Quantum interference in external gravitational fields beyond General Relativity

Luca Buoninfante,1, ∗ Gaetano Lambiase,2, 3, † and Luciano Petruzziello2, 4, ‡

1Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan2INFN - Sezione di Napoli, Gruppo collegato di Salerno, I-84084 Fisciano (SA), Italy

3Dipartimento di Fisica “E.R. Caianiello”, Universita degli Studi di Salerno, I-84084 Fisciano (SA), Italy4Dipartimento di Ingegneria Industriale, Universita degli Studi di Salerno, Fisciano (SA) 84084, Italy

In this paper, we study the phenomenon of quantum interference in the presence of externalgravitational fields described by alternative theories of gravity. We analyze both non-relativisticand relativistic effects induced by the underlying curved background on a superposed quantumsystem. In the non-relativistic regime, it is possible to come across a gravitational counterpart of theBohm-Aharonov effect, which results in a phase shift proportional to the derivative of the modifiedNewtonian potential. On the other hand, beyond the Newtonian approximation, the relativisticnature of gravity plays a crucial role. Indeed, the existence of a gravitational time dilation betweenthe two arms of the interferometer causes a loss of coherence that is in principle observable inquantum interference patterns. We work in the context of generalized quadratic theories of gravityto compare their physical predictions with the analogous outcomes in general relativity. In so doing,we show that the decoherence rate strongly depends on the gravitational model under investigation,which means that this approach turns out to be a promising test bench to probe and discriminateamong all the extensions of Einstein’s theory in future experiments.

I. INTRODUCTION

Einstein’s General Relativity (GR) has gone throughmany challenges in the last century, but it has alwaysbeen confirmed by high-precision experiments which haveverified many of its predictions [1]. The recent observa-tion of gravitational waves from binary merger representsone of the most astonishing of its achievements [2].

Despite its great success, there are conceptual prob-lems which have not found a definite answer yet. Forinstance, by focusing on galactic and cosmological scales,self-consistent and complete descriptions for dark matterand dark energy (which are both compatible with ex-perimental data) are still missing. Furthermore, in theshort-distance (ultraviolet) regime, GR turns out to beclassically incomplete due to the presence of cosmologi-cal and black hole singularities, whereas from a quantumpoint of view it is a non-renormalizable theory, whichthus lacks predictability at high energies. From an ex-perimental point of view, what we can say is that ourknowledge about short-distance gravity is extremely lim-ited; indeed, Newton’s law has been tested only up tomicrometer scales [3] and the smallest masses for whichthe gravitational coupling has been measured are of theorder of 100 milligrams [4].

In the past years, these fundamental open issues havechanneled a huge amount of efforts towards the questfor a consistent ultraviolet completion of GR. One of themost straightforward approaches consists in generalizingthe Einstein-Hilbert action by including terms which arequadratic in the curvature invariants, i.e. R2, RµνRµν

∗Electronic address: buoninfante.l.aa@m.titech.ac.jp†Electronic address: lambiase@sa.infn.it‡Electronic address: lupetruzziello@unisa.it

and RµνρσRµνρσ. The first remarkable achievements inthe framework of quadratic gravity date back to 1977with the results obtained by Stelle [5], who proved thata gravitational theory described by the Einstein-Hilbertaction with the addition of the terms R2 and RµνRµν ispower-counting renormalizable. At the same time, how-ever, such a gravitational model hides undesirable fea-tures, such as the emergence of a massive spin-2 ghostdegree of freedom that violates unitarity (when standardquantization prescriptions are implemented [6]). Despitethe presence of the ghost field, the above theory can beregarded as an effective field theory valid at the energyscales below the cut-off represented by the mass of theghost. Another important accomplishment in the frame-work of quadratic gravity is given by the Starobinskimodel of inflation [7], which is in good agreement with thecurrent data, even though in this case the only quadraticpart of the action is R2. In addition to that, it is worthobserving that gravitational actions with quadratic cur-vature corrections were recently considered also in otherdifferent scenarios [8–18].

The results mentioned so far were obtained for localquadratic theories of gravity, whose corresponding La-grangians depend polynomially on the derivative opera-tor. Recently, also nonlocal quadratic modifications haveburst into the spotlight, as the presence of nonlocal (i.e.non-polynomial) form factors in the gravitational actioncan help both to solve the problem of ghosts and to im-prove the ultraviolet behavior of the quantized theory.For this vast topic, we remand the interested reader toRefs. [19–38].

In this paper, our aim is to investigate the differencesbetween GR and several extended theories of gravity byresorting to the phenomenon of quantum interference.Specifically, we will consider both non-relativistic andrelativistic effects induced by a modified gravitationalmodel onto a quantum interference experiment to com-

arX

iv:2

104.

1115

4v2

[gr

-qc]

22

Jun

2021

2

pare the results with the case of Eistein’s theory. Theinterplay between GR and quantum interference has al-ready been addressed, both from a theoretical and a phe-nomenological perspective. As a matter of fact, in thenon-relativistic regime (Newtonian approximation), therelevant effect is a Bohm-Aharonov-like phase shift thatis proportional to the derivative of the gravitational po-tential, as discovered for the first time in 1975 in a lab-oratory test devised by Colella, Overhauser and Werner,better known under the name of COW experiment [39].On the other hand, in the relativistic domain the exis-tence of time dilation entails more drastic effects on aquantum superposition. Indeed, in Refs. [40–42] it wasshown that a (gravitational) time dilation between thetwo arms of an interferometer can cause a loss of coher-ence in the interference pattern, thereby giving rise todecoherence. This quantum manifestation has not beendirectly observed up to now, since a detectable loss ofcoherence would require either a large travel-time or alarge distance between the two arms of the interferom-eter (see Sec. IV). However, a decoherence mechanismoriginated by time dilation was recently found out in asimilar experiment [43], where a Stern-Gerlach interfer-ometer under the influence of an external inhomogeneousmagnetic field was employed to simulate the effect of timedilation on the spin precession of the examined system(atom chip).

To comply with the aforementioned purposes, the pa-per is organized as follows: in Sec. II we analyze thequantum mechanical setup, focusing in particular on thephysics behind a Mach-Zehnder interferometer and onthe complementary concepts of interferometric visibilityand which-way information. In Sec. III we adapt theabove setting to a configuration in which the interferom-eter is embedded in a (classical) weak and static grav-itational field. Then, we derive the Hamiltonian of aquantum system in curved backgrounds and rely on atwo-level system as a simple realization of a quantummassive “clock”. Equipped with this knowledge, we man-age to discuss both COW and gravitational time dilationeffects associated with a generic, linearized and staticspacetime metric. Section IV is devoted to the intro-duction of several extended theories of gravity; for eachof them, we exhibit the corresponding modified Newto-nian potential which has to be exploited for the compu-tation of detection probabilities and interferometric vis-ibility. In Sec. V, we compare the new predictions re-lated to alternative theories of gravity with the currentexperimental data. In this respect, we point out thatthe decoherence rate triggered by time dilation stronglydepends on the gravitational model under investigation.Furthermore, we comment on the fact that such an in-triguing aspect can provide a valuable test bench to probeand discriminate among several alternative theories in fu-ture experiments. Finally, Section VI contains conclud-ing remarks and outlook. In Appendix A, we allocatethe mathematical details of the derivation of the Hamil-tonian for a quantum system in an external, linearized

and static spacetime metric, whilst in Appendix B webriefly review the linearized (weak-field) limit of general-ized quadratic gravitational theories and their modifiedNewtonian potentials.

Before ending this preliminary Section, let us stressthat in this paper we only work with external and classi-cal gravitational fields (namely, we study quantum sys-tems on classical backgrounds). In other terms, we followa semi-classical approach and reasonably assume that theself-gravity of a given quantum system is negligible in theanalyzed physical setting. On the other hand, it must besaid that several articles recently appeared in literaturehave accounted for self-gravity effects, both at the clas-sical [44–46] and quantum level [47–50].

Throughout this work, we adopt the positive con-vention for the metric signature, that is η =diag(−1,+1,+1,+1).

II. QUANTUM COMPLEMENTARITY

In quantum mechanics, there are physical proper-ties that cannot be simultaneously accessed with arbi-trary precision; these quantity are addressed as comple-mentary, and mathematically they correspond to non-commuting operators. One particular example of comple-mentarity is realized when observing interference versusthe availability of which-way information: these notionsare mutually exclusive in any interference experiment, asthe double-slit or the Mach-Zehnder setup.

For later convenience, we need to briefly review themain characteristics of a Mach-Zehnder interferometer;in so doing, we closely follow Refs. [42, 51].

