Flavor vacuum entanglement in boson mixing
Massimo Blasone, Fabrizio Illuminati, Giuseppe Gaetano Luciano, Luciano Petruzziello
FFlavor vacuum entanglement in boson mixing
Massimo Blasone ∗ , , Fabrizio Illuminati † , , Giuseppe Gaetano Luciano ‡ , and Luciano Petruzziello § , Dipartimento di Fisica, Universit`a degli Studi di Salerno,Via Giovanni Paolo II, 132 I-84084 Fisciano (SA), Italy. INFN, Sezione di Napoli, Gruppo collegato di Salerno, Italy. Dipartimento di Ingegneria Industriale, Universit`a degli Studi di Salerno,Via Giovanni Paolo II, 132 I-84084 Fisciano (SA), Italy. (Dated: January 13, 2021)Mixing transformations in quantum field theory are non-trivial, since they are intimately relatedto the unitary inequivalence between Fock spaces for fields with definite mass and fields with definiteflavor. Considering the superposition of two neutral scalar (spin-0) bosonic fields, we investigatesome features of the emerging condensate structure of the flavor vacuum. In particular, we quantifythe flavor vacuum entanglement in terms of the von Neumann entanglement entropy of the reducedstate. Furthermore, in a suitable limit, we show that the flavor vacuum has a structure akin tothe thermal vacuum of Thermo Field Dynamics, with a temperature dependent on both the mixingangle and the particle mass difference.
I. INTRODUCTION
The analysis of quantum correlations in the context of particle physics (and in particular for neutrino and mesonsystems) is currently gaining a rising attention. The initial observation that flavor mixing and oscillations can beassociated with (single-particle) entanglement [1, 2] served as a basis for several studies [3–11] in which violations ofBell, Leggett-Garg and Mermin-Svetchlichny inequalities, non-locality, gravity/acceleration degradation effects andother similar occurrences have been investigated both theoretically and experimentally.Most of the above studies have been carried out within the framework of Quantum Mechanics (QM). The extensionto Quantum Field Theory (QFT) has been later considered in Ref. [12], thus leading to the discovery of non-trivialproperties of the mixing transformation [13]. Indeed, whilst in QM such a transformation acts as a simple rotationbetween flavor and mass states [14], its QFT counterpart behaves as a rotation nested into a non-commuting Bo-goliubov transformation [13, 15]. As a result, the vacua for fields with definite mass and fields with definite flavorbecome orthogonal to each other, with the latter acquiring the structure of a SU (2) coherent state [16] and turninginto a condensate of massive particle/antiparticle pairs [13, 17]. This gives rise to a deeeper understanding of particlemixing, since the Fock spaces for flavor and mass fields are found to describe unitarily inequivalent representations.Corrections to the standard QM predictions also appear in the oscillation probability formula, as explicitly shown inRef. [18].Originally developed for neutrinos propagating in flat spacetime, the above considerations have been later extendedto bosonic fields [19, 20] as well as to non-trivial spacetime backgrounds (see Refs. [21, 22] for more details). Recently,a further evidence for the complex structure of the flavor vacuum has been provided in Ref. [23], where it has beenestablished that the Fock space for flavor fields cannot be obtained by the direct product of the spaces for massivefields. Therefore, entanglement connected with flavor mixing appears to be an exquisite concept, boiling down tothe non-factorizability of the flavor states in terms of the ones with definite masses. In other words, it is possibleto come across flavor entanglement already at the level of the vacuum state. More generally, the phenomenon ofa non-vanishing entanglement even for free fields is ultimately related to the quotient space structure of the tensorHilbert spaces [24].In fact, the investigation of the properties of the QFT vacuum is one of the most significant (albeit difficult) tasks ina wide variety of physical scenarios. For instance, the Bardeen-Cooper-Schrieffer ground state plays a pivotal rˆole incondensed matter, being a condensate of Cooper pairs which underpins the phenomenon of superconductivity [25]. Inthe same fashion, vacuum is crucially important to explain the spontaneous symmetry breaking [26] and the ensuingappearance of Nambu-Jona Lasinio [27], Goldstone [28, 29] and pseudo-Goldstone bosons [30], both in low and highenergy regimes. On the other hand, in QFT vacuum energy is notoriously responsible for the existence