- Email: [email protected]
Berry phase for oscillating neutrinos
4 November 1999
Physics Letters B 466 Ž1999. 262–266
Berry phase for oscillating neutrinos Massimo Blasone
a,c,d,1
, Peter A. Henning
b,2
, Giuseppe Vitiello
c,3
a Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2BZ, UK Institut fur ¨ Kernphysik, TH Darmstadt, Schloßgartenstraße 9, D-64289 Darmstadt, Germany c Dipartimento di Fisica dell’UniÕersita` and INFN, Gruppo Collegato, I-84100 Salerno, Italy d Unita` INFM di Salerno, I-84100 Salerno, Italy
b
Received 25 February 1999; received in revised form 10 May 1999; accepted 23 September 1999 Editor: L. Alvarez-Gaume´
Abstract We show the presence of a topological ŽBerry. phase in the time evolution of a mixed state. For the case of mixed neutrinos, the Berry phase is a function of the mixing angle only. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 03.65.Bz; 11.10.-z; 14.60.Pq
1. Introduction Particle mixing and oscillations play a relevant role in high energy physics. In particular, in recent years a growing interest in neutrino mixing and oscillations w1x has been developed which manifests itself both in a strong experimental effort and in a renewed theoretical research activity. Contributions towards the correct theoretical understanding of the field mixing have been recently presented w2,3x. In the present paper we show how the notion of Berry phase w4x enters the physics of mixing by considering the example of neutrino oscillations. Since its discovery w4x, the Berry phase has attracted much interest w5x at theoretical as well as at experimental level. This interest arises because the
1
E-mail: [email protected] E-mail: [email protected] 3 E-mail: [email protected] 2
Berry phase reveals geometrical features of the systems in which it appears, which go beyond the specific dynamical aspects and as such contribute to a deeper characterization of the physics involved. The successful experimental findings in many different quantum systems w5x stimulate further search in this field. Aimed by these motivations, we show that the geometric phase naturally appears in the standard Pontecorvo formulation of neutrino oscillations. Our result shows that the Berry phase associated to neutrino oscillations is a function of the mixing angle only. We emphasize that such a result has phenomenological relevance: since geometrical phases are observable, the mixing angle can be Žat least in principle. measured directly, i.e. independently from dynamical parameters as the neutrino masses and energies. Although in the following we treat the neutrino case, we stress that our result holds in general, also in the case of mixed bosons ŽKaons, hX s, etc...
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 1 1 3 7 - 5
M. Blasone et al.r Physics Letters B 466 (1999) 262–266
2. Berry phase for oscillating neutrinos
where we have used the notation ² ne Ž t . < i E t < ne Ž t . : s ² ne Ž t . < H < ne Ž t . :
Let us first consider the two flavor case w1x:
s v 1 cos 2u q v 2 sin2u
< ne : s cos u < n 1 : q sin u < n 2 : , < nm : s ysin u < n 1 : q cos u < n 2 : .
' ve e .
Ž 1.
² nm Ž t . < i E t < nm Ž t . : s ² nm Ž t . < H < nm Ž t . :
< ne Ž t . : ' eyi H t < ne Ž 0 . :
s v 1 sin2u q v 2 cos 2u
s eyi v 1 t Ž cos u < n 1 : qeyi Ž v 2y v 1 . t sin u < n 2 : . ,
Ž 2.
where H < n i : s v i < n i :, i s 1, 2. Our conclusions will also hold for the muon neutrino state, with due changes which will be explicitly shown when necessary. The state < ne Ž t .:, apart from a phase factor, reproduces the initial state < ne Ž0.: after a period T s 2p v2y v1 :
fsy
2pv 1
v2 y v1
.
Ž 3.
We now show how such a time evolution does contain a purely geometric part, i.e. the Berry phase. It is a straightforward calculation to separate the geometric and dynamical phases following the standard procedure w6x:
be s f q sy
T
H0
² ne Ž t . < i E t < ne Ž t . : dt
2pv 1
v2 y v1
2p q
v2 y v1
Ž v 1 cos 2u q v 2 sin2u .
s 2p sin2u .
