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 quantum 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

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 experiments 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 = ± . Furthermore, exploiting Eq. (1), we find i v( ) between observables in the description and the quantities ± L vL measured by the detectors. t ( ) ≈ n ∓ αn2. (2) i c c2 We start by recalling the quantization procedure in an = However, this is different from the actual time it takes the inertial reference frame, i.e., with 0. In our case, we will signals to reach the detectors [see Fig. 1(b2)]: We need to take use the polar chart for the laboratory inertial reference frame. into account also the motion of the detectors. In particular, Specifically, the line element in an inertial reference frame (±) (±) = ± (±) expressed in the polar chart is given by we have the condition v ta L vta , which after some algebra readily gives ds2 = c2dt2 − r2dφ2, (5) L L Lv t (±) = ≈ n ± . (3) where r is a constant in our case (we have restricted the analy- a (±) ∓ 2 v v c c sistoa1+ 1 spacetime). From the line element in Eq. (5)itis

In this way we immediately recover the classic Sagnac delay then possibleμ to immediately read the time-evolution Killing = (+) − (−) = 2Lv 1 ∂ given by ts ta ta c2 . Speciﬁcally, to ﬁnd the usual vector ( c t ) [21]. The time-evolution Killing vector (more

043837-2 REVEALING AND CONCEALING ENTANGLEMENT WITH … PHYSICAL REVIEW A 101, 043837 (2020) precisely, a vector field) is the tangent vector to the flow lines Exploiting Eqs. (13)–(15), we can now finally write the time- of a system in free motion. Using the comoment map we can evolution operator, then map the time-evolution Killing vector to the Hilbert space ˆ = ˆ + ˆ, Hamiltonian [19,20], HBorn H r P (16) which we name the Born Hamiltonian. Note that Eq. (16) ∂t → Hˆ , (6) captures the idea that the dynamics on a rotating platform where Hˆ is the representation of the time-evolution generator can be fully explained in terms of the noninertial motion of on the considered Hilbert space. For example, for a single the detector [18]: The term ∼Pˆ keeps track of the noninertial modea ˆ we would have Hˆ = h¯ωaˆ†aˆ, where ω is the frequency motion of the detector. Pˆ, which is the generator of transla- of the oscillator. tions, changes the relative distance between the detector and To apply the quantization procedure in a noninertial refer- the system, e.g., in a time δt the relative distance changes by ence frame, i.e., with = 0, we have to make an additional rδt. step: We have to relate the corotating reference frame to the We note that the transformation in Eq. (7) leaves the time laboratory inertial reference frame. The reason is simple: We coordinate unchanged. This can be seen as a Galilean-type do not know how to directly quantize in a noninertial reference transformation on a circle, which we now generalize to a frame, but only in the inertial reference frame. We denote the Lorentz-type transformation. In particular, we consider laboratory (polar) and the corotating (Born) coordinates by the μ μ r2 unprimed [x = (ct, rφ)] and primed [x = (ct , rφ )] labels, dt = dt + A dφ, (17) respectively. In particular, we have the following relation [22], c2 dφ = B dφ + dt, (18) dt = dt, (7) − 1 r 2 2 where = [1 − ( ) ] .IfwesetA = 1 and B = 1, the dφ = dφ + dt . (8) c transformation is formally equivalent to a Lorentz boost with − It is straightforward to find the corresponding transformation speed v = r, while if we set A = 0 and B = 1, we obtain matrix, the transformation considered by Post [13]. We find that the Hamiltonian is insensitive to the value of A, but depends on ∂ μ x = 10, the chosen value of B. In the following, we set B = 1, which ∂ μ r (9) x c 1 leads to the Hamiltonian and to express the Minkowski metric in the two coordinates ˆ rel = ˆ + ˆ . HBorn (H r P) (19) systems, i.e., Equation (19) can be seen as a relativistic Born Hamiltonian, ds2 = c2dt2 − r2dφ2 (10) which generalizes Eq. (16). 2r2 We can also analyze nonuniform rotations using the above = c2 1 − dt2 − 2r2dtdφ − r2dφ2. (11) c2 formalism by considering a time-dependent angular frequency t (we remark that the time-evolution vector does not need to From Eqs. (10) and (11) we can immediately find the be generally a Killing vector). We repeat the derivation in this μ μ 1 ∂ = , 1 ∂ = , section with the formal replacements relevant Killing vectors ( c t ) (1 0) ,(c t ) (1 0) , 1 ∂ μ = , and ( r φ ) (0 1) .UsingEq.(9) we can then express → , μ t (20) 1 ∂ ( c t ) in the laboratory coordinates, i.e., − 1 μ μ 2 2 ∂ μ t r 1 x 1 → = 1 − . (21) ∂ = ∂ , (12) t c t ∂xμ c t c 1 μ rω At the end we obtain in place of Eq. (19) the following which gives ( ∂t ) = (1, ) , and thus c c Hamiltonian, 1 ∂ = ∂ + ∂ . rel t t r φ (13) Hˆ (t ) = (Hˆ + r Pˆ). (22) r Born t t We have now expressed the time-evolution Killing vector In this case one expects two physical effects, one related to the ∂t , which generates the dynamics in the corotating reference (geometrical) Sagnac phase, and a possible new contribution ∂ 1 ∂ related to a dynamical phase, which typically emerges in situ- frame, in terms of the Killing vectors t and r φ, which generate time evolution and space translation in the inertial ations where there is a time dependence in the Hamiltonian. laboratory reference frame, respectively. We can now map the latter Killing vectors to operators on a Hilbert space using the IV. PHOTON-INTERFEROMETRY EXPERIMENTS usual prescription, We now apply the Hamiltonian in Eq. (19) to photon ∂t → Hˆ , (14) interferometry. We use the Abraham relation between kinetic momentum and energy [23,24], 1 ∂φ → Pˆ. (15) ˆ = | ˆ|. r H nc P (23)

