Methods of Entangling Large Numbers of Photons for ... · Optimizing the Multi-Photon Absorption Properties of N00N States (submitted to Phys. Rev. A) William

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








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"


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

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








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








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!


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








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








Microwave to Optical TransducerKT Kapale and JPD, in preparation

Outline








Documents

Methods of Entangling Large Numbers of Photons for ... · Optimizing the Multi-Photon Absorption Properties of N00N States (submitted to Phys. Rev. A) William