of Casimireffect [31, 32], which has been largely studied both in flat [33–37] and curved [38–43] spacetime in recent years. ∗ [email protected] † fi[email protected] ‡ [email protected] § [email protected] a r X i v : . [ h e p - t h ] J a n Moreover, the study of vacua in the presence of gravity leads to the loss of the absolute concept of particle and theemergence of distinctive phenomena such as the black hole radiation [44] and the akin Unruh effect [45], which findapplication in many research areas [46].Starting from the above premises, in the present work we analyze some relevant and yet unexplored features ofthe flavor vacuum condensate, with a particular focus on its entanglement structure. For this purpose, we considerthe case of mixing neutral bosonic fields. Besides the intrinsic importance (i.e. as in the case of meson mixing),this choice allows to reduce several technical complications and render the physical insight as transparent as possible.Given the mixing of two bosonic fields, in the following we will quantify the entanglement content of the flavor vacuumby computing von Neumann entanglement entropy of its reduced density matrix in the limit of small mass differenceand/or mixing angle. We will then compare this result with the entanglement of the thermal vacuum of Thermo FieldDynamics (TFD) [47, 48], which exhibits the paradigmatic structure occurring in the case of black holes [49] andUnruh effect [45], namely the doubling of Fock space and the ensuing correlation between two different sets of modes(the particle states inside and outside the horizon for Hawking-Unruh effect, the physical and auxiliary modes in TFD).It must be stressed that entanglement in TFD has been previously discussed in Ref. [50] for both equilibrium andnon-equilibrium states. From the comparison between our results and the ones on TFD entanglement, we find that thecondensate in the flavor vacuum is in general richer than the one in TFD, as it exhibits all types of contributions, boththermal and non-thermal. Finally, we determine a suitable limit in which non-thermal contributions are subdominant,thereby allowing us to recognize a TFD-like structure in the flavor vacuum, with an effective temperature proportionalto the mixing angle and the mass difference between the two fields.The paper is organized as follows: in Sec. II we review the main aspects of boson field mixing, focusing on the case ofneutral scalar fields. In Sec. III we quantify the entanglement content of the flavor vacuum by computing the reducedvon Neumann entropy, and we discuss the condensate structure of this state in connection with the thermal vacuumof TFD. Conclusions and outlook are provided in Sec. IV. Throughout the work, we use natural units c = (cid:126) = 1 whilekeeping the Boltzmann constant k B explicit. II. MIXING OF BOSONIC FIELDS
In this Section, we review the crucial aspects associated with the mixing of two scalar (spin-0) neutral fields.Clearly, the same considerations can be extended to the case of three boson generations. To this aim, we closelyfollow Refs. [19, 20] and write down the mixing relations between fields with definite mass and flavor, that is φ A ( x ) = cos θ φ ( x ) + sin θ φ ( x ) , (1) φ B ( x ) = − sin θ φ ( x ) + cos θ φ ( x ) , (2)with an analogous set of equations for the conjugate momenta π ( x ) = ∂ t φ ( x ). The subscripts A and B indicate thefields in the flavor basis, whereas 1 and 2 the ones in the mass basis. Consequently, the expansions for φ and φ takethe form φ j ( x ) = (cid:90) d k (cid:113) π ) ω k,j (cid:16) a k,j e − iω k,j t + a †− k,j e iω k,j t (cid:17) e i k · x , j = 1 , , (3)where ω k,j = (cid:113) k + m j and a k,j ( a † k,j ) are the bosonic annihilation (creation) operators of field quanta with momen-tum k and mass m j .By requiring that the fields and the conjugate momenta obey the canonical commutation relation (CCR) at equaltimes, (cid:104) φ i ( x ) , π j ( x (cid:48) ) (cid:105) t = t (cid:48) = iδ ij δ ( x − x (cid:48) ) , (4)it follows that the only non-trivial commutator between the ladder operators is (cid:104) a k,i , a † k (cid:48) ,j (cid:105) = δ ij δ ( k − k (cid:48) ) . (5) Strictly speaking, in the case of bosons we should refer to the mixing of some other quantum number, such as the strangeness or theisospin rather than the flavor. However, with an abuse of notation, in the following we keep on denoting such intrinsic properties as“flavor” and the corresponding mixed fields as “flavor fields”. Furthermore, we work in a simplified two-flavor model.
