PHYSICAL REVIEW RESEARCH 3, 033065 (2021)

Decoherence-free entropic : Model and experimental tests

Alex J. Schimmoller,1,* Gerard McCaul,1,† Hartmut Abele ,2,‡ and Denys I. Bondar 1,§ 1Tulane University, New Orleans, Louisiana 70118, USA 2Technische Universitat Wien, Atominstitut, Stadionallee 2, 1020 Wien, Austria

(Received 19 December 2020; accepted 12 July 2021; published 19 July 2021)

Erik Verlinde’s theory of entropic gravity [E. Verlinde, J. High Energy Phys. 04 (2011) 029], postulating that gravity is not a fundamental force but rather emerges thermodynamically, has garnered much attention as a possible resolution to the problem. Some have ruled this theory out on grounds that entropic forces are by nature noisy and entropic gravity would therefore display far more decoherence than is observed in ultracold neutron experiments. We address this criticism by modeling linear gravity acting on small objects as an open quantum system. In the strong coupling limit, when the model’s unitless free parameter σ goes to infinity, the entropic master equation recovers conservative gravity. We show that the proposed master equation is fully compatible with the qBOUNCE experiment for ultracold neutrons as long as σ  250 at 90% confidence. Furthermore, the entropic master equation predicts energy increase and decoherence on long time scales and for large masses, phenomena that tabletop experiments could test. In addition, comparing entropic gravity’s energy increase to that of the Diósi-Penrose model for gravity-induced decoherence indicates that the two theories are incompatible. These findings support the theory of entropic gravity, motivating future experimental and theoretical research.

DOI: 10.1103/PhysRevResearch.3.033065

I. INTRODUCTION is unknown whether the particle still exists or has been de- stroyed. So, the particle has gone from being in a pure “exists” The theory of entropic gravity challenges the assumption state to either an “exists” or “destroyed” state with equal that gravity is a conservative force, i.e., one that is propor- probabilities. Hence, the ’s has increased tional to the gradient of a potential energy. Entropic gravity by S = k ln(2). Newton’s second law F = ma immedi- instead postulates that gravity is an that points b ately follows from the entropic force definition F = T S/x in the direction of maximum entropy [1]. after substituting (i) the amended form of Bekenstein’s for- The definition of entropic forces follows from the first mula S = 2πk ; (ii) the Compton wavelength x; and law of thermodynamics, δQ = dU + δW , which equates heat b (iii) Unruh’s formula [3–5], k T = ha¯ /(2πc), connecting ac- supplied to a system δQ to the change in the system’s internal b celeration with temperature. Such a derivation of Newton’s energy dU plus work done δW . If there is a change in entropy second law is valid for a black hole—an extreme concentra- dS = δQ/T with no change in internal energy, then there is tion of mass. Verlinde postulates this conclusion to be valid work done δW = TdS. The entropic force is the one per- for all masses, which should be represented by holographic forming the work F = δW/dx = TdS/dx due to the entropy screens [6]. gradient. Verlinde’s theory has undergone scrutiny, especially over While Newtonian gravity is conservative, Verlinde’s pro- the invocation of holographic screens and the Unruh formula posal that gravity is entropic in nature [1] has garnered much [7–10], although these criticisms acknowledge a connection attention. A simple argument in favor of this hypothesis between thermodynamics and gravity [11–13]. Recently, an goes as as follows: Bekenstein [2] argued that a particle extension to nonholographic screens has been established of mass m held by a string just outside a black hole will [14]. effectively be absorbed once the particle approaches within The aim of this paper is to refute another prevailing crit- one Compton wavelength, x = h¯/(mc), of the event hori- icism of entropic gravity [7–10] that entropic forces are by zon. Since the particle is so close to the event horizon, it nature too noisy and thus destroy quantum . In particular, it has been argued in [10] that if gravity were an entropic force, then it could be modeled as an environment in an open quantum system. Brownian motion is not observed for *[email protected] small masses inside the environment, so these small objects †[email protected] ‡[email protected] must be very strongly coupled to the gravity environment. But §[email protected] the strong coupling must lead to ample wave function collapse and quantum decoherence. However, such decoherence is not Published by the American Physical Society under the terms of the observed in cold neutron experiments [15]. Thus, entropic Creative Commons Attribution 4.0 International license. Further gravity cannot be true according to [7,8]. distribution of this work must maintain attribution to the author(s) We disprove this argument by constructing (Sec. II)a and the published article’s title, journal citation, and DOI. nonrelativistic model [Eq. (5)] for quantum particles (e.g.,

