sign in sign up
<<
Home , Rupert Ursin, Thomas precession, World line

Relativistic frame-dragging and the Hong-Ou-Mandel dip — a primitive to gravitational effects in multi- quantum-interference

Anthony J. Brady∗ and Stav Haldar Hearne Institute for Theoretical , Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana, 70803, USA (Dated: June 9, 2020) We investigate the Hong-Ou-Mandel (HOM) effect – a two-photon quantum-interference effect – in the - of a rotating spherical mass. In particular, we analyze a common-path HOM setup restricted to the surface of the earth and show that, in principle, general-relativistic frame-dragging induces observable shifts in the HOM dip. For completeness and correspondence with current literature, we also analyze the emergence of gravitational time-dilation effects in HOM interference, for a dual-arm configuration. The formalism thus presented establishes a basis for encoding general- relativistic effects into local, multi-photon, quantum-interference experiments. Demonstration of these instances would signify genuine observations of quantum and general relativistic effects, in tandem, and would also extend the domain of validity of , to the arena of quantized electromagnetic fields.

I. INTRODUCTION i.e., in Hong-Ou-Mandel (HOM) interference [11–13]– for various interferometric configurations. We show, General relativity and constitute in general, how to encode general-relativistic effects the foundation of modern physics, yet at seemingly dis- (frame-dragging and gravitational time-dilation) into parate scales. On the one hand, general relativity pre- multi-photon quantum-interference phenomena, and we dicts deviations from the Newtonian concepts of abso- show, in particular, that the background structure of lute space and time – due to the mass-energy distribu- -time induces observable changes in a HOM- tion of nearby matter and by way of the equivalence interference signature. principle [1,2] – which appear observable only at large- The motivation behind our investigation is twofold. distance scales or with high-precision measurement de- For one, HOM interference is simple to analyze, yet is vices [3]. On the other hand, quantum mechanics pre- a primitive to bosonic many-body quantum-interference dicts deviations from Newtonian concepts of determin- phenomena [13–15] and even “lies at the heart of linear istic reality and locality [4] and appears to dominate in optical quantum computing” [16]. Furthermore, there regimes in which general relativistic effects are typically exists a simple yet sharp criterion partitioning the suf- and safely ignored. This dichotomous paradigm of mod- ficiency of classical and quantum explanations. Namely, ern physics is, however, rapidly changing due to a) the if the visibility V of the HOM-interference experiment ever-increasing improvement of quantum measurement is greater than 1/2, a quantum description for the fields strategies and precise control of quantum technologies is sufficient, whereas a classical description fails to ad- [5,6] and b) current efforts to extend quantum mechan- here to observations ([12, 13]; though, see [17] for recent ical demonstrations to large-distance scales – like, e.g., criticism of the visibility criterion). For two, the emer- bringing the quantum to space [7,8]. gence of gravitational effects in photonic demonstra- The maturation of quantum technologies complements tions beckons general-relativistic explanations – whereas, the theoretical maturation of quantum fields in curved- for massive , the boundary between general- space – a formalism describing the evolution of quantum relativistic and Newtonian explanations is oft-blurred. fields on the (classical) background space-time of gen- As rightly pointed-out and emphasized by the authors of eral relativity – which has been finely developed over refs. [18, 19], within any quantum demonstration claim- arXiv:2006.04221v1 [quant-ph] 7 Jun 2020 last fifty-odd years [9, 10]. Alas, even with such develop- ing the emergence of general-relativistic effects, a concur- ments, there has not yet been a physical observation re- rent, clear-cut distinction between classical and quantum quiring the principles of both general relativity and quan- explanations as well as between Newtonian and general- tum mechanics for its explanation. With these consider- relativistic explanations must exist, in order to assert ations in mind, the present thus seems a ripe time to fur- that principles of general relativity and quantum me- nish potential proof-of-principle experiments capable of chanics are at play, in tandem. and unambiguously exploring the cohesion of general rel- To place our work in a sharper context, observe that ativity and quantum mechanics, in the near-term. Such one may broadly decompose the mutual arena of gravi- is the aim of our work. tation/relativity (beyond rectilinear motion) and quan- In this report, we investigate the emergence of grav- tum mechanics into four classes: (i) classical Newtonian itational effects in two-photon quantum interference – in quantum mechanics [20–23], (ii) non-inertial reference frames in quantum mechanics [24–28], (iii) clas- sical general relativity in quantum mechanics [18, 19, 29– 34], and (iv) the quantum of gravity [35–39]. Al- ∗ Corresponding author: [email protected] though there has been a great deal of theoretical investi- 2 gation in these areas, there has only been observational about. The wave equation (II.1) in the geometric op- evidence for (i) and (ii) (see, e.g., refs. [20–23] and [24– tics approximation yields transport equations along the 0 27], respectively), which indeed has supported consis- vector field ∇µSk, tency between these formalisms when they are concur- rently considered – e.g., consistency between Newton’s µ 0 0 theory of gravitation and standard non-relativistic quan- (∇ Sk)(∇µSk) = 0 (II.3) tum theory, in regimes where both are prevalent. Our 1 (∇µS0 )(∇ ln α0 ) = − ∇µ∇ S0. (II.4) work serves as a contribution to (iii) – where one must k µ k 2 µ describe the gravitational field by a classical metric the- 0 ory of gravity and physical probe-systems via quantum The first equation implies that ∇µSk is a null vector. mechanical principles. One can show that it also satisfies a equation We compartmentalize our paper in the following way. (∇µS0 ) ∇ (∇ S0 ) = 0. In SectionIIA, we provide a model for the electro- k µ ν k magnetic field in curved space-time, in the weak-field Thus, the transport equations, (II.3) and (II.4), deter- regime and proceed to quantize the field under suitable mine how the eikonal and amplitude, respectively, evolve 0 approximations. One can consider this formalism, in along the null ray drawn out by ∇µSk. Observe that the succinct terms, as geometric quantum-optics in curved evolution of the eikonal along the ray is sufficient to de- space-time, in the weak-field regime. In SectionIIB, termine the field φ entirely. These are the main results we discuss the metric local to an observer and its re- of geometric optics. lation to the background Post-Newtonian (PN) metric, Lastly, from the transport equations, one may derive thus prescribing the space-time upon which the quan- a conservation equation tum field of sectionIIA propagates. In SectionIIC, µ  0 0 2 we provide some background on HOM interference and ∇ ∇µSk|αk| = 0, (II.5) comment on interferometric noise and gravitational de- coherence in context. In SectionIII, we combine and which, via Gauss’s theorem, leads to a conserved ‘charge’ apply the formalism of previous sections to various inter- ferometric configurations, explicitly showing that gravi- µ 0 0 2 Qk := dΣ (∇µSk) |αk| , (II.6) tational effects, in principle, induce observable changes ˆΣ in HOM-interference signatures. We also provide order- with Σ being a spacelike hypersurface. Physically, the of-magnitude estimates for these effects and briefly dis- conservation equation is a conservation of photon flux cuss observational potentiality. Finally, in SectionIV, such that the number of Q is a constant for all we summarize our work. k time. The equations thus posed are precisely that of geo- II. METHODS metric optics in curved space-time, excepting a transport equation for the polarization vector of the field (see, e.g., refs. [1, 40]). A. Modeling quantum optics in curved space-time

