1 / 52

Quantum Lithography Theory: What's New with N00N States?

Explore the latest advancements in quantum lithography theory, including entangled photons beating diffraction limits, lithography with long wavelengths, dispersion cancellation, masking techniques, and N-photon resists.

rmccartney
Download Presentation

Quantum Lithography Theory: What's New with N00N States?

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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 QuantumScience & 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

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

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

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

  7. High-N00N Meets Quantum Computing

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

  9. (3) PBS 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). 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 .

  10. Cavity QED 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.

  11. 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

  12. 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

  13. Single-Photon Quantum Non-Demolition Cross-Kerr Hamiltonian: HKerr=a†ab†b Kerr medium b |in |1 |1 a D2 D1 Again, with  = 10–22, this is impossible. “1” 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).

  14. D0 |1 D1 D2 /2 |1 /2 |0 |1 2 |in = cn |n  n = 0 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

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

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

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

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

  19. Note the Square Root! Classical Lithography:  = kx 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)2is given by:N(1cos2)  The unknown phase can be estimated with accuracy: A 1  = =  |dA/d | N This is the standard shot-noise limit. P Kok, SL Braunstein, and JP Dowling, Journal of Optics B 6, (2004) S811

  20. Quantum Lithography & Metrology Quantum Lithography Effect: N = Nkx 1 N Heisenberg Limit — No Square Root! Now we consider the state and we measure N |AN|N = cos N QuantumLithography:  AN H = = QuantumMetrology: |dAN/d |  P. Kok, H. Lee, and J.P. Dowling, Phys. Rev. A 65, 052104 (2002).

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

  22. 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  = 10–22 ! *C. Gerry, and R.A. Campos, Phys. Rev. A64, 063814 (2001).

  23. single photon detection at each detector a a’ b b’ Best we found: Probability of success: Projective Measurements to the Rescue H. Lee, P. Kok, N.J. Cerf, and J.P. Dowling, Phys. Rev. A 65, R030101 (2002).

  24. a a’ c N cascade 1 2 3 2 d PS b b’ |N,N  |N-2,N + |N,N-2 |N,N  |N,0 + |0,N the consecutive phases are given by: 2 k k = 1 1 N/2 with T = (N–1)/N and R = 1–T p1 = N (N-1) T2N-2R2 2 2e2 N Inefficient High-N00N Generator Not Efficient! P Kok, H Lee, & JP Dowling, Phys. Rev. A65 (2002) 0512104

  25. High-N00N Experiments!

  26. |10::01> |10::01> |20::02> |20::02> |30::03> |30::03> |40::04>

  27. quant-ph/0511214 |10::01> |60::06>

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

  29. The Lowdown on High-N00N

  30. 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

  31. 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! |1001> Banaszek, Wodkiewicz, PRL 82 2009, (1999) CF Wildfeuer, AP Lund and JP Dowling, quant-ph/0610180

  32. NOON-States Violate Bell’s Inequalities Probabilities of correlated clicks and independent clicks Building a Clauser-Horne Bell inequality from the expectation values CF Wildfeuer, AP Lund and JP Dowling, quant-ph/0610180

  33. Wigner Function for NOON-States 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! CF Wildfeuer, AP Lund and JP Dowling, quant-ph/0610180

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

  35. linear optical processing Quantum P00Per Scooper! H Cable, R Glasser, & JPD, in preparation, see posters. 2-mode squeezing process χ beam splitter How to eliminate the “POOP”? quant-ph/0608170 G. S. Agarwal, K. W. Chan, R. W. Boyd, H. Cable and JPD

  36. 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!)

  37. 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)

  38. Linear Optical Quantum State Generator (LOQSG)N VanMeter, P Lougovski, D Uskov, JPD, in preparation. • Forward Problem for the LOQSGDetermine a set of output stateswhich can be generatedusing 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

  39. 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!

  40. 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).

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

  42. High-N00N Meets Phaseonium

  43. With sufficiently high cross-Kerr nonlinearity N00N generation possible. Implementation via Phaseonium Quantum Fredkin Gate (QFG) N00N GenerationKT Kapale and JPD, in preparation. Gerry and Campos, PRA 64 063814 (2001)

  44. 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 N00N generation via the QFGKT Kapale and JPD, in preparation.

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

  46. 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?

  47. 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 N00N Generation via Phaseonium Based Cross-Kerr Nonlinearity In the presence of single signal photon, andthe strong drive a weak probe field experiences a phase shift

  48. 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

  49. 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. Low-N00N via Entanglement swapping: The N00N gun

More Related