PHYSICAL REVIEW RESEARCH 3, 023024 (2021)

Frame dragging and the Hong-Ou-Mandel dip: Gravitational effects in multiphoton interference

Anthony J. Brady * and Stav Haldar Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USA

(Received 30 June 2020; revised 17 February 2021; accepted 19 March 2021; published 8 April 2021)

Applying quantum ﬁeld theory on a classical, curved space-time has led to the enthralling theoretical predictions of, e.g., black hole evaporation and particle creation in the early, expanding universe. However, the prospects of observing such in the near to mid future seem far from practical. This is discouraging because extrapolation of general relativity “down to the Hilbert space level,” though expected, carries no empirical support as of yet. On the other hand, fueled by the continuing developments in quantum technology, there has been growing interest in the prospect of observing more modest effects of general relativity at the level of quantum interference phenomena. Motivated by these recent developments, in this work, we investigate the Hong-Ou-Mandel (HOM) effect—a two-photon quantum-interference effect—in the space-time of a rotating spherical mass. In particular, we propose and analyze a common-path setup restricted to the surface of the Earth and show that, in principle, gravitational frame dragging induces observable shifts in the two-photon HOM dip. For completeness and correspondence with current literature, we also analyze the emergence of gravitational time dilation, for a dual-arm interferometer. The formalism thus presented establishes a basis for encoding general-relativistic effects into local, multiphoton, quantum-interference experiments. We also consider signatures of noninertial reference-frame effects on two-photon interference such as, e.g., the Sagnac effect induced by Earth’s rotation, which seems detectable with current quantum-optical technology. This can be viewed as a stepping-stone towards the more ambitious goal of observing gravitational frame dragging at the level of individual photons.

DOI: 10.1103/PhysRevResearch.3.023024

I. INTRODUCTION The maturation of quantum technologies complements the theoretical maturation of quantum ﬁelds in curved space— General relativity and quantum mechanics constitute the a formalism describing the evolution of quantum ﬁelds on foundation of modern physics, yet at seemingly disparate the (classical) background space-time of general relativity— scales. On the one hand, general relativity predicts devia- which has been ﬁnely developed over the last 50-odd tions from the Newtonian concepts of absolute space and years [9,10]. Alas, there has not yet been a physical ob- time—due to the mass-energy distribution of nearby matter servation requiring the principles of both general relativity and by way of the equivalence principle [1,2]—which appear and quantum mechanics for its explanation. With these con- observable only at large-distance scales or with high-precision siderations in mind, the present thus seems a ripe time measurement devices [3]. On the other hand, quantum me- to furnish potential proof-of-principle experiments capable chanics predicts deviations from Newtonian concepts of of and exploring the cohesion of general relativistic and deterministic reality and locality [4] and appears to dominate quantum-mechanical principles, in the near term. Such is the in regimes in which general relativistic effects are typically aim of our work. and safely ignored. This dichotomous paradigm of modern In this paper, we investigate the emergence of gravita- physics is, however, rapidly changing due to (a) the ever- tional effects in two-photon quantum interference [i.e., in increasing improvement of quantum measurement strategies Hong-Ou-Mandel (HOM) interference [11–13]] for various and precise control of quantum technologies [5,6] and (b) interferometric conﬁgurations (Fig. 1). We show, in general, current efforts to extend quantum-mechanical demonstrations how to encode frame dragging and gravitational time dilation to large-distance scales, such as bringing the quantum to into multiphoton quantum-interference phenomena, and we space [7,8]. show, in particular, that a background curved space-time in- duces observable changes in a two-photon HOM-interference signature. Observing these signatures in practice would vin- *[email protected] dicate the extrapolation of general relativity “down to the Hilbert space level.” Published by the American Physical Society under the terms of the HOM interference is simple to analyze, yet is a pre- Creative Commons Attribution 4.0 International license. Further cursor to bosonic many-body quantum-interference phenom- distribution of this work must maintain attribution to the author(s) ena [13–15] and even “lies at the heart of linear optical and the published article’s title, journal citation, and DOI. quantum computing” [16]. Furthermore, there exists a simple

2643-1564/2021/3(2)/023024(13) 023024-1 Published by the American Physical Society ANTHONY J. BRADY AND STAV HALDAR PHYSICAL REVIEW RESEARCH 3, 023024 (2021)