1. Geometric optics in curved space: a brief review 2. Geometric optics in the weak-field regime

We model the electromagnetic field by a free, massless, We work with laboratory or near-earth experiments scalar field φ satisfying the (minimally coupled) Klein- in mind. Therefore, it is sufficient to consider general Gordon equation relativity in the weak-field regime. We assume local gravitational fields, , etc. µ are sufficiently weak and consider first-order perturba- ∇ ∇µφ = 0, (II.1) tions hµν about the Minkowski metric ηµν such that the space-time metric g , in some local region in space, can where ∇ is the metric compatible covariant derivative of µν µ be written as general relativity. Such a model disregards polarization- dependent effects but accurately accounts for the ampli- gµν = ηµν + hµν . (II.7) tude and phase of the field. We assume the field to be quasi-monochromatic in the The metric perturbation induce perturbations of the geometric optics approximation [1, 40] so that it can be eikonal and amplitude about the Minkowski values rewritten (in complex form) as (Sk, αk), i.e.

0 0 0 iSk S = S + δS (II.8) φk = αke , (II.2) k k k 0 αk = αk(1 + εk), (II.9) with S0 the eikonal (or phase), α0 a slowly-varying am- plitude with respect to variations of the eikonal, and with (δSk, εk) being the leading perturbations of O(h). k denoting the wave-vector mode which φk is centered Defining kµ := ∂µSk as the Minkowski wave-vector, we 3

FIG. 1. Schematic of a Hong-Ou-Mandel interferometer, in a common-path configuration, located on the earth’s surface. The gray boxes just after the photon source are fiber-coupler and beamsplitter systems, with high transmissivity, used to ensure common-path propagation (cf. [27, 28]). show that the general transport equations, (II.3) and transport equations (II.4), imply the O(1) transport equations 1 µ ν δSk = − dx k hµν (II.14) µ µ 2 ˆ k kµ = 0 =⇒ k ∂µkν = 0 (II.10) γ 1 µ 1 µ ε = − dλ (∂ ∂µδS + Γµ kν ) . (II.15) k ∂µαk = − ∂µk , (II.11) k µ k µν 2 2 ˆγ which are the geometric optics equations in Minkowski Observe that, to first non-trivial order,   space-time , and O(h) transport equations 1 µ µ ν (1 + εk) = exp − dλ (∂µ∂ δSk + Γ µν k ) . 1 2 ˆγ kµ∂ δS = − kµkν h (II.12) µ k 2 µν Combining this with the O(1) amplitude and using the µ 1 µ µ ν  fact that k ∂µεk = − ∂µ∂ δSk + Γµν k , (II.13) 2 µ 0 µ µ µ ν ∇µ∇ Sk = ∂µk + ∂µ∂ δSk + Γ µν k , with to O(h), we obtain

µ 1 µκ  1  Γ βν = η (∂βhνκ + ∂ν hβκ − ∂κhβν ) , 0 µ 0 2 αk = exp − dλ (∇µ∇ Sk) , (II.16) 2 ˆγ being the torsion-free connection coefficients of general where the initial amplitude is taken as unity, for sim- relativity to O(h)[1]. Therefore, one observes that the plicity. This is analogous to the generic solution for the µ null ray k , together with the metric perturbation hµν , amplitude in curved space-time – excepting that the ge- 0 0 uniquely determines the eikonal Sk, the amplitude αk ometric , over which the integral is evaluated, is [per eqs. (II.10)-(II.13)], and hence, the field φ [eq. the Minkowski null geodesic and not the general null (II.2)]. 0 geodesic generated by ∇µS . a. Generic solutions: Let us suppose that the null k curve γ is parameterized by parameter λ and has (Minkowski) coordinate expression xµ(λ) such that kµ = 3. Approximately orthonormal modes and quantization dxµ/dλ. This implies dλ = dx0/k = d~x · ~k/k2, with k2 := ~k · ~k the Euclidean scalar product. With this sup- We wish to investigate local (e.g. in a finite region position, we find general solutions to the perturbative of space) quantum optics experiments on a curved back- 4

ground space-time, in the weak-field regime. Hence, we with (t, ~x) being Lorentz coordinates on Σ and f(h) a must produce a quantum description for the field. To function of O(h) that varies over the length scale `. It is do so, we introduce a set of approximately orthonormal (1) then sufficient to show that (uk, uk0 )KG only has support solutions to the Klein-Gordon equation (II.1), which – at k = k0. This is done in two steps. For k−k0  `−1, the upon quantization – allows us to define creation and an- phase rapidly oscillates with respect to f(h) over the en- nihilation operators of the quantum field obeying the tire integration volume. Thus, by the Reimann-Lebesgue usual commutation relations [10]. From this construc- lemma, the integral averages to zero. On the other side, tion, a Fock space can be built and quantum optics ex- for k − k0 ∼ `−1, we have (k − k0)x ∼ O(h) ∀ x ∈ Σ. periments analyzed per usual. We note that the follow- Thus, we may set k = k0 in the integral, to accuracy ing formalism describes ‘local’ field solutions which are Oh2. Therefore, the integral (II.21) only has support valid descriptions of the field, in some finite region of at k = k0 to Oh2, QEI. space (of size  `, see below), to some specified accu- 2 b. Canonical quantization: We decompose the field racy [to O h , see below]. in the basis set {u }, a. Approximately orthonormal modes: From the k generic solutions, define the mode uk which satisfies the Klein-Gordon equation in the weak-field regime 3~ φ = d k (akuk + c.c) , (II.22) µ ˆ 0 i(k xµ+δSk) uk := Nkαke , (II.17) with basis coefficients, ak, found from the Klein-Gordon where Nk is a normalization constant. We prove that, inner product to some error, the set {uk} define an orthonormal set of classical solutions to the Klein-Gordon equation, with ak ≈ (uk, φ)KG (II.23) respect to the Klein-Gordon inner product [10] ∗ ak ≈ (φ, uk)KG. (II.24)

