in a Penning trapJohn Bollinger

NIST ion storage groupBoulder, CO

Aspen ProgramWorkshop on Quantum Simulation/Computation with Cold Atoms and Molecules

Overarching Themes:I: What are the key outstanding problems from condensed matter physics which ultracold atoms and molecules can address?

II: What new many-body aspects of ultracold atoms and molecules require new techniques and new perspectives, in comparison to “traditional” solid state systems? What new insight can we obtain into issues in fundamental quantum mechanics and quantum information processing?

III: What are the main challenges for simulating quantum systems and using ultracold atoms and molecules for quantum information processing? What new simulation techniques on classical computers can be brought to bear on these challenges?

IV: What is the best way to perform a quantum computation in ultracold atoms and molecules with the appropriate fidelity? How does one then interrogate such a quantum simulation or “read out” the answer from such a quantum computer?

Aspen ProgramWorkshop on Quantum Simulation/Computation with Cold Atoms and Molecules

Overarching Themes:I: What are the key outstanding problems from condensed matter physics which ultracold atoms and molecules can address?

II: What new many-body aspects of ultracold atoms and molecules require new techniques and new perspectives, in comparison to “traditional” solid state systems? What new insight can we obtain into issues in fundamental quantum mechanics and quantum information processing?

III: What are the main challenges for simulating quantum systems and using ultracold atoms and molecules for quantum information processing? What new simulation techniques on classical computers can be brought to bear on these challenges?

IV: What is the best way to perform a quantum computation in ultracold atoms and molecules with the appropriate fidelity? How does one then interrogate such a quantum simulation or “read out” the answer from such a quantum computer?

Trapped Ions

Rf or Paul TrapRF & DCVoltages

good for tight confinement and laser cooling small numbers of particles;quantum computing;optical and microwave clocks; 1-d ion arrays

Penning Trap(or Penning-Malmberg trap)DC Voltages & Static B-Field

good for laser cooling larger numbers ;microwave clocks; cold plasma studies; 2-d and 3-d ion arrays

quantum simulation experiment: T. Schaetz etal., Nature Physics 4, 757 (2008)

Trapped Ions

Rf or Paul TrapRF & DCVoltages

good for tight confinement and laser cooling small numbers of particles;quantum computing;optical and microwave clocks; 1-d ion arrays

Penning Trap(or Penning-Malmberg trap)DC Voltages & Static B-Field

good for laser cooling larger numbers ;microwave clocks; cold plasma studies; 2-d and 3-d ion arrays

quantum simulation experiment: T. Schaetz etal., Nature Physics 4, 757 (2008)

Quantum simulation possibilities with trapped ions

Effective spin systems with trapped ionsPorras and Cirac, PRL 92, 207901 (2004)

Effective spin quantum phases in systems of trapped ionsDeng, Porras, and Cirac, PRA 72, 063407 (2005)

Quantum manipulation of trapped ions in 2-d Coulomb crystalsPorras and Cirac, PRL 96, 250501 (2006)

Quantum phases on interacting phonons in an ion trapDeng, Porras, and Cirac, PRA 77, 033403 (2008)

Mesoscopic spin-boson models of trapped ionsPorras, Marquardt, von Delft, and Cirac, PRA 78, 010101 (2008)

simulation of magnetic spin

systems

+

BxJmax

J

J

Bx=|J(t)|

ferromagnetic order

Simulating a quantum magnet with trapped ions, Friedenauer, Schmitz, Glueckert, Porras, Schaetz, Nature Physics 4, 757 (2008)

simulation of a 2-ion Ising chain in a transverse field

1mm

Ωrf ~2π x 60 MHzωradial

~2π x 7 MHzro ~400µm

1mm

|↓〉

|↑〉mf=1

mf=-2F=2

F=3

S1/2 (s=1/2, L=0)

P1/2 (S=1/2, L=0)

P3/2 (S=1/2, L=1)

25Mg+: I=5/2

[F=3,2]

[F=4,3,2]

mf=3mf=2

mf=1mf=0

mf=-1mf=-2

mf=-3

mf=-1mf=0

mf=2

mf=-4(mf=-4,….,+4)

(mf=-3,….,+3)