A. Mach-Zehnder interferometer

Let us consider a two-dimensional Hilbert space H1

with an orthonormal basis |+〉 , |−〉 . Such states can beused to describe a superposed quantum system travelingalong the two arms of an interferometer, as the Mach-Zehnder setup shown in Fig. 1.

The two detectors D± in Fig. 1 quantify the degree ofinterference by measuring the two physical observables σxand σz (x and z Pauli matrix respectively). The formeris defined as

σx := |+〉 〈−|+ |−〉 〈+| , (1)

and we can say that it is measured when the detection ofthe interfering system in D± is associated with an out-come ±1. Indeed, one can easily prove that the availableoutput states |+〉 ± |−〉 are eigenstates of σx with eigen-values ±1, namely σx(|+〉 ± |−〉) = ±(|+〉 ± |−〉).

On the other hand, the second observable quantifiesthe which-path information and it is given by

σz := |+〉 〈+| − |−〉 〈−| . (2)

3

FIG. 1: This figure illustrates a two-way interferometer knownas Mach-Zehnder setup. It is made up of two beam splittersBS1, BS2 and two detectors D±. The superposed systemtravels along the two paths that are labeled by |±〉 , and it cangenerally acquire a relative (controllable) phase shift φ. Thetwo detectors D± measure the degree of interference whichappears on the screen as an interference pattern.

Such quantity has eigenstates |±〉 and respective eigen-values ±1, which means σz |±〉 = ± |±〉 .

Therefore, the observables σx and σz are two anti-commuting operators that take into account the quan-tum complementarity between interference and which-way information. To move forward, we still have to tacklean important aspect, that is reflected into the main dif-ferences between quantum interferometry for pure andmixed states, as well as interferometry for single and com-posite systems.

B. Pure states

If the interfering system is in a pure state, we can write

|ψ〉 = a |+〉+ eiφb |−〉 , (3)

where a, b ∈ R satisfy a2 + b2 = 1, whilst φ ∈ [0, 2π) is acontrollable phase shift between the two different paths.The probabilities of obtaining the two values ± whenmeasuring σx are

Pσx=±1(φ) =

∣∣∣∣〈ψ|( |+〉 ± |−〉√2

)∣∣∣∣2 =1

2± a b cosφ . (4)

For the observable σz, instead, one has

Pσz=+1 = |〈ψ| +〉|2 = a2 , (5)

Pσz=−1 = |〈ψ| −〉|2 = b2 . (6)

Now, we can introduce two fundamental quantities thatturn out to be convenient to properly describe quantumcomplementarity. The interferometric visibility (or sim-ply visibility) is defined as

V :=maxφPσx=± −minφPσx=±1

maxφPσx=± + minφPσx=±1. (7)

Since maxφ cosφ = 1 and minφ cosφ = −1, for purestates the interferometric visibility reads

V = 2ab . (8)

Furthermore, the predictability of a measurement for theobservable σz is identified with

P := |Pσz=+1 − Pσz=−1| = |a2 − b2| . (9)

The pure state (3) is normalized; hence, it is straightfor-ward to observe that

V2 + P2 = 1 . (10)

The physical meaning of this last equation lies in the factthat a non-zero predictability of the two paths necessarilyimplies a non-maximal interferometric visibility of theinterference pattern and vice-versa. In other words, anon-maximal predictability entails the appearance of avisible interference pattern.

C. Mixed states

In the case of mixed states, Eq. (10) is generalized to

V2 + P2 ≤ 1 , (11)

consistently with the interpretation according to whichmixed states are characterized by an incomplete knowl-edge about the physical state. In order to reach Eq. (11),we have to rely on the density matrix formalism. Startingfrom

ρ = a2 |+〉 〈+|+ b2 |−〉 〈−|

+ c∗e−iφ |+〉 〈−|+ ceiφ |−〉 〈+| , (12)

we compute ρ2 and impose1 Trρ2≤ 1, which yields

|c|2 ≤ ab . (13)

The probabilities to measure |±〉 for the observable σzdo not depend on the off-diagonal elements, thus leavingthe expression for the predictability (9) untouched. Dif-ferently from the above picture, the probabilities Pσx=±1

to detect an interference pattern for the mixed state out-lined by the density matrix ρ is given by

Pσx=±1 =

(〈+| ± 〈−|√

2

)ρ

(|+〉 ± |−〉√

2

)=

1

2± |c| cos(φ+ α) , (14)

1 Recall that the trace operation is given by Trρ2

=⟨+|ρ2|+

⟩+⟨

−|ρ2|−⟩

and it defines the purity of a state. The inequality

Trρ2≤ 1 holds true for any mixed state, and it is saturated

only for pure states that indeed satisfy Trρ2

= 1.

4

FIG. 2: This figure illustrates the Mach-Zehnder interferom-eter in Fig. 1 with the addition of an extra degree of freedomthat is able to encode the which-way information. Apart fromthe two beam splitters BS1, BS2 and the two detectors D±,we now have a which-way detector indicated by WWD. Themain difference with respect to the simpler configuration ofFig. 1 lies in the possibility that the interfering system andWWD can become entangled, thereby allowing to obtain thewhich-way information from WWD. At the same time, theinterferometric visibility V is reduced, which tells us that thecomplementarity principle still holds even for a composite sys-tem.

where we have used c = |c|eiα. From the last equationand Eq. (7), we can deduce the interferometric visibilityfor a mixed state:

V = 2|c| . (15)

In general, the interferometric visibility coincides withtwice the amplitude of the off-diagonal component of thedensity matrix. As a matter of fact, for a pure state|c| = ab, which recovers Eq. (8).

Hence, by putting the relations (9) and (15) together,we obtain the inequality V2 + P2 ≤ 1, as claimed inEq. (11).

D. Composite systems

Quantum complementarity requires that precise mea-surements of σx and σz cannot be performed simulta-neously on the same system, as they correspond to twoanti-commuting operators. However, so far we have onlystudied a system made up of a single degree of freedom.We now want to demonstrate that quantum complemen-tarity also applies to systems made up of multiple de-grees of freedom. To this aim, we must add a secondsystem in our interferometric setup that is called which-way detector (WWD) (see Fig. 2 for this configuration).Practically speaking, WWD can be taken as a two-levelsystem.

The Hilbert space of the composite system is labeledas H = H1 ⊗ H2, where H1 is the Hilbert space of theinterfering system and H2 is the Hilbert space of WWD.

At the initial reference time t = 0, the which-way detec-tor is in some state |τ0〉 uncorrelated with the interferingsystem, which implies that the starting state is a prod-uct state. The crucial point is that, for t > 0, WWDbecomes excited and performs a transition to one of thetwo normalized states |τ±〉 depending on the arm trav-eled by the system. Consequently, the total state of thesystem becomes entangled and reads

|ψ〉 = a |+〉 |τ+〉+ eiφb |−〉 |τ−〉 . (16)

Since the (composite) state is now entangled, we canmeasure the two observables σx and σz with absoluteprecision simultaneously. Indeed, we can summarize themeasurements in two steps:

• we access the which-way information by measuringthe observable 11 ⊗ σz on WWD;

• we detect the interfering system at D± to measurethe observable σx ⊗ 12.

In the previous steps, 11 and 12 are the two identityoperators acting on the states of the Hilbert spaces H1

and H2, respectively.At this point, we need to determine the probabilities

Pσz=±1 and Pσx=±1 for the case of the composite sys-tem described by the entangled state (16). Given thetotal density matrix ρψ = |ψ〉 〈ψ| , one can obtain thedensity matrix for one of the two subsystems by partialtracing with respect to the other, i.e. ρi = Trj ρψ withi, j = 1, 2, thereby remaining with a mixed state. It isworth stressing that the states |τ+〉, |τ−〉 are in generalnot orthogonal (i.e. 〈τ+ |τ−〉 6= 0); thence, it comes inhandy to consider an orthogonal basis in H2 such that

|τ±〉 = A± |u〉+B± |v〉 ,

〈u |v〉 = δuv , |A±|2 + |B±|2 = 1 . (17)

By using the above orthonormal basis, we can readilytrace over the Hilbert space H2 and show that the re-duced density matrix ρ1 for the interfering system is

ρ1 = a2 |+〉 〈+|+ b2 |−〉 〈−|

−ab |〈τ+ |τ−〉|(e−iφ−iα |+〉 〈−|+ eiφ+iα |−〉 〈+|

), (18)

where we have used 〈τ+ |τ−〉 = |〈τ+ |τ−〉| eiα.By resorting to the computations already made explicit

in the previous Subsection, we arrive at the followingexpression for the detection probabilities:

Pσx=±1 =1

2± ab |〈τ+ |τ−〉| cos(φ+ α) , (19)

which translates into the formula

V = 2ab |〈τ+ |τ−〉| . (20)

As for the probabilities Pσz=±1 and P, we essentially re-cover the same result of Eq. (9). Clearly, if |τ±〉 are or-thogonal, then V = 0, which entails that, by measuring

5

11 ⊗ σz, one could have a maximal access to the which-way information. However, it is opportune to empha-size that the reduction of visibility V does not dependon whether the measurements at D± have been carriedout or not, and thus whether the two paths are distin-guishable. On the other hand, if the states |τ±〉 are notorthogonal, then not even in principle the two paths canbe perfectly distinguished.

