Relativistic frame-dragging and the Hong-Ou-Mandel dip | a primitive to gravitational effects in multi-photon quantum-interference Anthony J. Brady∗ and Stav Haldar Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana, 70803, USA (Dated: June 9, 2020) We investigate the Hong-Ou-Mandel (HOM) effect { a two-photon quantum-interference effect { in the space-time of a rotating spherical mass. In particular, we 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 general relativity, 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 quantum mechanics 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- curved space-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 particles, 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 { gravity 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 nature 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 rµSk, 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- (r Sk)(rµSk) = 0 (II.3) tum theory, in regimes where both are prevalent. Our 1 (rµS0 )(r ln α0 ) = − rµr 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 rµSk is a null vector. mechanical principles. One can show that it also satisfies a geodesic equation We compartmentalize our paper in the following way. (rµS0 ) r (r S0 ) = 0: In Section IIA, 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 rµSk. 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 Section IIB, 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 section IIA propagates. In Section IIC, µ 0 0 2 we provide some background on HOM interference and r rµSkjαkj = 0; (II.5) comment on interferometric noise and gravitational de- coherence in context. In Section III, 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 (rµSk) jαkj ; (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 photons 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, accelerations, etc. µ are sufficiently weak and consider first-order perturba- r rµφ = 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 r 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 + "k); (II.9) with S0 the eikonal (or phase), α0 a slowly-varying am- plitude with respect to variations of the eikonal, and with (δSk;"k) 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.