Simulating a quantum magnet with trapped ions, Friedenauer, Schmitz, Glueckert, Porras, Schaetz, Nature Physics 4, 757 (2008)

internal states of 25Mg+

2P1/2

2P3/2

2S1/2

|↓〉 1.79 GHz

~ 200 GHz

|↓〉 ≡ |F = 3, mF = -3〉 |↑〉 ≡ |F = 2, mF = -2〉

|↑〉

quantum state of motion(harmonic oscillator)

280 nm

|↓〉|1〉|↓〉|0〉

|↑〉|1〉|↑〉|0〉

~ 2750 GHz

Simulating a quantum magnet with trapped ions, Friedenauer, Schmitz, Glueckert, Porras, Schaetz, Nature Physics 4, 757 (2008)

25Mg+ laser transitions

2P1/2

2P3/2

2S1/2

|↓〉 1.79 GHz

~ 200 GHz

|↓〉 ≡ |F = 3, mF = -3〉 |↑〉 ≡ |F = 2, mF = -2〉

|↑〉

Detection (σ−)quantum state of motion(harmonic oscillator)

280 nm

|↓〉|1〉|↓〉|0〉

|↑〉|1〉|↑〉|0〉

~ 2750 GHz

Simulating a quantum magnet with trapped ions, Friedenauer, Schmitz, Glueckert, Porras, Schaetz, Nature Physics 4, 757 (2008)

25Mg+ laser transitions

2P1/2

2P3/2

2S1/2

|↓〉 1.79 GHz

~ 200 GHz

|↓〉 ≡ |F = 3, mF = -3〉 |↑〉 ≡ |F = 2, mF = -2〉

|↑〉

Detection (σ−)

Raman

quantum state of motion(harmonic oscillator)

280 nm

|↓〉|1〉|↓〉|0〉

|↑〉|1〉|↑〉|0〉

~ 2750 GHz

Simulating a quantum magnet with trapped ions, Friedenauer, Schmitz, Glueckert, Porras, Schaetz, Nature Physics 4, 757 (2008)

25Mg+ laser transitions

2P1/2

2P3/2

2S1/2

|↓〉 1.79 GHz

~ 200 GHz

|↓〉 ≡ |F = 3, mF = -3〉 |↑〉 ≡ |F = 2, mF = -2〉

|↑〉

Detection (σ−)

Raman

quantum state of motion(harmonic oscillator)

280 nm

|↓〉|1〉|↓〉|0〉

|↑〉|1〉|↑〉|0〉

~ 2750 GHz

repump

Simulating a quantum magnet with trapped ions, Friedenauer, Schmitz, Glueckert, Porras, Schaetz, Nature Physics 4, 757 (2008)

25Mg+ laser transitions

quantum-Ising model:

eff. magnetic field through resonant rf

Bx|↓〉

|↑〉

1.7 GHz rf

Simulating a quantum magnet with trapped ions, Friedenauer, Schmitz, Glueckert, Porras, Schaetz, Nature Physics 4, 757 (2008)

quantum-Ising model:

eff. magnetic field through resonant rf

Bx|↓〉

|↑〉

1.7 GHz rf

pulse duration

P

Simulating a quantum magnet with trapped ions, Friedenauer, Schmitz, Glueckert, Porras, Schaetz, Nature Physics 4, 757 (2008)

e.g. quantum-Ising model:

eff. magnetic field (global qubit-rotation)

1.7 GHz rf

Simulating a quantum magnet with trapped ions, Friedenauer, Schmitz, Glueckert, Porras, Schaetz, Nature Physics 4, 757 (2008)

e.g. quantum-Ising model:

eff. magnetic field (global qubit-rotation)

P3/2P1/2

S1/2

eff. spin-spin Interaction J(conditional optical dipole force)

1.7 GHz rf

Simulating a quantum magnet with trapped ions, Friedenauer, Schmitz, Glueckert, Porras, Schaetz, Nature Physics 4, 757 (2008)

e.g. quantum-Ising model:

eff. magnetic field (global qubit-rotation)

P3/2P1/2

S1/2

eff. spin-spin Interaction J(conditional optical dipole force)