formalism describes passive transformations (phase shifts) on quantum states of the electromagnetic ﬁeld, induced by curved space-time. Hence, particle production is beyond the regime that we consider here. One can consider this formalism as geometric quantum optics in a locally curved space-time. In Sec. II B, we discuss the metric local to an observer and its relation to the background post-Newtonian (PN) metric, thus prescribing the space-time upon which the quantum ﬁeld of Sec. II A propagates. In Sec. II C, we provide some back- ground on HOM interference and comment on interferometric noise and gravitational decoherence in context. In Sec. III, we combine and apply the formalism of previous sections to FIG. 1. Schematic of a Hong-Ou-Mandel interferometer, in a various interferometric conﬁgurations, explicitly showing that common-path conﬁguration, located on the Earth’s surface. The gray gravitational effects, in principle, induce observable changes boxes just after the photon source are ﬁber-coupler and beamsplit- in HOM-interference signatures. We also provide order-of- ter systems, with high transmissivity, used to ensure common-path magnitude estimates for these effects and brieﬂy discuss propagation (cf. [27,28]). observational potentiality. Finally, in Sec. IV, we summarize our work. We utilize Einstein’s summation convention throughout yet sharp criterion partitioning the sufﬁciency of classical and assume a metric signature (−, +, +, +). and quantum explanations. Namely, if the visibility V of the / HOM-interference experiment is greater than 1 2, a quantum II. METHODS description for the ﬁelds is sufﬁcient, whereas a classical description fails to adhere to observations ([12,13]; though, A. Modeling quantum optics in curved space-time see [17] for recent criticism of the visibility criterion). Fur- 1. Geometric optics in curved space: A brief review thermore, the emergence of gravitational effects in photonic We model the electromagnetic ﬁeld by a free, massless, demonstrations necessitates a general-relativistic explanation; scalar ﬁeld φ satisfying the (minimally coupled) Klein- whereas, for massive particles, the boundary between general- Gordon equation relativistic and Newtonian explanations is more intricate and μ oft-blurred. As rightly pointed out and emphasized by the ∇ ∇μφ = 0, (2.1) authors of Refs. [18,19], within any quantum demonstration claiming the emergence of general-relativistic effects, a clear- where ∇μ is the metric compatible covariant derivative of cut distinction between classical and quantum explanations as general relativity. Such a model disregards polarization- well as between Newtonian and general-relativistic explana- dependent effects but accurately accounts for the amplitude tions must exist, in order to assert that principles of general and phase of the ﬁeld. relativity and quantum mechanics are at play, in tandem. We assume the ﬁeld to be quasimonochromatic in the geo- To place our work in a sharper context, observe metric optics approximation [1,41] so that it can be rewritten that one may roughly decompose the mutual arena of (in complex form) as / gravitation relativity (beyond rectilinear motion) and quan- iS φ = α e k , (2.2) tum mechanics into four classes: (i) classical Newtonian k k gravity in quantum mechanics [20–23], (ii) noninertial ref- with S the eikonal (or phase), α a slowly varying envelope erence frames in quantum mechanics [24–28], (iii) classical with respect to variations of the eikonal, and k denoting the general relativity in quantum mechanics [18,19,29–34], and wave-vector mode which φk is centered about. The wave (iv) the quantum nature of gravity [35–40]. Although there equation (2.1) in the geometric optics approximation yields has been a great deal of theoretical investigation in these areas, ∇ transport equations along the vector ﬁeld μSk in the slowly there has only been observational evidence for (i) and (ii) (see, varying envelope approximation, e.g., Refs. [20–23] and [24–27], respectively), which indeed ∇μ ∇ = , has supported consistency between these formalisms when ( Sk )( μSk ) 0 (2.3) they are concurrently considered (e.g., consistency between μ 1 μ ∇ ∇μ α =− ∇ ∇μ . Newton’s theory of gravitation and nonrelativistic quantum ( Sk )( ln k ) 2 S (2.4) mechanics [20–23]). Our work serves as a contribution to ∇ (iii), where one must describe the gravitational ﬁeld by a The ﬁrst equation implies that μSk is a null vector. One can classical metric theory of gravity and physical probe systems show that it also satisﬁes a geodesic equation via quantum-mechanical principles. μ (∇ S )∇μ(∇ν S ) = 0. We compartmentalize our paper in the following way. In k k Sec. II A, we provide a model for the electromagnetic ﬁeld in Thus, the transport equations, (2.3) and (2.4), determine how curved space-time, in the weak-gravity regime, and proceed the eikonal and amplitude, respectively, evolve along the null ∇ to quantize the ﬁeld under suitable approximations. We work ray drawn out by μSk. Observe that the evolution of the to ﬁrst order in small perturbations about the Minkowski eikonal along the ray is sufﬁcient to determine the ﬁeld φ space-time metric. In this approximation, the corresponding entirely. These are the main results of geometric optics.

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Lastly, from the transport equations, one may derive a product. With this supposition, we ﬁnd general solutions to the conservation equation perturbative transport equations μ 2 ∇ (∇μS |α | ) = 0, (2.5) 1 μ ν k k δSk =− dx k hμν , (2.14) 2 γ which, via Gauss’s theorem, leads to a conserved “charge” 1 μ μ ν ε =− dλ ∂μ∂ δS + μν k . (2.15) = μ ∇ |α |2, k k Qk : d ( μSk ) k (2.6) 2 γ Observe that, to ﬁrst nontrivial order, with being a spacelike hypersurface. Physically, the conser- vation equation is a conservation of photon ﬂux such that the 1 μ μ ν (1 + εk ) = exp − dλ ∂μ∂ δSk + μν k . number of photons Qk is a constant for all time. 2 γ The equations thus posed are precisely that of geometric Combining this with the O(1) amplitude and using the fact optics in curved space-time, excepting a transport equation for that the polarization vector of the ﬁeld (see, e.g., Refs. [1,41]). ∇ ∇μ = ∂ μ + ∂ ∂μδ + μ ν , μ Sk μk μ Sk μν k 2. Geometric optics in the weak-ﬁeld regime to O(h), we obtain We work with laboratory or near-Earth experiments in 1 mind. Therefore, it is sufﬁcient to consider general relativity α = − λ ∇ ∇μ , k exp d ( μ Sk ) (2.16) in the weak-ﬁeld regime. 2 γ We assume local gravitational ﬁelds, accelerations, etc., where the initial amplitude is taken as unity, for simplicity. are sufﬁciently weak and consider ﬁrst-order perturbations This is analogous to the generic solution for the amplitude η hμν about the Minkowski metric μν such that the space-time in curved space-time, excepting that the geometric curve, γ , metric gμν , in some local region in space, can be written as over which the integral is evaluated, is the ﬂat-space geodesic generated by ∂μS and not the null geodesic generated by gμν = ημν + hμν . (2.7) k ∇ μSk. The metric perturbation induces perturbations of the eikonal and amplitude about the Minkowski values (Sk,αk ), i.e., 3. Approximately orthonormal modes and quantization S = S + δS , (2.8) We wish to investigate local (e.g., in a ﬁnite region of k k k space) quantum optics experiments on a curved background α = α (1 + ε ), (2.9) space-time, in the weak-ﬁeld regime. Hence, we must produce k k k a quantum description for the ﬁeld. To do so, we introduce with (δSk,εk ) being the leading perturbations of O(h). Deﬁn- a set of approximately orthonormal solutions to the Klein- ing kμ := ∂μSk as the Minkowski wave vector, we show that Gordon equation (2.1), which, upon quantization, allows us the general transport equations, (2.3) and (2.4), imply the O(1) to deﬁne creation and annihilation operators of the quantum transport equations ﬁeld obeying the usual commutation relations [10]. From this μ μ construction, a Fock space can be built and quantum optics k kμ = 0 ⇒ k ∂μkν = 0, (2.10) experiments analyzed per usual. We note that the following formalism describes “local” ﬁeld solutions which are valid μ 1 μ ∂μα =− ∂μ , k k 2 k (2.11) descriptions of the ﬁeld, in some ﬁnite region of space (of 2 which are the geometric optics equations in Minkowski space- size ; see below), to some speciﬁed accuracy [to O(h ); time, and O(h) transport equations see below]. Approximately orthonormal modes. From the generic so- μ 1 μ ν ∂μδ =− μν , lutions, deﬁne the mode uk which satisﬁes the Klein-Gordon k Sk 2 k k h (2.12) equation in the weak-ﬁeld regime μ μ 1 μ μ ν i(k xμ+δSk ) ∂με =− ∂μ∂ δ + , u := N α e , (2.17) k k 2 Sk μνk (2.13) k k k with where Nk is a normalization constant. We prove that, to some error, the set {uk} deﬁne an orthonormal set of classical μ 1 μκ βν = η ∂β νκ + ∂ν βκ − ∂κ βν , 2 ( h h h ) solutions to the Klein-Gordon equation, with respect to the being the torsion-free connection coefﬁcients of general rela- Klein-Gordon inner product [10] μ tivity to O(h)[1]. Therefore, one observes that the null ray k , , = μ ∗∂ − ∂ ∗ , ( f g)KG : i d ( f μg g μ f ) (2.18) together with the metric perturbation hμν , uniquely determines α the eikonal Sk, the amplitude k [per Eqs. (2.10)–(2.13)], and hence, the ﬁeld φ [Eq. (2.2)]. where ia a suitable normalization volume, i.e., a spatial Generic solutions. Let us suppose that the null curve γ hypersurface of sufﬁcient size. The sketch of the proof goes is parametrized by parameter λ and has (Minkowski) coor- as follows. dinate expression xμ(λ) such that kμ = dxμ/dλ. This implies We wish to show that λ = 0/ = · / 2 2 = · , ≈ δ − , d dx k dx k k , with k : k k the Euclidean scalar (uk uk )KG (k k ) (2.19)