2643-1564/2021/3(3)/033065(11) 033065-1 Published by the American Physical Society SCHIMMOLLER, MCCAUL, ABELE, AND BONDAR PHYSICAL REVIEW RESEARCH 3, 033065 (2021) neutrons) interacting with gravity represented by an environ- In the simplest case, a linear gravitational potential can ment. According to this model, the stronger the coupling to be treated as a single dissipative environment and the free- the reservoir, the lower the decoherence. Moreover, arbitrarily fall dynamics (2) are satisfied by the master equation of the low decoherence can be achieved by simply increasing the Lindblad form [20] positive dimensionless coupling constant σ, which is a free   dρˆ i pˆ2 parameter of this model. In the limit σ →∞, the model =− , ρˆ + D(ˆρ), (5) recovers Newtonian gravity as a potential force [Eq. (3)]. A dt h¯ 2m    σ comparison of our model with data from the recent qBOUNCE D ρ = mgx0 − ixˆ ρ + ixˆ − ρ , σ  (ˆ) exp ˆ exp ˆ (6) experiment [16] provides a lower bound 500 (Secs. h¯ x0σ x0σ III and IV). We discuss some of entropic gravity’s phys- where ical implications including monotonic energy increase and  1/3 mass-dependent decoherence in Sec. V. A relationship to the h¯2 x = (7) Diósi-Penrose gravitational model is also discussed. 0 2m2g is a characteristic length and σ is a unitless, positive coupling II. A MODEL OF ENTROPIC GRAVITY ACTING NEAR constant, which is a free parameter in the model [21]. Note EARTH’S SURFACE that the Hamiltonian in Eq. (5) only contains the kinetic- In this section, we develop a near-Earth model of entropic energy term, and the linear gravitational potential is replaced gravity acting on quantum particles. Consider a particle of by the dissipator (6). We propose to use Eq. (5) as the model mass m a small distance x above Earth’s surface in free-fall. for entropic gravity acting on quantum particles near Earth’s In the classical case, the particle’s dynamics are dictated by surface. Newton’s equations of motion To elucidate how the dissipator (6) mimics a linear gravita- d 1 d tional potential, we employ the Hausdorff expansion with the x = p, p =−mg, (1) assumption σ →∞to obtain dt m dt     dρˆ i pˆ2 mg 1 1 where p is the particle’s momentum and g is the gravitational =− + mgxˆ, ρˆ + xˆρˆ xˆ − xˆ2ρˆ − ρˆ xˆ2 acceleration. In the quantum regime, however, these equations dt h¯ 2m x0h¯σ 2 2   must be recast in the language of operators and expectation 1 + O . (8) values. This is accomplished via the Ehrenfest theorems [17] σ 2 d 1 d xˆ = pˆ, pˆ =−mg. (2) Thus, utilizing large values of the coupling constant σ ,the dt m dt master equation for entropic gravity (5) can approximate the Free fall of a quantum particle, whose state is represented conservative equation (3) with an arbitrarily high precision. by the density matrixρ ˆ , in a linear gravitational potential is The argument put forth in Refs. [7,8] against the entropic described by the Liouville equation [18] gravity has the following fault: It is based on the assumption   that the evolution of a neutron’s initial pure state to a mixed dρˆ i pˆ2 =− + mgxˆ, ρˆ . (3) one is generated by a non-Hermitian translation operator (see dt h¯ 2m Eq. (12) of [7]) leading to the Schrodinger equation with a non-Hermitian Hamiltonian (see Eq. (16) of [7]). While Recalling that the expectation value for an observable Oˆ is non-Hermitian corrections to the Schrodinger equation have given by Oˆ=Tr(Oˆρˆ ), it can easily be shown that Eq. (3) been historically used to incorporate some aspects of dissipa- satisfies the free-fall Ehrenfest theorems (2). Equation (3)is tion, such an approach suffers from physical inconsistencies the conservative model for free fall. The purity of a quantum [22] and has been abandoned in the modern theory of open stateρ ˆ is given by Tr(ρ ˆ 2 ). The purity reaches its maximum quantum systems. Hence, instead of Eq. (12) from Ref. [7] value of unity if and only if the density matrix corresponds to that reads the sate representable by a wave function. It is an important feature of Eq. (3) that it preserves the purity, i.e., Eq. (3) ρˆ (z + z) ≡ Uˆ ρˆ (z)Uˆ †, Uˆ Uˆ † = 1, (9) maintains coherence. Equation (3) is not the only one capturing free-fall dy- the Kraus representation (see, e.g, Ref. [19]) for the evolution ρˆ (z) → ρˆ (z + z) should have been used namics (2). In fact, within the language of open quantum   systems [19], there are an infinite number of master equations, ρˆ (z + z) ≡ Kˆ ρˆ (z)Kˆ †, Kˆ †Kˆ = 1. (10) which satisfy the above Ehrenfest theorems [20]. It has been n n n n n n shown in [20] that for arbitrary G(p) and F (x), the Ehrenfest theorems The Kraus representation furnishes the most general de- scription for evolution of open quantum systems. The only d d xˆ = G(ˆp), pˆ = F (ˆx) (4) requirement used to arrive at Eq. (10) is that the mapping dt dt ρˆ (z) → ρˆ (z + z) should be completely positive. The latter can be satisfied by coupling a closed system with the usual is a stronger requirement than the fact that physical evolution Hamiltonian Hˆ = pˆ2/(2m) + U (ˆx) to a series of tailored en- preserves the positivity of a density matrix. Finally, we note vironments. We take advantage of this fact to model gravity as that a Lindblad master equation [such as, e.g., Eq. (5)] can be an environment in an open quantum system fashion [19]. recast in a Kraus form.

