Kaonic Quantum Erasers Gianni Garbarino University of ...personalpages.to.infn.it/~garbarin/vaxjo.pdf · Kaonic Quantum Erasers Gianni Garbarino University of Torino, Italy Quantum

Kaonic Quantum Erasers

Gianni GarbarinoUniversity of Torino, Italy

Quantum Theory: Reconsideration of Foundations – 4 (QTRF4)

Vaxjo (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.





Introduction

F Bohr’s Complementarity

– Quantum systems have properties which are equally real but mutuallyexclusive =⇒ Wave–Particle duality: depending on the experimentalconditions, 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 andparticle 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— whichway the particle took.But, if that which way mark is erased by a suitable measurement —QuantumErasure—, 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 —doesthe superposition of amplitudes work or doesn’t it?— that is it”





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|





=⇒ 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 bya 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





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 andspin 1/2 particles

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

F Produced in strangeness conserving Strong Interactions [pp→ K−π+K0,pp→ K+π−K0, e+e− → φ(1020) → K0K0]Decay through strangeness changing Weak Interactions [KS → 2π, KL → 3π]





F Strangeness K0 = ds K0 = ds

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

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

H |KS(L)〉 = λS(L)|KS(L)〉 H = M − i

2Γ λS(L) = mS(L) −

i

2ΓS(L)

〈KS |KL〉 = 〈KL|KS〉 = 2Re ε/(1 + |ε|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 + |ε|2)[

(1 + ε)|K0〉 + (1 − ε)|K0〉]

' 1√2

[

|K0〉 + |K0〉]

|KL〉 =1

√

2(1 + |ε|2)[

(1 + ε)|K0〉 − (1 − ε)|K0〉]

' 1√2

[

|K0〉 − |K0〉]

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





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 → K0; t] ≡∣

∣〈K0|K0(t)〉∣

∣

2=

1

2{1 − V0(t) cos(∆mt)}

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





wS(t) ≡∣

∣〈KS |K0(t)〉∣

∣

2=

1

1 + e−∆Γ twL(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 quantitativecomplementarity relation





Entangled Neutral Kaons

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

F Strangeness and Lifetime measurements are completely destructive =⇒ useEntanglement

From e+e− → φ(1020) → K0K0 or pp→ K0K0 one starts at t = 0 with amaximally entangled state

|φ(0)〉 =1√2

{

|K0〉l|K0〉r − |K0〉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)e12∆Γ(tl−tr)|KS〉l|KL〉r

}

m

|Ψ(∆φ)〉 =1√2

{

|H〉l|V 〉r − ei∆φ|V 〉l|H〉l}





φ

t l

object meter

t r0

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)+V2

M(tl) = 1





F M = LP (Kl,Kr) with Kl = K0 or K0 and Kr = KS or KL show no strangenessoscillations:

VL(tl) = 0

KL(tl) ≡∣

∣P (KS(tl),KS(t0r)) − P (KL(tl),KS(t0r))∣

∣

+∣

∣P (KS(tl),KL(t0r)) − P (KL(tl),KL(t0r))∣

∣ = 1

=⇒ K2L(tl) + V2

L(tl) = 1

F M = SP (Kl,Kr) with Kl,r = K0 or K0 show the tl–dependent strangenessoscillations with visibility:

VS(tl) = 1/ cosh(

∆Γ (tl − t0r)/2)

KS(tl) ≡∣

∣P [KS(tl),K0(t0r)] − P [KL(tl),K

0(t0r)]∣

∣

+∣

∣P [KS(tl), K0(t0r)] − P [KL(tl), K

0(t0r)]∣

∣

=∣

∣tanh(

∆Γ(tl − t0r)/2)∣

∣

=⇒ K2S(tl) + V2

S(tl) = 1





The Quantum Eraser

F Proposed in[M.O. Scully and K. Druhl, Phys. Rev. A 25 (1982) 2208],[M.O. Scully, B.G. Englert and H. Walter, Nature 351 (1991) 111]

F Implemented experimentally using:

– Entangled Photons[T.J. Herzog, P.G. Kwiat, H. Weinfurter and A. Zeilinger, Phys. Rev. Lett. 75

(1995) 3034],[Yoon-Ho Kim, R. Yu, S.P. Kulik, Y. Shih and M.O. Scully, Phys. Rev. Lett. 84

(2000) 1],[T. Tsegaye and G. Bjork, Phys. Rev. A 62 (2000) 032106],[A. Trifonov, G. Bjork, J. Soderholm and T. Tsegaye, Eur. Phys. J. D 18 (2002)

