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.
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 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”
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
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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 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
Quantum Theory: Reconsideration
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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 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]
Quantum Theory: Reconsideration
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Kaonic Quantum Erasers (page 6) Gianni Garbarino
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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
Quantum Theory: Reconsideration
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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 → K0; t] ≡∣
∣〈K0|K0(t)〉∣
∣
2=
1
2{1 − V0(t) cos(∆mt)}
Time–dependent Visibility V0(t) = 1cosh (∆Γ t/2)
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Kaonic Quantum Erasers (page 8) Gianni Garbarino
University of Torino, Italy
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
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 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}
Quantum Theory: Reconsideration
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Kaonic Quantum Erasers (page 10) Gianni Garbarino
University of Torino, Italy
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
Quantum Theory: Reconsideration
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Kaonic Quantum Erasers (page 11) Gianni Garbarino
University of Torino, Italy
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
Quantum Theory: Reconsideration
of Foundations – 4 (QTRF4)
Kaonic Quantum Erasers (page 12) Gianni Garbarino
University of Torino, Italy
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]
Quantum Theory: Reconsideration
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Kaonic Quantum Erasers (page 13) Gianni Garbarino
University of Torino, Italy
Time and the Quantum: Erasing the Past and Impacting the FutureY. Aharonov and M. S. Zubairy, Science 307 (2005) 875
Quantum Theory: Reconsideration
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Kaonic Quantum Erasers (page 14) Gianni Garbarino
University of Torino, Italy
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
Quantum Theory: Reconsideration
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Kaonic Quantum Erasers (page 15) Gianni Garbarino
University of Torino, Italy
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 Theory: Reconsideration
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Kaonic Quantum Erasers (page 16) Gianni Garbarino
University of Torino, Italy
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
Quantum Theory: Reconsideration
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Kaonic Quantum Erasers (page 17) Gianni Garbarino
University of Torino, Italy
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.
Quantum Theory: Reconsideration
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Kaonic Quantum Erasers (page 18) Gianni Garbarino
University of Torino, Italy
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.
Quantum Theory: Reconsideration
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Kaonic Quantum Erasers (page 19) Gianni Garbarino
University of Torino, Italy
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
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Kaonic Quantum Erasers (page 20) Gianni Garbarino
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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
Quantum Theory: Reconsideration
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Kaonic Quantum Erasers (page 21) Gianni Garbarino
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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
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Kaonic Quantum Erasers (page 22) Gianni Garbarino
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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
Quantum Theory: Reconsideration
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Kaonic Quantum Erasers (page 23) Gianni Garbarino
University of Torino, Italy
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
Quantum Theory: Reconsideration
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Kaonic Quantum Erasers (page 24) Gianni Garbarino
University of Torino, Italy