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2 to O(h ). Consider hμν to be time-independent and to vary a basis decomposition − over a characteristic length scale ∂h ∼ 1, where, for ex- 2 φˆ = 3 + † ∗ , ample, ∼ c /g. We further assume the integration volume d k(ˆakuk aˆk uk ) (2.26) to obey 1/3 , i.e., h 1 within the spatial region of interest (the laboratory). Then, analyzing the Klein-Gordon with (ˆa, aˆ†) the annihilation and creation operators, which are inner product perturbatively, we see that found per above: (0) (1) , ≈ , + , , (uk uk )KG (uk uk )KG (uk uk )KG (2.20) aˆk = (uk, φˆ)KG, (2.27) (0) u , u ∝ δ k − k † where ( k k )KG ( ) is the ﬂat-space inner product aˆ = (φ,ˆ u ) . (2.28) (1) k k KG , for the plane-wave modes and (uk uk )KG is the inner product containing terms to O(h), which can be written as Given the basis decomposition and canonical commutation relation (2.25), one can show that (1) −i(ω −ω )t 3 i(k−k )·x , ∝ k k , † (uk uk )KG e d xe f (h) (2.21) a , a = u , u [ˆk ˆq] ( k q )KG (2.29) with (t, x) being Lorentz coordinates on and f (h) a function ≈ δ(k − q). (2.30) of O(h) that varies over the length scale . It is then sufﬁcient (1) Fock space and wave packets. The (approximate) orthonor- to show that (uk, uk ) only has support at k = k .Thisis KG − mality of the classical mode solutions, together with the done in two steps. For k − k 1, the phase rapidly oscil- commutation relations, is sufﬁcient to build a Fock space lates with respect to f (h) over the entire integration volume. spanned by states of the form [10] Thus, by the Reimann-Lebesgue lemma, the integral aver- −1 ages to zero. On the other hand, for k − k ∼ ,wehave † nk (ˆak ) k − k x ∼ O h ∀x ∈ k = k √ |0 , (2.31) ( ) ( ) . Thus, we may set in the n ! integral, to accuracy O(h2 ). Therefore, the integral (2.21) only k k has support at k = k to O(h2 ), which was to be shown. with Canonical quantization. We decompose the ﬁeld in the |0 := |0k , basis set {uk}, k 3 φ = d k(akuk + c.c), (2.22) being the vacuum state on τ . Thus, the usual interpretation of † the creation operatora ˆk follows (e.g., as creating a single pho- with the basis coefﬁcients, ak, found from the Klein-Gordon ton in the mode uk). Note, however, that the Fock states above inner product are non-normalizable (or normalizable up to a Dirac-delta = ,φ , distribution). We remedy this difﬁculty by constructing wave ak (uk )KG (2.23) packets. As an example, we construct a positive frequency ∗ = φ, . wave packet, ak ( uk )KG (2.24) = 3 ∗ , As a technical aside, we note that the basis expansion of f d kfk uk (2.32) the ﬁeld, Eq. (2.22), is only an approximate description of the ﬁeld to accuracy O(h2 ). Technically, one should ﬁrst deﬁne with fk ∈ C, usually peaked around some wave vector, but ∗ more suitably a normalized L2 function in k space. From this, the mode coefﬁcients, ak and ak , per Eqs. (2.23) and (2.24), which can be deﬁned for any function uk whatsoever. One then we deﬁne the annihilation wave-packet operator associated supposes that the ﬁeld φ can be expanded as in (2.22). The for- with f , mer deﬁnition and the latter supposition are then consistent to 2 = , φˆ ≈ 3 . accuracy O(h ), by virtue of the approximate orthonormality aˆ f : ( f )KG d kfkaˆk (2.33) of the mode functions, uk. We proceed to canonically quantize the ﬁeld by introduc- Commutation relations for the wave-packet operators then follow ing the canonical momentum for the classical Klein-Gordon ﬁeld [10] † 3 2 [ˆa f , aˆ ] = ( f , f )KG ≈ d k| fk| = 1, (2.34) 1/2 f π =|h| ∂τ φ, where approximate orthonormality of the mode functions was which is deﬁned on a future-oriented spatial hypersurface τ , 2 assumed and the L property of fk was used. Per above, with |h| the determinant of the induced spatial metric on τ . one can build a Fock space of wave-packet states which are Note that τ is simply the surface of simultaneity at time τ. properly normalizable. Furthermore, this construction permits The ﬁeld φ and canonical momentum π are then promoted the interpretation ofa ˆ† as creating a photon occupying the to Hermitian operators, φ → φˆ and π → πˆ , set to satisfy the f wave packet f . equal-time canonical commutation relation

