The Effect of Stationary Axisymmetric Spacetimes in Interferometric Visibility

The eﬀect of stationary axisymmetric spacetimes in interferometric visibility

Marcos L. W. Basso1, ∗ and Jonas Maziero1, † 1Departamento de Física, Centro de Ciências Naturais e Exatas, Universidade Federal de Santa Maria, Avenida Roraima 1000, Santa Maria, Rio Grande do Sul, 97105-900, Brazil In this article, we consider a scenario in which a spin-1/2 quanton goes through a superposition of co-rotating and counter-rotating circular paths, which play the role of the paths of a Mach-Zehnder interferometer in a stationary and axisymmetric spacetime. Since the spin of the particle plays the role of a quantum clock, as the quanton moves in a superposed path it gets entangled with the momentum (or the path), and this will cause the interferometric visibility (or the internal quantum coherence) to drop, since, in stationary axisymmetric spacetimes there is a diﬀerence in proper time elapsed along the two trajectories. However, as we show here, the proper time of each path will couple to the corresponding local Wigner rotation, and the eﬀect in the spin of the superposed particle will be a combination of both. Besides, we discuss a general framework to study the local Wigner rotations of spin-1/2 particles in general stationary axisymmetric spacetimes for circular orbits. Keywords: Interferometric Visibility; Stationary and Axisymmetric Spacetime; Local Wigner rotation

I. INTRODUCTION Wouthuysen Hamiltonian. The analogy consists in the realization that the difference of the periods of oscillation of two states with two different total angular momentum The effects of rotation in space-time are the sources of quantum numbers has an analog form of the classical important and stimulating research as well for the scien- clock effect found in general relativity. However, it is tific debates which have permeated the field of physics worth pointing out that the Schwarzschild metric repre- in the last centuries. In the beginning of the 19th cen- sents a static and spherically symmetric spacetime, while tury, Sagnac predicted and experimentally verified that the GCE requires a stationary and axisymmetric space- there exists a shift of the interference pattern when an time. Therefore, in this work, we take another approach interferometric apparatus is rotating, compared to what to study the effect of stationary and axisymmetric space- is observed when the device is at rest [1,2]. General times in spin-1/2 particles. relativity predicts that two freely counter revolving test Zych et al. [16] considered a Mach-Zehnder interfer- particles in the exterior field of a central rotating body ometer placed in a gravitational potential, with a ‘clock’ take different periods of time to complete the same or- used as an interfering particle, i.e., an evolving internal bit. This time difference leads to the gravitomagnetic degree of freedom of a particle. Due to the difference in clock effect (GCE) [3]. In fact, the influence of the an- proper time elapsed along the two trajectories, the ‘clock’ gular momentum of the field also appears in the form of evolves to different quantum states for each path of the frame dragging of local inertial frames, i.e., the Lense- interferometer. In this article, we present an analogue Thirring effect [4], which was experimentally verified in scenario in which a spin-1/2 quanton goes through a su- Refs. [5,6]. A time difference like in the gravitomagnetic perposition of co-rotating and counter-rotating geodetic clock effect can be easily converted into a phase differ- circular paths, which play the role of the paths of a Mach- ence due to the difference of proper time elapsed in each Zehnder interferometer in a stationary and axisymmetric path. Besides, it is the same mechanism which produces, spacetime, since there exists a difference in proper time in special relativity, the Sagnac effect. However, when elapsed along the two trajectories. Here the spin of the arXiv:2102.09623v2 [gr-qc] 21 Jul 2021 a gravitational field is present the situation is essentially particle plays the role of a quantum clock, and as the the same, except for corrections produced by curvature quanton [17] moves in a superposed path, it gets entan- and angular momentum of the rotating body [7]. Until gled with the momentum (or the path), and this will now, several approaches has been taken to derive and to cause the interferometric visibility (or the internal quan- propose experiments to verify the gravitomagnetic clock tum coherence) to drop, since, due to the wave-particle effect [8–15]. duality, the interferometric visibility will decrease by an In particular, Faruque [14] found a quantum analogue amount given by the which-way information accessible of the classical gravitomagnetic clock effect, by present- from the final state of the clock, which gets entangled ing an approximation to the solution of the Dirac equa- with the external degree of freedom of the particle. How- tion in the Schwarzschild field through the use of Foldy- ever, as we will show, the proper time of each path will couple to the corresponding local Wigner rotation, and the effect in the spin of the superposed particle will be a combination of both. Thus, our work follows the line ∗Electronic address: [email protected] of research towards probing general relativistic effects in †Electronic address: [email protected] quantum phenomena [16, 18–20]. It is worth mentioning 2

