wave-packet treatment of neutrinos and its quantum-mechanical implications

Wave-packet treatment of neutrinos and itsquantum-mechanical implications

Li, Cheng-Hsien

Cosmology Lunchtime Seminar

Feb. 8th, 2016

Li, Cheng-Hsien Feb. 8th, 2016 1 / 25

Outline

1 Introduction

2 Overlap of neutrino wave functions

3 Quantum-mechanical implications

4 Summary


Introduction– Motivation

Motivation:

Quantum particles are treated as Wave Packets (WPs).

Finite size of the WP introduces intrinsic momentum uncertainty.∆θ ∼ ∆k⊥/k0 ∼ 1/2Eνat

It’s assumed that ν-WPs do not overlap in experiments.

Only 1 ν is involced in each detection process

If wave functions of neutrinos overlap in the detector,indistinguishability may require proper theoretical treatment.


Introduction– Assumptions

Assumptions:

Neglect neutrino mass and thus no neutrino oscillations.

ν-WPs are simple Gaussian WPs.

At t = 0, the initial WP with average momentum k0z is described by

Ψ(~r , 0) =1

(2π)3/4 ata1/2l

exp

(− ρ2

4a2t

− z2

4a2l

+ ik0z

)Initial longitudinal width al and transverse width at .Sharp momentum distribution: Eνal 1 and Eνat 1.Spinor part of neutrino wave function is neglected.


Introduction– Paraxial Solution from Optics

Paraxial solution from laser optics:

Ψk(~r , t) =A

σt(z)exp

i

[kρ2

2R(z)− ξ(z)

]− ρ2

4σt(z)2+ ik (z − t)

An approximate solution to(−∂2

t +∇2)Ψ = 0 with theassumption (kat)

2 1.

Rayleigh range LR ≡ 2ka2t .

σt(z)2 ≡ a2t

(1 + z2

L2R

)ξ(z) ≡ tan−1(z/LR)

R(z) ≡ z(

1 +L2Rz2

).

Re[Ψk] at t=0 assuming kat = 5

0 1 2 3z/LR

-10

-5

0

5

10

ρ/a

t

-1.0

-0.5

0.0

0.5

1.0


Introduction– Constructing Gaussian WP solution from Paraxial Solution

To obtain a localized solution in the z-direction, superpose paraxialsolutions with different longitudinal momenta

Ψ(~r , t) = σ−1k

∫ ∞−∞

dk exp

[−(k − k0)2

4σ2k

]Ψk(~r , t), where σk ≡ (2al)

−1

The integration reduces to Gaussian integrals under both near-fieldlimit z LR and far-field limit z LR .

Ψnear(~r , 0) ≈[(2π)3a4

t a2l

]− 14 exp

(− z2

4a2l

− ρ2

4a2t

+ ik0z

)

Ψfar(~r , t) ≈[(2π)3σt(z)4a2

l

]− 14 exp

−(z + ρ2

2z − t)2

4a2l

− ρ2

4σt(z)2

× exp

[+ik0

(z +

ρ2

2z− t

)− iπ

2

]Li, Cheng-Hsien Feb. 8th, 2016 6 / 25

Introduction– Visualization of the Far-field Solution

Re[Ψfar(~r , t)] plotted by assuming at = al ≡ a and Eνat = 10

Wave-particle duality is manifested by the spherical wavefront of Ψfar

68 72 76 80 84 88 92

z/a

-12

-8

-4

0

4

8

12

ρ/a

-0.06

-0.03

0.00

0.03

0.06

t = 80 a


Overlap of Neutrino Wave Functions– WP Volume

Define the ”physical” volume of a WP via probability interpretationThe following plots |Ψfar|2 assuming at = al = a and Eνat = 10.The 90%-probability volume is enclosed between gray dashed sectors.The volume has radial width 4al and angular size θf ≈ 1.22/(Eνat)

0 20 40 60 80 100 120

z/a

-60

-40

-20

0

20

40

60ρ/a

0.000

0.001

0.002

0.003

0.004

t = 120 a

t = 80 a

2θf

4a

4a


Overlap of Neutrino Wave Functions– Definition of Overlap

Two WPs have ”similar” energies if their mean energy difference isless than the intrinsic energy uncertainty, ∆Eν ∼ 2/al , in a WP.

Eν = |~k | and Ψ(~k) ∝ exp[−a2

t

(k2x + k2

y

)− a2

l (kz − k0)2]

Consider an auxiliary sphere centered at the (point) sourceThe center of the reference WP (gray) passes the sphere at time tref

To overlap, another WP (blue) must be in the neighborhood.Criteria: tref − τ < t < tref + τ and cos−1(r · rref) < 2θfτ = 4al is the radial width of the fan-shaped strips.

