Jonathan P. Dowling

LINEAR OPTICAL QUANTUM INFORMATION PROCESSING, IMAGING, AND SENSING:

WHAT’S NEW WITH N00N STATES?

quantum.phys.lsu.edu

Hearne Institute for Theoretical PhysicsLouisiana State UniversityBaton Rouge, Louisiana

14 JUNE 2007 ICQI-07, Rochester

H.Cable, C.Wildfeuer, H.Lee, S.Huver, W.Plick, G.Deng, R.Glasser, S.Vinjanampathy, K.Jacobs, D.Uskov, JP.Dowling, P.Lougovski,

N.VanMeter, M.Wilde, G.Selvaraj, A.DaSilva

Not Shown: M.A. Can, A.Chiruvelli, GA.Durkin, M.Erickson, L. Florescu, M.Florescu, M.Han, KT.Kapale, SJ. Olsen, S.Thanvanthri, Z.Wu, J.Zuo

Hearne Institute for Theoretical Physics

Quantum Science & Technologies Group

Outline

1.1.Quantum Computing & Projective Quantum Computing & Projective

MeasurementsMeasurements

2.2.Quantum Imaging, Metrology, & Quantum Imaging, Metrology, &

SensingSensing

3.3.Showdown at High N00N!Showdown at High N00N!

4.4.Efficient N00N-State Generating Efficient N00N-State Generating

SchemesSchemes

5.5.ConclusionsConclusions

CNOT with Optical Nonlinearity

The Controlled-NOT can be implemented using a Kerr medium:

Unfortunately, the interaction (3) is extremely weak*: 10-22 at the single photon level — This is not practical!

*R.W. Boyd, J. Mod. Opt. 46, 367 (1999).

R is a /2 polarization rotation,followed by a polarization dependentphase shift .

R is a /2 polarization rotation,followed by a polarization dependentphase shift .

(3)

Rpol

PBS

z

|0= |H Polarization|1= |V Qubits|0= |H Polarization|1= |V Qubits

Two Roads to Optical CNOT

Cavity QED

I. Enhance Nonlinearity with Cavity, EIT — Kimble, Walther, Haroche, Lukin, Zubairy, et al.

I. Enhance Nonlinearity with Cavity, EIT — Kimble, Walther, Haroche, Lukin, Zubairy, et al.

II. Exploit Nonlinearity of Measurement — Knill, LaFlamme, Milburn, Franson, et al.

II. Exploit Nonlinearity of Measurement — Knill, LaFlamme, Milburn, Franson, et al.

210 γβα ++ 210 γβα −+

Linear Optical Quantum Computing

Linear Optics can be Used to Construct CNOT and a Scaleable Quantum Computer:

Knill E, Laflamme R, Milburn GJNATURE 409 (6816): 46-52 JAN 4 2001

Knill E, Laflamme R, Milburn GJNATURE 409 (6816): 46-52 JAN 4 2001

Franson JD, Donegan MM, Fitch MJ, et al.PRL 89 (13): Art. No. 137901 SEP 23 2002

Franson JD, Donegan MM, Fitch MJ, et al.PRL 89 (13): Art. No. 137901 SEP 23 2002

Milburn

Road to Entangled- Particle

Interferometry:

An Early Example of

Entanglement Generation by Erasure of Which-Path Information Followed by Detection!

Road to Entangled- Particle

Interferometry:

An Early Example of

Entanglement Generation by Erasure of Which-Path Information Followed by Detection!

Photon-PhotonXOR Gate

Photon-PhotonNonlinearity

Kerr Material

Cavity QEDEIT

Cavity QEDEIT

ProjectiveMeasurement

  LOQC  KLM

  LOQC  KLM

WHY IS A KERR NONLINEARITY LIKE A PROJECTIVE MEASUREMENT?

GG Lapaire, P Kok, JPD, JE Sipe, PRA 68 (2003) 042314

GG Lapaire, P Kok, JPD, JE Sipe, PRA 68 (2003) 042314

KLM CSIGN Hamiltonian Franson CNOT Hamiltonian

NON-Unitary Gates Effective Unitary GatesNON-Unitary Gates Effective Unitary Gates

A Revolution in Nonlinear Optics at the Few Photon Level:No Longer Limited by the Nonlinearities We Find in Nature! 

A Revolution in Nonlinear Optics at the Few Photon Level:No Longer Limited by the Nonlinearities We Find in Nature! 

Projective Measurement Yields Effective “Kerr”!

Nonlinear Single-Photon Quantum Non-

DemolitionYou want to know if there is a single photon in mode b, without destroying it.You want to know if there is a single photon in mode b, without destroying it.

*N Imoto, HA Haus, and Y Yamamoto, Phys. Rev. A. 32, 2287 (1985).

Cross-Kerr Hamiltonian: HKerr = a†a b†b

Again, with  = 10–22, this is impossible.

Kerr medium

“1”

a

b|in|1

|1

D1

D2

Linear Single-PhotonQuantum Non-Demolition

The success probability is less than 1 (namely 1/8).