043837-3 MARKO TOROŠ et al. PHYSICAL REVIEW A 101, 043837 (2020)

We combine Eqs. (19) and (23) to find The state |ψ f then interferes at a beam splitter and one ± measures the outputs using two detectors: The input modes are Hˆ ( ) = (1 ± β)Hˆ , (24) Born aˆ and bˆ and we denote the output modes byc ˆ and dˆ. Here, we where the positive (negative) superscript denotes the counter- consider the following relation between the input and output rotating (corotating) motion to be measured by the detector modes, D1 (D2), and β = r . We note that Eq. (24) suggests a simple nc c ω 1 a ω physical interpretation of the Hamiltonian as the relativistic ˆ( ) = √ 11ˆ( ) . ˆ ω − ˆ ω (27) Doppler-shifted energy of the system. d( ) 2 1 1 b( ) We can finally write the total Hamiltonian of the system, In particular, we are interested in the probability of detecting ˆ = ˆ (+) ⊗ + ⊗ ˆ (−) , photons in the modesc ˆ or dˆ. To this end, it is convenient to HBorn HBorn I I HBorn (25) define the temporal modes [29], where we have assumed that the photons moving in opposite directions do not interact, and I denotes the identity operators. cˆ(t ) = Ft [ˆc(ω)], dˆ(t ) = Ft [dˆ(ω)], (28) H = In summary, the total Hilbert space can be written as 1 −iωt + − + − where F [ · ] = √ dω · e . In particular, we define the H( ) ⊗ H( ), where H( ) (H( )) denotes the Hilbert space of t 2π single-photon probability of detection as the counter-rotating (corotating) modes. The time parameter t that keeps track of the dynamics (1) = ψ | † |ψ , through Schrödinger’s equation is ticking as a clock following Pc dt f cˆ (t )ˆc(t ) f (29) the detectors’ motion; this is a direct consequence of the quan- (1) tization procedure that leads to the Hamiltonian in Eq. (25). with a similar definition for the probability Pd for the output However, here we are mainly interested in the dominant ef- mode dˆ. In addition, we also define the two-photon probability fects where one can approximate the Lorentz factor as ∼ 1 of detection, and the distances and times coincide to those measured by a (2) † † ruler and a clock in the inertial laboratory frame. P = dt1 dt2 ψ f |dˆ (t1)ˆc (t2 )ˆc(t2 )dˆ(t1)|ψ f , (30) We now further develop the model by adopting Glauber’s theory of photodetection [25]. Here, we will focus on the which gives the coincidence probability. For the case P(2) < experimental situation of photons with a coherence time that 0.5 we speak of coalescence or HOM photon bunching and for is short compared to the time resolution of the detectors, but P(2) > 0.5 we speak of photon anticoalescence or antibunch- temporal aspects could be easily taken into account [26,27]. ing. Classically, one is limited to values 0.25 < P(2) < 0.5, In other words, here we are not concerned about when the making coincidence probabilities a valuable tool to assess the detectors click, but only which detector clicks. We assume that quantum nature of the electromagnetic field. Importantly, an- in the detectors’ time window only one photon or one photon tisymmetrization and photon anticoalescence reveals hidden pair has been measured. entanglement, as has already been demonstrated experimen- To analyze photon-interferometry experiments we will tally in a nonrotating setup [30]. work in the Schrödinger picture [28], where we will denote We consider first the experimental situation with a generic the initial (final) state with the subscripts i ( f ). We consider single-photon input state, the experimental situation where at time ti = 0 the photon is |ψ prepared in a state i , and then constrained to move in a |ψ = ω ψ ω † ω + ψ ω ˆ† ω | , = L |ψ f d [ a, f ( )â ( ) b, f ( )b ( )] 0 (31) circular motion for a time t f c n resulting in a state f , where L is the traveled distance. Although one can always where ψ , (ω),ψ , (ω) are one-photon wave functions. From postulate a given initial state, it is nonetheless instructive to a f b f |ψ Eqs. (29) and (31), exploiting Eqs. (27) and (28), we find compare the state i , which is assumed to be generated by the apparatus corotating with the platform, with the state (1) 1 1 ∗ P = ± dω[ψ , (ω)ψ , (ω) + c.c.], (32) generated by the same apparatus when the platform is not c,d 2 2 a f b f rotating, i.e., when = 0. In particular, it is reasonable to assume that the frequencies of the initial states generated in the where we have imposed the normalization of the state, i.e., ∼ ψ f |ψ f =1. As an example let us consider the quantum two experimental situations differ by at most 2π . However, such a difference in the initial state produces only subleading Sagnac experiment [16]: A photon is prepared in a superpo- effects which are not amplified during time evolution, as can sition of counterpropagating modes before interfering at the be explicitly verified using the formulas we will develop. beam splitter [see Fig. 2(a)]. Specifically, we consider the One can thus approximate the initial states generated on the initial state in Eq. (31) with the one-photon wave functions ψ ω = ψ ω = ω ω rotating platform with the states that would be generated at a,i( ) b,i( ) g( ), where g( ) is a Gaussian with μ σ = 0. mean frequency and bandwidth . After the time evolution using Eq. (26) we have the state in Eq. (31) with ψa, f (ω) = The time evolution is given by the usual Schrödinger ωβ − ωβ ω i t f ψ ω = ω i t f equation with the Hamiltonian in Eq. (25), i.e., g( )e and b, f ( ) g( )e .UsingEq.(32) we find the single-photon detection probability, (2) (+) † (−) † Hˆ =h¯ dω[ω aˆ (ω)â(ω) + ω bˆ (ω)bˆ(ω)], (26) (1) − σts 2 = 1 ± ( 2 ) μ , Pc,d 2 [1 e cos( ts )] (33) (±) where we have defined ω = (1 ± β)ω, anda ˆ (bˆ)isthe where ts is the classical Sagnac delay introduced in Eq. (4). If counter (corotating) mode. we now make the further approximation |g(ω)|2 ∼ δ(μ − ω),

