Methods of Entangling Large
Numbers of Photons for Enhanced
Phase Resolution
Jonathan P. Dowling
Hearne Institute for Theoretical Physics
Quantum Science and Technologies Group
Louisiana State University
Baton Rouge, Louisiana USA
quantum.phys.lsu.edu
Entanglement Beyond the Optical Regime
ONR Workshop, Santa Ana, CA 10 FEB 2010
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Hearne Institute for Theoretical Physics
Quantum Science & Technologies Group
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.DaSilva
Not 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,
Outline
1. Nonlinear Optics vs. Projective
Measurements
2. Quantum Imaging vs. Precision
Measurements
3. Showdown at High N00N!
4. Efficient N00N Generators
5. Mitigating Photon Loss
Optical C-NOT with Nonlinearity
The Controlled-NOT can be implemented using a Kerr medium:
|0= |H Polarization
|1= |V Qubits
(3)
PBS
R is a /2 polarization rotation,
followed by a polarization dependent
phase shift .
Rpol
z
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).
Two Roads to
Optical Quantum Computing
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.
Cavity QED
WHY IS A KERR NONLINEARITY LIKE
A PROJECTIVE MEASUREMENT?
LOQC
KLM
Photon-Photon
XOR Gate
Cavity QED
EIT
Photon-Photon
Nonlinearity
Projective
Measurement
Kerr Material
Projective Measurement
Yields Effective “Kerr”!
G. G. Lapaire, P. Kok,
JPD, J. E. Sipe, PRA 68
(2003) 042314
A Revolution in Nonlinear Optics at the Few Photon Level:
No Longer Limited by the Nonlinearities We Find in Nature!
NON-Unitary Gates 
KLM CSIGN Hamiltonian
Effective Unitary Gates
Franson CNOT Hamiltonian
Single-Photon Quantum
Non-Demolition
You want to know if there is a single photon in mode b,
without destroying it.
Cross-Kerr Hamiltonian: HKerr =
|in b
|1 a
 a †a b †b
Kerr medium
|1
D2
D1
“1”
Again, with  = 10–22, this is impossible.
*N. Imoto, H.A. Haus, and Y. Yamamoto, Phys. Rev. A. 32, 2287 (1985).
Linear Single-Photon
Quantum Non-Demolition
The success probability is less
than 1 (namely 1/8).
D0
|1
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.
D1
D2
 /2
|1
 /2
Effective  = 1/8
 22 Orders of
Magnitude
Improvement!
|0
|in =
2
cn |n

n=0
|1
P. Kok, H. Lee, and JPD, PRA 66 (2003) 063814
Outline
1. Nonlinear Optics vs. Projective
Measurements
2. Quantum Imaging vs. Precision
Measurements
3. Showdown at High N00N!
4. Efficient N00N Generators
5. Mitigating Photon Loss
Quantum Metrology
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.
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Low N00N
2 0  ei 2  0 2
HL
SNL
AN Boto, DS Abrams,
CP Williams, JPD,
PRL 85 (2000) 2733
Super-Resolution
a† N a N
Sub-Rayleigh
New York Times
Discovery
Could Mean
Faster
Computer
Chips
Quantum Lithography Experiment
Low N00N
2 0  ei 2  0 2
|20>+|02
>
|10>+|01
>
Canonical Metrology
Suppose we have an ensemble of N states | = (|0 + ei |1)/2,
and we measure the following observable: A = |0 1| + |1 0|


The expectation value is given by: |A| = N cos 
and the variance (A)2 is given by: N(1cos2)

The unknown phase can be estimated with accuracy:
A
1
 =
=
| d A/d | N

This is the standard shot-noise limit.
note the
square-root
P Kok, SL Braunstein, and JP Dowling, Journal of Optics B 6, (2004) S811
Quantum
Lithography & Metrology
 N   N,0  0,N
Now we consider the state
AN  0,N N,0  N,0 0,N
Quantum Lithography*:

Quantum Metrology:
N |AN|N = cos N
High-Frequency
Lithography
Effect
AN
1
H =
=
| d AN/d |
N



and we measure

Heisenberg Limit:
No Square Root!
P. Kok, H. Lee, and J.P. Dowling, Phys. Rev. A 65, 052104 (2002).
Recap: Super-Resolution

N=1 (classical)
N=5 (N00N)