µ ∗ ∗ (f, g)KG := i dΣ (f ∂µg − g∂µf ) , (II.18) We proceed to canonically quantize the field by intro- ˆΣ ducing the canonical momentum for the classical Klein- where Σ ia a suitable normalization volume, i.e. a spatial Gordon field [10]

hypersurface of sufficient size. We assume plane-waves 1 2 for the Minkowski modes. The sketch of the proof goes π = |h| ∂τ φ, as follows. We wish to show that which is defined on a future-oriented spatial hypersur- face Στ , with |h| the determinant of the induced spatial 0 0 ~ ~ metric on Στ . Note that Στ is simply the surface of (uk, uk )KG ≈ δ(k − k ), (II.19) simultaneity at time τ. 2 to O h . Consider hµν to be time-independent and to The field φ and canonical momentum π are then pro- vary over a characteristic length scale ∂h ∼ `−1, where, moted to Hermitian operators, φ → φˆ and π → πˆ, set to for example, ` ∼ c2/g ∼ 1ly for a terrestrial gravitational satisfy the equal-time canonical commutation relation g ≈ 9.8 m/s2. We further assume the inte- h i gration volume Σ to obey Σ1/3  `, i.e. h  1 within φˆ(~x), πˆ(~y) := iδ(~x − ~y), (II.25) the spatial region of interest (the laboratory). Then, an- alyzing the Klein-Gordon inner product perturbatively, per the correspondence principle. Here, (~x,~y) are spatial we see that coordinates on Στ . Under this prescription, the field (0) (1) operator has a basis decomposition (uk, uk0 )KG ≈ (uk, uk0 )KG + (uk, uk0 )KG , (II.20)  †  (0) ˆ 3~ ∗ ~ ~ 0 φ = d k aˆkuk +a ˆkuk , (II.26) where (uk, uk0 )KG ∝ δ(k − k ) is the flat-space inner ˆ II.1 (1) product for the plane-wave modes and (uk, uk0 )KG is the inner product containing terms to O(h), which can with (ˆa, aˆ†) the annihilation and creation operators, be written as which are found per above ˆ aˆk ≈ (uk, φ)KG (II.27) ~ ~ 0 (1) −i(ωk−ω 0 )t 3 i(k−k )·~x (u , u 0 ) ∝ e k d ~x e f(h), † ˆ k k KG aˆ ≈ (φ, uk)KG. (II.28) ˆΣ k (II.21) Given the basis decomposition and canonical commuta- tion relation (II.25), one can show that

aˆ , aˆ† = (u , u ) (II.29) II.1 We approximate a continuum for the normalization volume, k q k q KG whilst maintaining the constraint Σ1/3  `. ≈ δ(~k − ~q). (II.30) 5

c. Fock space and wave packets: The (approximate) 1. Proper reference frame: the local metric orthonormality of the classical mode solutions, together with the commutation relations, is sufficient to build a Consider an ideal observer moving along their world- Fock space spanned by states of the form [10] line C(t), where t is the (proper) time the observer mea-

† sures with a good clock. The observer also ‘carries’ with O (ˆa )nk √k |0i , (II.31) them a space-time tetrad {eˆ(µ)}, associated with coor- n ! (µ) k k dinates x , such that: i) the time coordinate satisfies (0) x = t, which impliese ˆ(0) := ∂(0)t is the observer’s four- with velocity (tangent to C), and ii) the set {eˆ(k)} is a rigid O set of spatial axes ‘attached’ to the observer (FIG.2). |0i := |0ki , Thus, the observer measures the passage of time, spatial k displacements, angles, local accelerations, rotations, etc. with respect to these basis vectors. The tetrad is cho- being the vacuum state on Στ . Thus, the usual inter- sen in such a manner that: i) the local metric, g(µ)(ν), pretation of the creation operatora ˆ† follows – e.g., as k reduces to the Minkowski metric on C,II.2 creating a single-photon in the mode uk. Note, how- ever, that the Fock states above are non-normalizable g = η , (or normalizable up to a Dirac-delta distribution). We (µ)(ν) C (µ)(ν) remedy this difficulty by constructing wave-packets. As example, we construct a positive frequency wave-packet, and ii) the metric, in a neighborhood of C, describes that of an accelerating and rotating reference frame in flat space-time, which, to linear displacements in x(k), f = d3~k f ∗u , (II.32) ˆ k k takes the form   with fk ∈ C, usually peaked around some wave-vector, ds2 = − 1+2~γ ·~x dt2 +2 (~ω0 × ~x)·d~xdt+d~x2, (II.35) but more suitably an L2 function in k-space. From thus, we define the annihilation wave-packet operator associ- 0 ated with f, where (~γ, ~ω ) are the acceleration and that the observer measures with, e.g., accelerometers and . Note that, these accelerations and angu- aˆ := (f, φˆ) ≈ d3~k f aˆ . (II.33) f KG ˆ k k lar velocities are independent of spatial coordinates x(k); though, in general, they may depend on the observer’s Commutation relations for the wave-packet operators , t. then follow At this juncture, some general observations and com- ments about the proper reference frame should be made. h † i aˆf , aˆf = (f, f)KG First, one should apparently associate the (small) linear (II.34) displacements from the observer’s world-line, as consid- ≈ d3~k |f |2 = 1, ered in this section, with the “weak field regime”, con- ˆ k sidered in previous parts of the paper. Second, in the where approximate orthonormality of the mode func- context of general relativity, accelerations and rotations 2 are coordinate-dependent quantities. Therefore, for a tions was assumed and the L property of fk was used. Per above, one can build a Fock space of wave-packet free-falling (parallel-transported) observer, such quanti- ties vanish entirely. Then, the first non-trivial compo- states which are properly normalizable. Furthermore, 2 † nents in the metric are at O x and depend on Reimann this construction permits the interpretation ofa ˆ as cre- f [41] (see also Ch. 13 of [1]). All this is not to ating a photon occupying the wave-packet f. say that the above construction is invalid or not use- ful. On the contrary, the proper reference frame is ex- tensive enough to accommodate for local gravitational B. The Metric time-dilation and general-relativistic frame-dragging ef- fects. Thirdly and lastly, the proper reference frame is In order to analyze gravitational effects in quantum quite generic and does not, necessarily, relate to gravi- optics experiments in a laboratory environment, we must tational effects. It is only once Einstein’s field equations prescribe the metric local to a laboratory observer, which are considered and a background space-time prescribed can be done in the so-called proper reference frame – a therefrom that the local , observed about C, reference frame naturally adapted to an arbitrary ob- can be attributed to gravitational phenomena. server in curved space-time. We follow references [1, 41] in the construction of the proper reference frame and its relation to the background PN frame, by providing key equations and brief explanations, whilst leaving explicit details to the references therein. II.2 A representation of the notion that space-time is locally flat [1]. 6

and J~ = I~ω, with I being the moment of inertia and ~ω the rotation rate of the earth in the PN frame. It follows that the off diagonal metric components are of O3. From here, one can relate the PN space-time metric to physical quantities that an (proper) observer locally measures, i.e. to the accelerations and rotations (~γ, ~ω0). One accomplishes this in the following manner (see [41] for explicit details). Let parenthetic subscripts indicate tensor components in the proper reference frame of an observer constrained to the surface of the earth. Thus, the spatial triad {ˆe(k)} physically correspond to rigid Cartesian axes stuck to the earth’s surface – where, e.g., two basis vectors may point along increasing longitudinal and latitudinal lines at the observer’s position and the third along the radial normal. Then, the following transport equation governs the evolution of the observer’s tetrad along their world- FIG. 2. Schematic of the observer’s world-line C through line, space-time, with ˆe(0) parallel to the observer’s four-velocity and {ˆe(k)} designating the spatial triad attached to the ob- server. All measurements are made, by the observer, with Dˆeµ respect to these basis vectors. (α) = −ˆeµ Ω(β) , (II.39) dt (β) (α)