T
H0
' vmm ,
s 12 Ž v 2 y v 1 . sin2 u ' vm e ,
with v e m s vm e . In order to better understand the meaning of Eqs. Ž4. – Ž6., we observe that, as well known, < ne : is not eigenstate of the Hamiltonian, and ² ne Ž 0 . < ne Ž t . : s eyi v 1 t cos 2u q eyi v 2 t sin2u .
Ž 10 .
Thus, as an effect of time evolution, the state < ne : ‘‘rotates’’ as shown by Eq. Ž10.. However, at t s T, ² ne Ž 0 . < ne Ž T . : s e i f s e i b e eyi v e e T ,
Ž 11 .
i.e. < ne ŽT .: differs from < ne Ž0.: by a phase f , part of which is a geometric ‘‘tilt’’ Žthe Berry phase. and the other part is of dynamical origin. In general, for t s T q t , we have ² ne Ž 0 . < ne Ž t . : s e i f ² ne Ž 0 . < ne Ž t . : 2
s e i2 p sin u eyi v e e T = Ž eyi v 1t cos 2u q eyi v 2t sin2u . .
Ž 12 . Also notice that ² nmŽ t .< ne Ž t .: s 0 for any t. However, ² nm Ž 0 . < ne Ž t . :
for t s T q t ,
Ž 5. Note that b e q bm s 2p . We can thus rewrite Eq. Ž3. as 2
Ž 9.
s 12 e i f eyi v 1t sin2 u Ž eyi Ž v 2y v 1 . t y 1 . ,
² nm Ž t . < i E t < nm Ž t . : dt s 2p cos 2u .
< ne Ž T . : s e i2 p sin u eyi v e e T < ne Ž 0 . : ,
Ž 8.
² nm Ž t . < i E t < ne Ž t . : s ² nm Ž t . < H < ne Ž t . :
Ž 4.
We thus see that there is indeed a non-zero geometrical phase b , related to the mixing angle u , and that it is independent from the neutrino energies v 1 , v 2 and masses m1 , m 2 . In a similar fashion, we obtain the Berry phase for the muon neutrino state:
bm s f q
Ž 7.
We will also use
The electron neutrino state at time t is w1x
< ne Ž T . : s e i f < ne Ž 0 . : ,
263
Ž 6.
Ž 13 .
which is zero only at t s T. Eq. Ž13. expresses the fact that < ne Ž t .: ‘‘oscillates’’, getting a component of muon flavor, besides getting the Berry phase. At t s T, neutrino states of different flavor are again each other orthogonal states.
M. Blasone et al.r Physics Letters B 466 (1999) 262–266
264
Generalization to n-cycles is also interesting. Eq. Ž4. Žand Eq. Ž5.. can be rewritten for the n-cycle case as
b eŽ n. s
nT
H0
² ne Ž t . < i E t y v 1 < ne Ž t . : dt
s 2p n sin2u , and Eq. Ž12. becomes
Ž 14 .
ss
² ne Ž 0 . < ne Ž t . : s e i n f ² ne Ž 0 . < ne Ž t . : , for t s nT q t . Ž 15 . Similarly Eq. Ž13. gets the phase e i n f instead of e i f . Eq. Ž14. shows that the Berry phase acts as a ‘‘counter’’ of neutrino oscillations, adding up 2p sin2u to the phase of the Želectron. neutrino state after each complete oscillation. Eq. Ž14. is interesting especially because it can be rewritten as nT
b eŽ n. s
H0
s
H0
nT
² ne Ž t . < Uy1 Ž t . i E t Ž U Ž t . < ne Ž t . : . dt &
We thus understand that Eq. Ž16. directly gives us the& geometric phase because &the quantity & i ²neŽ t .
&
²ne Ž t . < i E t
Ž 16 . with UŽ t . s eyi f Ž t ., where f Ž t . s f Ž0. y v 1 t, and &
nT
H0
vm e dt s p nsin2 u .