We can better formalize the notion of distinguishablepaths by introducing the distinguishability as the tracenorm distance2 between the final states of WWD, thatis

D :=1

2Tr |τ+〉 〈τ+| − |τ−〉 〈τ−| . (21)

For the setup analyzed in Fig. 2, it can be shown that3

D =

√1− |〈τ+ |τ−〉|2 . (22)

In a nutshell, the distinguishability D of the two paths isthe probability to correctly guess which path was takenby making a measurement on the subsystem WWD.

At this point, by combining Eqs. (20) and (22) andusing the inequalities D,P ≤ 1, we obtain

D2 + V2 = 1− (1−D2)P2 ≤ 1 . (23)

We can now understand the physical meaning of the twoinequalities (11) and (23).

• If P 6= 0, we can have access to the which-wayinformation regardless of the outcome contained inthe which-way detectors, namely even if D = 0 thevisibility V has to be limited;

• if P = 0, then D2 + V2 = 1 for pure states (whilefor mixed states one has ≤ 1), i.e. although thepredictability is zero, the presence of WWD im-plies that the interferometric visibility V cannot bemaximal because D is non-vanishing.

Let us conclude this Subsection by summarizing whatwe have learned about composite systems with multi-ple degrees of freedom. The main message is that thecomplementarity principle holds also for composite sys-tems. Even if one manages to encode the informationin the correlations between the subsystems, these quan-tum correlations necessarily correspond to entanglement,which means that the state of each subsystem is mixed

2 The trace norm distance between two matrices (operators) σ and

ρ is defined as T (σ, ρ) :=√

(σ − ρ)†(σ − ρ). In the case of densitymatrices, σ and ρ are hermitian (but not necessarily positive)

and hence T (σ, ρ) = 12

√(σ − ρ)2 = 1

2

∑i λi, with λi being the

eigenvalues of the difference matrix σ − ρ.3 For a detailed derivation which includes the treatment of mixed

states (for which the equality is replaced by an inequality ≤ 1),see Ref. [51].

and so V cannot be maximal. If the subsystems werepure states, there would not be any entanglement be-tween them and no quantum correlation that could inprinciple reveal which-way information (in this case wewould have had D = 0). Finally, let us remark againthat the reduction or loss of visibility does not dependon whether the degrees of freedom encoded in WWD aremeasured or not.

Since we have reviewed the concept of quantum com-plementarity and the physical properties of the interfero-metric setup under consideration, we are ready to studythe phenomenon of quantum interference in an externalgravitational field.

III. QUANTUM INTERFERENCE OF MASSIVEQUANTUM CLOCKS

In this Section, we will analyze a physical setup inwhich a Mach-Zehnder interferometer is embedded in aweak gravitational field, as for instance the one belongingto Earth (see Fig. 3). As we will see below, in sucha setting the presence of a non-vanishing gravitationalpotential induces a phase shift in the final state detectedat D±. Furthermore, GR effects are responsible for a timedilation between the two arms of the interferometer, asthey are located at two different heights with respect toEarth’s surface. Quantum clock interferometry has beenintensively studied in the context of Einstein’s GR [40,41, 52–55].

Before continuing, let us point out that in this casethe trajectories through the interferometer are supportedagainst gravity. Although it is important to specify themechanism with which the levitation is achieved (e.g.with a harmonic trap), here we just assume that a similarconfiguration exists and leave the aspects related to theprecise experimental details of the apparatus for futureworks.

In this scenario, the role of WWD is played by timedilation effects associated with the internal degrees offreedom of the interfering system. Therefore, we assumethat |τ1,2〉 are internal states that work as clocks, where 1and 2 refer to the upper and lower arm of the interferom-eter, respectively. Thus, we consider a quantum versionof the time dilation phenomenon in which a single clockis superposed along two paths having different propertimes because of the gravitational potential of Earth. Re-markably, we are dealing with a quantum version of thetwin paradox with only a “quantum child” whose “ages”(proper times) are superposed, so that he/she is becom-ing older and younger than himself/herself at the sametime.

Let us now understand what happens in an interfer-ence experiment with a similar clock system. The states|+〉, |−〉 related to the external degrees of freedom arenow labeled by |γ1〉, |γ2〉, where γ1 and γ2 denote thetwo paths traveled by the system in the interferometricapparatus (see Fig. 3). We suppose that γ1 lies farther

6

FIG. 3: This figure illustrates the Mach-Zehnder interferome-ter in Fig. 1 placed in an external gravitational field, i.e. underthe influence of a gravitational acceleration ~g. The informa-tion about the which-way detector WWD is now encoded inan internal time-evolving degree of freedom that works as aclock. Due to time dilation effects, an initial internal state |τ0〉can evolve into a linear combination of the two states |τ1,2〉characterized by two different proper times, ∆τ = τ1−τ2 6= 0.The phases φ1,2 depend on the path and on the dynamics ofthe internal degree of freedom, whereas φ is a controllablephase shift.

from Earth’s surface than γ2, so that the former feels aweaker gravitational potential with respect to the latter.

Following the mathematical formalism introduced inthe previous Section, we can write the quantum state ofthe composite clock system inside the interferometer asfollows [40, 42]

|ψ〉 =1√2

(e−iφ1 |γ1〉 |τ1〉+ e−iφ2+iφ |γ2〉 |τ2〉

), (24)

where the phases φ1,2 are path-dependent and their val-ues are intimately connected with the dynamics of theinternal clock, whereas φ is some controllable phase shift.For the sake of simplicity, we work in the context of zeropredictability, since a = b = 1/

√2⇒ P = 0.

As already mentioned before, by reasonably accountingfor the gravitational interaction on the clock in superposi-tion, the internal degrees of freedom must evolve with dif-ferent rates (i.e. proper times) along each path because ofrelativistic effects. This implies that such degrees of free-dom enclose the information about which path is taken,thus acting as WWD.

As we have chosen the predictability to be zero, wecan quantify the quantum complementarity by exploitingthe concepts of interferometric visibility (20) and distin-guishability (22), which in this case read

V = |〈τ1 |τ2〉| , (25)

and

D =

√1− |〈τ1 |τ2〉|2 , (26)

respectively. Furthermore, Pσz=±1 = 1/2; instead, byfollowing the steps contained in Sec. II D, one can show

that the probabilities to detect an outcome ±1 at D± aregiven by

Pσx=±1 =1

2± 1

2|〈τ1 |τ2〉| cos(∆φ+ φ+ α) , (27)

where ∆φ := φ1 − φ2 and 〈τ1 |τ2〉 = |〈τ1 |τ2〉| eiα.Let us recall one more time that the visibility is lim-

ited as a consequence of the complementarity principle;indeed (for total pure states) V2 + D2 = 1. We nownote that, if the system is in a stationary internal state(namely, if the clock is switched off), then the which-wayinformation is zero and V becomes maximal. In fact, sucha complementarity induced by time dilation can only beobserved with a non-stationary internal state that is al-lowed to evolve along the two paths (i.e. with a switched-on clock). Another comment is in order here: the distin-guishability of the final states of the clock strictly de-pends on whether the difference ∆τ := τ1 − τ2 is largeror smaller than the time interval required by the inter-nal states to evolve between two consecutive distinguish-able states, that is the so-called orthogonalization timet⊥ [40, 42].

In the next Subsections, we will derive the Hamiltonianof both the internal and the external degrees of freedom.Specifically, for the internal dynamics we will consider atwo-level clock system as a model and perform precisecomputations of detection probabilities (27), interfero-metric visibility (25) and distinguishability (26) in thepresence of a generic external, static and weak gravita-tional field.

A. Hamiltonian of a quantum system in agravitational field

To comply with the above requests, in addition to theprevious assumptions we also work under the conditionthat the evolution of the composite quantum system canbe described in a low-energy regime, and that the relativedistances among the internal constituents are sufficientlysmall so as to neglect the variations of the metric overthe size of the system. This hypothesis conveys the ideathat we can assign a single position degree of freedomto the center-of-mass of the system. Consequently, thequantum system can be described as a point-like objectwith internal degrees of freedom, and we can still definea world line along which the proper time is measured.

By accounting for these considerations, the totalHilbert space of the system is given by H = Hext ⊗Hint,where Hext includes states related to external (center-of-mass) degrees of freedom, whilst Hint incorporates thestates describing internal degrees of freedom, or in otherwords the states of the clock. Concerning the gravita-tional sector, the spacetime background can be describedby a generic linearized static metric expressed in isotropiccoordinates

ds2 =−(

1 +2Φ

c2

)c2dt2+

(1− 2Ψ

c2

)(dr2+r2dΩ2), (28)

7

where r =√x2 + y2 + z2, c is the speed of light and Φ(r)

and Ψ(r) are two generic metric potentials; in the caseof GR, we have Φ = Ψ = −GM/r, with G being theNewton’s constant and M the mass of the source (in ourcase M = M⊕).

Moreover, for simplicity and for consistency withRefs. [40, 42], as a preliminary investigation we neglectany complexity stemming from the spinor nature of theexamined system; therefore, we study a real scalar field ϕof rest mass mr in curved spacetime, whose field equationis given by (

g −m2rc

2

~2

)ϕ(x) = 0 , (29)

where g = gµν∇µ∇ν is the curved d’Alembertian,which acts on scalar quantities as follows:

gϕ(x) =1√−g

∂µ(√−ggµν∂ν

)ϕ(x) . (30)

In the linearized regime (i.e. up to linear order in G), byusing the form of the metric given in Eq. (28) we obtain(

+ 2c−2Φ∂20 + 2c−2Ψ∇2

+c−2~∇(Φ−Ψ) · ~∇− m2rc

2

~2

)ϕ(x) = 0 , (31)

where = −∂20 + ∇2 is the flat d’Alembertian, with

∂20 = c−2∂2

t and ∇2 = δij∂i∂j .By thoroughly relying on the procedure introduced in

Refs. [56, 57], we perform a non-relativistic expansion tocompute the Schrodinger equation of a quantum systemin the external spacetime metric (28)

i~∂tψ(t, ~x) = Hψ(t, ~x) , (32)

where ψ(t, ~x) represents the quantum wave function andH is the Hamiltonian

H = mrc2 +

p2

2mr− p4

8m3rc

2+mrΦ +

1

mrc2

(Φ

2+ Ψ

)p2

− 1

4mrc2[p2Φ]− 1

2mrc2[~pΨ] · ~p , (33)

where [· · · ] indicates that the momentum operator actsonly on the object inside the brackets. In the case of GR,as Ψ and Φ are equal to the Newtonian potential, we re-cover the result contemplated in Ref. [57]. The algebraicdetails on the derivation of Eqs. (32) and (33) can befound in Appendix A.

Let us emphasize that the Hamiltonian (33) acts onthe product space H, thereby governing the dynamics ofboth the external and the internal degrees of freedom ofthe composite quantum system. Another crucial aspectto pinpoint is that in general the rest mass mr containstwo distinct contributions [40], which are

mr = m1int + c−2H0 , (34)

where m is the static rest mass, c−2H0 is the dynamicalcontribution due to the internal degrees of freedom and1int is the identity operator on Hint.

If we ignore higher-order terms in c−2, we can writethe Hamiltonian (33) in the following compact form:

H = Hcm +H0

(1 +

Γ(x, p)

c2

), (35)

with

Hcm := mc2 +p2

2m+mΦ +Hcorr , (36)

and

Γ(x, p) := Φ(x)− p2

2m2. (37)

The term Hcorr includes special and general relativisticcorrections that will correspond to a mere phase shift inthe interference patter, which is not relevant for our pur-poses and so we can exclude them henceforth. Instead,the term Γ(x, p) is extremely important and grants accessto the time dilation effect. It is worth stressing that, upto the considered approximation, the gravitational po-tential Ψ does not contribute to the time dilation, butonly to Hcorr; hence, in this regime its only implicationresults in an additional phase shift, which we do not showexplicitly.

Now, using the expression for the Hamiltonian (35), wecan evaluate the evolution of the state inside the inter-ferometer. First of all, let us look at the following totalpure state:

|ψ〉 =1√2

(|ψ1〉+ eiφ |ψ2〉

). (38)

The states |ψ1,2〉 are associated with the two paths γ1,2

and can be determined by acting with the evolution op-erator on the initial state |xin〉 |τ0〉 , thus yielding

|ψi〉 = e− i

~´γi

dt[Hcm+H0(1+ Γc2

)] |xin〉 |τ0〉 , i = 1, 2 . (39)

Note that´γi

dt(1+Γ/c2) =´γi

dτi, where τi is the proper

time along the path γi. By further assuming that H0 istime-independent, we can write

|ψi〉 =[e− i

~´γi

dtHcm |xin〉] [e−

i~H0τi |τ0〉

]≡ |γi〉 |τi〉 , (40)

where we have used

|γi〉 = e− i

~´γi

dtHcm |xin〉 , (41)

and

|τi〉 = e−i~H0τi |τ0〉 . (42)

From the above equation, it is straightforward to calcu-late the interferometric visibility (25), that is

V = | 〈τ1 |τ2〉 | = | 〈τ0| ei~H0∆τ |τ0〉 | , (43)

8

where we have denoted ∆τ = τ1 − τ2 as the proper timedifference between the two paths γ1 and γ2. The inter-nal contribution to the Hamiltonian (35) not only is re-sponsible for time dilation effects, but it also gives riseto entanglement between internal and external degreesof freedom. In turn, the interferometric visibility is notequal to one but it decreases, thereby signaling a loss ofcoherence.

By virtue of the scheme depicted so far, we have ex-tended the results of Refs. [40, 42] to the case in whichthe spacetime metric is not described by GR, as in gen-eral we may have Φ 6= Ψ 6= −GM/r. However, we haveseen that Ψ only contributes to an irrelevant phase shift,whereas Φ is the source of the dominant contribution tothe phase shift and of time dilation. For each extendedtheory of gravity beyond Einstein’s GR, there is a cor-responding modified Newtonian potential Φ which willcause different phase shifts and losses of coherence. Inthe next Subsection, we consider a two-level quantumsystem as a clock to find a simple expression for H0, bymeans of which we can explicitly evaluate the interfero-metric visibility (43).

B. Two-level system as a quantum clock

A two-level system is the most immediate setup wecan think of to construct a quantum clock and to betterunderstand how time dilation can induce a loss of coher-ence. Let us assume that Hint is two-dimensional and isspanned by the basis of two energy eigenstates |1〉 , |2〉of the operator H0, with eigenvalues E1 and E2, respec-tively. Therefore, we can cast the operator H0 as

H0 = E1 |1〉 〈1|+ E2 |2〉 〈2| . (44)

In light of this, the initial internal state can be expressedas

|τ0〉 =1√2

(|1〉+ |2〉) , (45)

so that the evolved internal state is given by

|τi〉 =1√2

(e−

i~E1τ1 |1〉+ e−

i~E2τ2 |2〉

). (46)

With this knowledge, we now have all the ingredientsto explicitly evaluate the interferometric visibility for aninitial pure state in Eq. (43), which gives

V =

∣∣∣∣cos

(∆E∆τ

2~

)∣∣∣∣ , (47)

where ∆E := |E2 − E1|. By introducing the orthogo-nalization time t⊥ = π~

∆E [40] we can rephrase the lastequation in a different shape, that is

V =

∣∣∣∣cos

(∆τ

t⊥

π

2

)∣∣∣∣ . (48)

As long as ∆τ ≥ t⊥, there will be an amount of accessi-ble which-way information encoded in the internal statesof the interfering system, which is traduced in a loss ofcoherence.