∗ ν that the motion of spinning particles, either classical or cotangent space Tp (M), given by {∂µ} and {dx }, re- ν ν ν quantum, does not follow geodesics because the spin and spectively, such that dx (∂µ) := ∂µx = δµ . Therefore, µ ν curvature couple in a non-trivial manner [21]. However, the metric can be expressed as g = gµν (x)dx ⊗ dx , and the corrections, which are of order O(~), for quantum the elements of the metric, which encodes the gravita- particles, become important only when the motion of the tional field, are given by gµν (x) = g(∂µ, ∂ν ). Since the ν ∗ particle is ultra-relativistic or/and the source is a super- coordinate basis {∂µ} ⊂ Tp(M) and {dx } ⊂ Tp (M) are massive object [22]. In contrast, in this manuscript, we not necessarily orthonormal, it is always possible set up work only in slow-rotating and weak field approximation. any basis as we like. In particular, we can form an or- The effect of spin-curvature coupling on the interferomet- thonormal basis with respect to the pseudo-Riemannian ric visibility of a spin-1/2 quanton will be considered in manifold (spacetime) on which we are working. Following a future work. Ref. [26], let us consider the linear combination To accomplish this task, we use the method developed µ a a µ in Ref. [23] by Terashima and Ueda, who studied EPR ea = ea (x)∂µ, e = e µ(x)dx , (1) correlations and the violation of Bell’s inequalities in the a µ µ a ∂µ = e (x)ea, dx = e (x)e . (2) Schwarzschild spacetime, by considering a succession of µ a infinitesimal local Lorentz transformations and how the At each point p ∈ M, The Minkowski metric ηab = spin-1/2 representations of these local Lorentz transfor- diag(−1, 1, 1, 1) in the local frame and the spacetime met- mations (i.e. local Wigner rotations) affect the state of ric tensor gµν (x) are related by the tetrad field: the spin. Besides, we discuss a general framework to study the local Wigner rotations of spin-1/2 particles in µ ν gµν (x)ea (x)eb (x) = ηab, (3) general stationary axisymmetric spacetimes for circular a b orbits. The benefit of the approach taken here, through ηabe µ(x)e ν (x) = gµν (x), (4) the representation of the local Lorentz transformation, is that one does not need to know the (internal) Hamil- with tonian of the system and how it couples to the gravi- ea (x)e µ(x) = δa , ea (x)e ν (x) = δ ν . (5) tational field, as was the case in Refs. [13, 16, 20]. In µ b b µ a µ contrast, given an arbitrary spacetime metric, we only Above and in what follows, Latin letters a, b, c, d, ··· refer need to calculate the local Wigner rotations, which is a to coordinates in the local frame; Greek indices µ, ν, ··· straightforward procedure. run over the four general-coordinate labels; and repeated The organization of this article is as follows. In Sec.II, indices are to be summed over. The components of the we review the spin-1/2 dynamics in curved spacetimes. vielbein transform objects from the general coordinate In Sec.III, we describe the local Wigner rotations of system xµ to the local frame xa, and vice versa. There- spin-1/2 particles for circular geodesics in general station- fore, it can be used to shift the dependence of space- ary axisymmetric spacetimes. In Sec.IV, we consider a time curvature of the tensor fields to the tetrad fields. spin-1/2 quanton in a superposition of co-rotating and In addition, Eq. (4) informs us that the tetrad field counter-rotating geodetic circular paths and show how encodes all the information about the spacetime cur- the interferometric visibility is affected by spacetime ef- vature hidden in the metric. Besides, the tetrad field fects. Thereafter, in Sec.V, we give our conclusions. µ {ea (x), a = 0, 1, 2, 3} is a set of four 4-vector fields, which transforms under local Lorentz transformations in the local system. The choice of the local frame is not II. QUBIT DYNAMICS IN CURVED unique, since the local frame remains local under the lo- SPACETIMES cal Lorentz transformations. Therefore, a tetrad representation of a particular metric is not uniquely defined, A. Spin States in Local Frames and different tetrad fields will provide the same metric tensor, as long as they are related by local Lorentz trans- Coordinate transformations in general relativity are formations [27]. described through the group GLR(4), which is the set By constructing the local Lorentz transformation, we of all real regular 4 × 4 matrices. However, it is well can define a particle with spin-1/2 in curved space- known that GLR(4) does not have a spinor representa- times as a particle whose one-particle states furnish the tion. Therefore, to properly treat spins in the context spin-1/2 representation of the local Lorentz transforma- of curved spacetimes, the use of local frames of refer- tion, and not as a state of the diffeomorphism group ence is required. These frames are defined through an [23]. Therefore, if pµ(x) = muµ(x) represents the four- µ 2 orthonormal basis or tetrad field (or vielbein), which is momentum of such particle, with p (x)pµ(x) = −m , in a set of four linearly independent orthonormal 4-vector the general reference frame, then the momentum related a a µ fields defined at each point of spacetime [24]. The differ- to the local frame is given by p (x) = e µ(x)p (x), where ential structure of the spacetime, which is a differential m is the mass of the quanton, uµ(x) is the four-velocity manifold M [25], provides, in each point p, a coordinate in the general coordinate system, and we already putted basis for the tangent space Tp(M), as well as for the c = 1. Thus, in the local frame at point p ∈ M with 3