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If the O(σ −2 ) term is dropped in Eq. (8), then the resulting Eq. (8) describes a particle undergoing a continuous quan- tum measurement of its position [19,23]. The entropic master equation (5) interprets gravity as a continuous measurement process extracting information about the position of a massive particle. The extraction of information is responsible for the entropy creation [24]. As a result, the purity of the quantum system is no longer preserved. The rate of change of the purity induced by evolution governed by Eq. (5) is estimated as σ →∞,   d mg 1 Tr(ρ ˆ 2 ) =−2 Tr(ρ ˆ 2xˆ2 − (ˆρxˆ)2 ) + O . (11) σ σ 2 dt x0h¯ FIG. 1. Comparing the qBOUNCE experiment [16] with predic- It is shown in [20] that Tr(ρ ˆ 2xˆ2 − (ˆρxˆ)2 )  0; thus, the purity tions of the master equation for entropic gravity [Eq. (18)] as well σ as the conservative gravity [Eq. (17)]. All data points from the ex- is monotonically decreasing. Furthermore, the larger the , ω the more purity is preserved. Since we can elect to make σ periment are visible with corresponding frequency and oscillation strength aω data replaced with a single index on the horizontal arbitrarily large in our model, the original criticism of entropic axis. Twenty states are accounted for in numerical propagation of gravity not maintaining quantum coherence can no longer be Eqs. (17)and(18). considered valid. The proposed entropic master equation (5) obeys a variant of the (see, e.g., Refs. [25,26]). Ac- to results of the qBOUNCE experiment [16]. This experiment cording to [27], the strong equivalence principle states that was performed at the beam position for ultracold neutron at “all test fundamental physics is not affected, locally, by the the European neutron source at the Institut Laue-Langevin presence of a gravitational field.” Hence, dynamics induced in Grenoble and uses gravity resonance spectroscopy [55]to by a homogeneous gravitational field must be translationally induce transitions between quantum states of a neutron in invariant. Equation (5) is known to be translationally invariant the gravity potential of the earth. In region I of this experi- [28–33]. ment, neutrons are prepared in a known mixture of the first Since Verlinde’s theory treats gravity as a thermody- three quantum bouncer energy states (see Appendix A). These namically emergent force, it is not appropriate to quantize neutrons then traverse a 30 cm horizontal boundary, which os- gravity and talk about the existence of [34,35]. cillates with variable frequency ω and oscillation amplitude a, However, our entropic master equation (5) phenomelogically inducing Rabi oscillations between the “bouncing-ball” states hints at gravitons. Equations similar to Eq. (5) have long of neutrons. In Figs. 2 and 3 below, the oscillation strength been employed for the nonperturbative description of a quan- is defined as aω. Finally in region III, neutrons pass through tum system undergoing collisions with a background gas a state selector, leaving neutrons in an unknown mixture of of atoms or photons [31–33,36–38]. Transferring this mi- the three lowest energy states to be counted. To model this croscopic picture, the dissipator (6) can be interpreted as experiment, the free-fall master equation (5) must be amended describing colissions of a massive quantum particle with a to account for the oscillating boundary, and simulations must bath of gravitons; moreover,h ¯/(x0σ ) stands for the momen- account for variable neutron times-of-flight and the unknown tum of a . To preserve purity σ must be large, which selection of neutrons in region III. makes the momentum of a graviton infinitesimally small. This conclusion is compatible with the fact that detecting a graviton remains a tremendous challenge [39], which might become feasible [40]. A plethora of models for gravitation induced decoherence, which describe quantum matter interacting with a stochas- tic gravitational background, has been put forth [35]. It is worth pointing out that some of these models mathematically resemble the entropic master equation (5); in particular, the models of time fluctuations [41–43], spontaneous collapse [44–46], and the Diósi-Penrose model [47–52]. However, de- spite mathematical resemblance, they can make very different predictions from Eq. (5) (see Sec. V). We also note that Lindblad-like master equations have been recently emerged in post-quantum classical gravity [53,54], where a quantum FIG. 2. Comparing the qBOUNCE experiment [16] with predic- system interacts with classical space-time. tions of the master equation for entropic gravity [Eq. (18)] as well as the conservative gravity [Eq. (17)] by varying oscillation frequency ω ω / III. MODELING THE qBOUNCE EXPERIMENT ( ) when the oscillation strength (a ) is set to 2.05 mm s. Twenty states are accounted for in numerical propagation of Eqs. (17)and Now that the free-fall model for entropic gravity has been (18). σ is a free parameter in the entropic gravity master equation. established [Eq. (5)], it is desirable to see how it compares When σ  500 the experiment agrees well with entropic gravity.