Let us now observe that Eqs. (1) and (2) and the ones for momenta can also be rewritten as φ σ ( x ) = G − θ ( t ) φ j ( x ) G θ ( t ) , (6) π σ ( x ) = G − θ ( t ) π j ( x ) G θ ( t ) , (7)where ( σ, j ) = { ( A, , ( B, } and G θ ( t ) = exp { θ [ S + ( t ) − S − ( t )] } (8)is the mixing generator and an element of SU (2), whose algebra is built with the following operators S + ( t ) = − i (cid:90) d x π ( x ) φ ( x ) , (9) S − ( t ) = − i (cid:90) d x π ( x ) φ ( x ) (10) S ( t ) = − i (cid:90) d x [ π ( x ) φ ( x ) − π ( x ) φ ( x )] . (11)In light of the above equations, the flavor fields can then be expressed as φ σ ( x ) = (cid:90) d k (cid:113) π ) ω k,j (cid:16) a k,σ ( t ) e − iω k,j t + a †− k,σ ( t ) e iω k,j t (cid:17) e i k · x , σ = A, B, (12)and, with the aid of Eqs. (1) and (2), we recognize the Bogoliubov transformation between the flavor and mass ladderoperators a k,A ( t ) = cos θ a k, + sin θ (cid:104) U ∗ k ( t ) a k, + V k ( t ) a †− k, (cid:105) , (13) a k,B ( t ) = cos θ a k, − sin θ (cid:104) U k ( t ) a k, − V k ( t ) a †− k, (cid:105) . (14)The above relations exhibit the structure of rotations nested into Bogoliubov transformations with coefficients U k ( t )and V k ( t ) given by U k ( t ) = | U k | e i ( ω k, − ω k, ) t , V k ( t ) = | V k | e i ( ω k, + ω k, ) t , (15) | U k | = 12 (cid:18)(cid:114) ω k, ω k, + (cid:114) ω k, ω k, (cid:19) , | V k | = 12 (cid:18)(cid:114) ω k, ω k, − (cid:114) ω k, ω k, (cid:19) . (16)Accordingly, the flavor vacuum is provided with with a SU (2) coherent state structure [16] | t ) (cid:105) A,B = G − θ ( t ) | (cid:105) , , (17)with a condensation density given by A,B (cid:104) t ) | a † k,j a k,j | t ) (cid:105) A,B = sin θ | V k | , j = 1 , . (18)In the infinite volume limit [19, 20], the two sets of vacua become orthogonal, namely , (cid:104) | t ) (cid:105) A,B V →∞ −→ ∀ t , givingrise to physically inequivalent Fock spaces (i.e. unitarily inequivalent representations of the canonical commutationrelations for fields).Let us now cast the mixing generator in terms of mass definite annihilators and creators. Straightforward calcula-tions lead to | t ) (cid:105) A,B = exp (cid:110) − θ (cid:90) d k (cid:104) U ∗ k ( t ) a † k, a k, − U k ( t ) a k, a † k, + V k ( t ) a † k, a †− k, − V ∗ k ( t ) a k, a − k, (cid:105)(cid:111) | (cid:105) , . (19)Without harming the generality of our results, henceforth we perform calculations for t = 0. Thus, we have | t =0) (cid:105) A,B ≡ | (cid:105) A,B with | (cid:105) A,B = exp (cid:110) − θ (cid:90) d k (cid:104) U k (cid:16) a † k, a k, − a k, a † k, (cid:17) + V k (cid:16) a † k, a †− k, − a k, a − k, (cid:17)(cid:105)(cid:111) | (cid:105) , . (20)Equation (20) allows us to immediately identify the generator of the rotation (the operator in the brackets whichmultiplies U k ) and the one responsible for the Bogoliubov transformation (the operator in the brackets which multiplies V k ), in complete agreement with the fermion case [15].For later convenience, we can now further manipulate Eq. (20) by considering the case of a discrete set of modes.Apart from an irrelevant numerical factor, Eq. (20) becomes | (cid:105) A,B = exp (cid:110) − θ (cid:88) k (cid:104) U k (cid:16) a † k b k − a k b † k (cid:17) + V k (cid:16) a † k b †− k − a k b − k (cid:17)(cid:105)(cid:111) | (cid:105) , . (21)where we have introduced the shorthand notation a k, ≡ a k and a k, ≡ b k . In addition, since the transformationsEqs. (1) and (2) are valid for any value of the rotation angle, we can focus on the case of small values of θ to the O ( θ )-order, at which the first non-trivial contribution is expected. In this framework, we can use Zassenhaus formula e λ ( X + Y ) = e λX e λY e − λ [ X,Y ] , (22)up to O ( λ ) to approximate the flavor vacuum in Eq. (21). In fact, if we identify λ → − θ , X → (cid:80) k U k (cid:16) a † k b k − a k b † k (cid:17) and Y → (cid:80) k V k (cid:16) a † k b †− k − a k b − k (cid:17) , we are led to | (cid:105) A,B = e − θ (cid:80) k U k ( a † k b k − a k b † k ) e − θ (cid:80) k V k ( a † k b †− k − a k b − k ) e − θ (cid:80) k U k V k ( a † k a †− k − b † k b †− k − a k a − k + b k b − k ) | (cid:105) , (23)= e − θ (cid:80) k U k V k ( a † k a †− k − b † k b †− k ) e − θ (cid:80) k U k ( a † k b k − a k b † k ) e − θ (cid:80) k V k ( a † k b †− k − a k b − k ) | (cid:105) , , where in the second step we have made use of a simplification allowed only in the given approximation for θ . Fur-thermore, by applying the identity (22) to the two operators in Eq. (23) that depend on U k and V k , we get | (cid:105) A,B = e − θ (cid:80) k U k V k ( a † k a †− k − b † k b †− k ) e − θ (cid:80) k U k a † k b k e θ (cid:80) k U k a k b † k e θ (cid:80) k U k ( a k a † k − b k b † k ) (24) × e − θ (cid:80) k V k a † k b †− k e θ (cid:80) k V k a k b − k e − θ (cid:80) k V k ( a k a † k + b † k b k ) | (cid:105) , (cid:39) e − θ (cid:80) k U k V k ( a † k a †− k − b † k b †− k ) e − θ (cid:80) k ( V k b † k b k + U k b k b † k ) e − θ (cid:80) k V k a † k b †− k | (cid:105) , . In performing the second passage, we have omitted an unimportant constant factor and we have made use of thecurrent approximation to streamline the shape of the total operator. We will take advantage of the form (24) of theflavor vacuum in the next Section, when we compare the condensate structure of this state to the one of the thermalvacuum of Thermo Field Dynamics.
III. ENTANGLEMENT OF THE FLAVOR VACUUM
Let us now quantify the entanglement between the massive particle states in the flavor vacuum. As said earlier, weconsider the case t = 0. For notational convenience, it comes in handy to rewrite Eq. (20) as | (cid:105) A,B ≡ | ψ (cid:105) = exp (cid:110) − θ (cid:90) d k (cid:104) U k (cid:16) a † k b k − a k b † k (cid:17) + V k (cid:16) a † k b †− k − a k b − k (cid:17)(cid:105)(cid:111) | (cid:105) , (25)where | (cid:105) ≡ | (cid:105) , and we have used the same notation as in Eq. (21) for ladder operators. Interestingly, we observethat the above transformation is the result of a simultaneous coexistence of a beam splitter and a two-mode squeezingtransformation. Notice that, for t = 0, U k ( t ) = | U k | and V k ( t ) = | V k | (see Eq. (15)). As a preliminary analysis, we can perform a first-order approximation in θ and in the case of small mass difference,namely when m = m , m = m + δm and hence ε ≡ ( m − m ) /m = δm/m (cid:28)
1. This is in line with many worksregarding flavor mixing, and in particular with Ref. [15], in which the neutrino flavor vacuum is written up to thesecond order in θ and ε . Here, we make the same considerations with the purpose of seeking an analogous result.Starting from Eq. (25), it is immediate to derive that | ψ (cid:105) = exp {− θ (cid:90) d k (cid:20)(cid:16) a † k b k − a k b † k (cid:17) − ε m ω k (cid:16) a † k b †− k − a k b − k (cid:17)(cid:21) }| (cid:105) , (26)and thus expand the exponential operator as | ψ (cid:105) = ∞ (cid:88) n =0 n ! (cid:26) − θ (cid:90) d k (cid:20)(cid:16) a † k b k − a k b † k (cid:17) − ε m ω k (cid:16) a † k b †− k − a k b − k (cid:17)(cid:21)(cid:27) n | (cid:105) . (27)Since terms of the order O (cid:0) ε (cid:1) are neglected, we observe that it is possible to identify two recursive formulas in theexpression (27), one in front of the operator a † b † and another one next to (cid:0) a † a † − b † b † (cid:1) . The exact computation yields | ψ (cid:105) = (cid:40) + ε (cid:34) ∞ (cid:88) n =0 ( − n +1 θ n +2 n (2 n + 2)! (cid:35) (cid:90) d k m ω k (cid:104) a † k a †− k − b † k b †− k (cid:105) + ε (cid:34) ∞ (cid:88) n =0 ( − n θ n +1 n (2 n + 1)! (cid:35) (cid:90) d k m ω k a † k b †− k (cid:41) | (cid:105) . (28)It is straightforward to realize that the series in Eq. (28) converge to two simple analytic functions: ∞ (cid:88) n =0 ( − n +1 θ n +2 n (2 n + 2)! = − sin θ , (29) ∞ (cid:88) n =0 ( − n θ n +1 n (2 n + 1)! = sin 2 θ . (30)As a result, in the limit of small