251],[S.P. Walborn, M.O. Terra Cunha, S. Padua and C.H. Monken, Phys. Rev. A 65

(2002) 033818],[H. Kim, J. Ko and T. Kim, Phys. Rev. A 67 (2003) 054102]

– Atom Interferometers[S. Durr, T. Nonn and G. Rempe, Nature 395 (1998) 33],[S. Durr and G. Rempe, Optics Communications 179 (2000) 323]

– Neutron Interferometers[G. Badurek and H. Rauch, Physica B 276-278 (2000) 964]





Time and the Quantum: Erasing the Past and Impacting the FutureY. Aharonov and M. S. Zubairy, Science 307 (2005) 875





T.J. Herzog, P.G. Kwiat, H. Weinfurter and A. Zeilinger, Phys. Rev. Lett. 75 (1995) 3034

Pol. at +450

QWP

QWP

HWP 0 /90

+45 /-45

oriented at

or

0 0

0 0

meter system

object system

| + 45〉 = (|V 〉 + |H〉)/√

2 ⇐⇒ |K0〉 = (|KS〉 + |KL〉)/√

2

| − 45〉 = (|V 〉 − |H〉)/√

2 ⇐⇒ |K0〉 = (|KS〉 − |KL〉)/√

2

|Ψ(∆φ)〉 =1√2

{

|H〉object|V 〉meter − ei∆φ|V 〉object|H〉meter

}

m

|φ(∆t)〉 =1√

1 + e∆Γ∆t

{

|KL〉object|KS〉meter − ei∆m∆te∆Γ∆t/2|KS〉object|KL〉meter

}

∆t = tobject − tmeter ∆Γ = ΓL − ΓS ∆m = mL −mS





Yoon-Ho Kim, R. Yu, S. P. Kulik, Y. Shih and M. O. Scully, Phys. Rev. Lett. 84 (2000) 1

D0

D4

D3

D2

D1

BSA

BSB

Position A

Position B

BSy

meter system

object system

http://strangepaths.com/the-quantum-eraser-experiment/2007/03/20/en/





Quantum Eraser Protocol for Kaons

F STEP I Single (object) kaon evolution: Strangeness Oscillations

P [K0 → K0; t] =1

2

1 +cos(∆m t)

cosh(∆Γ t/2)

ff

P [K0 → K0; t] =1

2

1 − cos(∆mt)

cosh(∆Γ t/2)

ff

F STEP II Entangle the object kaon with a meter kaon ⇐⇒ Quantum Marking

|K0(t)〉 =1√

1 + e−∆Γ t

“

|KS〉 + e−∆Γ t/2e−i∆m t|KL〉”

=⇒

|φ(to, tm)〉 =

n

|KS〉object|KL〉meter − ei∆m(to−tm)e∆Γ(to−tm)/2|KL〉object|KS〉meter

o

√1 + e∆Γ(to−tm)

Strangeness Oscillations disappear: P [K0(to), ∗] = P [K0(to), ∗] = 12

F STEP III Object lifetime mark erased by a Strangeness Measurement on the

meter ⇐⇒ Quantum Erasure

Strangeness Oscillations restored:

P [K0(to), K0(tm)] =

1

4

1 +cos[∆m(to − tm)]

cosh[∆Γ(to − tm)/2]

ff

P [K0(to), K0(tm)] =

1

4

1 − cos[∆m(to − tm)]

cosh[∆Γ(to − tm)/2]

ff





Active vs Passive Measurement procedures for Neutral Kaons

F Active Measurement

The experimenter, exerting his/her free will, either places a slab of material(strangeness) or allows for free–space kaon propagation (lifetime).

– Strangeness =⇒ strangeness conserving strong interactions: K0p→ K+n,K0n→ K−p

– Lifetime =⇒ free–space weak decay: KS ’s (KL’s) are kaons decaying before(after) ' 4.8 τS . KS-KL misidentifications O(10−3)

F Passive Measurement

Exploit only the quantum dynamics of kaon decays in free space.

– Strangeness =⇒ assuming the ∆Q = ∆S rule:K0 → π−l+νl and K0 → π+l−νl (l = e, µ)

– Lifetime =⇒ neglecting CP violation (ε = 0), KS → 2π and KL → 3π

No control on the time when the measurement occurs, nor on the basis inwhich the measurement is performed.





