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

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 • 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/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.

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

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

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

High-N00N Meets Quantum Computing

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

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).

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.

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

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

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).

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) 063814

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

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

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

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

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

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, 052104 (2002).

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

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, 063814 (2001).

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, R030101 (2002).

Inefficient High-N00N Generator b’ 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) 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 Nonlinear Optics vs. Projective Measurements Quantum Imaging & Lithography Showdown at High N00N! Efficient N00N-State Generating Schemes Conclusions

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: 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

NOON-States Violate Bell’s 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

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!

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.

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/0608170 G. S. Agarwal, K. W. Chan, R. W. Boyd, H. Cable and JPD

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!)

(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)

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

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 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 m–2 modes is O(m2).

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

High-N00N Meets Phaseonium

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 64 063814 (2001)

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)

Phaseonium for High Index of Refraction Im Im Re 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 Nonlinear Optics vs. Projective Measurements Quantum Imaging & Lithography Showdown at High N00N! Efficient N00N-State Generating Schemes Conclusions