a a µ µ coordinates x = e µ(x)x , a momentum eigenstate of a where it was used the normalization condition for p (x) µ Dirac particle in a curved spacetime is given by [22] and the fact that p (x)aµ(x) = 0. Besides, ∇ν is the covariant derivative and aµ(x) := uν (x)∇ uµ(x) is the |pa(x), σ; xi := pa(x), σ; xa, ea (x), g (x) , (6) ν µ µν acceleration due to a non-gravitational force. Meanwhile, and represents the state with spin σ and momentum the variation of the tetrad field is due to spacetime ge- a a a µ p (x) as observed from the position x = e µ(x)x of ometry effects and it is given by the local frame defined by ea (x) in the spacetime M µ a ν a with metric gµν (x). It’s noteworthy that the description δe µ(x) = u (x)∇ν e µ(x)dτ of a Dirac particle state can only be provided regarding ν a b = −u (x)ων b(x)e µ(x)dτ, (11) the tetrad field and the local structure that it describes. By definition, the state |pa(x), σ; xi transforms as the a a λ λ a where ων b := e λ∇ν eb = −eb ∇ν e λ is the connection spin-1/2 representation under the local Lorentz transfor- 1-form (or spin connection) [31]. Collecting these results mation. In special relativity, a one-particle spin-1/2 state and substituting in Eq. (9), we obtain a a |p , σi transforms under a Lorentz transformation Λb as a a b [28] δp (x) = λ b(x)p (x)dτ, (12) a X a U(Λ) |p , σi = Dλσ(W (Λ, p)) |Λp , λi , (7) where λ 1 λa (x) = − (aa(x)p (x) − pb(x)a (x)) + χa where Dλ,σ(W (Λ, p)) is a unitary representation of the b m b a b Wigner’s little group, whose elements are Wigner rota- a a a a = −(a (x)ub(x) − u (x)ab(x)) + χ b (13) tions Wb (Λ, p) [29]. The subscripts can be suppressed and one can write U(Λ) |pa, σi = |Λpai⊗D(W (Λ, p)) |σi , a ν a with χ b := −u (x)ων b(x). The Eqs. (12) and (13) as sometimes we will do. In other words, under a Lorentz constitute an infinitesimal local Lorentz transformation a a transformation Λ, the momenta p goes to Λp , and the since, as the particle moves in spacetime during an in- spin transforms under the representation Dλσ(Λ, p) of the finitesimal proper time interval dτ, the momentum in Wigner’s little group [30]. The spin of the particle is then a a b the local frame transforms as p (x) = Λ b(x)p (x) where rotated by an amount that depends on the momentum a a a Λ b(x) = δ b + λ b(x)dτ [22]. In a similar manner, a spin of the particle. In contrast, in general relativity, single- state transforms via a representation of the local Lorentz particle states now form a local representation of the in- transformation. In other words, by using a unitary rep- homogeneous Lorentz group at each point p ∈ M, thus resentation of the local Lorentz transformation, the state we have |pa(x), σ; xi is now described as U(Λ(x)) |pa(x), σ; xi in a X a the local frame at the point x0µ, and Eq. (8) expresses U(Λ(x)) |p (x), σ; xi = Dλσ(W (x)) |Λp (x), λ; xi , λ how the state of the spin changes locally as the parti- (8) cle moves from xµ → x0µ along its world line. There- where W (x) := W (Λ(x), p(x)) is a local Wigner rotation. fore, one can see that spacetime tells quantum states how to evolve. For the infinitesimal Lorentz transformation, the infinitesimal Wigner rotation is given by a a a 0 i 0 B. Spin Dynamics W b(x) = δ b+ϑ bdτ, where ϑ 0(x) = ϑ 0(x) = ϑ i(x) = 0 with

Following Terashima and Ueda [23], different local i i i i λ 0(x)pj(x) − λj0(x)p (x) frames can be defined at each point of the curved space- ϑ j(x) = λ j(x) + 0 , (14) time. In the local frame at point p with coordinates p (x) + m a a µ x = e µ(x)x , the momentum of the particle is given by a a µ whereas all other terms vanish. In Ref. [32], the authors p (x) = e µ(x)p (x). After an infinitesimal proper time provided an explicit calculation of these elements. Hence, dτ, during which the quanton moves to the new point 0µ µ µ the two-spinor representation of the infinitesimal Wigner x = x + u dτ along its world line, the momentum of rotation is then given by the particle in the local frame at the new point becomes pa(x0) = pa(x) + δpa(x). Such infinitesimal change can 3 i X be decomposed as D(W (x)) = I + ϑ (x)σ dτ 2×2 4 ijk ij k δpa(x) = ea (x)δpµ(x) + δea (x)pµ(x). (9) i,j,k=1 µ µ i µ = I + ϑ · σdτ, (15) The variation δp (x) in the first term on the right hand 2×2 2 side of the last equation is due to an external non- 3 gravitational force where I2×2 is the identity matrix, {σk}k=1 are the Pauli µ ν µ µ matrices, and ijk is the Levi-Civita symbol. Moreover, δp (x) = u (x)∇ν p (x)dτ = ma (x)dτ (10) the Wigner rotation for a quanton that moves over a 1 = − (aµ(x)p (x) − pµ(x)a (x))pν (x)dτ, finite proper time interval can be obtained by iterating m ν ν the expression for the infinitesimal Wigner rotation [23], 4 and the spin-1/2 representation for a finite proper time where ω = −gtφ/gφφ. The form of the metric given by can be obtained by iterating the Eq. (15): Eq. (18) is useful to define world lines of locally nonrotating observers which carries an orthonormal frame i R τ ϑ·σdτ 0 D(W (x, τ)) = T e 2 0 , (16) with himself, i.e., it is possible to define the following where T is the time-ordering operator [23], since, in gen- tetrad fields q eral, the Wigner rotation varies at different points along 0 2 1 √ 2 √ e = −(gtt − ω gφφ), e = grr, e = gθθ, the trajectory. t r θ 3 √ 3 √ e t = − gφφω, e φ = gφφ, (19) with all the other components being zero. Also, only the III. WIGNER ROTATION FOR CIRCULAR GEODESICS IN STATIONARY AXISYMMETRIC nonzero components will be shown from now on. The SPACETIMES inverse of these elements are given by q q t 2 φ 2 e = 1/ −(gtt − ω gφφ), e = ω/ −(gtt − ω gφφ), In this section, we will describe the local Wigner rota- 0 0 r √ θ √ φ √ tions of spin-1/2 particles in general stationary axisym- e1 = 1/ grr, e2 = 1/ gθθ, e3 = 1/ gφφ, (20) metric spacetimes for circular geodesics. Perhaps, the with uµ = e µ being the four-velocity of the local non- most important of such type of spacetime is the Kerr 0 rotating observer in the coordinate frame. It is important metric [33, 34], which represents an exact solution of a to realize that this observer is rotating in respect to the stationary axisymmetric spacetime produced by a rotat- coordinate frame. This type of observer is called ZAMO ing body of mass M and angular momentum J . How- observer [36]. Meanwhile, the other useful form of the ever, a rotating frame in Minkowski spacetime will also metric is given by describe the spacetime as a stationary and axisymmetric 2 2 2 2 one, since the rotation axis defines a special direction. ds =gtt(dt − αdφ) + grrdr + gθθdθ Once again, it is worth pointing out that the motion of 2 2 + (gφφ − α gtt)dφ , (21) spinning particles, either classical or quantum, does not follow geodesics because the spin and curvature couples from which it is possible to define ‘hovering observers’ in a non-trivial manner [21]. However, the corrections, [37] from the tetrad fields √ √ which are of order O( ), become important only when 0 0 1 √ ~ e t = −gtt, e φ = − −gttα, e r = grr, the motion of the particle is ultra-relativistic or/and the q 2 √ 3 2 source is a supermassive object [22]. Therefore, this ef- e θ = gθθ, e φ = gφφ − α gtt, (22) fect can be safely ignored in the regime explored in this article. Finally, the description of local Wigner rotation while the inverse non-null components are given by: √ for circular non-geodetic orbits is easily extended within t r √ θ √ e0 = 1/ −gtt, e1 = 1/ grr, e2 = gθθ, (23) the framework developed in the Sec.II, as argued in the q q t 2 φ 2 end of this section. e3 = α/ gφφ − α gtt, e3 = 1/ gφφ − α gtt, A stationary spacetime is characterized by the fact that µ there exists a one-parameter group of isometries whose where α = −gtφ/gtt. Since we can define e0 as the orbits are timelike curves, which implies the existence of four-velocity of the hovering observer, one can see that a timelike Killing vector field. Analogously, a spacetime such observer stands still with respect to the coordinate is axisymmetric if there exists a one-parameter group of frame. It is worth mentioning that both local observers isometries whose orbits are closed spacelike curves, which are not inertial, as one can see by calculating the four- implies the existence of a spacelike Killing vector field acceleration [27]. whose integral curves are closed. Thus, a spacetime is Since we shall deal only with circular geodesic orbits stationary and axisymmetric if it possesses both of these for massive particles, let us restrict ourselves to the equa- symmetries and the actions of both of these groups com- torial plane characterized by θ = π/2. In this case, the mute, i.e., the rotations commute with the time trans- non-vanishing metric components are functions only of lations [24]. Therefore, we can choose a coordinate sys- r, i.e., gµν = gµν (r) and dθ = 0. Because of the time tem with x0 = t and x3 = φ such that the metric is independence and axial symmetry of the metric, the en- independent of them. The general form of a stationary ergy per unit mass K and the angular momentum per unit mass J about the axis of symmetry are constants axisymmetric metric is given by [35] 1 µ ν of the motion. From the Lagrangian L = 2 gµν x˙ x˙ and 2 2 2 2 2 µ ds = gttdt +2gtφdtdφ+grrdr +gθθdθ +gφφdφ , (17) the normalization condition u uµ = −1, the equations of motion for a massive particle confined to the equatorial with g = g (r, θ) and g < 0. Besides, the metric µν µν tt plane are above can be rewritten in two different ways, with one of them being t˙ = −gtt(K − ωJ), (24) 2 2 2 2 2 ˙ −1 tt ds =(gtt − ω gφφ)dt + grrdr + gθθdθ φ = gφφ J − ωg (K − ωJ), (25) 2 2 tt 2 φφ 2 tt 2 + gφφ(dφ − ωdt) , (18) grrr˙ = −g (K − ωJ) − (g − ω g )J − 1, (26) 5

