Jonathan P' Dowling

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QUANTUM LITHOGRAPHY THEORY WHATS NEW WITH N00N
STATES?
Jonathan P. Dowling
Hearne Institute for Theoretical Physics Quantum
Science and Technologies Group Louisiana State
University Baton Rouge, Louisiana USA

quantum.phys.lsu.edu
Quantum Imaging MURI Annual Review, 23 October
2006, Ft. Belvoir
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Hearne Institute for Theoretical Physics Quantum
Science Technologies Group
H.Cable, C.Wildfeuer, H.Lee, S.Huver, W.Plick,
G.Deng, R.Glasser, S.Vinjanampathy,
K.Jacobs, D.Uskov, J.P.Dowling, P.Lougovski,
N.VanMeter, M.Wilde, G.Selvaraj, A.DaSilva Not
Shown R.Beaird, J. Brinson, M.A. Can,
A.Chiruvelli, G.A.Durkin, M.Erickson, L.
Florescu, M.Florescu, M.Han, K.T.Kapale, S.J.
Olsen, S.Thanvanthri, Z.Wu, J. Zuo
3
Quantum Lithography Theory
Objective Entangled Photons Beat Diffraction
Limit Lithography With Long-Wavelengths
Dispersion Cancellation Masking Techniques
N-Photon Resists
Accomplishments Investigated Properties of
N00N States GA Durkin JPD, quant-ph/0607088 CF
Wildfeuer, AP Lund JPD, quant-ph/0610180
First Efficient N00N Generators H Cable, R
Glasser, JPD, in preparation (posters). N
VanMeter, P Lougovski, D Uskov, JPD in prep. CF
Wildfeuer, AP Lund, JPD, in prep.
Approach Investigate Which States are
Optimal Design Efficient Quantum State
Generators Investigate Masking Systems
Develop Theory of N-Photon Resist Integrate
into Optical System Design
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Quantum Lithography A Systems Approach
Non-Classical Photon Sources
N-Photon Absorbers
Imaging System
Ancilla Devices
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Outline

• Nonlinear Optics vs. Projective Measurements
• Quantum Imaging Lithography
• Showdown at High N00N!
• Efficient N00N-State Generating Schemes
• Conclusions

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The Quantum Interface
Quantum Imaging
Quantum Sensing
Quantum Computing
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High-N00N Meets Quantum Computing
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Outline

• Nonlinear Optics vs. Projective Measurements
• Quantum Imaging Lithography
• Showdown at High N00N!
• Efficient N00N-State Generating Schemes
• Conclusions

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Optical C-NOT with Nonlinearity
The Controlled-NOT can be implemented using a
Kerr medium
0? H? Polarization 1? V? Qubits
R is a ?/2 polarization rotation, followed by a
polarization dependent phase shift ?.
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Two Roads to C-NOT
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.
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WHY IS A KERR NONLINEARITY LIKE A PROJECTIVE
MEASUREMENT?
Photon-Photon XOR Gate
Cavity QED EIT
  LOQC   KLM
Photon-Photon Nonlinearity
???
Kerr Material
Projective Measurement
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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 ?? Effective Unitary Gates
Franson CNOT Hamiltonian
KLM CSIGN Hamiltonian
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Single-Photon Quantum Non-Demolition
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).
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Linear Single-Photon Quantum 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.
Effective ?  1/8 ? 22 Orders of Magnitude
Improvement!
P. Kok, H. Lee, and JPD, PRA 66 (2003) 063814
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Outline

• Nonlinear Optics vs. Projective Measurements
• Quantum Imaging Lithography
• Showdown at High N00N!
• Efficient N00N-State Generating Schemes
• Conclusions

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H.Lee, P.Kok, JPD, J Mod Opt 49, (2002) 2325.
Quantum Metrology
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AN Boto, DS Abrams, CP Williams, JPD, PRL 85
(2000) 2733
N-Photon Absorber
a N a N
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Quantum Lithography Experiment
20gt02gt
10gt01gt
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Classical Metrology Lithography
Suppose we have an ensemble of N states ??
(0? ei? 1?)/?2,
?
A 0? 1 1? 0
?
and we measure the following observable
?A?? N cos ?
?
The expectation value is given by
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.
P Kok, SL Braunstein, and JP Dowling, Journal of
Optics B 6, (2004) S811
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Quantum Lithography Metrology
Now we consider the state
and we measure
?N AN?N? cos N?
Quantum Lithography
?
?AN
??H
Quantum Metrology
d AN?/d?
?
P. Kok, H. Lee, and J.P. Dowling, Phys. Rev. A
65, 052104 (2002).
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Outline