mass difference the flavor vacuum takes the form | ψ (cid:105) = (cid:26) − ε θ (cid:90) d k m ω k (cid:104) a † k a †− k − b † k b †− k (cid:105) + ε θ (cid:90) d k m ω k a † k b †− k (cid:27) | (cid:105) . (31)Before proceeding with the computation of von Neumann entropy, we pause to compare the above form of the flavorvacuum with the thermal vacuum of Thermo Field Dynamics defined in the Appendix (see in particular Eq. (A9)).The similarity between these two states sinks its roots in the fact that in TFD there is the need to double the physicalHilbert space by introducing a dual space (and hence an auxiliary field) whose excitations are holes from the pointof view of the physical field (a detailed mathematical explanation of these notions can be found in the Appendix).As a consequence, the thermal vacuum appears as a condensate of excitations of the physical and auxiliary fields,which thus resembles the structure of the flavor vacuum being a condensate of particle/antiparticle pairs with differentmasses.In this vein, we emphasize that a first attempt to perform a comparison between the flavor and thermal vacua wascarried out in Ref. [51] and later in [15], arguing that the two states cannot be exactly matched, since the would-beentropy operator defined for mixed fields does not possess the same properties as the one introduced in TFD (seeEq. (A11)). However, here we tackle this problem from a different perspective; indeed, we directly look at the inherentstructure of the two vacua. Even though the result of Ref. [15] remains in general valid since the thermal state (A9)does not contain terms of the form a † k a †− k , b † k b †− k which instead appear in the flavor vacuum (31), for small values ofthe mixing angle we can approximate Eq. (31) to the leading order as | ψ (cid:105) = (cid:26) + ε θ (cid:90) d k m ω k a † k b †− k (cid:27) | (cid:105) . (32)Then, from comparison with the TFD vacuum in Eq. (A13), | ϑ ) (cid:105) (cid:39) + (cid:88) k a † k ˜ a † k ϑ k | (cid:105) , one can recognize in the state (32) a thermal-like vacuum with an inverse temperature β = ( k B T ) − given by (up toa scaling factor due to the conversion of the k -integral into a discrete sum, as viewed in Eq. (21) and following) β (cid:39) ω k ln (cid:18) ω k εθm (cid:19) . (33)Clearly, for θ and/or ε = 0, we have β → ∞ , or equivalently T →
0. This is somehow expected since, for vanishingmixing, the flavor and mass vacua coincide with each other, which corresponds in the TFD language to the case wherethe doubling of degrees of freedom (and thus the temperature) disappears.Let us now come back to the quantification of von Neumann entropy of the flavor vacuum. To this aim, we focuson the k = 0 mode, for which the condensation density reaches its maximal value in the case of boson mixing (seeEq. (18)) [20]. Therefore, we expect that the effects of the mixing transformation are maximally non-trivial in thiscase. However, for this purpose we need to go beyond the linear order approximation in the mass difference, as it canbe easily shown that the linear term in ε gives a vanishing contribution. Then, by resorting to Eq. (27) and retainingonly the factors that do not exceed O ( ε ), we find a recurrence series of the form | ψ (cid:105) = (cid:110) + (cid:18) ε − ε (cid:19) (cid:34) ∞ (cid:88) n =0 ( − n θ n +1 n (2 n + 1)! (cid:35) a † b † + (cid:18) ε − ε (cid:19) (cid:34) ∞ (cid:88) n =0 ( − n +1 θ n +2 n (2 n + 2)! (cid:35) (cid:2) ( a † ) − ( b † ) (cid:3) (34)+ ε (cid:34) ∞ (cid:88) n =0 ( − n θ n +2 n (2 n + 2)! (cid:35) (cid:2) ( a † ) ( b † ) − (cid:3) + 3 ε (cid:40) ∞ (cid:88) n =0 (cid:34) ( − n +1 θ n +3 n (2 n + 3)! n (cid:88) m =0 (cid:16) √ √ (cid:17) m (cid:35)(cid:41) (cid:2) ( a † ) b † − a † ( b † ) (cid:3) + 3 ε (cid:40) ∞ (cid:88) n =0 (cid:34) ( − n θ n +4 n (2 n + 4)! n (cid:88) m =0 (cid:16) √ √ (cid:17) m (cid:35)(cid:41) (cid:104) ( a † ) − √ a † ) ( b † ) + ( b † ) (cid:105)(cid:111) | (cid:105) . The series appearing