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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
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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
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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
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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()(
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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
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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)()(
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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
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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
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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
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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
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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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