Interferometric Visibility in Curved Spacetimes
IInterferometric Visibility in Curved Spacetimes
Marcos L. W. Basso ∗ and Jonas Maziero † Departamento de Física, Centro de Ciências Naturais e Exatas,Universidade Federal de Santa Maria, Avenida Roraima 1000, Santa Maria, RS, 97105-900, Brazil
In [M. Zych et al., Nat. Commun. 2, 505 (2011)], the authors predicted that the interferometricvisibility is affected by a gravitational field in way that cannot be explained without the generalrelativistic notion of proper time. In this work, we take a different route and derive the sameresult using the unitary representation of the local Lorentz transformation in the Newtonian Limit.In addition, we show that the effect on the interferometric visibility due to gravity persists indifferent spacetime geometries. However, the influence is not necessarily due to the notion of propertime. For instance, in Schwarzschild spacetime the influence on the interferometric visibility canbe due to another general relativistic effect, the geodetic precession. Besides, by using the unitaryrepresentation of the local Lorentz transformation, we show that this behavior of the interferometricvisibility is general for an arbitrary spacetime, provided that we restrict the motion of the quantonto a two-dimensional spacial plane.
Keywords: Interferometric Visibility; Curved spacetimes; General relativistic effects
I. INTRODUCTION
Recently, there has been an increasingly effort in prob-ing the interplay between gravity and quantum mechan-ics or, more specifically, to witness the quantumness ofgravity [1–4], as well as to probe general relativistic ef-fects in quantum phenomena [5–9]. For instance, in [6]the authors predicted a quantum effect that cannot be ex-plained without the general relativistic notion of propertime. They considered a Mach-Zehnder interferometerplaced in a gravitational potential, with a ‘clock’ used asan interfering particle, i.e., an evolving internal degree offreedom of a particle. Due to the difference in proper timeelapsed along the two trajectories, the ‘clock’ evolves todifferent quantum states for each path of the interferom-eter. Because of the wave-particle duality, the interfero-metric visibility will decrease by an amount given by thewhich-way information accessible from the final state ofthe clock, which gets entangled with the external degreeof freedom of the particle.Moreover, concerns about how entanglement behavesin relativistic scenarios has grown more and more [10].For example, the authors of Refs. [11, 12] showed thatthe entanglement of Bell states depends on the velocityof the observer. On the other hand, the authors in Ref.[13] argued that the overall entanglement of a Bell stateremains invariant for a Lorentz boosted observer, the en-tanglement is just shuffled between the different degreesof freedom. However, it was demonstrated by Peres etal. [14] that the entropy of a single massive spin-1/2particle does not remain invariant under Lorentz boosts.These apparently conflicting results involve systems con-taining different particle states and boost geometries [15].Therefore, entanglement under Lorentz boosts is highly ∗ Electronic address: [email protected] † Electronic address: [email protected] dependent on the boost scenario in question [16]. Moregenerally, the entanglement for observers constantly ac-celerated in a flat space-time was considered in Refs.[17–19]. A step forward in the investigations of theserelativistic scenarios was taken by Terashima and Ueda[20], who studied EPR correlations and the violation ofBell’s inequalities in curved spacetimes, by considering asuccession of infinitesimal local Lorentz transformations.In addition, the same authors, in Ref. [21], studied thedecoherence of spin states due to the presence of a grav-itational field, using the same method.In this work, based on the method developed in [20],we derive the same result of [6]. We use the unitary repre-sentation of the local Lorentz transformation in the New-tonian Limit. In addition, as we’ll show here, the effecton the interferometric visibility due to gravity persistsin different spacetime geometries. However, the oscilla-tion is not necessarily due to the difference of the propertime elapsed in each path. For instance, in Schwarzschildspacetime, the influence on the interferometric visibilityis due to another general relativistic effect, i.e., geodeticprecession. Besides, we show that by using the unitaryrepresentation of the local Lorentz transformation, thisbehavior of the interferometric visibility in an arbitraryspacetime is general, provided that we restrict the motionof the quanton to a two-dimensional spacial plane. Thebenefit of the approach taken here, through the repre-sentation of the local Lorentz transformation, is that onedoes not need to know the (internal) Hamiltonian of thesystem and how it couples to the gravitation field, as in[6, 9]. In contrast, given an arbitrary spacetime metric,we only need to calculate the Wigner rotation, which isa mechanic and straightforward procedure.The organization of this article is as follows. In Sec. II,we review the spin- / dynamics in curved spacetimes.In Sec. III, we study the behavior of interferometric vis-ibility in different spacetime geometries. Thereafter, inSec. IV, we give our conclusions. a r X i v : . [ g r- q c ] F e b II. SPIN DYNAMICS IN CURVEDSPACETIMESA. Spin States in Local Frames