043837-4 REVEALING AND CONCEALING ENTANGLEMENT WITH … PHYSICAL REVIEW A 101, 043837 (2020)

(a) 1 3 c (b) c C b 2 a PD2 PD1 Ω Ω b BS a d d Ω BS b a FIG. 2. (a) Quantum Sagnac experiment [16]. The element C denotes the circulator which allows only the paths 1 to 2 and 2 to BBO 3. A photon enters through the path 1, is directed into path 2, and then interferes for the first time with the beam splitter. After evolving laser in counterprogating directions, the photon then interferes again at the beam splitter, after which it is detected. (b) Hong-Ou-Mandel experiment on a rotating platform [17]. Two identical photons coun- terpropagate before interfering at a beam splitter. One detects the FIG. 3. Layout of proposed quantum Sagnac/Hong-Ou-Mandel arrival of the photons and extracts the coincidence probability P(2). interferometer on a rotating platform. Two entangled photons are In both cases, the setups are placed on a rotating platform. emitted from the BBO crystal: Photon a (purple arrow) enters a Sagnac interferometer using the lower 50/50 beam splitter (BS) / σ → and exits towards the upper 50 50 BS where two-photon HOM i.e., 0, we then recover the results reported in Ref. [16], interference occurs with photon b (green arrow) that circles around (1) = 1 ± μ i.e., Pc,d 2 [1 cos ( ts )]. the setup (in order to maintain the same overall path length as photon We next consider the two-photon state, a). Coincidence counts are measured between detectors PD1 and † † PD2 as a function of the rotation frequency . |ψ f = dω1 dω2 ψ f (ω1,ω2 )â (ω1)bˆ (ω2 )|0 , (34) where ψ f (ω1,ω2 ) is the two-photon spectrum. From Eqs. (30) and (34), exploiting Eqs. (27) and (28), we find In particular, for a completely antisymmetric spectrum, i.e., ψ(ω ,ω ) =−ψ(ω ,ω ), one obtains perfect anticoales- 1 1 1 2 2 1 (2) = − ω ω ψ∗ ω ,ω ψ ω ,ω , cence, but even with a partially antisymmetric spectrum one P d 1 d 2 f ( 1 2 ) f ( 2 1) (35) 2 2 can have P(2) > 0.5, thus witnessing entanglement. As we where we have imposed the normalization ψ f |ψ f =1. As show below, this manifestation of entanglement may be sus- an example we consider the Hong-Ou-Mandel experiment on ceptible to the motion of the interferometer. a rotating platform [17]: Two identical photons counterprop- We consider the experimental setup depicted in Fig. 3. agate before interfering at a beam splitter [see Fig. 2(b)]. As an initial state we consider a SPDC type-I two-photon The experimentalist controls the initial time delay δt of the state (i.e., two photons with the same polarization) with the modea ˆ; the initial state is given by Eq. (34) with the two- spectrum −iω1δt photon spectrum ψi(ω1,ω2 ) = g(ω1)g(ω2 )e .Afterthe time evolution we find the final state in Eq. (34) with ψi(ω1,ω2 ) = δ(ω1 + ω2 − 2μ)g(ω1)g(ω2 ), (38) −iω1δt iβ(ω1−ω2 ) ψ f (ω1,ω2 ) = g(ω1)g(ω2 )e e . (36) Using Eq. (35) we then immediately find the coincidence where we have omitted the normalization. We note that the probability, initial spectrum in Eq. (38) is completely symmetric, i.e., (2) −σ 2 t −δt 2 ψ(ω ,ω ) = ψ(ω ,ω ), and gives P = 0at = 0. P(2) = 1 − 1 e ( s ) , (37) 1 2 2 1 2 2 We now consider the same setup with the interferometer in where ts is the classical Sagnac delay. a constant rotational motion with frequency = 0. The final Above, we have considered identical photons with a sep- spectrum of the two-photon state changes to arable spectrum [7], but one could also consider identical frequency-entangled photons. For example, if we consider − βω ψ ω ,ω = ψ ω ,ω βω i 2t f , spontaneous parametric down-conversion (SPDC) type-I two- f ( 1 2 ) i( 1 2 ) cos( 1t f )e (39) photon generation [31], we again find the coincidence probability in Eq. (37). It would thus seem that entanglement in where the factor cos(βω t ) results from the interference of combination with rotational motion leaves no trace on the 1 f the modea ˆ; by “final state” we again mean the state that photon coincidence rate P(2) measurement. We now further arrives at the last beam splitter. Using Eq. (35) we then find explore this question. the coincidence probability,

