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Faculty of Physics, University of Vienna, Austria Institute for Quantum Optics and Quantum Information Austrian Academy of Sciences Macrorealism, the freedom-of-choice loophole, and an EPR-type BEC experiment Johannes Kofler Max Planck Institute for Quantum Optics Garching, July 12th 2011 Double slit experiment With photons, electrons, neutrons, molecules etc. With cats? |cat left + |cat right ? When and how do physical systems stop to behave quantum mechanically and begin to behave classically? Macroscopic superpositions Two schools: - Decoherence uncontrollable interaction with environment; within quantum physics - Objective collapse models (GRW, Penrose, etc.) forcing superpositions to decay; altering quantum physics Alternative answer: - Coarse-grained measurements measurement resolution is limited; within quantum physics A. Peres, Quantum Theory: Concepts and Methods, Kluver (2002) Macrorealism Leggett and Garg (1985): Macrorealism per se “A macroscopic object, which has available to it two or more macroscopically distinct states, is at any given time in a definite one of those states.” Non-invasive measurability “It is possible in principle to determine which of these states the system is in without any effect on the state itself or on the subsequent system dynamics.” Q(t1) Q(t2) t t=0 t1 t2 The Leggett-Garg inequality Dichotomic quantity: t t=0 t Temporal correlations t1 t2 t3 t4 All macrorealistic theories fulfill the Leggett-Garg inequality Violation macrorealism per se or/and non-invasive measurability failes Violation of the inequality Rotating spin ½ particle (eg. electron) ½ Rotating classical spin vector (e.g. torque) Precession around axis with frequency (through manetic field or external force) Measurement along orthogonal axis K > 2: Violation of the Leggett-Garg inequality K 2: Classical time evolution, no violation classical limit Violation for arbitrary Hamiltonians t Initial state t t State at later time t t1 = 0 t2 Measurement ! t3 ? ? Survival probability Leggett-Garg inequality classical limit Choose can be violated for any E J. Kofler and Č. Brukner, PRL 101, 090403 (2008) Why no violation in everyday life? Coarse-grained measurements Model system: Spin j macroscopic: j ~ 1020 Arbitrary state: - Measure Jz, outcomes: m = – j, –j+1, ..., +j (2j+1 levels) - Assume measurement resolution is much weaker than the intrinsic uncertainty such that neighbouring outcomes are bunched together into “slots” m. m = –j m= m = +j 1 2 3 4 Example: Rotation of spin j Sharp measurement of spin z-component Coarse-grained measurement –j 1 3 5 7 ... Q = –1 –j +j +j 2 4 6 8 ... Q = +1 classical limit Fuzzy measurement Violation of Leggett-Garg inequality for arbitrarily large spins j Classical physics of a rotating classical spin vector J. Kofler and Č. Brukner, PRL 99, 180403 (2007) Coarse-graining Coarse-graining Sharp parity measurement Neighbouring coarse-graining (two slots) (many slots) 1 3 5 7 ... 2 4 6 8 ... Slot 1 (odd) Slot 2 (even) Violation of Leggett-Garg inequality Note: Classical physics Superposition vs. mixture To see the quantumness of a spin j, you need to resolve j1/2 levels Non-classical Hamiltonians Hamiltonian: Produces oscillating Schrödinger cat state: Under fuzzy measurements it appears as a statistical mixture at every instance of time: But the time evolution of this mixture cannot be understood classically: time J. Kofler and Č. Brukner, PRL 101, 090403 (2008) Non-classical Hamiltonians are complex Oscillating Schrödinger cat Rotation in real space “non-classical” rotation in Hilbert space “classical” Complexity is estimated by number of sequential local operations and two-qubit manipulations Simulate a small time interval t O(N) sequential steps 1 single computation step all N rotations can be done simultaneously Monitoring by an environment Exponential decay of survival probability - Leggett-Garg inequality is fulfilled (despite the non-classical Hamiltonian) - However: Decoherence cannot account for a continuous spatiotemporal description of the spin system in terms of classical laws of motion. - Classical physics: differential equations for observable quantitites (real space) - Quantum mechanics: differential equation for state vector (Hilbert space) Relation quantum-classical A brief history of hidden variables Quantum mechanics and realism 1927 Kopenhagen interpretation (Bohr, Heisenberg) 1932 von Neumann’s (wrong) proof of non-possibility of hidden variables 1935 Einstein-Podolsky-Rosen paradox 1952 De Broglie-Bohm (nonlocal) hidden variable theory 1964 Bell’s theorem on local hidden variables 1972 First successful Bell test (Freedman & Clauser) Bohr and Einstein, 1925 Bell’s assumptions λ Realism: [J. F. Clauser & A. Shimony, Rep. Prog. Phys. 41, 1881 (1978)] Hidden variables λ determine outcome probabilities: p(A,B|a,b,λ) [J. S. Bell, Physics 1, 195 (1964)] Locality: (OI) Outcome Independence: p(A|a,b,B,λ) = p(A|a,b,λ) & vice versa (SI) Setting Independence: p(A|a,b,λ) = p(A|a,λ) & vice versa [J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, p. 243 (2004)] Freedom of Choice: (FC) p(a,b|λ) = p(a,b) p(λ|a,b) = p(λ) Bell’s theorem Realism + Locality + Freedom of Choice Bell‘s Inequality CHSH form: |E(a1,b2) + E(a2,b1) + E(a2,b1) - E(a2,b2)| 2 The original Bell paper (1964) implicitly assumes freedom of choice: explicitly: A(a,b,B,λ) locality (outcome and setting independence) freedom of choice implicitly: (λ|a,b) A(a,λ) B(b,λ) – (λ|a,c) A(a,λ) B(c,λ) Loopholes Locality loophole: There may be a communication from the setting or outcome on one side to the outcome on the other side Closed by Aspect et al., PRL 49, 1804 (1982) & Weihs et al., PRL 81, 5039 (1998) Fair-sampling loophole: The measured events stem from an unrepresentative subensemble Closed by Rowe et al., Nature 409, 791 (2001) Freedom-of-choice loophole: The setting choices may be correlated with the hidden variables Closed by Scheidl et al., PNAS 107, 10908 (2010) [this talk] Geography Space-time diagram t l l l l l B l l A l b a x E La Palma Tenerife Locality: Freedom of choice: A is space-like separated from B (OI) and b (SI) a and b are random and B is space-like separated from A (OI) and a (SI) space-like separated from E Geographic details Tenerife La Palma 144 km free-space link 144 km free-space link NOT Source 6 km fiber channel Alice OGS 1.2 km RF link QRNG Bob QRNG Experimental results Coincidence rate detected: 8 Hz Measurement time: 2400 s Number of total detected coincidences: 19200 Polarizer settings a, b 0°, 22.5° 0, 67.5° 45°, 22.5° 45°, 67.5° Correlation E(a,b) 0.62 ± 0.01 0.63 ± 0.01 0.55 ± 0.01 –0.57 ± 0.01 Obtained Bell value Sexp 2.37 ± 0.02 T. Scheidl, R. Ursin, J. Kofler, S. Ramelow, X. Ma, T. Herbst, L. Ratschbacher, A. Fedrizzi, N. Langford, T. Jennewein, and A. Zeilinger, PNAS 107, 19708 (2010) Important remarks • In a fully deterministic world, neither the locality nor the freedom-ofchoice loophole can be closed: Setting choices would be predetermined and could not be space-like separated from the outcome at the other side (locality) or the particle pair emission (freedom-of-choice). • Thus, we need to assume stochastic local realism: There, setting choices can be created randomly at specific points in space-time. • We have to consistently argue within local realism: The QRNG is the best candidate for producing stochastic settings. • Practical importance: freedom of choice can be seen as a resource for device-independent cryptography and randomness generation/amplification Acknowledgments Thomas Scheidl Rupert Ursin Lothar Ratschbacher Alessandro Fedrizzi Sven Ramelow Xiao-Song Ma Thomas Herbst Nathan Langford Thomas Jennewein Anton Zeilinger Colliding BECs Cigar-shaped BEC of metastable He4 (high internal energy) Three laser beams kick the atoms: Recoil velocity: Two freely falling species are produced and undergo s-wave scattering Momentum-entangled particle pairs are produced, lying on a shell in velocity space: A. Perrin, H. Chang, V. Krachmalnicoff, M. Schellekens, D. Boiron, A. Aspect, and C. I. Westbrook, PRL 99, 150405 (2007) Proposal: The double double slit If the condensate is too small, there is a product of one-particle interference patterns: If the condensate is sufficiently large, one obtains two-particle interference (conditional interference fringes): Experimental conditions (I) Sufficiently large source size Sx to achieve well defined momentum correlation (px Sx–1) and wash out the single-particle interference pattern: (II) Sufficiently small source to not wash out the two-particle interference pattern: (III) Resolution of interference fringes: (IV) Ability to identify pairs, i.e. coincidences: In preparation (2011) Two-particle interference In preparation (2011) Acknowledgments Michael Keller Maximilian Ebner Mateusz Kotyrba Mandip Singh Anton Zeilinger Summary • Coarse-grained measurements are a way to understand the quantum-to-classical transition (complementary to decoherence) • We simultaneously closed the locality and the freedom-of-choice loophole; a loophole-free Bell test is still missing • Proposal: A BEC double double slit experiment can show EPR-type entanglement of massive particles Thank you for your attention! Appendix Macrorealism per se Probability for outcome m can be computed from an ensemble of classical spins with positive probability distribution: Coarse-grained measurements: any quantum state allows a classical description This is macrorealism per se. Experimental setup