super resolution and beating shot noise limit by entangled ... · f. sciarrino, c. vitelli, f. d....
TRANSCRIPT
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∆
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 State: CATU↓↓↓ ↓ →L
2i2: e
xSOperation U
π
=
Heisenberg Limited Measurements
)(2
1 2/4i ↑↑↑↑+↓↓↓↓=
↓↓↓↓=Ψ
−LLLL
LL
ππ
iNeie
U
↓↓L
2
2
ixS
eUπ
=CAT
Interferometer
φMeasurementMeasurement
Output +UzSie φ
Input
Heisenberg Limited Measurements
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
Heisenberg Limited Measurements
� 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 ˆa 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 ˆa b b a⟨ ⟩
visibility of 3 3 3 3ˆ ˆˆ ˆa 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:
Super Resolution via Newer Observables
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.
Super Resolution via Newer Observables
• 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
Super Resolution via Newer Observables
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
Super Resolution via Newer Observables
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:
(θ is the angle of rotation)
Achievable Angularsensitivity:
A. K. Jha, G. S. Agarwal, and R. W. Boyd, Phys. Rev. A 83, 053829 (2011)
Supersensitive Angular-Displacement Measurements
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
(θ is the angle of rotation)
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:
Supersensitive Angular-Displacement Measurements
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 – 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