Aug 20, 2020

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

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