V. MANIFESTATION OF ENTANGLEMENT THROUGH − 1 σ 2t 2 1 − 1 σ 2t 2 ROTATION 1 cos(μts )e 8 s + 1 + e 2 s P(2) = − 2 . (40) − 1 σ 2t 2 Entanglement can manifest itself in a HOM coincidence 2 2 1 + cos(μts )e 8 s rate measurement through anticoalescence, i.e., P(2) > 0.5.

043837-5 MARKO TOROŠ et al. PHYSICAL REVIEW A 101, 043837 (2020)

ω1 ↔ ω2. In frequency space these time delays then change the final biphoton spectrum from symmetric to antisymmetric and vice versa, resulting in oscillations between bunching and antibunching, respectively, as seen in Fig. 4. The paths a and b in Fig. 3 forming the asymmetric HOM setup can be also seen as reminiscent of an unbalanced Mach-Zehnder setup [32]. The asymptotic value for the coincidence probability in Fig. 4 can also be intuitively understood in terms of quantum Sagnac and the HOM setups. At high rotation frequency the counterpropagating mode (blue arrow) no longer interferes with the corotating modes (purple and green arrows) due to nonoverlapping frequency spectra. Loosely speaking, the counterpropagating mode (blue arrow) can be associated with half of the initial modea ˆ, while the other half of the modea ˆ corotates (purple arrow) and interferes with the other FIG. 4. Coincidence plot as a function of the angular fre- ˆ quency = 2π f . We have set the interferometer area A = 22.7m2, corotating mode b (green arrow). From Eq. (40) we find at (2) ∼ / μ = 2.36 × 1015 Hz, corresponding to a typical photon carrier high rotation frequency the value P 1 4; this is halfway (2) wavelength of 800 nm. Two curves are shown for two differ- between a full dip, P ∼ 0, and the case without bunching (2) ent bandwidths, σ = 1.47 × 1013 Hz (blue solid curve) and σ = or antibunching, P ∼ 1/2. As the frequency spectrum of 1.18 × 1014 Hz (dashed red curve), corresponding to 5- and 40- the two corotating modes is identical (blue and green arrow), nm bandwidths, respectively. The shaded region corresponding to i.e., no time delay exists between them, they bunch together P(2) > 0.5 indicates the region where measurements imply photon at the output ports of the beam splitter. Hence the counter- entanglement. rotating (corotating) part of the modea ˆ is associated with P(2) ∼ 1/2(P(2) ∼ 0). The coincidence probability at high rotation frequency can be thus seen as the average behavior, We have plotted P(2) as a function of the angular fre- i.e., P(2) ∼ (0 + 1/2)/2 = 1/4. quency in Fig. 4 with the interferometer area A = 22.7m2 The experimental setups shown in Fig. 2 and the one shown (assuming the photons travel through a 100-m-long fiber, in Fig. 3 are substantially different, both experimentally and wound 35 times along a 0.9-m-diameter loop) and μ = conceptually. In particular, the HOM on a rotating platform 2.36 × 1015 Hz, corresponding to a typical photon carrier [17] is a symmetric setup, which is unable to induce the wavelength of 800 nm. Two curves are shown for two dif- antibunching behavior of photons, but produces only a coinci- σ = . × 13 (2) < 1 ferent photon bandwidths, 1 47 10 Hz (blue solid dence dip. Although photon bunching below P 4 implies curve) and σ = 1.18 × 1014 Hz (dashed red curve), corre- quantum