Decoherence-Free Entropic : Model and Experimental Tests

Alex J. Schimmoller,1, ∗ Gerard McCaul,1, † Hartmut Abele,2, ‡ and Denys I. Bondar1, § 1Tulane University, New Orleans, LA 70118, USA 2Technische Universitat Wien, Atominstitut, Stadionallee 2, 1020 Wien, Austria (Dated: July 23, 2021) ’s theory of [JHEP 2011, 29 (2011)], 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 ultra-cold 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 ultra-cold 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 which tabletop experiments could test. In addition, comparing entropic gravity’s energy increase to that of the Di´osi-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.

I. INTRODUCTION the amended form of Bekenstein’s formula ∆S = 2πkb, ii) the Compton wavelength ∆x, and iii) Unruh’s for- The theory of entropic gravity challenges the assump- mula [3–5], kbT = ~a/(2πc), connecting acceleration with tion that gravity is a conservative force, i.e., one that is temperature. Such a derivation of Newton’s second law proportional to the gradient of a potential energy. En- is valid for a – an extreme concentration of tropic gravity instead postulates that gravity is an en- mass. Verlinde postulates this conclusion to be valid for tropic force that points in the direction of maximum en- all masses, which should be represented by holographic tropy [1]. screens [6]. The definition of entropic forces follows from the first Verlinde’s theory has undergone scrutiny, especially law of thermodynamics, δQ = dU + δW , which equates over the invocation of holographic screens and the Unruh heat supplied to a system δQ to the change in the sys- formula [7–10], although these criticisms acknowledge a tem’s internal energy dU plus work done δW . If there connection between thermodynamics and gravity [11–13]. is a change in dS = δQ/T with no change in Recently, an extension to non-holographic screens has internal energy, then there is work done δW = T dS. been established [14]. The is the one performing the work F = The aim of this work is to refute another prevailing δW/dx = T dS/dx due to the entropy gradient. criticism of entropic gravity [7–10] that entropic forces While Newtonian gravity is conservative, Verlinde’s are by nature too noisy and thus destroy quantum co- proposal that gravity is entropic in nature [1] has gar- herence. In particular, it has been argued in [10] that if nered much attention. A simple argument in favor of this gravity were an entropic force, then it could be modeled hypothesis goes as as follows: Bekenstein [2] argued that as an environment in an open quantum system. Brow- a particle of mass m held by a just outside a black nian motion is not observed for small masses inside the hole will effectively be absorbed once the particle ap- environment, so these small objects must be very strongly proaches within one Compton wavelength, ∆x = ~/(mc), coupled to the gravity environment. But the strong cou- of the event horizon. Since the particle is so close to the pling must lead to ample wavefunction collapse and quan- arXiv:2012.10626v2 [quant-ph] 22 Jul 2021 event horizon, it is unknown whether the particle still ex- tum decoherence. However, such decoherence is not ob- ists or has been destroyed. So, the particle has gone from served in cold neutron experiments [15]. Thus, entropic being in a pure “exists” state to either an “exists” or “de- gravity cannot be true according to [7, 8]. stroyed” state with equal probabilities. Hence, the black We disprove this argument by constructing (Sec. II) hole’s entropy has increased by ∆S = kb ln(2). Newton’s a non-relativistic model [Eq. (5)] for quantum particles second law F = ma immediately follows from the en- (e.g., neutrons) interacting with gravity represented by tropic force definition F = T ∆S/∆x after substituting i) an environment. According to this model, the stronger the coupling to the reservoir, the lower the decoherence. Moreover, arbitrarily low decoherence can be achieved ∗ [email protected] by simply increasing the positive dimensionless coupling † [email protected] constant σ, which is a free parameter of this model. In ‡ [email protected] the limit σ → ∞, the model recovers Newtonian grav- § [email protected] ity as a potential force [Eq. (3)]. A comparison of our 2 model with data from the recent qBounce experiment In the simplest case, a linear gravitational potential [16] provides a lower bound σ & 500 (Secs. III and IV). can be treated as a single dissipative environment and the We discuss some of entropic gravity’s physical impli- free-fall dynamics (2) are satisfied by the master equation cations including monotonic energy increase and mass- of the Lindblad form [20] dependent decoherence in Sec. V. A relationship to the dρˆ i  pˆ2  Di´osi-Penrose gravitational model is also discussed. = − , ρˆ + D(ˆρ), (5) dt ~ 2m mgx σ   ixˆ   ixˆ   D(ˆρ) = 0 exp − ρˆexp + − ρˆ , II. A MODEL OF ENTROPIC GRAVITY x σ x σ ACTING NEAR EARTH’S SURFACE ~ 0 0 (6) In this section, we develop a near-Earth model of en- where tropic