Since vm e is the energy shift from the level v e e caused by the flavor interaction term in the Hamiltonian w1x, it is easily seen that
vm2 e s D E 2 ' ² ne Ž t . < H 2 < ne Ž t . : y ² ne Ž t . < H < ne Ž t . :2 ,
Ž 21 . and then we recognize that Eq. Ž20. gives the geometric invariant discussed in Ref. w7x, where it is defined quite generally as s s HD EŽ t . dt. It has the advantage to be well defined also for systems with non-cyclic evolution. We now consider the case of three flavor mixing. Consider again the electron neutrino state at time t w2x: < ne Ž t . : s eyi v 1 t Ž cos u 12 cos u 13 < n 1 : q eyi Ž v 2y v 1 . t sin u 12 cos u 13 < n 2 :
s eyi f Ž0. Ž cos u < n 1 : q eyi Ž v 2y v 1 . t sin u < n 2 : . . Ž 17 . Eq. Ž16. actually provides an alternative way for defining the Berry phase w6x, which makes use of the & state
&
&
&
²ne Ž 0 .
for t s nT q t , Ž 18 .
which is to be compared with Eq. Ž15.. From Eq. Ž17. we also see that time evolution only affects the & < n 2 : component of the state
i E t
s Ž H y v 1 .
Ž 19 .
Ž 20 .
q eyi Ž v 3y v 1 . t e i d sin u 13 < n 3 : . ,
Ž 22 .
where d is the analogous of the CP violating phase of the CKM matrix. Let us consider the particular case in which the two frequency differences are proportional: v 3 y v 1 s q Ž v 2 y v 1 ., with q a rational number. In this case the state ŽEq. Ž22.. is periodic over a period T s v 22yp v 1 and we can use the previous definition of Berry phase:
b s fq
T
H0
² ne Ž t . < H < ne Ž t . : dt
s 2p Ž sin2u 12 cos 2u 13 q qsin2u 13 . ,
Ž 23 .
which of course reduces to the result Ž4. for u 13 s 0. Eq. Ž23., however, shows that b is not completely free from dynamical parameters since the appearance in it of the parameter q. Although because of this, b is not purely geometric, nevertheless it is interesting that it does not depend on the specific frequencies v i , i s 1, 2, 3, but
M. Blasone et al.r Physics Letters B 466 (1999) 262–266
on the ratio of their differences only. This means that we have now Žgeometric. classes labelled by q. It is in our plan to calculate the geometric invariant s for the three flavor neutrino state: this requires consideration of the projective Hilbert space in the line of Refs. w7,8x.
3. Final remarks and conclusions The geometric phase is generally associated with a parametric dependence of the time evolution generator. In such cases, the theory exhibits a gauge-like structure which may become manifest and characterizing for the physical system, e.g. in the BohmAharonov effect w9x. It is then natural to ask the question about a possible gauge structure in the case considered in this paper. Let us see how, indeed, a covariant derivative may be here introduced. Let us consider the evolution of the mass eigenstates iEt < n i Ž t . : s H < n i Ž t . : ,
Ž 24 .
where i s 1, 2. These equations are invariant under the following Žlocal in time. gauge transformation &
< n i Ž t . : ™
Ž 26 .
This suggests that, by rewriting Eq. Ž24. as
Ž iEt y H . < n i Ž t . : s 0 ,
Ž 27 .
we can consider Dt ' E t q iH as the ‘‘covariant derivative’’: Dt ™ DXt s U Ž t . Dt Uy1 Ž t . .
