Michael G. Jabbour , Nicolas J. We explore the multiparticle transition probabilities in Gaussian unitaries effected by a two-mode Bogoliubov bosonic transformation on the mode annihilation and creation operators. We show that the transition probabilities can be characterized by remarkably simple, yet unsuspected recurrence equations involving a linear combination of probabilities.
The recurrence exhibits an interferometric suppression term - a negative probability - which generalizes the seminal Hong-Ou-Mandel effect to more than two indistinguishable photons impinging on a beam splitter of rational transmittance.
Unexpectedly, interferences thus originate in this description from the cancellation of probabilities instead of amplitudes. Our framework, which builds on the generating function of the non-Gaussian matrix elements of Gaussian unitaries in Fock basis, is illustrated here for the most common passive and active linear coupling between two optical modes driven by a beam splitter or a parametric amplifier.
Hence, it also allows us to predict unsuspected multiphoton interference effects in an optical amplifier of rational gain. In particular, we confirm the newly found two-photon interferometric suppression effect in an amplifier of gain 2 originating from timelike indistinguishability [Proc.
Front Matter Pages i-xi. Front Matter Pages Pages Back Matter Pages Authors and affiliations Jian-Xin Zhu 1 1. Parinov, zbMATH Buy options. The paper is organized as follows. In Section II we investigate the compati- bility of the mixing transformation at level of states and fields, and show that a Bogoliubov transformations is required. The possibility of a thermodynamical interpretation of such a condensate is considered in Section IV. Finally, in Section V we draw our conclusions. The paper is completed with three Appendices.
Extension to three neutrinos is in our plans. However, we have good reasons to believe that the present results are general, since our arguments are of algebraic nature. It has been shown [3] that this is not the case and indeed a deep conceptual difference is present between mixing of states and mixing of fields.
The results also extend to the mixing phenomenon of any particle, and are not limited to the case of Dirac neutrinos. See Appendix for other useful relations. Observe that Eqs. This is a well-known feature of QFT [24] reflecting into the non-unitary nature in the infinite volume limit of the generator of Bogoliubov transformations. We now consider the action of the rotation Eq. However, it is possible to recover the wanted expression by means of a suitable Bogoliubov transformation, which implements a mass shift.
Vacuum structure and non-commutativity In the previous Section, we have shown the incompatibility of the mixing transformation as mere rotations both for states and fields, and the necessity of implementing a mass shift for reproducing the correct relations for fields: such an operation is highly non-trivial and indeed requires infinite energy in the infinite volume limit. It thus arises the problem of the decomposition of such generator in terms of rotation and Bogoliubov transformations; a prelimi- nary solution to this problem has been presented in [29].
From Eq. More- over, considering that the Bogoliubov coefficients Uk and Vk appearing in Eq. Moreover, the last term shows the explicit dependance on the true physical parameters of the mixing transformation, i.
This feature can be further understood by looking at the tilde vacuum, defined as cf. Another interesting feature of this phenomenon appears as one analyses more closely the parameter a, which in order to exist needs at least two fermion families to be present.
In fact, with just one family the only adimensional parameter one can form is k m , which however depends on k and thus cannot be extracted from the integrals. Finally, let us express the flavor vacuum by means of the full finite decom- 4 The complete operatorial structure of the flavor vacuum Eq. Withal, as a result of the non vanishing commutator in Eq. Thermodynamical properties In this Section we investigate the possibility of a thermodynamical interpre- tation for the condensate structure of the flavor vacuum.
We have the following relations - cf. Nf p is plotted for different values of a. This is indeed sufficient for the following considerations. This results in an impossibility to introduce a well defined temperature or equiva- lently in a deviation from the Fermi distribution, due to the non diagonal pairs in the condensate structure of the flavor vacuum. From Fig.
Conclusions We have discussed the algebraic structure of the mixing generator for two Dirac neutrino fields with different masses. We have shown that such a generator can be decomposed in terms of a rotation depending only on the mixing angle and a Bogoliubov transformation depending only on the neutrino masses.
These two transformations do not commute among themselves and this fact produces important effects on the vacuum structure. It is interesting to observe that the Bogoliubov transformations are indeed responsible for the mass shift and thus the results of this paper can lead to fur- ther insight in the interplay between mixing phenomenon and mass generation in a dynamical perspective as recently discussed in Refs.
Moreover, the condensate structure of the vacuum suggests a thermodynam- ical interpretation which we investigated, showing peculiarities in the thermal behavior due to the character of the particle-antiparticle condensate involved in the flavor vacuum.
Such an issue will be further investigated in a future work. Finally, we observe that the algebraic mechanism discussed in the present paper appears to be of quite general nature and thus we expect it to hold, with the due differences, also for the mixing among other kinds of fields.
For Majorana fields [32], the mixing generator has essentially the same form as the one for Dirac fields Eq. The case of bosonic fields will be discussed in a separate publication together with the extension of the present work to three flavor mixing. One may recast Eqs. It is, indeed, possible to disentangle the two dependances, mass and angle, of the mixing generator.
These relations can be easily verified. References References [1] M. Bilenky and B. Pontecorvo, Phys. Giunti and C. Press, Oxford, Blasone and G. Vitiello, Ann. Blasone, P. Henning and G. Vitiello, Phys. B Fujii, C.
Habe and T. Yabuki Phys. D59, ; ibid.
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