gravity acting on quantum particles. Consider a  2 1/3 particle of mass m a small distance x above Earth’s sur- x = ~ (7) face in free-fall. In the classical case, the particle’s dy- 0 2m2g namics are dictated by Newton’s equations of motion is a characteristic length and σ is a unitless, positive cou- d 1 d pling constant, which is a free parameter in the model x = p, p = −mg (1) dt m dt [21]. Note that the Hamiltonian in Eq. (5) only contains the kinetic energy term, and the linear gravitational po- where p is the particle’s momentum and g is the gravi- tential is replaced by the dissipator (6). We propose to tational acceleration. In the quantum regime, however, use Eq. (5) as the model for entropic gravity acting on these equations must be recast in the language of oper- quantum particles near Earth’s surface. ators and expectation values. This is accomplished via To elucidate how the dissipator (6) mimics a linear the Ehrenfest theorems [17] gravitational potential, we employ the Hausdorff expan- sion with the assumption σ → ∞ to obtain d 1 d hxˆi = hpˆi , hpˆi = −mg. (2) dρˆ i  pˆ2  mg  1 1  dt m dt = − + mgx,ˆ ρˆ + xˆρˆxˆ − xˆ2ρˆ − ρˆxˆ2 dt 2m x σ 2 2 Free fall of a quantum particle, whose state is repre- ~ 0~  1  sented by the density matrixρ ˆ, in a linear gravitational + O . (8) potential is described by the Liouville equation [18] σ2

dρˆ i  pˆ2  Thus, utilizing large values of the coupling constant σ, = − + mgx,ˆ ρˆ . (3) the master equation for entropic gravity (5) can approx- dt 2m ~ imate the conservative equation (3) with an arbitrarily Recalling that the expectation value for an observable high precision. Oˆ is given by hOˆi = Tr(Oˆρˆ), it can easily be shown The argument put forth in Refs. [7, 8] against entropic that Eq. (3) satisfies the free-fall Ehrenfest theorems (2). gravity has the following fault: It is based on the as- Equation (3) is the conservative model for free-fall. The sumption that the evolution of a neutron’s initial pure purity of a quantum stateρ ˆ is given by Tr(ˆρ2). The pu- state to a mixed one is generated by a non-Hermitian rity reaches its maximum value of unity if and only if the translation operator (see Eq. (12) of [7]) leading to the density matrix corresponds to the sate representable by a Schrodinger equation with a non-Hermitian Hamiltonian wave function. It is an important feature of Eq. (3) that (see Eq. (16) of [7]). While non-Hermitian corrections it preserves the purity, i.e., Eq. (3) maintains . to the Schrodinger equation have been historically used Equation (3) is not the only one capturing free-fall to incorporate some aspects of dissipation, such an ap- dynamics (2). In fact, within the language of open proach suffers from physical inconsistencies [22] and has quantum systems [19], there are an infinite number of been abandoned in the modern theory of open quantum master equations which satisfy the above Ehrenfest the- systems. Hence, instead of Eq. (12) from Ref. [7] that orems [20]. It has been shown in [20] that for arbitrary reads G(p) and F (x), the Ehrenfest theorems ρˆ(z + ∆z) ≡ Uˆρˆ(z)Uˆ †, UˆUˆ † = 1, (9) d d hxˆi = hG(ˆp)i , hpˆi = hF (ˆx)i (4) the Kraus representation (see, e.g, Ref. [19]) for the evo- dt dt lutionρ ˆ(z) → ρˆ(z + ∆z) should have been used can be satisfied by coupling a closed system with the X X ρˆ(z + ∆z) ≡ Kˆ ρˆ(z)Kˆ † , Kˆ † Kˆ = 1. (10) usual Hamiltonian Hˆ =p ˆ2/(2m) + U(ˆx) to a series of n n n n tailored environments. We take advantage of this fact n n to model gravity as an environment in an open quantum The Kraus representation furnishes the most general system fashion [19]. description for evolution of open quantum systems. The 3 only requirement used to arrive at Eq. (10) is that the [41–43], spontaneous collapse [44–47], and the Di´osi- mappingρ ˆ(z) → ρˆ(z+∆z) should be completely positive. Penrose model [48–53]. However, despite mathematical The latter is a stronger requirement than the fact that resemblance, they can make very different predictions physical evolution preserves the positivity of a density from Eq. (5) (see Sec. V). We also note that Lindblad- matrix. Finally, we note that a Lindblad master equation like master equations have been recently emerged in post- [such as, e.g., Eq. (5)] can be recast in a Kraus form. quantum classical gravity [54, 55], where a quantum sys- If the O σ−2 term is dropped in Eq. (8), then the tem interacts with classical space-time. resulting Eq. (8) describes a particle undergoing a con- tinuous quantum measurement of its position [19, 23]. The entropic master equation (5) interprets gravity as a III. MODELING THE QBOUNCE continuous measurement process extracting information EXPERIMENT about the position of a massive particle. The extraction of information is responsible for the entropy creation [24]. Now that the free-fall model for entropic gravity has As a result, the purity of the quantum system is no longer been established [Eq. (5)], it is desirable to see how it preserved. compares to results of the qBounce experiment [16]. The rate of