Summary Introduction: the EPR argument and the J. Bell test EPR - - PowerPoint PPT Presentation
Measurement of EPR- type flavour entanglement in _ (4S) B 0 B 0 decays A. Bay Ecole Polytechnique Fdrale de Lausanne quant-ph/0702267 accepted for publication in PRL 1 TaM August 2007 Summary Introduction: the EPR
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quant-ph/0702267 accepted for publication in PRL
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Introduction: the EPR argument and the J. Bell test EPR correlations with neutral B mesons (and Kaons)
EPR correlation studies by the BELLE experiment
states
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to the other
rule for non-commuting observables.
The EPR argument (1935) Ψ
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Bohm analysis of the EPR : two spin 1/2 particles from singlet state
|Ψ1,2 〉 = (1/√2) ( |↑〉1 |↓〉2 − |↓〉1 |↑〉2 )
Source
Particle 2 Particle 1
spin analyser
y x z
* The two spin are entangled: a measurement Sx = +1/2 of the spin projection //x for particle 1 implies that we can predict the outcome of a measurement for 2: Sx = −1/2. * This will happen even if the decision to orient the polarizer for particle 1 is done at the very last moment => no causal connection. How to explain this ?
the two sub-systems ...
the particle knows how to behave. => QM is incomplete. I II
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This problem was revitalized in 1964, when Bell suggested a way to distinguish QM from local models featuring hidden variables (J. S. Bell, Physics 1, 195 (1964)). The Bell theorem: Local hidden variable theories cannot reproduce all possible (statistical) results of QM. Extended by J. Clauser, M. Horne, A. Shimony, and R. Holt,
Several experiments have been done with photon pairs, atoms,...
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a, b (and b',c) : orientations of the two analyzers λ : additional information, i. e. hidden variables A(a,λ), B(b,λ) : results of the measurements (+1 or −1) Correlation function : E(a,b) ≡ A(a,
Γ
λ)B(b,λ)ρ(λ)dλ ρ(λ) is the normalized probability distribution . . . with products like B(b,λ)B(c,λ) . . . gives E(a,b) − E(a,c) + E(b',b) + E(b',c) ≤ 2 If E(a,b) depends only on α = a − b, with β = c − b, γ = b − b'
E(α) − E(α + β) + E(γ) + E(β + γ) ≤ 2
can be violated by QM
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Experiment by A. Aspect, P. Grangier, G. Roger, PRL 49, 91 (1981): measurement of the correlations of the linear polarization of two photons from a Ca40 source: the probabilities of obtaining a ±1 result along direction a (particle 1) and b (particle 2). is the correlation coefficient of the measurements on the two photons with
E r a , r b
( ) = P
++
r a , r b
( ) + P
−−
r a , r b
( ) − P
+−
r a , r b
( ) − P
−+
r a , r b
( )
P
±±
r a , r b
( )
r a , r b
the directions of the two analyzers
−2 ≤ S ≤ 2 S = E(r a , r b ) − E(r a , r b ') + E(r a ', r b ) + E(r a ', r b ')
where Assuming local realism we have QM predicts E(δ) = cos(2δ), with δ the angle between the two directions
r a , r b
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Two polarimeters with orientations a and b perform dichotomic measurements
The apparatus registers single rates and coincidence rates.
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The experiment gives: S(φ=22.5o)=3E(22.5ο)−Ε(67.5ο) = 2.697±0.015 > 2 E(δ) = (R++ + R−−) − (R+− + R−+) (R++ + R−−) + (R+− + R−+) (δ) Aspect et al. estimate the correlations for a given angle (δ=φ or δ=3φ):
(r a , r b ) = (r a ', r b ) = (r a ', r b ') ≡ φ (r a , r b ') = 3φ
They choose the following optimal configuration
r a r b r a ' r b '
22.50
giving S(φ) = 3E(φ) − E(3φ) The QM maximal value is S(22.5o) = 2√2=2.83 The LR limit is |S| ≤ 2 φ [deg] S
2 2
QM Needs to account for detection efficiency: LR limit
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* Tests have been carried out on correlated
Apostolakis et al., CPLEAR collab., Phys. Lett. B 422, 339 (1998) Ambrosino et al., KLOE collab., Phys. Lett. B 642, 315 (2006).
K0 K
In B factories we have the opportunity to study the flavour entanglement in B0 pairs from ϒ(4S) → B0 B0 In the '60 Lee and Yang recognized the EPR behaviour of the neutral Kaon doublet in a JPC = −1 state: here the "strangeness" S=+1 or S=−1 of the two Kaons plays the role of spin up or down.
