methods of entangling large numbers of photons for ... · optimizing the multi-photon absorption...
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Jonathan P. Dowling
Methods of Entangling LargeNumbers of Photons for Enhanced
Phase Resolution
quantum.phys.lsu.edu
Hearne Institute for Theoretical PhysicsQuantum Science and Technologies Group
Louisiana State UniversityBaton Rouge, Louisiana USA
Entanglement Beyond the Optical RegimeONR Workshop, Santa Ana, CA 10 FEB 2010
H.Cable, C.Wildfeuer, H.Lee, S.D.Huver, W.N.Plick, G.Deng, R.Glasser, S.Vinjanampathy,K.Jacobs, D.Uskov, J.P.Dowling, P.Lougovski, N.M.VanMeter, M.Wilde, G.Selvaraj, A.DaSilvaNot Shown: P.M.Anisimov, B.R.Bardhan, A.Chiruvelli, G.A.Durkin, M.Florescu, Y.Gao, B.Gard,
K.Jiang, K.T.Kapale, T.W.Lee, D.J.Lum, S.B.McCracken, S.J.Olsen, G.M.Raterman, C.Sabottke,K.P.Seshadreesan, S.Thanvanthri, G.Veronis
Hearne Institute for Theoretical PhysicsQuantum Science & Technologies Group
Outline
1.1. Nonlinear Optics Nonlinear Optics vsvs. Projective Measurements. Projective Measurements
2.2. Quantum Imaging Quantum Imaging vsvs. Precision Measurements. Precision Measurements
3.3. Showdown at High N00N!Showdown at High N00N!
4.4. Efficient N00N GeneratorsEfficient N00N Generators
5.5. Mitigating Photon LossMitigating Photon Loss
6.6. Microwave-Entangled SQUID QubitsMicrowave-Entangled SQUID Qubits
7.7. Microwave to Optical Photon TransducerMicrowave to Optical Photon Transducer
Optical C-NOT with 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 toOptical Quantum Computing
Cavity QED
I. Enhance NonlinearInteraction with aCavity or EIT —Kimble, Walther,Lukin, et al.
II. ExploitNonlinearity ofMeasurement — Knill,LaFlamme, Milburn,Franson, et al.
Photon-PhotonXOR Gate
Photon-PhotonNonlinearity
Kerr Material
Cavity QEDEIT
ProjectiveMeasurement
LOQC KLM
WHY IS A KERR NONLINEARITY LIKE APROJECTIVE MEASUREMENT?
G. G. Lapaire, P. Kok,JPD, J. E. 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”!
Single-Photon QuantumNon-Demolition
You want to know if there is a single photon in modeb, without destroying it.
*N. Imoto, H.A. 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→ 22 Orders of
MagnitudeImprovement! P. Kok, H. Lee, and JPD, PRA 66 (2003) 063814
Outline
1.1. Nonlinear Optics Nonlinear Optics vsvs. Projective Measurements. Projective Measurements
2.2. Quantum Imaging Quantum Imaging vsvs. Precision Measurements. Precision Measurements
3.3. Showdown at High N00N!Showdown at High N00N!
4.4. Efficient N00N GeneratorsEfficient N00N Generators
5.5. Mitigating Photon LossMitigating Photon Loss
6.6. Microwave-Entangled SQUID QubitsMicrowave-Entangled SQUID Qubits
7.7. Microwave to Optical Photon TransducerMicrowave to Optical Photon Transducer
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
Super-Resolution
Sub-Rayleigh
New York Times
DiscoveryCould MeanFasterComputerChips
Quantum Lithography Experiment
|20>+|02>
|10>+|01>
Low!N00N2 0 + ei2! 0 2
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( )
Recap: Super-Resolution
λ
λ/Ν
N=1 (classical)N=5 (N00N)
Recap: Super-Sensitivity
dP1/dϕ
dPN/dϕ!" =!P
d P / d"
N=1 (classical)N=5 (N00N)
Outline
1.1. Nonlinear Optics Nonlinear Optics vsvs. Projective Measurements. Projective Measurements
2.2. Quantum Imaging Quantum Imaging vsvs. Precision Measurements. Precision Measurements
3.3. Showdown at High N00N!Showdown at High N00N!
4.4. Efficient N00N GeneratorsEfficient N00N Generators
5.5. Mitigating Photon LossMitigating Photon Loss
6.6. Microwave-Entangled SQUID QubitsMicrowave-Entangled SQUID Qubits
7.7. Microwave to Optical Photon TransducerMicrowave to Optical Photon Transducer
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
N00N & Linear Optical Quantum Computing
For proposals* to exploit a non-linear photon-photon interactione.g. cross-Kerr interaction ,
the required optical non-linearity not readily accessible.