change of the purity induced by evolution This experiment was performed at the beam position for governed by Eq. (5) is estimated as σ → ∞, ultra-cold neutron at the European neutron source at the   Institut Laue-Langevin in Grenoble and uses gravity res- d 2 mg 2 2 2 1 onance spectroscopy [56] to induce transitions between Tr(ˆρ ) = −2 Tr ρˆ xˆ − (ˆρxˆ) + O 2 . dt x0~σ σ quantum states of a neutron in the gravity potential of (11) the earth. In region I of this experiment, neutrons are prepared in a known mixture of the first three quantum It is shown in [20] that Tr ρˆ2xˆ2 − (ˆρxˆ)2 ≥ 0; thus, bouncer energy states (see Appendix A). These neutrons the purity is monotonically decreasing. Furthermore, the then traverse a 30 cm horizontal boundary which oscil- larger the σ, the more purity is preserved. Since we can lates with variable frequency ω and oscillation amplitude elect to make σ arbitrarily large in our model, the origi- a, inducing Rabi oscillations between the “bouncing-ball” nal criticism of entropic gravity not maintaining quantum states of neutrons. In Figs. 2 and 3 below, the oscillation coherence can no longer be considered valid. strength is defined as aω. Finally in region III, neutrons The proposed entropic master equation (5) obeys a pass through a state selector, leaving neutrons in an un- variant of the (see, e.g., Refs. [25, known mixture of the three lowest energy states to be 26]). According to [27], the strong equivalence principle counted. To model this experiment, the free-fall master states that “all test fundamental physics is not affected, equation (5) must be amended to account for the oscillat- locally, by the presence of a gravitational field.” Hence, ing boundary, and simulations must account for variable dynamics induced by a homogeneous gravitational field neutron times-of-flight and the unknown selection of neu- must be translationally invariant. Equation (5) is known trons in region III. to be translationally invariant [28–33]. The simplest way to model the boundary is by mod- Since Verlinde’s theory treats gravity as a thermody- ifying the Ehrenfest theorems. For a system with the namically emergent force, it is not appropriate to quan- general Hamiltonian tize gravity and talk about the existence of [34, 35]. However, our entropic master equation (5) phe- Hˆ =p ˆ2/(2m) + U(ˆx), (12) nomelogically hints at gravitons. Equations similar to Eq. (5) have long been employed for the nonperturbative and the boundary condition hx = 0|ψi = 0, the second description of a quantum system undergoing collisions Ehrenfest theorem reads with a background gas of atoms or photons [31–33, 36– 2     38]. Transferring this microscopic picture, the dissipa- d 0 ~ d d hpˆi = h−U (ˆx)i + hx|ψi hψ|xi tor (6) can be interpreted as describing colissions of a dt 2m dx x=0 dx x=0 massive quantum particle with a bath of gravitons; more- 2 0 ~ 00 over, /(x σ) stands for the momentum of a . To = h−U (ˆx)i + hδ (ˆx)i , (13) ~ 0 4m preserve purity σ must be large, which makes the momen- tum of a graviton infinitesimally small. This conclusion where δ(x) is the Dirac delta function, defined as is compatible with the fact that detecting a graviton re- Z ∞ mains a tremendous challenge [39], which might become dxδ(n)(x − x0)f(x) = (−1)(n)f (n)(x0). (14) feasible [40]. −∞ A plethora of models for gravitation induced decoher- ence, which describe quantum matter interacting with a Thus modifying the Hamiltonian Hˆ to include the bound- stochastic gravitational background, has been put forth ary term, [35]. It is worth pointing out that some of these mod- pˆ2 2 els mathematically resemble the entropic master equa- Hˆ = + U(ˆx) − ~ δ0(ˆx). (15) tion (5); in particular, the models of time fluctuations 2m 4m 4 recovers the desired Ehrenfest theorem (13). Now that proper master equations have been estab- In order to make the boundary oscillate, one simply lished for region II, how long must they run? The time-of- needs to add a sinusoidal term inside of the Dirac delta flight tf for each neutron is determined by its horizontal function: velocity v = 0.30(mgx0)/(~τf ), ultimately determining final state populations P (τ ) = Tr (ˆρ(τ )|E ihE |). In pˆ2 2 j f f j j Hˆ = + U(ˆx) − ~ δ0(ˆx − a sin(ωt)). (16) this experiment, neutrons are measured to have horizon- 2m 4m tal velocities v between 5.6 and 9.5 m/s. We elect to Here, a is the oscillation amplitude. make the horizontal neutron velocity v an additional free In the particular case of potential gravity [Eq. (3)], a parameter in the model confined to this range. While this neutron’s dynamics while inside the qBounce apparatus choice in modeling does not capture the range of veloc- is described by the Liouville equation: ities contributing to the overall transmission, results in Sec. IV indicate that this assumption does not diminish  2 2  dρˆ i pˆ ~ 0 the overall point of the paper. = − + mgxˆ − δ (ˆx − a sin(ωt)) , ρˆ . Finally, a full model of the q experiment [16] dt ~ 2m 4m Bounce (17) requires modeling the state selection in region III. The