Transcript What`s new with NOON States

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
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
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
Accomplishments:
• Investigated Properties of N00N States
GA Durkin & JPD, quant-ph/0607088
• Design Efficient Quantum State Generators
CF Wildfeuer, AP Lund & JPD, quant-ph/0610180
• Investigate Masking Systems
• First Efficient N00N Generators
• Develop Theory of N-Photon Resist
H Cable, R Glasser, JPD, in preparation (posters).
• Integrate into Optical System Design
N VanMeter, P Lougovski, D Uskov, JPD in prep.
CF Wildfeuer, AP Lund, JPD, in prep.
Quantum Lithography:
A Systems Approach
NonClassica
l Photon
Sources
Imaging
System
Ancilla
Devices
N-Photon
Absorbers
Outline
1. Nonlinear Optics vs. Projective
Measurements
2. Quantum Imaging & Lithography
3. Showdown at High N00N!
4. Efficient N00N-State Generating Schemes
The Quantum Interface
Quantum
Imaging
Quantum
Computing
You are here!
Quantum
Sensing
High-N00N Meets Quantum Computing
Outline
1. Nonlinear Optics vs. Projective
Measurements
2. Quantum Imaging & Lithography
3. Showdown at High N00N!
4. Efficient N00N-State Generating Schemes
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 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.
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 & Lithography
3. Showdown at High N00N!
4. Efficient N00N-State Generating Schemes
Quantum Metrology
H.Lee, P.Kok, JPD,
J Mod Opt 49,
(2002) 2325.
AN Boto, DS
Abrams, CP
Williams, JPD, PRL
85 (2000) 2733
NPhoton
Absorbe
r
a† N a N
Quantum Lithography Experiment
|20>+|02
>
|10>+|01
>
Classical Metrology &
Lithography
Suppose we have an ensemble of N states | = (|0 + ei |1)/2,

A = |0 1| + |1 0|
|A| = N cos 

The expectation value is given by:

and we measure the following observable:
and the variance (A)2 is given by: N(1cos2)
The unknown phase can be estimated with accuracy:
Classical
Lithography:
 = kx
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
Quantum
Lithography
Effect:
N = Nkx
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).
Outline
1. Nonlinear Optics vs. Projective
Measurements
2. Quantum Imaging & Lithography
3. Showdown at High N00N!
4. Efficient N00N-State Generating Schemes
Showdown at High-N00N!
How do we make N00N!?
|N,0 + |0,N
With a large Kerr nonlinearity!*
|1
|0
|N
|0
This is not practical! —
need  =  but  = 10–22 !
*C. Gerry, and R.A. Campos, Phys. Rev. A 64, 063814 (2001).
Projective Measurements
to the Rescue
single photon detection
at each detector
a’
a
3a3
b
b’
b
6
4
2
0
a
a
a
a
0
2
4
6
b
b
3a1b
b
1a3b
4
a'
0
b'
0
a'
4
b'
b
Probability of success:
3
64
Best we found:
3
16
H. Lee, P. Kok, N.J. Cerf, and J.P. Dowling, Phys. Rev. A 65, R030101 (2002).
Inefficient High-N00N Generator
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!
P Kok, H Lee, & JP Dowling, Phys. Rev. A 65 (2002) 0512104
High-N00N Experiments!
|10::01>
|10::01>
|20::02>
|20::02>
|30::03>
|40::04>
|30::03>
quant-ph/0511214
|10::01>
|60::06>
Outline
1. Nonlinear Optics vs. Projective
Measurements
2. Quantum Imaging & Lithography
3. Showdown at High N00N!
4. Efficient N00N-State Generating Schemes
The Lowdown on High-N00N
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
NOON-States Violate Bell’s Inequalities!
CF Wildfeuer, AP Lund and JP Dowling, quant-ph/0610180
|1001> Banaszek, Wodkiewicz,
PRL 82 2009, (1999)
Unbalanced homodyne
tomography setup:
T 1
 
Beam splitters act as
ˆ ( 1 T  )
D
displacement operators


Local oscillators serve as a
reference frame 
with amplitudes
  1 T  a   1 T  b
Measuring clicks with respect to

parameters

,
Binary result: click
1
no click  0

NOON-States Violate Bell’s Inequalities
CF Wildfeuer, AP Lund and JP Dowling, quant-ph/0610180
Probabilities of correlated clicks and independent clicks
Pab (,),Pa (),Pb ()
Building a Clauser-Horne Bell inequality from the expectation
values Pab (,),Pa (),Pb ()


1 Pab (,)  Pab (,)  Pab (,)  Pab (,)  Pa ()  Pb ()  0

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!
N 1

N3
N 2


Efficient Schemes for
Generating N00N States!
|N>|0>
Constrained
|N0::0N>
Desired
|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.
Quantum P00Per Scooper!
H Cable, R Glasser, & JPD, in preparation, see posters.
2-mode
squeezing process
linear
optical
processing
χ
beam
splitter
How to eliminate
the “POOP”?
U(50:50)|4>|4>
0.3
|amplitude|^2
0.25
0.2
0.15
0.1
0.05
0
|0>|8>
|2>|6>
|4>|4>
Fock basis state
|6>|2>
|8>|0>
quant-ph/0608170
G. S. Agarwal, K. W. Chan,
R. W. Boyd, H. Cable
and JPD
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!)
Linear Optical Quantum State Generator (LOQSG )
N VanMeter, P Lougovski, D Uskov, JPD, in preparation.
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)
Linear Optical Quantum State Generator (LOQSG )
N VanMeter, P Lougovski, D Uskov, JPD, in preparation.
• Forward Problem for the LOQSG out
which can be
Determine a set of output states
generated using different ancilla resources.
• Inverse Problem for the LOQSG
 U generating required
Determine linear optical matrix
out
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
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!
LOQSG: A N00N-State Example




2
2
2
0
0.03
2
U
1
0



( 50  05 )
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).
Numerical Optimization
Optimizing “success probability” for the
non-linear sign gate by steepest ascent
method
Starting point
U
0
Patch of local coordinates
U
Manifold of unitary matrices
U opt
An optimal unitary
High-N00N Meets 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 64 063814 (2001)
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)
Phaseonium for High Index of Refraction
Re
Im
Im
Re
N  1015 cm-3
Re(  )  100 cm-3
n  10cm-3
With larger density high index of refraction can be obtained
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?
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
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
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.
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.
Conclusions
1. Nonlinear Optics vs. Projective
Measurements
2. Quantum Imaging & Lithography
3. Showdown at High N00N!
4. Efficient N00N-State Generating Schemes