2. Relation to background PN metric where D/dt is the covariant derivative along the ob- µ servers world-line and ˆe(α) are the vector components One may use the PN formalism (see [2] and Ch. 39 of of the observer’s tetrad basis in the PN frame. The anti- [1]) to describe the space-time metric near massive bod- symmetric transport tensor Ω(α)(β) encodes gravitational ies in the weak field, slowly moving, and slowly evolving and non-inertial effects and is found via regime – which is sufficient to analyze, e.g., solar-system experiments aiming to test metric theories of gravity. From the viewpoint of general relativity, the idea is to ν Dˆeµ ! 1 µ Dˆe(β) ν (α) expand, in successive orders of a small parameter , and Ω(α)(β) = gµν ˆe − ˆe . (II.40) 2 (α) dt (β) dt solve Einstein’s field equations, thereby obtaining a suf- C ficiently accurate background space-time metric. The small parameter  is set by physical quantities – the New- tonian potential U and the velocity v of the body in the The components of the transport tensor correspond pre- PN frame – such that cisely to the locally measured accelerations and rota- tions, i.e. U, v2 ∼ O2.

The condition of “slowly evolving”, for all quantities Q, Ω(0)(j) := γ(j) 0 is set by ∂tQ/∂iQ ∼ O(). Ω(i)(j) := ε(i)(j)(k)ω(k), Now, for example, letting bare subscripts indicate ten- sor components in the PN frame, the space-time metric for a solid rotating sphere (the earth) in the PN formal- with εijk being the Levi-Civita symbol. From thus, one ism is finds [41]  g = − (1 − 2U) + O4  00 5 gµν : g0i = −4Vi + O  (II.36) ~γ = ~a − ∇U, (II.41)  4 gij = (1 + 2U) δij + O  , ~ where with a being the coordinate (e.g., centrifugal) accelera- tion. Note that, for an earthbound observer ∇U = ~g, GM where g ≈ 9.8m/s2 is the gravitational acceleration on U = (II.37) r earth’s surface. J~ × ~r V~ = , (II.38) One also finds an explicit expression for the rotation 2r3 vector, 7

 1  1 3  ~ω0 = ~ω 1 + v2 + U + ~v × ~γ − ~v × ∇U + 2∇ × V~ . (II.42) 2 2 2

The first term is the rotation rate of the earth, as mea- where subscripts indicate spectral dependence. Intro- sured by an earthbound observer. The second term is the ducing a relative time delay δt in, say, mode a, followed Thomas term. Finally, the third and fourth by the 50/50 beamsplitter transformation, terms are the geodetic and Lense-Thirring terms, respec- tively. Considering the coordinate velocity (and acceler- 1   aˆ† −→ √ aˆ† + ˆb† ation, a) as being due solely to the rotational motion of 2 the earth (hence, ~v = ~ω ×R~ with R being the earth’s ra- 1   ˆb† −→ √ ˆb† − aˆ† , dius) one finds the centrifugal component of the Thomas 2 precession term, ~v ×~a, to be 1000 smaller than the gravitational terms. We thus ignore this term hereafter. leads to the state For brevity, we shall also let ~ω denote the rotational rate as measured by an earthbound observer. 1 |Ψ i = dν dν f g eiν1δt δt 2 ˆ 1 2 1 2 h   i × aˆ†ˆb† − aˆ†ˆb† − aˆ†aˆ† − ˆb†ˆb† |0i , C. Two-photon interference and the 1 2 2 1 1 2 1 2 Hong-Ou-Mandel dip (II.44)

The HOM effect is a two-photon quantum-interference with the first parenthetic term causing coincident de- effect, which embodies the general phenomena of boson tection events and the second term embodying pho- (photon) bunching – the inclination for identical bosons ton bunching. Note that, for identical monochromatic to congregate – as a consequence of the commutation re- wavepackets (e.g., equivalent Dirac-delta distributions lations for a bosonic field, and can be used to quantify for the spectral functions), the first term vanishes – in- the distinguishability of single photons [14]. For exam- dependent of any delay – and both photons come out in ple, a primitive HOM configuration consists of a two- mode a or mode b. photon source, which creates independent single-photon To calculate the likelihood of single-photon coinci- wavepackets, and a 50/50 beamsplitter, where the pho- dence events, we introduce the single-photon projection tons interfere. At the output ports of the beamsplitter, operators one positions photodetectors, records coincident detec- tion events, and quantifies the probability of a coincident ˆ † Πa = dν aˆν |0ih0| aˆν (II.45) detection – i.e., the likelihood that both detectors regis- ˆ ter a single-photon . For identical single-photons – Πˆ = dν ˆb† |0ih0| ˆb . (II.46) identical polarization, spectral and temporal profiles etc. b ˆ ν ν – the probability of a coincident detection is zero, due to the tendency of the photons to bunch in one output port The coincidence detection probability pc, as a function of the beamsplitter or the other. If, however, we con- of the time delay, is then given by sider spectral wavepackets, we can induce distinguisha- bility by introducing a relative time delay between the pc(δt) = hΨδt|Πˆ a ⊗ Πˆ b|Ψδti . (II.47) wavepackets prior to the beamsplitter, ceteris paribus. As a consequence, the likelihood of a coincident detec- For identical, Gaussian wave-packets (fk = gk) with tion event increases, as the relative time delay departs spectral width σ, the coincident probabilityII.3 reduces from zero. The transition of the coincidence probability, to from zero to a nonzero value, leads to a dip-like structure in its functional behavior, with respect to the time delay 1   1  p (δt) = 1 − exp − δt2σ2 . (II.48) (see, e.g., inset of FIG.4). This dip in coincidence events c 2 2 is the well-known HOM dip [11, 12]. We mathematically describe this contrivance as follows. For δt = 0, the coincident probability vanishes, since the Consider a two-photon source, which generates inde- single-photons jointly exit from one port of the 50/50 pendent single photons, in spatial modes (a, b) and oc- beamsplitter or the other, however this is the ideal case cupying wavepackets (f, g), such that initial two-photon – without noise and with unit visibility. state is

† ˆ† |Ψi =a ˆf bg |0i   (II.43) II.3 See also refs. [13, 42] for meticulous calculations of this inde- = dν dν f g aˆ†ˆb† |0i , ˆ ˆ 1 2 1 2 1 2 pendent wave-packet scenario and many more. 8