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Here, the kinetic and boundary terms are inside the commu- tator and D(ˆρ) is the gravity environment (6). Because D(ˆρ) is translationally invariant, the oscillating boundary does not alter the dissipator (6). For simulations of the qBOUNCE experiment, we transform the equations of motion into the reference frame of the oscil- lating boundary (see Appendix B). After applying the change of variablesx ˜ = x − a sin(ωt ) and translatingx ˜ → x, the con- servative model’s Liouville equation (17) becomes [56]   dρˆ i pˆ2 h¯2 =− + mgxˆ − δ(ˆx) − aω cos(ωt )ˆp, ρˆ , dt h¯ 2m 4m FIG. 3. Comparing the qBOUNCE experiment [16]withpredic- (19) tions of the master equation for entropic gravity [Eq. (18)] as well as the conservative gravity [Eq. (17)] by varying oscillation strength and the entropic Lindblad equation (18) reads ω ω   (a ) with the oscillation frequency ( ) set to the the transition dρˆ i pˆ2 h¯2 between the ground and third excited states of the “bouncing ball” =− − δ(ˆx) − aω cos(ωt )ˆp, ρˆ + D(ˆρ). dt h¯ 2m 4m problem [ω = ω03 = (E3 − E0 )/h¯ = 4.07 kHz]. σ is a free param- eter in the entropic gravity master equation. When σ  500 the (20) experiment agrees well with entropic gravity. Differentiating with respect to the unitless time τ = tmgx0/h¯ yields the unitless conservative Liouville equation The simplest way to model the boundary is by modifying dρˆ the Ehrenfest theorems. For a system with the general Hamil- =−i[hˆ + ξˆ + wˆ , ρˆ ], (21) tonian dτ Hˆ = pˆ2/(2m) + U (ˆx), (12) along with the unitless entropic Lindblad equation  = |ψ= dρˆ and the boundary condition x 0 0, the second Ehren- =−i[hˆ + wˆ , ρˆ ] + σ (Dˆ ρˆ Dˆ † − ρˆ ). (22) fest theorem reads dτ     2 ˆ d   = −   + h¯ d  |ψ d ψ|  Here, h represents the kinetic energy and boundary terms, pˆ U (ˆx) x x ξˆ dt 2m dx = dx = gives the potential energy term,w ˆ accounts for the ac- x 0 x 0 ˆ 2 celerating frame and D gives the first exponential inside the  h¯  D ρ = −U (ˆx) + δ (ˆx), (13) (ˆ) term. Matrix elements for these operators are given in 4m Appendix C. Equations (21) and (22) are used in the following where δ(x) is the Dirac delta function, defined as simulations. ∞ Now that proper master equations have been established dxδ(n)(x − x) f (x) = (−1)(n) f (n)(x). (14) for region II, how long must they run? The time-of-flight −∞ t f for each neutron is determined by its horizontal veloc- = . / τ Thus modifying the Hamiltonian Hˆ to include the boundary ity v 0 30(mgx0 ) (¯h f ), ultimately determining final state τ = ρ τ |  | term, populations Pj ( f ) Tr( ˆ ( f ) E j E j ). In this experiment, neutrons are measured to have horizontal velocities v between 2 2 pˆ h¯  5.6 and 9.5 m/s. We elect to make the horizontal neutron Hˆ = + U (ˆx) − δ (ˆx). (15) 2m 4m velocity v an additional free parameter in the model confined recovers the desired Ehrenfest theorem (13). to this range. While this choice in modeling does not capture In order to make the boundary oscillate, one simply needs the range of velocities contributing to the overall transmis- to add a sinusoidal term inside of the Dirac delta function: sion, results in Sec. IV indicate that this assumption does not diminish the overall point of the paper. pˆ2 h¯2 Hˆ = + U (ˆx) − δ(xˆ − a sin(ωt )). (16) Finally, a full model of the qBOUNCE experiment [16] 2m 4m requires modeling the state selection in region III. The state Here, a is the oscillation amplitude. selector consists of an upper mirror positioned just above the In the particular case of potential gravity [Eq. (3)], a attainable height of a ground state neutron. However, higher neutron’s dynamics while inside the qBOUNCE apparatus is states leak into the detector as well. We thus define relative described by the Liouville equation: transmission (neutron count rate with the oscillating boundary   divided by the count rate without oscillation) to be a linear dρˆ i pˆ2 h¯2 =− + mgxˆ − δ(xˆ − a sin(ωt )), ρˆ . (17) combination of the three lowest energy state populations: dt h¯ 2m 4m T = c0P0 + c1P1 + c2P2, (23) Meanwhile, the entropic case [Eq. (5)] gives   where c , c , and c are unknown, positive coefficients to be dρˆ i pˆ2 h¯2 0 1 2 =− − δ(xˆ − a sin(ωt )), ρˆ + Dˆ (ˆρ). (18) determined from experimental data as explained in Appendix dt h¯ 2m 4m D. Since the state selector is designed to scatter away excited

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σ = 500, and approach 1.28, 0.55, and 0.55, respectively, as σ →∞.

V. DISCUSSION AND FUTURE DIRECTIONS We have shown that a linear gravitational potential can be modeled by an environment coupled to neutrons. This entropic gravity model overcomes the criticism put forth in Ref. [10] since the master equation (5) is capable of maintain- ing both strong coupling and negligible decoherence and is fully compatible with the qBOUNCE experiment [16]. More- over, the entropic model recovers the conservative gravity FIG. 4. χ 2 as a function of σ . The gray area represents the (3)asσ →∞. Our findings provide support for the entropic χ 2 χ 2 90% confidence interval. min is the minimum value among the gravity hypothesis, which may spur further experimental and simulated results. When σ  250, entropic gravity falls within this theoretical inquires. region. When σ  500, entropic gravity fits experimental data as Let us compare the predictions of the entropic master equa- well as conservative gravity. tion (5) and the Diósi-Penrose (D-P) model [35,57]. Consider the total-energy operator Hˆ = pˆ2/(2m) + mgxˆ. While the ex-  ˆ  neutrons, the physical and engineering consideration leads to pected total energy H remains constant in the conservative case (3), the entropic model’s rate of the expected energy the constraint c0  c1  c2. change is given by