In order to compute ∆τ , we recall that we have setthe interferometer so that the gravitational accelerationis only present along the z-direction and that the velocityof the quantum system has only y-component. At thisstage, we further make the reasonable ansatz according towhich the velocities are the same for both paths, in sucha way that the existence of time dilation is solely due tothe presence of the external gravitational field. Now, byexpanding the gravitational potential in the limit ∆hR, with ∆h being the distance between the two arms ofthe interferometer and R the Earth’s radius as well asthe height of the path γ2 (see Fig. 3), namely

Φ(R+ ∆h) ' Φ(R) + Φ′(R)∆h , (49)

we can write

∆τ = c−2

ˆ ∆T

0

dτ [Φ(R+ ∆h)− Φ(R)]

= c−2Φ′(R)∆h∆T , (50)

where ∆T is the coordinate travel time as seen fromthe laboratory frame, which expresses the time exertedby the quantum system to travel through the two armsof the interferometer at two constant heights R andR + ∆h. Note that we have used the shorthand nota-tion f ′ ≡ df/dr. Therefore, the interferometric visibilitycan be reformulated as

V =

∣∣∣∣cos

(Φ′(R)∆h∆T∆E

2~c2

)∣∣∣∣ . (51)

Furthermore, we can also compute the probabilitiesPσx=±1, which turn out to be equal to

Pσx=±1 =1

2± 1

2

∣∣∣∣cos

(Φ′(R)∆h∆T∆E

2~c2

)∣∣∣∣× cos

(mΦ′(R)∆h∆T

~+ φ+ β

), (52)

with β being a phase which includes both special and gen-eral relativistic contributions and whose explicit form isnot relevant for our purposes (see Ref. [40] for its deriva-tion in the context of GR).

To grasp the meaning of the behavior of the detectionprobabilities and of the interferometric visibility, we canstudy the following difference

Pσx=+1 − Pσx=−1 =

∣∣∣∣cos

(Φ′(R)∆h∆T∆E

2~c2

)∣∣∣∣× cos

(mΦ′(R)∆h∆T

~+ φ+ β

). (53)

Since we are interested in theories beyond Einstein’s GR,we can write the gravitational potential as

Φ(r) = ΦGR(r) + χ(r) , (54)

9

where ΦGR = −GM/r while χ takes into account cor-rections to Newton’s potential; the same separation canbe done also for β = βGR + δ. The corrections due tonew physics beyond GR affect both the interferometricvisibility and the phase shift. This feature opens a newwindow of opportunities to test, constrain and discrimi-nate extended theories of gravity in tabletop laboratoryexperiments based upon quantum interference.

C. Non-relativistic regime: COW effect

In the non-relativistic regime, the termΦ′(R)∆h∆T∆E/2~c2 ∼ O(c−2) is negligible (i.e.no time dilation effect appears), thereby allowing thedetection probability to take the form

P nrσx=+1 − P nr

σx=−1 = cos

(mΦ′(R)∆h∆T

~+ φ

). (55)

The term ∆ϕ := mΦ′(R)∆h∆T/~ ∼ O(c0) is the non-relativistic phase shift due to the gravitational potential,and it has been measured for the first time in the COWexperiment [39]. The phenomenon is also known underthe name “COW effect”, and it can be regarded as thegravitational analogue of the Bohm-Aharonov effect [58].This was the first-ever gravitational effect measured ona quantum system. The blue dashed curve in Fig. 4 por-traits the qualitative behavior of the detection probabil-ities in the case of the COW effect.

The first measurement of COW effect [39] was per-formed with neutrons, but many experimental improve-ments have been made in the last decades [59]. To testand constrain gravitational theories beyond GR, we mustcompare the magnitude of the phase shift induced by thecorrected Newtonian potential (54) with the experimen-tal error. From Refs. [60, 61], one can see that the cur-rent experimental error ∆errϕ on the determination ofthe gravitational phase shift is of the order of

∆errϕ = ±0.001 rad . (56)

Thus, for consistency with the experimental data, anypredicted correction ∆χϕ to the phase shift inducedby alternative theories beyond Einstein’s GR such that∆ϕ = ∆GRϕ+∆χϕ must satisfy the following constraint:

|∆χϕ| < |∆errϕ| = 0.001 rad . (57)

D. General relativistic regime: time dilation

In the fully relativistic regime, the termΦ′(R)∆h∆T∆E/2~c2 ∼ O(c−2) is no longer negli-gible, thereby giving rise to drastic differences withrespect to the non-relativistic scenario. Because of thisterm, the absolute value of the cosine function (53) isfundamental, as it physically causes a loss of coherenceas depicted in Fig. 4.

0.0 0.5 1.0 1.5 2.0 2.5 3.0

-1.0

-0.5

0.0

0.5

1.0

ΔT [sec]

P+-P-

FIG. 4: Quantum interference in GR. The blue dashed linecorresponds to the coherent oscillation due to the COW effect,whereas the red solid line represents the interference patternin the presence of time dilation. One can explicitly noticethat, because of time dilation, there is a loss of coherence.For the sake of argument, we have set Φ′∆h∆E/(2~c2) = 1Hz, mΦ′∆h/~ = 15 Hz, φ = β = 0.

This intriguing effect has not been observed yet in apurely gravitational experiment. However, in Ref. [43]the very same decoherence induced by time dilation fora two-level system was experimentally simulated witha Bose-Einstein condensate. In this test, the authorsconsider a Stern-Gerlach type matter-wave interferom-eter under the influence of an external inhomogeneousmagnetic field which mimics the effect of gravitationaltime dilation. In a similar framework, the quantum clockis simply represented by the spin precession of the sys-tem. The final statement of Ref. [43] asserts that theclaimed result may potentially lead to a deeper study ofself-interacting clocks in a laboratory on Earth. More-over, there are good chances that, in the near future, itwill be possible to achieve the high sensitivity requiredto perform a proper gravitational experiment [40].

IV. EXTENDED THEORIES OF GRAVITYBEYOND GENERAL RELATIVITY

In this Section, we introduce a wide class of alternativetheories of gravity whose action contains quadratic cur-vature terms in addition to the Einstein-Hilbert part. Weshow the expressions for the corresponding gravitationalpotentials in the weak-field approximation, which will beemployed to evaluate the relevant physical quantities forour quantum interference setup. In particular, we startfrom the following generic gravitational action [22, 26]:

S =1

2κ2

ˆd4x√−gR+

1

2

[RF1(g)R

+RµνF2(g)Rµν], (58)

10

where κ ≡√

8πG/c4, and the differential operatorsFi(g) can be either analytic or non-analytic functionsof g

Fi(g) =

N∑n=0

fi,nng , i = 1, 2. (59)

Positive (negative) powers of the d’Alembertian, that is,n > 0 (n < 0), correspond to ultraviolet (infrared) gen-eralizations of Einstein’s GR. When N < ∞ and n > 0,we have a local (polynomial) theory of gravity, whosederivative order is 2N + 4; when N = ∞ and/or n < 0we have a nonlocal (non-polynomial) theory of gravity,which means that the two form factors Fi(g) can benon-polynomial differential operators of g.

In what follows, we list several theories of gravity andthe ensuing modified Newtonian potentials. Let us re-call that we deal with static and spherically symmetricbackgrounds; for this reason, the general form of the lin-earized spacetime metric we investigate is the one givenby Eq. (28). In Appendix B, we review quadratic theoriesof gravity and their linearized regime more accurately, in-cluding a derivation of the modified Poisson equations forthe metric potentials.

A. R2-gravity

Let us first start from the simplest quadratic exten-sion of the Einstein-Hilbert action which involves a Ricciscalar squared term with constant form factor

F1 = α , F2 = 0 . (60)

This choice belongs to the wide class of f(R)-theories,where the Lagrangian can be a generic function of theRicci scalar.

For such a model, the two metric potentials Φ and Ψdefined in Eq. (28) are

Φ(r) = −GMr

(1 +

1

3e−m0r

),

Ψ(r) = −GMr

(1− 1

3e−m0r

), (61)

with m0 = 1/√

3α being the inverse of a length and themass of an extra massive spin-0 degree of freedom.

The potential Φ is the only term that appears in theregime we are interested in (see Eq. (53)). From its ex-pression, we can compute the correction χ(r) to the New-tonian potential introduced in Eq. (54), which reads

χ(r) = −GM3r

e−m0r . (62)

B. Stelle gravity

For Stelle four-derivative gravity [5], the differentialoperators of the action are chosen to be

F1 = α , F2 = β . (63)

Consequently, the two metric potentials are

Φ(r) = −GMr

(1 +

1

3e−m0r − 4

3e−m2r

),

Ψ(r) = −GMr

(1− 1

3e−m0r − 2

3e−m2r

), (64)

where m0 = 2/√

12α+ β and m2 =√

2/(−β) are theinverse of a length and the masses of an extra spin-0 anda spin-2 massive degree of freedom, respectively. Notethat we must impose β < 0 in order to avoid tachyonicmodes.

In this case, the correction χ(r) is

χ(r) = −GM3r

e−m0r +4GM

3re−m2r . (65)

C. Analytic nonlocal gravity

As an analytic nonlocal theory of gravity, we select thefollowing form factors [24]:

F1 = −1

2F2 =

1− e−`2g2g

, (66)

where ` is the fundamental length scale of nonlocality atwhich new gravitational physics should become manifest.For this theory, the only propagating degree of freedomaround Minkowski background is the massless transversespin-2 graviton with ±2 helicities.