The study of the dynamics of spin- / particles in grav-itational fields requires the use of local frames of refer-ence defined at each point of spacetime. These framesare defined through an orthonormal basis or tetrad field(or vielbein), which is a set of four linearly independentorthonormal 4-vector fields [22]. The differential struc-ture of the spacetime, which is a differential manifold M [23], provides, in each point p , a coordinate basis for thetangent space T p ( M ) , as well as for the cotangent space T ∗ p ( M ) , given by { ∂ µ } and { dx ν } , respectively, such that dx ν ( ∂ µ ) := ∂ µ x ν = δ νµ . Therefore, the metric can beexpressed as g = g µν ( x ) dx µ ⊗ dx ν , and the elementsof the metric, which encodes the gravitational field, aregiven by g µν ( x ) = g ( ∂ µ , ∂ ν ) . Since the coordinate basis { ∂ µ } ⊂ T p ( M ) and { dx ν } ⊂ T ∗ p ( M ) are not necessarilyorthonormal, it’s always possible set up any basis as welike. In particular, we can form an orthonormal basiswith respect to the pseudo-Riemannian manifold (space-time) on which we are working. Following Ref. [24], letus consider the linear combination e a = e µa ( x ) ∂ µ , e a = e aµ ( x ) dx µ , (1) ∂ µ = e aµ ( x ) e a , dx µ = e µa ( x ) e a . (2)To define a local frame at each point p ∈ M , we require { e a } to be orthonormal in the following sense g ( e a , e b ) := η ab , g := η ab e a ⊗ e b , (3)where η ab = diag ( − , , , is the Minkowski metric.Equivalently, we can define the tetrad field in terms ofits components g µν ( x ) e µa ( x ) e νb ( x ) = η ab , (4) η ab e aµ ( x ) e bν ( x ) = g µν ( x ) , (5)with e aµ ( x ) e µb ( x ) = δ ab , e aµ ( x ) e νa ( x ) = δ νµ . (6)In what follows, Latin letters a, b, c, d, · · · refer to co-ordinates in the local frame; Greek indices µ, ν, · · · runover the four general-coordinate labels; and repeated in-dices are to be summed over. The components of thetetrad field and its inverse transforms a tensor in the gen-eral coordinate system into one in the local frame, andvice versa. Therefore it can be used to shift the depen-dence of spacetime curvature of the tensor fields to thetetrad fields. In addition, Eq. (5) informs us that thetetrad field encodes all the information about the space-time curvature hidden in the metric. Besides, the tetradfield { e µa ( x ) , a = 0 , , , } is a set of four 4-vector fields,which transforms under local Lorentz transformations inthe local system. The choice of the local frame is not unique, since the local frames remains local under thelocal Lorentz transformations. Therefore, a tetrad repre-sentation of a particular metric is not uniquely defined,and different tetrad fields will provide the same metrictensor, as long as they are related by local Lorentz trans-formations [25].By constructing the local Lorentz transformation, wecan define a particle with spin- / in curved space-times as a particle whose one-particle states furnish thespin- / representation of the local Lorentz transforma-tion [20]. Thus, let’s consider a massive spin- / parti-cle moving with four-momentum p µ ( x ) = mu µ ( x ) with p µ ( x ) p µ ( x ) = − m , where m is the mass of the quanton, u µ ( x ) is the four-velocity in the general coordinate sys-tem, and we already putted c = 1 . Now, we can use thetetrad field e aµ ( x ) to project the four-momentum p µ ( x ) into the local frame, i.e., p a ( x ) = e aµ ( x ) p µ ( x ) . Thus,in the local frame at point p ∈ M with coordinates x a = e aµ ( x ) x µ , a momentum eigenstate of a Dirac parti-cle in a curved spacetime is given by [26] | p a ( x ) , σ ; x (cid:105) := (cid:12)(cid:12) p a ( x ) , σ ; x a , e aµ ( x ) , g µν ( x ) (cid:11) , (7)and represents the state with spin σ and momentum p a ( x ) as observed from the position x a = e aµ ( x ) x µ ofthe local frame defined by e aµ ( x ) in the spacetime M with metric g µν ( x ) . By definition, the state | p a ( x ) , σ ; x (cid:105) transforms as the spin- / representation under the localLorentz transformation. In the case of special relativity,a one-particle spin- / state | p a , σ (cid:105) transforms under aLorentz transformation Λ ab as [27] U (Λ) | p a , σ (cid:105) = (cid:88) λ D λσ ( W (Λ , p )) | Λ p a , λ (cid:105) , (8)where D λ,σ ( W (Λ , p )) is a unitary representation of theWigner’s little group, whose elements are Wigner rota-tions W ab (Λ , p ) [28]. The subscripts can be suppressedand one can write U (Λ) | p a , σ (cid:105) = | Λ p a (cid:105)⊗ D ( W (Λ , p )) | σ (cid:105) , as sometimes we’ll do. In other words, under a Lorentztransformation Λ , the momenta p a goes to Λ p a , and thespin transforms under the representation D σ,λ (Λ , p ) ofthe Wigner’s little group [29]. Meanwhile, in a curvedspacetime everything above remains essentially the same,except by the fact that single-particle states now form alocal representation of the inhomogeneous Lorentz groupat each point p ∈ M , i.e., U (Λ( x )) | p a ( x ) , σ ; x (cid:105) = (cid:88) λ D λσ ( W ( x )) | Λ p a ( x ) , λ ; x (cid:105) , (9)where W ( x ) := W (Λ( x ) , p ( x )) is a local Wigner rotation. B. Spin Dynamics
Following Terashima and Ueda [20], let us consider howthe spin changes when the quanton moves from one pointto another in curved spacetime. In the local frame atpoint p with coordinates x a = e aµ ( x ) x µ , the momentumof the particle is given by p a ( x ) = e aµ ( x ) p µ ( x ) . Afteran infinitesimal proper time dτ , the quanton moves toa new point with general coordinates x (cid:48) µ = x µ + u µ dτ .Then, the momentum of the particle in the local frameat the new point becomes p a ( x (cid:48) ) = p a ( x ) + δp a ( x ) , wherethe variation of the momentum in the local frame canbe described by the combination of changes due to non-gravitational external forces δp µ ( x ) , and spacetime ge-ometry effects δe aµ ( x ) : δp a ( x ) = e aµ ( x ) δp µ ( x ) + δe aµ ( x ) p µ ( x ) . (10)The variation δp µ ( x ) in the first term on the right handside of the last equation is simply given by δp µ ( x ) = u ν ( x ) ∇ ν p µ ( x ) dτ = ma µ ( x ) dτ, (11)where ∇ ν is the covariant derivative and a µ ( x ) := u ν ( x ) ∇ ν u µ ( x ) is the acceleration due to a non-gravitational force. Once p µ ( x ) p µ ( x ) = − m and p µ ( x ) a µ ( x ) = 0 , Eq. (11) can be rewritten as δp µ ( x ) = − m ( a µ ( x ) p ν ( x ) − p µ ( x ) a ν ( x )) p ν ( x ) dτ. (12)Meanwhile, the variation of the tetrad field is given by δe aµ ( x ) = u ν ( x ) ∇ ν e aµ ( x ) dτ = − u ν ( x ) ω aν b ( x ) e bµ ( x ) dτ, (13)where ω aν b := e aλ ∇ ν e λb = − e λb ∇ ν e aλ is the connection1-form (or spin connection) [30]. Collecting these resultsand substituting in Eq. (11), we obtain δp a ( x ) = λ ab ( x ) p b ( x ) dτ, (14)where λ ab ( x ) = − m ( a a ( x ) p b ( x ) − p b ( x ) a a ( x )) + χ ab = − ( a a ( x ) u b ( x ) − u a ( x ) a b ( x )) + χ ab (15)with χ ab := − u ν ( x ) ω aν b ( x ) . It can be shown that Eqs.