Jonathan P. Dowling Quantum Optical Computing, Imaging, and Metrology quantum.phys.lsu.edu Hearne Institute for Theoretical Physics Quantum Science and Technologies Group Louisiana State University Baton Rouge, Louisiana USA AQIS, 31 AUG 10, University of Tokyo Dowling JP, “Quantum Optical Metrology — The Lowdown On High-N00N States,” Contemporary Physics 49 (2): 125-143 (2008).
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Jonathan P. Dowling
Quantum Optical Computing,Imaging, and Metrology
quantum.phys.lsu.edu
Hearne Institute for Theoretical PhysicsQuantum Science and Technologies Group
Louisiana State UniversityBaton Rouge, Louisiana USA
AQIS, 31 AUG 10, University of Tokyo
Dowling JP, “Quantum Optical Metrology — The Lowdown On High-N00NStates,” Contemporary Physics 49 (2): 125-143 (2008).
Sanders, PRA 40, 2417 (1989).Boto,…,Dowling, PRL 85, 2733 (2000).Lee,…,Dowling, JMO 49, 2325 (2002).
The Abstract Phase-Estimation ProblemEstimate , e.g. path-length, field strength, etc. withmaximum sensitivity given samplings with a total ofN probe particles.
Phase Estimation
Prepare correlationsbetween probes
Probe-systeminteraction DetectorN single
particles
Kok, Braunstein, Dowling, Journal of Optics B 6, (27 July 2004) S811
Strategies to improve sensitivity:
1. Increase — sequential (multi-round) protocol.
2. Probes in entangled N-party state and one trial
To make as large as possible —> N00N!
Theorem: Quantum Cramer-Rao bound
optimal POVM, optimal statistical estimator
Phase Estimation
S. L. Braunstein, C. M. Caves, and G. J. Milburn, Annals of Physics 247, page 135 (1996)V. Giovannetti, S. Lloyd, and L. Maccone, PRL 96 010401 (2006)
independent trials/shot-noise limit
!H
Optical N00N states in modes a and b ,Unknown phase shift on mode b so .
Cramer-Rao bound “Heisenberg Limit!”.
Phase Estimation
mode a
mode b phaseshift
paritymeasurement
Deposition rate:
Classical input :
N00N input :
Quantum Interferometric Lithography
source of two-modecorrelated
light
mirror
N-photonabsorbingsubstrate
phase difference along substrate
Boto, Kok, Abrams, Braunstein, Williams, and Dowling PRL 85, 2733 (2000)
Super-resolution, beating the single-photon diffraction limit.
!N "( ) = a† + e# i"b†( )N a + e+ i"b( )N
!N "( ) = cos2N " / 2( )
!N "( ) = cos2 N" / 2( )
NOONGenerator
a
b
Quantum MetrologyH.Lee, P.Kok, JPD,J Mod Opt 49,(2002) 2325
Shot noise
Heisenberg
Sub-Shot-Noise Interferometric MeasurementsWith Two-Photon N00N States
A Kuzmich and L Mandel; Quantum Semiclass. Opt. 10 (1998) 493–500.
Low!N00N2 0 + ei2! 0 2
SNL
HL
a† N a N
AN Boto, DS Abrams,CP Williams, JPD, PRL85 (2000) 2733
Loss in Quantum SensorsSD Huver, CF Wildfeuer, JP Dowling, Phys. Rev. A 78 # 063828 DEC 2008
!N00N
Generator
Detector
Lostphotons
Lostphotons
La
Lb
Visibility:
Sensitivity:
! = (10,0 + 0,10 ) 2
! = (10,0 + 0,10 ) 2
!
SNL---
HL—
N00N NoLoss —
N00N 3dBLoss ---
Super-LossitivityGilbert, G; Hamrick, M; Weinstein, YS; JOSA B 25 (8): 1336-1340 AUG 2008
!" =!P
d P / d"
3dB Loss, Visibility & Slope — Super Beer’s Law!