Due to the peculiar choice for F1 and F2, the two met-ric potentials coincide and are given by

Φ(r) = Ψ(r) = −GMr

Erf( r

2`

), (67)

where Erf(x) = 2√π

´ x0e−t

2

dt is the so-called error func-

tion.Now, the corresponding correction to the Newtonian

gravitational field is

χ(r) =GM

rErfc

( r2`

), (68)

where Erfc(x) = 1 − Erf(x) is the complementary errorfunction.

D. Non-analytic nonlocal gravity

Finally, we consider a non-analytic nonlocal modelwhich is realized with the following choice of form fac-tors:

F1 =α

, F2 = 0 . (69)

11

The two metric potentials contain an infrared modifica-tion of the standard Newtonian one, as it can be seenfrom their expression, namely

Φ(r) = −GMr

(4α− 1

3α− 1

),

Ψ(r) = −GMr

(2α− 1

3α− 1

). (70)

In this case, the correction to the metric potential reads

χ(r) = − α

3α− 1

GM

r. (71)

V. DISCUSSION & EXPERIMENTALCONSTRAINTS

After the calculation of the quantity χ(r) for differentquadratic models of gravity, we can now compute the de-tection probability (53) for each gravitational theory tounderstand how quantum interference is affected by thepresence of a corrected Newtonian potential with respectto standard GR. To this aim, we work with both the non-relativistic COW effect and the relativistic decoherencearising from time dilation to constrain and explore (clas-sical) modified gravity via its interplay with quantummechanics.

Let us start from the COW effect and recall that thecorrections χ(r) to the Newtonian potential must satisfythe experimental bound (57), which can be rephrased as

|χ′(R⊕)| < (0.001)× ~m∆h∆T

, (72)

where R⊕ is Earth’s radius, m is the mass of the interfer-ing quantum system, ∆h is the distance between the twoarms of the interferometer and ∆T is the travel time, orin other words the duration of the quantum superposi-tion.

In what follows, we consider neutrons as the super-posed quantum system for which the spin precession actsas a clock; see Ref. [40] for more details on the experi-mental apparatus and data. By using the neutron massm ' 1.67 × 10−27kg, ~ ' 1.05 × 10−34 m2 kg/s and theachieved experimental value ∆h∆T ' 10−6 m · s, we canrewrite the bound (72) as

|χ′(R⊕)| < 6.5× 10−5 m

s2. (73)

By virtue of this bound, we can constrain the gravita-tional theories introduced in the previous Section usingR⊕ ' 6.37× 106 m and GM⊕ ' 4× 1014 m3/s2.

• In the case of R2-gravity, the inequality reads

e−R⊕m0(1 +R⊕m0) . 6.4× 10−6 , (74)

from which we get the following constraint on thefree parameter

1

m0. 4.3× 105 . (75)

• For Stelle gravity, we have∣∣e−R⊕m0(1 +R⊕m0)

−4e−R⊕m2(1 +R⊕m2)∣∣ . 6.4× 10−6 ,(76)

from which we obtain constraints on m0 and m2 ofthe same order of the one in Eq. (75).

• In the context of analytic nonlocal gravity, the con-straint yields∣∣∣∣∣e−R

2⊕/4`

2

R⊕√π`

+ Erfc

(R⊕2`

)∣∣∣∣∣ . 6.4× 10−6 , (77)

from which it follows

` . 8.7× 105 . (78)

• Finally, in the framework of non-analytic nonlocalgravity we obtain∣∣∣∣ α

3α− 1

∣∣∣∣ . 6.4× 10−6 , (79)

and the constraint on the free parameter reads

|α| . 6.4× 10−6 . (80)

Interestingly, these constraints are of the same order ofthe ones coming from Gravity Probe B [62], which is thebest satellite experiment so far. On the other hand, weshould point out that the best laboratory constraint ondeviations from Newton’s law still comes from torsion-balance experiments performed on Earth. To give anidea on the precise order of magnitude, we can rely onthe results coming from the Eot-Wash experiment, whichgives 1/m0, ` . 10µm [3].

It is common belief that, in the near future, a huge de-velopment will be made in this sector of quantum inter-ference by bringing heavier systems in superposition andby simultaneously increasing the travel time (i.e. thelength of the arms). For instance, there are promisingindications towards the feasibility of superposing heavymasses of the order of 10−16kg (see Ref. [47] for re-lated discussions) and at the same time achieving values∆h∆T ' 106 m · s [40]. Together with a smaller ex-perimental error ∆errϕ, this technological enhancementwould allow us to significantly decrease the magnitude ofthe r.h.s. of the inequality constraints (75,78,80).

So far, we have only discussed the non-relativisticregime, but interesting outcomes emerge especially whenrelativistic effects are accounted for. Although an ex-perimental verification of any effect beyond GR wouldrequire a non-trivial improvement of the current experi-mental status, we can still make a qualitative prediction.Specifically, we can understand how the decoherence as-sociated with time dilation is dramatically influenced bythe modification of the Newtonian potential. As a matter

12

0.0 0.5 1.0 1.5 2.0 2.5 3.0

-1.0

-0.5

0.0

0.5

1.0

ΔT

P+-P-

FIG. 5: Behavior of the detection probabilities (53) in Ein-stein’s GR (dashed red line), R2-gravity (orange solid line)and analytic nonlocal gravity (blue solid line). Since we areonly interested in the qualitative behavior of the decoher-ence, in the plot we have magnified the differences betweenthe three cases under examination; for this reason we haverescaled ∆T so as to render it dimensionless. Moreover, toease the comparison we have chosen the same phase for allthe models. Thus, we can clearly see that for R2-gravity thedecoherence process is faster as compared to GR, while inanalytic nonlocal gravity it is slower.

of fact, the phenomenon of decoherence significantly de-pends on the strength of gravity: the weaker (stronger)the gravitational field interacts, the slower (faster) theloss of coherence becomes. In Sec. IV, we have intro-duced both these types of theories, namely theories inwhich gravity turns out to be weaker at short distanceswith respect to GR and theories in which gravity becomesincreasingly stronger. Hence, it is now clear that the lossof coherence heavily depends on the specific theory withwhich to describe the gravitational interaction.

To better illustrate this concept qualitatively, we con-siderR2-gravity with the gravitational potential (61) andthe analytic nonlocal gravity with the related metric po-tential (67). It is easy to understand that, for the formermodel, the gravitational interaction becomes stronger atsmaller scales, whereas for the latter it becomes weaker.Thus, we would expect that in R2-gravity the decoher-ence effect goes faster, whilst in analytic nonlocal gravityit goes slower. In Fig. 5, we show the magnified behaviorof the detection probability (53) for these two theories incomparison with the GR case. Since we are interested incomparing the interferometric visibility for several gravi-tational theories, we assume that some controllable phaseshift is introduced, so that no difference in the oscillatoryphase appears between GR and the various alternatives.

It is indeed evident that the presence of an extra at-tractive contribution tends to render the decoherence ef-fect faster, while nonlocality tends to weaken gravity andslow down the loss of coherence. This analysis might befurther substantiated in the near future if the experimen-tal status undergoes a non-trivial improvement. In fact,if the gravitational decoherence process is verified in a

laboratory test, then one could experimentally discrimi-nate among several extended theories of gravity, and setthe stage for a brand-new series of experimental tests toprobe new physics beyond GR.

VI. CONCLUDING REMARKS & OUTLOOK

In this paper, we have studied the phenomenon ofquantum interference in an external gravitational fieldbeyond Einstein’s GR. After the description of our the-oretical framework and the experimental apparatus, wehave determined the relevant physical quantities whichare measured in these laboratory tests, among which wehave seen the detection probabilities and the interfero-metric visibility. Subsequently, we have computed thesequantities in the presence of several weak gravitationalfields related to distinct extended theories of gravity.

We have discussed both non-relativistic (COW effect)and relativistic (decoherence) implications, with an equalemphasis on both quantitative and qualitative features.We have noticed that, by working in the setup of theCOW laboratory experiment, we can get constraints thatare of the same order of the ones coming from GravityProbe B, which is the best satellite experiment conceivedso far. We have also pointed out that there are promisingsignals towards significant experimental improvements inthe near future, which could hopefully allow to exceed theperformance of torsion-balance laboratory experimentsand thus give stronger constraints on departures fromGR. In particular, we have observed that, when relativis-tic effects are non-negligible, deviations from GR couldbe probed by looking at the rate of the loss of coher-ence induced by time dilation effects. Indeed, each the-ory of gravity predicts a different time scale over whichthe decoherence occurs, and this could be deemed as anunambiguous signature to discriminate among several ex-tended gravitational models in a laboratory, thereby pro-viding a unique test bench to challenge Einstein’s theory.

