Jonathan P. Dowling LINEAR OPTICAL QUANTUM INFORMATION PROCESSING, IMAGING, AND SENSING: WHAT’S NEW WITH N00N STATES? quantum.phys.lsu.edu Hearne Institute for Theoretical Physics Louisiana State University Baton Rouge, Louisiana 14 JUNE 2007 ICQI-07, Rochester

H.Cable, C.Wildfeuer, H.Lee, S.Huver, W.Plick, G.Deng, R.Glasser, S.Vinjanampathy, K.Jacobs, D.Uskov, JP.Dowling, P.Lougovski, N.VanMeter, M.Wilde, G.Selvaraj, A.DaSilva Not Shown: M.A. Can, A.Chiruvelli, GA.Durkin, M.Erickson, L. Florescu, M.Florescu, M.Han, KT.Kapale, SJ. Olsen, S.Thanvanthri, Z.Wu, J.Zuo Hearne Institute for Theoretical Physics Quantum Science & Technologies Group

Outline 1.Quantum Computing & Projective Measurements 2.Quantum Imaging, Metrology, & Sensing 3.Showdown at High N00N! 4.Efficient N00N-State Generating Schemes 5.Conclusions

CNOT with Optical Nonlinearity The Controlled-NOT can be implemented using a Kerr medium: Unfortunately, the interaction  (3) is extremely weak*: at the single photon level — This is not practical! *R.W. Boyd, J. Mod. Opt. 46, 367 (1999). R is a  /2 polarization rotation, followed by a polarization dependent phase shift . R is a  /2 polarization rotation, followed by a polarization dependent phase shift .  (3) R pol PBS zz |0  = |H  Polarization |1  = |V  Qubits |0  = |H  Polarization |1  = |V  Qubits

Two Roads to Optical CNOT Cavity QED I. Enhance Nonlinearity with Cavity, EIT — Kimble, Walther, Haroche, Lukin, Zubairy, et al. II. Exploit Nonlinearity of Measurement — Knill, LaFlamme, Milburn, Franson, et al.

Linear Optical Quantum Computing Linear Optics can be Used to Construct CNOT and a Scaleable Quantum Computer: Knill E, Laflamme R, Milburn GJ NATURE 409 (6816): JAN Knill E, Laflamme R, Milburn GJ NATURE 409 (6816): JAN Franson JD, Donegan MM, Fitch MJ, et al. PRL 89 (13): Art. No SEP Franson JD, Donegan MM, Fitch MJ, et al. PRL 89 (13): Art. No SEP Milburn

Road to Entangled- Particle Interferometry: An Early Example of Entanglement Generation by Erasure of Which-Path Information Followed by Detection!

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

GG Lapaire, P Kok, JPD, JE Sipe, PRA 68 (2003) KLM CSIGN Hamiltonian Franson CNOT Hamiltonian NON-Unitary Gates   Effective Unitary Gates A Revolution in Nonlinear Optics at the Few Photon Level: No Longer Limited by the Nonlinearities We Find in Nature! A Revolution in Nonlinear Optics at the Few Photon Level: No Longer Limited by the Nonlinearities We Find in Nature! Projective Measurement Yields Effective “Kerr”!

Nonlinear Single-Photon Quantum Non-Demolition You want to know if there is a single photon in mode b, without destroying it. *N Imoto, HA Haus, and Y Yamamoto, Phys. Rev. A. 32, 2287 (1985). Cross-Kerr Hamiltonian: H Kerr =  a † a b † b Again, with  = 10 –22, this is impossible. Kerr medium “1”“1” a b |  in  |1|1 |1  D1D1 D2D2

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 D 1 and D 2. |1|1 |1|1 |1  D1D1 D2D2 D0D0  /2 |  in  = c n |n   n = 0 2 |0  Effective  = 1/8  21 Orders of Magnitude Improvement! Effective  = 1/8  21 Orders of Magnitude Improvement! P Kok, H Lee, and JPD, PRA 66 (2003)

Outline 1.Quantum Computing & Projective Measurements 2.Quantum Imaging, Metrology, & Sensing 3.Showdown at High N00N! 4.Efficient N00N-State Generating Schemes 5.Conclusions

Quantum Metrology with N00N States H Lee, P Kok, JPD, J Mod Opt 49, (2002) Supersensitivity! Shotnoise to Heisenberg Limit

a † N a N AN Boto, DS Abrams, CP Williams, JPD, PRL 85 (2000) 2733 Superresolution!

