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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(1cos2) 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 N3 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 is in transformation R unitary • 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 ( 50 05 ) 2 U 1 0 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