1. Digression on noise: visibility, fluctuations, and superpositions of gravitational waves and/or artifacts of gravitational decoherence other gravitational sources scattered about the . One can model such fluctuations as stochastic waves, The expressions in the previous section dealt with with unequal time correlation functions of metric com- ideal HOM interference, where one has perfect control ponents, which one characterizes with spectral distribu- over all distinguishability parameters. In reality, how- tions. Then, for example, the geometry of an interfer- ever, such is not the case, and one is led to the more ometer along with these spectral distributions determine general expression for the coincidence probability the visibility of an interference signature and can thus, in principle, have observable manifestations [45]. 1   1  On the quantum side, however, a fully determined p (δt) = 1 − V exp − δt2σ2 , (II.49) c 2 2 and consistent understanding of the origins of quantum space-time fluctuations is far from satisfactory. Nev- where V is the visibility of the interference experiment, ertheless, several phenomenological models have been defined by proposed, including treatments where one links the col- lapse of the wave-function to gravitational decoherence pmax − pmin : c c or semi-classical treatments, which replace the sources V = min , (II.50) pc in Einstein’s field equations with the expectation value of the stress-energy tensor operator. These models gen- with erally rely on parameters determining the scale at which

max quantum-gravity effects become relevant, i.e. the Planck pc := lim pc(δt) δt→∞ scale, however this scale is discouragingly minuscule −35 min (Planck length, `P ∼ 10 m). Therefore, it is difficult pc := lim pc(δt). δt→0 to imagine that quantum-gravitational fluctuations will become prevalent in interferometry scenarios, in the near Systematic control allows one to maintain the visibil- to mid-term. Though, with the continuing development ity criterion V > 1/2, which demarcates the boundary of large-scale interferometers and ever-improving sensi- between sufficient quantum and/or classical descriptions tivities, perhaps bounds may be put on these scales, in for the fields [12, 13]. the future. One source of noise – pertinent to this work – is tempo- ral fluctuations arising from, say, random changes in the relative paths taken by the photons. One can model this III. SETUP AND RESULTS as a background Gaussian-noise, where the time delay is a random variable with mean δt and fluctuation σ. The e A. Relativistic frame-dragging and the HOM dip mathematical particulars are inconsequential (see, e.g., [43] for various noise models in optical interferometry); however, since we concern ourselves with gravitationally Consider a HOM-interference experiment, wherein induced temporal shifts in the HOM dip, we require the two single-photon wavepackets counter-propagate through a common-path interferometer constrained to size of the fluctuations σe to be smaller than the average size of the shift, in order for the gravitational phenom- the earth’s surface (FIG.1). Relativistic effects – e.g., ena to be resolvable in practice. Such is easier for, e.g., the Sagnac, Lense-Thirring, and geodetic effects – in- common-path (SectionIIIA) versus dual-arm (Section duce distinguishability between the counter-propagating IIIB) interferometry. paths, leading to a temporal shift in the HOM dip, A more fascinating noise source is that of which would otherwise be at δt = 0 if relativistic effects gravitational/space-time fluctuations, which could, were absent. in principle, measurably affect the visibility of opti- The off-diagonal terms of the metric perturbation – cal interference signatures. General relativity treats physically corresponding to rotational motion – are the space-time as a dynamical variable, hence, gravitational sole contributors to the time delay, ceteris paribus, which fluctuations – either of classical or quantum origin – one can calculate via concurrent use of equations (II.14) lead to a loss of coherence in quantum systems.II.4 and (II.35), in the proper reference frame of an observer. Unlike other sources of noise, which can be eliminated For a single path around the interferometer loop – say, by lowering temperatures and creating extreme vacua, the right-handed path – one finds it is impossible to get rid of gravitational decoherence. On the classical side, space-time fluctuations can have well-determined and various origins – from the seismic δt = c−1 dx(i)h = c−2 d~x · (~ω0 × ~x) RH ‰ (0)(i) ‰ activity of nearby masses, to the more exotic incoherent = c−2 dA~ · ∇ × (~ω0 × ~x) ˆ 2~ω0 · A~ II.4 = , For all details omitted here and for a more thorough discussion, c2 see ref. [44]. (III.1) 9

where A~ is the areal vector, perpendicular to the inter- ferometry plane, and we have used the spatial-coordinate independence of ω0. A similar relation holds for the left- handed path, albeit with opposite sign, such that

4~ω0 · A~ δt := δt − δt = , (III.2) RH LH c2

with ~ω0 given by equation (II.42).

To evaluate the above expression, for an observer constrained to the earth’s surface, we choose an earth-centered spherical- (ˆer, ˆeθ, ˆeφ) co-moving with the observer, with ~ω aligned along the polar axis.III.1 The earth’s radius R and the observer’s latitude θ establishes the earthbound constraint, since – due to the intrinsic axisymmetric structure of the back- ground space-time – characterization of the observer’s FIG. 3. Space-time diagram of counter-propagating null rays. longitude is superfluous. Furthermore, we assume the Two electromagnetic signals (red and blue ) counter- areal vector to take the form A~ = A(ˆer cos α − ˆeθ sin α), propagate along a common path in space. Relativistic frame- with α being the angle between the observer’s normal dragging effects cause the pitch of the space-time helices to and the areal vector of the interferometer. Under these differ, which leads to differing arrival times at a stationary considerations, the coordinate expression for the time detector system. All motion is constrained to a world-tube delay is then of spatial size  ` ∼ min(c2/g, c/ω0).

4ωA 4ωA 2GM GI  δt = cos(θ − α) + sin θ sin α + (2 cos θ cos α − sin θ sin α) , (III.3) c2 c2 c2R c2R3 | {z } | {z } Sagnac geodetic + Lense-Thirring

which agrees with [46]. We note that the general rela- relative to the physical size of the interferometer; thus, −9 tivistic contributions are on the order of rS/R ∼ 10 define times smaller than the Sagnac contribution, where rS = δt? 2GM/c2 is the Schwarzschild radius of the earth. F := , (III.4) We now substitute the above expression into the coin- A cidence probability formula (II.48) and plot the results which has s/km2 and where δt? is crudely for various parameter values (θ, A, α). See FIG.4 for the difference between the maximum and minimum time analysis. delay. This quantity is of practical interest as e.g., given some physical constraint on the areal dimensions, one can see what level of precision is required in order to 1. Order-of-magnitude estimates observe relativistic effects. As example, the quantity F, for the Sagnac effect, goes Perhaps more insightful than a full-blown analysis is as an order-of-magnitude estimate for these effects. A key 4ω F ∼ ∼ 10−15 s/km2, quantity to then consider is the size of the temporal shift Sag c2 and since the Lense-Thirring and geodetic effects are a billionth of the size of the Sagnac effect, we have