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m~0.5 GeV/c2 m~5 GeV/c2
K0 = ds K
0 = d
s
KS
0 =
1 Normalization p K0 − q K
[ ]
KL
0 =
1 Normalization p K0 + q K
[ ]
p =1+ ε q =1− ε
ε is the CP violating parameter ~10−3 lifetime K0
Short τ = 0.9 10−10 s
K0
Long τ = 5.2 10−8 s
B0 = db B
0 = d
b
~identical lifetimes τ = 1.5 10−12 s cτ ~ 500 µm quite fast oscillation:
Δmd = mBH
0 − mBL 0 ≈ 3×10−10MeV
~ 0.5 1/ps flavour oscillation parameter:
ΔmK = mK L
0 − mKS 0 ≈ 3×10−12MeV
~ 0.005 1/ps ~ ~ ~ ~
BH
0 = ...
BL
0 = ...
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Suppose to produce a pure beam of B0 at t=0, then
B0 B0(t)
2
= 1 4 e−ΓHt + e−ΓLt + 2e−Γt cos(tΔmd)
{ } ≈ 1
2 e−Γt 1+ cos(tΔmd)
{ }
B
0 B0(t) 2
= 1 4 e−ΓHt + e−ΓLt − 2e−Γt cos(tΔmd)
{ } ≈ 1
2 e−Γt 1− cos(tΔmd)
{ }
Γ = 1 2 ΓL + ΓH
( ) ΓL ≈ ΓH
d
B0 B
b W+ W+ t t d b
Δmd = mBH
0 − mBL 0 ≈ 3×10−10MeV
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The analogy proposed (by many authors) is
Meson spin1/2 photon B0 ⇑ z V B ⇓ z H BH ⇒ y L =
1 2
V − i H
{ }
BL ⇐ y R =
1 2
V + i H
{ }
Differences with photons: K and B are unstable, and we cannot test for an arbitrary superposition state a B0 + b B Kaon case will be presented by Di Domenico, and Sozzi, ... K or B
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8 GeV electrons 3.5GeV positrons
Interaction Point extension: Interaction Point extension: σ σx
x≈
≈77 77
µ
µm m σ σy
y ≈
≈ 2 2
µ
µm m σ σz
z ≈
≈ 4 4
mm
mm ⇒ production of Υ(4S) (10.58GeV/c2) βγ = 0.425 Υ(4S) → B0 B0 → B+ B− 24% Y(4S) 76% continuum
e+e− → Υ(4S) → B0B
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From Y(4S) is a resonance JCP = 1− − . The strong decay conserves C which is transferred to the final state:
e+e− → Υ(4S) → B0B |Ψ〉 = (1/√2) ( |B0〉1 |B0〉2 − |B0〉1 |B0〉2 )
Because of this a measurement of the flavour (B0 or B0) of particle 1, tells with 100% probability that, at the same proper time, particle 2 is of the opposite flavour (B0 or B0). Experimentally, the flavour of a B can only be determined when it decays, and only if it decays into a “flavour-specific” mode, like
B0 → D*−l+ν
d
l+ ν
D*− B0
b c W+ the sign of the lepton tells us the neutral B flavour
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ϒ(4S) z z1 z2 Δz e, µ l+ → B0 l− → B
QM: region of B0 & B0 coherent evolution
Than the other B0 oscillates freely before decaying after a time given by
Δt ≈Δz /cβγ
B0 and B0 oscillate coherently. When the first decays, the other is known to be of the opposite flavour, at the same proper time D ϒ(4S) produced with βγ = 0.425 by the asymmetric collider
N.B. : production vertex position z0 not very well known : only Δz is available !
z0
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Time (Δt)-dependent decay rate into two Opposite Flavour (OF) states
ROF ∝e−Γ(t1 +t 2 )(1+ cos(ΔmdΔt)) RSF ∝e−Γ(t1 +t 2 )(1− cos(ΔmdΔt))
idem, into two Same Flavour (SF) states => we obtain the time-dependent asymmetry
Δmd is the mass difference
eigenstates
AQM(Δt) = ROF − RSF ROF + RSF (Δt) = = cos(ΔmdΔt)
( ignoring CP violation effects O(10-4), and taking ΓH = ΓL )
Δmd = 0.507ps−1
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Bramon and Nowakowsi, Bertlmann and Hiesmayr, Gisin and Go:
AQM(Δt) = ROF − RSF ROF + RSF (Δt) = cos(ΔmdΔt)≡
?