µ µ µ α 1 ασ where x˙ = dx /dτ = u . By defining Q(r, K, J) = where Γµν = 2 g (∂µgσν +∂ν gµσ −∂σgµν ) are Christoffel 2 0 1 1 3 grrr˙ , the circular geodesics of radius rc are characterized symbols and λ 1 = λ 0, λ 3 = −λ 1. Thus, one can see by the following conditions that the local Lorentz transformation consists of a boost in the 1 or r-axis directions and rotations about the 2 Q(r, K, J)| = 0 and ∂ Q(r, K, J)| = 0. (27) r=rc r r=rc or θ-axis. The infinitesimal Wigner rotation associated For a given value of r, these are two simultaneous equa- with the rotation over the 2-axis is given by tions for the energy K and angular momentum J that 2 1 can be solved. In addition, ∂ Q(r, K, J)| = 0 gives λ u3 r r=rc ϑ1 = λ1 + 0 . (36) the innermost circular orbit (ISCO). Therefore, for each 3 3 u0 + 1 circular orbit we have two specific values of K and J, 1 1 which characterizes the co-rotating and counter-rotating The difference between ϑ 3 and λ 3 gives rise to a pre- orbits [38]. Since the values of K and J are determined, cession of the spin, which is the general relativistic ver- for each circular orbit, from the conditions above, t˙ and φ˙ sion of the Thomas precession [23]. In addition, it is 1 are also settled for each circular orbit such that the angu- noteworthy that ϑ 3 is only a function of r and of the lar velocity of the particle with respect to the coordinate properties of the rotating source. After the test particle frame is given by has moved in the circular orbit across some proper time τ, the total angle is given by φ˙ J Ω = = ω − , (28) ˙ tt φφ Z Z dτ dτ t g g (K − ωJ) Θ = ϑ1 dτ = ϑ1 dφ = ϑ1 Φ, (37) 3 3 dφ 3 dφ which also possesses two specific values for each orbit, corresponding to the co-rotating and counter-rotating since, for circular orbits, r is fixed and, therefore, motion. One can see this by considering the geodesic 1 ϑ 3, dτ/dφ are constant. The angle Φ is the angle tra- equation for r and imposing r˙ =r ¨ = 0 [39]. Thus versed by the particle during the proper time τ. For p 2 instance, in Ref. [37] the local Wigner rotations in the −∂rgtφ ± (∂rgtφ) − ∂rgtt∂rgφφ Ω = , (29) Kerr spacetime geometry were studied. Besides, the same ± ∂ g r φφ analysis can be done if one chooses the ZAMO observers. where the +(−) refers to the co-rotating (counter- It is noteworthy that the angle Θ reflects all the ro- rotating) circular orbits. Besides, it is important to note tations suffered by the spin of the qubit as it moves in that Ω± is only a function of r and the properties of the circular orbit, which means that there are two con- the rotating source. Since a circular orbit also implies tributions: The “trivial rotation” Φ and the rotation due that φ˙ = Ωt˙, solving for t˙ in the normalization condition to gravity [23]. Therefore, to obtain the Wigner rota- µ u uµ = −1 with r˙ = 0, and substituting in Eqs. (24) tion angle that is produced solely by spacetime effects, and (25) gives it is necessary to compensate for the trivial rotation angle Φ, i.e., ϕ := Θ − Φ is the total Wigner rotation of −gtt − gtφΩ± K± = , (30) the spin exclusively due to the spacetime curvature. Be- q 2 1 −gtt − 2gtφΩ± − gφφΩ± sides, the local Wigner rotation, ϑ 3, as well as the proper time along the geodesic will depend if the particle is co- gtφ + gφφΩ± J± = . (31) rotating or counter-rotating with respect to the source. q 2 −gtt − 2gtφΩ± − gφφΩ± By denoting τ+ the proper time of the particle along the co-rotating geodesic and τ− the proper time of the parti- The four-velocity in both local frames is given by cle along the counter-rotating geodesic, and analogously 0 0 µ 0 t 0 φ for the azimuthal angle φ, we have u = e µu = e tu + e φu , (32) 3 3 µ 3 t 3 φ s u = e µu = e tu + e φu . (33) dτ± dt± 2 dt± = −gtt( ) − 2gtφ( ) − gφφ For instance, if we choose the tetrad field corresponding dφ± dφ± dφ± 3 to the hovering observer, then e t = 0. From Eq. (13), q 1 2 one can see that, for any type of geodesic motion, the = −gtt − 2gtφΩ± − gφφΩ , (38) Ω ± change of the tetrad field is equal to the local Lorentz ± transformations. In particular, for the hovering observer where Ω± = dφ±/dτ±. For one closed orbit, we obtain in circular geodesics, the non-zero local Lorentz transfor- the integrated proper time as mations are expressed by q 0 t r t 0 φ 0 φ r t 0 φ 0 2π 2 λ 1 = −u e1 Γtre t + Γtre φ − u e1 Γφre t + Γφre φ , τ± = ± −gtt − 2gtφΩ± − gφφΩ±, (39) Ω± (34)