The input state is constrained to be a superposition of 0, 1, and 2 photons only.

Conditioned on a detector coincidence in D1 and D2.

|1

|1

|1

D1

D2

D0

/2

/2

|in = cn |nn = 0

2

|0Effective  = 1/8

21 Orders of Magnitude Improvement!

Effective  = 1/8

21 Orders of Magnitude Improvement!

P Kok, H Lee, and JPD, PRA 66 (2003) 063814

P Kok, H Lee, and JPD, PRA 66 (2003) 063814

Outline

1.1.Quantum Computing & Projective Quantum Computing & Projective

MeasurementsMeasurements

2.2.Quantum Imaging, Metrology, & Quantum Imaging, Metrology, &

SensingSensing

3.3.Showdown at High N00N!Showdown at High N00N!

4.4.Efficient N00N-State Generating Efficient N00N-State Generating

SchemesSchemes

5.5.ConclusionsConclusions

+NA0BeiNϕ0ANB

1 + cos N ϕ

2

1 + cos ϕ

2

ABϕ|N⟩A |0⟩B N Photons

N-PhotonDetector

ϕ = kxΔϕ: 1/√N →1/ΝuncorrelatedcorrelatedOscillates N times as fast!N-XOR GatesN-XOR Gatesmagic BSMACH-ZEHNDER INTERFEROMETERApply the Quantum Rosetta Stone!

Quantum Metrology with N00N StatesH Lee, P Kok,

JPD, J Mod Opt 49, (2002) 2325.

H Lee, P Kok, JPD,

J Mod Opt 49, (2002) 2325.

Supersensitivity!

Shotnoise to Heisenberg

Limit

N Photons

N-PhotonDetectorϕ = kx+NA0BeiNϕ0ANB

1 + cos N ϕ

2

1 + cos ϕ

2

uncorrelatedcorrelatedOscillates in REAL Space!N-XOR Gatesmagic BSFROM QUANTUM INTERFEROMETRYTO QUANTUM LITHOGRAPHY

Mirror

N

2A

N

2B

LithographicResist

ϕ →Νϕ λ →λ/Ν

ψ a†

a†

aa ψa† N a

N

AN Boto, DS Abrams, CP Williams, JPD, PRL 85 (2000) 2733

AN Boto, DS Abrams, CP Williams, JPD, PRL 85 (2000) 2733

Superresolution!

Quantum Lithography Experiment

|20>+|02>

|10>+|01>

Canonical Metrology

note the square-root

P Kok, SL Braunstein, and JP Dowling, Journal of Optics B 6, (2004) S811

P Kok, SL Braunstein, and JP Dowling, Journal of Optics B 6, (2004) S811

Suppose we have an ensemble of N states | = (|0 + ei |1)/2,and we measure the following observable:

The expectation value is given by: and the variance (A)2 is given by: N(1cos2)

A = |0 1| + |1 0|

|A| = N cos

The unknown phase can be estimated with accuracy:

This is the standard shot-noise limit.

= = A

| d A/d |

N1

Quantum Lithography & Metrology

Now we consider the state

and we measure

High-FrequencyLithographyEffect

Heisenberg Limit:No Square Root!

P. Kok, H. Lee, and J.P. Dowling, Phys. Rev. A 65, 052104 (2002).

P. Kok, H. Lee, and J.P. Dowling, Phys. Rev. A 65, 052104 (2002).

Quantum Lithography*:

Quantum Metrology:

N |AN|N = cos N

H = = AN

| d AN/d |

N1

€

AN = 0,N N,0 + N,0 0,N

€

ϕ N = N,0 + 0,N( )

Outline

1.1.Quantum Computing & Projective Quantum Computing & Projective

MeasurementsMeasurements

2.2.Quantum Imaging, Metrology, & Quantum Imaging, Metrology, &

SensingSensing

3.3.Showdown at High N00N!Showdown at High N00N!

4.4.Efficient N00N-State Generating Efficient N00N-State Generating

SchemesSchemes

5.5.ConclusionsConclusions

Showdown at High-N00N!

|N,0 + |0,NHow do we make High-N00N!?

*C Gerry, and RA Campos, Phys. Rev. A 64, 063814 (2001).

With a large cross-Kerr nonlinearity!* H =  a†a b†b

This is not practical! — need  =  but  = 10–22 !

|1

|N

|0

|0

|N,0 + |0,N

N00N StatesIn Chapter 11

ba33

a

b

a’

b’

ba06

ba24

ba42

ba60

Probability of success:

64

3 Best we found:

16

3

Solution: Replace the Kerr with Projective Measurements!

H Lee, P Kok, NJ Cerf, and JP Dowling, Phys. Rev. A 65, R030101 (2002).

H Lee, P Kok, NJ Cerf, and JP Dowling, Phys. Rev. A 65, R030101 (2002).

ba13

ba31

single photon detection at each detector

''''4004

baba−

CascadingNotEfficient!