interference phenomena, one cannot assert anything sponding to 5 and 40 nm, respectively. The shaded region about entanglement. On the other hand, the setup in Fig. 3 corresponding to P(2) > 0.5 indicates the presence of en- is asymmetric, and as shown in Fig. 4 is able to invert a (2) < 1 (2) > 1 tanglement that manifests as photon anticoalescence. Short coincidence dip, i.e., P 2 , to a spike, i.e., P 2 .Its (2) > 1 bandwidths, i.e., long coherence photons, show a periodic importance lies in that photon antibunching P 2 implies series of revivals of entanglement with increasing rotation fre- entanglement. quency. For larger photon bandwidths, i.e., shorter coherence The antisymmetrization of the photon spectrum, which lengths, increasing the relative photon delay by increasing leads to a modification of the coincidence probability, is a the rotation speed leads to a reduction of the coincidence direct consequence of the noninertial motion of the platform. peak values and of the overall fringe visibility. This is a More generally, this shows that rotational motion can activate result of the loss of mutual coherence between the two dormant asymmetries in the experimental setup leading to an interfering photons. antisymmetric spectrum. It is also interesting to consider an The proposal shown in Fig. 3 can be also seen as a combi- initial antisymmetric spectrum ψi; the proposed experiment nation of the setups shown in Fig. 2, which can be exploited to shows that ψi can become symmetrized during time evolution, gain an intuitive understanding of the results. As in the case of fully concealing the anticoalescence signature of entangle- the quantum Sagnac and the HOM setup, also in this case the ment. wave packets traveling in the corotating and counter-rotating These effects can be traced to the impossibility of clock directions acquire a relative time delay during time evolution synchronization along a closed loop on the rotating platform. which is mathematically represented by a phase factor. Specif- In particular, the Hamiltonian in Eq. (26) can be linked to ically, the paths denoted by the purple and blue arrows in the effect of clock desynchronization [10]. It is important to Fig. 3 can be related to the quantum Sagnac and contribute an note that this is a genuine relativistic effect, which is not −iβω1t f +iβω1t f interference factor e + e ∼ cos(βω1t f ). Together expected to arise in a Newtonian theory, although it imprints these paths, i.e., the purple and blue arrows, form half of the a non-negligible experimental trace in the regime typically HOM setup, while the other half is depicted by the corotating associated with the latter. This is different from the observer- − βω green path which in turn contributes a time delay e i 2t f .For dependent entanglement effect in noninertial reference frames −iβω2t f the whole setup we thus find a factor cos(βω1t f )e which [33], expected to arise as a consequence of the Unruh radiation is not symmetric under exchange of the frequencies, i.e., [34,35] which vanishes at low accelerations.

043837-6 REVEALING AND CONCEALING ENTANGLEMENT WITH … PHYSICAL REVIEW A 101, 043837 (2020)