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

Canonical Metrology note the square-root P Kok, SL Braunstein, and JP Dowling, Journal of Optics B 6, (2004) S811 Suppose we have an ensemble of N states |  = (|0  + e i  |1  )/  2, and we measure the following observable: The expectation value is given by: and the variance (  A) 2 is given by: N(1  cos 2  ) A = |0  1| + |1  0|    |A|  = N cos   The unknown phase can be estimated with accuracy: This is the standard shot-noise limit.  = = AA | d A  /d  |  NN 1

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

Outline 1.Quantum Computing & Projective Measurements 2.Quantum Imaging, Metrology, & Sensing 3.Showdown at High N00N! 4.Efficient N00N-State Generating Schemes 5.Conclusions

Showdown at High-N00N! |N,0  + |0,N  How do we make High-N00N!? *C Gerry, and RA Campos, Phys. Rev. A 64, (2001). With a large cross-Kerr nonlinearity!* H =  a † a b † b This is not practical! — need  =  but  = 10 –22 ! |1  |N|N |0  |N,0  + |0,N  N00N States In Chapter 11

a b a’ b’ Probability of success:Best we found: Solution: Replace the Kerr with Projective Measurements! H Lee, P Kok, NJ Cerf, and JP Dowling, Phys. Rev. A 65, R (2002). single photon detection at each detector Cascading Not Efficient! OPO

These Ideas Implemented in Recent Experiments!

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

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, in a passive MZI, the Heisenberg limit is the true upper limit for local phase sensitivity — and Only N00N States Reach It! 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, in a passive MZI, the Heisenberg limit is the true upper limit for local phase sensitivity — and Only N00N States Reach It! N00N Local and Global Distinguishability in Quantum Interferometry GA Durkin & JPD, quant-ph/

NOON-States Violate Bell’s Inequalities Building a Clauser-Horne Bell inequality from the expectation values Probabilities of correlated clicks and independent clicks CF Wildfeuer, AP Lund and JP Dowling, quant-ph/ Shared Local Oscillator Acts As Common Reference Frame! Bell Violation!

Outline 1.Quantum Computing & Projective Measurements 2.Quantum Imaging, Metrology, & Sensing 3.Showdown at High N00N! 4.Efficient N00N-State Generating Schemes 5.Conclusions

Efficient Schemes for Generating N00N States! Question: Do there exist operators “U” that produce “N00N” States Efficiently? Answer: YES! H Cable, R Glasser, & JPD, quant-ph/ Linear! N VanMeter, P Lougovski, D Uskov, JPD, quant-ph/ Linear! KT Kapale & JPD, quant-ph/ (Nonlinear.) Question: Do there exist operators “U” that produce “N00N” States Efficiently? Answer: YES! H Cable, R Glasser, & JPD, quant-ph/ Linear! N VanMeter, P Lougovski, D Uskov, JPD, quant-ph/ Linear! KT Kapale & JPD, quant-ph/ (Nonlinear.) Constrained Desired |N > |0 > |N0::0N > |1,1,1 > Number Resolving Detectors

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

Spinning glass wheel. Each segment a different thickness. N00N is in Decoherence-Free Subspace! Spinning glass wheel. Each segment a different thickness. N00N is in Decoherence-Free Subspace! Generates and manipulates special cat states for conversion to N00N states. First theoretical scheme scalable to many particle experiments! Generates and manipulates special cat states for conversion to N00N states. First theoretical scheme scalable to many particle experiments! “Pizza Pie” Phase Shifter Feed-Forward-Based Circuit Quantum P00Per Scoopers! H Cable, R Glasser, & JPD, quant-ph/

Linear-Optical Quantum-State Generation: A N00N-State Example N VanMeter, D Uskov, P Lougovski, K Kieling, J Eisert, JPD, quant-ph/ 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(m 2 ). Upper bound theorem: The maximal size of a N00N state generated in m modes via single photon detection in m–2 modes is O(m 2 ).

Conclusions 1.Quantum Computing & Projective Measurements 2.Quantum Imaging & Metrology 3.Showdown at High N00N! 4.Efficient N00N-State Generating Schemes 5.Conclusions