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  • Kaonic Quantum Erasers

    Gianni Garbarino University of Torino, Italy

    Quantum Theory: Reconsideration of Foundations – 4 (QTRF4)

    Växjö (Sweden), June 11–16, 2007

    Quantum Theory: Reconsideration

    of Foundations – 4 (QTRF4)

    Kaonic Quantum Erasers (page 1) Gianni Garbarino

    University of Torino, Italy

  • OUTLINE

    F Introduction

    F Quantitative Complementarity

    F The Quantum Eraser

    F Conclusions

    Albert Bramon (Universitat Autonoma de Barcelona)

    Beatrix C. Hiesmayr (University of Vienna)

    [1] Quantum marking and quantum erasure for neutral kaons, A. Bramon, G. G. and B. C. Hiesmayr, Phys. Rev. Lett. 92 (2004) 020405.

    [2] Active and passive quantum erasers for neutral kaons, A. Bramon, G. G. and B. C. Hiesmayr, Phys. Rev. A 69 (2004) 062111.

    [3] Quantitative duality and neutral kaon interferometry, A. Bramon, G. G. and B. C. Hiesmayr, Eur. Phys. J C 32 (2004) 377.

    Quantum Theory: Reconsideration

    of Foundations – 4 (QTRF4)

    Kaonic Quantum Erasers (page 2) Gianni Garbarino

    University of Torino, Italy

  • Introduction

    F Bohr’s Complementarity

    – Quantum systems have properties which are equally real but mutually exclusive =⇒ Wave–Particle duality: depending on the experimental conditions, a quantum system behaves either like a wave (interference fringes) or like a particle (“which way” information)

    – Here we will also consider intermediate cases with simultaneous wave and particle knowledge: Quantitative Complementarity

    F Feynman’s Lectures on Physics

    – On the Double–Slit Experiment : “In reality, it contains the only mystery”

    Interference patterns are observed if and only if it is impossible to know, even in principle, which way the particle took. Interference disappears if there is a way to know —Quantum Marking— which way the particle took. But, if that which way mark is erased by a suitable measurement —Quantum Erasure—, interference reappears.

    – On the Neutral Kaon System : “If there is any place where we have a chance

    to test the main principles of quantum mechanics in the purest way —does the superposition of amplitudes work or doesn’t it?— that is it”

    Quantum Theory: Reconsideration

    of Foundations – 4 (QTRF4)

    Kaonic Quantum Erasers (page 3) Gianni Garbarino

    University of Torino, Italy

  • Quantitative Complementarity

    Quantum system in a two–path interferometer

    |Ψ(φ)〉 = a|ψI〉 + b eiφ|ψII〉

    a, b ≥ 0 a2 + b2 = 1 〈ψI |ψII〉 = 0

    Interference patterns: I±(φ) ≡ |〈ψ±|Ψ(φ)〉|2 = 1

    2 [1 ± V0 cosφ]

    |ψ±〉 = 1√ 2

    [|ψI〉 ± |ψII〉]

    Fringe Visibility V0 ≡ Imax − Imin Imax + Imin

    = 2 a b

    Path Predictability [D. Greenberger and A. Yasin, Phys. Lett. A 128 (1988) 391]

    P ≡ |wI − wII | = |a2 − b2|

    Quantum Theory: Reconsideration

    of Foundations – 4 (QTRF4)

    Kaonic Quantum Erasers (page 4) Gianni Garbarino

    University of Torino, Italy

  • =⇒ Quantitative Complementarity Relation

    P2 + V20 = 1

    F Modern and Quantitative statement of Bohr’s Complementarity

    F Wave vs Particle information not governed by an Uncertainty Principle but by a single parameter (a)

    F Even with P = 0.98 a non–negligible visibility, V0 = 0.20, is observable

    F Symmetric interferometer: a = b = 1/ √

    2, V0 = 1, P = 0

    F V0 = 0 and P = 1 ⇐⇒ either a = 0 or a = 1

    Quantum Theory: Reconsideration

    of Foundations – 4 (QTRF4)

    Kaonic Quantum Erasers (page 5) Gianni Garbarino

    University of Torino, Italy

  • The Neutral Kaon System

    F Kaons are “strange” mesons, discovered in “V” events (1946) – θ-τ puzzle =⇒ violation of Parity conservation in weak interactions (1957) – weak interactions also violate CP (Charge Conjugation × Parity) (1964) – signals of physics beyond the Standard Model

    – tests of Quantum Mechanics and Local Realism (Bell’s inequalities)

    F Particle physics two–level quantum system analogous to polarized photons and spin 1/2 particles

    F Differences due to kaon time evolution (decay), strangeness oscillations, internal symmetries, only two alternative measurement bases: Strangeness {K0, K̄0} and Lifetime {KS ,KL}

    F Produced in strangeness conserving Strong Interactions [pp̄→ K−π+K0, pp̄→ K+π−K̄0, e+e− → φ(1020) → K0K̄0] Decay through strangeness changing Weak Interactions [KS → 2π, KL → 3π]

    Quantum Theory: Reconsideration

    of Foundations – 4 (QTRF4)

    Kaonic Quantum Erasers (page 6) Gianni Garbarino

    University of Torino, Italy

  • F Strangeness K0 = ds̄ K̄0 = d̄s

    S|K0〉 = +1|K0〉 S|K̄0〉 = −1|K̄0〉 〈K0|K̄0〉 = 0

    F Lifetime KS and KL short– and long–lived states Eigenstates of a non–Hermitian weak interaction Hamiltonian