• Nonlinear Optics vs. Projective Measurements
• Quantum Imaging Lithography
• Showdown at High N00N!
• Efficient N00N-State Generating Schemes
• Conclusions

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Showdown at High-N00N!
How do we make N00N!?
N,0? 0,N?
With a large Kerr nonlinearity!
1?
0?
N?
N,0? 0,N?
0?
This is not practical! need ?  p but
?  1022 !
C. Gerry, and R.A. Campos, Phys. Rev. A 64,
063814 (2001).
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Projective Measurements to the Rescue
H. Lee, P. Kok, N.J. Cerf, and J.P. Dowling,
Phys. Rev. A 65, R030101 (2002).
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Inefficient High-N00N Generator
Not Efficient!
P Kok, H Lee, JP Dowling, Phys. Rev. A 65
(2002) 0512104
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High-N00N Experiments!
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1001gt
1001gt
2002gt
2002gt
3003gt
3003gt
4004gt
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quant-ph/0511214
1001gt
6006gt
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Outline

• Nonlinear Optics vs. Projective Measurements
• Quantum Imaging Lithography
• Showdown at High N00N!
• Efficient N00N-State Generating Schemes
• Conclusions

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The Lowdown on High-N00N
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Local and Global Distinguishability in Quantum
Interferometry Gabriel A. Durkin JPD,
quant-ph/0607088
A statistical distinguishability based on
relative entropy characterizes the fitness of
quantum states for phase estimation. This
criterion is used to interpolate between two
regimes, of local and global phase
distinguishability. The analysis demonstrates
that the Heisenberg limit is the true upper limit
for local phase sensitivity and Only N00N
States Reach It!
N00N
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NOON-States Violate Bells Inequalities!

• CF Wildfeuer, AP Lund and JP Dowling,
  quant-ph/0610180

1001gt Banaszek, Wodkiewicz, PRL 82 2009, (1999)
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NOON-States Violate Bells Inequalities

• CF Wildfeuer, AP Lund and JP Dowling,
  quant-ph/0610180

Probabilities of correlated clicks and
independent clicks
Building a Clauser-Horne Bell inequality from the
expectation values
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Wigner Function for NOON-States

• CF Wildfeuer, AP Lund and JP Dowling,
  quant-ph/0610180

The two-mode Wigner function has an operational
meaning as a correlated parity measurement
(Banaszek, Wodkiewicz)
Calculate the marginals of the two-mode Wigner
function to display nonlocal correlations of two
variables!
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Efficient Schemes for Generating N00N States!
Question Do there exist operators U that
produce N00N States Efficiently? Answer
YES! H Cable, R Glasser, JPD, in preparation,
see posters. N VanMeter, P Lougovski, D Uskov,
JPD, in preparation. KT Kapale JPD, in
preparation.
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Quantum P00Per Scooper!
H Cable, R Glasser, JPD, in preparation, see
posters.
2-mode squeezing process
?
beam splitter
How to eliminate the POOP?
quant-ph/0608170 G. S. Agarwal, K. W. Chan, R.
W. Boyd, H. Cable and JPD
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Quantum P00Per Scooper!
H Cable, R Glasser, JPD, in preparation, see
posters.
Pie Phase Shifter
Spinning wheel. Each segment a different
thickness. N00N is in Decoherence-Free Subspace!
Feed Forward based circuit
Generates and manipulates special cat states for
conversion to N00N states.First theoretical
scheme scalable to many particle experiments.
(In preparation SEE POSTERS!)
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Linear Optical Quantum State Generator (LOQSG)
N VanMeter, P Lougovski, D Uskov, JPD, in
preparation.