IV. SIMULATING THE qBOUNCE EXPERIMENT d gh¯ Hˆ  = . σ (24) With the results of Sec. III, we can effectively simulate dt 2x0 the qBOUNCE experiment [16]. In region I of the experiment, That is, under entropic gravity, the test particle’s total energy neutrons are prepared initially as an incoherent mixture with increases at a rate ∝ 1/σ regardless of the initial state. Hence, 59.7% population in the ground state, 34.0% in the first the entropic model avoids a thermal catastrophe in the large excited state, 6.3% in the second excited state and no pop- coupling limit (σ →∞), unlike the D-P model. According ulation in higher states. Thus, the initial state of simulated to the latter, the rate of energy√ increase (given by Eq. (94) 3 neutrons is the incoherent mixtureρ ˆ (0) = 0.597|E0E0|+ in Ref. [35]) equals mGh¯/(4 πR ), where G is the gravita- . |  |+ . |  | 0 0 340 E1 E1 0 063 E2 E2 . In region II, neutrons inter- tional constant and R0 is a coarse-graining parameter set to act with gravity and the oscillating boundary. The density the nucleon’s radius, 10−15 m. For a neutron, the D-P model matrix evolves according to either the conservative (21)or predicts the rate of energy increase to be 1.66 × 10−27 W entropic (22) unitless master equations, with frequency ω (= 10.4 neV/s), while the entropic model prediction is sig- and oscillation strength aω determined by the experimental nificantly lower: 1.76 × 10−31 W(= 1.1 peV/s) assuming setup. After the interaction time τ f (determined by the free- σ = 500 (see Sec. IV). For the entropic model to display as velocity parameter v), simulated neutrons have effectively much energy increase as the D-P model predicts, σ would passed through region II of the experiment. We calculate the need to be 0.05, much less than what is permitted by the final populations P0, P1, and P2. qBOUNCE experiment as shown in Sec. IV. Moreover, for a We perform minimization of χ 2 over the space of the five 1 kg mass, the D-P model predicts a rate of energy increase parameters: c0, c1, c2, v, and σ (see Appendix D for details). ≈1 Watt! Such a significant quantity should be readily notice- An agreement between the theory and experiment can be able. Comparatively, the entropic model predicts the rate of observed in Figs. 1, 2, and 3.AsEq.(8) predicts, transmission energy increase of only 0.125 pW when σ = 500. Raising R0 values for entropic simulations approach those of the con- can significantly reduce the D-P model’s energy increase, but σ servative model as increases. This is to say, conservative there is no physical justification for larger values of R0.We gravity can be recovered with large enough σ in the entropic also note that recent extensions to the D-P model to include model, and decoherence effects are therefore unnoticed. In the gravitational backreaction [58] suffer from the same issue. particular, a good agreement of the experimental data with the As shown in Appendix F, the additional terms arising from the entropic model is observed when σ equals 500. Furthermore, inclusion of a semiclassical field serve only to double the rate χ 2 analysis shown in Fig. 4 reveals that simulations with of energy increase. Comparatively, there is no known upper σ  250 fit the experimental data with 90% confidence. Note bound on σ , and energy increase vanishes as σ →∞. that at σ = 500 the values of χ 2 for conservative and entropic We believe that the lower bound σ = 500, deduced in gravity coincide. In conclusion, we take 500 to be the lower Sec. IV from the qBOUNCE experiment, is highly likely to be bound for σ . an underestimation. A more realistic lower bound should be In total, the entropic model of the qBOUNCE experiment σ  4.6 × 105. Let us describe how the latter value could be consists of five free parameters: σ, v, c0, c1, and c2.For confirmed experimentally. According to Eq. (24), a neutron entropic simulations with σ  250, the best-fit velocity hovers will gain energy E within a time t, around the lower limit of 5.6 m/s. As σ →∞, the best-fit 2x σ velocity approaches 6.58 m/s. The transmission coefficients t = 0 E. (25) c0, c1, and c2 equal to 1.46, 0.50, and 0.50, respectively, for gh¯

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Assume the neutron is initially prepared in the ground state potential barrier at x = 0. We wish to find the the eigenvalues |E0 of the “bouncing ball”. Then, we let it evolve for the E and eigenvectors |E such that time approaching the neutron’s lifetime t = 881.5 s and ˆ |  = | , measure the final state. If it jumped to the first excited state Hc E E E where (A1) |  E1 , then according to Eq. (25), the neutron must have gained 2   − σ  . × 5 pˆ E E1 E0 implying that 4 6 10 . If the neutron Hˆc = + mgxˆ. (A2) 5 2m does not reach |E1, then σ>4.6 × 10 . Storage experiments with neutrons might provide these limits [56]. Applying x| to Eq. (A1), the equation can be rewritten as The entropic master equation (5) predicts gravity induced   d2 2m decoherence albeit at a much lower rate than, e.g., the D-P − [mgx − E] x|E = 0, (A3) 2 2 model. In Appendix E, we show that if td is the decoherence dx h¯  time for a particle of mass m, then the decoherence time td for and the infinite potential barrier manifests itself in the bound-  = / −1/3 mass M is td (M m) td . Hence, the larger the mass, the ary condition faster the decoherence. Moreover, measuring the decoherence times would also directly identify σ. The recent experiment x = 0|E = 0. (A4) [59] that observed optomechanical nonclassical correlations It is easy to confirm that the solutions to Eq. (A3)aregiven involving a nanoparticle could perform such a test. by Although the proposed entropic gravity model is limited to     the low-energy, near-Earth regime, its physical implications  |  = ξ − E + ξ − E , provide a glimpse into several open cosmological questions. x E c1Ai c2Bi (A5) mgx0 mgx0 As Ref. [35] mentions regarding collapse gravitational mod- els, entropic gravity’s nonunitarity dynamics could resolve where the black hole information paradox [60,61], and its runaway  1/3 h¯2 energy (24) could pose solutions to the [62], x0 = , (A6) cosmological inflation, and quantum measurement problems 2m2g σ [63]. With greater restriction of from precision experiments ξ = x/x0, (A7) and better understanding of its physical implications at all , time and energy scales, entropic gravity can be further ex- and c1 c2 are constants, and Ai and Bi are the two linearly- plored as a feasible gravitational theory. independent solutions to the Airy equation   d2 − y w(y) = 0, w = Ai(y), Bi(y). (A8) 2 ACKNOWLEDGMENTS dy Considering the normalization condition H.A. and D.I.B. are grateful to Wolfgang Schleich and Marlan Scully for inviting us to the PQE-2019 conference, ∞ | | |2 = , where this collaboration was conceived. H.A. thanks T. Jenke x E dx 1 (A9) 0 for fruitful discussions. A.J.S. and D.I.B. wish to acknowledge →∞ →∞ the Tulane Honors Summer Research Program for funding we exclude Bi since Bi(x) as x [65]. Applying = this project. G.M. and D.I.B. are supported by the Army (A9)to(A3) with c2 0, we get our normalization coeffi- Research Office (ARO) (Grant No. W911NF-19-1-0377), De- cient:    fense Advanced Research Projects Agency (DARPA) (Grant ∞ −1/2 2 E No. D19AP00043), and Air Force Office of Scientific Re- c = c(E ) = x0 dξAi ξ − . (A10) m gx search (AFOSR) (Grant No. FA9550-16-1-0254). The views 0 g 0 and conclusions contained in this document are those of the Thus, solutions in the coordinate representation are given by authors and should not be interpreted as representing the of- Ai ξ − E ficial policies, either expressed or implied, of ARO, DARPA, x|E =  mgx0  . (A11) AFOSR, or the U.S. Government. The U.S. Government is au- ∞ 2 E 1/2 x0 dξAi ξ − thorized to reproduce and distribute reprints for Government 0 mgx0 purposes notwithstanding any copyright notation herein. H.A. Applying the boundary condition (A4) yields eigenvalues gratefully acknowledges support from the Austrian Fonds zur En =−mgx0an+1, where n = 0, 1, 2, ... and a j denotes the jth Förderung der Wissenschaftlichen Forschung (FWF) under zero of Ai. By convention, the energy eigenstates of a system Contract No. P 33279-N. are numbered beginning with zero to signify the ground state, whereas the zeros of a function are numbered beginning with one, hence the nth energy state corresponding to the (n + 1)th APPENDIX A: SOLVING THE SCHRÖDINGER EQUATION zero of the Airy function. Corresponding eigenfunctions are FOR A BOUNCING BALL given by

