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Effetto Zenone quantistico e controllo della decoerenza Milano, 23 marzo 2007 Paolo Facchi Dipartimento di Matematica, Università di Bari

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Quantum Zeno effect: fundamentals

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QZE: fundamentals..…. P U P U P U P U PP Quantum Zeno effect: repeated observation in succession inhibit transitions outside

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Theorem Misra and Sudarshan, J. Math. Phys. 18, 756 (1977) Quantum Zeno paradox: an observed particle does not evolve.

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Zeno of Elea At any given moment it is in a space equal to its own length, and therefore is at rest at that moment. So, it's at rest at all moments. The sum of an infinite number of these positions of rest is not a motion. Zeno was an Eleatic philosopher, a native of Elea in Italy, son of Teleutagoras, and the favorite disciple of Parmenides. He was born about 488 BC, and at the age of forty accompanied Parmenides to Athens The flying arrow is at rest.

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Peres, Am. J. Phys. 48, 931 (1980) Example: spin

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Neutron spin Pascazio, Namiki, Badurek, Rauch, Phys. Lett. A 169, 155 (1993) Increasing N

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History von Neumann,1932 Beskow and Nilsson,1967 Khalfin 1968 Friedman 1972 Misra and Sudarshan, 1977 Experiments (Cook 1988) Itano, Heinzen, Bollinger, and Wineland 1990 Nagels, Hermans, and Chapovsky 1997 Balzer, Huesmann, Neuhauser, and Toschek, 2000 Wunderlich, Balzer, and Toschek, 2001 Wilkinson, Bharucha, Fischer, Madison, Morrow, Niu, Sundaram, and Raizen, Fischer, B. Gutierrez-Medina, and Raizen, 2001 Theory and interesting Mathematics (not quoted)

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VESTA ISIS Jericha, Schwab, Jakel, Carlile, Rauch, Physica B 283, 414 (2000); Rauch, Physica B 297, 299 (2001).

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Are projections à la von Neumann necessary? No: a dynamical explanation can be given, in terms of the Schroedinger equation (and involving no projection operators). Wigner, Am. J. Phys Petrosky, Tasaki, Prigogine, PLA 1990; Physica 1991 Pascazio, Namiki, PRA 1994 This is best proven by looking at some examples.

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Zeno What really provokes Zeno

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Incomplete measurements

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Nonselective measurements P. F. and Pascazio, Phys. Rev. Lett. 89, (2002)

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Unitary kicks..…. UcUc U UcUc U UcUc U UcUc U UcUc UcUc Quantum maps Berry, Balazs, Tabor,Voros (1979) Casati, Chirikov, Ford and Izrailev (1979) Viola and Lloyd, Phys. Rev. A 58, 2733 (1998) Viola, Knill and Lloyd, Phys. Rev. Lett. 82, 2417 (1999) Byrd and Lidar, Quant. Inf. Proc. 1, 19 (2002) P.F., Lidar, Pascazio, Phys. Rev. A 69, (2004) Dynamical decoupling “Bang-bang” control

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Continuous coupling P. F. and Pascazio, Phys. Rev. Lett. 89, (2002) system“apparatus” K coupling

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Coupling or N Dynamical superselection sectors Quantum Zeno subspaces

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The quantum Zeno dynamics Is the Zeno dynamics unitary? Answer: under general hypotheses YES (Misra and Sudarshan 1977: semigroup) Friedman 1972 Facchi, Gorini, Marmo, Pascazio, Sudarshan 2000 Facchi, Pascazio, Scardicchio, Schulman 2002 Exner, Ichinose 2003 ?

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Free particle in D dimensions How does the particle move inside ? Does it leaks out? Free particle in a box with perfectly reflecting hard walls P.F., Pascazio, Scardicchio, Schulman 2002 P.F., Marmo, Pascazio, Scardicchio, Sudarshan 2003 …although there is NO wall!

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Measurements à la von Neumann (projections) Unitary kicks (bang-bang control) Different manifestations of the same physical phenomenon Continuous coupling

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Projections Zeno limit

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Kicks Zeno limit

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Continuous coupling Zeno limit Simonius 1978 Harris & Stodolsky 1982

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Main objective: understand and suppress decoherence Decoherence-free subspaces Palma, Suominen and Ekert (1996) Duan and Guo (1997) Zanardi and Rasetti (1997) Lidar, Chuang and Whaley (1998) Viola, Knill and Lloyd (1999) Vitali and Tombesi (1999, 2001) Beige, Braun, Tregenna and Knight (2000) … Tasaki, Tokuse, P.F., Pascazio (2002) P.F:, Lidar and Pascazio (2004) Benenti, Casati, Montangero, Shepelyansky (2001) Vitali, Tombesi, Milburn (1997, 1998) Fortunato, Raimond, Tombesi, Vitali (1999) Kofman, Kurizki (2001) Calarco, Datta, Fedichev, Pazy, Zoller, “Spin-based all-optical quantum computation with quantum dots: understanding and suppressing decoherence” (2003) Falci (2003)

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Relevant timescales P.F., Nakazato and Pascazio, Phys. Rev. Lett. 86, 2699 (2001) Experiment by Fischer, Gutièrrez- Medina and Raizen, Phys. Rev. Lett. 87, (2001) IZE How frequent or strong must be the coupling? QZE UNDISTURBED NOTICE:

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Main idea coupling K frequency N Enhancement of decoherence Control of decoherence ?

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The problem ? Unstable systems Unstable systems and Inverse Zeno

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Framework decoherence Liouvillian Hilbert spaces

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Framework (cont’d) initial state of total system evolved state of system Decoherence ifnot unitarily equivalent to

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Computational subspace

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System-bath interaction (Gardiner & Zoller) form factor

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Polynomial and exponential case

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Form factors Full line: exponential; dashed line: polynomial form factor. inverse temperature bandwidth

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Controlled Dynamics important !! control (protection) enhancement P.F., Tasaki, Pascazio, Nakazato,Tokuse, Lidar, Phys. Rev. A 71, (2005)

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Measurements

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Measurements (small times)

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“Kicks”

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kicks (small times)

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Continuous coupling

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(strong) continuous coupling

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Comparison

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Comparison (small times 1/N -- strong coupling K)

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Remarkable differences Zeno control (non-unitary) “bang bang” control (kicks) (unitary) control via continuous coupling (unitary) For unitary controlsfeels the “tail” of the form factor

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(partial) CONCLUSIONS BEST strategy: kicks/continuous rather than measurements GAIN: factor 10 Hopefully: more to come Qdots, cavities, JJ

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