VI. CONCLUSIONS asymmetries in the setups of complex experiments as well as rotations, both controlled and spurious, are ubiquitous. It We have developed a formalism for describing quantum is thus likely that these effects will have ramifications for experiments on rotating platforms, both matter-wave and quantum-sensing applications. photon interferometry, exploiting the map between symme- The proposed experiment can be also viewed as a test of the try generators and Hilbert space operators. The dynamics underlying Poincaré symmetry group for entangled system. assumes a simple form with the time evolution given by the The proposal is based on a theoretical model that relies heavily Born Hamiltonian. The developed theory can be represented on intrinsic features of both relativity and quantum mechanics. on a generic Hilbert space and is valid in regimes where Although we do not suspect deviations from the predicted the particle number is conserved, as is the case of a typical behavior, we nonetheless note that entanglement in noniner- table-top laboratory experiment. tial reference frames has not yet been fully experimentally We have further developed the theory to analyze two re- scrutinized. We conclude by recalling that classical tests of cent photon-interferometry experiments, namely, the quantum spacetime symmetries such as the Michelson-Morley and the Sagnac and the HOM experiment on a rotating platform. We Sagnac experiment were also initially devised with the goal have also proposed a modified HOM interferometer where of reaffirming the established theories, but ended profoundly entanglement can be revealed or concealed depending on the affecting the elementary notions of space and time. rotational frequency. Specifically, we have shown that rotations together with an asymmetry of the experimental setup can strongly affect the bunching and antibunching behavior, and hence the manifestation of entanglement. ACKNOWLEDGMENT The interferometer shown in Fig. 3 can generate both bunching as well as antibunching solely through rotation, a The authors acknowledge support from the EU H2020FET feature which distinguishes it from the previously considered project TEQ (Grant No. 766900), from the EPSRC (U.K., HOM interferometer on a rotating platform [17]. Furthermore, Grant No. EP/M009122/1), and from the European Union’s the strength of the effect at low rotational speed is suggestive Horizon 2020 research and innovation programme under that such phenomena are all but rare. Indeed, we note that Grant Agreement No. 820392.