(14) and (15) constitute an infinitesimal local Lorentztransformation since, as the particle moves in spacetime,the momentum in the local frame will transform underan infinitesimal local Lorentz transformation p a ( x ) =Λ ab ( x ) p b ( x ) where Λ ab ( x ) = δ ab + λ ab ( x ) dτ [26]. If theparticle moves in a geodesic in spacetime, then a µ ( x ) = 0 and the infinitesimal Lorentz transformation in the lo-cal frame reduces to λ ab ( x ) = − u ν ( x ) ω aν b ( x ) . Classi-caly, in this case, the spin will be affected according to u ν ∇ ν s µ = 0 , i.e., the spin is parallel transported alongthe geodesic. It’s well known that, due to the curvatureeffects, when considering an arbitrary closed path thespin will not return to its initial state after being paral-lel transported along such path, even when the path ofthe particle corresponds to a geodesic. Moreover, sincegeodesic paths preserves the magnitude of vectors, the spin magnitude remains constant along the path, hencethe change of the spin state is associated with the spinprecession. This phenomena is known as geodetic preces-sion [25], and quantum mechanically, can be associatedwith the action of successive local Wigner rotations [31].Now, given the local Lorentz transformation, we canconstruct the local Wigner rotation that affects the spinof the particle. In other words, by using a unitary repre-sentation of the local Lorentz transformation, the state | p a ( x ) , σ ; x (cid:105) is now described as U (Λ( x )) | p a ( x ) , σ ; x (cid:105) inthe local frame at the point x (cid:48) µ , and Eq. (9) expresseshow the spin of the quanton rotates locally as the parti-cle moves from x µ → x (cid:48) µ along its world-line. Therefore,one can see that spacetime tells quantum states how toevolve. For the infinitesimal Lorentz transformation, theinfinitesimal Wigner rotation is given by W ab ( x ) = δ ab + ϑ ab dτ, (16)where ϑ ( x ) = ϑ i ( x ) = ϑ i ( x ) = 0 and ϑ ij ( x ) = λ ij ( x ) + λ i ( x ) p j ( x ) − λ j ( x ) p i ( x ) p ( x ) + m . (17)In [32], the authors provided an explicitly calculation ofthese elements, and the two-spinor representation of theinfinitesimal Wigner rotation is then given by D ( W ( x )) = I × + i (cid:88) i,j,k =1 (cid:15) ijk ϑ ij ( x ) σ k dτ = I × + i ϑ · σ dτ, (18)where I × is the identity matrix, { σ k } k =1 are the Paulimatrices, and (cid:15) ijk is the Levi-Civita symbol. Moreover,the Wigner rotation for a quanton that moves over afinite proper time interval can be obtained by iteratingthe expression for the infinitesimal Wigner rotation [20],and the spin- / representation for a finite proper timecan be obtained by iterating the Eq. (18): D ( W ( x, τ )) = T e i (cid:82) τ ϑ · σ dτ (cid:48) , (19)where T is the time-ordering operator [20], since, in gen-eral, the Wigner rotation varies at different points alongthe trajectory. III. INTERFEROMETRIC VISIBILITY INCURVED SPACETIME
In this section, we’ll study the behavior of the inter-ferometric visibility of a spin- / quanton (or a qubit) ina Mach-Zehnder interferometer in curved spacetime. Be-cause we are interested in qubits, it’s worth pointing outthat the motion of spinning particles, either classical orquantum, does not follow geodesics because the spin andcurvature couples in a non-trivial manner [33]. However,since the deviation from geodetic motion is very small, itcan be safely ignored in the cases explored here. A. Interferometric Visibility in the NewtonianLimit
The Newtonian limit is an approximation applicableto physical scenarios exhibiting: weak gravitation field;objects moving slowly compared to the speed of light;and static gravitational fields [23]. For instance, in thevicinity of Earth, and in a small region such that thegravitation field is uniform, the spacetime metric can beapproximately expressed in the coordinate basis as ds = − (1 + 2 gx ) dt + dx + dy + dz , (20)where g = GM/R denotes the value of the Earth’s grav-itational acceleration in the origin of a laboratory frame(x = 0), which is at distance R from the centre of Earthand the coordinate x measures the different heights in thegravitational field, as in Fig. 1. Considering an interfer-ometric setup where a spin- / particle goes through aMach-Zehnder interferometer, we assume that the quan-ton is moving with a constant velocity u µ = dx µ /dτ . Un-der this assumption, the particle does not move along ageodesic, hence it must be subject to a non-gravitationalforce to maintain the quanton on its path. Therefore, inthis case, the Wigner rotation arises from the externalforce as well from the spacetime geometry effects. More-over, if the spin degree of freedom can be considered asa ‘clock’, according to general relativity, the proper timeshould evolve differently along the two arms of the inter-ferometer in the presence of gravity. Because of Bohr’scomplementarity principle, the interferometric visibilitywill decrease by an amount given by the which-way infor-mation accessible from the final state of the spin, whichgets entangled with the path of the quanton, as showedin [6]. The separation between the horizontal arms of thepaths γ, κ of the interferometer is h .Here, we will derive the same result using the unitaryrepresentation of the local Lorentz transformation, onceEq. (9) expresses how the spin of the quanton rotatesand couples with its momentum as the particle movesalong its world-line in the interferometer. In order tocalculate the Wigner rotation, we consider the followingtetrad field e t = (1 + 2 gx ) / , e x = e y = e z = 1 , (21)and all the other components are zero. Also, only nonzerocomponents will be shown from now