F↓= -1.5 F↑

1.7 GHz rf

Simulating a quantum magnet with trapped ions, Friedenauer, Schmitz, Glueckert, Porras, Schaetz, Nature Physics 4, 757 (2008)

e.g. quantum-Ising model:

eff. magnetic field (global qubit-rotation)

Bx

eff. spin-spin Interaction J(conditional optical dipole force)

1.7 GHz rf

Simulating a quantum magnet with trapped ions, Friedenauer, Schmitz, Glueckert, Porras, Schaetz, Nature Physics 4, 757 (2008)

e.g. quantum-Ising model:

eff. magnetic field (global qubit-rotation)

Bx

eff. spin-spin Interaction J(conditional optical dipole force)

1.7 GHz rf

Simulating a quantum magnet with trapped ions, Friedenauer, Schmitz, Glueckert, Porras, Schaetz, Nature Physics 4, 757 (2008)

e.g. quantum-Ising model:

eff. magnetic field (global qubit-rotation)

Bx

eff. spin-spin Interaction J(conditional optical dipole force)

1.7 GHz rf

Simulating a quantum magnet with trapped ions, Friedenauer, Schmitz, Glueckert, Porras, Schaetz, Nature Physics 4, 757 (2008)

e.g. quantum-Ising model:

eff. magnetic field (global qubit-rotation)

Bx

eff. spin-spin Interaction J(conditional optical dipole force)

all parameters to be chosen individually:(e.g. amplitude, range, anti- or ferromagnetic phase …)

1.7 GHz rf

Simulating a quantum magnet with trapped ions, Friedenauer, Schmitz, Glueckert, Porras, Schaetz, Nature Physics 4, 757 (2008)

= + + +

ampl

itude

duration

?

Simulating a quantum magnet with trapped ions, Friedenauer, Schmitz, Glueckert, Porras, Schaetz, Nature Physics 4, 757 (2008)

+fidelity( )= 88%+- 3%

scaling up the number of ions- many ion strings in linear rf traps: Schaetz (MPI), Monroe (U. Md.), Leibfried (NIST), …

Large scale quantum computation in an anharmonic linear ion trap, Lin, Zhu, Islam, Kim, Chang, Korenblit, Monroe, Duan, arXiv:0901.0579

- rf trap arrays: Chiaverini (LANL), Leibfried (NIST), Chuang (MIT)…

Chiaverini and Lybarger, PRA 77, 022324 (2008)

- Penning trapoptimized electrode for a bilayer honeycomb lattice; Schmeid, Wesenberg, Leibfried, arXiv:0902.1686v1

NIST Penning trap

B=4.5 T

9Be+

νc ~ 7.6 MHzνz ~ 800 kHzνm ~ 50 kHz

NIST Penning trap

B=4.5 T

9Be+

νc ~ 7.6 MHzνz ~ 800 kHzνm ~ 50 kHz

NIST Penning trap

QuartzVacuum envelope

P < 10-10 Torr

4.5 Tesla Super Conducting

Solenoid

● thermal equilibrium ⇒ rigid rotation ωr

Equilibrium plasma propertiesDubin and O’Neil, Rev. Mod. Phys. 71, 87 (1999)

● thermal equilibrium ⇒ rigid rotation ωr

● T ∼ 0 ⇒ constant plasma density, no = 2εomωr(Ωc-ωr)/e2, Ωc = cyclotron frequency plasma density → 0 over a Debye length λD = [kBT/(4πnoe2)]1/2

Equilibrium plasma propertiesDubin and O’Neil, Rev. Mod. Phys. 71, 87 (1999)

● thermal equilibrium ⇒ rigid rotation ωr

● T ∼ 0 ⇒ constant plasma density, no = 2εomωr(Ωc-ωr)/e2, Ωc = cyclotron frequency plasma density → 0 over a Debye length λD = [kBT/(4πnoe2)]1/2 ● quadratic trap potential, eφT ~ mωz

2(z2-r2/2) ⇒ plasma shape is a spheroid

Equilibrium plasma properties

r

z

2ro

2zo

ωr

B

Dubin and O’Neil, Rev. Mod. Phys. 71, 87 (1999)

● thermal equilibrium ⇒ rigid rotation ωr

● T ∼ 0 ⇒ constant plasma density, no = 2εomωr(Ωc-ωr)/e2, Ωc = cyclotron frequency plasma density → 0 over a Debye length λD = [kBT/(4πnoe2)]1/2 ● quadratic trap potential, eφT ~ mωz