in Eq. (34) converge to trigonometric functions in θ . More precisely, we have | ψ (cid:105) = (cid:20) − ε θ (cid:21) | (cid:105) − (cid:18) ε − ε (cid:19) sin θ (cid:16) | (cid:105) − | (cid:105) (cid:17) + (cid:18) ε − ε (cid:19) sin 2 θ | (cid:105) (35)+ ε (cid:20) sin θ − √ χ ( θ ) (cid:21) | (cid:105) + 3 ε χ ( θ ) (cid:16) | (cid:105) + | (cid:105) (cid:17) + 3 ε ξ ( θ ) (cid:16) | (cid:105) − | (cid:105) (cid:17) , where χ ( θ ) = (cid:0) √ (cid:1) (1 − cos 2 θ ) − √ (cid:104) − cos (cid:16) (cid:112) √ θ (cid:17)(cid:105) (cid:0) √ (cid:1) (cid:0) √ √ − (cid:1) , (36) ξ ( θ ) = 2 sin (cid:16) (cid:112) √ θ (cid:17) − (cid:112) √ θ (cid:112) √ (cid:0) √ √ − (cid:1) . (37)The state in Eq. (35) is not normalized, since (cid:104) ψ | ψ (cid:105) (cid:54) = 1. For this reason, before computing the projector ρ = | ψ (cid:105)(cid:104) ψ | ,we must divide the above state by a suitable normalization factor, which is explicitly given by N = (cid:112) (cid:104) ψ | ψ (cid:105) = (cid:115) − ε (cid:20) sin θ − sin θ − sin θ (cid:21) . (38)Accordingly, the normalized density matrix reads ρ = | ψ (cid:105)(cid:104) ψ | N = (cid:20) − ε θ − ε
16 sin θ (cid:21) | (cid:105)(cid:104) | + (cid:18) ε − ε (cid:19) sin 2 θ (cid:16) | (cid:105)(cid:104) | + | (cid:105)(cid:104) | (cid:17) (39) − (cid:18) ε − ε (cid:19) sin θ (cid:16) | (cid:105)(cid:104) | − | (cid:105)(cid:104) | + | (cid:105)(cid:104) | − | (cid:105)(cid:104) | (cid:17) + ε
16 sin θ | (cid:105)(cid:104) | + ε (cid:20) sin θ − √ χ ( θ ) (cid:21) (cid:16) | (cid:105)(cid:104) | + | (cid:105)(cid:104) | (cid:17) + 3 ε χ ( θ ) (cid:16) | (cid:105)(cid:104) | + | (cid:105)(cid:104) | + | (cid:105)(cid:104) | + | (cid:105)(cid:104) | (cid:17) + ε
16 sin θ (cid:16) | (cid:105)(cid:104) | − | (cid:105)(cid:104) | − | (cid:105)(cid:104) | + | (cid:105)(cid:104) | (cid:17) + 3 ε ξ ( θ ) (cid:16) | (cid:105)(cid:104) | − | (cid:105)(cid:104) | + | (cid:105)(cid:104) | − | (cid:105)(cid:104) | (cid:17) − ε
16 sin θ sin 2 θ (cid:16) | (cid:105)(cid:104) | − | (cid:105)(cid:104) | + | (cid:105)(cid:104) | − | (cid:105)(cid:104) | (cid:17) . By partial tracing ρ with respect to either one of the two subsystems, say S , we obtain the reduced density matrix ρ (2) r = Tr S ρ = (cid:20) − ε
16 sin θ − ε
16 sin θ (cid:21) | (cid:105)(cid:104) | + (cid:18) ε − ε (cid:19) sin θ (cid:16) | (cid:105)(cid:104) | + | (cid:105)(cid:104) | (cid:17) + ε
16 sin θ | (cid:105)(cid:104) | (40)+ 3 ε χ ( θ ) (cid:16) | (cid:105)(cid:104) | + | (cid:105)(cid:104) | (cid:17) + ε
16 sin θ | (cid:105)(cid:104) | , with the same result for ρ (1) r = Tr S ρ . From Eq. (40), we can compute von Neumann entropy up to O ( ε ). Noticethat the matrix ρ (1) r = ρ (2) r ≡ ρ r , given by ρ r = − ε sin θ − ε sin θ (cid:16) ε − ε (cid:17) sin θ ε χ ( θ )0 ε sin θ (cid:16) ε − ε (cid:17) sin θ ε sin θ ε χ ( θ ) 0 0 0 , (41)has eigenvalues λ i = (cid:26) O ( ε ) , O ( ε ) , − ε
32 (1 − cos 4 θ ) , ε
16 sin θ (cid:27) . (42)Therefore, to order ε , von Neumann entanglement entropy reads S V = − (cid:88) i λ i log λ i = − (cid:20) − ε
32 (1 − cos 4 θ ) (cid:21) log (cid:20) − ε
32 (1 − cos 4 θ ) (cid:21) − ε
16 sin θ log (cid:18) ε
16 sin θ (cid:19) . (43)As expected, in the limit of either ε → θ →
0, we recover S V = 0, as it should correctly be, because in thesecases the flavor vacuum reduces to the tensor product | (cid:105) , = | (cid:105) ⊗ | (cid:105) . In Fig. 1 we report the behavior of vonNeumann entropy as a function of both ε and θ . In order to better convey the dependence of S V on the mixing angle,it is also appropriate to exhibit how it varies by keeping ε fixed. This is done in Fig. 2. In particular, we can noticethat for arbitrary ε the maximum of S V always occurs at the perfectly balanced mixing angle θ max = π/
4. Increasing ε only results in an upper shift in the magnitude of S V , with an essentially invariant shape. The value of θ max isexactly the one that could have been initially predicted, since π/ IV. CONCLUSIONS
In this paper, we have explored some non-trivial features of the flavor vacuum in the case of mixing of two neutralscalar fields. A first preliminary analysis has shown that we cannot quantify entanglement by resorting to a directcomparison between | (cid:105) A,B and the free thermal vacuum introduced in Thermo Field Dynamics, since the nature of
FIG. 1: Behavior of von Neumann entanglement entropy S V as a function of ε ∈ (cid:2) , − (cid:3) and θ ∈ (cid:2) , π (cid:3) . As explicitly shownby the level curves, S V monotonically decreases with decreasing ε . the two states is not exactly the same. Note that this is consistent with the result found in Ref. [15] in the contextof neutrino mixing. A similar outcome has also been exhibited in Refs. [21], where it has been shown that the flavorvacuum for an accelerated observer is not strictly a thermal state, in contrast with the case of the standard Unruheffect. However, in spite of these differences, we have found that, in a proper limit, the flavor vacuum can be identifiedwith the vacuum of TFD, with a temperature dependent on the mixing angle and the mass difference of the twomixed fields.To further investigate the condensate structure of the flavor vacuum, we have restricted our attention to the mode k = 0, for which the condensation density in | (cid:105) A,B is maximal. In this setting, we have quantified von Neumannentropy for small values of the mass difference. The shape exhibited in Fig. 1 is the one derived from our analysis,and the picture of Fig. 2 shows that there is an angle in correspondence of which von Neumann entropy is maximalfor a given value of the small mass difference. It is worth observing that such angle is precisely the one responsiblefor maximal mixing, namely θ max = π/
4. Clearly, it would be interesting to go beyond the single-mode analysisand derive the full expression for the von Neumann entropy. However, we expect that the result will exhibit someultraviolet divergence of the same kind as in Ref. [52], where the contribution of the flavor vacuum energy to thecosmological constant was shown to diverge only logarithmically, in contrast with the standard asymptotic behaviorof free-field vacuum energy.It is worth remarking that entanglement entropy and particle mixing have already been analyzed together inseveral papers in recent years [1, 2], but only in the context of flavor transitions and for one-particle flavor states.In this respect, it is interesting to compare the field-theoretical approach to flavor entanglement developed in thepresent work with the exact methods for the quantification of entanglement introduced in the framework of quantuminformation and quantum optics of non-relativistic continuous variable systems (see for instance Refs. [53–56] andreferences therein). Such methods revolve around the transformation of the quadrature operators (namely, positionand momentum) under a given operation, which in our case is the mixing transformation. Since at the level of theseoperators mixing acts merely as a rotation, we would not achieve the desired result, thus preventing us from reachingan accurate evaluation of the entanglement entropy of the flavor vacuum via the procedures adopted for quantumoptical systems [53–55].Finally, we emphasize that the present analysis may have several implications in a wide range of diversified contexts.For instance, once fully extended to non-inertial frames [57–59] and, more generally, to gravity scenarios [60–70],one could investigate how the information content of the flavor vacuum is degraded for increasing values of theacceleration/gravity. Note that a similar analysis has been carried out in Ref. [71] for the case of the entanglementbetween two free modes of a scalar field as seen by an inertial observer detecting one of the modes and a uniformlyaccelerated observer detecting the other one. These and other aspects will be thoroughly investigated in future works.