on. The inverse ofthese elements are given by e t = (1 + 2 gx ) − / , e x = e y = e z = 1 , (22)This vierbein represents a static local frame at each point.In addition, at each point, the − , − , − , and − axesare parallel to the t, x, y, and z directions, respectively.The components of velocity in the local frame are u = − u = (cid:112) u , (23) u = u = u x , u = u = u z , (24) BS1 BS2 g zxh ϕ D+ D- γκ Figure 1: Spin- / particle in a Mach-Zehnder interferometerin the vicinity of Earth. The setup consists of two beam split-ters BS and BS , a phase shifter, which gives a controllablephase φ to the path κ , and two detectors D ± . A uniformgravitational field g is oriented antiparallel to the x -direction. where u = u x + u z and u y = 0 since the quantonis restricted to move in the x − z plane. Therefore,the non-zero components of the acceleration due to non-gravitational external forces is given by: a = gu x √ u gx , a = g (1 + u )1 + 2 gx . (25)Meanwhile, the change of the local frames along the worldline is characterized by χ ab := − u ν ( x ) ω aν b ( x ) , which hasonly one non-zero component given by χ = − g (1 + u ) / gx . (26)Using the above equations, the non-zero infinitesimalLorentz transformations λ ab can be obtained, and astraightforward calculation similar to those in [34] showsthat λ = g (1 + u ) / u z gx , (27) λ = − g (1 + u ) / u x u z gx , (28) λ = − g (1 + u ) u z gx , (29)which express that the change in the local frame consistsof a boosts along the and -axis and a rotation aboutthe -axis. It’s noteworthy that δp a = λ ab p b dτ = 0 ,which is consistent with the assumption that the particlefollows straight paths in the interferometer.Therefore, the Wigner rotation corresponds to a rota-tion over the 2-axis and is given by: ϑ = λ + λ p − λ p p + m = − gp z (cid:112) p + m (1 + 2 gx ) m , (30)where p = (cid:112) p x + p z . One can see that the Wigner rota-tion depends on g , if g = 0 there is no rotation. It alsodepends on the height x . Now, let’s suppose that theinitial state of the quanton before BS in Fig. 1 is givenby | Ψ i (cid:105) = | p i (cid:105) ⊗ | τ i (cid:105) = 1 √ | p i (cid:105) ⊗ ( |↑(cid:105) + |↓(cid:105) ) , (31)with the local quantization spin axis along the 1-axis.Right after the BS1, we have a coherent superposition ofthe path γ and κ states such that | Ψ (cid:105) = 12 ( | p γ (cid:105) + i | p κ (cid:105) ) ⊗ ( |↑(cid:105) + |↓(cid:105) ) . (32)Therefore, the state before the BS is affected by thegravitational field and is given by U (Λ) | Ψ (cid:105) = 12 | p γ (cid:105) ⊗ D ( W ( γ ))( |↑(cid:105) + |↓(cid:105) )+ ie iφ | p κ (cid:105) ⊗ D ( W ( κ ))( |↑(cid:105) + |↓(cid:105) ) , (33)where W ( η ) is the total Wigner rotation due to the path η = γ, κ . Besides, we choose to maintain the notation | p η (cid:105) for the momentum states before BS , because thevelocity of quanton is constant along the paths. Now,the spin- / representation of the Wigner rotation forthe paths is given by D ( W ( η, τ )) = T e i σ y (cid:82) τ ϑ ( η ) dτ (cid:48) , η = γ, κ. (34)Thus, one can see that D ( W ( η, τ )) depends of the propertime elapsed in each path η = γ, κ .Since part of the paths are in different heights, gen-eral relativity predicts that the amount of the elapsedproper time is different along the two paths, and there-fore the Wigner rotation for each path will be differ-ent, which implies that the spin-state will be differentand path-information can be accessed. It’s worthwhilementioning that the part of the total Wigner rotation Θ( η ) := (cid:82) τ ϑ ( η ) dτ (cid:48) of each path η that will affect dif-ferently the spin-state is due to the horizontal arm ofeach path of the interferometer. The Wigner rotationdue to the vertical route is the same for both paths τ, γ and therefore it cancels out. Since ϑ ( η ) is constantalong both horizontal paths, the time-ordering opera-tor is not needed, and the integration is straightforward.After BS , the states of the momentum are given by | p γ (cid:105) → ( | p + (cid:105) + i | p − (cid:105) ) / √ and | p κ (cid:105) → ( i | p + (cid:105) + | p − (cid:105) ) / √ ,and the final state of the quanton is given by | Ψ f (cid:105) = 12 √ | p + (cid:105) + i | p − (cid:105) ) ⊗ e i σ y Θ( γ ) ( |↑(cid:105) + |↓(cid:105) ) ie iφ √ i | p + (cid:105) + | p − (cid:105) ) ⊗ e i σ y Θ( κ ) ( |↑(cid:105) + |↓(cid:105) ) . (35)Tracing out the spin states in equation (35) gives thedetection probabilities P ± = 12 ∓
12 cos (cid:18) Θ( κ ) − Θ( γ )2 (cid:19) cos φ. (36) When the controllable phase shift φ is varied, the proba-bilities P ± oscillate with amplitude V , which defines thevisibility of the interference pattern: V := (cid:12)(cid:12)(cid:12)(cid:12) max φ P ± − min φ P ± max φ P ± + min φ P ± (cid:12)(cid:12)(cid:12)(cid:12) , (37)where we included the absolute value because the cosineterm of the Wigner angles can be positive or negative.Without the entanglement between the internal degreeof freedom and the momentum, the expected visibility isalways maximal, i.e., V = 1 . Whereas, in the case of Eq.(36) it reads [6] V = |(cid:104) τ γ | τ κ (cid:105)| = (cid:12)(cid:12)(cid:12) (cid:104) τ i | e i σ y (Θ( κ ) − Θ( γ )) | τ i (cid:105) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) cos (cid:18) Θ( κ ) − Θ( γ )2 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , (38)where | τ η (cid:105) is the spin-state in the path η before BS . Changing the integration variable of Θ( η ) := (cid:82) τ ϑ ( η ) dτ (cid:48) to the time coordinate of the laboratoryframe, it follows that Θ( κ ) − Θ( γ ) = (cid:90) τ ( ϑ ( κ ) − ϑ ( γ )) dτ (cid:48) (39) = (cid:90) ∆ T ( ϑ ( κ ) − ϑ ( γ )) dτ (cid:48) dt dt (40) = α (cid:32) √ gx − (cid:112) g ( x + h ) (cid:33) ∆ T, where α = gp z (cid:112) p + m /m and ∆ T is the time mea-sured in the laboratory frame, for which the particle trav-els along its world line throughout the interferometer ina superposition of two trajectories at constant heights[6]. Using the approximation (1 + x ) − / ≈ − x/ , thevisibility of the interference pattern is given by V = cos (cid:18) α ∆ V ∆ T (cid:19) , (41)where ∆ V = gh is the difference in the gravitationalpotential between the paths. The introduction of theinternal degree of freedom and its entanglement withthe momentum due to the fact that the Wigner rota-tion depends on the momentum of the particle results inthe change of the interferometric visibility. In this case,the difference of the Wigner rotation for each path canbe attributed to the difference between the proper timeelapsed in each path, as already showed in [6] using adifferent route. However, it’s worth pointing out that wederived the same result using the unitary representationof the local Lorentz transformation, and it’s noteworthythe similarities between Eq. (41) and the Eq. (13) of[6]. Besides, in this case, we can defined E = 1 − V asa measure of entanglement between the momentum andspin degrees of freedom such that the visibility and en-tanglement are complementary quantities, as we can see T ( s ) ES vn Figure 2: Interferometric visibility, V , linear momentum-spin entanglement, E , and the spin von Neumann entropy, S vn ( ρ s ) , as a function of the time ∆ T of the laboratory framefor α ∆ V = 1 s − . in Fig. 2. Also, we plotted the von Neumann entropy ofthe spin reduced state, S vn ( ρ s ) , for comparison. Besides,as pointed out in [9], a clock (evolving internal degree offreedom) with a finite dimensional Hilbert space has aperiodic time evolution and thus it’s expected that thevisibility oscillates periodically as a function of the dif-ference of the proper times elapsed in the two paths. Aswe’ll see, this behavior persists when one studies the in-terferometric visibility in different spacetime geometries.However, the oscillation is not necessarily due to the dif-ference between the proper time elapsed in each path. B. Interferometric Visibility in the SchwarzschildSpacetime
In this section, we’ll study the behavior of the visi-bility of a spin- / quanton which is in motion in theSchwarzschild spacetime. In the physical scenario ex-plored in this section, as we’ll see, the effect of the grav-itational field in the interferometric visibility is due togeodetic precession, instead of time dilation.The Schwarzschild solution describes the spacetimeoutside of a static and spherically symmetric body ofmass M, which constitutes a vacuum solution. Becauseof its symmetries, the Schwarzschild metric describes astatic and spherically symmetric gravitational field [35].In the spherical coordinates system ( t, r, θ, φ ) , the lineelement of the Schwarzschild metric is given by ds = g µν ( x ) dx µ dx ν (42) = − f ( r ) dt + f − ( r ) dr + r ( dθ + sin θdφ ) , where f ( r ) = 1 − r s /r , with r s = 2 GM being theSchwarzschild radius. It’s straightforward to observe thatthe metric diverges in two distinct points, at r = r s and BS1rBS2D+ D-+ φ - φϒ κ γ Figure 3: Spin- / particle in a ’astronomical’ Mach-Zehnderinterferometer. The setup consists of two beam splitters BS and BS , a phase shifter, which gives a controllable phase Υ to the path κ , and two detectors D ± . The paths consists ofa clock and counterclockwise circular geodesic centered in astatic and spherically symmetric body of mass M. at r = 0 . However, it is important to distinguish the dif-ferent nature of both singularities. It’s well known thatthe singularity at r = r s is not an intrinsic singularity,since it can be shown that all curvature scalars are fi-nite at r = r s , while r = 0 is an intrisic singularity thatcannot be removed by changing the coordinate system[23]. To make the Schwarzschild metric reduce to theMinkowski metric, it is possible to choose the followingtetrad field e t ( x ) = (cid:112) f ( r ) , e r ( x ) = 1 (cid:112) f ( r ) ,e θ ( x ) = r, e φ ( x ) = r sin θ, (43)and all the other components are zero. The inverse ofthese elements are given by e t ( x ) = 1 (cid:112) f ( r ) , e r ( x ) = (cid:112) f ( r ) ,e θ ( x ) = 1 r , e φ ( x ) = 1 r sin θ . (44)This vierbein represents a static local frame at each point.Therefore it can used to represent an observer in the as-sociated local frame [20]. In addition, at each point, the − , − , − , and − axes are parallel to the t, r, θ, and φ directions, respectively.Now, let’s consider the case of a free-falling test spin- / quanton moving around the source of the gravita-tional field in a superposition of a clock and counter-clockwise geodetic circular orbit, which plays the roleof the paths of a Mach-Zehnder interferometer. Thefour-velocity of these circular geodesics in the equatorialplane, θ = π/ , are given by: u t = Kf ( r ) , u r = 0 , (45) u θ = 0 , u φ = Jr , (46)where K, J are integration constants related to the en-ergy and angular momentum of the required orbit, re-spectively, and are given by K = 1 − r s /r (cid:113) − r s r , J = 12 rr s − r s r . (47)The energy of the spin- / quanton of rest mass m in acircular orbit of radius r is then given by E = Km . Fur-thermore, the value of J implies that the angular velocityis given by u φ = ± (cid:114) r s r (1 − r s r ) , (48)which means that stable circular geodesic orbits are onlypossible when r > r s . The non-zero infinitesimalLorentz transformations in the local frame defined by thetetrad field are given by [26] λ = λ = − Kr s r f ( r ) , (49) λ = − λ = J (cid:112) f ( r ) r , (50)which corresponds to a boost in the direction of the 1-axes and a rotation over the 2-axis, respectively. While,the four-velocity in the local frame is found to be u a = e aµ ( x ) = (cid:32) K (cid:112) f ( r ) , , , Jr (cid:33) . (51)Therefore, the Wigner angle that corresponds to the ro-tation over the 2-axis is given by: ϑ ( x ) = J (cid:112) f ( r ) r (cid:32) − Kr s rf ( r ) 1 K + (cid:112) f ( r ) (cid:33) . (52)After the test particle has moved in the circular orbitacross some proper time τ , the total angle is given by Θ = (cid:90) ϑ ( x ) dτ = (cid:90) ϑ ( x ) dτdφ dφ (53) = ϑ ( x ) r J Φ , (54)since, for a circular orbit, r is fixed and ϑ ( x ) , K, and J are constants. The angle Φ is the angle traversed bythe particle during the proper time τ . It is noteworthythat the angle Θ reflects all the rotation suffered by thespin of the qubit as it moves in the circular orbit, which means that are two contributions: The “trivial rotation” Φ and the rotation due to gravity [20]. Therefore, to ob-tain the Wigner rotation angle that is produced solely byspacetime effects, it’s necessary to compensate the triv-ial rotation angle Φ , i.e., Ω := Θ − Φ is the total Wignerrotation of the spin exclusively due to the spacetime cur-vature, which only depends on the radius of the circulargeodesic r and the mass of the source of the gravitationalfield expressed by r s .In Fig. 3, we represent a spin- / quanton in a ‘as-tronomical’ Mach-Zehnder interferometer. The physicalscenario consists of two beam splitters BS and BS , aphase shifter, which gives a controllable phase Υ to thepath κ , and two detectors D ± . The paths consist of aclockwise and a counterclockwise circular geodesic cen-tered in a static and spherically symmetric body of massM. The initial state of the quanton, before BS1, is givenby | Ψ i (cid:105) = | p i (cid:105) ⊗ | τ i (cid:105) = √ | p i (cid:105) ⊗ ( |↑(cid:105) + |↓(cid:105) ) , with the localquantization of the spin axis along the 1-axis. Right afterBS1, the state is | Ψ (cid:105) = 12 (cid:16) | p γ ; 0 (cid:105) + i | p κ ; 0 (cid:105) ) ⊗ ( |↑(cid:105) + |↓(cid:105) ) , (55)where φ = 0 is the coordinate of the point wherethe quanton was putted in a coherent superposition inopposite directions with constant four-velocity u a ± =( K/ (cid:112) f ( r ) , , , ± J/r ) . After some proper time τ = r Φ /J , the particle travelled along its circular paths andthe spinor representation of the finite Wigner rotationdue only to gravitation effects is given by D ( W ( ± Φ)) = e ± i σ Ω . (56)Since ϑ ( x ) is constant along the path, the time-orderingoperator is not necessary. Therefore, the state of thequanton in the local frame at points φ = π before BS is given U (Λ) | Ψ (cid:105) = 12 | p γ ; π (cid:105) ⊗ e i σ y Ω ( |↑(cid:105) + |↓(cid:105) )+ ie i Υ | p κ ; π (cid:105) ⊗ e − i σ y Ω ( |↑(cid:105) + |↓(cid:105) ) . (57)The detection probabilities corresponding to equation(57) after BS2 is given by P ± = 12 (cid:16) ∓ cos Ω cos Υ (cid:17) , (58)such that the interferometric visibility is given by V = | cos Ω | . (59)In addition, V = C l ( ρ s ) , where C l ( ρ s ) is a measure ofquantum coherence [36] of the spin-state in the interfer-ometer obtained by tracing over the momentum states.It’s noteworthy that the proper time elapsed in bothpaths is the same, since dτ = r dφ/J . Therefore the r s / r E (a) V , E as a function of r s /r . r s r = 0 r s r = 0.1 r s r = 0.25 r s r = 0.5 r s r = 0.66 (b) The evolution of the visibility along the‘astronomical’ Mach-Zehnder interferometer fordifferent values of r , since τ ∝ Φ . Figure 4: Interferometric visibility and linear momentum-spinentanglement for the Schwarzschild spacetime. influence on the interferometric visibility is due to an-other general relativistic effect, i.e., the geodetic preces-sion [25]. Since the spin is assumed to be parallel trans-ported along the geodesic, it’s well known that, whenconsidering an arbitrary closed path, the spin will not re-turn to its initial state due to the curvature effects. Thechange of the spin state is associated with the spin pre-cession, once that magnitude of any vector is conservedin the geodesic. Here, the spin is in a superposition ofclock and counterclockwise circular path, and thereforethe spin precession of each path is given in opposite di-rections. Thus, the difference between the total Wignerrotation of the paths ends summing up, which affects theinterferometric visibility, given that the local Wigner ro-tation depends on the momentum, which in turn ends upcoupling with the spin.Once again, E = 1 − V can be used to measure theentanglement between the momentum and the spin, asone can see in Fig. 4(a) and 4(b). Specifically, in Fig.4(a), we plotted V , E as a function of r s /r , when theparticle reaches the detectors. Hence, for each value of r ∈ ( r s , ∞ ) , we have a specific value for V , E . While, inFig. 4(b), we plotted the ‘evolution’ of the visibility (orthe l -norm quantum coherence) along the interferometerfor different values of r , since τ ∝ Φ . For instance, when V , Υ = 0 , the global state is maximally entangled and isgiven by | Ψ (cid:105) = 1 √ (cid:16) | p γ , ↑(cid:105) + i | p κ , ↓(cid:105) (cid:17) , (60)which implies that the path information is accessible inthe internal degree of freedom. Besides, it’s possible toextend the orbits closer to r s by considering non-geodeticcircular orbits. In order for the particle to maintain suchnon-geodetic circular orbit, it’s necessary to apply anexternal radial force against gravity and the centrifugalforce, allowing the quanton to travel in the circular orbitwith the specific angular velocity at a given distance r from the source. In this case, as r → r s , Ω varies veryrapidly such that, at the event horizon, in the strong fieldlimit lim r → r s Ω = −∞ [20, 26]. This fact will cause thevisibility of the quanton to oscillate very rapidly near theSchwarzschild radius, given the choice of the tetrad field,and therefore the local frames. C. Interferometric Visibility in an ArbitraryCurved Spacetime
