LINEAR OPTICAL QUANTUM INFORMATION PROCESSING, IMAGING ...phys.lsu.edu/~jdowling/talks/Rochester07.pdf · LINEAR OPTICAL QUANTUM INFORMATION PROCESSING, IMAGING, AND SENSING:

Jonathan P. Dowling

LINEAR OPTICAL QUANTUM INFORMATIONPROCESSING, IMAGING, AND SENSING:

WHAT’S NEW WITH N00N STATES?

quantum.phys.lsu.edu

Hearne Institute for Theoretical PhysicsLouisiana State University

Baton 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 PhysicsQuantum Science & Technologies Group

Outline

1.1. Quantum Computing & Projective MeasurementsQuantum Computing & Projective Measurements

2.2. Quantum Imaging, Metrology, & SensingQuantum Imaging, Metrology, & Sensing

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

4.4. Efficient N00N-State Generating SchemesEfficient N00N-State Generating Schemes

5.5. ConclusionsConclusions

CNOT with Optical Nonlinearity

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

Unfortunately, the interaction χ(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 π/2 polarization rotation,followed by a polarization dependentphase shift π.

χ(3)

Rpol

PBS

σz

|0〉= |H〉 Polarization|1〉= |V〉 Qubits

Two Roads to Optical CNOT

Cavity QED

I. EnhanceNonlinearity withCavity, EIT — Kimble,Walther, Haroche,Lukin, Zubairy, et al.

II. ExploitNonlinearity ofMeasurement — Knill,LaFlamme, Milburn,Franson, et al.

210 !"# ++ 210 !"# $+

Linear Optical Quantum Computing

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

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

Milburn

Road toEntangled-Particle

Interferometry:

An Early Exampleof EntanglementGeneration by

Erasure ofWhich-PathInformationFollowed byDetection!

Photon-PhotonXOR Gate

Photon-PhotonNonlinearity

Kerr Material

Cavity QEDEIT

ProjectiveMeasurement

LOQC KLM

WHY IS A KERR NONLINEARITY LIKE APROJECTIVE MEASUREMENT?

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

KLM CSIGN Hamiltonian Franson CNOT Hamiltonian

NON-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!

Projective MeasurementYields Effective “Kerr”!

Nonlinear Single-PhotonQuantum Non-Demolition

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

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

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

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

Kerr medium

“1”

a

b|ψin〉

|1〉|1〉

D1

D2

Linear Single-PhotonQuantum Non-Demolition

The success probability isless than 1 (namely 1/8).

The input state isconstrained to be asuperposition of 0, 1, and 2photons only.

Conditioned on a detectorcoincidence in D1 and D2.

|1〉

|1〉

|1〉D1

D2

D0

π /2

π /2

|ψin〉 = cn |n〉Σn = 0

2

|0〉Effective κ = 1/8→ 21 Orders of

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

Outline






Quantum Metrology with N00N StatesH Lee, P Kok, JPD,

J Mod Opt 49,(2002) 2325.

Supersensitivity!

Shotnoise toHeisenberg Limit

a† N a N

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

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

The expectation value is given by: and the variance (ΔA)2 is given by: N(1−cos2ϕ)

A = |0〉 1| + |1〉 0|〉〉

ϕ|A|ϕ〉 = N cos ϕ〉The unknown phase can be estimated with accuracy:

This is the standard shot-noise limit.

Δϕ = = ΔA

| d A〉/dϕ |〉

√N1

QuantumLithography & Metrology

Now we consider the state

and we measureHigh-FrequencyLithographyEffect

Heisenberg Limit:No Square Root!

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

Quantum Lithography*:

Quantum Metrology:

ϕN |AN|ϕN〉 = cos Nϕ〉

ΔϕH = = ΔAN

| d AN〉/dϕ |〉 N1

!

AN

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

!

"N

= N,0 + 0,N( )

Outline






Showdown at High-N00N!

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

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

With a large cross-Kerrnonlinearity!* H = κ a†a b†b

This is not practical! — need κ = π but κ = 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 withProjective Measurements!

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

ba13

ba31

single photon detection at each detector

''''4004baba

!

CascadingNotEfficient!

OPO

These Ideas Implemented inRecent Experiments!

|10::01>

|20::02>

|40::04>

|10::01>

|20::02>

|30::03>

|30::03>

A statistical distinguishability based on relative entropy characterizesthe fitness of quantum states for phase estimation. This criterion isused to interpolate between two regimes, of local and global phasedistinguishability.

The analysis demonstrates that, in a passive MZI, the Heisenberg limit isthe true upper limit for local phase sensitivity — and Only N00N StatesReach 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 expectationvalues

!

Pab(",#),P

a("),P

b(#)

!

"1# Pab($,%) " P

ab($, & % ) + P

ab( & $ ,%) + P

ab( & $ , & % ) " P

a( & $ ) " P

b(%) # 0

Probabilities of correlated clicks and independent clicks

!

Pab(",#),P

a("),P

b(#)

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

Shared Local Oscillator ActsAs Common Reference Frame!

Bell Violation!

Outline






Efficient Schemes forGenerating 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.)

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

|am

pli

tud

e|^

2

How to eliminatethe “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 adifferent thickness.

N00N is in Decoherence-Free Subspace!

Generates and manipulates specialcat states for conversion to N00N

states.

First theoretical scheme scalableto

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-StateGeneration: 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 disprovesthe N00N Conjecture: That NModes Required for N00N.

The upper bound on the resources scales quadratically!

Upper bound theorem:

The maximal size of a N00Nstate generated in m modes viasingle photon detection in m–2modes is O(m2).

Conclusions


2.2. Quantum Imaging & MetrologyQuantum Imaging & Metrology




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LINEAR OPTICAL QUANTUM INFORMATION PROCESSING, IMAGING ...phys.lsu.edu/~jdowling/talks/Rochester07.pdf · LINEAR OPTICAL QUANTUM INFORMATION PROCESSING, IMAGING, AND SENSING: