Super Resolution and Beating Shot Noise Limit by Entangled ... · F. Sciarrino, C. Vitelli, F. D. Martini, R. Glasser, H. Cable, and J. P. Dowling, Phys. Rev. A 77 , 012324 (2008)

The shot noise of photons is a big barrier in a range of applications likethe coherent processing of signals, quantum communication andquantum metrology. Use of squeezed light would be a way out andgreat progress has been made towards producing light with largesqueezing. In this talk I would discuss a newer possibility i.e. the use of

Super Resolution and Beating Shot NoiseLimit by Entangled Photons

G S AgarwalPhysics Department, OSU

squeezing. In this talk I would discuss a newer possibility i.e. the use ofsources of light which produce entangled photon pairs. I would describethe progress made using entangled photon pairs and the challengesahead.

Key collaborators: Aziz Kolkiran (Ph.D. thesis), OSUBoyd`s group, RochesterBill Plick and J Dowling, LSUKenan Qu, OSU

� Using entangled photon sources like Parametric Down Conversion (PDC) to produce superresolution and supersensitivity

� Magneto-optical Rotation (MOR) of light and sensing of magnetic fields

� Measurement of rotations as in Sagnac Interferometers, gyroscopes

Application Areas

� Measurement of rotations as in Sagnac Interferometers, gyroscopes

� Sensitive measurement of refractive index

� Improving the sensitivity of light scattering measurements

� Quantum sensors, target detection

1~N∆⋅∆ϕ

Quantum Sensing with Nonclassical and Entangled Light

Quantum Noise of Measurement

Coherent Sources NN ~∆

Traditional Measurements

Heisenberg Limited Measurements

N

1~ϕ∆

N

1~ϕ∆

Quantum Sensing with Nonclassical and Entangled Light

First introduced by E. Schrödinger

[Naturwissenschaften, 23, 807 (1935)]

To explain intriguing features of a composite system

“Any state of a composite system is said to be entangled if it cannot be

expressed as a direct product of the state of each subsystem .”

AB A BΨ = Φ ⊗ χ ≡Ψ = Φ ⊗ χ ≡Ψ = Φ ⊗ χ ≡Ψ = Φ ⊗ χ ≡ Not entangled

∑∑∑∑

Quantum Entanglement

(((( ))))

(((( ))))A B A BA B

A B A BA B

10 1 1 0

21

0 0 1 12

±±±±

±±±±

Ψ = ±Ψ = ±Ψ = ±Ψ = ±

Φ = ±Φ = ±Φ = ±Φ = ±

Basic bipartite entangled states :

i jAB ij A B

ij

CΨ = Φ ⊗ χ ≡Ψ = Φ ⊗ χ ≡Ψ = Φ ⊗ χ ≡Ψ = Φ ⊗ χ ≡∑∑∑∑ Entangled jiCCC if ij ≠

BEC; y;,circular/xleft right,-onPolarizati ;Spin ↑↓

(Bell States)

0 0k , ωr

1 1k , ωr

2 2k , ωr

(2); χ

PDC as a Source of Entanglement

(2)

210210

g .);.(

,

χωωω

∝+=

+=+=++ chaagahH

kkkrrr

( Sum over all k’s; frequencies ; Polarizations etc.)

0 0

0 0

, 0 , 0 , 1 , 1

, 0 , 0 , 1 , 1

H α α

ψ µ α ν α

→

→ +

0a : Coherent pump

(2)210 g .);.( χ∝+= ++ chaagahH

0

1| ( tanh ) | |

coshi n

H Vn

e g n ng

θψ∞

=

⟩ = ⟩ ⟩∑

ˆHaˆVa

Type-II collinear

Type-II non-collinear

Types of PDC

ˆ ˆ,H Va a

ˆ ˆ,H Vb b

20 0

1| ( ) (tanh ) | | | |

cosh H V H Va

ni m n

a b bn mg

e g n m m m n mθψ∞

= =⟩ = − ⟩ ⟩ ⟩ − ⟩∑∑

Entangled Photon Pairs

Entangled State: CATU↓↓↓ ↓ →L

2i2: e

xSOperation U

π

=


)(2

1 2/4i ↑↑↑↑+↓↓↓↓=

↓↓↓↓=Ψ

−LLLL

LL

ππ

iNeie

U

↓↓L

2

2

ixS

eUπ

=CAT

Interferometer

φMeasurementMeasurement

Output +UzSie φ

Input


φMeasurementMeasurement e

2exp2

cos ( ) sin2 2

zi Sout x

N

i S e CAT

N Ni

ϕπ

ϕ ϕ

Ψ = −

= ↓↓↓ ↓ + ↑↑↑ ↑

LL LL

)cos(2z φNN

S =

)(sin4

)( 22

2 φNN

Noise =� Noise

� Accuracy of measurement


� Signal

� Accuracy of measurement

222

2

22

2

2 1~

)(sin4

)(sin4

)/(

)(

NNN

N

NN

S

Noise

z φ

φ

φ=

∂∂ N

1~φ∆

1 2

1| ( tanh ) | |

coshi ne g n n

gθψ

∞

⟩ = − ⟩ ⟩∑Input state:

Entangled Photon Pairs from PDC

1 20

| ( tanh ) | |cosh n

e g n ng

ψ=

⟩ = − ⟩ ⟩∑Input state:

Fig. 5

Can We Improve Resolution Further?

4

4 1 2 1 22

tanh 1[11 12cos(2 ) 9cos(4 )]

cosh 8

gP TT R R

gφ φ= + +

Multi-Photon Entanglement!

Fig. 6

Can We Get Fringes with a Constant Contrast?

42

4 1 1 2 22

tanh 9[1 cos(4 )]

cosh 8

gP T R T R

gφ= −

Kolkiran and Agarwal, Opt Exp 15, 6798 (2007);T. Nagata, et al. Science 316, 726-729, (2007).

Fig. 6

[ ]after the first BS and Phase shifter after the second BS

2| 20 | 02 sin|11 | 20 | 02 cos |11

2 2

ie φ φ φ⟩ + ⟩⟩ → → − ⟩− ⟩ + ⟩

1442443 1444442444443

43 | 40 | 04 1ie φ ⟩ + ⟩

| ie φα ⟩

phase shifter

|n⟩ |i ne nφ ⟩classical field

Non-classical Fock state

|α⟩

phase shifter

How does this work?

[ ] [ ]

after the first BS and Phase shifter

2 2

after the second BS

243 | 40 | 04 1

| 22 | 224 2 4

6 6sin | 40 | 04 sin 2 | 31 |13 (2 3sin ) | 22

4 4

iie

e φφ

φ φ φ

⟩ + ⟩⟩ → + ⟩

→ ⟩+ ⟩ + ⟩− ⟩ + − ⟩

14444444244444443

14444444444444244444444444443

All interferometric schemes using the maximally correlated entangled states show the phase sensitivity approaching to 1/N, which is the Heisenberg limit.

Only this part contributes to the detection in the setup given by Fig. 6

This term leads to unequal fringes in the detection setup given by Fig. 5

Fig. 1. An optical interferometer for beating the SQL. (Inset) Aschematic of a Mach-Zehnder (MZ) interferometer consisting oftwo 50:50 beam splitters (BS1 and BS2). Photons are input inmodesa and/orb, and detected in modese and/orf, after a phaseshift (PS) is applied to moded. (Main panel) A schematic of theintrinsically stable displaced-Sagnac architecture usedto ensurethat the optical path lengths in modesc andd are subwavelength(nm) stable. A frequency-doubled 780-nm fs-pulsed laser(repetition interval 13 ns) pumps a type I phase-matched betabarium borate (BBO) crystal to generate the state |22>ab viaspontaneousparametricdown-conversion. Interferencefilters (not

Beating the Standard Quantum Limit with Four-Entangled Photons

spontaneousparametricdown-conversion. Interferencefilters (notshown) with a 4-nm bandwidth were used. The photons are guidedvia polarization-maintaining fibers (PMFs) to the interferometer,which has the same function as the MZ interferometer in the inset.A variable phase shift in moded is realized by changing the angleof the phase plate (PP) in the interferometer. Photons are collectedin single-mode fibers (SMFs) at the output modes and detectedwith a single-photon counting module (SPCM, detectionefficiency 60% at 780 nm) in modef and three cascaded SPCMsin modee.

T. Nagata, R. Okamoto, J. L. O'Brien, K. Sasaki, S. Takeuchi ,Science 316,726 (2007).

Fig. 3.Beating the SQL with four-entangled photons. (A) Single-photon count rate in mode e as a function of phase plate (PP) angle with single-photon input |10>ab. (B) Two-photon

Beating the Standard Quantum Limit with Four-Entangled Photons

single-photon input |10>ab. (B) Two-photon count rate in modes e and f for input state |11>ab. (C) Four-photon count rate of three photons in mode e and one photon in mode f for the input state |22>ab. Accumulation times for one data point were (A) 1s, (B) 300s, and (C) 300s.

1 1| a bψ ⟩ =

1 1

1( tanh ) | ,

coshi n

a bn

e g n ng

θ ⟩∑

Beating the Standard Quantum Limit – High Gain OPA

mean photon number per mode: g2sinh~

G. S. Agarwal, R. W. Boyd, E. M. Nagasako, and S. J. Bentley, Phys. Rev. Lett. 86, 1389 (2001).

In the present paper we experimentally demonstratethat the output of a high-gain optical parametricamplifier can be intense yet exhibits quantum features,namely, sub-Rayleigh fringes, as proposed by[Agarwal et al., Phys. Rev. Lett. 86, 1389 (2001)]. Weinvestigate multiphoton states generated by a high-gain optical parametric amplifier operating with aquantum vacuum input for gain values up to 2.5.

Experimental sub-Rayleigh Resolution by an Unseeded OPA for Quantum Lithography

F. Sciarrino, C. Vitelli, F. D. Martini, R. Glasser, H. Cable, and J. P. Dowling, Phys. Rev. A 77, 012324 (2008)

BS BSSPDC

0a

b

1a

b

3a

3b

aD

coincidencecounter

pump2a

b

|α⟩

Large gain – How to improve visibility

φ

Enhancement of phase measurement via the interference of stimulated and spontaneous parametric processes

0b 1b3b

bD2b

† *3 0 0

† *3 0 0

ˆˆ ˆ( ) ( )1 1 0 11 1ˆ ˆ1 0 12 2 ˆ( ) ( )

i

a a bi i

i e ib b aφ

µ α ν α

µ α ν α

+ + + = + + +

|α⟩

cosh

sinhi

g

e gψ

µν

==

, ,g φ ψ : Interaction Parameter, Pump Phase, Interferometric Phase

| | ie θα α= : coherent beam amplitude

φ

BS BSSPDC

0a

0b

1a

1b

3a

3b

aD

bD

coincidencecounter

pump2a

2b|α⟩

|α⟩

3

† 2 2 2 2 2 23 3ˆ ˆ sinh | | [cosh sinh 2sinh cosh cos( 2 )][1 sin( )]aI a a g g g g gα ψ θ φ≡ ⟨ ⟩ = + + + − −

3

† 2 2 2 2 2 23 3ˆ ˆ sinh ( ) | | [cosh sinh 2sinh cosh cos( 2 )][1 sin( )]bI b b g g g g gα ψ θ φ≡ ⟨ ⟩ = + + + − +

visibility of † †3 3 3 3

ˆ ˆˆ â b b a⟨ ⟩

φ

Mean intensity and two-photon counts

, , :ψ θ φ pump phase, coh. beam phase,interferometer phase

two-photon coincidence counts

† †3 3 3 3

ˆ ˆˆ â b b a⟨ ⟩

visibility of 3 3 3 3ˆ ˆˆ â b b a⟨ ⟩

dashed line: coherent beam offsolid line: coherent beam on with phase at pi/2g: the interaction (squeezing) parameter

(a) Stimulated emission enhanced two-photon counts for various

phases of the coherent field at the gain g = 0.5. The horizontal

line shows the interferometric phase. The pump phase

is fxed at . The

ψπ counts are in units of two-photon coincidence

rates coming from spontaneous down-conversion process.

The modulus of the coherent | | is chosen such that the

coincidences coming from SPDC and the coher

αent fields are

The magnitude of two-photon counts

equal to each other. The dashed line shows the two-photon

counts for the case of spontaneous process. (b) The same with

(a) at the gain g = 2.0. Here, the counts for the case of

sponta3

neous process (dashed line) is multiplied by a factor

of 10 .

A. Kolkiran and G. S. Agarwal, Optics Express 16, 6479 (2008).

A New Class of Interferometers

SU[1,1] Interferometers SU[1,1] Interferometers

W. N. Plick, J. P. Dowling and G. S. Agarwal, New J. Phys. 12, 083014 (2010)

SU[1,1] Interferometers

A strong laser beam pumps the first OPA. The beam (which is assumed to be undepleted)undergoes a π-phase shift and then pumps the second OPA. The input modes of the first OPAare fed with coherent light. After the first OPA, one of the outputs interacts with the phase to beprobed. Both outputs are then brought back together as the inputs for the second OPA.Measurements are taken on the second OPA’s outputs.

statescoherent input theof phase (the /4 and 0When πθφ ==

2

22

2

ˆ

ˆˆ

∂

∂

−=∆

φ

φT

TT

N

NN

ffffT bbaaN ˆˆˆˆˆ ++ +=

SU[1,1] Interferometers

CohOPAOPA

2 1

)2(

1

NNN +=∆φ

,22

Coh βα +=N),(sinh2 Where 2OPA rN = .βα =

equal)), be taken toare(which

statescoherent input theof phase (the /4 and 0When πθφ ==

Scattering particle

Classical

light

Back scattering generally

the polarization changes

Resolution: scatterer Fourier components (Born & Wolf) p.702

4πλ

≤

Lidar with entangled photons based on Mie scattering

Entangled | ,H V ⟩photons

Target 1D

2D

C

Two detectors

Measure cross correlations; detectors are set to measure certain polarizations

Cross correlations expected to

yield better resolution.