As noticed in [9], a ‘clock’ (in our case the quanton’sspin) with a finite dimensional Hilbert space has a pe-riodic time evolution which causes periodic losses andrebirths of the visibility with increasing time dilation be-tween the two arms of the interferometer. For a clock im-plemented in a two-level system (which evolves betweentwo mutually orthogonal states), the visibility will be acosine function. However, as showed in Sec. III B, theeffect on the interferometric visibility is due to anothergeneral relativistic effect, i.e., the geodetic precession. Inthis section, we show that by using the unitary represen-tation of the local Lorentz transformation, this effect onthe interferometric visibility due to spacetime effects isgeneral and have the same form, i.e., it’s a cosine func-tion, provided that we restrict the motion of the quantonto a spacial plane.As depicted in Fig. 5, let’s consider an arbitrary curvedspacetime M with metric g µν ( x ) , where a spin- / parti-cle in a superposition of paths γ and κ is located. Giventhat the motion of the quanton is restricted to a two-dimensional spacial plane, it’s possible to choose localframes (a tetrad field) such that Wigner rotation takesplace along a single direction. For instance, let’s con-sider the -axis, perpendicular to the plane. In addition,the quantization axis is along the or -axis located atthe plane of motion. Therefore, the corresponding -levelunitary representation of the Wigner rotation in the path η = γ, κ is given by D ( W ( η )) = e i σ y (cid:82) τ ϑ ( η ) dτ (cid:48) = e i σ y Θ( η ) , (61)for paths in which the time ordering operator is not nec-essary. Therefore, if the initial state of the system beforethe superposed paths is | Ψ i (cid:105) = √ | p i (cid:105) ⊗ ( |↑(cid:105) + |↓(cid:105) ) , the g μν γκ spinspin Figure 5: An arbitrary curved spacetime M with metric g µν ( x ) where a spin- / particle in a superposition of paths γ and κ is located. state of the quanton before the end of the superposedpath is given by | Ψ (cid:105) = 12 | p γ (cid:105) ⊗ e i σ y Θ( γ ) ( |↑(cid:105) + |↓(cid:105) )+ 12 | p κ (cid:105) ⊗ e i σ y Θ( κ ) ( |↑(cid:105) + |↓(cid:105) ) (62)and the interferometric visibility related to state (62),seen by a hovering observer located at point where theoverlapping paths meet, is given by V = (cid:12)(cid:12)(cid:12) (cid:104) τ ini | e i σ y (Θ( κ ) − Θ( γ ) | τ ini (cid:105) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) cos (cid:18) (Θ( κ ) − Θ( γ )2 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) cos (cid:32) (cid:82) τ ( ϑ ( κ ) − ϑ ( γ )) dτ (cid:48) (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (63)which has again the form of a cosine function for the vis-ibility, as suggested in [9]. However, the effect on thevisibility is not necessarily given by the time dilation be-tween the two superposed paths. As we showed before,it can be due to another general relativistic effect, for in-stance, the geodetic precession. Besides, it’s always pos-sible to reparametrize the integral in Eq.(63) in termsof another convenient coordinate variable. These resultscan be easily generalized to states with spread momen-tum (i.e., a wave packet). Therefore, the gravitationalfield (the spacetime geometry) entangles the momentumand spin of the quanton, such that the momentum statecan be, in principle, accessed via the internal degree offreedom, thus affecting the interferometric visibility. Atlast, if the superposed path is not restrict to a plane, butthe Wigner angle does not depend on the paths, we can see that V = (cid:12)(cid:12)(cid:12) (cid:104) τ ini | e i ϑ · σ ( (cid:82) κ − (cid:82) γ ) dτ | τ ini (cid:105) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) (cid:104) τ ini | e i ϑ · σ ∆ τ | τ ini (cid:105) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ (cid:88) n =0 ( i ∆ τ n (cid:104) ( ϑ · σ ) n (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (64)i.e., the influence in the visibility is fully described by themoments (cid:104) ( ϑ · σ ) n (cid:105) := (cid:104) τ ini | ( ϑ · σ ) n | τ ini (cid:105) . ExpandingEq.(64) up to second order in ∆ τ , it follows that V = (cid:114) − (cid:16) ∆ τ ∆( ϑ · σ )2 (cid:17) ≈ − (cid:16) ∆ τ ∆( ϑ · σ )2 (cid:17) , (65)which is similar to an expression already obtained in [9],where, in this case, ∆( ϑ · σ ) = (cid:10) ( ϑ · σ ) (cid:11) − (cid:104) ϑ · σ (cid:105) isvariance of ( ϑ · σ ) and not the variance of the internalHamiltonian of the system. From Eq. (65), we can seethat the initial decrease (or decoherence) of the visibilitydue do spacetime effects is a general phenomena. Finally,for the most general case, i.e., in situations wherein onecannot disregard the time ordering operator, it’s possi-ble to construct a Dyson series for D ( W ( η )) from Eq.(19). The benefit of the approach taken here, throughthe representation of the local Lorentz transformation,is that we do not need to know the (internal) Hamilto-nian of the system and how it couples to the gravitationfield, as in [6, 9]. In contrast, here what is needed is thecalculation of the Wigner rotation. IV. CONCLUSIONS
In this article, we extended the work of [6] for differentspacetime geometries. First, we derived the same resultof [6] using the unitary representation of the local Lorentztransformation in the Newtonian Limit. In addition, weshowed that the effect in the interferometric visibilitydue to gravity persists in different spacetime geometries.However the oscillation is not necessarily due to the dif-ference between the proper times elapsed in each path.For instance, in Schwarzschild spacetime, the influenceon the interferometric visibility is due to another generalrelativistic effect, i.e., geodetic precession. Besides, weshowed that by using the unitary representation of thelocal Lorentz transformation, this behavior of the inter-ferometric visibility in an arbitrary spacetime is general,provided that we restrict the motion of the quanton to atwo-dimensional spacial plane. However, we did not tookinto account the spin-curvature coupling, which is rele-vant in the case of supermassive compact objects and/orultra-relativistic test particles. Hence, our work helps inthe understanding of how the interferometric visibility ofa quantum system is affected due to general relativistic0effects as well it opens the possibility for different studies.For instance, it would be interesting to study the behav-ior of the interferometric visibility in the Kerr geometry,where one takes into account the frame dragging.