[1] H. Minkowski, Raum und zeit, Jahresber. Dtsch. Math. Ver. 18, [13] E. J. Post, Sagnac effect, Rev. Mod. Phys. 39, 475 (1967). 75 (1909). [14] M. Fink, A. Rodriguez-Aramendia, J. Handsteiner, A. Ziarkash, [2] Minkowski Spacetime: A Hundred Years Later,editedbyV. F. Steinlechner, T. Scheidl, I. Fuentes, J. Pienaar, T. C. Ralph, Petkov, Vol. 6 (Springer, Berlin, 2010). and R. Ursin, Experimental test of photonic entanglement [3] W. A. Von Ignatowsky, Einige allgemeine Bemerkungen zum in accelerated reference frames, Nat. Commun. 8, 15304 Relativitätsprinzip, Verh. Dtsch. Phys. Ges. Berlin 12, 788 (2017). (1910). [15] M. Fink, F. Steinlechner, J. Handsteiner, J. P. Dowling, T. [4] S. Liberati, Tests of Lorentz invariance: A 2013 update, Scheidl, and R. Ursin, Entanglement-enhanced optical gyro- Classical Quantum Gravity 30, 133001 (2013). scope, New J. Phys. 21, 053010 (2019). [5] H. P. Robertson, Postulate versus observation in the special [16] G. Bertocchi, O. Alibart, D. B. Ostrowsky, S. Tanzilli, and P. theory of relativity, Rev. Mod. Phys. 21, 378 (1949). Baldi, Single-photon Sagnac interferometer, J. Phys. B: At., [6] A. A. Michelson and E. W. Morley, On the relative motion of Mol. Opt. Phys. 39, 1011 (2006). the earth and of the luminiferous ether, Sidereal Messenger 6, [17] S. Restuccia, M. Toroš, G. M. Gibson, H. Ulbricht, D. Faccio, 306 (1887). and M. J. Padgett, Photon Bunching in a Rotating Reference [7] C.-K. Hong, Z.-Y. Ou, and L. Mandel, Measurement of Subpi- Frame, Phys. Rev. Lett. 123, 110401 (2019). cosecond Time Intervals between Two Photons by Interference, [18] A. Tartaglia and M. L. Ruggiero, The Sagnac effect and pure Phys.Rev.Lett.59, 2044 (1987). geometry, Am.J.Phys.83, 427 (2015). [8] L. Mandel, Quantum effects in one-photon and two-photon [19] P. Woit, Quantum Theory, Groups and Representations interference, in More Things in Heaven and Earth (Springer, (Springer, Berlin, 2017). Berlin, 1999), pp. 460–473. [20] A. Cannas Da Silva and A. Cannas Da Salva, Lectures on [9] A. Lyons, G. C. Knee, E. Bolduc, T. Roger, J. Leach, Symplectic Geometry, Vol. 3575 (Springer, Berlin, 2001). E. M. Gauger, and D. Faccio, Attosecond-resolution [21] E. Poisson, A Relativist’s Toolkit: The Mathematics of Black- Hong-Ou-Mandel interferometry, Sci. Adv. 4, eaap9416 Hole Mechanics (Cambridge University Press, Cambridge, (2018). U.K., 2004). [10] É. Gourgoulhon, Special Relativity in General Frames [22] G. Rizzi and M. L. Ruggiero, Space geometry of rotating (Springer, Berlin, 2016). platforms: An operational approach, Found. Phys. 32, 1525 [11] G. Sagnac, L’éther lumineux démontré par l’effet du vent relatif (2002). d’éther dans un interféromètre en rotation uniforme, C. R. Acad. [23] M. J. Padgett, On diffraction within a dielectric medium as an Sci. 157, 708 (1913). example of the Minkowski formulation of optical momentum, [12] G. Sagnac, Sur la preuve de la réalité de l’éther lumineux par Opt. Express 16, 20864 (2008). l’expérience de l’interférographe tournant, C. R. Acad. Sci. 157, [24] S. M. Barnett, Resolution of the Abraham-Minkowski 1410 (1913). Dilemma, Phys.Rev.Lett.104, 070401 (2010).