cos(φ) = E(φ)
The time-dependent asymmetry is formally equivalent to the expression for the EPR correlation experiment with spin 1/2 particles: Then Gisin and Go (Am. J. Phys. 69, 3 (2001)) suggest to Bell-CHSH test with S(φ) = 3E(φ) − E(3φ)
to Δt~2.6 ps
The experimental result (A. Go, Journal of Modern Optics 51, 991 (2004)) 2.73 ± 0.19 was >3σ above the Local Realism . QM S φ LR limit
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* Bertlmann, Bramon, Garbarino, and Hiesmayr, Phys. Lett. A 332 (2004) 355 1) the measurement is passive => cannot enforce locality; 2) non unitary evolution of the system because of decay. Bell-CHSH inequalities refer to dichotomic measurements, "Are you a B0 or a B0?" and "Are you a B0 or not ?" are two different questions. * Bramon, Escribano, and Garbarino, Journ. of Modern Optics 12 (2005) 1681 Published results are a "convincing proof of quantum entanglement" but "the test is not a genuine Bell's inequality and thus cannot discriminate between QM and local realistic approaches". Proof : a local realistic model is considered which contains the information (i. e. hidden variables) needed to "deterministically specifying ab ovo the future decay times and decay modes of its two members". (Only know implementation so far is a quite artificial model by E. Santos in quant-ph/0703206). Let's consider a simple model with locality ...
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Δmd = 0.507ps−1
Just after the decay into opposite flavor states, we considers an independent evolution for the B0 pair
≠ QM
Only Δt is known: need to integrate
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PUnlike(t1,t 2) = 1 4 e−Γ(t1 +t 2 ) 1+ cos(Δmdt1)cos(Δmdt 2)
[ ]
PLike(t1,t 2) = 1 4 e−Γ(t1 +t 2 ) 1− cos(Δmdt1)cos(Δmdt 2)
[ ]
E(t1,t 2) = PUnlike − PLike PUnlike + PLike (t1,t 2) ≡ ASD(t1,t 2) = cos(Δmdt1)cos(Δmdt 2)
S(Δt) = 3E(Δt) − E(3Δt)
This local realistic model has a region with |S|≤ 2 ! At which level is all this wrong ? Is E not appropriate ? Is the analogy flavour oscillation ≡ rotation of the polarization angle OK ? ~ experiment with photons SD: After integration for fixed Δt we can compute S:
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All this suggests to forget the Bell-CHSH test, and look for some more modest goal:
We decided to to check if specific local realistic models are compatible with our experimental results at Y(4S).
We didn't find too many local realistic models on the market (QM is difficult to imitate !)
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Local Realism, each B has "elements of reality” (hidden variables) λ1 : CP = +1 or -1 λ2 : Flavour = +1 or -1 indexed by i = 1, 2, 3, 4
_ _ B0
H, B0 H, B0 L, B0 L
=> 4 basic states
* Mass states are stable in time, simultaneous anti-correlated flavor jumps.
The model works with probabilities pij(t|0) = prob for a B to be in the state j at proper time t=t, conditional of having been in state i at t=0.
* pij set to be consistent with single B0 evolution ~ exp{(Γ/2 + im)t}.
* PS build a model with a minimal amount of assumptions ⇒They only determine upper and lower limits for combined probabilities ...
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=> analytical expressions for A corresponding to the limits. The Amax is PSmin
AQM>APS in the Δt region below ~5 ps
APSmax (t1,t 2) =1− {1− cos(ΔmdΔt)}cos(Δmdt min) + sin(ΔmdΔt)sin(Δmdt min)
Δmd = 0.507ps−1 t min = min(t1,t 2)
≠ QM
Only Δt is known: need to integrate
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We want to provide FULLY CORRECTED time-dependent flavour asymmetry for the events For this, we will
procedure => the result can then be directly compared to the models: We will use our data to test
against QM predictions
e+e− → Υ(4S) → B0B
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.