 
P̂
Recap: Super-Sensitivity
d P̂ / d
dP1/d
N=1 (classical)
N=5 (N00N)
dPN/d
Outline
1. Nonlinear Optics vs. Projective
Measurements
2. Quantum Imaging vs. Precision
Measurements
3. Showdown at High N00N!
4. Efficient N00N Generators
5. Mitigating Photon Loss
Showdown at High-N00N!
How do we make High-N00N!?
|N,0 + |0,N
With a large cross-Kerr
nonlinearity!* H =  a†a b†b
|1
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|0
|N
|0
This is not practical! —
need  =  but  = 10–22 !
N00N States
In Chapter 11
*C Gerry, and RA Campos, Phys. Rev. A 64, 063814 (2001).
N00N & Linear Optical Quantum Computing
For proposals* to exploit a non-linear photon-photon interaction
e.g. cross-Kerr interaction Ĥ   â †âb̂ †b̂ ,
the required optical non-linearity not readily accessible.
Nature 409,
page 46,
(2001).
*C. Gerry, and R.A. Campos, Phys. Rev. A 64, 063814 (2001).
Measurement-Induced Nonlinearities
G. 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.
Scheme conditions on the detection
of one photon at each detector
mode a
e.g.
component of
light from an
optical
parametric
oscillator
mode b
H. Lee, P. Kok, N. J. Cerf and J. P. Dowling, PRA 65, 030101 (2002).
Towards High-N00N!
a
c
a’
cascade
b
d
1
PS
2
3
N
2
b’
|N,N  |N-2,N + |N,N-2
p1 = 1 N (N-1) T2N-2 R2  1 2
N 2e
2
with T = (N–1)/N and R = 1–T
|N,N  |N,0 + |0,N
the consecutive phases are given by:
2 k
 k = N/2
Not Efficient!
Kok, Lee, & Dowling, Phys. Rev. A 65 (2002) 0512104
Implemented in Experiments!
|10::01>
|10::01>
|20::02>
|20::02>
|30::03>
|40::04>
|30::03>
N00N State Experiments
1990
2-photon
Rarity, (1990)
Ou, et al. (1990)
Shih, Alley (1990)
….
Mitchell,…,Steinberg
Nature (13 MAY)
Toronto
2004
3, 4-photon
Super6-photon
resolution
Super-resolution
only
Only!
Resch,…,White
PRL (2007)
Queensland
2007
4-photon
Super-sensitivity
&
Super-resolution
Walther,…,Zeilinger
Nature (13 MAY)
Vienna
Nagata,…,Takeuchi,
Science (04 MAY)
Hokkaido & Bristol
N00N
Physical Review 76, 052101 (2007)
N00N
LHV
Outline
1. Nonlinear Optics vs. Projective
Measurements
2. Quantum Imaging vs. Precision
Measurements
3. Showdown at High N00N!
4. Efficient N00N Generators
5. Mitigating Photon Loss
Efficient Schemes for
Generating N00N States!
|N>|0>
Constrained
|1,1,1>
|N0::0N>
Desired
Number
Resolving
Detectors
Question: Do there exist operators “U” that produce “N00N” States Efficiently?
Answer: YES!
Phys. Rev. Lett. 99, 163604 (2007)
Linear Optical N00N Generator II
Terms & Conditions
M-port photocounter
• Only disentangled inputs are allowed
(

 n
 ...  n
)
1 1
R isR unitary
in transformation
• Modes
(U is a set of beam splitters)
•Number-resolving photodetection
(single photon detectors)
Linear optical device
(Unitary action on modes)
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




0.03
2
2
2
0
U
2
1
0
( 50  05 )



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).
Entanglement Seeded Dual OPA
1  2 tanh

Prob(n, m) 
2
r  tanh r 
4
4
2
tanh 2( n  m ) r[(n  1)(n  2)  (m  1)(m  2)]
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)
We present a theoretical analysis of the properties of an unseeded
optical parametric amplifier (OPA) used as the source of
entangled photons.
OPA Scheme
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.
G.S.Agarwal, et al., J. Opt. Soc. Am. B 24, 270 (2007).
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.
State Before Projection
Visibility Saturates
at 20% with
105 Counts Per
Second!
Lovely, Fresh, Data!
F.Sciarrino, et al., Phys. Rev. A 77, 012324 (2008)
This is an Experiment!
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.
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Return of the Kerr Nonlinearity!
How do we make High-N00N!?
|N,0 + |0,N
With a large cross-Kerr
nonlinearity!* H =  a†a b†b
|1
|0
|N
|0
This is not practical! —
need  =  but  = 10–22 !
*C Gerry, and RA Campos, Phys. Rev. A 64, 063814 (2001).
Implementation of QFG via Cavity QED
Kapale, 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
1. Nonlinear Optics vs. Projective
Measurements
2. Quantum Imaging vs. Precision
Measurements
3. Showdown at High N00N!
4. Efficient N00N Generators
5. Mitigating Photon Loss
Computational Optimization of
Quantum LIDAR
Noise
Target
INPUT
Nonclassical
Light
Source
inverse problem solver
“find
min(
)“