In this section, we solve the quantum bouncing ball prob- Ai(ξ + an+1) x|En =   . (A12) lem (as is done in [64]). Consider the time-independent ∞ 2 1/2 x dξAi (ξ + a + ) Schrödinger equation for a particle of mass m experiencing 0 0 n 1 = {| }∞ a linear gravitational potential U (ˆx) mgxˆ and an infinite The set of eigenvectors En n=0 forms an orthonormal basis.

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APPENDIX B: THE QUANTUM BOUNCER WITH AN the original Schrodinger equation (B7) from the mirror’s ref- OSCILLATING BOUNDARY: CHANGE OF VARIABLES erence frame: In this section, the Schrödinger equation used to model the ∂ ih¯ |ψ(t ) = {Hˆ0 + Wˆ }|ψ(t ), (B12) qBOUNCE experiment is converted to the reference frame of ∂t the oscillating boundary. The following treatment closely fol- 2 2 pˆ h¯  lows Ref. [56]. Consider the 1D time-dependent Schrödinger Hˆ0 = + U (ˆx) − δ (ˆx), (B13) 2m 4m equation for a particle with potential energy U (ˆx), along with an infinite potential barrier, which oscillates with a frequency Wˆ = U (a sin ωt ) − aω cos(ωt )ˆp. (B14) ω and amplitude a about the point x = 0: Note that thep ˆ operator in this last equation is only present − ∂  | d so as to be transformed into ih¯ ∂ when x is reapplied. ih¯ |ψ(t ) = Hˆ |ψ(t ) (B1) x dt When we convert our Schrodinger equation (B12) into the density matrix formalism, we get that where   2 2 dρˆ i pˆ h¯  pˆ2 h¯2 =− + U (ˆx) − δ (ˆx) − aω cos(ωt )ˆp, ρˆ . Hˆ = + U (ˆx) − δ(xˆ − a sin(ωt )). (B2) dt h¯ 2m 4m 2m 4m (B15) When x| is applied on the left to both sides of (B1), one gets Furthermore, consider the D(ˆρ) operator in the entropic the Schrödinger equation in the coordinate representation: =  model given by Eq. (6). Under the change of variablesx ˜ d h¯2 ∂2 x − a sin(ωt ), D(ˆρ) is invariant under the change of variables ih¯ x|ψ(t ) = − + U (ˆx) since dt 2m ∂x2      ˆ + ω ˆ + ω 2 −i(x˜ a sin t ) ρ +i(x˜ a sin t ) − h¯ δ − ω  |ψ . exp ˆ exp (x a sin( t )) x (t ) (B3) x0σ x0σ 4m     ˆ ˆ = − ix˜ ρ + ix˜ . Given the infinite potential barrier, one can impose the bound- exp σ ˆ exp σ (B16) ary condition x0 x0