ACC Silicon Vertex Detector SVD resolution on Δz ~ 100 µm Central Drift Chamber CDC (σPt/Pt)2 = (0.0019 Pt)2 + (0.0030)2 K/π separation : dE/dx in CDC σdE/dx =6.9% TOF σTOF = 95ps Aerogel Cerenkov ACC Efficiency = ~90%, Fake rate = ~6% →3.5GeV/c γ, e± : ECL (CsI crystals) σE/E ~ 1.8% @ E=1GeV
e± : efficiency > 90% ~0.3% fake for p > 1GeV/c
KL and µ± : KLM (RPC) µ± : efficiency > 90%
<2% fake at p > 1GeV/c
~ 8 m this study considers
152 106 B0B0 pairs
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B0 → D*−l+ν
K+π− K+π−π0 K+π−π+π−
D
0πslow −
* First B measured via
0.14 0.15 0.16 0.17 GeV/c2
M(D*)−M(D0) Signal Sideband
* Remaining tracks are used to guess the flavour of second B, from the standard Belle flavour tagging procedure From a total of 152 106 B0B0 pairs: − 6718 OF and 1847 SF events after selection. − Δz is obtained from track fit of the two vertices and converted into a Δt value
OF SF
0 5 10 15 20 Nevents/ps
OF SF Δt [ps]
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0 2 4 6 8 10 12 14 16 18 20 Δt [ps] 1 0.5
syst stat after events selection background subtracted
We correct bin by bin the OF and SF distributions for ♦ Fake D* background ♦ Uncorrelated D*-leptons, mainly D* and leptons from different B0 ♦ B± → D** l v background ♦ ~1.5% fraction of wrong flavour associations
A(Δt)
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Deconvolution is performed using response matrices for OF and SF distributions. The two 11x11 matrices are build from GEANT MC
Höcker and V. Kartvelishvili, NIM A 372, 469 (1996). Toy MC of the 3 models (QM, PS and SD) have been used to study the method and to estimate the associated systematic error.
The result is given here: Window Asymmetry
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Add OF+SF distributions and fit for τB0 + data
τB0 = 1.532±0.017(stat) ps χ2 = 3/11 bins => consistent with PDG value
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Least-square fits including a term taking the world-average Δmd into account. To avoid bias we discard BaBar and BELLE measurements, giving <Δmd> = ( 0.496 ± 0.013 ) ps-1
fitted value:
Δmd = (0.501±0.009) ps-1
χ2 = 5.2 (11 dof)
=> Data fits QM
Data QM (error from Δmd)
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Fit data to PS model, using the closest boundary. We conservatively assign a null deviation when data falls between boundaries => Data favors QM over PS at the level of 5.1σ PS
fitted value:
Δmd=(0.447±0.010)ps-1
χ2=31.3 PS (error from Δmd) Data
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LR
χ2=74/11 bins fitted value:
Δmd=(0.419±0.008)ps-1
χ2=174
=> Data favors QM over SD model by 13σ.
Data SD (error from Δmd)
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"Entanglement, Belle Inequalities, and Decoherence in Particle Physics" . . .
Example: Decoherence can originate from the "interaction" with a foamy space-time => a modified Liouville equation describes the time evolution
dρ(t) dt = i[ρ,H]+ Δ[ρ]
Simplest parametrization: Δ[ρ] = λD[ρ] with the decoherence parameter λ O(λ) = m2/mPl (at the best) => time dependent decoherence
RateOF
SF
(t1,t 2) ~ e−Γ(t1 +t 2 ) 1+ (1− e−λt min )cos ΔmdΔt
( )
{ }
−
=> amplitude reduction of the time dependent asymmetry
A(Δt) ~ (1−ζ)AQM(Δt)
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Our setup does not permit time-dependent studies. In practice, we limit our decoherence studies to the two simplest possibilities:
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2) Decoherence in B0, B0: A ~ (1-ζB°B°)AQM+ζB°B°ASD
Measurements in K0 system:
ζB°B°= 0.029±0.057
1) Decoherence in BH, BL : A ~ (1−ζBHBL)AQM
ζBHBL = 0.004 ± 0.017 (preliminary)
Measurements in K0 system:
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We have measured the time-dependent asymmetry due to flavour oscillation. The asymmetry has been corrected for the experimental effects and can be used directly to compare with the different theoretical models.
* The time dependent asymmetry is consistent with QM predictions, while the local realistic model of Pompili and Selleri is disfavoured at the level of 5.1σ. A model with immediate disentanglement into flavour eigenstates is excluded by 13σ. This EPR state can be used for searches of New Physics, by the measurement of the decoherence. * The time dependent asymmetry measured by us is consistent with no decoherence. We have performed an experimental study of the EPR-type flavour entanglement in ϒ (4S)B0B0 decays.