N=1 (classical)N=5 (N00N)
dP1 /d!
dPN /d!
ei! " eiN! e#$L " e#N$L
Loss in Quantum SensorsS. Huver, C. F. Wildfeuer, J.P. Dowling, Phys. Rev. A 78 # 063828 DEC 2008
!N00N
Generator
Detector
Lostphotons
Lostphotons
La
Lb
!
Q: Why do N00N States Do Poorly in the Presence of Loss?
A: Single Photon Loss = Complete “Which Path” Information!
N A 0 B + eiN! 0 A N B " 0 A N #1 B
A
B
Gremlin
Towards A Realistic Quantum SensorS. Huver, C. F. Wildfeuer, J.P. Dowling, Phys. Rev. A 78 # 063828 DEC 2008
Try other detection scheme and states!
M&M Visibility
!M&M
Generator
Detector
Lostphotons
Lostphotons
La
Lb
! = ( m,m' + m',m ) 2M&M state:
! = ( 20,10 + 10,20 ) 2
! = (10,0 + 0,10 ) 2
!
N00N Visibility
0.05
0.3
M&M’ Adds Decoy Photons
!M&M
Generator
Detector
Lostphotons
Lostphotons
La
Lb
! = ( m,m' + m',m ) 2M&M state:
!
M&M State —N00N State ---
M&M HL —M&M HL —
M&M SNL ---
N00N SNL ---
A FewPhotons
LostDoes Not
GiveComplete
“Which Path”
Towards A Realistic Quantum SensorS. Huver, C. F. Wildfeuer, J.P. Dowling, Phys. Rev. A 78 # 063828 DEC 2008
Optimization of Quantum Interferometric Metrological Sensors In thePresence of Photon Loss
PHYSICAL REVIEW A, 80 (6): Art. No. 063803 DEC 2009
Tae-Woo Lee, Sean D. Huver, Hwang Lee, Lev Kaplan, Steven B. McCracken,Changjun Min, Dmitry B. Uskov, Christoph F. Wildfeuer, Georgios Veronis,
Jonathan P. Dowling
We optimize two-mode, entangled, number states of light in the presence ofloss in order to maximize the extraction of the available phase information in aninterferometer. Our approach optimizes over the entire available input Hilbertspace with no constraints, other than fixed total initial photon number.
Lossy State ComparisonPHYSICAL REVIEW A, 80 (6): Art. No. 063803 DEC 2009
Here we take the optimal state, outputted by the code, ateach loss level and project it on to one of three knowstates, NOON, M&M, and “Spin” Coherent.
The conclusion from thisplot is that the optimalstates found by thecomputer code are N00Nstates for very low loss,M&M states forintermediate loss, and“spin” coherent states forhigh loss.
Super-Resolution at the Shot-Noise Limit with Coherent Statesand Photon-Number-Resolving Detectors
J. Opt. Soc. Am. B/Vol. 27, No. 6/June 2010
Y. Gao, C.F. Wildfeuer, P.M. Anisimov, H. Lee, J.P. Dowling
We show that coherent light coupled with photon numberresolving detectors — implementing parity detection —produces super-resolution much below the Rayleighdiffraction limit, with sensitivity at the shot-noise limit.
Quantum Metrology with Two-Mode Squeezed Vacuum: Parity Detection Beats the Heisenberg Limit
PRL 104, 103602 (2010)PM Anisimov, GM Raterman, A Chiruvelli, WN Plick, SD Huver, H Lee, JP Dowling
We show that super-resolution and sub-Heisenberg sensitivity isobtained with parity detection. In particular, in our setup, dependenceof the signal on the phase evolves <n> times faster than in traditionalschemes, and uncertainty in the phase estimation is better than 1/<n>.
SNL HL
TMSV& QCRB
HofL
SNL ! 1 / n HL ! 1 / n TMSV ! 1 / n n + 2 HofL ! 1 / n2