265
We thus see that the time dependent canonical transformation of the Hamiltonian Eq. Ž26. and Eq. Ž25. play the role of a local Žin time. gauge transformation. Note that the state < n˜e Ž t .: of Eq. Ž17. is a superposition of the states < n˜ i Ž t .:. The role of the ‘‘diabatic’’ force arising from the term Uy1 Ž t . i E t UŽ t . has been considered in detail elsewhere w10x. Summarizing, we have shown that there is a Berry phase built in in the neutrino oscillations, we have explicitly computed it in the cyclic two-flavour case and in a particular case of three flavor mixing. The result also applies to other Žsimilar. cases of particle oscillations. We have noticed that a measurement of this Berry phase would give a direct measurement of the mixing angle independently from the values of the masses. On the other hand, the phenomenological relevance of the Berry phase for oscillating neutrinos has been also discussed in Ref. w11x in connection with neutrino oscillation in matter. Our analysis however is different from the one in Ref. w11x. The geometrical phase here discussed is deeply rooted in the same mechanism of mixing at a fundamental level. For this reason our results go even beyond the neutrino case and apply to any case of evolution of entangled states. The above analysis in terms of ‘‘tilting’’ of the state in its time evolution, parallel transport and covariant derivative also suggests that field mixing may be seen as the result of a curvature in the state space. The Berry phase appears to be a manifestation of such a curvature. Finally, we remark that the recognition of the geometric phase associated to mixed states also suggests to us that a similar geometric phase also occurs in entangled quantum states which can reveal to be relevant in completely different contexts than particle oscillations, namely in quantum computation w12x.
Ž 28 .
We have indeed &
Acknowledgements
iDXt
Ž 29 .
which in fact expresses the invariance of Eq. Ž24. under Eq. Ž25..
We thank M. Nowak and D. Brody for fruitful discussions. M.B. and G.V. acknowledge INFN, INFM, MURST and ESF for support.
266
M. Blasone et al.r Physics Letters B 466 (1999) 262–266
References w1x S.M. Bilenky, B. Pontecorvo, Phys. Rep. 41 Ž1978. 225. w2x For a recent review see for example: M. Zralek, hepphr9810543; M. Blasone, G. Vitiello, Ann. Phys. ŽNY. 244 Ž1995. 283; M. Blasone, P.A. Henning, G. Vitiello, Phys. Lett. B 451 Ž1999. 140. w3x M. Blasone, P.A. Henning, G. Vitiello, in: Results and Perspectives in Particle Physics, M. Greco ed., INFN Frascati 1996, p.139; M. Binger and C-R. Ji, hep-phr9901407. w4x M.V. Berry, Proc. Roy. Soc. London A 392 Ž1984. 45.
w5x J. Anandan, J. Christian, K. Wanelik, Am. J. Phys. 65 Ž1997. 180. w6x Y. Aharonov, J. Anandan, Phys. Rev. Lett. 58 Ž1987. 1593. w7x J. Anandan, Y. Aharonov, Phys. Rev. Lett. 65 Ž1990. 1697. w8x D.C. Brody, L.P. Hughston, J. Math. Phys. 39 Ž1998. 1. w9x Y. Aharonov, D. Bohm, Phys. Rev. 115 Ž1959. 485. w10x Y.N. Srivastava, G. Vitiello, A. Widom, quant-phr9810095. w11x V.A. Naumov, Phys. Lett. B 323 Ž1994. 351. w12x J. Preskill, in Introduction to Quantum Computation, H.K. Lo, S. Popescu, T.P. Spiller eds., quant-phr9712048; P. Zanardi, M. Rasetti, Phys. Rev. Lett. 79 Ž1997. 3306; quantphr9710041;
Physics Letters B 466 Ž1999. 262–266
Berry phase for oscillating neutrinos Massimo Blasone
a,c,d,1
, Peter A. Henning
b,2
, Giuseppe Vitiello
c,3
a Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2BZ, UK Institut fur ¨ Kernphysik, TH Darmstadt, Schloßgartenstraße 9, D-64289 Darmstadt, Germany c Dipartimento di Fisica dell’UniÕersita` and INFN, Gruppo Collegato, I-84100 Salerno, Italy d Unita` INFM di Salerno, I-84100 Salerno, Italy
b
Received 25 February 1999; received in revised form 10 May 1999; accepted 23 September 1999 Editor: L. Alvarez-Gaume´