It is worth mentioning that the proper-time differencein a Mach-Zender interferometer involving light pulseswas proven to vanish in the case of a linear gravitationalpotential, so that the loss of coherence is only visible atnon-linear order in the potential [64]. The same featuredoes not seem to become manifest in the case of neutroninterferometry, but more detailed and quantitative futurestudied are surely needed.

Before concluding, let us remark that our study maypotentially unravel a new and unexplored path to testand constrain new physics beyond Einstein’s GR by re-sorting to the physics of quantum interference. In thisrespect, it is worth mentioning that recently Bell-type ex-periments were also regarded as a novel and fertile groundwhere to examine gravitational physics [63].

13

Acknowledgments

L.B. acknowledges support from JSPS and KAKENHIGrant-in-Aid for Scientific Research No. JP19F19324.

Appendix A: Schrodinger equation in an externalgravitational field

In this Appendix, we derive the Schrodinger equationfor a quantum system in an external gravitational fieldand the ensuing Hamiltonian. In particular, we computethe important equations (32) and (33) that were usedin the main text. Let us recall that, by working withthe linearized spacetime metric (28), we can expand thecurved d’Alembertian (30) and write the Klein-Gordonequation (29) up to the linear order in Newton’s constant,thus obtaining Eq. (31). We closely follow the proceduredisplayed in Refs. [56, 57] and generalize it to the case oftwo distinct metric potentials Φ and Ψ.

As a first step, we cast the scalar field in a WKB-likeform

ϕ(x) = ei~ ϕ(x) , (A1)

and consider a relativistic expansion

ϕ(x) = c2ϕ0(x) + ϕ1(x) +1

c2ϕ2(x) , (A2)

where ϕ0(x) is taken as a real function. We can now solvethe Klein-Gordon equation perturbatively in powers ofthe speed of light c.

At the order c4, the only contribution to the Klein-Gordon equation (31) is(

~∇ϕ0

)2

= 0 ⇒ ϕ0 ≡ ϕ0(t) . (A3)

At the order c2, we have

(∂tϕ0)2 −m2 = 0 ⇒ ϕ0(t) = ±mt+ const. , (A4)

and we select the ”minus” sign which corresponds to par-ticles at rest with positive energy. As a consequence, upto the order c2 the scalar field reads

ϕ(x) ' e−imc2

~ t . (A5)

At the order c0, we see that

∂tϕ1 = −mΦ +i~2m∇2Ψ− 1

2m

(~∇ϕ1

)2

. (A6)

By defining ϕ1 = ei~ ϕ1 we can rewrite the previous equa-

tion as

i~∂tϕ1 = − ~2

2m∇2ϕ1 +mΦϕ1 , (A7)

which is the non-relativistic Schrodinger equation for aquantum field ϕ1 in an external gravitational field Φ.

At the next order, relativistic contributions to theSchrodinger equation begin to appear. Indeed, at c−2

the Klein-Gordon equation (31) gives

−2m∂tϕ2 − i~∂2t ϕ1 + (∂tϕ1)2 + 4mΦ∂tϕ1 + i~∇2ϕ2

−2~∇ϕ2 · ~∇ϕ1 + 2Ψ[i~∇2ϕ1 − (~∇ϕ1)2]

+i~~∇(Φ−Ψ) · ~∇ϕ1 = 0 . (A8)

By recalling the definition of ϕ1 and denoting ψ =

ϕ1ei

~c2ϕ2 , after some algebra we can cast the previous

equation in terms of the function ψ, that is

i~∂tψ(t, ~x) =

[− ~2

2m∇2 − ~4

8m3c2∇4 +mΦ +

~2

4mc2∇2Φ

− ~2

mc2

(Φ

2+ Ψ

)∇2 +

~2

2mc2~∇Ψ · ~∇

]ψ(t, ~x)

≡ Hψ(t, ~x) , (A9)

where we have neglected orders higher than c−2 and wehave defined the Hamiltonian of a relativistic quantumsystem in an external gravitational field

H = − ~2

2m∇2 − ~4

8m3c2∇4 +mΦ +

~2

4mc2∇2Φ

− ~2

mc2

(Φ

2+ Ψ

)∇2 +

~2

2mc2~∇Ψ · ~∇ . (A10)

By using the momentum representation ~p = −i~~∇, wecan check that the Hamiltonian becomes

H = mrc2 +

p2

2mr− p4

8m3rc

2+mrΦ +

1

mrc2

(Φ

2+ Ψ

)p2

− 1

4mrc2[p2Φ]− 1

2mrc2[~pΨ] · ~p , (A11)

which is precisely the one employed in the main text (33).

Appendix B: Linearized quadratic theories of gravity

In what follows, we briefly review several propertiesof quadratic theories of gravity in the linearized regimeand provide a generic integral expression for the modifiedNewtonian potentials.

Since we are interested in the weak-field approxima-tion, we can expand the action (58) around Minkowski

gµν = ηµν + κhµν , (B1)

where we have introduced the small metric perturbationhµν . In so doing, we obtain [24]

S=1

4

ˆd4x

[1

2hµνf()hµν−hσµf()∂σ∂νh

µν− 1

2h g()h

14

+h g()∂µ∂νhµν+

1

2hλσ

f()−g()

∂λ∂σ∂µ∂νh

µν

], (B2)

where h ≡ ηµνhµν and

f() = 1 +1

2F2() , (B3)

g() = 1− 2F1()− 1

2F2() . (B4)

By varying the action in Eq. (B2), we get the field equa-tions

f()(hµν − ∂σ∂νhσµ − ∂σ∂µhσν

)+ g() (ηµν∂ρ∂σh

ρσ + ∂µ∂νh− ηµνh)

+f()− g()

∂µ∂ν∂ρ∂σh

ρσ = −2κ2Tµν , (B5)

with Tµν being the stress-energy tensor for the matteraction Sm

Tµν = − 2√−g

δSm

δgµν' 2

δSm

δhµν. (B6)

For our purposes, it suffices to consider a configurationin which the source can be well-approximated by a staticand pressure-less point-like object. The stress-energytensor associated with such a source is

Tµν = Mc2δ0µδ

0νδ

(3)(~r). (B7)

In light of this choice, by resorting to the generic shapefor the line element given in Eq. (28), we have κh00 =2Φ/c2 and κhij = 2Ψδij/c

2, and hence the modified Pois-son equations for the two metric potentials read

f(∇2)[f(∇2)− 3g(∇2)]

f(∇2)− 2g(∇2)∇2Φ(r) = 8πGMδ(3)(~r) , (B8)

f(∇2)[f(∇2)− 3g(∇2)]

g(∇2)∇2Ψ(r) = −8πGMδ(3)(~r) , (B9)

where we assume ' ∇2 due to the staticity require-ment.

The differential equations (B8) and (B9) can be solvedby using the Fourier transform method. Indeed, by usingspherical coordinates we can exhibit the solutions for themetric potentials in the following integral form:

Φ(r) = −4GM

πr

ˆ ∞0

dkf − 2g

f(f − 3g)

sin(kr)

k,

Ψ(r) =4GM

πr

ˆ ∞0

dkg

f(f − 3g)

sin(kr)

k,

(B10)

where f = f(−k2) and g = g(−k2) are functions of theFourier momentum squared k2. These last two formu-las were exploited in Sec. IV to compute the modifiedgravitational potentials in each extended theory of grav-ity. As a consistency check, note that, as f = g = 1(F1 = F2 = 0), we recover the Newtonian potentialΦ = Ψ = −GM/r.

[1] C. M. Will, Living Rev. Rel. 17, 4 (2014).[2] B. P. Abbott et al. [LIGO Scientific and Virgo], Phys.

Rev. Lett. 116 (2016) no.6, 061102.[3] D. J. Kapner, T. S. Cook, E. G. Adelberger, J. H. Gund-

lach, B. R. Heckel, C. D. Hoyle and H. E. Swanson, Phys.Rev. Lett. 98 (2007), 021101.

[4] T. Westphal, H. Hepach, J. Pfaff and M. Aspelmeyer,Nature 591 (2021) no.7849, 225-228.

[5] K. S. Stelle, Phys. Rev. D 16, 953 (1977); K. S. Stelle,Gen. Rel. Grav. 9, 353 (1978).

[6] D. Anselmi, JHEP 1706, 086 (2017).[7] A. A. Starobinski, Phys. Lett. B 91, 99 (1980); A. A.