state selector consists of an upper mirror positioned just Meanwhile, the entropic case [Eq. (5)] gives above the attainable height of a ground state neutron. However, higher states leak into the detector as well. We dρˆ i  pˆ2 2  thus define relative transmission (neutron count rate with = − − ~ δ0 (ˆx − a sin(ωt)) , ρˆ + Dˆ(ˆρ). (18) dt ~ 2m 4m the oscillating boundary divided by the count rate with- out oscillation) to be a linear combination of the three Here, the kinetic and boundary terms are inside the com- lowest energy state populations: mutator and D(ˆρ) is the gravity environment (6). Be- cause D(ˆρ) is translationally invariant, the oscillating T = c0P0 + c1P1 + c2P2, (23) boundary does not alter the dissipator (6). For simulations of the qBounce experiment, we trans- where c0, c1, and c2 are unknown, positive coefficients form the equations of motion into the reference frame of to be determined from experimental data as explained in the oscillating boundary (see Appendix B). After apply- Appendix D. Since the state selector is designed to scat- ing the change of variablesx ˜ = x − a sin(ωt) and trans- ter away excited neutrons, the physical and engineering latingx ˜ → x, the conservative model’s Liouville equa- consideration leads to the constraint c0 ≥ c1 ≥ c2. tion (17) becomes [57]

dρˆ i  pˆ2 2  IV. SIMULATING THE QBOUNCE = − + mgxˆ − ~ δ0(ˆx) − aω cos(ωt)ˆp, ρˆ , EXPERIMENT dt ~ 2m 4m (19) With the results of Sec. III, we can effectively sim- and the entropic Lindblad equation (18) reads ulate the qBounce experiment [16]. In region I of the experiment, neutrons are prepared initially as an incoher-  2 2  dρˆ i pˆ ~ 0 ent mixture with 59.7% population in the ground state, = − − δ (ˆx) − aω cos(ωt)ˆp, ρˆ + D(ˆρ). 34.0% in the first excited state, 6.3% in the second excited dt ~ 2m 4m (20) state and no population in higher states. Thus, the ini- tial state of simulated neutrons is the incoherent mixture Differentiating with respect to the unitless time τ = ρˆ(0) = 0.597|E0ihE0|+0.340|E1ihE1|+0.063|E2ihE2|. In tmgx0/~ yields the unitless conservative Liouville equa- region II, neutrons interact with gravity and the oscillat- tion ing boundary. The density matrix evolves according to either the conservative (21) or entropic (22) unitless mas- dρˆ h i = −i hˆ + ξˆ+w, ˆ ρˆ , (21) ter equations, with frequency ω and oscillation strength dτ aω determined by the experimental setup. After the in- along with the unitless entropic Lindblad equation teraction time τf (determined by the free velocity pa- rameter v), simulated neutrons have effectively passed dρˆ h i   through region II of the experiment. We calculate the = −i hˆ +w, ˆ ρˆ + σ DˆρˆDˆ † − ρˆ . (22) dτ final populations P0, P1, and P2. We perform minimization of χ2 over the space of the ˆ Here, h represents the kinetic energy and boundary five parameters: c0, c1, c2, v, and σ (see Appendix D for terms, ξˆ gives the potential energy term,w ˆ accounts for details). An agreement between the theory and experi- the accelerating frame and Dˆ gives the first exponential ment can be observed in Figs. 1, 2, and 3. As Eq. (8) inside the D(ˆρ) term. Matrix elements for these opera- predicts, transmission values for entropic simulations ap- tors are given in Appendix C. Equations (21) and (22) proach those of the conservative model as σ increases. are used in the following simulations. This is to say, conservative gravity can be recovered with 5

large enough σ in the entropic model, and decoherence ef- 1.4 fects are therefore unnoticed. In particular, a good agree- 1.2 ment of the experimental data with the entropic model is observed when σ equals 500. Furthermore, χ2 analysis 1.0 shown in Fig. 4 reveals that simulations with σ & 250 fit 0.8 the experimental data with 90% confidence. Note that 0.6 2 at σ = 500 the values of χ for conservative and entropic = 100 0.4 = 500 gravity coincide. In conclusion, we take 500 to be the Relative Transmission = 103 lower bound for σ. 0.2 Conservative Gravity Qbounce Experiment In total, the entropic model of the q experi- 0.0 Bounce 2500 3000 3500 4000 4500 ment consists of five free parameters: σ, v, c0, c1, and Oscillation Frequency (Hz) c2. For entropic simulations with σ . 250, the best-fit velocity hovers around the lower limit of 5.6 m/s. As FIG. 2. Comparing the qBounce experiment [16] with pre- σ → ∞, the best-fit velocity approaches 6.58 m/s. The dictions of the master equation for entropic gravity [Eq. (18)] as well as the conservative gravity [Eq. (17)] by varying os- transmission coefficients c0, c1, and c2 equal to 1.46, 0.50 and 0.50, respectively, for σ = 500, and approach 1.28, cillation frequency (ω) when the oscillation strength (aω) is 0.55, and 0.55, respectively, as σ → ∞. set to 2.05 mm/s. 20 states are accounted for in numerical propagation of Eqs. (17) and (18). σ is a free parameter in the entropic gravity master equation. When σ & 500 the 1.6 = 100 experiment agrees well with entropic gravity. = 500 = 103 1.4 Conservative Gravity Qbounce Experiment 1.2 = 100 1.2 = 500 = 103 1.0 1.0 Conservative Gravity Qbounce Experiment 0.8 0.8 0.6 Relative Transmission 0.6 0.4