Abstract We show the presence of a topological ŽBerry. phase in the time evolution of a mixed state. For the case of mixed neutrinos, the Berry phase is a function of the mixing angle only. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 03.65.Bz; 11.10.-z; 14.60.Pq
1. Introduction Particle mixing and oscillations play a relevant role in high energy physics. In particular, in recent years a growing interest in neutrino mixing and oscillations w1x has been developed which manifests itself both in a strong experimental effort and in a renewed theoretical research activity. Contributions towards the correct theoretical understanding of the field mixing have been recently presented w2,3x. In the present paper we show how the notion of Berry phase w4x enters the physics of mixing by considering the example of neutrino oscillations. Since its discovery w4x, the Berry phase has attracted much interest w5x at theoretical as well as at experimental level. This interest arises because the
1
E-mail: [email protected] E-mail: [email protected] 3 E-mail: [email protected] 2
Berry phase reveals geometrical features of the systems in which it appears, which go beyond the specific dynamical aspects and as such contribute to a deeper characterization of the physics involved. The successful experimental findings in many different quantum systems w5x stimulate further search in this field. Aimed by these motivations, we show that the geometric phase naturally appears in the standard Pontecorvo formulation of neutrino oscillations. Our result shows that the Berry phase associated to neutrino oscillations is a function of the mixing angle only. We emphasize that such a result has phenomenological relevance: since geometrical phases are observable, the mixing angle can be Žat least in principle. measured directly, i.e. independently from dynamical parameters as the neutrino masses and energies. Although in the following we treat the neutrino case, we stress that our result holds in general, also in the case of mixed bosons ŽKaons, hX s, etc...
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 1 1 3 7 - 5
M. Blasone et al.r Physics Letters B 466 (1999) 262–266
2. Berry phase for oscillating neutrinos
where we have used the notation ² ne Ž t . < i E t < ne Ž t . : s ² ne Ž t . < H < ne Ž t . :
Let us first consider the two flavor case w1x:
s v 1 cos 2u q v 2 sin2u
< ne : s cos u < n 1 : q sin u < n 2 : , < nm : s ysin u < n 1 : q cos u < n 2 : .
' ve e .
Ž 1.
² nm Ž t . < i E t < nm Ž t . : s ² nm Ž t . < H < nm Ž t . :
< ne Ž t . : ' eyi H t < ne Ž 0 . :
s v 1 sin2u q v 2 cos 2u
s eyi v 1 t Ž cos u < n 1 : qeyi Ž v 2y v 1 . t sin u < n 2 : . ,
Ž 2.
where H < n i : s v i < n i :, i s 1, 2. Our conclusions will also hold for the muon neutrino state, with due changes which will be explicitly shown when necessary. The state < ne Ž t .:, apart from a phase factor, reproduces the initial state < ne Ž0.: after a period T s 2p v2y v1 :
fsy
2pv 1
v2 y v1
.
Ž 3.
We now show how such a time evolution does contain a purely geometric part, i.e. the Berry phase. It is a straightforward calculation to separate the geometric and dynamical phases following the standard procedure w6x:
be s f q sy
T
H0
² ne Ž t . < i E t < ne Ž t . : dt
2pv 1
v2 y v1
2p q
v2 y v1
Ž v 1 cos 2u q v 2 sin2u .
s 2p sin2u .
T
H0
' vmm ,
s 12 Ž v 2 y v 1 . sin2 u ' vm e ,
with v e m s vm e . In order to better understand the meaning of Eqs. Ž4. – Ž6., we observe that, as well known, < ne : is not eigenstate of the Hamiltonian, and ² ne Ž 0 . < ne Ž t . : s eyi v 1 t cos 2u q eyi v 2 t sin2u .
Ž 10 .
Thus, as an effect of time evolution, the state < ne : ‘‘rotates’’ as shown by Eq. Ž10.. However, at t s T, ² ne Ž 0 . < ne Ž T . : s e i f s e i b e eyi v e e T ,
Ž 11 .
i.e. < ne ŽT .: differs from < ne Ž0.: by a phase f , part of which is a geometric ‘‘tilt’’ Žthe Berry phase. and the other part is of dynamical origin. In general, for t s T q t , we have ² ne Ž 0 . < ne Ž t . : s e i f ² ne Ž 0 . < ne Ž t . : 2
s e i2 p sin u eyi v e e T = Ž eyi v 1t cos 2u q eyi v 2t sin2u . .