Starobinski, Proceedings of the Second Seminar “Quan-tum Theory of Gravity”, Moscow, 13-15 October 1981,INR Press, Moscow, 58, (1982); A. A. Starobinsky, Sov.Astron. Lett. 9, 302 (1983).

[8] S. Capozziello, G. Lambiase, M. Sakellariadou and An.Stabile, Phys. Rev. D 91, 044012 (2015).

[9] G. Lambiase, M. Sakellariadou, A. Stabile and An. Sta-bile, JCAP 1507, 003 (2015).

[10] G. Lambiase, M. Sakellariadou and A. Stabile, JCAP1312, 020 (2013).

[11] L. Buoninfante, G. Lambiase and A. Stabile, Eur. Phys.J. C 80, no.2, 122 (2020).

[12] G. Lambiase, M. Sakellariadou and A. Stabile,arXiv:2012.00114 [gr-qc] (2020).

[13] M. Asorey, J. L. Lopez and I. L. Shapiro, Int. J. Mod.Phys. A 12, 5711 (1997).

[14] M. Blasone, G. Lambiase, L. Petruzziello and A. Stabile,Eur. Phys. J. C 78, no. 11, 976 (2018).

[15] G. Lambiase, A. Stabile and An. Stabile, Phys. Rev.D 95, 084019 (2017); L. Buoninfante, G. Lambiase,L. Petruzziello and A. Stabile, Eur. Phys. J. C 79, no. 1,41 (2019).

[16] L. Buoninfante and B. L. Giacchini, Phys. Rev. D 102no.2, 024020 (2020).

[17] L. Buoninfante, G. G. Luciano, L. Petruzziello andL. Smaldone, Phys. Rev. D 101, no.2, 024016 (2020).

[18] L. Petruzziello and F. Wagner, arXiv:2101.05552 [gr-qc](2021).

[19] N. V. Krasnikov, Theor Math. Phys. 73 1184, 1987, Teor.Mat. Fiz. 73, 235 (1987).

[20] Yu. V. Kuzmin, Yad. Fiz. 50, 1630-1635 (1989).[21] E. T. Tomboulis, hep-th/9702146.[22] T. Biswas, A. Mazumdar and W. Siegel, JCAP 0603,

009 (2006),[23] L. Modesto, Phys. Rev. D 86, 044005 (2012).[24] T. Biswas, E. Gerwick, T. Koivisto and A. Mazumdar,

Phys. Rev. Lett. 108, 031101 (2012).[25] T. Biswas, A. Conroy, A. S. Koshelev and A. Mazumdar,

Class. Quant. Grav. 31, 015022 (2014); Erratum: Class.Quant. Grav. 31, 159501 (2014).

15

[26] T. Biswas, A. S. Koshelev and A. Mazumdar, Fundam.Theor. Phys. 183, 97 (2016). T. Biswas, A. S. Koshelevand A. Mazumdar, Phys. Rev. D 95, 043533 (2017).

[27] J. Edholm, A. S. Koshelev and A. Mazumdar, Phys. Rev.D 94, no. 10, 104033 (2016).

[28] A. S. Koshelev, K. Sravan Kumar and A. A. Starobinsky,JHEP 03, 071 (2018).

[29] A. S. Koshelev, K. Sravan Kumar, A. Mazumdar andA. A. Starobinsky, JHEP 06, 152 (2020).

[30] L. Buoninfante, A. S. Koshelev, G. Lambiase andA. Mazumdar, JCAP 1809 no.09, 034 (2018).

[31] L. Buoninfante, A. S. Koshelev, G. Lambiase, J. Martoand A. Mazumdar, JCAP 1806, no. 06, 014 (2018).

[32] L. Buoninfante, G. Harmsen, S. Maheshwari andA. Mazumdar, Phys. Rev. D 98 no.8, 084009 (2018).

[33] L. Buoninfante, A. S. Cornell, G. Harmsen,A. S. Koshelev, G. Lambiase, J. Marto and A. Mazum-dar, Phys. Rev. D 98, no. 8, 084041 (2018).

[34] V. P. Frolov, Phys. Rev. Lett. 115 (2015) no.5, 051102,[arXiv:1505.00492 [hep-th]].

[35] J. Boos, V. P. Frolov and A. Zelnikov, Phys. Rev. D 97(2018) no.8, 084021, [arXiv:1802.09573 [gr-qc]].

[36] I. Kolar and A. Mazumdar, Phys. Rev. D 101 (2020)no.12, 124005, [arXiv:2004.07613 [gr-qc]].

[37] S. Dengiz, E. Kilicarslan, I. Kolar and A. Mazumdar,Phys. Rev. D 102 (2020) no.4, 044016, [arXiv:2006.07650[gr-qc]].

[38] J. Boos and I. Kolar, [arXiv:2103.10555 [gr-qc]].[39] R. Colella, A. W. Overhauser and S. A. Werner, Phys.

Rev. Lett. 34, 1472 (1975).[40] M. Zych, F. Costa, I. Pikovski and C. Brukner, Nature

Commun. 2, 505 (2011).[41] I. Pikovski, M. Zych, F. Costa and C. Brukner, Nature

Phys. 11, 668 (2015).[42] M. Zych, “Quantum Systems Under Gravitational Time

Dilation”, Springer thesis (2017).[43] Y. Margalit, Z. Zhou, S. Machluf, D. Rohrlich, Y. Japha

and R. Folman, Science 349, 1205 (2015).[44] M. Bahrami, A. Großardt, S. Donadi and A. Bassi, New

J. Phys. 16, no.11, 115007 (2014).[45] L. Buoninfante, G. Lambiase and A. Mazumdar, Nucl.

Phys. B 931, 250 (2018).[46] L. Buoninfante, G. Lambiase and A. Mazumdar, Eur.

Phys. J. C 78, no.1, 73 (2018).[47] S. Bose, A. Mazumdar, G. W. Morley, H. Ulbricht,

M. Toros, M. Paternostro, A. Geraci, P. Barker,M. S. Kim and G. Milburn, Phys. Rev. Lett. 119, no.24,240401 (2017).

[48] C. Marletto and V. Vedral, Phys. Rev. Lett. 119, no.24,240402 (2017).

[49] A. Belenchia, R. M. Wald, F. Giacomini, E. Castro-Ruiz,C. Brukner and M. Aspelmeyer, Phys. Rev. D 98, no.12,126009 (2018).

[50] R. J. Marshman, A. Mazumdar and S. Bose, Phys. Rev.A 101, no.5, 052110 (2020).

[51] B. G. Englert, Phys. Rev. Lett. 77, 2154 (1996).[52] S. Sinha and J. Samuel, Class. Quant. Grav. 28 (2011),

145018 [erratum: Class. Quant. Grav. 32 (2015) no.23,239501].

[53] S. Dimopoulos, P. W. Graham, J. M. Hogan andM. A. Kasevich, Phys. Rev. D 78 (2008), 042003.

[54] C. Ufrecht, F. Di Pumpo, A. Friedrich, A. Roura,C. Schubert, D. Schlippert, E. M. Rasel, W. P. Schle-ich and E. Giese, Phys. Rev. Res. 2 (2020) no.4, 043240.

[55] F. D. Pumpo, C. Ufrecht, A. Friedrich, E. Giese,W. P. Schleich and W. G. Unruh, [arXiv:2104.14391[quant-ph]].

[56] C. Kiefer and T. P. Singh, Phys. Rev. D 44, 1067 (1991).[57] C. Lammerzahl, Phys. Lett. A 203, 12 (1995).[58] Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959).[59] H. Abele and H. Leeb, New J. Phys. 14, 055010 (2012).[60] J. L. Staudenmann, S. A. Werner, R. Colella and

A. W. Overhauser, Phys. Rev. A 21, no.5, 1419 (1980).[61] G. van der Zouw, “Gravitational and Aharonov-Bohm

Phases in Neutron Interferometry”, Ph.D. Thesis (2000).[62] C. W. F. Everitt, B. Muhlfelder, D. B. DeBra,

B. W. Parkinson, J. P. Turneaure, A. S. Silbergleit,E. B. Acworth, M. Adams, R. Adler and W. J. Bencze,et al. Class. Quant. Grav. 32 (2015) no.22, 224001.

[63] V. A. S. V. Bittencourt, M. Blasone, F. Illuminati,G. Lambiase, G. G. Luciano and L. Petruzziello, Phys.Rev. D 103, no.4, 044051 (2021).

[64] S. Loriani, A. Friedrich, C. Ufrecht, F. Di Pumpo,S. Kleinert, S. Abend, N. Gaaloul, C. Meiners, C. Schu-bert and D. Tell, et al. Sci. Adv. 5 (2019) no.10,eaax8966.