III.1 Thus, ~v = ~ω × R~ is the observer’s coordinate velocity −9 −24 2 FGR ∼ 10 FSag ∼ 10 s/km , 10

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

m) 0.2

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

FIG. 4. Photon path delay (cδt, where c 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 counter-propagating 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) whilst performing the experiment at several interferometer-orientation settings (various α). Right: Path delay due to the combined Lense-Thirring and geodetic effects.

where the subscript GR concurrently signifies the Lense- of radius r and N turns, the effective area of the inter- Thirring and geodetic effects. ferometer is A = Nπr2 or A = lr/2, where l = 2πNr is The smallness of the Lense-Thirring and geodetic ef- the length of the fiber. Taking l ≈ 2km and r ≈ 1m im- −3 2 fects is immediately concerning, as it implies stringent plies ASag ∼ 10 km , which in turn implies a required experimental constraints (high-precision and control of time-delay precision at the attosecond (1as = 10−18s) a large-area interferometer), however, this is not to say scale – on a par with current experiment [49]. Proof-of- that observation of such is impractical. On the contrary, principle demonstrations like these are quite intriguing, there is ongoing research towards terrestrially measuring as they constitute quantum fields in non-inertial refer- these frame-dragging effects with classical-optical inter- ence frames. ference [46, 47]. With regards to a similar undertaking in HOM interferometry, a full feasibility analysis is duly wanting.III.2 On the other hand, observing signatures of the Sagnac effect (induced by the earth’s rotation) via HOM in- B. Gravitational time-dilation and the HOM dip terference appears approachable. Recently, phenomena akin to this were observed by Restuccia et al. [27] We now consider a HOM-interference experiment, for (and analyzed further in [28]), wherein the authors con- a dual arm configuration, and calculate the time delay structed a HOM configuration (FIG.1) upon a rotat- due to the effects of the uniform gravitational acceler- ing table with tunable rotation rate, and subsequently ation and centrifugal acceleration measured by an ob- discovered a rotation-induced shift in the HOM dip server fixed upon earth’s surface (see FIG.5). We ignore (proportional to the rotation frequency), as one would the off-diagonal rotation-induced contributions, as they predict with the formalism presented here. To mea- are equivalent (up to a factor of 2) to the calculations of sure an analogous effect induced by the earth’s rota- the previous section. tion, one requires much greater time-delay precision, due to the minute rotational frequency of the earth The unit vectors, along each arm of the interferometer, ω ∼ 10−5s−1, which seems presently achievable with are given by modest resources. For example, considering a fiber-loop

nˆBA =n ˆCD = (sin α sin β, cos α, sin α cos β)

nˆDA =n ˆCB = (cos α sin β, − sin α, cos α cos β). III.2 Perhaps at the level of [48]. One should also leverage parameter estimation techniques to optimize operating conditions (see ref. [49]), but such is beyond the scope of this work. The coordinate equations for the lines along the arms 11

gravitational acceleration on the earth’s surface, we find the phase along the segment BA to be

A (0) ~ δSBA = gk (dx · nˆBA)z, ˆB

wheren ˆBA is the along BA. Integrating along the each arm of the interferometer, we then obtain g δS = k(0) d2 sin α cos β BA 2c3 g l2  δS = k(0) cos α cos β + ld sin α cos β CB c3 2 g d2  δS = k(0) sin α cos β + ld cos α cos β CD c3 2 g δS = k(0) l2 cos α cos β. DA 2c3

FIG. 5. A dual-arm configuration with a two-photon source Since the observer is fixed to the earth’s surface (a non- (or, a beamsplitter and two photon-detectors) at C, mirrors inertial reference frame), they also measure a centrifugal at B/D, and a beamsplitter and two photon-detectors (or, a acceleration two-photon source) at the origin A. The is fixed to the earth’s surface at a latitude θ. The z-axis is 2 ~a = ω R sin θ(sin θˆez + cos θˆey). parallel to the radial vector ˆer, and the y-axis points along the longitudinal line ˆeφ at the observer’s position. The locally measured angles, α and β, determine the orientation of the Then, setting ~γ = ~a in equation (III.5), we find addi- interferometer, with respect to the horizontal and vertical tional phase differences along the different interferometer planes. The area of the interferomter is A = ld. arms,

ω2R δS = k(0) d2 sin θ(cos θ cos α + sin θ sin α cos β) are thus, BA 2c3 " x y z ω2R l2  BA: = = δS = k(0) sin θ sin θ cos β cos α + ld sin α d sin α sin β d cos α d sin α cos β CB c3 2 x − l cos α sin β y + l sin α z − l cos α cos β # CD: = = l2  d sin α sin β d cos α d sin α cos β − cos θ sin α + ld cos α x y z 2 DA: = = l cos α sin β −l sin α l cos α cos β 2 "  2  (0) ω R d x − d sin α sin β y − d cos α z − d sin α cos β δSDC = k sin θ cos θ cos α − ld sin α CB: = = c3 2 l cos α sin β −l sin α l cos α cos β # d2  By concurrent use of equations (II.14) and (II.35), we + sin θ cos β sin α + ld cos α 2 calculate the phase difference, δS, accrued along a path due to the uniform acceleration ~γ, ω2R δS = k(0) l2 sin θ (sin θ cos α cos β − cos θ sin α) . DA 2c3 1 (0) (0) δS = − dx k h(0)(0) 2 ˆ Therefore, considering identical single-photons travers- (III.5)   ing separate paths ABC and CDA, we obtain the to- = d~x · ~k ~γ · ~x, ˆ tal time delay between the paths (after dividing by the mean-frequency k(0)), due to both the gravitational and 2 For example, setting ~γ = gˆez, where g ≈ 9.8m/s is the centrifugal accelerations,

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)], (III.6) c3 c3 | {z } | {z } gravitational, g centrifugal, a

where A = ld is the interferometer area. We note that a/g ∼ 10−2. Plots for these results are shown in FIG.5. 12

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

m) = /2 2 = /2 2

0 0 -2

-2

-4 Photon path delay ( delay path Photon

-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 ( delay path Photon -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 × 10−2m/s2, as a function of the latitude θ, for various orientation angles α (β) and with β = 0 (α = 0) for all curves.