0.2 Relative Transmission 0.4 0 10 20 30 40 Data Point Index 0.2 0 1 2 3 4 5 FIG. 1. Comparing the qBounce experiment [16] with pre- Oscillation Strength (mm/s) dictions of the master equation for entropic gravity [Eq. (18)] as well as the conservative gravity [Eq. (17)]. All data points FIG. 3. Comparing the qBounce experiment [16] with pre- from the experiment are visible with corresponding frequency dictions of the master equation for entropic gravity [Eq. (18)] ω and oscillation strength aω data replaced with a single in- as well as the conservative gravity [Eq. (17)] by varying oscilla- dex on the horizontal axis. 20 states are accounted for in tion strength (aω) with the oscillation frequency (ω) set to the numerical propagation of Eqs. (17) and (18). the transition between the ground and third excited states of the “bouncing ball” problem [ω = ω03 = (E3 − E0)/~ = 4.07 kHz]. σ is a free parameter in the entropic gravity master equation. When σ & 500 the experiment agrees well with entropic gravity. V. DISCUSSION AND FUTURE DIRECTIONS

We have shown that a linear gravitational potential expected energy change is given by can be modeled by an environment coupled to neutrons. d D E g This entropic gravity model overcomes the criticism put Hˆ = ~ . (24) forth in Ref. [10] since the master equation (5) is ca- dt 2x0σ pable of maintaining both strong coupling and negligible That is, under entropic gravity, the test particle’s total decoherence and is fully compatible with the qBounce energy increases at a rate ∝ 1/σ regardless of the ini- experiment [16]. Moreover, the entropic model recovers tial state. Hence, the entropic model avoids a thermal the conservative gravity (3) as σ → ∞. Our findings pro- catastrophe in the large coupling limit (σ → ∞), un- vide support for the entropic gravity hypothesis, which like the D-P model. According to the latter, the rate of may spur further experimental and theoretical inquires. energy increase√ (given by Eq. (94) in Ref. [35]) equals Let us compare the predictions of the entropic master 3 mG~/(4 πR0), where G is the equation (5) and the Di´osi-Penrose (D-P) model [35, 58]. and R0 is a coarse-graining parameter set to the nu- Consider the total energy operator Hˆ =p ˆ2/(2m) + mgxˆ. cleon’s radius, 10−15 m. For a neutron, the D-P model While the expected total energy hHˆ i remains constant in predicts the rate of energy increase to be 1.66 × 10−27 W the conservative case (3), the entropic model’s rate of the (= 10.4 neV/s), while the entropic model prediction is 6

75 Entropic Gravity the D-P model. In Appendix E, we show that if td is Conservative Gravity 2 2 the decoherence time for a particle of mass m, then the = min + 2.7 70 0 0 −1/3 decoherence time td for mass M is td = (M/m) td. Hence, the larger the mass, the faster the decoherence. 65 2 Moreover, measuring the decoherence times would also directly identify σ. The recent experiment [60] that ob- 60 served optomechanical nonclassical correlations involving

55 a nanopartcile could perform such a test. Although the proposed entropic gravity model is lim- 50 200 400 600 800 1000 ited to the low-energy, near-Earth regime, its physical im- plications provide a glimpse into several open cosmolog- ical questions. As Ref. [35] mentions regarding collapse 2 FIG. 4. χ as a function of σ. The gray area represents gravitational models, entropic gravity’s non-unitarity dy- 2 2 the 90% confidence interval. χmin is the minimum χ value namics could resolve the black hole information paradox among the simulated results. When σ & 250, entropic gravity [61, 62], and its runaway energy (24) could pose solu- falls within this region. When σ 500, entropic gravity fits & tions to the [63], cosmological inflation, and experimental data as well as conservative gravity. quantum measurement problems [64]. With greater re- striction of σ from precision experiments and better un- significantly lower: 1.76 × 10−31 W (= 1.1 peV/s) as- derstanding of its physical implications at all time and suming σ = 500 (see Sec. IV). For the entropic model energy scales, entropic gravity can be further explored as to display as much energy increase as the D-P model a feasible gravitational theory. predicts, σ would need to be 0.05, much less than what is permitted by the qBounce experiment as shown in Sec. IV. Moreover, for a 1 kg mass, the D-P model pre- dicts a rate of energy increase ≈ 1 Watt! Such a sig- ACKNOWLEDGMENTS nificant quantity should be readily noticeable. Compar- atively, the entropic model predicts the rate of energy H.A. and D.I.B. are grateful to Wolfgang Schleich increase of only 0.125 pW when σ = 500. Raising R0 can and Marlan Scully for inviting us to the PQE-2019 con- significantly reduce the D-P model’s energy increase, but ference, where this collaboration was conceived. H.A. there is no physical justification for larger values of R0. thanks T. Jenke for fruitful discussions. A.J.S. and We also note that recent