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Computational and Nonlinear Quantum Optics CNQO at Strathclyde

Computational and Nonlinear Quantum Optics CNQO at Strathclyde. http://cnqo.phys.strath.ac.uk. Graeme McCartney Kieran Hunter Eng-Kian (Peter) Tan Nick Whitlock Erika Andersson Sonja Franke-Arnold Damia Gomila-Villalonga Gordon Robb Andrew Scroggie Roberta Zambrini Patrik Ohberg

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Computational and Nonlinear Quantum Optics CNQO at Strathclyde

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  1. Computational and Nonlinear Quantum Optics CNQO at Strathclyde http://cnqo.phys.strath.ac.uk Graeme McCartney Kieran Hunter Eng-Kian (Peter) Tan Nick Whitlock Erika Andersson Sonja Franke-Arnold Damia Gomila-Villalonga Gordon Robb Andrew Scroggie Roberta Zambrini Patrik Ohberg John Jeffers Francesco Papoff Steve Barnett Willie Firth Gian-Luca Oppo Alain Aspect (Carnegie prof.) Quantum information students BEC and cold gases postdocs Optical angular momentum : Quantum images lect. Visiting profs: Rodney Loudon, Essex Simon Phoenix, BT David Pegg, Griffith, Aus Maxi San Miguel, Palma prof.

  2. BEC and cold gases • Slow light • Low-dimensional cold gases • Quantum information with cold atoms & slow light • Matter flux • Quantum Hall effect • in cold gases • Coherence • Solitons in BECs .. Nick Whitlock, Sonja Franke-Arnold, Patrik Ohberg, John Jeffers, Steve Barnett, Willie Firth, Alain Aspect, Luis Santos (Hannover) Experimental: First Scottish condensate, “MacBEC”, June 24! http://www.photonics.phys.strath.ac.uk Craig Garvie, Aidan Arnold, Erling Riis (Photonics)

  3. Quantum images Optical patterns in non-linear media Cavity solitons Optical sprinklers Control of spatial and temporal disorder in optical devices Quantum imaging Graeme McCartney, Damia Gomila Villalonga, Gordon Robb, Andrew Scroggie, John Jeffers, Francesco Papoff, Steve Barnett, Willie Firth, Gian-Luca Oppo, Maxi San Miguel (Palma de Mallorca)

  4. Quantum information Retrodiction Generalised measurements Quantum coding, communication Implementations of quantum information theory Experimental realisation of POM measurements! Non-Markovian master equations Kieran Hunter, Eng-Kian (Peter) Tan, Erika Andersson, Andrew Scroggie, John Jeffers, Stephen Barnett, Erling Riis (exp) Collaborators and visitors: Jim Cresser (Macquarie, Aus), Igor Jex (Prague), David Pegg (Griffith, Aus), John Vaccaro and Tony Chefles (Hertfordshire), Masahide Sasaki (CRL, Japan)…

  5. Simultaneous measurements of spin, signal locality, and uncertainty Erika Andersson, Steve Barnett, and Alain Aspect University of Strathclyde, Glasgow Thanks: EPSRC, Scottish Executive Education and Lifelong Learning Department, EU Marie Curie scheme, Carnegie =>Talk in Oxford (Thursday) on quantum comparison<=

  6. R L Entanglement and signal locality EPR-Bohm paradox Singlet: 1 2 - 1 2 |Y> = (|+ >|- > - |- >|+ > ) = (|+ >|- > - |- >|+ >) z z z z x x x x L R L R L R L R Right and left measurement results correlated, butthiscannot be used to communicate. p (+)= 1/2, p (-)=1/2. Measurement on the left particle: L L 1 2 x • = (|+ > <+ | + |- > <- |) x x x x R RR RR 1 2 z = (|+ > <+ | + |- > <- |) = r . z z z z RR RR R ris the same no matter what is done at L! R

  7. Signal locality Ghirardi, Rimini and Weber, Lett. Nuov. Cim. 27, 293 (1980). B U A V L observable <L > independent of of A only what is done to B. A A No local operation acting on only one of a pair of systems can change the reduced density matrix of the other system. This can be used to derive bounds on quantum operations! QM Signal locality Bruss, D’Ariano, Macchiavello, Sacchi: PRA 62, 062302 (2000).

