Jonathan P. Dowling Methods of Entangling Large Numbers of Photons for Enhanced Phase Resolution quantum.phys.lsu.edu Hearne Institute for Theoretical

Jonathan P. Dowling

Methods of Entangling Large Numbers 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

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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 vs. Projective Nonlinear Optics vs. Projective

MeasurementsMeasurements

2.2. Quantum Imaging vs. Precision Quantum Imaging vs. Precision


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. Microwave-Entangled SQUID Qubits6. Microwave-Entangled SQUID Qubits

7. Microwave to Optical Photon 7. Microwave to Optical Photon

TransducerTransducer

Optical C-NOT with 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 Quantum

Computing

Cavity QED

I. Enhance Nonlinear Interaction with a Cavity or EIT — Kimble, Walther, Lukin, et al.

I. Enhance Nonlinear Interaction with a Cavity or EIT — Kimble, Walther, Lukin, et al.

II. Exploit Nonlinearity of Measurement — Knill, LaFlamme, Milburn, Franson, et al.

II. Exploit Nonlinearity of Measurement — Knill, LaFlamme, Milburn, Franson, et al.

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?

G. G. Lapaire, P. Kok, JPD, J. E. Sipe, PRA 68 (2003) 042314

G. G. Lapaire, P. Kok, JPD, J. E. 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”!

Single-Photon Quantum Non-Demolition

You 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, H.A. 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

22 Orders of Magnitude

Improvement!

Effective = 1/8

22 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











+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 MetrologyH.Lee, P.Kok, JPD, J Mod Opt 49, (2002) 2325

H.Lee, P.Kok, JPD, J Mod Opt 49, (2002) 2325

Shot noise

Heisenberg

Sub-Shot-Noise Interferometric Measurements With Two-Photon N00N States

A Kuzmich and L Mandel; Quantum Semiclass. Opt. 10 (1998) 493–500.





Low N00N

2 0 + ei2ϕ 0 2

Low N00N

2 0 + ei2ϕ 0 2

SNL

HL

N Photons

N-PhotonDetector

ϕ = 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

Super-Resolution

Sub-Rayleigh

New York Times

Discovery Could Mean Faster Computer Chips

Quantum Lithography Experiment

|20>+|02>

|10>+|01>

Low N00N

2 0 + ei2ϕ 0 2

Low N00N

2 0 + ei2ϕ 0 2

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( )

Recap: Super-Resolution

N=1 (classical)N=5 (N00N)

Recap: Super-Sensitivity

dP1/d

dPN/dϕ =

P

d P / dϕ


Outline











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

N00N & Linear Optical Quantum Computing

For proposals* to exploit a non-linear photon-photon interaction e.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 =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

optical parametric oscillator

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) 0512104Kok, Lee, & Dowling, Phys. Rev. A 65 (2002) 0512104

the consecutive phases are given by:

k = 2 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-photon Super-

resolutionOnly!

Resch,…,WhitePRL (2007)Queensland

19902-photon

Nagata,…,Takeuchi,

Science (04 MAY)Hokkaido & Bristol

20074-photon

Super-sensitivity&

Super-resolution

Mitchell,…,Steinberg

Nature (13 MAY)Toronto

20043, 4-photon

Super-resolution

only

Walther,…,Zeilinger

Nature (13 MAY)Vienna

N00N

Physical Review 76, 052101 (2007)

N00N

LHV

Outline











Efficient Schemes for Generating N00N

States!

Question: Do there exist operators “U” that produce “N00N” States Efficiently?

Answer: YES!

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-port photocounter

Linear optical device(Unitary action on modes)

Terms & Conditions

• Only disentangled inputs are allowed

( )

• 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 quantum state generation scheme: Applications to maximally path-entangled states, Physical Review A, 76 (6): Art. No. 063808 DEC 2007.