Source


Overlap of Neutrino Wave Functions– Overlap Factor

For every chosen reference WP, the expected number of overlappingWPs in the neighborhood:

η ≡ dΦ

dEν×∆Eν × 2τ ×

∆Ωoverlap

∆Ωsource

dΦ/dEν is the energy-differential rate of the source.∆Ωoverlap ≈ 6π(Eνat)

−2 is the solid angle covered by r satisfyingcos−1(r · rref) < 2θf∆Ωsource is the solid angle over which the source emits neutrinos.Dependence on al cancels: ∆Eν × 2τ ∼ 16.

η depends on the source intensity, Eνat , and a geometric factor:

η ∼ dΦ

dEν× 96π

(Eνat)2∆Ωsource


Overlap of Neutrino Wave Functions– Visualization

Schematic illustration of different degree of overlap by varying at .Emission time and angle are given by random number generator.Overlapped WPs are marked in blue.

η = 0.17

η = 1.0


Overlap of Neutrino Wave Functions– Estimate of η of Various Neutrino Sources

Estimate of η using typical values of source intensities

The estimate is subject to the unknown atNegligible overlap in accelerator and reactor neutrino experiments.dΦ/dEν of line sources requires proper estimate of spectral broadeningHBT geometry is discussed the next slide.

Source ηpoint source Distance Energy HBT×(Eνat)2 Scale Scale Geometry Spectrum

1. Accelerator 10−1 106 m GeV No continuous2. Reactor core 100 103 − 105 m MeV No continuous3. Radioactive 102 − 103 1 m MeV No line4. The Sun 109 − 1019 1 AU 0.1− 10 MeV No continuous5. Supernova 1032 − 1036 100 − 101 kpc 10 MeV Yes continuous


Quantum-mechanical Implications– When a Pair Is Detected Simultaneously

Hanbury Brown and Twiss effectA method proposed to measure the angular size of a star in 1956Coincident counts require

1 E × rab,⊥rcd,⊥/R . 1 (HBT geometry)2 Wave functions of the two photons overlap in the detection region

Requirements 1. and 2. ⇒ bosons tend to be found in the same phasespace cell: ∆kx∆ky∆kz∆x∆y∆z ∼ h3

HBT effect for fermions realized in experiments recently

Production Detection

a

b

c

d

E. Bocquillon et al. Nature 339, 1054 (2013)


Quantum-mechanical Implications– Is HBT Effect for Supernova Neutrinos Possible?

Most likely, the next Milky Way SN explosion happens at 1− 20 kpcaway from the Earth.

HBT geometry is satisfied with HyperK or JUNO detector size.

With luck, the explosion of the nearest possible SN progenitor at 0.2kpc away will result in 107 events in JUNO detector.

Assume all events are recorded within a ∆t = 0.1s time window.Kersten & Smirnov recently estimates al ∼ 10−11 cm for SN-ν.These ”to-be-detected” wave trains spans c∆t ∼ 107m.The probability of one wave train overlapping with another is

P ∼ 107 × alc∆t

∼ 10−13


Quantum-mechanical Implications– When Only One of the Overlapping Neutrinos is Detected

Most likely, one ν is measured at a time in a detector.

A pair of overlapping neutrinos from production processes a and bcan be described as

Ψ2-p(~r1, ~r2, t) =Ψa(~r1, t)Ψb(~r2, t)−Ψa(~r2, t)Ψb(~r1, t)√

2− 2|〈Ψa|Ψb〉|2

〈Ψa|Ψb〉 6= 0 for two WP-states that overlap in both position andmomentum spaces


Quantum-mechanical Implications– When Only One of the Overlapping Neutrinos is Detected

Probability of finding one ν at d3~r regardless of the other

dP = d3~r ×[∫

d3~r2 |Ψ2-p(~r , ~r2, t)2|+∫

d3~r1 |Ψ2-p(~r1, ~r , t)|2]

= d3~r ×|Ψa(~r , t)|2 + |Ψb(~r , t)|2 − 2Re[Ψa(~r , t)Ψ∗b(~r , t)〈Ψa|Ψb〉]

1− |〈Ψa|Ψb〉|2

Total probability is conserved:∫dP = 2

Interference pattern in probability due to 〈Ψa|Ψb〉 6= 0 is washed awayby the isotropy of neutrino source


More Questions

Any subtle QM effect due to WP overlap that affects neutrinooscillations?

How to estimate at for neutrinos from different sources?

