linear optical quantum information processing, imaging...
TRANSCRIPT
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
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
Quantum Metrology with N00N StatesH Lee, P Kok, JPD,
J Mod Opt 49,(2002) 2325.
Supersensitivity!
Shotnoise toHeisenberg Limit
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
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
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
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
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
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
1.1. Quantum Computing & Projective MeasurementsQuantum Computing & Projective Measurements
2.2. Quantum Imaging & MetrologyQuantum Imaging & Metrology
3.3. Showdown at High N00N!Showdown at High N00N!
4.4. Efficient N00N-State Generating SchemesEfficient N00N-State Generating Schemes
5.5. ConclusionsConclusions