VanMeter NM, Lougovski P, Uskov DB, et al., General linear-optical quantum state 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

€

0.03

2( 50 + 05 ) This example disproves

the N00N Conjecture: “That it Takes At Least N Modes to Make N00N.”

This example disproves the N00N Conjecture: “That it Takes At Least N Modes to Make 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).

Linear Optical N00N Generator II

Entanglement Seeded Dual OPAEntanglement Seeded Dual OPA

( )22 4

2( )1 2 tanh tanh

Prob( , ) tanh [( 1)( 2) ( 1)( 2)]4

n mr r

n 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.

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 unseeded optical parametric amplifier (OPA) used as the source of entangled photons.

The idea is to take known bright sources of entangled photons coupled to number resolving detectors and see if this can be used in LOQC, while we wait for the single photon sources.

The idea is to take known bright sources of entangled photons coupled to number resolving detectors and see if this can be used in LOQC, while we wait for the single photon sources.

OPA SchemeOPA Scheme

Quantum States of Light From a High-Gain OPA (Experiment)

In the present paper we experimentally demonstrate that the output of a high-gain optical parametric amplifier can be intense yet exhibits quantum features, namely, a bright source of Bell-type polarization entanglement.

In the present paper we experimentally demonstrate that the output of a high-gain optical parametric amplifier can be intense yet exhibits quantum features, namely, a bright source of Bell-type polarization entanglement.

F.Sciarrino, et al., Phys. Rev. A 77, 012324 (2008)This is an Experiment!This is an Experiment!

State Before ProjectionState Before Projection

Visibility Saturates at 20% with105 Counts Per Second!

Lovely, Fresh, Data!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 path entangled number states. We consider two cases. The first involves the N-photon absorption properties of the ideal N00N state, one that does not include spectral information. We study how the N-photon absorption probability of this state scales with N. We compare this to the absorption probability of various other states. The second case is that of two-photon absorption for an N = 2 N00N state generated from a type II spontaneous down conversion event. In this situation we find that the absorption 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 states of light may be partially mitigated by adjusting the spectral parameters to improve their two-photon absorption rates.





Return of the Kerr Nonlinearity!

|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

Implementation of QFG via Cavity QEDKapale, KT; Dowling, JP, PRL, 99 (5): Art. No. 053602 AUG 3 2007.

Ramsey Interferometry for atom initially in state b.

Dispersive coupling between the atom and cavity gives required conditional phase shift

Outline











Computational Optimization of Quantum LIDAR

€

Ψin =

c i N − i, ii= 0

N

∑

€

δ

forward problem solver

€

δ= f ( Ψin , φ ; loss A, loss B)

INPUT

“find min( )“

€

δ

FEEDBACK LOOP:Genetic Algorithm

inverse problem solver

OUTPUT

€

min(δφ) ; Ψin(OPT ) = c i

(OPT ) N − i, i , φOPT

i= 0

N

∑

N: photon number

loss A

loss B

Lee, TW; Huver, SD; Lee, H; et al.PHYSICAL REVIEW A, 80 (6): Art. No. 063803 DEC 2009Lee, TW; Huver, SD; Lee, H; et al.PHYSICAL REVIEW A, 80 (6): Art. No. 063803 DEC 2009

NonclassicalLight

Source

DelayLine

Detection

Target

Noise

04/21/23 43

Loss in Quantum SensorsSD Huver, CF Wildfeuer, JP Dowling, Phys. Rev. A 78 # 063828 DEC 2008

€

N00N

Generator

Detector

Lost photons

Lost photons

La

Lb

Visibility:

Sensitivity:

€

=(10,0 + 0,10 ) 2

€

=(10,0 + 0,10 ) 2

€

ϕ

SNL---

HL—

N00N NoLoss —

N00N 3dB Loss ---

Super-LossitivityGilbert, G; Hamrick, M; Weinstein, YS; JOSA B 25 (8): 1336-1340 AUG 2008

ϕ =P

d P / dϕ

3dB Loss, Visibility & Slope — Super Beer’s Law!


