The effect of stationary axisymmetric spacetimes in interferometric visibility Marcos L. W. Basso1, ∗ and Jonas Maziero1, y 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 yElectronic 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 f@µg and fdx g, 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 f@µg ⊂ Tp(M) and fdx g ⊂ 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 2 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 fea (x); a = 0; 1; 2; 3g 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.