Modification of Schrödinger–Newton Equation Due to Braneworld Models

Physics Letters B 770 (2017) 325–330

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Physics Letters B

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Modiﬁcation of Schrödinger–Newton equation due to braneworld models with minimal length ∗ Anha Bhat a, Sanjib Dey b,c, , Mir Faizal d,e, Chenguang Hou f, Qin Zhao f a Department of Metallurgical and Materials Engineering, National Institute of Technology, Srinagar 190006, India b Institut des Hautes Études Scientiﬁques, Bures-sur-Yvette 91440, France c Institut Henri Poincaré, Paris 75005, France d Irving K. Barber School of Arts and Sciences, University of British Columbia–Okanagan, 3333 University Way, Kelowna, BC V1V 1V7, Canada e Department of Physics and Astronomy, University of Lethbridge, Lethbridge, AB T1K 3M4, Canada f Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore a r t i c l e i n f o a b s t r a c t

Article history: We study the correction of the energy spectrum of a gravitational quantum well due to the combined Received 21 March 2017 effect of the braneworld model with inﬁnite extra dimensions and generalized uncertainty principle. The Received in revised form 19 April 2017 correction terms arise from a natural deformation of a semiclassical theory of quantum gravity governed Accepted 1 May 2017 by the Schrödinger–Newton equation based on a minimal length framework. The two fold correction in Available online 4 May 2017 the energy yields new values of the spectrum, which are closer to the values obtained in the GRANIT Editor: N. Lambert experiment. This raises the possibility that the combined theory of the semiclassical quantum gravity and the generalized uncertainty principle may provide an intermediate theory between the semiclassical and the full theory of quantum gravity. We also prepare a schematic experimental set-up which may guide to the understanding of the phenomena in the laboratory. © 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction existence of nonzero minimal uncertainty in position coordinate, which is familiar as the minimal length in the literature. An inti- The fact that the minimal observable length can be useful to mate connection between the gravitation and the existence of the impose an effective cut-off in the ultraviolet domain in order to fundamental length scale was proposed in [8]. The string theory, make the theory of quantum fields renormalizable was suggested which is the most popular approach to quantum gravity, also sup- very early by Heisenberg. It was Snyder, who formalized the idea ports the presence of such minimal length [9–13], since the strings for the first time in the form of an article, and showed that the are the smallest probes that exist in perturbative string theory, noncommutative structure of space–time characterizes the mini- and so it is not possible to probe space–time below the string mal measurable length in a very natural way [1]. Since then, the length scale. In loop quantum gravity, the existence of a minimum notion of noncommutativity has evolved from time to time and length has a very interesting consequence, as it turns the big bang revealed its usefulness in different contexts of modern physics [2]. into a big bounce [14,15]. Furthermore, the arguments from black Some natural and desirable possibilities arise when the canonical hole physics suggest that any theory of quantum gravity must be space–time commutation relation is deformed by allowing gen- equipped with a minimum length scale [16,17], due to the fact that eral dependence of position and momentum [3–7]. In such sce- the energy required to probe any region of space below the Plank narios, the Heisenberg uncertainty relation necessarily modifies to length is greater than the energy required to create a mini black a generalized version to the so-called generalized uncertainty prin- hole in that region of space. The minimal length exists in other ciple (GUP). Over last two decades, it is known that within this subjects too; such as, path integral quantum gravity [18,19], spe- framework, in particular, where the space–time commutation re- cial relativity [20], doubly special relativity [21–23], etc. In short, lation involves higher powers of momenta, explicitly lead to the the existence of minimal measurable length, by now, has become a universal feature in almost all approaches of quantum gravity. Moreover, some Gedankenexperiments and thought experiments Corresponding author. * [8,24,25] in the spirit of black hole physics have also supported E-mail addresses: [email protected] (A. Bhat), [email protected] (S. Dey), [email protected] (M. Faizal), [email protected] (C. Hou), the idea. For further informations on the subject one may follow [email protected] (Q. Zhao). some review articles devoted to the subject, for instance, [26–28]. http://dx.doi.org/10.1016/j.physletb.2017.05.005 0370-2693/© 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 326 A. Bhat et al. / Physics Letters B 770 (2017) 325–330