1 t 1 r t r φ φ 1 r t r φ since τ+ 6= τ−, the difference τ+ − τ− gives rise to the λ 3 = −u e r Γtte3 + Γtφe3 − u e r Γφte3 + Γφφe3 , so called gravitomagnetic clock effect. The name comes (35) from the fact that, in the weak field approximation, the 6 gravitational field may be decomposed into a gravitoelec- than the variance of the Gaussian distributions. There- tric field and a gravitomagnetic field in analogy with elec- fore, both distributions are distinguishable and centered tromagnetism [42]. It is worth emphasize that this effect around different momentum values such that it is possible presupposes that the time difference of the two clocks to represent them by orthogonal state vectors. is taken with respect to a fixed angle φ, i.e., after each As well, as pointed out in Ref. [40], it is possible to clock has covered an azimuthal interval of 2π. However, assume that the quanton wave-packet has a mean cen- by taking into account the Wigner rotation, the proper troid that, in a semiclassical approach, one can regard time of each path dτ± will couple to the correspond- to describe the motion of the center of mass in each su- 1 ing infinitesimal Wigner rotation ϑ 3(±). For instance, perposed path. The momentum of the particle can be if we consider the quanton in a superposed path corre- assumed to be distributed properly around the momen- sponding to a co-rotating and counter-rotating circular tum of the centroid. Assuming that the centroid moves geodesic, then the effect of the total Wigner rotation along a specified path xµ(τ) in the curved spacetime. So, in the spin of the quanton will not be only due to the the four-momentum of the centroid as measured in a local difference of proper time elapsed in each path, but ac- a a µ frame will be p (x) = e µ(x)p (x). Therefore, the local tually a difference taking into account the infinitesimal observer, defined in each point of the circular geodesic 1 1 Wigner rotation: ϑ 3(+)τ+ − ϑ 3(−)τ−. As noticed in by the tetrad field, deals only with the centroid, i.e., the Ref. [20], a ‘clock’ (in our case the quanton’s spin) with mean value of the four-momentum which is connected to a finite dimensional Hilbert space has a periodic time the four-velocity of the corresponding circular geodesic, evolution which causes periodic losses and rebirths of the and with the centroid of the center of mass in the position visibility with increasing time dilation between the two basis which is also a Gaussian distribution with the mean arms of the interferometer. For a clock implemented in value corresponding to the coordinates xa of the circular a two-level system (which evolves between two mutually geodesic. Therefore, our calculation of the Wigner ro- orthogonal states), the visibility will be a cosine function. tation in this section is with regard to these two mean Therefore, for stationary and axisymmetric spacetimes, values, even though both wave-packets have some width. the argument of the cosine function will be given by the However, when considering the full wave-packet, we have 1 1 difference ϑ 3(+)τ+ − ϑ 3(−)τ−. to deal with congruence of geodesics, which is also not Finally, it is possible to extend this procedure for non- a problem, since the wave-packets of momenta and posi- geodetic circular orbits. However, in order for the par- tion are not distorted along the geodesic, and therefore ticle to maintain such non-geodetic circular orbit, it is the mean value of the position and the momenta remains necessary to apply an external radial force against grav- in the same geodesic (which is part of a geodesic congru- ity, allowing the quanton to travel in the circular orbit ence). with a specific angular velocity at a given distance r from 0 To show this, it is enough to work in the ~ order of the source. The analysis above would still be valid in this the WKB expansion for the four-spinor ψ(x) of the Dirac a a case, except for the fact that in this case χ 6= λ , as 0 b b equation in curved spacetime [22]. In order ~ , the Dirac one can see from Eq. (13). For instance, one can use this µ equation in curved spacetimes gives (γ (x)∂µS(x) + approach to study the Sagnac effect for spin-1/2 particles m)ψ0(x) = 0, which is a homogeneous system of four in circular orbits. algebraic equations and therefore have non-trivial solu- µ tions if and only if det[γ (x)∂µS(x) + m] = 0, implying 2 µ that m = −∂ S(x)∂µS(x). Thus, it is straightforward IV. INTERFEROMETRIC VISIBILITY IN AN to see that pµ = ∂µS(x). Therefore, at zero-th order in ~, EARTH-LIKE ENVIRONMENT it is possible to identify S(x) with the classical action for R µ a particle in a gravitational field: S(x) = pµdx , which In this section, we will study the behavior of interfer- also corresponds to the phase of a quantum mechanical ometric visibility under local Wigner rotations as a spin- particle in curved spacetime [22]. These equations define 1/2 quanton goes through a superposition of co-rotating a family of integral curves xµ(τ) of uµ which is the four- and counter-rotating geodetic circular paths, which play velocity vector field (which in our case will be circular the role of the paths of a Mach-Zehnder interferometer geodesics). Therefore, the computation of S(x) at this in stationary and axisymmetric spacetimes given by an approximation order can be accomplished by assuming Earth like environment, as depicted in Fig.1. To make that the Dirac particle is a classical particle moving with our investigation easier, we began by assuming that mo- momentum pµ in curved spacetime. The associated four- menta can be treated as discrete variables, as in Refs. velocity field uµ can be obtained by solving the geodesic [16, 23]. This can be justified once we can consider nar- equations for a classical test particle in the specific met- row distributions centered around different momentum ric under consideration, as we did in the previous section. values such that it is possible to represent them by or- However, as the extended state of the quantum particle thogonal state vectors. For instance, if we consider nar- takes on a path superposition, it should be understood row Gaussian distributions for each path, it is a reason- that uµ actually represents a geodesic congruence. able assumption to assume that both distributions do not Besides, it can be shown that the general zeroth order overlap since the separation between the paths is bigger solution to the Dirac equation in curved spacetime ψ0(x) 7 can be written as an envelope function times a normalized BS1 spinor ψ0(x) = f(x)φ0(x), where the envelope function f(x) (which expresses the form of the wave-packet) sat- 2 µ - +φ isfies the equation ∇µ(f (x)u ) = 0, which represents the propagation equation for the envelope function [41]. r If we assume that the divergence of the velocity field uµ vanishes (since we are dealing with geodesics), then

