1. Jonathan P. Dowling QUANTUM LITHOGRAPHY THEORY:
WHAT’S 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

2. 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
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
Accomplishments:
• Investigated Properties of N00N States
GA Durkin & JPD, quant-ph/
CF Wildfeuer, AP Lund & JPD, quant-ph/
• 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.

5. Quantum Lithography: A Systems Approach
Non-Classical Photon Sources
Imaging System
N-Photon Absorbers
Ancilla
Devices

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

7. The Quantum Interface You are here! Quantum Imaging Quantum Quantum
Sensing
Quantum
Computing

8. High-N00N Meets Quantum Computing

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

10. Optical C-NOT with Nonlinearity
The Controlled-NOT can be implemented using a Kerr medium:
(3)
Rpol
PBS z
|0= |H Polarization
|1= |V Qubits
R is a /2 polarization rotation,
followed by a polarization dependent
phase shift .
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).

11. Two Roads to C-NOT
Cavity QED
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.

12. WHY IS A KERR NONLINEARITY LIKE A PROJECTIVE MEASUREMENT?
Photon-Photon
XOR Gate
  LOQC  
KLM
Cavity QED
EIT
Photon-Photon
Nonlinearity
???
Kerr Material
Projective
Measurement

13. Projective Measurement Yields Effective “Kerr”!
G. G. Lapaire, P. Kok, JPD, J. E. Sipe, PRA 68 (2003)
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

14. 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 =  a†a b†b
Again, with  = 10–22, this is impossible.
Kerr medium
“1” a b
|in
|1 D1 D2
*N. Imoto, H.A. Haus, and Y. Yamamoto, Phys. Rev. A. 32, 2287 (1985).

15. Quantum Non-Demolition  22 Orders of Magnitude Improvement!
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.
|1 D1 D2 D0
 /2
|in = cn |n 
n = 0 2
|0
Effective  = 1/8
 22 Orders of Magnitude Improvement!
P. Kok, H. Lee, and JPD, PRA 66 (2003)

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

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

18. a† N a N N-Photon Absorber
AN Boto, DS Abrams, CP Williams, JPD, PRL 85 (2000) 2733
N-Photon Absorber
a† N a N

19. Quantum Lithography Experiment
|20>+|02>
|10>+|01>

20. Classical Metrology & Lithography
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 
Classical
Lithography:
 = kx
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
Note the
Square Root!
This is the standard shot-noise limit.
P Kok, SL Braunstein, and JP Dowling, Journal of Optics B 6, (2004) S811

21. Lithography & Metrology
Quantum
Lithography & Metrology
Now we consider the state
and we measure
Quantum
Lithography
Effect:
N = Nkx
Quantum Lithography: 
N |AN|N = cos N
AN N 1
Quantum Metrology:
H = =
| d AN/d | 
Heisenberg Limit —
No Square Root!
P. Kok, H. Lee, and J.P. Dowling, Phys. Rev. A 65, (2002).

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

23. Showdown at High-N00N! |N,0 + |0,N How do we make N00N!?
With a large Kerr nonlinearity!*
|1
|0
|N
|N,0 + |0,N
|0
This is not practical! —
need  = p but  = 10–22 !
*C. Gerry, and R.A. Campos, Phys. Rev. A 64, (2001).

24. Projective Measurements
to the Rescue
single photon detection
at each detector a b a’ b’
Probability of success:
Best we found:
H. Lee, P. Kok, N.J. Cerf, and J.P. Dowling, Phys. Rev. A 65, R (2002).

25. Inefficient High-N00N Generatorb’ b c d
|N,N  |N-2,N + |N,N-2 PS
cascade 1 2 3 N
|N,N  |N,0 + |0,N
p1 = N (N-1) T2N-2 R2 
N 1 2
2e2
with T = (N–1)/N and R = 1–T
the consecutive phases are given by:
 k =
2 k
N/2
Not Efficient!
P Kok, H Lee, & JP Dowling, Phys. Rev. A 65 (2002)

26. High-N00N Experiments!

27. |10::01>
|10::01>
|20::02>
|20::02>
|30::03>
|40::04>
|30::03>

28. quant-ph/
|10::01>
|60::06>

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

30. The Lowdown on High-N00N

31. Local and Global Distinguishability in Quantum Interferometry
Gabriel A. Durkin & JPD, quant-ph/
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

32. NOON-States Violate Bell’s Inequalities!
CF Wildfeuer, AP Lund and JP Dowling, quant-ph/
|1001> Banaszek, Wodkiewicz, PRL , (1999)
Unbalanced homodyne tomography setup:
Beam splitters act as displacement operators
Local oscillators serve as a reference frame with amplitudes
Measuring clicks with respect to parameters
Binary result: click
no click

33. NOON-States Violate Bell’s Inequalities
CF Wildfeuer, AP Lund and JP Dowling, quant-ph/
Probabilities of correlated clicks and independent clicks
Building a Clauser-Horne Bell inequality from the expectation values

34. Wigner Function for NOON-States
CF Wildfeuer, AP Lund and JP Dowling, quant-ph/
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!

35. Efficient Schemes for Generating N00N States!
Constrained
Desired
|N>|0>
|N0::0N>
|1,1,1>
Number
Resolving
Detectors
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.

36. Quantum P00Per Scooper! χ 2-mode squeezing process linear optical
H Cable, R Glasser, & JPD, in preparation, see posters.
2-mode
squeezing process
linear
optical
processing χ
beam
splitter
How to eliminate
the “POOP”?
quant-ph/
G. S. Agarwal, K. W. Chan,
R. W. Boyd, H. Cable
and JPD

37. Quantum P00Per Scooper! “Pie” Phase Shifter Feed Forward based circuit
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!)

38. (Unitary action on modes)
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)

39. Linear Optical Quantum State Generator (LOQSG) N VanMeter, P Lougovski, D Uskov, JPD, in preparation.
Forward Problem for the LOQSG Determine a set of output states which can be generated using different ancilla resources.
Inverse Problem for the LOQSG Determine 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

40. 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!

41. LOQSG: A N00N-State ExampleU
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 m–2 modes is O(m2).

42. Numerical Optimization
Optimizing “success probability” for the non-linear sign gate by steepest ascent method
Manifold of unitary matrices
Starting point
Patch of local coordinates
An optimal unitary

43. High-N00N Meets Phaseonium

44. Implementation via Phaseonium
Quantum Fredkin Gate (QFG) N00N Generation KT Kapale and JPD, in preparation.
With sufficiently high cross-Kerr nonlinearity N00N generation possible.
Implementation via Phaseonium
Gerry and Campos, PRA (2001)

45. As a high-refractive index material to obtain the large phase shifts
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)

46. Phaseonium for High Index of RefractionIm Im Re
With larger density high index of refraction can be obtained

47. 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?

48. 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 N=30 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

49. 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

50. Low-N00N via Entanglement swapping: The N00N gun
Single photon gun of Rempe PRL (2000) and Fock state gun of Whaley group quant-ph/ 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 (1997) and (II) Zheng PRL (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.

51. 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 N=30 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.

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