In spite of having several serious proposals for quantizing For further information on the minimization procedure, one may the general relativity, unfortunately we do not have a fully con- refer, for instance [3,6,43]. A possible representation of the algebra sistent quantum theory of gravity yet. This has motivated the (1) in terms of the canonical position and momentum operators study of semi-classical quantum gravity (SCQG), where the gravi- x, p is given by [3,4] tational filed is treated as a background classical field, and the = = + 2 matter fields are treated quantum mechanically. Under such ap- Xi xi, P i pi(1 β p ), (4) proximation, if |ψ is the wave function of the matter field, the ∂ where p j =−ih¯ . Certainly, the representation (4) is not unique Einstein tensor G can be obtained in terms of the quantum ∂x j μν [3], as for instance; see [45], where the authors explore four pos- mechanical energy–momentum tensor for matter fields T μν as sible representations of the algebra (1) with some of them being G = 8π Gψ|T μν |ψ/c4. However, Newtonian gravity has been μν Hermitian. However, it is easy to show that (4) satisfies (1) up to observed to be the correct approximation to general relativity till the first order of β (hence, we neglect higher order terms of β). the smallest length scale (0.4 mm) to which general relativity has Physically, the notations are understood in the sense that p repre- been tested [29]. Thus, at small distances, it is expected that the i sents the momenta at low energy, while P correspond to those at semi-classical approximation can be described by a Schrödinger– i high energy. Newton equation [30–34], which is the nonrelativistic limit of the Dirac equation and the Klein–Gordon equation with a clas- 3. The gravitational quantum well sical Newtonian potential [35]. It has been proposed that the Schrödinger–Newton equation can be utilized to explore various The gravitational quantum well (GQW) is characterized by the interesting properties of gravitational systems at the given length motion of a nonrelativistic object of mass m in a gravitational field scale [36–38]. g =−ge with a restriction imposed by a mirror placed at the Some interesting consequences may follow from the combina- z origin z = 0, such that the potential turns out to be tion of the above two frameworks, namely, the GUP and SCQG, and it is allowed since both of the effects occur at small scales. +∞ ≤ = , z 0, Technically, this can be achieved by imposing the minimal length V 0(z) (5) mgz, z > 0, structure into the semi-classical scheme of gravity by means of deforming the Schrödinger–Newton equation in accordance with with g being the gravitational acceleration. The experimental set- the laws of GUP. The most interesting fact is that the new the- up of the corresponding potential (5) has already been studied ory resulting from the combination of GUP and SCQG may provide [46]. Theoretically, the problem resemble a quantum mechani- an intermediate theory between a full quantum theory of gravity cal particle moving in a potential well given by (5) subjected and the SCQG and, this is precisely the issue that we address in to a boundary condition ψ 0(0) = 0at z = 0. If we consider the present manuscript. Recently, it has been suggested that both ψ 0(x) = ψ 0(z)ψ 0(y), then the solutions of the corresponding time- the Schrödinger–Newton equation [39,40] and the deformation of independent Schrödinger equation along the z direction quantum mechanical structure by the GUP [41] can be tested ex- 2 perimentally by using the opto-mechanical systems. Therefore, it 0 0 0 p H0ψ (z) = E ψ (z), H0 = + mgz, (6) becomes important to understand the effects of the combined the- 2m ory, especially when there is a strong viability that the theory may are well-known and, are given by [47] (Problem 40) be tested by using a similar type of experimental set-up in near 1/3 future. mg 2m2 g 0 =− αn = En ,θ , (7) θ h¯ 2 2. GUP and minimal length 0 = [ + ] ψn (z) NnAi θ z αn , (8)