FIG. 2: Behavior of von Neumann entanglement entropy S V as a function of θ ∈ (cid:2) , π (cid:3) for different values of ε . Appendix A: Thermo Field Dynamics
This Appendix is devoted to review the basics of Thermo-Field-Dynamics (TFD) [47], which represents one of theapproaches to QFT at finite temperature and density. The reason for this study lies in the fact the TFD and theformalism of mixing analyzed in Sec. II exhibit some non-trivial conceptual similarities, the most prominent one beingthe doubling of the degrees of freedom of the original system.In order to introduce the TFD from a more quantitative point of view, let us consider the Hamiltonian of a freephysical system written in the usual form H = (cid:88) k (cid:15) k a † k a k . (A1)We now define a completely identical fictitious system, which we name “tilde” system, sharing the same properties ofthe physical one. The former can then be written as˜ H = (cid:88) k (cid:15) k ˜ a † k ˜ a k . (A2)The above relation allows us to establish a new set of “thermal” operators via the Bogoliubov transformations a k ( ϑ ) = a k cosh ϑ k − ˜ a † k sinh ϑ k , (A3)˜ a k ( ϑ ) = ˜ a k cosh ϑ k − a † k sinh ϑ k , (A4)with θ k encoding the information about the original and thermal ladder operators. Notice that the transforma-tions (A3) and (A4) leave the total Hamiltonian of the system H tot = H − ˜ H invariant, provided that the Hamilto-nians (A1) and (A2) have the same spectrum.The Bogoliubov transformation (A3) and (A4) can also be recast as a k ( ϑ ) = e − iG a k e iG , (A5)˜ a k ( ϑ ) = e − iG ˜ a k e iG , (A6)with G = i (cid:88) k ϑ k (cid:16) a † k ˜ a † k − ˜ a k a k (cid:17) , (A7)0being the generator of the transformation, which is also a conserved quantity (namely [ G, H tot ] = 0) since the totalHamiltonian is conserved under its action. By means of G , we can now build a new vacuum state for the operators a k ( ϑ ) and ˜ a k ( ϑ ) defined as [47] | ϑ ) (cid:105) = e − iG | (cid:105) , (A8)where | (cid:105) is the vacuum associated with a k and ˜ a k . If we explicitly act with the above operator on | (cid:105) , we are leftwith | ϑ ) (cid:105) = (cid:89) k e a † k ˜ a † k tanh ϑ k cosh ϑ k | (cid:105) . (A9)The above expression conveys the idea that the “theta” vacuum is actually a condensate of a - and ˜ a -particles, whichsomehow resembles the situation we have already encountered with mixing. However, in order to construct an effectivecomparison with the latter framework, we need to reformulate Eq. (A9) as | ϑ ) (cid:105) = e − K e (cid:80) k a † k ˜ a † k | (cid:105) = e − ˜ K e (cid:80) k a † k ˜ a † k | (cid:105) , (A10)where K = − (cid:88) k (cid:16) a † k a k ln sinh ϑ k − a k a † k ln cosh ϑ k (cid:17) , (A11)is called the entropy operator [47], because its vacuum expectation value multiplied by k B yields precisely the entropyof the physical system. As a matter of fact, in order to make contact with a feasible physical picture and describeQFT for free fields at finite temperature, it is possible to prove that the arbitrary factors ϑ k should satisfy the relation β ω k = − ln tanh ϑ k , (A12)with β = ( k B T ) − being the inverse of the emerging temperature T of the thermal vacuum | ϑ ) (cid:105) and ω k = (cid:15) k − µ ,where µ is the chemical potential. Remarkably, in the limit θ k → T small enough), Eq. (A9) reads | ϑ ) (cid:105) (cid:39) + (cid:88) k a † k ˜ a † k ϑ k | (cid:105) . 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