*C. Gerry, and R.A. Campos, Phys. Rev. A 64, 063814 (2001).
Nature 409,page 46,(2001).
H = !! a†ab†b
Measurement-Induced NonlinearitiesG. G. Lapaire, Pieter Kok, JPD, J. E. Sipe, PRA 68 (2003) 042314
First linear-optics based High-N00N generator proposal:
Success probability approximately 5% for 4-photon output.
e.g.component oflight from an
opticalparametricoscillator
Scheme conditions on the detection of one photon at each detector
mode a
mode b
H. Lee, P. Kok, N. J. Cerf and J. P. Dowling, PRA 65, 030101 (2002).
Towards High-N00N!
Kok, Lee, & Dowling, Phys. Rev. A 65 (2002) 0512104
the consecutive phases are given by:
ϕ k = 2π kN/2
a’a
b’b
c
d
|N,N〉 → |N-2,N〉 + |N,N-2〉
PS
cascade 1 2 3 N2
|N,N〉 → |N,0〉 + |0,N〉
p1 = N (N-1) T2N-2 R2 → N→∞
12
12e2
with T = (N–1)/N and R = 1–T
Not Efficient!
Implemented in Experiments!
|10::01>
|20::02>
|40::04>
|10::01>
|20::02>
|30::03>
|30::03>
N00N State Experiments
Rarity, (1990)Ou, et al. (1990)Shih, Alley (1990)
….
6-photonSuper-resolution
Only!Resch,…,White
PRL (2007)Queensland
19902-photon
Nagata,…,Takeuchi,Science (04 MAY)Hokkaido & Bristol
20074-photon
Super-sensitivity&
Super-resolution
Mitchell,…,SteinbergNature (13 MAY)
Toronto
20043, 4-photon
Super-resolution
only
Walther,…,ZeilingerNature (13 MAY)
Vienna
N00N
Physical Review 76, 052101 (2007)
N00N
LHVΘ
Outline
1.1. Nonlinear Optics Nonlinear Optics vsvs. Projective Measurements. Projective Measurements
2.2. Quantum Imaging Quantum Imaging vsvs. Precision Measurements. Precision Measurements
3.3. Showdown at High N00N!Showdown at High N00N!
4.4. Efficient N00N GeneratorsEfficient N00N Generators
5.5. Mitigating Photon LossMitigating Photon Loss
6.6. Microwave-Entangled SQUID QubitsMicrowave-Entangled SQUID Qubits
7.7. Microwave to Optical Photon TransducerMicrowave to Optical Photon Transducer
Efficient Schemes forGenerating N00N States!
Question: Do there exist operators “U” that produce “N00N” States Efficiently?
Answer: YES!
Constrained Desired
|N>|0> |N0::0N>
|1,1,1> NumberResolvingDetectors
Phys. Rev. Lett. 99, 163604 (2007)
M-portphotocounter
Linear optical device(Unitary action on modes)
Terms & Conditions• Only disentangled inputs areallowed
( )
• Modes transformation is unitary
(U is a set of beam splitters)
•Number-resolving photodetection
(single photon detectors)
! in = n1 1 " ..." nR R
VanMeter NM, Lougovski P, Uskov DB, et al., General linear-optical quantumstate generation scheme: Applications to maximally path-entangled states,Physical Review A, 76 (6): Art. No. 063808 DEC 2007.
Linear Optical N00N Generator II
U
2
2
2
0
1
0
0.032( 50 + 05 ) This example disproves the
N00N Conjecture: “That itTakes At Least N Modes toMake N00N.”
The upper bound on the resources scales quadratically!
Upper bound theorem:The maximal size of aN00N state generatedin m modes via singlephoton detection in m-2modes is O(m2).
Linear Optical N00N Generator II
Entanglement Seeded Dual OPAEntanglement Seeded Dual OPA
( )22 42( )
1 2 tanh tanhProb( , ) tanh [( 1)( 2) ( 1)( 2)]
4n m
r rn m r n n m m+
! += + + + + +
Idea is to look at sources which access a higher dimensional Hilbert space(qudits) than canonical LOQC and then exploit number resolving detectors,while we wait for single photon sources to come online.