€

dP1 /dϕ

€

dPN /dϕ

€

e−γL → e−NγL

Loss in Quantum SensorsS. Huver, C. F. Wildfeuer, J.P. Dowling, Phys. Rev. A 78 # 063828 DEC 2008

€

N00N

Generator

Detector

Lost photons

Lost photons

La

Lb

€

ϕ

Q: Why do N00N States Do Poorly in the Presence of Loss?A: Single Photon Loss = Complete “Which Path” Information!

€

N A 0 B + e iNϕ 0 A N B → 0 A N −1 B

A

B

Gremlin

Towards A Realistic Quantum Sensor

S. Huver, C. F. Wildfeuer, J.P. Dowling, Phys. Rev. A 78 # 063828 DEC 2008Try other detection scheme and states!

M&M Visibility

€

M&M

Generator

Detector

Lost photons

Lost photons

La

Lb

€

=( m,m' + m',m ) 2M&M state:

€

=( 20,10 + 10,20 ) 2

€

=(10,0 + 0,10 ) 2

€

ϕ

N00N Visibility

0.05

0.3

M&M’ Adds Decoy Photons

Try other detection scheme and states!

€

M&M

Generator

Detector

Lost photons

Lost photons

La

Lb

€

=( m,m' + m',m ) 2M&M state:

€

ϕ

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 Sensor

S. Huver, C. F. Wildfeuer, J.P. Dowling, Phys. Rev. A 78 # 063828 DEC 2008

Optimization of Quantum Interferometric Metrological Sensors In the Presence 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 of loss in order to maximize the extraction of the available phase information in an interferometer. Our approach optimizes over the entire available input Hilbert space 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, at each loss level and project it on to one of three know states, NOON, M&M, and Generalized Coherent.

The conclusion from this plot is thatThe optimal states found by the computer code are N00N states for very low loss, M&M states for intermediate loss, and generalized coherent states for high loss.

This graph supports the assertion that a Type-II sensor with coherent light but a non-classical number resolving detection scheme is optimal for very high loss.

Super-Resolution at the Shot-Noise Limit with Coherent States and 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 to metrology, sub-wavelength imaging, and remote sensing, such as in quantum laser radar (LADAR). For quantum LADAR, atmospheric absorption rapidly degrades any quantum state of 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 which provides sensitivity at the shot-noise limit. This approach leaves open the question -- what is the 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 can provide a super-resolution much below the Rayleigh diffraction limit, with sensitivity no worse than shot-noise in terms of the detected photon power.


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Classical

Quantum

Waves 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-Zehnder interferometer with two-mode squeezed vacuum (<n> photons on average). We show that super-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 in traditional schemes, and uncertainty in the phase estimation is better than 1/<n>.



SNL HL

TMSV &Parity

Resolution and Sensitivity of a Fabry-perot Interferometer With 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 fringes with increasing photon numbers for a Fabry-Perot interferometer. This feature provides a higher precision in determining the position of the interference maxima compared to a classical detection strategy. We also theoretically show supersensitivity if N-photon states are sent into the interferometer and a photon-number resolving measurement is performed.





Outline











Heisenberg Limited Magnetometer:N00N in Microwave Photons Transferred into N00N in SQUID Qubits and Reverse!

NQ xg /1∝Φ∝δδ

Entangled “SQUID” Qubit MagnetometerA Guillaume & JPD, Physical Review A Rapid 73 (4): Art. No. 040304 APR 2006.











Quantum Magnetometer Arrays Can Detect Submarines From Orbit

Outline











Microwave to Optical TransducerKT Kapale and JPD, in preparation







Outline











Documents

Jonathan P. Dowling Methods of Entangling Large Numbers of Photons for Enhanced Phase Resolution quantum.phys.lsu.edu Hearne Institute for Theoretical