1. Order-of-magnitude estimates (1as=10−18s) – closely approaching the scale implied by Fg – but with a much smaller interferometer area than Consider the photon time delay per unit area, char- required here. Nevertheless, a tabletop HOM experi- acterized by the quantity F [eq. (III.4)] and induced ment of this sort seems achievable with present or near- by gravitational and centrifugal accelerations. For the term technology. Indeed, Hilweg et al. [48] investigated gravitational acceleration, we have the feasibility of a similar experiment aimed at measur- ing gravitational time-dilation effects via single-photon g −19 2 interference, with a variant Mach-Zehnder interferome- Fg ∼ ∼ 10 s/km , c3 ter, and found that, though challenging, a tabletop ex- periment is possible with long-fiber spools and active which is 103 times smaller than the Sagnac effect and 6 phase stabilisation. An analogous feasbility analysis for 10 times larger than the minute geodetic and Lense- a HOM experiment – leveraging parameter estimation Thirring effects considered in Section IIIA1. Similarly, techniques, like those used in [49] – should be pursued. for the centrifugal acceleration, we have We leave this for future work. ω2R F ∼ ∼ 10−21 s/km2. a c3 Note that Lyons et al. [49] recently acheived HOM time-delay resolution at the attosecond scale 13

IV. SUMMARY AND CONCLUSION have been pursued and found feasible with current tech- nology. Further inspiration for this potentiality comes from concurrent consideration of recent experimental en- deavors: a HOM-interference experiment in a rotating, In this work, we showed how to encode general- non-inertial reference frame [27]; high-precision HOM relativistic effects – e.g., frame-dragging and gravita- time-delay resolution [49]; and current attempts to mea- tional time-dilation effects – into multi-photon quantum- sure relativistic frame-dragging effects with a classical, interference phenomena, for various interferometer con- optical, earthbound system [46, 47]. figurations. This was theoretically achieved by quantiz- From historical considerations, one may view our re- ing a massless scalar-field in a weak gravitational field sult as a quantum analog to the seminal proposal of (geometric quantum-optics in curved space-time, in the Scully et al. [50] – made nearly forty years ago – weak field regime; SectionIIA). Applying this formal- who originally investigated the potentiality of observing ism to a terrestrial laboratory setting (SectionIIB), we PN (e.g., general-relativistic frame-dragging) effects with showed that, in principle, gravitational effects can in- classical optical interferometry. In a similar fashion, duce observable changes in quantum-interference signa- a proof-of-principle demonstration of our results would tures, using HOM interference as an exemplar (Sections serve as a genuine quantum PN test of general relativity IIC andIII). Non-inertial effects, due to the earth’s – extending the domain of validity of Einstein’s classical rotation and centrifugal acceleration, were also consid- theory of gravitation, to the arena of quantized electro- ered, and the potentiality of practical demonstrations magnetic fields. was briefly analyzed. Though the latter analysis was un- fulfilling (a full feasibility, in this regard, is still wanting), the landscape of potential proof-of-principle demonstra- ACKNOWLEDGMENTS tions, like these, appears promising – the reason being that analogous enterprises exploring, e.g., gravitational AJB and SH acknowledge Jonathan P. Dowling for time-dilation via single-photon interference, with table- many fruitful conversations and his ever-endearing sup- top long-fiber spools [48] or with quantum satellites [32], port. AJB acknowledges financial support from the NSF.

[1] Charles W Misner, Kip S Thorne, and John A Wheeler, stein and the Changing Worldviews of Physics (Springer, Gravitation (Macmillan, 1973). 2012) pp. 317–331. [2] Clifford M Will, “The confrontation between general rel- [10] Ted Jacobson, “Introduction to quantum fields in curved ativity and experiment,” Living Reviews in Relativity and the Hawking effect,” in Lectures on Quan- 17, 4 (2014). tum Gravity (Springer, 2005) pp. 39–89. [3] Chin-Wen Chou, David B Hume, Till Rosenband, and [11] Chong-Ki Hong, Zhe-Yu J Ou, and Leonard Mandel, David J Wineland, “Optical clocks and relativity,” Sci- “Measurement of subpicosecond time intervals between ence 329, 1630–1633 (2010). two photons by interference,” Physical Review Letters [4] Anton Zeilinger, “Experiment and the foundations of 59, 2044 (1987). quantum physics,” Rev. Mod. Phys. 71, S288–S297 [12] Zhe-Yu J Ou, “Quantum theory of fourth-order interfer- (1999). ence,” Physical Review A 37, 1607 (1988). [5] Jonathan P Dowling and Gerard J Milburn, “Quantum [13] Zhe-Yu J Ou, Multi-Photon Quantum Interference, technology: the second quantum revolution,” Philosoph- Vol. 43 (Springer, 2007). ical Transactions of the Royal Society of London. Se- [14] Jian-Wei Pan, Zeng-Bing Chen, Chao-Yang Lu, Har- ries A: Mathematical, Physical and Engineering Sciences ald Weinfurter, Anton Zeilinger, and Marek Zukowski,˙ 361, 1655–1674 (2003). “Multiphoton entanglement and interferometry,” Re- [6] Stefano Pirandola, Bhaskar R Bardhan, Tobias Gehring, views of Modern Physics 84, 777 (2012). Christian Weedbrook, and Seth Lloyd, “Advances in [15] Mattia Walschaers, “Signatures of many- inter- photonic quantum sensing,” Nature Photonics 12, 724– ference,” Journal of Physics B: Atomic, Molecular and 733 (2018). Optical Physics 53, 043001 (2020). [7] Yu-Hao Deng, Hui Wang, Xing Ding, Z-C Duan, Jian [16] Pieter Kok, William J Munro, Kae Nemoto, Timo- Qin, M-C Chen, Yu He, Yu-Ming He, Jin-Peng Li, Yu- thy C Ralph, Jonathan P Dowling, and Gerard J Mil- Huai Li, et al., “Quantum interference between light burn, “Linear optical quantum computing with photonic sources separated by 150 million kilometers,” Physical qubits,” Reviews of Modern Physics 79, 135 (2007). Review Letters 123, 080401 (2019). [17] Simanraj Sadana, Debadrita Ghosh, Kaushik Joarder, [8] Ping Xu, Yiqiu Ma, Ji-Gang Ren, Hai-Lin Yong, Timo- A Naga Lakshmi, Barry C Sanders, and Urbasi Sinha, thy C Ralph, Sheng-Kai Liao, Juan Yin, Wei-Yue Liu, “Near-100% two-photon-like coincidence-visibility dip Wen-Qi Cai, Xuan Han, et al., “Satellite testing of a with classical light and the role of complementarity,” gravitationally induced quantum decoherence model,” Physical Review A 100, 013839 (2019). Science 366, 132–135 (2019). [18] Magdalena Zych, Fabio Costa, Igor Pikovski, and Caslavˇ [9] Robert M Wald, “The history and present status of Brukner, “Quantum interferometric visibility as a wit- quantum field theory in curved spacetime,” in Ein- ness of general relativistic proper time,” Nature Com- 14