µ d u ∇µf(x) = f(x) = 0. (40) dτ ω

Thus, the shape of the envelope in the local frame re- - + mains unchanged during the evolution. However, of course, because of the uncertainty principle, if the wave packet has finite spatial extent it cannot simultaneously BS2 have a sharp momentum, and therefore the divergence in D+ D- velocity cannot be exactly zero. Nevertheless, we can re- µ lax our assumption since, for geodesic congruences, ∇µu Figure 1: A spin-1/2 particle in an ‘astronomical’ Mach- measures the rate of change of the local-frame volume [24] Zehnder interferometer. The setup consists of two beam split- µ and assume that the expectation value of ∇µu is small ters BS1 and BS2, a phase shifter, which gives a controllable enough: phase Υ to the counter-rotating path −, and two detectors D±. The paths consist of a co-rotating and a counter-rotating µ 1 circular geodesic centered in a stationary and axisymmetric h∇µu i << , (41) τΓ body of mass M and angular momentum J . where τΓ is the proper time along some geodesic Γ. Therefore, we can assume that our initial localized wave- and all the other components are zero. The inverse of packet is rigidly transported along the geodesic congru- these elements are given by ence with the mean values of position and momentum fol- −r a/rf(r) lowing the same geodesic [41]. This discussion is enough e t = 1/pf(r), e t = s (44) 0 3 p 2 2 2 2 to validate the following calculations. r + rs a /r f(r) Let us consider the situation in the surroundings of e r = pf(r), e φ = 1/pr2 + r2a2/r2f(r). (45) our planet (or the sun). The appropriate (approximate) 1 3 s metric is the one corresponding to a weak axisymmetric At each point, the 1− and 2−axis are parallel to the r field, and it is given by [42] and θ directions, respectively. In particular, we choose this tetrad field because the gravitomagnetic effect as- −1 2 gtt = −f(r), grr = f (r), gθθ ' r , (42) sumes that the time difference of the two clocks is taken r a g ' − s sin2 θ, g ' r2 sin2 θ, with respect to a fixed angle φ, and therefore hovering tφ r φφ observers are necessary in each point. Now, let us consider the case of a free-falling test where f(r) = 1 − rs/r, with rs = 2GM being the spin-1/2 particle moving around the source of the grav- Schwarzschild radius, and a = J /M, with J being the itational field in a superposition of a co-rotating and angular momentum and M the mass of the rotating body. counter-rotating geodetic circular orbit, which play the One can see that, for J = 0, the metric reduces to the role of the paths of a Mach-Zehnder interferometer, as Schwarzschild metric. Besides, we use the Schwarzschild- depicted in Fig.1. The four-velocity of these circular like coordinates (t, r, θ, φ) and assume that the angular geodesics in the equatorial plane are given by: momentum is orthogonal to the equatorial plane θ = π/2. The Earth-like body is assumed to be spherical and the 1 ut = , ur = 0, (46) low rotating approximation is given by the term contain- ± q −g − 2g Ω − g Ω2 ing J , which, for instance, in the terrestrial environment tt tφ ± φφ ± is six orders of magnitude smaller than the mass terms φ Ω± θ [42]. u± = q , u = 0, (47) −g − 2g Ω − g Ω2 By restricting ourselves to the equatorial plane, dθ = 0, tt tφ ± φφ ± the relevant components of tetrad field corresponding to where the angular velocity of the particle with respect to hovering observers are given by the coordinate frame is obtained directly from Eq. (29), 0 p 0 rsa 1 p and is given by e t = f(r), e φ = p , e r = 1/ f(r), r f(r) r 2 rsa rsa rs 3 p 2 2 2 2 Ω± = − ± + , (48) e φ = r + rs a /r f(r), (43) 2r3 2r3 2r3 8 where the plus sign refers to the co-rotating orbit, while 3 0 the minus sign refers to the counter-rotating path, since 1 2 1 the angular velocity of the source is given by ω = 3 3 2 −gtφ/gφφ = rsa/r . Ignoring the term gφφ of order Ω± 1 in Eq. (38), and integrating over φ± [43], we have

