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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 , Department of Physics and , 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 on a classical, curved - has led to the enthralling theoretical of, e.g., hole evaporation and creation in the early, expanding . However, the prospects of observing such in the near to mid future seem far from practical. This is discouraging because extrapolation of “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 , 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 , we investigate the Hong-Ou-Mandel (HOM) effect—a two- quantum-interference effect—in the space-time of a rotating spherical . In particular, we propose and analyze a common-path setup restricted to the surface of the and show that, in principle, gravitational frame dragging induces shifts in the two-photon HOM dip. For completeness and correspondence with current literature, we also analyze the of gravitational , for a dual-arm interferometer. The formalism thus presented establishes a 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 induced by Earth’s , 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 .

DOI: 10.1103/PhysRevResearch.3.023024

I. INTRODUCTION The maturation of quantum technologies complements the theoretical maturation of quantum fields in — General relativity and quantum constitute the a formalism describing the of quantum fields on foundation of , 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- distribution of nearby servation requiring the principles of both general relativity and by way of the [1,2]—which appear and 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 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. ” [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 in a locally curved space-time. In Sec. II B, we discuss the 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 (−, +, +, +). 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 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 , 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 of cut distinction between classical and quantum explanations as general relativity. Such a model disregards - well as between Newtonian and general-relativistic explana- dependent effects but accurately accounts for the 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 /  relativity (beyond rectilinear ) and quan-  iS φ = α e k , (2.2) tum mechanics into four classes: (i) classical Newtonian k k 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 -vector mode which φk is centered about. The wave (iv) the quantum 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 . One can classical metric theory of gravity and physical probe systems show that it also satisfies a 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 “” 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, , 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 , 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 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 , 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 - ; 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 + , := 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 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 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 -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 . We decompose the field in the |0 := |0k , basis set {uk}, k 3  φ = d k(akuk + c.c), (2.22) being the 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 ∗ = 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 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 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 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 . 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 line, as considered in this sec- tion, with the “weak-field regime,” considered in previous parts of the paper. , 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 [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 , ob- served about C, can be attributed to gravitational phenomena. FIG. 2. Schematic of the observer’s C through space- 2. Relation to background PN metric time, withe ˆ(0) parallel to the observer’s four- 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 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 of quantities—the Newtonian potential U and the velocity v of 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 (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 (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 in flat space-time, 2r which, to linear displacements in x(k), takes the form and J = Iω, with I being the moment of 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 and that can relate the PN space-time metric to physical quantities that the observer measures with, e.g., and gyro- an (proper) observer locally measures, i.e., to the accelerations scopes. Note that these accelerations and angular 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 , 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 ). For identical single photons creasing longitudinal and latitudinal lines at the observer’s (identical polarization, spectral, and temporal profiles, etc.) 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., 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 term, v × a, to be 1000 ˆ = ν †|  | , 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 p (δt ) =δ |ˆ ⊗ ˆ |δ . (2.47) (photon) bunching—the inclination for identical 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 ) 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 ); 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 -energy . 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 scale; however, this scale is discouragingly minuscule  ∼ −35 principle, measurably affect the visibility of optical (, 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 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 , to the more exotic incoherent super- Consider a HOM-interference experiment, wherein two positions of gravitational 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 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 ) 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- (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 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 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 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 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

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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, 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/, 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 of our results would signify the dragging of space-time as Foundation through Grant No. CCF-1838435.

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