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Revealing and Concealing Entanglement with Noninertial Motion

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Revealing and Concealing Entanglement with Noninertial Motion PHYSICAL REVIEW A 101, 043837 (2020) Revealing and concealing entanglement with noninertial motion Marko Toroš ,1,* Sara Restuccia,2 Graham M. Gibson,2 Marion Cromb ,2 Hendrik Ulbricht,3 Miles Padgett,2 and Daniele Faccio2,† 1Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom 2School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom 3Department of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, United Kingdom (Received 14 November 2019; revised manuscript received 24 March 2020; accepted 30 March 2020; published 27 April 2020) Photon interference and bunching are widely studied quantum effects that have also been proposed for high-precision measurements. Here, we construct a theoretical description of photon interferometry on rotating platforms, specifically exploring the relation between noninertial motion, relativity, and quantum mechanics. On the basis of this, we then propose an experiment where photon entanglement can be revealed or concealed solely by controlling the rotational motion of an interferometer, thus providing a route towards studies at the boundary between quantum mechanics and relativity. DOI: 10.1103/PhysRevA.101.043837 I. INTRODUCTION setup and the relevant classical effect, namely, the Sagnac effect (Sec. II). We then give the theoretical description of The notions of space and time are at the core of modern quantum experiments on a rotating platform by deriving the physics and remain an area of intense research [1,2]. A Hamiltonian on a generic Hilbert space (Sec. III). We apply striking example of how elementary notions of space and time the theoretical description to study photon-interferometry ex- lead to surprising consequences is the derivation of Lorentz periments and show that the developed model recovers the transformations, a cornerstone of quantum field theory, utiliz- results for the Sagnac effect in the quantum regime [16]as ing only basic assumptions [3,4]. well as for the recent demonstration of photon bunching in a The exploration of the special-relativistic regime is histor- rotating reference frame [17] (Sec. IV). Finally, we propose ically strongly linked to investigations of the propagation of a modified HOM interferometer which can be readily imple- light [5], e.g., the Michelson-Morley experiment [6]. More mented using currently available technology. In particular, we recent experiments have also started to probe the quantum discover that simply setting the apparatus in rotational motion nature of light, e.g., the Hong-Ou-Mandel (HOM) experiment can change bunching to antibunching and vice versa, i.e., one [7], indirectly testing the underpinning spacetime symmetries. can reveal and conceal entanglement (Sec. V). Quantum optical interference effects, either one photon or two The overall purpose of this work is thus threefold. The photon, are thus of fundamental importance [8], as well as first goal is to give a theoretical foundation for quantum providing paths to technological applications [9]. interferometry experiments on rotating platforms, both for A further test of special relativity, moving towards the matter-wave and photon interferometry. The second is to show domain and ideas of general relativity, is possible in situations that there are nontrivial consequences for the manifestation where linear acceleration or rotational motion is present [10]. of entanglement already at low rotation speeds which should A notable example is the classical Sagnac experiment, where be taken into account for state-of-the-art sensing. Finally, the an interferometer is placed on a rotating platform [11–13]. third is that photon interferometry can be used to test the More recent experiments include experimental tests of pho- validity of the underlying spacetime symmetries in noninertial tonic entanglement in accelerated reference frames [14], the and genuinely quantum mechanical experiments. demonstration of how to overcome the shot-noise limit using an entanglement-enhanced optical gyroscope [15], the quan- tum Sagnac interferometer [16], and the extension of HOM II. EXPERIMENTAL SETUP interference to rotating platforms [17]. We consider an experimental platform rotating at angular In this article, we theoretically explore noninertial experi- frequency depicted in Fig. 1(a). The system is confined to ments which require both a quantum mechanical as well as a move on a circle of radius r in the equatorial plane normal fully relativistic analysis. In particular, we explore the relation to the rotation axis. We further suppose there is a corotating between interference, entanglement, and noninertial rotational medium with refractive index n. For the corotating observer, motion. We first provide a brief overview of the conceptual the light propagation speed is c/n in both directions as can be deduced from symmetry considerations. It is