Source ηpoint source Distance Energy HBT×(Eνat)2 Scale Scale Geometry Spectrum

3. Radioactive 102 − 103 1 m MeV No line4. The Sun 109 − 1019 1 AU 0.1− 10 MeV No continuous5. Supernova 1032 − 1036 100 − 101 kpc 10 MeV Yes continuous


Summary

We define the volume ”occupied” by a 3D Gaussian WP according toits probability distribution.

The overlap factor is constructed to quantify how likely ν-WPsoverlap. It depends on the source intensity, Eνat , and a geometricfactor.

The overlap is potentially significant for neutrinos from radioactivesources, if Eνat is not too large. For astrophysical sources, theoverlap seems overwhelming with any microscopic at .

Neglecting ν-oscillations, indistinguishability of neutrinos does notaffect experimental observables.


Backup Slides– 90% Probability Volume of Ψfar(~r , t)

The WP asymptotically spans a small angle, so r ≈ z + ρ2/2z .

The probability density in the far-field limit can be expressed as|Ψfar(~r , t)|2 = R(r , t) Θ(θ), where

R(r , t) ≈ 1

r2al√

2πexp

[−(r − t)2

2a2l

]Θ(θ) ≈ (2k0at)

2

2πexp

[− θ2

2 · (2k0at)−2

]The volume occupied by a WP can be defined via probability density.It corresponds to

95% probability from R(r , t): t − 2al ≤ r ≤ t + 2al95% probability from Θ(θ): 0 ≤ θ ≤ θf , where θf ≈ 1.22/(Eνat)Overall, the volume corresponds to 90% probability.


Backup Slides– Estimate of η for accelerator neutrinos

Accelerator experiment (e.g. LE configuration of NuMI beam)

νµ or νµ are produced mostly from pion decays.Instantaneous POT rate 4× 1019POT · s−1 within ∼ 1 nsec window.Pions and subsequent muon neutrinos retain similar time structure.Each POT results in 10−3GeV−1 of 3 GeV νµ intercepted by the ND.The ND subtends an effective solid angle ∼ 10−5π.

Estimated overlap factor η ∼ 0.3× (Eνat)−2 at Eν = 3 GeV.


Backup Slides– Estimate of η for reactor neutrinos

Fission reactor core at typical 3 GW thermal power.

Thermal power ∝ fission rate ∝ neutrino production rateIsotropic source of νe with dΦ

dEν∼ 1020 MeV−1 · s−1 at Eν = 3.6 MeV.

Estimated overlap factor η ∼ 1.6× (Eνat)−2 at the peak energy.

Vogel, P. et al. Nature Communications 6, 6935 (2015)


Backup Slides– Estimate of η for neutrinos from radioactive sources

Radioactive sources: 51Cr and 37Ar.

Used in GALLEX and SAGE for calibrating neutrino detectors.Both decay via electron capture (X→ X′ + νe).Source intensity decreases exponentially over time.Initial intensities: ΦGALLEX ≈ 6.6× 1016s−1& ΦSAGE ≈ 1.7× 1016s−1.The neutrino spectrum is broadened by thermal Doppler effect.

J. N. Abdurashitov et al. PRC 59, 2246 (1999) J. N. Abdurashitov et al. PRC 73, 045805(2006)


Backup Slides– Estimate of η for neutrinos from radioactive sources

Thermal Doppler broadening:

Random thermal motion of the decaying atom creates relative motion.Thermal velocity is of order vth ∼ 10−6, from 1

2Mv2th ∼ kBT .

∆Espectral ∼ Eν × vth and dΦ/dEν ∼ Φ/∆Espectral.

At the beginning of GALLEX measurementsdΦdEν≈ 9× 1022 MeV−1 · s−1.

η ∼ 1400× (Eνat)−2.

At the beginning of SAGE measurementsdΦdEν≈ 2× 1022 MeV−1 · s−1.

η ∼ 300× (Eνat)−2.


Backup Slides– Estimate of η for solar neutrinos

The Sun:dΦdEν∼ (3× 1028 − 6× 1038) MeV−1 · s−1 at 0.2 MeV < Eν < 20 MeV.

η ∼ (109 − 1019)× (Eνat)−2

W. C. Haxton et al. Ann. Rev. Astron. Astrophys. 51, 21 (2013)


Backup Slides– Estimate of η for supernova neutrinos

At 0.1− 10 seconds after the onset of collapse of a 20 M supernova,dΦdEν∼ (1052 − 1056) MeV−1 · s−1 within 4 MeV < Eν < 40 MeV.

η ∼ (1032 − 1036)× (Eνat)−2.

T. Totani et al. ApJ 496, 216 (1998)


wave-packet treatment of neutrinos and its quantum-mechanical implications

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