[φˆ(x), πˆ (y)] := iδ(x − y), (2.25) B. The metric per the correspondence principle. Here, (x, y) are spatial coor- In order to analyze gravitational effects in quantum optics dinates on τ . Under this prescription, the ﬁeld operator has experiments in a laboratory environment, we must prescribe

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from the observer’s world line, as considered in this sec- tion, with the “weak-ﬁeld regime,” considered in previous parts of the paper. Second, in the context of general rela- tivity, accelerations and rotations are coordinate-dependent quantities. Therefore, for a free-falling (parallel-transported) observer, such quantities vanish entirely. Then, the ﬁrst non- trivial components in the metric are at O(x2 ) and depend on Reimann curvature [42] (see also Chap. 13 of [1]). All this is not to say that the above construction is invalid or not use- ful. On the contrary, the proper reference frame is extensive enough to accommodate for local gravitational time dilation and general-relativistic frame-dragging effects. Thirdly and lastly, the proper reference frame is quite generic and does not, necessarily, relate to gravitational effects. It is only once Einstein’s ﬁeld equations are considered and a background space-time prescribed therefrom that the local dynamics, ob- served about C, can be attributed to gravitational phenomena. FIG. 2. Schematic of the observer’s world line C through space- 2. Relation to background PN metric time, withe ˆ(0) parallel to the observer’s four-velocity and {eˆ(k)} designating the spatial triad attached to the observer. All measure- One may use the PN formalism (see [2] and Chap. 39 ments are made, by the observer, with respect to these basis vectors. of [1]) to describe the space-time metric near massive bodies in the weak-ﬁeld, slowly moving, and slowly evolving regime, the metric local to a laboratory observer, which can be done in which is sufﬁcient to analyze, e.g., solar-system experiments the so-called proper reference frame—a reference frame nat- aiming to test metric theories of gravity. From the viewpoint urally adapted to an arbitrary observer in curved space-time. of general relativity, the idea is to expand, in successive We follow Refs. [1,42] in the construction of the proper ref- orders of a small parameter , and solve Einstein’s ﬁeld equa- erence frame and its relation to the background PN frame, by tions, thereby obtaining a sufﬁciently accurate background providing key equations and brief explanations, while leaving space-time metric. The small parameter is set by physical explicit details to the references therein. We set the speed of quantities—the Newtonian potential U and the velocity v of light to c = 1inwhatfollows. the body in the PN frame—such that U, v2 ∼ O(2 ). 1. Proper reference frame: The local metric Q Consider an ideal observer moving along their world line The condition of “slowly evolving,” for all quantities ,isset ∂ Q/∂ Q ∼ C(t ), where t is the (proper) time the observer measures with by t i O( ). a good clock. The observer also “carries” with them a space- Now, for example, letting bare subscripts indicate tensor (μ) components in the PN frame, the space-time metric for a solid time tetrad {eˆ(μ)}, associated with coordinates x , such that (i) the time coordinate satisﬁes x(0) = t, which impliese ˆ := rotating sphere (the Earth) in the PN formalism is (0) ⎧ ∂ t is the observer’s four-velocity (tangent to C), and (ii) the 4 (0) ⎨g00 =−(1 − 2U ) + O( ) set {eˆ(k)} is a rigid set of spatial axes “attached” to the observer 5 gμν : g =−4V + O( ) (2.36) (Fig. 2). Thus, the observer measures the passage of time, ⎩ 0i i = + δ + 4 , spatial displacements, angles, local accelerations, rotations, gij (1 2U ) ij O( ) etc., with respect to these basis vectors. The tetrad is chosen where in such a manner that (i) the local metric, g μ ν , reduces to ( )( ) GM the Minkowski metric on C [43], U = , (2.37) r g(μ)(ν)|C = η(μ)(ν), J× r and (ii) the metric, in a neighborhood of C, describes that of V = , (2.38) 3 an accelerating and rotating reference frame in ﬂat space-time, 2r which, to linear displacements in x(k), takes the form and J = Iω, with I being the moment of inertia and ω the rotation rate of the earth in the PN frame. It follows that the ds2 =−(1 + 2γ·x)dt2 + 2(ω × x) · xdt + dx2, (2.35) off-diagonal metric components are of O(3). From here, one where (γ, ω ) are the acceleration and angular velocity that can relate the PN space-time metric to physical quantities that the observer measures with, e.g., accelerometers and gyro- an (proper) observer locally measures, i.e., to the accelerations scopes. Note that these accelerations and angular velocities and rotations (γ, ω ). One accomplishes this in the following are independent of spatial coordinates x(k); though, in general, manner (see [42] for explicit details). they may depend on the observer’s proper time, t. Let parenthetic subscripts indicate tensor components in At this juncture, some general observations and comments the proper reference frame of an observer constrained to about the proper reference frame should be made. First, one the surface of the Earth. Thus, the spatial triad {eˆ(k)} physi- should apparently associate the (small) linear displacements cally corresponds to rigid Cartesian axes stuck to the Earth’s

