We study a relativistic spin-1/2 fermion subjected to a Dirac oscillator coupling and a constant magnetic field. An interplay between opposed chirality interactions culminates in the appearance of a relativistic quantum phase transition, which can be fully characterized. We obtain analytical expressions for the energy gap, order parameter, and canonical quantum fluctuations across the critical point. Moreover, we also discuss the effect of this phase transition on the statistics of the chiral bosonic ensemble, where its super-or subPoissonian nature can be controlled by means of external parameters. Finally, we study the entanglement properties between the degrees of freedom in the relativistic ground state, where an interesting transition between a biseparable and a genuinely tripartite entangled state occurs.

The problem of a correct quantum description of the phase difference is examined from the perspective of parameter estimation theory. It is shown that an optimum phase-shift measurement defines a phase difference operator which coincides with other approaches to the same problem. We also study the fundamental limit to the accuracy of a phase difference shift detection. We show that this limit can be reached by a measurement having countable outcomes despite the fact that a phase shift can take any value. We show that this is the case of the phase difference operator defined by an optimum phase-shift measurement. ͓S1050-2947͑96͒09911-8͔ PACS number͑s͒: 42.50.Dv

Luis and Sánchez-Soto Reply: We would like to point out that in our article [1] we did not consider arbitrary which-way detectors, but we restricted our analysis to arrangements that neither modify the state vectors jc 1 ͘, jc 2 ͘ associated with each path nor affect the path probabilities. Our demonstration about the enforcement of complementarity by random classical phase kicks applies just to such a class of arrangements. Our article does not exactly imply the statement labeled as (i) in the preceding Comment [2], and accordingly we avoided suggesting the kind of universality associated with "always."Leaving aside this minor precision, we disagree with the claim that statements (i) and (ii) [or more precisely Eqs. (4) and (6)] are opposite. On the contrary, we think they are fully equivalent, as argued in Ref. [3]. In fact, we have demonstrated that Eq. (6) implies Eq. (4). The link between these two equations is established by the eigenstates and eigenvalues of the operator V y

We introduce a unitary operator representing the exponential of the phase difference between two modes of the electromagnetic field. The eigenvalue spectrum has a discrete character that is fully analyzed. We relate this operator with a suitable polar decomposition of the Stokes parameters of the field, obtaining a natural classical limit. The cases of weakly and highly excited states are considered, discussing to what extent it is possible to talk about the phase for a single-mode field. This operator is applied to some interesting two-mode fields.

No abstract