OPO

These Ideas Implemented in Recent

Experiments!

|10::01>

|20::02>

|40::04>

|10::01>

|20::02>

|30::03>

|30::03>

A statistical distinguishability based on relative entropy characterizes the fitness of quantum states for phase estimation. This criterion is used to interpolate between two regimes, of local and global phase distinguishability.

The analysis demonstrates that, in a passive MZI, the Heisenberg limit is the true upper limit for local phase sensitivity — and Only N00N States Reach It!

A statistical distinguishability based on relative entropy characterizes the fitness of quantum states for phase estimation. This criterion is used to interpolate between two regimes, of local and global phase distinguishability.

The analysis demonstrates that, in a passive MZI, the Heisenberg limit is the true upper limit for local phase sensitivity — and Only N00N States Reach It!

N00N

Local and Global Distinguishability in Quantum InterferometryGA Durkin & JPD, quant-ph/0607088

NOON-States Violate Bell’s Inequalities

Building a Clauser-Horne Bell inequality from the expectation values

€

Pab (α ,β ),Pa (α ),Pb (β )

€

−1≤ Pab (α ,β ) − Pab (α , ′ β ) + Pab ( ′ α ,β ) + Pab ( ′ α , ′ β ) − Pa ( ′ α ) − Pb (β ) ≤ 0

Probabilities of correlated clicks and independent clicks

€

Pab (α ,β ),Pa (α ),Pb (β )

CF Wildfeuer, AP Lund and JP Dowling, quant-ph/0610180

Shared Local Oscillator Acts As Common Reference Frame!

Bell Violation!

Outline

1.1.Quantum Computing & Projective Quantum Computing & Projective

MeasurementsMeasurements

2.2.Quantum Imaging, Metrology, & Quantum Imaging, Metrology, &

SensingSensing

3.3.Showdown at High N00N!Showdown at High N00N!

4.4.Efficient N00N-State Generating Efficient N00N-State Generating

SchemesSchemes

5.5.ConclusionsConclusions

Efficient Schemes for Generating N00N

States!

Question: Do there exist operators “U” that produce “N00N” States Efficiently?

Answer: YES!

H Cable, R Glasser, & JPD, quant-ph/0704.0678. Linear!N VanMeter, P Lougovski, D Uskov, JPD, quant-ph/0612154. Linear!KT Kapale & JPD, quant-ph/0612196. (Nonlinear.)

Question: Do there exist operators “U” that produce “N00N” States Efficiently?

Answer: YES!

H Cable, R Glasser, & JPD, quant-ph/0704.0678. Linear!N VanMeter, P Lougovski, D Uskov, JPD, quant-ph/0612154. Linear!KT Kapale & JPD, quant-ph/0612196. (Nonlinear.)

Constrained Desired

|N>|0> |N0::0N>

|1,1,1> NumberResolvingDetectors

linear optical processing

U(50:50)|4>|4>

0

0.05

0.1

0.15

0.2

0.25

0.3

|0>|8> |2>|6> |4>|4> |6>|2> |8>|0>

Fock basis state

|amplitude|^2

How to eliminate the “POOP”?

beamsplitter

quant-ph/0608170 G. S. Agarwal, K. W. Chan,

R. W. Boyd, H. Cable and JPD

Quantum P00Per Scooper!

χ

2-mode squeezing process

H Cable, R Glasser, & JPD, quant-ph/0704.0678.

OPO

Old Scheme

New Scheme

Spinning glass wheel. Each segment a different thickness. N00N is in Decoherence-Free

Subspace!

Spinning glass wheel. Each segment a different thickness. N00N is in Decoherence-Free

Subspace!

Generates and manipulates special

cat states for conversion to N00N states.

First theoretical scheme scalable to

many particle experiments!

Generates and manipulates special

cat states for conversion to N00N states.

First theoretical scheme scalable to

many particle experiments!

“PizzaPie”Phase Shifter

Feed-Forward-Based Circuit

Quantum P00Per Scoopers!H Cable, R Glasser, & JPD, quant-ph/0704.0678.

Linear-Optical Quantum-State Generation: A N00N-State Example

N VanMeter, D Uskov, P Lougovski, K Kieling, J Eisert, JPD, quant-ph/0612154

U

€

2

€

2

€

2

€

0

€

1

€

0

€

0.03

2( 50 + 05 )

This counter example disproves the N00N Conjecture: That N Modes Required for N00N.

This counter example disproves the N00N Conjecture: That N Modes Required for N00N.

The upper bound on the resources scales quadratically!

Upper bound theorem:

The maximal size of a N00N state generated in m modes via single photon detection in m–2 modes is O(m2).

Upper bound theorem:

The maximal size of a N00N state generated in m modes via single photon detection in m–2 modes is O(m2).

Conclusions

1.1.Quantum Computing & Projective Quantum Computing & Projective

MeasurementsMeasurements

2.2.Quantum Imaging & MetrologyQuantum Imaging & Metrology

3.3.Showdown at High N00N!Showdown at High N00N!

4.4.Efficient N00N-State Generating Efficient N00N-State Generating

SchemesSchemes

5.5.ConclusionsConclusions