Lecture 40: MON 27 APR Physics 2102 Jonathan Dowling Ch. 36: Diffraction.
Jonathan P. Dowling
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Transcript of Jonathan P. Dowling
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