Ž 12 . Also notice that ² nmŽ t .< ne Ž t .: s 0 for any t. However, ² nm Ž 0 . < ne Ž t . :
for t s T q t ,
Ž 5. Note that b e q bm s 2p . We can thus rewrite Eq. Ž3. as 2
Ž 9.
s 12 e i f eyi v 1t sin2 u Ž eyi Ž v 2y v 1 . t y 1 . ,
² nm Ž t . < i E t < nm Ž t . : dt s 2p cos 2u .
< ne Ž T . : s e i2 p sin u eyi v e e T < ne Ž 0 . : ,
Ž 8.
² nm Ž t . < i E t < ne Ž t . : s ² nm Ž t . < H < ne Ž t . :
Ž 4.
We thus see that there is indeed a non-zero geometrical phase b , related to the mixing angle u , and that it is independent from the neutrino energies v 1 , v 2 and masses m1 , m 2 . In a similar fashion, we obtain the Berry phase for the muon neutrino state:
bm s f q
Ž 7.
We will also use
The electron neutrino state at time t is w1x
< ne Ž T . : s e i f < ne Ž 0 . : ,
263
Ž 6.
Ž 13 .
which is zero only at t s T. Eq. Ž13. expresses the fact that < ne Ž t .: ‘‘oscillates’’, getting a component of muon flavor, besides getting the Berry phase. At t s T, neutrino states of different flavor are again each other orthogonal states.
M. Blasone et al.r Physics Letters B 466 (1999) 262–266
264
Generalization to n-cycles is also interesting. Eq. Ž4. Žand Eq. Ž5.. can be rewritten for the n-cycle case as
b eŽ n. s
nT
H0
² ne Ž t . < i E t y v 1 < ne Ž t . : dt
s 2p n sin2u , and Eq. Ž12. becomes
Ž 14 .
ss
² ne Ž 0 . < ne Ž t . : s e i n f ² ne Ž 0 . < ne Ž t . : , for t s nT q t . Ž 15 . Similarly Eq. Ž13. gets the phase e i n f instead of e i f . Eq. Ž14. shows that the Berry phase acts as a ‘‘counter’’ of neutrino oscillations, adding up 2p sin2u to the phase of the Želectron. neutrino state after each complete oscillation. Eq. Ž14. is interesting especially because it can be rewritten as nT
b eŽ n. s
H0
s
H0
nT
² ne Ž t . < Uy1 Ž t . i E t Ž U Ž t . < ne Ž t . : . dt &
We thus understand that Eq. Ž16. directly gives us the& geometric phase because &the quantity & i ²neŽ t .
&
²ne Ž t . < i E t
Ž 16 . with UŽ t . s eyi f Ž t ., where f Ž t . s f Ž0. y v 1 t, and &
nT
H0
vm e dt s p nsin2 u .
Since vm e is the energy shift from the level v e e caused by the flavor interaction term in the Hamiltonian w1x, it is easily seen that
vm2 e s D E 2 ' ² ne Ž t . < H 2 < ne Ž t . : y ² ne Ž t . < H < ne Ž t . :2 ,
Ž 21 . and then we recognize that Eq. Ž20. gives the geometric invariant discussed in Ref. w7x, where it is defined quite generally as s s HD EŽ t . dt. It has the advantage to be well defined also for systems with non-cyclic evolution. We now consider the case of three flavor mixing. Consider again the electron neutrino state at time t w2x: < ne Ž t . : s eyi v 1 t Ž cos u 12 cos u 13 < n 1 : q eyi Ž v 2y v 1 . t sin u 12 cos u 13 < n 2 :
s eyi f Ž0. Ž cos u < n 1 : q eyi Ž v 2y v 1 . t sin u < n 2 : . . Ž 17 . Eq. Ž16. actually provides an alternative way for defining the Berry phase w6x, which makes use of the & state
&
&
&
²ne Ž 0 .
for t s nT q t , Ž 18 .
which is to be compared with Eq. Ž15.. From Eq. Ž17. we also see that time evolution only affects the & < n 2 : component of the state
i E t
s Ž H y v 1 .
Ž 19 .