023024-5 ANTHONY J. BRADY AND STAV HALDAR PHYSICAL REVIEW RESEARCH 3, 023024 (2021) surface, where, e.g., two basis vectors may point along in- register a single-photon event). For identical single photons creasing longitudinal and latitudinal lines at the observer’s (identical polarization, spectral, and temporal proﬁles, etc.) position and the third along the radial normal. Then, the the probability of a coincident detection is zero, due to the following transport equation governs the evolution of the ob- tendency of the photons to bunch in one output port of the server’s tetrad along their world line, beamsplitter or the other. If, however, we consider spectral μ wave packets, we can induce distinguishability by introducing Deˆ (α) μ (β) a relative time delay between the wave packets prior to the =−eˆ β (α), (2.39) dt ( ) beamsplitter, ceteris paribus. As a consequence, the likeli- where D/dt is the covariant derivative along the observers hood of a coincident detection event increases, as the relative μ time delay departs from zero. The transition of the coinci- world line ande ˆ(α) are the vector components of the observer’s tetrad basis in the PN frame. The antisymmetric transport dence probability, from zero to a nonzero value, leads to a tensor (α)(β) encodes gravitational and noninertial effects diplike structure in its functional behavior, with respect to the and is found via time delay (see, e.g., inset of Fig. 4). This dip in coincidence μ events is the well-known HOM dip [11,12]. We mathemati- Deˆν Deˆ = 1 μ (β) − ν (α) . cally describe this contrivance as follows. (α)(β) gμν eˆ(α) eˆ(β) (2.40) 2 dt dt C Consider a two-photon source, which generates indepen- dent single photons, in spatial modes (a, b) and occupying The components of the transport tensor correspond precisely , to the locally measured acceleration and rotational velocity, wave packets ( f g), such that initial two-photon state is i.e., |=aˆ† bˆ† |0= dν dν f g aˆ†bˆ† |0 , (2.43) = γ , = ε ω , f g 1 2 1 2 1 2 (0)( j) : ( j) (i)( j) : (i)( j)(k) (k) with εijk being the Levi-Civita symbol. From this, one where subscripts indicate spectral dependence. Introducing ﬁnds [42] a relative time delay δt in mode a, such thata ˆ → eiνδt aˆ, followedbythe50/50 beamsplitter transformation, γ = a − ∇U, (2.41) † 1 † † † 1 † † with a being the coordinate (e.g., centrifugal) acceleration. aˆ −→ √ (ˆa + bˆ ), bˆ −→ √ (bˆ − aˆ ), Note that, for an earthbound observer ∇U = g, where g ≈ 2 2 . / 2 9 8m s is the gravitational acceleration on the Earth’s sur- leads to the state face. 1 ν δ | = ν ν i 1 t One also ﬁnds an explicit expression for the rotation vector, δt d 1d 2 f1g2e 2 ω = ω 1+ 1 v2 + U + 1 v × γ − 3 v × ∇U + 2∇ × V . × † ˆ† − † ˆ† − † † − ˆ† ˆ† | , 2 2 2 [(ˆa1b2 aˆ2b1) (ˆa1aˆ2 b1b2 )] 0 (2.44) (2.42) with the ﬁrst parenthetic term causing coincident detection The ﬁrst term is the rotation rate of the Earth, as measured events and the second term embodying photon bunching. by an earthbound observer. The second term is the Thomas Note that, for identical monochromatic wave packets (e.g., precession term. Finally, the third and fourth terms are the equivalent Dirac-delta distributions for the spectral functions), geodetic and Lense-Thirring terms, respectively. Considering the ﬁrst term vanishes, independent of any delay, and both the coordinate velocity (and acceleration, a) as being due photons come out in mode a or mode b. solely to the rotational motion of the Earth (hence, v = ω × R To calculate the likelihood of single-photon coincidence with R being the Earth’s radius) one ﬁnds the centrifugal events, we introduce the single-photon projection operators component of the Thomas precession term, v × a, to be 1000 ˆ = ν †| | , times smaller than the gravitational terms. We thus ignore a d aˆν 0 0 aˆν (2.45) this term hereafter. For brevity, we shall also let ω denote the † rotational rate as measured by an earthbound observer. ˆ b = dν bˆν |00|bˆν . (2.46)

C. Two-photon interference and the Hong-Ou-Mandel dip The coincidence detection probability pc, as a function of the The HOM effect is a two-photon quantum-interference time delay, is then given by effect, which embodies the general phenomena of boson p (δt ) =δ |ˆ ⊗ ˆ |δ . (2.47) (photon) bunching—the inclination for identical bosons to c t a b t congregate—as a consequence of the commutation relations For identical, Gaussian wave packets ( fk = gk) with spectral for a bosonic ﬁeld, and can be used to quantify the distin- width σ , the coincident probability [44] reduces to guishability of single photons [14]. For example, a primitive 1 1 2 2 HOM conﬁguration consists of a two-photon source, which pc(δt ) = 1 − exp − σ δt . (2.48) creates independent single-photon wave packets, and a 50/50 2 2 beamsplitter, where the photons interfere. At the output ports For δt = 0, the coincident probability vanishes, since the of the beamsplitter, one positions photodetectors, records co- single photons jointly exit from one port of the 50/50 beam- incident detection events, and quantiﬁes the probability of a splitter or the other; however, this is the ideal case, without coincident detection (i.e., the likelihood that both detectors noise and with unit visibility.

