Optical Quantum Imaging, Computing, and Metrology: WHAT S NEW WITH N00N STATES? Jonathan P. Dowling

Transcription

1 Optical Quantum Imaging, Computing, and Metrology: WHAT S NEW WITH N00N STATES? Jonathan P. Dowling Hearne Institute for Theoretical Physics Louisiana State University Baton Rouge, Louisiana quantum.phys.lsu.edu 07 JUNE 2007 DAMOP-07, Calgary

2 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, 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

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

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

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

6 Linear Optical Quantum Computing Linear Optics can be Used to Construct CNOT and a Scaleable Quantum Computer: # 0 + " 1 +! 2 # 0 + " 1 $! 2 Milburn Knill E, Laflamme R, Milburn GJ NATURE 409 (6816): JAN Franson JD, Donegan MM, Fitch MJ, et al. PRL 89 (13): Art. No SEP

7 Road to Entangled- Particle Interferometry: First Example of Entanglement Generation by Erasure of Which-Path Information Followed by Detection!?

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

9 Projective Measurement Yields Effective Kerr! GG Lapaire, P Kok, JPD, JE Sipe, PRA 68 (2003) 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 KLM CSIGN Hamiltonian Franson CNOT Hamiltonian

10 Nonlinear Single-Photon Quantum Non-Demolition You want to know if there is a single photon in mode b, without destroying it. Cross-Kerr Hamiltonian: H Kerr = κ a a b b ψ in 1 b a Kerr medium 1 D 2 D 1 1 Again, with κ = 10 22, this is impossible. *N Imoto, HA Haus, and Y Yamamoto, Phys. Rev. A. 32, 2287 (1985).

11 Linear Single-Photon Quantum Non-Demolition The success probability is less than 1 (namely 1/8). D 0 1 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 π /2 D 1 D 2 Effective κ = 1/8 21 Orders of Magnitude 0 2 ψ in = Σ c n n n = 0 Improvement! P Kok, H Lee, and JPD, PRA 66 (2003) π /2

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

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

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

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

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

17 Quantum Lithography & Metrology Now we consider the state and we measure Quantum Lithography*: " N = ( N,0 + 0,N ) A N = 0,N N,0 + N,0 0,N ϕ N A N ϕ N = cos Nϕ High-Frequency Lithography Effect Quantum Metrology: ΔA Δϕ H = N = d A N /dϕ 1 N Heisenberg Limit: No Square Root! P. Kok, H. Lee, and J.P. Dowling, Phys. Rev. A 65, (2002).

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

19 Showdown at High-N00N! How do we make High-N00N!? N,0 + 0,N With a large cross-kerr nonlinearity!* H = κ a a b b 1 0 N 0 N,0 + 0,N This is not practical! need κ = π but κ = 10 22! *C Gerry, and RA Campos, Phys. Rev. A 64, (2001).

20 Solution: Replace the Kerr with Projective Measurements! OPO 3 a 3 a b b single photon detection at each detector a b a b 2 a b 4 a b 6 a b Probability of success: a b a 3 b 4 0! 0 4 a ' b' a' b ' 3 Best we found: Cascading Not Efficient! H Lee, P Kok, NJ Cerf, and JP Dowling, Phys. Rev. A 65, R (2002).

21 These Ideas Implemented in Recent Experiments!

22 10::01> 10::01> 20::02> 20::02> 30::03> 40::04> 30::03>

23 Local and Global Distinguishability in Quantum Interferometry GA Durkin & JPD, quant-ph/ 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

24 NOON-States Violate Bell s Inequalities CF Wildfeuer, AP Lund and JP Dowling, quant-ph/ Probabilities of correlated clicks and independent clicks P ab (",#),P a ("),P b (#) Building a Clauser-Horne Bell inequality from the expectation values P ab (",#),P a ("),P b (#) "1# P ab ($,%) " P ab ($,% &) + P ab ( $ &,%) + P ab ( $ &,% &) " P a ( $ &) " P b (%) # 0 Shared Local Oscillator Acts As Common Reference Frame! Bell Violation!

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

26 Efficient Schemes for Generating N00N States! N> 0> Constrained Desired 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, quant-ph/ Linear! N VanMeter, P Lougovski, D Uskov, JPD, quant-ph/ Linear! KT Kapale & JPD, quant-ph/ (Nonlinear.)

27 Quantum P00Per Scooper! H Cable, R Glasser, & JPD, quant-ph/ mode squeezing process χ OPO beam splitter U(50:50) 4> 4> Old Scheme linear optical processing New Scheme How to eliminate the POOP? amplitude ^ > 8> 2> 6> 4> 4> 6> 2> 8> 0> Fock basis state quant-ph/ G. S. Agarwal, K. W. Chan, R. W. Boyd, H. Cable and JPD

28 Quantum P00Per Scoopers! H Cable, R Glasser, & JPD, quant-ph/ Pizza Pie Phase Shifter Spinning glass 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!

29 Linear-Optical Quantum-State Generation: A N00N-State Example N VanMeter, D Uskov, P Lougovski, K Kieling, J Eisert, JPD, quant-ph/ U ( ) 2 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 ).

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