    H |KS(L)〉 = λS(L)|KS(L)〉 H = M − i

    2 Γ λS(L) = mS(L) −

    i

    2 ΓS(L)

    〈KS |KL〉 = 〈KL|KS〉 = 2Re �/(1 + |�|2) = 3.2 × 10−3

    Time evolution: |KS(L)(t)〉 = e−iλS(L)t|KS(L)〉

    Lifetimes: τS = 0.9 × 10−10 s τL = 5.2 × 10−8 s ΓSΓL ' 579

    Decay modes: KS → 2π KL → 3π

    F The two complementary observables are maximally incompatible:

    |KS〉 = 1

    2(1 + |�|2) [

    (1 + �)|K0〉 + (1 − �)|K̄0〉 ]

    ' 1√ 2

    [

    |K0〉 + |K̄0〉 ]

    |KL〉 = 1

    2(1 + |�|2) [

    (1 + �)|K0〉 − (1 − �)|K̄0〉 ]

    ' 1√ 2

    [

    |K0〉 − |K̄0〉 ]

    The CP violating parameter � can be safely neglected in our discussion

    Quantum Theory: Reconsideration

    of Foundations – 4 (QTRF4)

    Kaonic Quantum Erasers (page 7) Gianni Garbarino

    University of Torino, Italy

  • Quantitative Complementarity for Neutral Kaons

    Suppose a K0 is produced (π−p→ K0Λ) at time t = 0:

    |K0〉 = 1√ 2

    (|KS〉 + |KL〉) =⇒ |K0(t)〉 = 1√ 2

    (

    e−iλSt|KS〉 + e−iλLt|KL〉 )

    Time evolution, normalizing to surviving kaons:

    |K0(t)〉 = 1√ 1 + e−∆Γ t

    (

    |KS〉 + e−∆Γ t/2e−i∆m t|KL〉 )

    ∆Γ = ΓL − ΓS ∆m = mL −mS |∆Γ|/∆m ' 2.0

    Strangeness Oscillations

    P [K0 → K0; t] ≡ ∣

    ∣〈K0|K0(t)〉 ∣

    2 =

    1

    2 {1 + V0(t) cos(∆mt)}

    P [K0 → K̄0; t] ≡ ∣

    ∣〈K̄0|K0(t)〉 ∣

    2 =

    1

    2 {1 − V0(t) cos(∆mt)}

    Time–dependent Visibility V0(t) = 1cosh (∆Γ t/2)

    Quantum Theory: Reconsideration

    of Foundations – 4 (QTRF4)

    Kaonic Quantum Erasers (page 8) Gianni Garbarino

    University of Torino, Italy

  • wS(t) ≡ ∣

    ∣〈KS |K0(t)〉 ∣

    2 =

    1

    1 + e−∆Γ t wL(t) ≡

    ∣〈KL|K0(t)〉 ∣

    2 =

    1

    1 + e∆Γ t

    “Width Predictability” P(t) ≡ |wS(t) − wL(t)| = |tanh (∆Γ t/2)|

    Quantitative Complementarity Relation P2(t) + V20 (t) = 1

    F Strangeness Oscillations ⇐⇒ interference patterns (wave–like) F KS and KL free–space propagation ⇐⇒ two interferometric paths (particle–like)

    0

    0.2

    0.4

    0.6

    0.8

    1

    0 1 2 3 4 5 6

    t/τS

    V20 (t)

    P2(t)

    F CPLEAR (CERN) data admit an interpretation in terms of the quantitative complementarity relation

    Quantum Theory: Reconsideration

    of Foundations – 4 (QTRF4)

    Kaonic Quantum Erasers (page 9) Gianni Garbarino

    University of Torino, Italy

  • Entangled Neutral Kaons

    F To improve the “a priori” knowledge on the kaon lifetime, P(t), a measurement must be performed on the kaon state [B.G. Englert, Phys. Rev. Lett. 77 (1996) 2154]

    F Strangeness and Lifetime measurements are completely destructive =⇒ use Entanglement

    From e+e− → φ(1020) → K0K̄0 or pp̄→ K0K̄0 one starts at t = 0 with a maximally entangled state

    |φ(0)〉 = 1√ 2

    {

    |K0〉l|K̄0〉r − |K̄0〉l|K0〉r }

    = 1√ 2 {|KL〉l|KS〉r − |KS〉l|KL〉r}

    Two–time state, after normalizing to surviving kaon pairs:

    |φ(tl, tr)〉 = 1√

    1 + e∆Γ(tl−tr)

    {

    |KL〉l|KS〉r − ei∆m(tl−tr)e 1 2∆Γ(tl−tr)|KS〉l|KL〉r

    }

    m

    |Ψ(∆φ)〉 = 1√ 2

    {

    |H〉l|V 〉r − ei∆φ|V 〉l|H〉l }

    Quantum Theory: Reconsideration

    of Foundations – 4 (QTRF4)

    Kaonic Quantum Erasers (page 10) Gianni Garbarino

    University of Torino, Italy

  • φ

    t l

    object meter

    t r 0

    S M = S or L

    For the object kaon we want to introduce a “Which Width Knowledge”:

    KM(tl) ≥ P(tl) ≡ | tanh(∆Γ tl/2)|

    M = S or L ⇐⇒ measurement on the meter kaon

    and a Visibility of the object kaon strangeness oscillations:

    VM(tl) ≤ V0(tl) ≡ 1/ cosh(∆Γ tl/2)

    such that they satisfy the Quantitative Complementarity Relation

    K2M(tl)+V2M(tl) = 1

    Quantum Theory: Reconsideration

    of Foundations – 4 (QTRF4)

    Kaonic Quan