r 0 Φ± rs rsa τ± ≈ 1 − + 2 Ω± Ω± r r 1 Φ± rs rsa ≈ 1 − + Ω± , (49) 2 Ω± 2r r where Φ± = ±Φ, which implies that the diﬀerence of 3 proper time between the two paths is given by 6 8 10 12 14 16 18 20 r

Ω− + Ω+ rs 2rsa 0 1 τ+ − τ− = 1 − Φ + 2 Φ (a) λ 1, λ 3 as a function of r, for the parameters rs = 3 Ω+Ω− r r and a = 0.1, with r ≥ 1.4604rs. = 2Φa. (50)

0.14 For one complete revolution, τ+ − τ− = 4πa = 4πJ/M, which is a result independent from both the radius of the 0.12 orbit and the gravitational constant G [44]. In the terrestrial environment, this proper time diﬀerence is of order 0.10 −7 3 0.08

10 s [9, 11]. Besides, it is possible to derive the same 1 result from the Kerr metric without any approximation 0.06 (see Ref. [8]). For the hovering observer in circular geodesics, the 0.04 non-zero local Lorentz transformation are determined by 0.02 Eqs. (34) and (35) and, in this case, given by 0.00 3 2 6 8 10 12 14 16 18 20 t 1 φ rs a t φ 0 −rsf(r)(u + 2 au ) − r4 (u − au ) r λ 1 = 2 2 2 2 (51) 2(r f(r) + rs a /r ) 1 (b) ϑ 3 as a function of r, for the parameters rs = 3 and t 2 2 1 u r a r a a = 0.1, with r ≥ 1.4604rs. λ1 = s − uφ s − rf(r) , 3 p 2 2 2 2 2 3 r f(r) + rs a /r 2r 2r (52) Figure 2: The local Lorentz transformations and the local Wigner rotation for the co-rotating circular orbits as a func- which for a = 0 reduces to the local Lorentz transfor- tion of r with the parameters rs = 3 and a = 0.1 mation in the Schwarzschild spacetime [22]. Thus, the local Wigner rotation is determined by Eq. (36). In Figs. 2(a) and 2(b), we plotted the local Lorentz trans- BS2, a phase shifter, which gives a controllable phase formations and the local Wigner rotation, respectively, Υ to the path −, and two detectors D±. The paths for the co-rotating circular orbits as a function of r with consist of a co-rotating and a counter-rotating circular geodesic centered in a stationary and axisymmetric the parameters rs = 3 and a = 0.1 in some units of body of mass M and angular momentum J . The ini- distance. For instance, rs = 3 km is the Schwarzschild radius of the Sun. The stable circular geodesics can be tial state of the quanton, before the BS1, is given by |Ψ i = |p i ⊗ |τ i = √1 |p i ⊗ (|↑i + |↓i), with the local obtained by requiring that the argument of the square i i i 2 i root of Eqs. (46) and (47) must be greater than or equal quantization of the spin axis along the 1-axis. Right after to zero, which can be solved numerically for each value the BS1, the state is of rs and a. In this case, for the co-rotating circular or- 1 bits, the inner most stable geodesic is characterized by |Ψi = (|p ; 0i + i |p ; 0i) ⊗ (|↑i + |↓i), (53) 2 + − r ' 1.4604rs, which agrees with the fact that, for co- rotating circular orbits, the ISCO is closer to rs than where φ = 0 is the coordinate of the point where in the Schwarzschild case [38]. While for the counter- the quanton was putted in a coherent superposition in a rotating circular orbits, the inner most stable geodesic is opposite directions with constant four-velocity u± = 0 µ 3 µ characterized by r ' 1.53755rs. Besides, from Fig. 2(b), (e µu±, 0, 0, e µu±). Here, we assumed that the beam- one can see that for r → ∞, the local Wigner rotation splitters BS1 and BS2 do not aﬀect the spin degree of tends to zero. freedom. However, it is possible to consider the beam- In Fig.1, we represent a spin- 1/2 quanton in a ‘as- splitter as a Stern-Gerlach apparatus, such that the spin tronomical’ Mach-Zehnder interferometer. The phys- d.o.f will gets entangled with momentum degree of free- dτ ical scenario consists of two beam splitters BS1 and dom. After some proper time τ = dφ Φ, the particle 9

