Gregynog QIP meeting QIP Experiments with ions, atoms and molecules Christopher Foot, University of Oxford c.foot@physics.ox.ac.uk.

Gregynog QIP meeting

QIP Experiments with ions, atoms and molecules

Christopher Foot, University of Oxfordc.foot@physics.ox.ac.uk

Objectives of these lectures:

Describe QIP experiments that use atomic and molecular physics (not including NMR)

• General requirements for a quantum gate between two qubits – spin-dependent interaction

• Review experimental techniques and some current experiments with ion in traps

• Neutral atoms in optical lattices: simulation of condensed matter physics and QIP applications

• Recent ideas for QIP with polar molecules = hybrid of atoms and ions

Quantum gate - controlled operation

• CNOT gate:

• CROT gate (or controlled-Z gate)

Exercise: Show how to construct a CNOT gate from a controlled-Z gate

and 2 Hadamard gates. (Ex 4.17 in Nielson & Chuang)

CROT gate (or controlled-Z gate) based on state-dependent (spin-dependent) interaction

or

Interaction

0 01 1

where

Qubit A Qubit B

Comment on quantum gates

Equivalent to

Difficult to implement since requires very precise controlof experimental timing.

`Pushing gate’– laser beam exerts state-dependent force on ions

Harmonic trapping potential

RepulsiveCoulomb force

1

0

E

Phase factor

Phase difference

Cf. `wobble gate’ discussed later

Repulsive

Attractive

Summary of Lecture 1: Ions

1. Requirements for QIP2. Ion trap principles3. Read out and Quantum jumps4. Manipulation of single qubit by Raman transitions5. Laser cooling to the lowest vibrational level6. Current experimental capability in Oxford and

elsewhere7. Survey of ideas being actively explored in

experimental groups8. How to make a computer,

Requirements for quantum computing

• Excellent quality qubits

Q = Tcoherence / Tgate = 106 now realised for ions

• move qubits around quickly and without error

• High precision gate between neighbours (error = 10-4 )• Single-qubit gates • Measurement of qubits (read out)

D. Deutsch, Proc R Soc Lond A 400, 97 (1985) & 425, 73 (1989).A.Steane, Phys. Rev. A 68, 042322 (2003) & Quant. Inf. Comp. 2, 297 (2002).

*

Basic ion trap methods

N.B.Most of the slides in this lecture come from the Ion trapping group in Oxford (part of the IRC)

Paul Trap

r.f. quadrupole+ ‘end caps’++

trapped ions1000 volts,10 MHz

end view

Vx

tim e‘=’

Axial confinement by electrostatic (quadratic) potential.Radial confinement by oscillating quadrupole potential.

Electrostatic trapping not possible, see Foot, Atomic Physics, OUP 2005

The trap

Axial motional freq. of order MHz

Ion-electrode distance = 0.1 to 1 mm

7 mm

Alkali-like ions

Choice of ion:Want simple energy level structure when singly-ionisedGroup II or other metals:X = Be, Ca, Sr, Ba, Yb, Cd, Hg

Ground configuration of X+ ion has electron spin s = ½.

Hyperfine structure arising from interaction of nuclear spin (nuclear magnetic moment) withmagnetic field created by the unpaired electron.

The hyperfine structure of Ca-43, the ion used in Oxford experiments, is inverted

S 1/2

F=3

F=4 m F=0

3.2 G H z

m F=+4

B field-independent

1st order Zeeman sensitive

• Use transition with no first-order Zeeman effect, which is therefore insensitive to magnetic field fluctuations,cf. atomic clocks.• Qubit coherence time of order seconds (see later)

Readout by fluorescence on cycling transition

2S1/2

F=1, MF=1

F=2, MF=21.25 GHz

2P3/2

2P1/2

F=3, MF=3

E.g. Be+Selection rules cycling transition

PM tube or CCD camera

Electron shelving or “quantum jumps”

0 1 2 3 4 5 6 7 8 90

100

200

300

400

500

600

700(d)

C

ount

s pe

r 8

ms

Time (s)

0 1 2 3 4 5 6 7 8 90

100

200

300

400(c)