x = a sin(ωt )|ψ(t ) = 0. (B4) APPENDIX C: qBOUNCE SIMULATION MATRIX ELEMENTS The goal is now to convert (B3) to the reference frame of the oscillating mirror. Given the change of variablesx ˜ = x − In order to simulate the qBOUNCE experiment using QuTiP a sin(ωt ), it is easy to show that [66], the master equations (19) and (20) must first be made unitless. This can be accomplished by differentiating with re- ∂2 ∂2 τ = / = , (B5) spect to unitless time (tmgx0 ) h¯. The conservative master ∂ 2 ∂ 2 x˜  x equation becomes   d ∂ ∂x˜ ∂ 2 2 = + dρˆ i pˆ h¯  dt ∂t ∂t ∂x˜ =− + mgxˆ − δ (ˆx) − aω cos(ωt )ˆp, ρˆ dτ mgx0 2m 4m ∂ ∂ (C1) = − aω cos(ωt ) . (B6) ∂t ∂x˜ and the entropic Lindblad equation becomes Thus, the equation of motion (B3) in the reference frame of   dρˆ i pˆ2 h¯2 D(ˆρ) the oscillating barrier becomes =− − δ(ˆx) − aω cos(ωt )ˆp, ρˆ + . dτ mgx 2m 4m mgx ∂ 0 0 ih¯ x˜|ψ˜ (t )={H + W (˜x, t )}x˜|ψ˜ (t ), (B7) (C2) ∂t 0 Sandwiching these master equations between E j| on the left where and |Ek [see Eq. (A12)] on the right yields the unitless con- h¯2 ∂2 h¯2 servative master equation H =− + U (˜x) − δ(˜x), (B8) 0 m ∂x2 m dρˆ 2 ˜ 4 =−i[hˆ + ξˆ + wˆ , ρˆ ], (C3) ∂ dτ W (˜x, t ) = U (a sin ωt ) + iha¯ ω cos(ωt ) , (B9) ∂x˜ along with the unitless entropic master equation U (x˜ˆ + a sin ωt ) = U (x˜ˆ) + U (a sin ωt ), (B10) dρˆ =−i[hˆ + wˆ , ρˆ ] + σ (Dˆ ρˆ Dˆ † − ρˆ ), (C4) dτ x˜|ψ˜ (t )=x|ψ(t ). (B11) where  Notice how when the time-dependent term W (˜x, t ) = 0, the ∞ ξξ ξ + ξ + 0 d Ai( a j+1 )Ai( ak+1) equation of motion reduces to the time-independent quantum h jk =−a j+1δ jk − , bouncing ball problem of Appendix A. To simplify notation, NjNk returnx ˜ → x and ψ˜ (t ) → ψ(t )inEqs.(B7)–(B9) and rewrite (C5)

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 ∞ dξξAi(ξ + a + )Ai(ξ + a + ) experimental measurements Texp(a,ω) by fixing the velocity ξ = 0 j 1 k 1 , jk (C6) v such that it minimizes χ 2(σ,v)[Eq.(D2)] for a given value NjNk   / of σ . 4m 1 3 w =+i (aω) cos(ωt ) jk hg¯  APPENDIX E: ENTROPIC GRAVITY MASS DEPENDENCE ∞ ξ ξ + d ξ + d Ai( a j+1) ξ Ai( ak+1) × 0 d , (C7) Let us answer the question: How does the entropic master  NjNk equation (5) change when a different mass is introduced, say ∞ ξ − ξ/σ ξ + ξ + M = κm? Substituting m → κm in Eq. (18) (explicitly and 0 d exp( i )Ai( a j+1)Ai( ak+1) D jk = , (C8) and also implicitly in x )gives N N 0  j k    ∞ 1/2 dρˆ i pˆ2 h¯2 = ξ 2 ξ + . =− − δ(ˆx), ρˆ Nj d Ai ( a j+1 ) (C9) κ κ 0 dt h¯ 2m 4m / / / σκ1 3  ixˆκ2 3 ixˆκ2 3  ˆ ξˆ = / mgx0 − σ + σ Here, h gives the boundary and kinetic-energy term. xˆ x0 + e x0 ρˆ e x0 − ρˆ . (E1) is the unitless position operator,w ˆ accounts for the accelerat- h¯ D ρ ing frame, Dˆ is the first exponential term in (ˆ), and Nj is However, eigenfunctions (A12) are also nontrivially depen- the normalization factor. In a similar fashion, we can show that dent on mass. In particular, eigenfunctions for objects of mass the matrix elements for the position and momentum operators M are given by  xˆ andp ˆ, along with δ (ˆx)inthe|Ei basis are given by  ξκ2/3 + κ1/3 ∞  |   = Ai( an+1) , dξξAi(ξ + a j+1)Ai(ξ + ak+1) x En (M)  ∞  / (E2) x = x 0 , ξ 2 ξ + 1 2 jk 0 (C10) x0 0 d Ai ( an+1) NjNk  ∞ d h¯2 1/3 dξAi(ξ + a j+1) Ai(ξ + ak+1 ) where again ξ = x/x and x = ( ) . Solving for matrix =−ih¯ 0 dξ , 0 0 2m2g p jk (C11) ˆ x0  N jNk  elements of D using the above eigenfunction definition yields d d ξ + + ξ + +   dξ Ai( a j 1) ξ= dξ Ai( ak 1 ) ξ= D =E |Dˆ |E  δ ξ = 0 0 . jk j k (E3) jk( ) (C12)   NjNk κ2/3   = −ix  |  |   × dxexp E j x x Ek (E4) In all our numerical simulations we use 20 20 matrices. x0σ