R.T.Glasser, H.Cable, JPD, in preparation.
Quantum States of Light From a High-Gain OPA (Theory)
G.S.Agarwal, et al., J. Opt. Soc. Am. B 24, 270 (2007).
We present a theoretical analysis of the properties of an unseededoptical parametric amplifier (OPA) used as the source ofentangled photons.
The idea is to take known bright sources ofentangled photons coupled to number resolvingdetectors and see if this can be used in LOQC,while we wait for the single photon sources.
OPA Scheme
Quantum States of Light From a High-Gain OPA (Experiment)
In the present paper weexperimentally demonstrate that theoutput of a high-gain opticalparametric amplifier can be intenseyet exhibits quantum features,namely, a bright source of Bell-typepolarization entanglement.
F.Sciarrino, et al., Phys. Rev. A 77, 012324 (2008)This is an Experiment!
State Before Projection
Visibility Saturatesat 20% with105 Counts PerSecond!
Lovely, Fresh, Data!
Optimizing the Multi-Photon Absorption Properties of N00N States<arXiv:0908.1370v1> (submitted to Phys. Rev. A)
William N. Plick, Christoph F. Wildfeuer, Petr M. Anisimov, Jonathan P. Dowling
In this paper we examine the N-photon absorption properties of "N00N" states, a subclass of pathentangled number states. We consider two cases. The first involves the N-photon absorptionproperties of the ideal N00N state, one that does not include spectral information. We study how theN-photon absorption probability of this state scales with N. We compare this to the absorptionprobability of various other states. The second case is that of two-photon absorption for an N = 2 N00Nstate generated from a type II spontaneous down conversion event. In this situation we find that theabsorption probability is both better than analogous coherent light (due to frequency entanglement)and highly dependent on the optical setup. We show that the poor production rates of quantum statesof light may be partially mitigated by adjusting the spectral parameters to improve their two-photonabsorption rates.
Return of the Kerr Nonlinearity!
|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〉
Implementation of QFG via Cavity QEDKapale, KT; Dowling, JP, PRL, 99 (5): Art. No. 053602 AUG 3 2007.
Ramsey Interferometryfor atom initially in state b.
Dispersive coupling between the atom and cavity givesrequired conditional phase shift
Outline
1.1. Nonlinear Optics Nonlinear Optics vsvs. Projective Measurements. Projective Measurements
2.2. Quantum Imaging Quantum Imaging vsvs. Precision Measurements. Precision Measurements
3.3. Showdown at High N00N!Showdown at High N00N!
4.4. Efficient N00N GeneratorsEfficient N00N Generators
5.5. Mitigating Photon LossMitigating Photon Loss
6.6. Microwave-Entangled SQUID QubitsMicrowave-Entangled SQUID Qubits
7.7. Microwave to Optical Photon TransducerMicrowave to Optical Photon Transducer
Computational Optimization ofQuantum LIDAR
!in =
ci N " i, ii= 0
N
#
!"
forward problem solver
!" = f ( #in , " ; loss A, loss B)
INPUT
“findmin( )“
!"
FEEDBACK LOOP:Genetic Algorithm
inverse problem solver
OUTPUT
min(!") ; #in(OPT ) = ci
(OPT ) N $ i, i , "OPTi= 0
N
%
N: photon number
loss Aloss B
Lee, TW; Huver, SD; Lee, H; et al.PHYSICAL REVIEW A, 80 (6): Art. No. 063803 DEC 2009
NonclassicalLight
Source
DelayLine
Detection
Target
Noise
3/28/11 43
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!
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
Try other detection scheme and states!
!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 Generalized Coherent.
The conclusion from this plot is thatThe optimal states found by thecomputer code are N00N states forvery low loss, M&M states forintermediate loss, and generalizedcoherent states for high loss.
This graph supports the assertionthat a Type-II sensor with coherentlight but a non-classical numberresolving detection scheme isoptimal for very high loss.