munications 2, 505 (2011). [34] Marco Rivera-Tapia, Aldo Delgado, and Guillermo F [19] Magdalena Zych, Fabio Costa, Igor Pikovski, Timothy C Rubilar, “Weak gravitational field effects on large-scale Ralph, and Caslavˇ Brukner, “General relativistic ef- optical interferometric Bell tests,” Classical and Quan- fects in quantum interference of photons,” Classical and tum Gravity (2020), 10.1088/1361-6382/ab8a60. 29, 224010 (2012). [35] Chiara Marletto and Vlatko Vedral, “Gravitationally in- [20] Roberto Colella, Albert W Overhauser, and Samuel A duced entanglement between two massive particles is suf- Werner, “Observation of gravitationally induced quan- ficient evidence of quantum effects in gravity,” Physical tum interference,” Physical Review Letters 34, 1472 Review Letters 119, 240402 (2017). (1975). [36] Sougato Bose, Anupam Mazumdar, Gavin W Morley, [21] Valery V Nesvizhevsky, Hans G B¨orner,Alexander K Hendrik Ulbricht, Marko Toroˇs,Mauro Paternostro, An- Petukhov, Hartmut Abele, Stefan Baeßler, Frank J drew A Geraci, Peter F Barker, MS Kim, and Gerard Rueß, Thilo St¨oferle, Alexander Westphal, Alexei M Milburn, “ entanglement witness for quantum grav- Gagarski, Guennady A Petrov, et al., “Quantum states ity,” Physical Review Letters 119, 240401 (2017). of neutrons in the earth’s gravitational field,” Nature [37] Sofia Qvarfort, Sougato Bose, and Alessio Serafini, 415, 297–299 (2002). :1812.09776 [quant-ph]. [22] Peter Asenbaum, Chris Overstreet, Tim Kovachy, [38] Tanjung Krisnanda, Guo Yao Tham, Mauro Paternos- Daniel D. Brown, Jason M. Hogan, and Mark A. Ka- tro, and Tomasz Paterek, “Observable quantum entan- sevich, “Phase shift in an interferometer due to glement due to gravity,” npj Quantum Information 6, spacetime curvature across its wave function,” Phys. 1–6 (2020). Rev. Lett. 118, 183602 (2017). [39] Richard Howl, Vlatko Vedral, Marios Christodoulou, [23] G Rosi, G D’Amico, L Cacciapuoti, F Sorrentino, Carlo Rovelli, Devang Naik, and Aditya Iyer, M Prevedelli, M Zych, Cˇ Brukner, and GM Tino, arXiv:2004.01189 [quant-ph]. “Quantum test of the for in [40] Max Born and Emil Wolf, Principles of Optics (Cam- coherent superposition of internal energy states,” Nature bridge University Press, 1999). Communications 8, 1–6 (2017). [41] Pacˆome Delva and Jan Gerˇsl, “Theoretical tools for [24] J-L Staudenmann, SA Werner, R Colella, and AW Over- relativistic gravimetry, gradiometry and chronomet- hauser, “Gravity and inertia in quantum mechanics,” ric geodesy and application to a parameterized post- Physical Review A 21, 1419 (1980). Newtonian metric,” Universe 3, 24 (2017). [25] Ulrich Bonse and Thomas Wroblewski, “Measurement [42] Agata M Bra´nczyk, arXiv:1711.00080 [quant-ph]. of neutron quantum interference in noninertial frames,” [43] Rafal Demkowicz-Dobrza´nski,Marcin Jarzyna, and Jan Phys. Rev. Lett. 51, 1401–1404 (1983). Ko lody´nski,“Quantum limits in optical interferometry,” [26] Matthias Fink, Ana Rodriguez-Aramendia, Johannes in Progress in Optics, Vol. 60 (Elsevier, 2015) pp. 345– Handsteiner, Abdul Ziarkash, Fabian Steinlechner, 435. Thomas Scheidl, , Jacques Pienaar, Tim- [44] Angelo Bassi, Andr´eGroßardt, and Hendrik Ulbricht, othy C Ralph, and Rupert Ursin, “Experimental test of “Gravitational decoherence,” Classical and Quantum photonic entanglement in accelerated reference frames,” Gravity 34, 193002 (2017). Nature Communications 8, 1–6 (2017). [45] Lorenzo Asprea, Angelo Bassi, Hendrik Ulbricht, and [27] Sara Restuccia, Marko Toroˇs,Graham M Gibson, Hen- Giulio Gasbarri, 1912.12732 [gr-qc]. drik Ulbricht, Daniele Faccio, and Miles J Padgett, [46] F Bosi, G Cella, A Di Virgilio, A Ortolan, A Porzio, “Photon bunching in a rotating reference frame,” Phys- S Solimeno, M Cerdonio, JP Zendri, Maria Allegrini, ical Review Letters 123, 110401 (2019). Jacopo Belfi, et al., “Measuring gravitomagnetic effects [28] Marko Toroˇs,Sara Restuccia, Graham M. Gibson, Mar- by a multi-ring-laser ,” Physical Review D 84, ion Cromb, Hendrik Ulbricht, Miles Padgett, and 122002 (2011). Daniele Faccio, “Revealing and concealing entanglement [47] Angela DV Di Virgilio, Filippo Bosi, Umberto Gi- with noninertial motion,” Phys. Rev. A 101, 043837 acomelli, Andrea Simonelli, Giuseppe Terreni, An- (2020). drea Basti, Nicolo’ Beverini, Giorgio Carelli, Donatella [29] David Rideout, Thomas Jennewein, Giovanni Amelino- Ciampini, Francesco Fuso, et al., 2003.11819 [astro- Camelia, Tommaso F Demarie, Brendon L Higgins, ph.IM]. Achim Kempf, Adrian Kent, Raymond Laflamme, Xian [48] Christopher Hilweg, Francesco Massa, Denis Martynov, Ma, Robert B Mann, et al., “Fundamental quantum Nergis Mavalvala, Piotr T Chru´sciel, and Philip optics experiments conceivable with satellites—reaching Walther, “Gravitationally induced phase shift on a single relativistic distances and velocities,” Classical and Quan- photon,” New Journal of Physics 19, 033028 (2017). tum Gravity 29, 224011 (2012). [49] Ashley Lyons, George C Knee, Eliot Bolduc, Thomas [30] Aharon Brodutch, Alexei Gilchrist, Thomas Guff, Roger, Jonathan Leach, Erik M Gauger, and Daniele Alexander RH Smith, and Daniel R Terno, “Post- Faccio, “Attosecond-resolution Hong-Ou-Mandel inter- Newtonian gravitational effects in optical interferome- ferometry,” Science Advances 4, eaap9416 (2018). try,” Physical Review D 91, 064041 (2015). [50] Marlan O Scully, M Suhail Zubairy, and MP Haugan, [31] Magdalena Zych and Caslavˇ Brukner, “Quantum for- “Proposed optical test of metric gravitation theories,” mulation of the Einstein equivalence principle,” Nature Physical Review A 24, 2009 (1981). Physics 14, 1027–1031 (2018). [32] Daniel R Terno, Francesco Vedovato, Matteo Schiavon, Alexander RH Smith, Piergiovanni Magnani, Giuseppe Vallone, and Paolo Villoresi, arXiv:1811.04835 [gr-qc]. [33] Albert Roura, “Gravitational in quantum-clock interferometry,” Phys. Rev. X 10, 021014 (2020).