extensions to the D-P model D.I.B. wish to acknowledge the Tulane Honors Summer to include the gravitational backreaction [59] suffer from Research Program for funding this project. G.M. and the same issue. As shown in Appendix F, the additional D.I.B. are supported by the Army Research Office (ARO) terms arising from the inclusion of a semiclassical field (grant W911NF-19-1-0377), Defense Advanced Research serve only to double the rate of energy increase. Com- Projects Agency (DARPA) (grant D19AP00043), and paratively, there is no known upper bound on σ, and Air Force Office of Scientific Research (AFOSR) (grant energy increase vanishes as σ → ∞. FA9550-16-1-0254). The views and conclusions contained We believe that the lower bound σ = 500, deduced in in this document are those of the authors and should Sec. IV from the qBounce experiment, is highly likely not be interpreted as representing the official policies, to be an underestimation. A more realistic lower bound either expressed or implied, of ARO, DARPA, AFOSR, 5 should be σ & 4.6 × 10 . Let us describe how the latter or the U.S. Government. The U.S. Government is au- value could be confirmed experimentally. According to thorized to reproduce and distribute reprints for Govern- Eq. (24), a neutron will gain energy ∆E within a time ment purposes notwithstanding any copyright notation ∆t, herein. H.A. gratefully acknowledges support from the Austrian Fonds zur F¨orderungder Wissenschaftlichen 2x σ ∆t = 0 ∆E. (25) Forschung (FWF) under contract no. P 33279-N. g~ Assume the neutron is initially prepared in the ground state |E0i of the “bouncing ball”. Then, we let it evolve for the time approaching the neutron’s lifetime Appendix A: Solving the Schr¨odingerEquation For ∆t = 881.5 s and measure the final state. If it jumped to a Bouncing Ball the first excited state |E1i, then according to Eq. (25), the neutron must have gained ∆E ≥ E1 − E0 implying In this section, we solve the quantum bouncing ball 5 that σ ≤ 4.6 × 10 . If the neutron does not reach |E1i, problem (as is done in [65]). Consider the time- then σ > 4.6 × 105. Storage experiments with neutrons independent Schr¨odingerequation for a particle of mass might provide these limits [57]. m experiencing a linear gravitational potential U(ˆx) = The entropic master equation (5) predicts gravity in- mgxˆ and an infinite potential barrier at x = 0. We wish duced decoherence albeit at a much lower rate than, e.g., to find the the eigenvalues E and eigenvectors |Ei such 7 that energy state corresponding to the (n + 1)th zero of the Airy function. Corresponding eigenfunctions are given ˆ Hc |Ei = E |Ei , where (A1) by pˆ2 Hˆc = + mgx.ˆ (A2) Ai(ξ + an+1) 2m hx|Eni = . (A12)  R ∞ 2 1/2 x0 dξAi (ξ + an+1) Applying hx| to equation (A1), the equation can be 0 ∞ rewritten as The set of eigenvectors {|Eni}n=0 forms an orthonormal basis.  d2 2m  − [mgx − E] hx|Ei = 0, (A3) dx2 ~2 Appendix B: The Quantum Bouncer with an and the infinite potential barrier manifests itself in the Oscillating Boundary: Change of Variables boundary condition

hx = 0|Ei = 0. (A4) In this section, the Schr¨odinger equation used to model the qBounce experiment is converted to the reference It is easy to confirm that the solutions to equation (A3) frame of the oscillating boundary. The following treat- are given by ment closely follows Ref. [57]. Consider the 1D time- dependent Schr¨odinger equation for a particle with po-  E   E  tential energy U(ˆx), along with an infinite potential bar- hx|Ei = c1Ai ξ − + c2Bi ξ − , (A5) mgx0 mgx0 rier, which oscillates with a frequency ω and amplitude a about the point x = 0: where d ˆ  2 1/3 i~ |ψ(t)i = H |ψ(t)i (B1) ~ dt x0 = , (A6) 2m2g where ξ = x/x , (A7) 0 pˆ2 2 Hˆ = + U(ˆx) − ~ δ0 (ˆx − a sin(ωt)) . (B2) and c1, c2 are constants, and Ai and Bi are the two 2m 4m linearly-independent solutions to the Airy equation When hx| is applied on the left to both sides of (B1), one gets the Schr¨odingerequation in the coordinate represen-  d2  − y w(y) = 0, w = Ai(y), Bi(y). (A8) tation: dy2 d  2 ∂2 i hx|ψ(t)i = − ~ + U(ˆx) Considering the normalization condition ~dt 2m ∂x2 Z ∞ 2  ~ 0 | hx|Ei |2dx = 1, (A9) − δ (x − a sin(ωt)) hx|ψ(t)i . (B3) 0 4m we exclude Bi since Bi(x) → ∞ as x → ∞ [66]. Apply- Given the infinite potential barrier, one can impose the ing (A9) to (A3) with c2 = 0, we get our normalization boundary condition coefficient: hx = a sin(ωt)|ψ(t)i = 0. (B4)  Z ∞  −1/2 2 E c = c(E) = x0 dξAi ξ − . (A10) The goal is now to convert (B3) to the reference frame 0 mggx0 of the oscillating mirror. Given the change of variables x˜ = x − a sin(ωt), it is easy to show that Thus, solutions in the coordinate representation are given by ∂2 ∂2 = , (B5)   ∂x˜2 ∂x2 Ai ξ − E   mgx0 d ∂ ∂x˜ ∂ hx|Ei = . (A11) 1/2 = + h R ∞ 2  E i dt ∂t ∂t ∂x˜ x0 dξAi ξ − 0 mgx0 ∂ ∂ = − aω cos(ωt) . (B6) Applying the boundary condition (A4) yields eigen- ∂t ∂x˜ values En = −mgx0an+1, where n = 0, 1, 2, ... and aj Thus, the equation of motion (B3) in the reference frame denotes the jth zero of Ai. By convention, the