ħω

M=1F=1

M=0

F=0

M=-1

gF µ B

σ + σ −

ε+

ε−

( )2

y

iε ε ε+ −−= −

Magneto-optical (Faraday) Rotation (MOR)

M=1M=0M=-1

( )klφ χ χ+ −= −

χ χ+ −≠l

B

/ 2θ

lε +

lε −

lyε

Bφ ∝ for weak fields

Joint Probability of Detecting N Photons and M photons

at Different Ports Parity Measurements

Super Resolution via Newer Observables

at Different Ports Parity Measurements

W. N. Plick, P. M. Anisimov, J. P. Dowling, H. Lee and G. S. Agarwal, New J.Phys. 12,113025 (2010)

• Parity detection is the measurement of the evenness or

oddness of the number of photons in an optical mode.

• It is represented by the operator:


where N is the number operator.

• Parity detection has been shown to perform very well

across a wide variety of input states to an MZI

• The power of parity measurement is highlighted in a

TMSV input MZI.


• In this setup parity detection reaches slightly below

the Heisenberg limit on phase sensitivity.

• See Phys. Rev. Lett. 104, 103602 (2010) for more.

• However practical measurement of the

parity operator is non-trivial.

• Parity may be simulated with number

-resolving detectors, but this is more than

is necessary.

• Furthermore number-resolving detectors

are difficult to build and operate, and are


are difficult to build and operate, and are

only accurate at small numbers of photons.

• However it can be shown that the Wigner function

at the origin is equivalent to parity.

• This can be seen easily in the case of number states.

a) n=0, b) n=1 c) n=5Img. By J S Lundeen

• Mathematically the statement is

• If we restrict ourselves to states with Gaussian Wigner

distributions we can write in general


Supersensitive Angular-Displacement Measurements

Question: Given N photons with orbital angular momentum(OAM)lћ per photon, how accurately can angular-displacementsbe measured?

N One-Photon Fock-State Input

(θ is the angle of rotation)

Dove prism mode transformation:


Achievable Angularsensitivity:

A. K. Jha, G. S. Agarwal, and R. W. Boyd, Phys. Rev. A 83, 053829 (2011)


Question: Given N photons with orbital angular momentum(OAM) lћ per photon, how accurately can angular-displacements be measured?

N One-Photon Fock-State Input


Dove prism mode transformation:

Achievable Angularsensitivity:

Post-selected two-photon state in modes a and b:

Two-Photon Entangled-State Input

Input State:

The Measurement Operator:

Angular Sensitivity:


Input State:

Four-Photon Entangled-State Input

Post-selected four-photon state in modes a and b:

Input State:

The Measurement Operator:

Angular Sensitivity:

N-Photon Entangled-State Input

Achievable Angular Sensitivity:

Suitable Measurement Scheme

Novel Ramsey SpectroscopyEmploying Joint-Detectionand Nonclassical Fields

Ramsey Spectroscopy

N. F. Ramsey, Molecular Beams, (Oxford Univ. Press, New York, 1956).M. M. Salour and C. Cohen-Tannoudji, Phys. Rev. Lett. 38, 757 (1977).J. C. Bergquist, S. A. Lee, and J. L. Hall, Phys. Rev. Lett. 38, 159 (1977).

Ramsey Spectroscopy – Transit Time Broadening

Probability of getting excited

The frequency resolution is limited by the time passing through the field.

How can we enhance the resolution by several orders?How can we enhance the resolution by several orders?

Ramsey Spectroscopy – Separate Fields

Ramsey’s fringes are sharper than traditional ones.

T

Ramsey Spectroscopy – Two Path Interferometer

T

Two amplitudes are summed up coherently.

Ramsey Spectroscopy – Go Further

Squeezed Fields

Detected Signal

Other possible improvement:

• Stimulated PDC

In the squeezed case, fringe oscillates twice as fast.

•Using entangled photon sources like Parametric Down Conversion (PDC) to produce superresolution and supersensitivity

•Magneto-optical Rotation (MOR) of light and sensing of magnetic fields

•Measurement of rotations as in Sagnac Interferometers; gyroscopes

Conclusions

•Improving the sensitivity of light scattering measurements

•Quantum sensors; target detection

•New types of Interferometers

•Newer Detection Schmes

•Atomic Interferometers

l∆

: path differencel∆

MACH-ZEHNDER Interferometer

2 2Outputs: cos , sin2 2

k l k l∆ ∆

Rayleigh criterion: 2 2

k l π∆ =2

lλ∆ �

2Super-resolution: cos , 2

Nk l

φ φ = ∆

Documents

Super Resolution and Beating Shot Noise Limit by Entangled ... · F. Sciarrino, C. Vitelli, F. D. Martini, R. Glasser, H. Cable, and J. P. Dowling, Phys. Rev. A 77 , 012324 (2008)