Acknowledgments
This work was supported by the Coordenação de Aper-feiçoamento de Pessoal de Nível Superior (CAPES), pro- cess 88882.427924/2019-01, and by the Instituto Nacionalde Ciência e Tecnologia de Informação Quântica (INCT-IQ), process 465469/2014-0. [1] S. Bose, A. Mazumdar, G. W. Morley, H. Ulbricht, M.Toro ˇ s , M. Paternostro, A. A. Geraci, P. F. Barker, M.S. Kim, and G. Milburn, Spin Entanglement Witness forQuantum Gravity, Phys. Rev. Lett. 119, 240401 (2017).[2] C. Marletto and V. Vedral, Gravitationally Induced En-tanglement between Two Massive Particles is SufficientEvidence of Quantum Effects in Gravity, Phys. Rev. Lett.119, 240402 (2017).[3] M. Christodoulou and C. Rovelli, On the possibility oflaboratory evidence for quantum superposition of geome-tries, Phys. Lett. B 792, 64 (2019).[4] R. Howl, R. Penrose, and I. Fuentes, Exploring the uni-fication of quantum theory and general relativity witha Bose-Einstein condensate, New J. Phys. 21, 043047(2019).[5] S. Wajima, M. Kasai, and T. Futamase, Post-Newtonianeffects of gravity on quantum interferometry, Phys. Rev.D 55, 1964 (1997).[6] M. Zych, F. Costa, I. Pikovski, and ˇ C . Brukner, Quan-tum interferometric visibility as a witness of general rel-ativistic proper time, Nat Commun. 2, 505 (2011).[7] M. Zych, F. Costa, I. Pikovski, T. C. Ralph, and ˇ C .Brukner, General relativistic effects in quantum inter-ference of photons, Class. Quantum Grav. 29, 224010(2012).[8] A. Brodutch, A. Gilchrist, T. Guff, A. R. H. Smith,and D. R. Terno, Post-Newtonian gravitational effects inquantum interferometry, Phys. Rev. D 91, 064041 (2015).[9] M. Zych, I. Pikovski, F. Costa, and ˇ C . Brukner, Generalrelativistic effects in quantum interference of "clocks", J.Phys.: Conf. Ser. 723, 012044 (2016).[10] A. Peres and D. R. Terno, Quantum Information andRelativity Theory, Rev. Mod. Phys. 76, 93 (2004).[11] R. M. Gingrich and C. Adami, Quantum entanglementof moving bodies, Phys. Rev. Lett. 89, 270402 (2002).[12] H. Terashima and M. Ueda, Relativistic Einstein-Podolsky-Rosen correlation and Bell’s inequality, Int. J.Quantum Inf. 01, 93 (2003).[13] P. M. Alsing and G. J. Milburn, Lorentz invariance ofentanglement, Quantum Inf. Comput. 2, 487 (2002).[14] A. Peres, P. F. Scudo, and D. R. Terno, Quantum entropyand special relativity, Phys. Rev. Lett. 88, 230402 (2002).[15] V. Palge and J. Dunningham, Entanglement of two rela-tivistic particles with discrete momenta, Ann. Phys. 363,275 (2015).[16] V. Palge and J. Dunningham, Generation of maximallyentangled states with sub-luminal Lorentz boost, Phys.Rev. A 85, 042322 (2012). [17] P. M. Alsing and G. J. Milburn, Teleportation with a uni-formly accelerated partner, Phys. Rev. Lett. 91, 180404(2003).[18] I. Fuentes-Schuller and R. B. Mann, Alice falls into ablack hole: Entanglement in non-inertial frames, Phys.Rev. Lett. 95, 120404 (2005).[19] P. M. Alsing, I. Fuentes-Schuller, R. B. Mann, and T.E. Tessier, Entanglement of Dirac fields in non-inertialframes, Phys. Rev. A 74, 032326 (2006).[20] H. Terashima and M. Ueda, Einstein-Podolsky-Rosencorrelation in gravitational field, Phys. Rev. A 69, 032113(2004).[21] H. Terashima and M. Ueda, Spin decoherence by space-time curvature, J. Phys. A: Math. Gen. 38, 2029 (2005).[22] R. M. Wald, General Relativity , (University of ChicagoPress, Chicago, 1984).[23] S. Carroll,
Spacetime and Geometry: An Introduction toGeneral Relativity , (Addison-Wesley, Reading, 2004).[24] M. Nakahara,
Geometry, Topology and Physics , (Insti-tute of Physics Publishing, Bristol, 1990).[25] C. W. Misner, K. S. Thorne, and J. A. Wheeler,
Gravi-tation (WH Freeman, San Francisco, 1973).[26] M. Lanzagorta,
Quantum Information in GravitationalFields (Morgan & Claypool Publishers, California, 2014).[27] S. Weinberg,
The Quantum Theory of Fields I (Cam-bridge University Press, Cambridge, 1995).[28] E. P. Wigner, On Unitary Representations of the Inho-mogeneous Lorentz Group, Ann. Math. 40, 149 (1939)[29] Y. Ohnuki,
Unitary Representations of the Poincarégroup and Relativistic Wave Equations (World Scientific,Singapore, 1988).[30] S. Chadrasekhar,
The Mathematical Theory of BlackHoles , (Oxford University Press, New York, 1983).[31] M. Lanzagorta, M. Salgado, Detection of gravitationalframe dragging using orbiting qubits, Class. QuantumGrav. 33, 105013 (2016).[32] P. M. Alsing, G. J. Stephenson Jr., and P. Kilian,Spin-induced non-geodesic motion, gyroscopic preces-sion, Wigner rotation and EPR correlations of massivespin 1/2 particles in a gravitational field, arXiv:0902.1396[quant-ph] (2009).[33] A. Papapetrou, Spinning test-particles in general relativ-ity. I, Proc. R. Soc. London A: Math. Phys. Sci. 209 248(1951).[34] Y. Dai, Y. Shi, Kinetic Spin Decoherence in a Gravita-tional Field, Int. J. Mod. Phys. D 28, 1950104 (2019).[35] M. P. Hobson, G. Efstathiou, A. N. Lasenby,
General Rel-ativity: An Introduction for Physicists (Cambridge Uni-1