Maulik Nariya

DHEP (INO)

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Outline

Density matrix formalism

Use density matrix to describe a neutrino beam

2 flavor case (νe-νμ)

Evolution of S: analogous to spin precession in magnetic field

Understanding oscillations with spin precession analogy

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Density Matrix Formalism Pure Ensemble: a collection of physical states characterized by

the same ket

Mixed Ensemble: a fraction of the members with relative

population wi are characterized by ket

The density matrix formalism helps in quantitative description

of physical situations with mixed ensembles.

Populations are constrained by

the kets need not be orthogonal.

)(i

1i

iw

)(i

3

Measurement of some observable A gives the expectation

value:

Probabilistic concepts enters twice

- quantum mechanical probability

- statistical wi weight for finding in the ensemble

We define a density operator as

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Ammmw

AwA

i

i m

i

i

i

i

i

ofket Eigen : 2

)(

)()(

2)()( ii A

)(i

)()( i

i

i

iw

Some properties Matrix element looks like

Expectation value of the observable A:

is Hermitian.

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nmwnm ii

i

i

)()(

)tr( A

mAnnmAm n

1

)tr(

)()(

)()(

ii

i

i

i

i m

i

i

w

mmw

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ensemble mixedfor 1

ensemble purefor 1

)tr(

2

)()()()(22

i

i

i

iii

m n

i

i

w

mnnmw

The time evolution of the density matrix is given by

],[

Ht

i

Use density matrix to describe a ν beam A neutrino beam can be described by this density matrix

operator

Special case of one initial flavor β can be obtained by

setting wα=δαβ

Probability of detecting a νβ at a distance x is give by

where ρF(x): density matrix in flavor basis

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)()()( xwxx

)()()( xxxP F

Evolution of ρ Evolution equation of ρF(x) is given by

Recall

UM: the effective mixing matrix in matter

The density matrix in effective mass basis in matter is

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FFF

Hdx

di

,

2

2

0

0

M

M

MMFMm

mHUHU

M

F

M

M UU

The evolution equation of ρM is

For adiabatic case second term is negligible and can be ignored

The diagonal elements of ρM remain constant and for k≠j

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ΜMM

M

M

M

dx

dUUiH

dx

di

,,

MkjjjMkkMM

kjHH

dx

di

xdxHxHix

x

jjkk

M

kj

M

kj

0

)()()0()(

2 flavor case (νe-νμ) A hermitian matrix X can be decomposed in terms of Pauli Spin

matrices as

Both HF and ρF are Hermitian. Tr(HF )=0, Tr(ρ

F )=1.

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k

kkXXI

X )Tr()(Tr2

zSySxSS

zByBxBB

zyx

SIBH

zyx

zyx

zyx

F

F

ˆˆˆ

ˆˆˆ

ˆˆˆ

vectors,with the

2

1 ,

2

1

Components of B and S in flavor basis are:

From the initial conditions for the density matrix, we have

The probabilities of detection of a νe or a νμ at a distance x is

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,,Im2 ,Re22

2cos

2

2sin

,2

2cos,0 ,

2

2sin

22

22

FF

eez

F

ey

F

ex

MMMM

Czyx

SSSE

m

E

mE

AmBB

E

mB

wwSSS ezyx )0( ,0)0( ,0)0(

zSSxxP

zSSxxP

z

F

z

F

eee

ˆ12

11

2

1)()(

ˆ12

11

2

1)()(

Evolution of the vector S

The evolution equation of the vector S is

Analogous to magnetic moment (with g=1), precessing around

a magnetic field. The precession frequency is given by

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BSdx

Sd

E

mB M

2

2

Angle between S and B

Precession becomes clear by rotating the reference frame by 2θM

in 1-3 plane

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MBS

BS2cos

)0(

)0(

zxz

yy

zxx

MM

MM

ˆ2cosˆ2sinˆ

ˆˆ

ˆ2sinˆ2cosˆ

In this rotated frame B lies along zʹ

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Em

BBB

BB

BBB

MzMxMz

yy

zMxx

22cos2sin

0

02sin2cos

2

Fig(1):Precession of S around B with constant matter density Ne

The evolution equation of S thus becomes

Using the initial conditions on S, the solution is

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0, ,

dx

SdS

dx

SdS

dx

Sd zx

y

yx

0,2

2

,

2

yx

yxS

dx

Sd

)(2cos)(

)sin()(2sin)(

)cos()(2sin)(

wwxS

xwwxS

xwwxS

eMz

eMy

eMx

The third component of the S in flavor basis becomes

The probabilities of detecting νe and νμ are

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2sin2sin21)()(2cos)(2sin)( 22

xwwxSxSxS MezMxMz

2sin2sin

2

1)(

2

1)(

2sin2sin

2

1)(

2

1)(

22

22

xwwxP

xwwxP

Me

Mee

Solar case (variable matter density) Initially pure νe beam

Very large matter density (Ne>>NeR)

θM ≈ π/2

νe almost coincides with ν2M

S describes the surface on a narrow

cone around the negative zʹ axis

When the ν passes through the resonance, Bx (and hence B)

changes its value.

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If the resonance region is crossed adiabatically, the speed of

rotation of S around B is much faster than the change in B.

The cone swept by S is dragged by B and is finally rotated

upside-down.

Thus for small θ, the probability of νeνμ conversion is large

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References C. Giunti and C. W. Kim , Fundamentals of Neutrino Physics and

Astrophysics, Oxford University Press, Oxford, 2007, pp.329-326

C. W. Kim, J. Kim, and W. K. Sze, Geometrical interpretations of

neutrino oscillations in vacuum and matter, Phys. Rev. D37 (1988)

Basudeb Dasgupta and Amol Dighe, Collective three-flavor

oscillation of supernova neutrinos, Phy. Rev. D77, 113002(2008)

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Thank You

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