• Terms Conditions
• Only disentangled inputs are allowed
• ( )
• Modes transformation is unitary
• (U is a set of beam splitters)
• Number-resolving photodetection
• (single photon detectors)

M-port photocounter
Linear optical device (Unitary action on modes)
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Linear Optical Quantum State Generator (LOQSG)
N VanMeter, P Lougovski, D Uskov, JPD, in
preparation.

• Forward Problem for the LOQSGDetermine a set of
  output states which can be generated
  using different ancilla resources.
• Inverse Problem for the LOQSGDetermine linear
  optical matrix U generating required target state
  .
• Optimization Problem for the Inverse Problem
• Out of all possible solutions of the Inverse
  Problem determine the one with the greatest
  success probability

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LOQSG Answers

• Theory of invariants can solve the inverse
  problem but there is no theory of invariants
  for unitary groups!
• The inverse problem can be formulated in terms of
  a system of polynomial equations then if
  unitarity conditions are relaxed we can find a
  desired mode transform U using Groebner Basis
  technique.
• Unitarity can be later efficiently restored using
  extension theorem.
• The optimal solution can be found analytically!

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LOQSG A N00N-State Example
U
This counter example disproves the N00N
Conjecture That N Modes Required for 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 m2 modes is O(m2).
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Numerical Optimization
Optimizing success probability for the
non-linear sign gate by steepest ascent method
An optimal unitary
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High-N00N Meets Phaseonium
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Quantum Fredkin Gate (QFG) N00N GenerationKT
Kapale and JPD, in preparation.

• With sufficiently high cross-Kerr nonlinearity
  N00N generation possible.
• Implementation via Phaseonium

Gerry and Campos, PRA 64 063814 (2001)
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Phaseonium for N00N generation via the QFG KT
Kapale and JPD, in preparation.

• Two possible methods
• As a high-refractive index material to obtain the
  large phase shifts
• Problem Requires entangled phaseonium
• As a cross-Kerr nonlinearity
• Problem Does not offer required phase shifts of
  ? as yet (experimentally)

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Phaseonium for High Index of Refraction
Re
Im
Im
Re
With larger density high index of refraction can
be obtained
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N00N Generation via Phaseonium as a Phase Shifter
The needed large phase-shift of ? can be obtained
via the phaseonium as a high refractive index
material.
However, the control required by the Quantum
Fredkin gate necessitates the atoms be in the GHZ
state between level a and b Which could be
possible for upto 1000 atoms.
Question Would 1000 atoms give sufficiently high
refractive index?
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N00N Generation via Phaseonium Based Cross-Kerr
Nonlinearity

• Cross-Kerr nonlinearities via Phaseonium have
  been shown to impart phase shifts of 7?controlled
  via single photon
• One really needs to input a smaller N00N as a
  control for the QFG as opposed to a single photon
  with N30 roughly to obtain phase shift as large
  as ?.
• This suggests a bootstrapping approach

In the presence of single signal photon, and the
strong drive a weak probe field experiences a
phase shift
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Implementation of QFG via Cavity QED
Ramsey Interferometry for atom initially in
state b.
Dispersive coupling between the atom and cavity
gives required conditional phase shift
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Low-N00N via Entanglement swapping The N00N gun

• Single photon gun of Rempe PRL 85 4872 (2000) and
  Fock state gun of Whaley group quant-ph/0211134
  could be extended to obtain a N00N gun from
  atomic GHZ states.
• GHZ states of few 1000 atoms can be generated in
  a single step via (I) Agarwal et al. PRA 56 2249
  (1997) and (II) Zheng PRL 87 230404 (2001)
• By using collective interaction of the atoms with
  cavity a polarization entangled state of photons
  could be generated inside a cavity
• Which could be out-coupled and converted to N00N
  via linear optics.

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Bootstrapping

• Generation of N00N states with N roughly 30 with
  cavity QED based N00N gun.
• Use of Phaseonium to obtain cross-Kerr
  nonlinearity and the N00N with N30 as a control
  in the Quantum Fredkin Gate to generate high N00N
  states.
• Strong light-atom interaction in cavity QED can
  also be used to directly implement Quantum
  Fredkin gate.

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Conclusions

• Nonlinear Optics vs. Projective Measurements
• Quantum Imaging Lithography
• Showdown at High N00N!
• Efficient N00N-State Generating Schemes
• Conclusions