2(z2-r2/2) ⇒ plasma shape is a spheroid

aspect ratio α ≡ zo/ro determined by ωr

Equilibrium plasma properties

r

z

2ro

2zo

ωr

B

Ωc − ωm

Den

sity

no

ωm Ωc / 2Rotation frequency ωr

nB

Dubin and O’Neil, Rev. Mod. Phys. 71, 87 (1999)

● thermal equilibrium ⇒ rigid rotation ωr

● T ∼ 0 ⇒ constant plasma density, no = 2εomωr(Ωc-ωr)/e2, Ωc = cyclotron frequency plasma density → 0 over a Debye length λD = [kBT/(4πnoe2)]1/2 ● quadratic trap potential, eφT ~ mωz

2(z2-r2/2) ⇒ plasma shape is a spheroid

aspect ratio α ≡ zo/ro determined by ωr

Equilibrium plasma properties

r

z

2ro

2zo

ωr

B

Ωc − ωm

Den

sity

no

ωm Ωc / 2Rotation frequency ωr

nB

● ωr precisely controlled by a rotating electric field (rotating wall)

Dubin and O’Neil, Rev. Mod. Phys. 71, 87 (1999)

Experimental techniques

B

Experimental techniques

axialcooling beam

B

Experimental techniques

axialcooling beam

top-viewcamera

B

Experimental techniques

axialcooling beam

top-viewcamera

B

Bragg scattering

Experimental techniques

axialcooling beam

top-viewcamera

B

strobed Bragg

scattering

Experimental techniques

axialcooling beam

top-viewcamera

B

imaging of the ion crystals

Experimental techniques

axialcooling beam

top-viewcamera

B

imaging of the ion crystals

side-viewcamera

Side-view image: N ~ 1.8 k ions, Rotating wall spin-up from ωr/(2π) = 30 kHz to 90 kHz

0.37 mm

Side-view image: N ~ 1.8 k ions, Rotating wall spin-up from ωr/(2π) = 30 kHz to 90 kHz

0.37 mm

Ωc − ωm

Den

sity

no

ωm Ωc / 2Rotation frequency ωr

nB

Real-space images with planar plasmas

with planar plasmas all the ions can reside within the depth of focus

Ωc − ωm

Den

sity

no

ωm Ωc / 2Rotation frequency ωr

nB

1 lattice plane, hexagonal order

65.70 kHz

Real-space images with planar plasmas

with planar plasmas all the ions can reside within the depth of focus

Ωc − ωm

Den

sity

no

ωm Ωc / 2Rotation frequency ωr

nB

1 lattice plane, hexagonal order

65.70 kHz

2 planes, cubic order

66.50 kHz

Real-space images with planar plasmas

with planar plasmas all the ions can reside within the depth of focus

y

z

x

z1z2

z3

increasing rotation frequency →

Mitchell, et al., Science 282, 1290 (98)

y

z

x

z1z2

z3

increasing rotation frequency →

Theoretical curve from Dan Dubin, UCSDMitchell, et al., Science 282, 1290 (98)

● stick-slip motion of the crystal rotation!● frequency offset (ωr-ωwall) due to creep of 2 -18 mHz● regions of phase-locked separated by sudden slips in the crystal orientation● stick-slip motion due to competition between ? laser and rotating wall torques● mean time between slips ~10 s; what triggers the slips?

Single ion addressing?

Present experimental effort: ● spin squeezing through global addressing of a single plane of ions with an optical dipole force

● first step towards an Ising model simulation with > 100 ions

our qubit:(+3/2, +1/2)=| ↑ ›

(+3/2,-1/2)=| ↓ ›(+1/2,-1/2)(-1/2,-1/2)(-3/2,-1/2)

F = 1

F = 22s 2S1/2

(mI , mJ)

124 GHz

2p 2P1/2

2p 2P3/2

(mI, -1/2)

(mI, +1/2)

(mI, -3/2)

(mI, -1/2)

(mI, +1/2)

(mI, +3/2)