energy of the oscillating barrier becomes eigenstates of a system are numbered beginning with zero ∂ to signify the ground state, whereas the zeroes of a func- i hx˜|ψ˜(t)i = {H + W (˜x, t)} hx˜|ψ˜(t)i, (B7) tion are numbered beginning with one, hence the nth ~∂t 0 8 where Sandwiching these master equations between hEj| on the 2 ∂2 2 left and |Eki [see Eq. (A12)] on the right yields the unit- H = − ~ + U(˜x) − ~ δ0(˜x), (B8) less conservative master equation 0 2m ∂x˜2 4m ∂ dρˆ h i W (˜x, t) = U(a sin ωt) + i~aω cos(ωt) , (B9) = −i hˆ + ξˆ+w, ˆ ρˆ , (C3) ∂x˜ dτ U(x˜ˆ + a sin ωt) = U(x˜ˆ) + U(a sin ωt), (B10) along with the unitless entropic master equation hx˜|ψ˜(t)i = hx|ψ(t)i . (B11) dρˆ h i   = −i hˆ +w, ˆ ρˆ + σ DˆρˆDˆ † − ρˆ , (C4) Notice how when the time-dependent term W (˜x, t) = 0, dτ the equation of motion reduces to the time-independent quantum bouncing ball problem of Appendix A. To where simplify notation, returnx ˜ → x and ψ˜(t) → ψ(t) R ∞ dξξAi(ξ + a )Ai(ξ + a ) in Eqs. (B7)-(B9) and rewrite the original Schrodinger 0 j+1 k+1 hjk = −aj+1δjk − , equation (B7) from the mirror’s reference frame: NjNk (C5) ∂ n o ∞ i |ψ(t)i = Hˆ0 + Wˆ |ψ(t)i , (B12) R ~ 0 dξξAi(ξ + aj+1)Ai(ξ + ak+1) ∂t ξjk = , (C6) 2 2 NjNk pˆ ~ 0 Hˆ0 = + U(ˆx) − δ (ˆx), (B13) 2m 4m 4m1/3 ˆ wjk = +i (aω) cos(ωt) W = U(a sin ωt) − aω cos(ωt)ˆp. (B14) ~g R ∞ d Note that thep ˆ operator in this last equation is only 0 dξAi (ξ + aj+1) dξ Ai (ξ + ak+1) ∂ , (C7) present so as to be transformed into −i~ ∂x when hx| is NjNk reapplied. When we convert our Schrodinger equation R ∞ 0 dξ exp(−iξ/σ)Ai(ξ + aj+1)Ai(ξ + ak+1) (B12) into the density matrix formalism, we get that Djk = , NjNk dρˆ i  pˆ2 2  = − + U(ˆx) − ~ δ0(ˆx) − aω cos(ωt)ˆp, ρˆ . (C8) dt ~ 2m 4m Z ∞ 1/2 2 (B15) Nj = dξAi (ξ + aj+1) . (C9) 0 Furthermore, consider the D(ˆρ) operator in the en- tropic model given by equation (6). Under the change Here, hˆ gives the boundary and kinetic energy term. ξˆ = of variablesx ˜ = x−a sin(ωt), D(ˆρ) is invariant under the x/xˆ 0 is the unitless position operator,w ˆ accounts for change of variables since the accelerating frame, Dˆ is the first exponential term in ! ! D(ˆρ), and N is the normalization factor. In a similar i(x˜ˆ + a sin ωt) i(x˜ˆ + a sin ωt) j exp − ρˆexp + fashion, we can show that the matrix elements for the x0σ x0σ position and momentum operatorsx ˆ andp ˆ, along with ! ! δ00(ˆx) in the |E i basis are given by ix˜ˆ ix˜ˆ i = exp − ρˆexp + . (B16) ∞ x0σ x0σ R 0 dξξAi(ξ + aj+1)Ai(ξ + ak+1) xjk = x0 , (C10) NjNk R ∞ dξAi(ξ + a ) d Ai(ξ + a ) Appendix C: qBounce Simulation Matrix Elements i~ 0 j+1 dξ k+1 pjk = − , x0 NjNk In order to simulate the qBounce experiment using (C11) QuTiP [67], the master equations (19) and (20) must h d i h d i dξ Ai (ξ + aj+1) dξ Ai (ξ + ak+1) first be made unitless. This can be accomplished by dif- 00 ξ=0 ξ=0 δjk(ξ) = . ferentiating with respect to unitless time τ = (tmgx0)/~. NjNk The conservative master equation becomes (C12) dρˆ i  pˆ2 2  = − + mgxˆ − ~ δ0(ˆx) − aω cos(ωt)ˆp, ρˆ In all our numerical simulations we use 20 × 20 matrices. dτ mgx0 2m 4m (C1) 2 and the entropic Lindblad equation becomes Appendix D: χ minimization dρˆ i  pˆ2 2  D(ˆρ) = − − ~ δ0(ˆx) − aω cos(ωt)ˆp, ρˆ + . To simulate region III of the qBounce experiment [16] dτ mgx0 2m 4m mgx0 using the entropic model (22), for each value of the neu- (C2) tron’s velocity 5.6 m/s ≤ v ≤ 9.5 m/s and each value 9 of the coupling constant 102 ≤ σ ≤ 103, we solve the definition yields following convex optimization problems 0 ˆ 0 Djk = hEj|D|Eki (E3) 2 Z  2/3  minimize χ (σ, v) ixκ 0 0 c0, c1, c2 = dx exp − E |x hx|E i (E4) (D1) x σ j k subject to c ≥ c ≥ c ≥ 0, 0 0 1 2 2/3 R 2/3 − iξκ 2/3 2/3 dξκ e σ Ai(ξκ + aj+1)Ai(ξκ + ak+1) where χ2(v) is the chi-square goodness of fit = . NjNk (E5) 2 2 X [Texp(a, ω) − Ttheor(a, ω; σ, v)] χ (σ, v) = , (D2) 2/3 2 Scaling the integration by ξκ → ξ will thus yield the exp(a, ω) a,ω original matrix elements (C8). In a similar fashion, ma- trix elements for the Hamiltonian (C5) are recovered. Texp(a, ω) is experimentally measured relative trans- Hence, the master equation becomes mission with corresponding error exp(a, ω), and 1/3 1/3 Ttheor(a, ω; v) is the theoretical transmission [Eq. (23)] dρˆ imgx κ h i mgx σκ   = − 0 h,ˆ ρˆ + 0 DˆρˆDˆ † − ρˆ , dt ~ ~ 2 () X  mgx0  T (a, ω; σ, v) = c P a, ω; σ, τ = 0.30 . theor j j f v j=0 ~ with matrix elements given by (C5) and (C8), exactly (D3) the same as with the original mass. Differentiating with 1/3 respect to τM = mgx0κ /~ gives Here, Pj(a, ω; σ, τf ) are the final population of state dρˆ h i   j = 0, 1, 2 as a function of the driving frequency and = −i h,ˆ ρˆ + σ DˆρˆDˆ † − ρˆ , () dτ strength. The summation in Eq. (D2) is done over mea- M sured data. We find c0, c1, and c2 in Eq. (D1) using the whose right hand side is equal to that of master equation optimizer CVXPY [68, 69]. As a convex optimization (C4) (wihout the oscillation termw ˆ) in which mass is task, the problem (D1) has a unique solution. Note that equal to m. the optimal solution (c0, c1, c2) depends on v and σ. In Consider the purity rate of change with respect to τM : Figs. 1, 2, and 3, we compare (D3) with the experimen- d 2  2 † tal measurements Texp(a, ω) by fixing the velocity v such Tr(ˆρ ) = −2σ Tr ρˆ − ρˆDˆρˆDˆ . () that it minimizes χ2(σ, v) [Eq. (D2)] for a given value of dτM σ. Employing the Hausdorff expansion with respect to σ to DˆρˆDˆ † gives d 2    1  Appendix E: Entropic Gravity Mass Dependence 2 2 ˆ2 ˆ 2 Tr(ˆρ ) = − Tr ρˆ ξ − (ˆρξ) + O 2 , (E9) dτM σ σ Let us answer the question: How does the entropic ˆ where ξ =x/x ˆ 0. master equation (5) change when a different mass is intro- Thus, purity decay for different masses follows the duced, say M = κm? Substituting m → κm in Eq. (18) same form. Only time scale is changed. If td is the time (explicitly and and also implicitly in x0) gives 0 scale for mass m, then the time scale td for mass M is 0 −1/3 t = κ td, where κ = M/m. dρˆ i  pˆ2 2  d = − − ~ δ0(ˆx), ρˆ As an illustration, let us select M to be the Planck dt ~ 2mκ 4mκ mass and m – the neutron’s mass, then κ = 1.30 × 1019. 1/3  2/3 2/3  Say some effect is observed for the neutron during its mgx0σκ − ixκˆ + ixκˆ + e x0σ ρeˆ x0σ − ρˆ . (E1) lifetime of about 881.5 s. Then the same effect is theo- ~ retically observable for the Planck mass, but at a time of However, eigenfunctions (A12) are also non-trivially de- 375 µs. Likewise, say the Planck mass experiences some pendent on mass. In particular, eigenfunctions for ob- purity decay within one second of interacting with the jects of mass M are given by gravity environment. Then to observe the same purity decay in the neutron, it would take 2.35 × 106 s (≈ 27.2 2/3  1/3 days) far beyond the neutron’s lifetime. 0 Ai ξκ + an+1 κ hx|Eni (M) = , (E2)  R ∞ 2 1/2 x0 0 dξAi (ξ + an+1) Appendix F: Spontaneous Localisation Models

 2 1/3 where again ξ = x/x and x = ~ . Solving 0 0 2m2g One of the principal efforts to combine classical gravi- for matrix elements of Dˆ using the above eigenfunction tational fields with quantum dynamics are spontaneous 10 localisation models. Such models introduce additional operator non-linear and stochastic terms to quantum dynamics, Z mˆ (y) so as to guarantee the spatial localisation of matter at Φ(ˆ x) = −G d3y , (F3) macroscopic scales while leaving the microscopic dynam- |x − y| ics unchanged [35]. This is achieved by specifying that the additional collapse operators correspond to the local leads to an extra term in Eq.(F1) of the form: P (3) mass densitym ˆ (x) = i miδ (x − xˆi), coupled to a Z 1 3 h h ii stochastic process. For a Markovian noise, the dynamics − d x ∇Φˆ R(x), ∇Φˆ R(x), ρˆ . () of the stochastically averaged density matrixρ ˆ are given 16πG by [35]: This gravitational backreaction is not only local, but of precisely the same form as the collapse term in Eq. (F1). This is most easily seen in the Fourier representation for d h i 1 Z Z ρˆ = −i H,ˆ ρˆ − d3x d3y K(x−y) [m ˆ (x), [m ˆ (y), ρˆ]] , each term. Usingm ˜ (k) = F[m ˆ (x)] we obtain: dt 4 (F1) 1 Z h h ii d3x ∇Φˆ (x), ∇Φˆ (x), ρˆ where K(x − y) is the kernel of the stochastic process 16πG R R and ~ ≡ 1. G Z exp −R2|k|2 = d3k 0 m˜ (k), m˜ †(k), ρˆ The connection of such spontaneous collapse models 8π2 |k|2 to gravity was made explicit in the Di´osi-Penrose (D-P) 1 Z Z = d3x d3y K(x − y) [m ˆ (x), [m ˆ (y), ρˆ]] . (F5) model [35, 58], where the stochastic kernel was chosen 4 G to be the Newtonian gravitational potential, K(x) = |x| . While such a choice of kernel provides a natural connec- Consequently, the addition of the gravitational back- tion to gravitation, it also necessitates a coarse-graining reaction term in Eq.(F4) amounts to a doubling of the procedure, without which the integral in the second term strength of the decoherence term in Eq.(F1) [59]. This will diverge. This is achieved by convolving the mass has important consequences when considering the ener- getic implications of these models. Without a dissipative density with a Gaussian of width R0: term, energy is conserved in neither model, and it is easy to show [35] that for a single particle of mass m in the D- dE (t) DP mG√ ~ Z  2  P model, the rate of change of energy is dt = 4 πR3 2−3/2 3 |x − y| 0 fR(x) = 2πR0 d y exp − 2 f(y). [35]. When including the backreaction, this rate is simply 2R0 (F2) doubled. In order to obtain a finite dissipator, it is critical that the In both cases, the rate of energetic change is strongly mass density operator is coarse-grained in this manner, determined by the chosen coarse-graining cut-off R0.A natural approach to choosing this cut-off is to argue that i.e.m ˆ (x) → mˆ R(x) it should correspond to the Compton wavelength of a −15 In more recent work, this model has been extended to nucleon, R0 ≈ 10 m. As shown in the main text include the backreaction of the quantised matter on the however, such a value leads to enormous rates of change, gravitational field [59]. Introducing the Newtonian field and therefore predicts a thermal catastrophe.

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