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Digression on noise: Visibility, ﬂuctuations, and gravitational decoherence The expressions in the previous section dealt with ideal HOM interference, where one has perfect control over all distinguishability parameters. In reality, however, such is not the case, and one is led to the more general expression for the coincidence probability δ = 1 − V − 1 σ 2δ 2 , pc( t ) 2 1 exp 2 t (2.49) where V is the visibility of the interference experiment, de- ﬁned by pmax − pmin V := c c , (2.50) max pc with max = δ , min = δ . pc : lim pc( t ) pc : lim pc( t ) δt→∞ δt→0 Systematic control allows one to maintain the visibility cri- terion V > 1/2, which demarcates the boundary between sufﬁcient quantum and/or classical descriptions for the ﬁelds [12,13]. One source of noise, pertinent to this work, is temporal FIG. 3. Space-time diagram of counterpropagating null rays. Two electromagnetic signals (red and blue curves) counterpropagate ﬂuctuations arising from, say, random changes in the relative along a common path in space. Relativistic frame-dragging effects paths taken by the photons. One can model this as a back- cause the pitch of the space-time helices to differ, which leads to ground Gaussian noise, where the time delay is a random δ σ differing arrival times at a stationary detector system. All motion is variable with mean t and ﬂuctuation . The mathematical constrained to a world tube of spatial size ∼ min(c2/g, c/ω ). particulars are inconsequential (see, e.g., [46] for various noise models in optical interferometry); however, since we concern ourselves with gravitationally induced temporal shifts phenomenological models have been proposed, including in the HOM dip, we require the size of the ﬂuctuations σ treatments where one links the collapse of the wave func- to be smaller than the average size of the shift, in order for tion to gravitational decoherence or semiclassical treatments, the gravitational phenomena to be resolvable in practice. Such which replace the sources in Einstein’s ﬁeld equations with the is easier for, e.g., common-path (Sec. III A) versus dual-arm expectation value of the stress-energy tensor operator. These (Sec. III B) interferometry. models generally rely on parameters determining the scale A more fascinating noise source is that of at which quantum-gravity effects become relevant, i.e., the gravitational/space-time ﬂuctuations, which could, in Planck scale; however, this scale is discouragingly minuscule ∼ −35 principle, measurably affect the visibility of optical (Planck length, P 10 m). Therefore, it is difﬁcult to interference signatures. General relativity treats space-time as imagine that quantum-gravitational ﬂuctuations will become a dynamical variable, hence, gravitational ﬂuctuations, either prevalent in interferometry scenarios, in the near to mid term. of classical or quantum origin, lead to a loss of coherence in Though, with the continuing development of large-scale inter- quantum systems [47]. Unlike other sources of noise, which ferometers and ever-improving sensitivities, perhaps bounds can be eliminated by lowering temperatures and creating may be put on these scales in the future. extreme vacua, it is impossible to get rid of gravitational decoherence. III. SETUP AND RESULTS On the classical side, space-time ﬂuctuations can have well-determined and various origins, from the seismic activity A. Frame dragging and the HOM dip of nearby masses, to the more exotic incoherent super- Consider a HOM-interference experiment, wherein two positions of gravitational waves and/or artifacts of other single-photon wave packets counterpropagate through a gravitational sources scattered about the universe. One can common-path interferometer constrained to the Earth’s model such ﬂuctuations as stochastic waves, with unequal surface (Fig. 1). Relativistic effects, e.g., the Sagnac, Lense- time correlation functions of metric components, which one Thirring, and geodetic effects, induce distinguishability be- characterizes with spectral distributions. Then, for example, tween the counterpropagating paths, leading to a temporal the geometry of an interferometer along with these spec- shift in the HOM dip, which would otherwise be at δt = 0 tral distributions determine the visibility of an interference if relativistic effects were absent (Fig. 3). signature and can thus, in principle, have observable mani- The off-diagonal terms of the metric perturbation are the festations [49]. sole contributors to the time delay, ceteris paribus, which one On the quantum side, however, a fully determined and can calculate via concurrent use of Eqs. (2.14) and (2.35), in consistent understanding of the origins of quantum space- the proper reference frame of an observer. For a single path time ﬂuctuations is far from satisfactory. Nevertheless, several around the interferometer loop, say, the right-handed path, one

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10-9 4 HOM dip shift ( = /2) 0.5 4 0.4 3 0.3

m) 0.2

2 Coincidence prob. 0.1 2 0 -2 0 2 Tunable time delay 1 0 0 = 0 = 0 -2 Photon path delay ( = /4 -1 = /4 = /2 = /2 -4 -2 0123 0 0.5 1 1.5 2 2.5 3 (rad) (rad)

FIG. 4. Photon path delay (cδt,wherec is the speed of light) as a function of the interferometer-orientation angle α, for a common-path conﬁguration, with A = 1km2. Solid (red), dotted (yellow), and dashed (black) lines represent various latitudes θ at which a detector is placed. Left: Path delay due to the Sagnac effect. The inset is a representative ﬁgure for the HOM-dip shift (the spectral widths have been exaggerated for visual clarity) due to the relative delay between the counterpropagating modes, at a ﬁxed orientation angle α = π/2 and at various latitudes θ. Similar results hold for, e.g., a ﬁxed latitude and various orientation angles; this latter scenario being more practical, as one would be at a ﬁxed position on earth (θ = constant) while performing the experiment at several interferometer-orientation settings (various α). Right: Path delay due to the combined Lense-Thirring and geodetic effects.

c ﬁnds (reintroducing factors of ) To evaluate the above expression, for an observer con- strained to the Earth’s surface, we choose an earth-centered δ = −1 (i) = −2 · ω × tRH c dx h(0)(i) c dx ( x) spherical-coordinate system (ˆer, eˆθ , eˆφ ) comoving with the observer, with ω aligned along the polar axis [50]. The − 2ω · A Earth’s radius R and the observer’s latitude θ establishes the = c 2 dA · ∇ × (ω × x) = , (3.1) c2 earthbound constraint, since due to the intrinsic axisymmet- where A is the areal vector, perpendicular to the inter- ric structure of the background space-time, characterization ferometry plane, and we have used the spatial-coordinate of the observer’s longitude is superﬂuous. Furthermore, we A = A α − independence of ω . A similar relation holds for the left- assume the areal vector to take the form (ˆer cos α α handed path, albeit with opposite sign, such that eˆθ sin ), with being the angle between the observer’s nor- mal and the areal vector of the interferometer. Under these ω · A 4 considerations, the coordinate expression for the time delay is δt := δtRH − δtLH = , (3.2) c2 then with ω given by Eq. (2.42).