Observed fluorescence signals from one (a,b), two (c) and three ions (d). (a): random telegraph (850 nm laser is left permanently on)(b-d): controlled shelving (850 nm laser pulsed on when desired)

Cou

nts

per

8 m

s

Time (s)

0 1 2 3 4 5 6 7 8 90

50

100

150

200

250

300(b)

Cou

nts

per

8 m

sTime (s)

0 1 2 3 4 5 6 7 8 90

100

200

300

400

500 (a)

Time (s)

Cou

nts

per

20 m

s

Time needed to measure a qubit at 99.9% fidelity

• Collection & photon detection efficiency = 0.02• excited state lifetime = 5 ns• required photon count for P(error) < 0.001 is 10

photons• time to count 10 photons = 2 x 10 / = 5 s

• Current experiments allow 100 s to 1 ms

Poissonian distribution of photon count

Scattering rate

The trap

Excellent signal-to-noise in detection of individual ions

7 mm

Comment: There is well-developed and efficient scheme for reading out the state of ions. This is not yet achieved for neutral atoms or molecules.

Single bit rotations

Microwaves:U on all qubits at once

Stimulated Raman transition:U on a chosen individual qubit

1-bit gate time' 1 micro-second

Raman transition

2S1/2

F=1, MF=1

F=2, MF=21.25 GHzhyperfine structure

313 nm

2P3/2

2P1/2

AOM: shift 1.25 GHZ

Very high-precisionand stable phase

Raman transition: effective 2-level system

1

2

100 GHz

500 MHz500 MHz

10 MHz

photon scattering = 10-4

= 1.25 MHz

(unwanted) photon scattering rate

2S1/2

F=1, MF=1

F=2, MF=2

2P3/2

2P1/2

Qubit decoherence per gate time

Photons scattered per gate time

PRL 95, 030403 (2005)

Trapped atom: quantum simple harmonic motion

z

z

12

0

z

12

0

0

0

Laser cooling of trapped ions: “Sideband cooling”

3

2

0

L 0 z

12

0

1

3

<< 1

Resultant thermal distribution:

Laser cooling a trapped ion very different to cooling of free atoms

Measure temperature of trapped ion

10

12

2

Pg

L

Compare excitation probability of first red sideband and first blue sideband

0Cf Fig. 12.10 in Foot (2005)

Laser cooling of trapped ions: “Sideband cooling”

3

2

0

L 0 z

12

0

1

3

Experimentalresults:

2005 Oxford 0.02

<< 1

Current experimental capability in Oxford

Dr David LucasProf. Andrew SteaneProf. Derek Stacey…………….N.B.Most of the slides in this lecture come from the Ion trapping group in Oxford (part of the IRC)

Main results

VERY LONG COHERENCE

DETERMINISTIC ENTANGLEMENTOF ION SPIN QUBITS

Very long (1s) qubit coherence time

S 1/2

F=3

F=4 m F=0

3.2 G H z

m F=+4

field-independent,prepare with ~15% efficiency

1st order Zeeman sensitive,prepare with 100% efficiency

• Readout by shelving with 95% efficiency

• Rabi flopping and Ramsey experiments using 3.2 GHz microwaves

Ca-43 hyperfine qubit

Rabi flopping

microwave pulse length (ms)

data fit

P(F

=4,

MF=

0)

Coherence (T2 ) time control experiment (small delay) Ramsey experiment (long delay)

Detuning (Hz) from 3,225,611,696

Fringe visibility vs. delay

Ramsey fringes

= 0.8(2) s

Ramsey gap (ms)

Long-lived memory qubit

p/2 p/2 p/2 p/2854

393

854

393

PMT

PMT0.1ms 300ms

Dop

ple r

cool

prep

are

shor

tR

amse

yex

pt

shel

ve

dete

ct A

Observe many (~270) Rabi flops on the m =0 ->0 field-independent transition, lasting >30ms.

F

A single spin-echo pulse can be used to protect the memory qubit from the residual field-sensitivity. We detect no decoherence in a 1 second experiment (Ramsey fringe , right), implying an effective coherence time of

.

contrast >99%=10s–100sT2

SE

Spin

-dow

n (F

=4) P

opul

atio

n (a

.u.)