  8. Simultaneous measurements of spin Measuring two non-commuting observables at the same time gives increased uncertainty. . . Measure both A = asand B = b s of a S=1/2 particle: 2 2 2 2 2 DA = <A > - <A > = 1 - a <A> s s s 2 2 2 2 2 S=1/2 DB = <B > - <B > = 1 - b <B> s s s Reduction in expectation values. Largest possible a, b?

  9. R L Measurement bound using locality 1 2 - Singlet: |Y> = (|+ >|- > - |- >|+ > ) L R L R On L, measure both A and B . s s On R, measure either C or D. p(A = B ) and p(A = -B ) have to be independent of whether C or D is measured to the right. ssss p(A=B )=p(A =B =C) + p(A =B =-C) ssssss > |p(A =B =C) - p(A =B =-C)| ssss 1 2 = |E(A ,C) + E(B ,C)|, where E(A,B)=AB. ss 1 2 Similarly p(A =-B ) > |E(A ,D) - E(B ,D)|. ssss

  10. Measurement bound using locality Adding these two inequalities, we obtain |E(A ,C) + E(B ,C)| + |E(A ,D) - E(B ,D)| £ 2. ssss Very similar to Bell inequality! This is because we have demanded that probabilities for triples, e.g. p(A = B = C), exist. ss . Use E(As,C)= - a a c for singlet to obtain . . |(aa +bb)c| + |(a a - bb) d| £ 2. Choose c and d to maximise LHS: Final bound: |a a +b b| + |aa - bb| £ 2.

  11. Measurement bound using locality |a a +b b| + |aa - bb| £ 2 a a • diagonals £ 2 a, b < 1 b b The bound has been derived before with POMs: P. Busch, PRD 33, 2253 (1986). P. Busch, M. Grabowski and P. J. Lahti, Operational Quantum Physics, Springer-Verlag, Berlin, 1995. Here we have assumed nothing about how to describe the measurement!

  12. Uncertainty relation D2AsD2Bs= (1- a2<A>2)(1- b2<B>2) • = (1-a2) (1-b2) + (1-a2)b2(1-<B>2) • + (1-b2)a2(1-<A>2) + a2 b2(1-<A>2)(1-<B>2) (1-<A>2), (1-<B>2)“internal” uncertainties (1-a2) , (1-b2)“external” measurement uncertainties a |a a +b b| + |aa - bb| £ 2 can be rewritten as q b (1-a2)(1-b2) > a2b2 sin2q . This is a tight bound on the “external” measurement uncertainty!

  13. Uncertainty relation Total uncertainty relation: D2AsD2Bs > a2b2sin2q (1+|<sz>|)2 This is stronger than the Arthurs-Goodman relation: (PRL 1988) D2AsD2Bs > a2b2|<A,B>|2 = 4 a2b2sin2q |<sz>|2. The difference is that we used a tight bound for the external measurement uncertainty. The Heisenberg uncertainty relation is not always tight!

  14. Three observables |a a +b b +g c| + |aa + bb -g c| + |a a -b b +g c| + |aa - bb -g c| £ 4 g c • diagonals £ 4 a, b,g < 1 a a b b . Can also be written a2 + b2 + g2 £ 1 + a2b2 (ab)2 + ... If a = x, b = y, c = z, then a2 + b2 + g2 £ 1. This means <Sx,s>2 + <Sy,s>2 + <Sz,s>2 £ 1/3 for a simultaneous measurement with x,y,z equally “dizzy”.

  15. Summary • The locality principle can be used to derive bounds on quantum measurements. • Bound on simultaneous spin measurements without measurement operators. • Uncertainty relations for simultaneous measurements of spin; tight bound for external measurement uncertainty. Barnett and Andersson, PRA 65, 044307 (2002) for bound on discrimination between non-orthogonal states. quant-ph/03? for simultaneous spin measurements. Thanks: EPSRC, Scottish Executive Education and Lifelong Learning Department, EU Marie Curie scheme, Carnegie =>Talk in Oxford (Thursday) on quantum comparison<=

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