1.0000 Without the entanglement between the internal degree of freedom and the momentum, the expected visibility 0.9998 is always maximal, i.e., V = 1. The introduction of the internal degree of freedom and its entanglement with the 0.9996 momentum, due to the fact that the Wigner rotation depends on the momentum of the particle, results in a 0.9994 change in the interferometric visibility. In this case, the difference of the Wigner rotation for each path can be 0.9992 1 1 attributed to the difference ϑ (+)τ+ − ϑ (−)τ−, which r = 1000rs 3 3 in part is due to the gravitomagnetic clock effect. In Fig. 0.9990 r = 100rs r = 50rs 3, we plotted the ‘evolution’ of the visibility along the interferometer, since τ ∝ Φ, for different values of r with 0.0 0.5 1.0 1.5 2.0 2.5 3.0 rs = 3 and a = 0.1. We can see that for large r, the effect on the interferometric visibility is tiny, since the spacetime effect does not strongly couples the momentum with Figure 3: The evolution of the visibility (or internal quantum the spin, and the states of the spin for the two paths are coherence) along the ‘astronomical’ Mach-Zehnder interfer- almost indistinguishable. However, as the circular orbit ometer, since τ ∝ Φ, for different values of r with r = 3 and s gets closer to rs, the effect on the visibility gets more no- a = 0.1 ticeable, since the visibility can reach zero, which means that the which-way information is accessible in the final state of the clock (spin). Actually, as pointed out in Ref. travelled along its circular paths and the spinor repre- [20], a clock with a finite dimensional Hilbert space has a sentation of the finite Wigner rotation due only to grav- periodic time evolution, and thus it is expected that the itational effects is given by visibility oscillates periodically as a function of the differ- − i σ ϕ ence of the proper times elapsed in the two paths. This D(W (±Φ)) = e 2 2 ± , (54) will also happen here if we let the quanton revolving along the superposed orbit long enough. Besides, it is expected where ϕ± = Θ± − Φ±, with Φ± = ±Φ and Θ± = dτ± 1 that the same effect occurs in the Kerr’s spacetime. Fi- ϑ31(±) Φ±. Since ϑ 3(x) is constant along the path, dφ± nally, it’s worth mentioning that, if the beam splitters en- the time-ordering operator is not necessary. Therefore, tangles the momentum and spin, then the visibility after the state of the quanton in the local frame at point φ = π BS1, in principle, is zero because the which-path infor- before BS2 is given mation is encoded in the spin d.o.f. Therefore, the action of U(Λ) in this case will be to degrade the entanglement 1 − i σ ϕ U(Λ) |Ψi = |p ; πi ⊗ e 2 y + (|↑i + |↓i) 2 + between both degrees of freedom, and increase the quan- iΥ tum coherence of both degrees of freedom between the ie i − σy ϕ− + |p−; πi ⊗ e 2 (|↑i + |↓i), (55) (BS1) and (BS2), since, in this scenario, entanglement 2 and coherence are complementary quantities, as already which, in general, is an entangled state. Besides, we shown by us in [45]. choose to maintain the notation |p±i for the momentum states before BS2, because the velocity of the quanton is constant along the paths. The detection probabilities V. CONCLUSIONS corresponding to Eq. (55), after the BS2, are given by In this article, we presented a scenario in which a spin- 1 ϕ+ − ϕ− P± = 1 ∓ cos cos Υ . (56) 1/2 quanton goes through a superposition of co-rotating 2 2 and counter-rotating geodetic circular paths, which play When the controllable phase shift Υ is varied, the prob- the role of the paths of a Mach-Zehnder interferometer abilities P± oscillate with amplitude V which is called in- in stationary and axisymmetric spacetimes. Since the terferometric visibility, and can be calculated using [16]: spin of the particle plays the role of a quantum clock, as the quanton moves in a superposed path it gets en- − i σ (ϕ −ϕ ) 2 y + − tangled with the momentum (or the path), and this will V = hτi| e |τii (57) cause the interferometric visibility (or the internal quan- ϕ+ − ϕ− tum coherence) to drop, since, in stationary axisymmet- = cos , (58) 2 ric spacetimes, there is a difference in proper time elapsed along the two trajectories. However, as showed here, the where |τ i = √1 (|↑i + |↓i) is the initial state of the clock. i 2 proper time of each path will couple to the corresponding Besides, from the Eq. (57), one can see that the vis- local Wigner rotation, and the effect in spin of the super- ibility will depend only in the initial state before BS1 posed particle will be a combination of both. Besides, we and the evolution of this state between BS1 and BS2. discuss a general framework to study the local Wigner ro- 10 tations of spin-1/2 particles in general stationary axisym- through the local Wigner rotation of each path, then, metric spacetimes for circular orbits. In addition, we did since entanglement is monogamous [46], the initial spin- not took into account the spin-curvature coupling, which spin entanglement has to be redistributed around as spin- need to be considered when investigating spinning parti- momentum entanglement of each particle. cles in the case of supermassive compact objects and/or ultra-relativistic test particles. Finally, it is worth mentioning another interesting physical scenario. Instead of considering a superposed Acknowledgments particle in a co-rotating and a counter-rotating circular orbit, as we did here, it is possible to consider a source This work was supported by the Coordenação de Aper- that emits two spin-1/2 particles, maximally entangled feiçoamento de Pessoal de Nível Superior (CAPES), pro- on the spin states, that go in opposite circular geode- cess 88882.427924/2019-01, and by the Instituto Nacional tic motion. Since the momentum of each particle gets de Ciência e Tecnologia de Informação Quântica (INCT- entangled with the respective internal degree of freedom IQ), process 465469/2014-0.

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