4ωA 4ωA 2GM GI δt = cos(θ − α) + sin θ sin α + (2 cos θ cos α − sin θ sin α) , (3.3) 2 2 2 2 3 c c c R c R Sagnac geodetic + Lense-Thirring which agrees with [51]. We note that the general relativistic to then consider is the size of the temporal shift relative to the −9 contributions are on the order of rS/R ∼ 10 times smaller physical size of the interferometer; thus, deﬁne 2 than the Sagnac contribution, where rS = 2GM/c is the Schwarzschild radius of the Earth. δt F := , We now substitute the above expression into the coin- A (3.4) cidence probability formula (2.48) and plot the results for various parameter values (θ,A,α). See Fig. 4 for analysis. which has dimensions s/km2 and where δt is crudely the dif- ference between the maximum and minimum time delay. This Order-of-magnitude estimates quantity is of practical interest as, e.g., given some physical Perhaps more insightful than a full-blown analysis is an constraint on the areal dimensions, one can see what level of order-of-magnitude estimate for these effects. A key quantity precision is required in order to observe relativistic effects.

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As an example, the quantity F, for the Sagnac effect, goes as 4ω F ∼ ∼ 10−15 s/km2, Sag c2 and since the Lense-Thirring and geodetic effects are a bil- lionth of the size of the Sagnac effect, we have −9 −24 2 FGR ∼ 10 FSag ∼ 10 s/km , where the subscript GR concurrently signiﬁes the Lense- Thirring and geodetic effects. The smallness of the Lense-Thirring and geodetic effects is immediately concerning, as it implies stringent experimental constraints (high precision and control of a large-area inter- ferometer); however, this is not to say that observation of such is impractical. On the contrary, there is ongoing research towards terrestrially measuring these frame-dragging effects with classical-optical interference [51,52]. With regard to a similar undertaking in HOM interferometry, a full feasibility analysis is duly wanting [53]. On the other hand, observing signatures of the Sagnac effect (induced by the eEarth’s rotation) via HOM interfer- FIG. 5. A dual-arm conﬁguration with a two-photon source (or, ence appears experimentally accessible. Recently, phenomena a beamsplitter and two photon-detectors) at C, mirrors at B/D,and akin to this were observed by Restuccia et al. [27] (and a beamsplitter and two photon-detectors (or, a two-photon source) at analyzed further in [28]), wherein the authors constructed a the origin A. The local reference frame is ﬁxed to the earth’s surface θ HOM conﬁguration (Fig. 1) upon a rotating table with tunable at a latitude .Thez-axis is parallel to the radial vector eˆr , and the rotation rate, and subsequently discovered a rotation-induced y-axis points along the longitudinal line eˆφ at the observer’s position. The locally measured angles, α and β, determine the orientation of shift in the HOM dip (proportional to the rotation frequency), the interferometer, with respect to the horizontal and vertical planes. as one would predict with the formalism presented here. To The area of the interferomter is A = ld. measure an analogous effect induced by the Earth’s rotation, one requires much greater time-delay precision, due to the ω ∼ −5 −1 x y z minute rotational frequency of the Earth 10 s .This DA : = = , seems presently achievable with modest resources. For ex- l cos α sin β −l sin α l cos α cos β ample, considering a ﬁber loop of radius r and N turns, the x − d sin α sin β y − d cos α z − d sin α cos β effective area of the interferometer is A = Nπr2 or A = lr/2, CB : = = . l cos α sin β −l sin α l cos α cos β where l = 2πNr is the length of the ﬁber. Taking l ≈ 2km −3 2 and r ≈ 1 m implies ASag ∼ 10 km , which in turn implies By concurrent use of Eqs. (2.14) and (2.35), we calculate the a required time-delay precision at the attosecond (1 as = phase difference, δS, accrued along a path due to the uniform 10−18 s) scale, on a par with current experiment [55]. Proof- acceleration γ, of-principle demonstrations like these are quite intriguing, as 1 they constitute quantum ﬁelds in noninertial reference frames. δS =− dx(0)k(0)h = k(0)c−3 dx · kˆ γ · x. 2c (0)(0) B. Gravitational time dilation and the HOM dip (3.5) 2 For example, setting γ = geˆz, where g ≈ 9.8m/s is the grav- We now consider a HOM-interference experiment, for itational acceleration on the Earth’s surface, we ﬁnd the phase a dual-arm conﬁguration, and calculate the time delay due along the segment BA to be to the effects of the uniform gravitational acceleration and centrifugal acceleration measured by an observer ﬁxed upon g A δS = k(0) (dx · nˆ )z, the Earth’s surface (see Fig. 5). We ignore the off-diagonal BA 3 BA c B rotation-induced contributions, as they are equivalent (up to a factor of 2) to the calculations of the previous section. wheren ˆBA is the unit vector along BA. Integrating along each The unit vectors, along each arm of the interferometer, are arm of the interferometer, we then obtain given by g δS = k(0) d2 sin α cos β, = = α β, α, α β , BA 2c3 nˆBA nˆCD (sin sin cos sin cos ) = = α β,− α, α β . g l2 nˆDA nˆCB (cos sin sin cos cos ) δS = k(0) cos α cos β + ld sin α cos β , CB c3 2 The coordinate equations for the lines along the arms are thus x y z g d2 BA : = = , δS = k(0) sin α cos β + ld cos α cos β , d sin α sin β d cos α d sin α cos β CD c3 2 − α β + α − α β g x l cos sin y l sin z l cos cos δ = (0) 2 α β. CD : = = , SDA k l cos cos d sin α sin β d cos α d sin α cos β 2c3

023024-9 ANTHONY J. BRADY AND STAV HALDAR PHYSICAL REVIEW RESEARCH 3, 023024 (2021)