Ž 20 .
q eyi Ž v 3y v 1 . t e i d sin u 13 < n 3 : . ,
Ž 22 .
where d is the analogous of the CP violating phase of the CKM matrix. Let us consider the particular case in which the two frequency differences are proportional: v 3 y v 1 s q Ž v 2 y v 1 ., with q a rational number. In this case the state ŽEq. Ž22.. is periodic over a period T s v 22yp v 1 and we can use the previous definition of Berry phase:
b s fq
T
H0
² ne Ž t . < H < ne Ž t . : dt
s 2p Ž sin2u 12 cos 2u 13 q qsin2u 13 . ,
Ž 23 .
which of course reduces to the result Ž4. for u 13 s 0. Eq. Ž23., however, shows that b is not completely free from dynamical parameters since the appearance in it of the parameter q. Although because of this, b is not purely geometric, nevertheless it is interesting that it does not depend on the specific frequencies v i , i s 1, 2, 3, but
M. Blasone et al.r Physics Letters B 466 (1999) 262–266
on the ratio of their differences only. This means that we have now Žgeometric. classes labelled by q. It is in our plan to calculate the geometric invariant s for the three flavor neutrino state: this requires consideration of the projective Hilbert space in the line of Refs. w7,8x.
3. Final remarks and conclusions The geometric phase is generally associated with a parametric dependence of the time evolution generator. In such cases, the theory exhibits a gauge-like structure which may become manifest and characterizing for the physical system, e.g. in the BohmAharonov effect w9x. It is then natural to ask the question about a possible gauge structure in the case considered in this paper. Let us see how, indeed, a covariant derivative may be here introduced. Let us consider the evolution of the mass eigenstates iEt < n i Ž t . : s H < n i Ž t . : ,
Ž 24 .
where i s 1, 2. These equations are invariant under the following Žlocal in time. gauge transformation &
< n i Ž t . : ™
Ž 26 .
This suggests that, by rewriting Eq. Ž24. as
Ž iEt y H . < n i Ž t . : s 0 ,
Ž 27 .
we can consider Dt ' E t q iH as the ‘‘covariant derivative’’: Dt ™ DXt s U Ž t . Dt Uy1 Ž t . .
265
We thus see that the time dependent canonical transformation of the Hamiltonian Eq. Ž26. and Eq. Ž25. play the role of a local Žin time. gauge transformation. Note that the state < n˜e Ž t .: of Eq. Ž17. is a superposition of the states < n˜ i Ž t .:. The role of the ‘‘diabatic’’ force arising from the term Uy1 Ž t . i E t UŽ t . has been considered in detail elsewhere w10x. Summarizing, we have shown that there is a Berry phase built in in the neutrino oscillations, we have explicitly computed it in the cyclic two-flavour case and in a particular case of three flavor mixing. The result also applies to other Žsimilar. cases of particle oscillations. We have noticed that a measurement of this Berry phase would give a direct measurement of the mixing angle independently from the values of the masses. On the other hand, the phenomenological relevance of the Berry phase for oscillating neutrinos has been also discussed in Ref. w11x in connection with neutrino oscillation in matter. Our analysis however is different from the one in Ref. w11x. The geometrical phase here discussed is deeply rooted in the same mechanism of mixing at a fundamental level. For this reason our results go even beyond the neutrino case and apply to any case of evolution of entangled states. The above analysis in terms of ‘‘tilting’’ of the state in its time evolution, parallel transport and covariant derivative also suggests that field mixing may be seen as the result of a curvature in the state space. The Berry phase appears to be a manifestation of such a curvature. Finally, we remark that the recognition of the geometric phase associated to mixed states also suggests to us that a similar geometric phase also occurs in entangled quantum states which can reveal to be relevant in completely different contexts than particle oscillations, namely in quantum computation w12x.
Ž 28 .
We have indeed &
Acknowledgements
iDXt
Ž 29 .
which in fact expresses the invariance of Eq. Ž24. under Eq. Ž25..
We thank M. Nowak and D. Brody for fruitful discussions. M.B. and G.V. acknowledge INFN, INFM, MURST and ESF for support.
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