also instructive to describe the same experiment from the inertial frame of the *[email protected] laboratory: One can formally map the circular trajectories to †[email protected] straight-line motions as shown in Fig. 1(b) [18]. In this latter 2469-9926/2020/101(4)/043837(8) 043837-1 ©2020 American Physical Society MARKO TOROŠ et al. PHYSICAL REVIEW A 101, 043837 (2020) D2 expression of the Sagnac delay, we set L = 2πrN, where N D11 (-) (+) (-) (+) t i (-) (+) t denotes the winding number, and define the encircled area as (a) H H (b1) v v i Ω D11 D2 A = Nπr2.Usingv = r = r2π f , we then immediately find [10,13] (b2) (-) (+) t a t c/n c/n v (-) (+) a D11 v v D2 v 8πAf counter co-rotating t = . (4) s 2 rotating c From this discussion it is clear why the description in the FIG. 1. Conceptual setup. (a) Description from the viewpoint corotating reference frame is slightly more convenient: There, of the corotating observer. The counter-rotating (corotating) quan- only the noninertial motion of the detectors has to be taken tities are shown on the left (right). The counter-rotating (corotating) into account (through the Hamiltonians). On the other hand, Hamiltonians are different, while the light speed in the two directions / in the laboratory inertial reference frame, one has to account is equal (c n). D1 (D2) indicate the detectors for the counter- for the motion of the detectors as well as of the medium (again rotating (corotating) direction, respectively. (b) Description from the through the Hamiltonians). In short, the advantage of the inertial laboratory frame represented in a straight line. Here, both corotating reference frame is the absence of the light-dragging the Hamiltonian and the speed of light differ in the two directions, (−) = (+) (±) = c effect. i.e., v v and v n . (b1) Only the light-drag effect is (+), (−) Finally, we note that the Sagnac delay can be obtained also taken into account; ti ti denote how long it would take to reach the detectors assuming they would not have moved; the subscript i from the perspective of the corotating observer where it arises stands for “initial.” (b2) Both the light-drag effect and the motion of from clock desynchronization [10]. (+), (−) detectors is taken into account. ta ta denote how long it takes to reach the detectors; the subscript a stands for “actual.” v = r is the III. DERIVATION OF THE HAMILTONIAN speed of the detectors as seen from the inertial laboratory reference frame. In this section we derive the Hamiltonian for the experi- ments depicted in Fig. 1(a): We consider a rotating platform, which spins at angular frequency , and we restrict to the case one has to account for the light-dragging effect [13], i.e., dynamics on a circle of radius r, centered on the symmetry the Fizeau effect. axis of the rotating platform. Specifically, we will adopt the We now first discuss the relation between the corotating methods of representation theory [19] and symplectic Hamil- reference frame and the inertial laboratory reference frame, tonian mechanics [20] to map the time-evolution generator focusing on the classical effects. We start by noting that the of the Poincaré algebra to a Hilbert space operator. One speed of light in the corotating reference frame, where the could of course make an ad hoc quantization in a noninertial corotating medium is stationary, is the same in the corotating reference frame, but the results might be inconsistent with and counter-rotating directions; specifically, we have that the basic symmetry requirements. Anyhow, we choose the former ± c velocities are n . Using the relativistic velocity addition for- method which constructs the Hamiltonian starting from basic ± mula we find the corresponding velocities u( ) in the inertial symmetry considerations of the Poincaré group. As argued in laboratory frame, Sec. II, we will for convenience describe the experiment in the ± c + corotating reference frame. ± v c u( ) = n ≈± + αv, (1) One typically starts describing the experiments by setting 1 ± c v n n c2 up a chart, e.g., a Cartesian chart. We note that the chosen = α = − 1 spacetime coordinates critically reflect the motion of the ob- where v r , and 1 n2 . The corresponding speeds (±) ≈ c ± α server which affects the resulting description of the dynamics. are given by v n v. See Fig. 1(b) for a graphical illustration of this light-dragging effect, which coincides with For example, two observers moving with different speeds the Fizeau effect, but could in principle differ [13]. or accelerations will use different charts, and hence ascribe We assume an initial spatial distance L between the sys- different energies to the same system, and hence care must tems and the detectors. We can convert L into a time distance, be taken with the choice of the coordinate system. In the i.e., the initial time distance [see Fig. 1(b1)], which is given following, we will assume that the detectors are stationary in (±) L the observer’s chart: With this choice there is a simple relation by t = ± .
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