A. Active Eraser with Active Measurements

S, tl

Source

S, tr0

T, tr0

object system

meter system

Pol. at +450

QWP

QWP

HWP 0 /90

+45 /-45

oriented at

or

0 0

0 0

meter system

object system

QUANTUM MARKING: “Which width” information

P [K0(tl),KS(t0r)] = P [K0(tl),KS(t0r)] =1

2(

1 + e∆Γ(tl−t0r))

P [K0(tl),KL(t0r)] = P [K0(tl),KL(t0r)] =1

2(

1 + e−∆Γ(tl−t0r))

QUANTUM ERASURE: Strangeness oscillations and anti–oscillations

P [K0(tl), K0(t0r)] = P [K0(tl),K

0(t0r)] =1

4

{

1 + V(tl − t0r) cos[∆m(tl − t0r)]}

P [K0(tl),K0(t0r)] = P [K0(tl), K

0(t0r)] =1

4

{

1 − V(tl − t0r) cos[∆m(tl − t0r)]}

V(tl − t0r) = 1/ cosh[∆Γ(tl − t0r)/2]

Analogous to the photon experiment by Zeilinger et al.





B. Partially Active Eraser with Active Measurements

S, tl

Source

S, tr0

T

object system

meter system D0

D4

D3

D2

D1

BSA

BSB

Position A

Position B

BSy

meter system

object system

F The meter kaon makes the “choice” to show full “which width” information ornot

F The eraser is partially active: instability of the meter kaon, the experimenteronly chooses t0r (conditional quantum eraser)

F No control over the Marking and Erasure operations for individual kaon pairs

Analogous to the photon experiment by Scully et al.: t0r ⇐⇒ BSA/B transmittivities

t0r = 0 ⇐⇒ TA = TB = 0t0r → ∞ ⇐⇒ TA = TB = 1





C. Passive Eraser with Passive Measurements on the Meter

S, tl

Source

T

object system

meter systemTS

F Strangeness K0 → π−l+νl K0 → π+l−νl (l = e, µ)Lifetime KS → 2π KL → 3π

F Kaon decays are spontaneous processes ⇐⇒ passive eraser

F Different quantum–mechanical calculation of the joint probabilities, but withthe same results of the previous erasers

No analog in any other considered two–level quantum system





D. Passive Eraser with Passive Measurements

Source

T TS S

F Same quantum–mechanical predictions of the previous erasers for the jointprobabilities

F Extreme case of a passive quantum eraser!: the experimenters have no controlover individual pairs, neither on which of the two complementary observablesare measured nor when they are measured

F Which is the object? Strictly speaking not a quantum eraser!

No analog in any other considered two–level quantum system





Passive Eraser at KLOE2 (DAΦNE φ–factory)

Time–dependent Asymmetry [A. Di Domenico and A. Go]

A(∆t) =P [K0,Kmeter; ∆t] − P [K0,Kmeter; ∆t]

P [K0,Kmeter; ∆t] + P [K0,Kmeter; ∆t]∆t ≡ tobject − tmeter

Toy Monte Carlo simulation with L = 50 fb−1

∆t/τS

Kmeter = K0

Kmeter = KL

Kmeter = K0

Kmeter = KS

AQM(∆t) = − cos(∆m∆t)

AQM(∆t) = 0

AQM(∆t) = +cos(∆m∆t)

AQM(∆t) = 0





Conclusions

F The Neutral Kaon System does not need any kind of double–slit device.It is a double–slit: |K0〉 = 1√

2(|KS〉 + |KL〉) !

F Easily illustrates Quantitative formulations of Complementarity

F Reveals to be suitable for an optimal demonstration of Quantum Erasure

– “which width” information carried by a system (the meter kaon) distinct andspatially separated from the interfering system (the object kaon)

– =⇒ the marking/erasure operation can be easily performed in the “delayedchoice” mode (tmeter > tobject)

– quantum erasure allows one to restore the same KS-KL interferencephenomenon (strangeness oscillations) exhibited by a single kaon produced asa K0 or K0

F Different versions, Active and Passive, of Kaonic Quantum Erasers. Thepassive ones have no analog to any other two–level quantum system consideredup to date

F Experimental implementation currently under study at DaΦne, the Frascati(Rome) φ–factory





Kaonic Quantum Erasers Gianni Garbarino University of ...personalpages.to.infn.it/~garbarin/vaxjo.pdf · Kaonic Quantum Erasers Gianni Garbarino University of Torino, Italy Quantum

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