F = 1,2

F = 0,1,2,3

~ 40 GHz

~ 80 GHz

cooling Raman

Rabi flopping on 124 GHz electron spin flip

B

-V0

axialcooling beam

6.35cm

124 GHz Microwave Feed Horn

Teflon lens

60 c

m

Rabi time

0 1 2 3 4Ramsey Precession time (ms)

(µs)

coherence can be extended to 10’s of ms with spin echo (dynamical decoupling)

Ramsey (T2) coherence on 124 GHz electron spin flip

Biercuk, Uys, VanDevender, Shiga, Itano, Bollinger, Nature 458, 996 (2009)

Spin squeezing through single axis twisting

1. prepare , Tmotional ~ 0.5 mK

2. π/2 pulse of 124 GHz microwaves

RepumpLaser

z

y

z

y

x

Microwave

Kitigawa and Ueda, PRA 47, 5138 (1993)

3. Apply exp(iχ {Jz}2 t) “push” gate on the axial center-of-mass mode of a single ion plane

Raman beams

4. Measure ΔJz as a function of rotation about x-axis (mean spin vector direction)

Microwave with 90°Phase shift

z

y

z

y

D.Leibfried, et al., Science 304, 1476 (2004)

Spin squeezing through single axis twisting

νL+ωz+δ

νL

θ ~1º

Experimental parameters for squeezing interaction exp(iχ{Jz}2 t)

θ ~ ±1o ⇒λeff ~ 10 µm

N=100, νz= 800 kHz, T~ 0.5 mK

For θ ~± 1o, actual squeezing will be limited by spontaneous emission

Evidence for optical dipole excitation of the axial COM mode with N~200 ions

Motion towards beam, fluorescence is brighter

Motion away from beam, fluorescence is dimmer

axial cooling beam

axial COMmode

Raman beamson

800 µs

coolingbeams

on

oscillations in cooling beam fluorescence triggered on Raman beat note

Jz2 interaction permits the realization of a long range Ising interaction ?

constant

5 kHz

This interaction can be solved analytically. Does it exhibit a quantum phase transition?

Simulating this Hamiltonian could serve as a useful test of the fidelity with which we can simulate Ising models in a Penning trap

Steps for increasing χ - larger Raman beam power (x10 reasonable) - larger Raman beam angles can help – Lamb Dicke requirement eventually means colder temperatures - smaller Raman beam detuning (spontaneous emission?)

Ising interaction with intermediate range (a computationally complex system) can be generated by optical dipole forces which couple to many modes of the planar array.

Effective spin systems with trapped ionsPorras and Cirac, PRL 92, 207901 (2004)

For planar arrays and an adiabatic push

With our current parameters Ji,j tiny (~1 Hz)

require larger Raman beam powers, Raman beam angles

What about Raman beam detuning Δ?

How small can Δ be?

Quantum simulation with ~100 ion planar arrays looks feasible – laser power requirements are challenging

Summary

Features of trapped ion simulation

- trapped ions naturally permit the realization of quantum spin Hamiltonians

- hexagonal 2-d lattice enables the study of spin frustration

- the state of individual ions can be measured with high signal-to-noise ratio which permits direct measurement of the spin correlation fuctions

- trapped ion simulations can be done with small ion numbers through several hundred ions; the basic building blocks of strongly interacting systems quantum systems can be studied and understood, helping with the measurement and interpretation of larger scale systems

- general 2-d lattice symmetries can be achieved with properly designed surface electrode rf trap arrays

Slowheatingatshorttimes:50-100mK/sduetoresidualgascollisions

Ion heating rate

How much time is there for entangling the ions?

longer ??

Jensen, et. al., PRL 94, Jan. 05; PRA 70, (2004)

The data agrees with the theoretical dispersion relationship for drumhead waves in a plasma slab of thickness 2Zp.

Dispersion relationship

drum head mode frequencies (laboratory frame, fluid model)D.H.E. Dubin, PRL 66, 2076 (1991)

Weimer, Bollinger, Moore, Wineland, PRA 49, 3842 (1994)

N=100, νz=870 kHz, near 1-2 plane transition

|ω1,0| = ωz

|ω2,1 | = ωz +/- ωr-(1/8)παωz

|ω3,0 | = ωz [1-(5/16)παωz]

2Rp

2Zp

α= Zp/Rp

= 0.075