Super-Resolution at the Shot-Noise Limit with Coherent Statesand Photon-Number-Resolving Detectors
<arXiv:0907.2382v2> (submitted to JOSA B)
Yang Gao, Christoph F. Wildfeuer, Petr M. Anisimov, Hwang Lee, Jonathan P. Dowling
There has been much recent interest in quantum optical interferometry for applications tometrology, sub-wavelength imaging, and remote sensing, such as in quantum laser radar(LADAR). For quantum LADAR, atmospheric absorption rapidly degrades any quantum stateof light, so that for high-photon loss the optimal strategy is to transmit coherent states of light,which suffer no worse loss than the Beer law for classical optical attenuation, and whichprovides sensitivity at the shot-noise limit. This approach leaves open the question -- what isthe optimal detection scheme for such states in order to provide the best possible resolution?We show that coherent light coupled with photon number resolving detectors canprovide a super-resolution much below the Rayleigh diffraction limit, with sensitivity noworse than shot-noise in terms of the detected photon power.
ClassicalQuantum
µWaves are Coherent!
MagicDetector!
Quantum Metrology with Two-Mode Squeezed Vacuum: Parity Detection Beats the Heisenberg Limit
<arXiv:0910.5942v1> (submitted to Phys. Rev. Lett.)
Petr M. Anisimov, Gretchen M. Raterman, Aravind Chiruvelli, William N.Plick, Sean D. Huver, Hwang Lee, Jonathan P. Dowling
We study the sensitivity and resolution of phase measurement in a Mach-Zehnderinterferometer with two-mode squeezed vacuum (<n> photons on average). We show thatsuper-resolution and sub-Heisenberg sensitivity is obtained with parity detection. In particular,in our setup, dependence of the signal on the phase evolves <n> times faster than intraditional schemes, and uncertainty in the phase estimation is better than 1/<n>.
SNL HL
TMSV &Parity
Resolution and Sensitivity of a Fabry-perot InterferometerWith a Photon-number-resolving Detector
Phys. Rev. A 80 043822 (2009)
Christoph F. Wildfeuer, Aaron J. Pearlman, Jun Chen, Jingyun Fan,Alan Migdall, Jonathan P. Dowling
With photon-number resolving detectors, we show compression of interference fringeswith increasing photon numbers for a Fabry-Perot interferometer. This featureprovides a higher precision in determining the position of the interferencemaxima compared to a classical detection strategy. We also theoretically showsupersensitivity if N-photon states are sent into the interferometer and a photon-numberresolving measurement is performed.
Outline
1.1. Nonlinear Optics Nonlinear Optics vsvs. Projective Measurements. Projective Measurements
2.2. Quantum Imaging Quantum Imaging vsvs. Precision Measurements. Precision Measurements
3.3. Showdown at High N00N!Showdown at High N00N!
4.4. Efficient N00N GeneratorsEfficient N00N Generators
5.5. Mitigating Photon LossMitigating Photon Loss
6.6. Microwave-Entangled SQUID QubitsMicrowave-Entangled SQUID Qubits
7.7. Microwave to Optical Photon TransducerMicrowave to Optical Photon Transducer
Heisenberg Limited Magnetometer:N00N in Microwave PhotonsTransferred into N00N in SQUIDQubits and Reverse!
NQ xg /1!"!##
Entangled “SQUID” Qubit MagnetometerA Guillaume & JPD, Physical Review A Rapid 73 (4): Art. No. 040304 APR 2006.
Quantum Magnetometer Arrays Can DetectSubmarines From Orbit
Outline
1.1. Nonlinear Optics Nonlinear Optics vsvs. Projective Measurements. Projective Measurements
2.2. Quantum Imaging Quantum Imaging vsvs. Precision Measurements. Precision Measurements
3.3. Showdown at High N00N!Showdown at High N00N!
4.4. Efficient N00N GeneratorsEfficient N00N Generators
5.5. Mitigating Photon LossMitigating Photon Loss
6.6. Microwave-Entangled SQUID QubitsMicrowave-Entangled SQUID Qubits
7.7. Microwave to Optical Photon TransducerMicrowave to Optical Photon Transducer
Microwave to Optical TransducerKT Kapale and JPD, in preparation
Outline
1.1. Nonlinear Optics Nonlinear Optics vsvs. Projective Measurements. Projective Measurements
2.2. Quantum Imaging Quantum Imaging vsvs. Precision Measurements. Precision Measurements
3.3. Showdown at High N00N!Showdown at High N00N!
4.4. Efficient N00N GeneratorsEfficient N00N Generators
5.5. Mitigating Photon LossMitigating Photon Loss
6.6. Microwave-Entangled SQUID QubitsMicrowave-Entangled SQUID Qubits
7.7. Microwave to Optical Photon TransducerMicrowave to Optical Photon Transducer