Spin-echo

p/2

p/2

( )

p

1s

T2

SE > 10s

Exercise

Show why the pulse sequence /2 - - /2 is more

robust than a Ramsey experiment (/2 - /2 sequence)

Also called Square root of NOT gate

Cf Exercise 7.3 in Foot, Atomic Physics

NOT gate:

Entanglement

Spin-dependent oscillating force

Dipole force in standing wave

B

50 m

Raman beams

F

• Laser standing wave produces an oscillating force on a pair of ions

• Robust and “fast”• To be described in detail later

+

( Explain how state-dependence of the force arises in nextlecture. Force/potential depends on the polarization of the light. )

*

Leibfried or “wobble” gate

D. Leibfried et. al. Nature 422, 412 (2003)

Equivalent to controlled-not

Really nice gate

1. Only the area, not the shape of the loop matters

2. Non-zero ion temperature?– just displace the starting point,– same loop no problem

3. Shift of laser standing wave phase?– rotate the loop about the initial point– same loop no problem

Deterministic entanglement by phase gate

spin qubit (40Ca ground state)

stretch mode freq. 866 kHzion separation 9 m (= 22

<n> = 0.35 , < 0.1

gate time 77 s(s = 67) fidelity 75 (5) %

cool

| ↓ ↓

2, )2

measure

2

analysis pulseprepare

gate

Wobble gate results

July 2005 data:

use twin loop,

fidelity 82(2)%

limited largely by photon scattering( = 30 GHz)and laser intensity noise

D. Lucas, M. McDonnell, S. Webster, J. Home, B. Keitch, D. Stacey, A. Ramos, A. Steane

Tomography

Deduced density matrix

hence entanglement of formation E = 0.52

Two-qubit gates

• Use laser-driven oscillatory motion of ions:• Current experiments (2 to 8 ions):

– fidelity » 90% (97% reported)– gate time » 10 to 100 s

• Future:– fidelity 99.99% (10-4 ) with good (bright and

stable) lasers

– time 100 ns to 1 s

Requirements for quantum computing

• Excellent quality qubits

Q = Tcoherence / Tgate = 106 now realised for ions

• move qubits around quickly and without error

• High precision gate between neighbours (error = 10-4 )• Single-qubit gates • Measurement of qubits (read out)D. Deutsch, Proc R Soc Lond A 400, 97 (1985) & 425, 73 (1989).

A.Steane, Phys. Rev. A 68, 042322 (2003) & Quant. Inf. Comp. 2, 297 (2002).

Moving information around the machine

m

Logical information encoded in large groups of ions.QEC uses a lot of parallel ops. (Animated version, Chuang website.)

Kielpinski, Monroe, Wineland, Nature 417,709 (2002)

Moving ions around

-Already mentioned by Prof Knight

7-zone trap, Oxford/Liverpool collaboration

ion-electrode distance = 0.7 mmtrap-trap separation = 0.8 mmtest open design conceptBuilt by University of Liverpool, S. Taylor

Moving quantum information around

• Array of ion traps with ions transported between traps

• Possibly use large numbers of ions in same trap

• Possibly use ion—photon coupling

Ion entangled with photon (2004)

March 2004Blinov,Moehring,DuanandMonroe

Scalable computer by cluster methods

Duan et al, Quant. Inf. Comp. 4, 165 (2004)

multi-qubit controlled entanglement

4,5,6- ion “cat” state |0000 + |1111F > 0.76, W< 0.51 |00000 + |11111 F > 0.60, W< 0.2 |000000 + |111111F > 0.51, W< 0.02NIST, Boulder, USA

3 to 8- ion “W” state

|001 + |010 + |100 F = 0.82, W= 0.53 |0001 + |0010+|0100 + |1000 F = 0.60, W= 0.46 ...

|00000001+|00000010+F = 0.72, W= 0.029

Innsbruck, Austria

Requirements for quantum computing

• Excellent quality qubits

Q = Tcoherence / Tgate = 106 now realised for ions

• move qubits around quickly and without error

• High precision gate between neighbours (error = 10-4 )• Single-qubit gates • Measurement of qubits (read out)

Conclusion:Ion trapping fulfils these requirements.No obvious `roadblock’ in the way of QIP with ions,`just’ requires development of technology

How to make a computer

quant-ph 2004