Delay
Line
Detection
forward problem solver
N: photon
number

in 
N
loss A
c
loss B
i
N  i, i
  f ( in ,  ; loss A, loss B)

i 0


FEEDBACK LOOP:

Genetic Algorithm
OUTPUT
min(  ) ;
N
)
in(OPT )   c (OPT
N  i, i , OPT
i
i 0

Lee, TW; Huver, SD; Lee, H; et al.
PHYSICAL REVIEW A, 80 (6): Art. No. 063803 DEC 2009
Loss in Quantum Sensors
SD Huver, CF Wildfeuer, JP Dowling, Phys. Rev. A 78 # 063828 DEC 2008
La

N00N

Detector
Generator

Lb
Visibility:
  ( 10,0  0,10 )
Lost
photons
2

Lost
photons
Sensitivity:   ( 10,0  0,10 )
2
N00N 3dB
Loss ---


N00N No
Loss —
SNL--3/14/2016
43
HL—
Super-Lossitivity
Gilbert, G; Hamrick, M; Weinstein, YS; JOSA B 25 (8): 1336-1340 AUG 2008
 
P̂
d P̂ / d
dPN /d
N=1 (classical)
N=5 (N00N)

e
L
e
NL
dP1 /d
3dB Loss, Visibility & Slope — Super Beer’s Law!
Loss in Quantum Sensors
S. Huver, C. F. Wildfeuer, J.P. Dowling, Phys. Rev. A 78 # 063828 DEC 2008
A
N00N

Lost
photons
La

Detector
B
Generator
Lb

Lost
photons

Gremlin
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
iN
0
A
N
B
0
A
N 1 B
Towards A Realistic Quantum Sensor
S. Huver, C. F. Wildfeuer, J.P. Dowling, Phys. Rev. A 78 # 063828 DEC 2008
Lost
photons
Try other detection scheme and states!
M&M
La


Detector
Generator
M&M state:
  ( m,m'  m',m )
M&M Visibility
  ( 20,10  10,20 )

Lb
N00N Visibility
  ( 10,0  0,10 )
2

0.3
Lost
photons
2

0.05
2
M&M’ Adds Decoy Photons
Towards A Realistic Quantum Sensor
S. Huver, C. F. Wildfeuer, J.P. Dowling, Phys. Rev. A 78 # 063828 DEC 2008
Try other detection scheme and states!
M&M
La


Detector
Generator
M&M state:
  ( m,m'  m',m )


Lost
photons
Lb
2
Lost
photons
N00N State --M&M State —
N00N SNL --M&M SNL --M&M HL —
M&M HL —
A Few
Photons
Lost
Does Not
Give
Complete
“Which Path”
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.
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Lossy State Comparison
PHYSICAL 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 that
The 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.
Quantum
Classical
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Magic
Detector!
Waves are Coherent!
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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
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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.
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Outline
1. Nonlinear Optics vs. Projective
Measurements
2. Quantum Imaging vs. Precision
Measurements
3. Showdown at High N00N!
4. Efficient N00N Generators
5. Mitigating Photon Loss
Entangled “SQUID” Qubit Magnetometer
A Guillaume & JPD, Physical Review A Rapid 73 (4): Art. No. 040304 APR 2006.
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Heisenberg Limited
Magnetometer:
N00N in Microwave Photons
Transferred into N00N in SQUID
Qubits and Reverse!
Qg   x  1 / N
Quantum Magnetometer Arrays Can Detect Submarines
From Orbit
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Outline
1. Nonlinear Optics vs. Projective
Measurements
2. Quantum Imaging vs. Precision
Measurements
3. Showdown at High N00N!
4. Efficient N00N Generators
5. Mitigating Photon Loss
Microwave to Optical Transducer
KT Kapale and JPD, in preparation
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Outline
1. Nonlinear Optics vs. Projective
Measurements
2. Quantum Imaging vs. Precision
Measurements
3. Showdown at High N00N!
4. Efficient N00N Generators
5. Mitigating Photon Loss