Since the observer is ﬁxed to the Earth’s surface (a non- ω2 2 (0) R d inertial reference frame), they also measure a centrifugal δSDC = k sin θ cos θ cos α − ld sin α c3 2 acceleration a = ω2R sin θ(sin θeˆ + cos θeˆ ). d2 z y + sin θ cos β sin α + ld cos α , Then, setting γ = a in Eq. (3.5), we ﬁnd additional phase 2 differences along the different interferometer arms, ω2R δS = k(0) l2 sin θ(sin θ cos α cos β − cos θ sin α). ω2R DA 2c3 δS = k(0) d2 sin θ(cos θ cos α + sin θ sin α cos β), BA 2c3 ω2 2 (0) R l δSCB = k sin θ sin θ cos β cos α + ld sin α Therefore, considering identical single photons traversing c3 2 separate paths ABC and CDA, we obtain the total time delay (0) l2 between the paths (after dividing by the mean frequency k ), − cos θ sin α + ld cos α , due to both the gravitational and centrifugal accelerations, 2

gA ω2RA δt = cos β(cos α − sin α) + sin θ[sin θ cos β(cos α − sin α) + cos θ(cos α + sin α)], (3.6) c3 c3 gravitational, g centrifugal, a where A = ld is the interferometer area. We note that a/g ∼ for various interferometer conﬁgurations. This was 10−2. Plots for these results are shown in Figs. 5 and 6. theoretically achieved by quantizing a massless scalar ﬁeld in a weak gravitational ﬁeld (geometric quantum optics in curved Order-of-magnitude estimates space-time, in the weak-ﬁeld regime; Sec. II A). Applying this formalism to a terrestrial laboratory setting (Sec. II B), Consider the photon time delay per unit area, characterized we showed that, in principle, gravitational effects can induce F by the quantity [Eq. (3.4)] and induced by gravitational and observable changes in quantum-interference signatures, centrifugal accelerations. For the gravitational acceleration, using HOM interference as an exemplar (Secs. II C and III). we have Noninertial effects, due to the Earth’s rotation and centrifugal g −19 2 Fg ∼ ∼ 10 s/km , acceleration, were also considered, and the potentiality of c3 practical demonstrations was brieﬂy analyzed. Though the which is 103 times smaller than the Sagnac effect and 106 latter analysis was unfulﬁlling (a full feasibility, in this regard, times larger than the minute geodetic and Lense-Thirring ef- is still wanting), the landscape of potential proof-of-principle fects considered in Sec. III A. Similarly, for the centrifugal demonstrations, like these, appears promising. The reason acceleration, we have being that analogous enterprises exploring, e.g., gravitational ω2 time dilation via single-photon interference, with tabletop R −21 2 Fa ∼ ∼ 10 s/km . long-ﬁber spools [54] or with quantum satellites [32], have c3 been pursued and found feasible with current technology. Note that Lyons et al. [55] recently achieved HOM time- Further inspiration for this potentiality comes from consider- delay resolution at the attosecond scale (1 as = 10−18 s), ation of recent experimental endeavors: a HOM-interference closely approaching the scale implied by Fg, but with a experiment in a rotating, noninertial reference frame [27]; much smaller interferometer area than required here. Nev- high-precision HOM time-delay resolution [55]; and current ertheless, a tabletop HOM experiment of this sort seems attempts to measure relativistic frame-dragging effects with a achievable with present or near-term technology. Indeed, classical, optical, earthbound system [51,52]. Hilweg et al. [54] investigated the feasibility of a similar Apart from fundamental physics considerations, we should experiment aimed at measuring gravitational time-dilation mention that quantum states of light and quantum measure- effects via single-photon interference, with a variant Mach- ment strategies could also be leveraged for more efﬁcient Zehnder interferometer, and found that, though challenging, estimation of gravitational parameters, such as, e.g., the PN a tabletop experiment is possible with long-ﬁber spools and parameters of general relativity or the gravitational parame- active phase stabilization. An analogous feasbility analysis ters of rotating massive bodies (see, e.g., [56,57]). The main for a HOM experiment, leveraging parameter estimation tech- idea is to encode general relativistic parameters into optical niques, like those used in [55], should be pursued. phases (as done here), using exotic, quantum states of light as a resource, while taking advantage of quantum parameter- estimation strategies, resulting in a “quantum advantage” over IV. SUMMARY AND CONCLUSION any classical measurement scheme (see, e.g., [58–60] for gen- In this work, we showed how to encode general-relativistic eral discussions in this regard). Indeed, gravitational wave effects (e.g., frame-dragging and gravitational time-dilation observatories are already employing such ideas for enhanced effects) into multiphoton quantum-interference phenomena, phase sensitivity [61]. It is thus probable that the “quantum

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10-4 10-4 4 4 = 0 = 0 = /4 = /4

m) = /2 2 = /2 2

0 0 -2

-2 -4 Photon path delay (

-4 -6 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5 (rad) (rad) 10-6 10-6 1.5 1.5

m) 1 1

0.5 0.5 0 0 -0.5 = 0 -0.5 = 0

Photon path delay ( -1 = /4 = /4 = /2 = /2 -1 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 3.5 (rad) (rad)

FIG. 6. Photon path delay (cδt), for a dual-arm conﬁguration, with A = 1km2. Top left (right): Path delay induced by the gravitational acceleration g ≈ 9.8m/s2, as a function of the orientation angle β (α). Bottom left (right): Path delay induced by the centrifugal acceleration a ≈ 3.4 × 10−2 m/s2, as a function of the latitude θ, for various orientation angles α (β) and with β = 0(α = 0) for all curves. advantage” that quantum states of light permit will also appear witnessed by individual quanta of the electromagnetic ﬁeld, in other tests of general relativity, in the near to mid future; extending the validity of general relativity down to the Hilbert though we do not delve into the details of such in this work. space level. Lastly, one may view our result as a quantum analog to the seminal proposal of Scully et al. [62], made nearly 40 years ago, who originally investigated the potentiality of observing ACKNOWLEDGMENT gravitational frame dragging with classical electromagnetic ﬁelds. In a similar fashion, a proof-of-principle demonstration A.J.B. acknowledges support from the National Science of our results would signify the dragging of space-time as Foundation through Grant No. CCF-1838435.

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