043837-7 MARKO TOROŠ et al. PHYSICAL REVIEW A 101, 043837 (2020)

[25] R. J. Glauber, The quantum theory of optical coherence, Phys. veals hidden entanglement, New J. Phys. 11, 103052 Rev. 130, 2529 (1963). (2009). [26] T. Legero, T. Wilk, A. Kuhn, and G. Rempe, Characterization [31] M. Barbieri, E. Roccia, L. Mancino, M. Sbroscia, I. Gianani, of single photons using two-photon interference, Adv. At., Mol., and F. Sciarrino, What Hong-Ou-Mandel interference says Opt. Phys. 53, 253 (2006). on two-photon frequency entanglement, Sci. Rep. 7, 7247 [27] H. P. Specht, J. Bochmann, M. Mücke, B. Weber, E. Figueroa, (2017). D. L. Moehring, and G. Rempe, Phase shaping of single-photon [32] D. V. Strekalov, T. B. Pittman, and Y. H. Shih, What we wave packets, Nat. Photonics 3, 469 (2009). can learn about single photons in a two-photon interference [28] K. Wang, Quantum theory of two-photon wavepacket interfer- experiment, Phys.Rev.A57, 567 (1998). ence in a beamsplitter, J. Phys. B: At., Mol. Opt. Phys. 39, R293 [33] I. Fuentes-Schuller and R. B. Mann, Alice Falls into a Black (2006). Hole: Entanglement in Noninertial Frames, Phys.Rev.Lett.95, [29] K. J. Blow, R. Loudon, S. J. D. Phoenix, and T. J. Shepherd, 120404 (2005). Continuum ﬁelds in quantum optics, Phys. Rev. A 42, 4102 [34] P. C. W. Davies, Scalar production in Schwarzschild and (1990). Rindler metrics, J. Phys. A: Math. Gen. 8, 609 (1975). [30] A. Fedrizzi, T. Herbst, M. Aspelmeyer, M. Barbieri, [35] W. G. Unruh, Notes on black-hole evaporation, Phys. Rev. D T. Jennewein, and A. Zeilinger, Anti-symmetrization re- 14, 870 (1976).

043837-8