 ξκ2/3 2/3 − i 2/3 2/3 dξκ e σ Ai(ξκ + a + )Ai(ξκ + a + ) APPENDIX D: χ2 MINIMIZATION = j 1 k 1 . NjNk To simulate region III of the qBOUNCE experiment [16] using the entropic model (22), for each value of the neutron’s (E5) .   . / velocity 5 6m/s v 9 5 m/s and each value of the cou- Scaling the integration by ξκ2 3 → ξ will thus yield the origi- 2  σ  3 pling constant 10 10 , we solve the following convex nal matrix elements (C8). In a similar fashion, matrix elements optimization problems for the Hamiltonian (C5) are recovered. Hence, the master minimize χ 2 σ, equation becomes , , ( v) c0 c1 c2 (D1)    , 1/3 1/3 Subject to c0 c1 c2 0 dρˆ imgx κ mgx σκ =− 0 [hˆ, ρˆ ] + 0 (Dˆ ρˆ Dˆ † − ρˆ ), (E6) where χ 2(v) is the chi-square goodness of fit dt h¯ h¯  ,ω − ,ω σ, 2 with matrix elements given by (C5) and (C8), exactly the same 2 [Texp(a ) Ttheor (a ; v)] τ = χ (σ,v) = , (D2) as with the original mass. Differentiating with respect to M (a,ω)2 κ1/3/ a,ω exp mgx0 h¯ gives ,ω dρˆ Texp(a ) is experimentally measured relative transmission =− ˆ, ρ + σ ˆ ρ ˆ † − ρ , ,ω ,ω i[h ˆ ] (D ˆ D ˆ ) (E7) with corresponding error exp(a ), and Ttheor (a ; v)isthe dτM theoretical transmission [Eq. (23)] whose right-hand side is equal to that of master equation (C4) 2   (without the oscillation termw ˆ ) in which mass is equal to m. ,ω σ, = ,ω σ,τ = . mgx0 . Ttheor (a ; v) c jPj a ; f 0 30 Consider the purity rate of change with respect to τM : = h¯v j 0 d (D3) Tr(ρ ˆ 2 ) =−2σ Tr(ρ ˆ 2 − ρˆ Dˆ ρˆ Dˆ †). (E8) dτM ,ω σ,τ = Here, Pj (a ; f ) are the final population of state j σ 0, 1, 2 as a function of the driving frequency and strength. The Employing the Hausdorff expansion with respect to to Dˆ ρˆ Dˆ † gives summation in Eq. (D2) is done over measured data. We find   c0, c1, and c2 in Eq. (D1) using the optimizer CVXPY [67,68]. d 2 1 Tr(ρ ˆ 2 ) =− Tr(ρ ˆ 2ξˆ 2 − (ˆρξˆ )2 ) + O , (E9) As a convex optimization task, the problem (D1) has a unique 2 dτM σ σ solution. Note that the optimal solution (c0, c1, c2 ) depends on v and σ .InFigs.1, 2, and 3, we compare (D3) with the where ξˆ = xˆ/x0.

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Thus, purity decay for different masses follows the same In order to obtain a finite integral, it is critical that the mass form. Only time scale is changed. If td is the time scale for density operator is coarse-grained in this manner, i.e.,m ˆ (x) →   = κ−1/3 mass m, then the time scale td for mass M is td td , mˆ R(x) where κ = M/m. In more recent work, this model has been extended to As an illustration, let us select M to be the Planck mass include the backreaction of the quantised matter on the grav- and m – the neutron’s mass, then κ = 1.30 × 1019. Say some itational field [58]. Introducing the Newtonian field operator effect is observed for the neutron during its lifetime of about

881.5 s. Then the same effect is theoretically observable for mˆ (y) the Planck mass, but at a time of 375 μs. Likewise, say the ˆ (x) =−G d3y , (F3) |x − y| Planck mass experiences some purity decay within one second of interacting with the gravity environment. Then to observe leads to an extra term in Eq. (F1)oftheform . × the same purity decay in the neutron, it would take 2 35 1 106 s(≈27.2 days) far beyond the neutron’s lifetime. − d3x [∇ˆ (x), [∇ˆ (x), ρˆ ]]. (F4) 16πG R R

APPENDIX F: SPONTANEOUS LOCALISATION MODELS This gravitational backreaction is not only local, but of pre- cisely the same form as the collapse term in Eq. (F1). This is One of the principal efforts to combine classical grav- most easily seen in the Fourier representation for each term. itational fields with quantum dynamics are spontaneous Usingm ˜ (k) = F[ˆm(x)] we obtain: localisation models. Such models introduce additional non- linear and stochastic terms to quantum dynamics, so as to 1 3 d x [∇ˆ R(x), [∇ˆ R(x), ρˆ ]] guarantee the spatial localisation of matter at macroscopic 16πG scales while leaving the microscopic dynamics unchanged G exp (−R2|k|2 ) [35]. This is achieved by specifying that the additional col- = d3k 0 [˜m(k), [˜m†(k), ρˆt]] 8π 2 |k|2 lapse operators correspond to the local mass densitym ˆ (x) = m δ(3) x − xˆ 1 i i ( i ), coupled to a stochastic process. For a = d3x d3yK(x − y)[m ˆ (x), [ˆm(y), ρˆ ]]. (F5) Markovian noise, the dynamics of the stochastically averaged 4 density matrixρ ˆ are given by [35] Consequently, the addition of the gravitational backreac- d 1 tion term in Eq. (F4) amounts to a doubling of the strength ρˆ =−i[Hˆ , ρˆ ]− d3x d3yK(x − y)[mˆ (x), [mˆ (y), ρˆ ]], dt 4 of the decoherence term in Eq. (F1)[58]. This has important (F1) consequences when considering the energetic implications of where K(x − y) is the kernel of the stochastic process and these models. Without a dissipative term, energy is conserved h¯ ≡ 1. in neither model, and it is easy to show [35] that for a single The connection of such spontaneous collapse models to particle of mass m in the D-P model, the rate of change of dEDP (t ) = √mGh¯ gravity was made explicit in the Diósi-Penrose (D-P) model energy is π 3 [35]. When including the backreac- dt 4 R0 [35,57], where the stochastic kernel was chosen to be the tion, this rate is simply doubled. = G Newtonian gravitational potential, K(x) |x| . While such a In both cases, the rate of energetic change is strongly de- choice of kernel provides a natural connection to gravitation, it termined by the chosen coarse-graining cut-off R0. A natural also necessitates a coarse-graining procedure, without which approach to choosing this cut-off is to argue that it should the integral in the second term will diverge. This is achieved correspond to the Compton wavelength of a nucleon, R0 ≈ −15 by convolving the mass density with a Gaussian of width R0: 10 m. As shown in the main text however, such a value   leads to enormous rates of change, and therefore predicts a − / |x − y|2 = π 2 3 2 3 − . fR(x) 2 R0 d y exp 2 f (x) (F2) thermal catastrophe. 2R0

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