Let us commence by introducing a particular version of mod- th where αn is the n zero of the regular Airy function Ai(z) [48], iﬁed commutation relation between the position and momentum and N = θ 1/2/|Ai (α )| is the normalization factor. Along the y operators X, P [3,4] n n axis, the particle is free and the corresponding wave function takes 2 the form [Xi, P j]=ih¯ (δij + βδijP + 2β P i P j), [Xi, X j]=[P i, P j]=0, ∞ (1) ψ 0(y) = g(k)eikydk, (9) with βhaving the dimension of inverse squared momentum and, 2 = 3 2 → −∞ P j=1 P j . As obviously, in the limit β 0, the deformed relations (1) reduce to the standard canonical commutation relations where g(k) determines the shape of the wave packet in momen- [xi , p j] = ih¯ δij. However, there exist many other similar type of tum space. In analogy to a classical object, a collision of a quantum deformations in the literature, which have been used to investi- particle with the impenetrable mirror along the z direction will gated many interesting phenomena in different contexts; see, for make it bounced at a critical height hn instance [6,7,28,42–44]. Nevertheless, the generalized uncertainty E0 relation or the Robertson–Schrödinger uncertainty relation corre- n αn hn = =− , (10) sponding to the deformed algebra (1) turns out to be mg θ 1 which is, naturally, quantized. In the GRANIT experiment [49], the ≥ [ ] XiP j Xi, P j (2) measured critical heights for the ﬁrst two states are 2 ¯ exp h 2 2 2 2 h = 12.2 μm ± 1.8 ± 0.7 , (11) XiP i ≥ 1 + β (P ) +P + 2β (P i) +P i , 1 syst stat 2 exp = μ ± ± (3) h2 21.6 m 2.2syst 0.7stat. (12) from which one can compute the exact expression of the minimal Whereas, the theoretical values for h1 and h2 can be obtained from observable length by using the standard minimization technique. (10) as A. Bhat et al. / Physics Letters B 770 (2017) 325–330 327

th = μ h1 13.7 m, (13) 4.1. Correction due to Earth–particle interaction th = μ h2 24.0 m, (14) Let us first derive the potential coming from the Earth–particle interaction. By considering our planet to be a spherical body with where m = 939 MeV/c2 and g = 9.81 m/s2. The variation of the mass density ρE , the total potential acting on the particle is heights h and h resulting from the theory and the experiment 1 2 3 th th th exp exp exp d r are, therefore, δ = h − h = 10.3 μm and δ = h − h = E =− 2 1 2 1 V b (r) mGρEkb + . (20) 9.4 μm ± 5.4 μm, respectively. Thus, the deviation of δ between |r − r |b 1 the theory and the experiment turns out to be 4.5 μm. The ar- E gument that we shall pursue here is that any correction in the In spherical coordinates which becomes Hamiltonian (6) will effectively reduce this deviation of δ between R the theory and the experiment, and thus the theoretical result will E 2 V (h) =−2πmGρEk r dr become closer to the experiment. A similar argument has already b b been used in [50–53]. However, we explore two types of correction 0 here. First, we consider a braneworld model studied in [54], which 1 du was discovered in the course of solving the hierarchy problem of × + , (21) the standard model. Their theory is based on the assumption that, [r2 − 2r(h + R)u + (h + R)2](b 1)/2 − while all the standard model fields, gauge and matter, are confined 1 in a (3 + 1) dimensional manifold, only the graviton can propagate where h is the altitude above Earth’s surface and R being the ra- freely in the extra dimensions which are considered to be infinite. dius of the Earth. For, b = 0, 1, 2, 3, one obtains [53] In presence of such infinite extra dimensions, the Newtonian po- GmM V E (h) =− , tential is modified to the following form 0 + h R + + E 3mgk1 h(h 2R) h 2R GMm kb b V (h) =− 2R − ln , V (r) =− 1 + , r >> = |k |, (15) 1 + b b 4R h R h r r (22) 3mgk2 h + 2R 2R V E (h) =− ln − , where is the length scale at which the correction due to the infi- 2 + 2R h h R nite extra dimensions becomes dominant. If we consider the New- + − = 3mgk3 2R 1 h 2R tonian potential GMm/r V 0(r), we can write the above equa- V E (h) =− − ln , b+1 3 + + tion (15) as V (r) = V 0(r) + V b(r), so that V b(r) =−GMmkb/r 4R h(h 2R) h R h can be considered as a perturbation, which will eventually con- while for b > 3, it becomes tribute to the correction over the theoretical values of δ. Therefore, th th 3mgkb δ will be changed to δ + , where V E (h) =− b 2(b − 1)(b − 2)(b − 3)R(h + R) 1 − − − + = 0 | | 0 − 0 | | 0 (b 3)R h (b 1)R h ψ2 (z) V b ψ2 (z) ψ1 (z) V b ψ1 (z) , (16) × + . (23) mg hb−2 (h + 2R)b−2 and | | ≤ 4.5 μm. The second correction emerges from the con- Notice that in the limit b → 0 (k0 = 1), we recover the Newtonian tribution of the GUP deformed Hamiltonian. If we replace the potential in an exact form. momentum p corresponding to the low energy in (6) by the momentum P coming from the GUP deformation (4), we obtain 4.2. Correction due to mirror–particle interaction

β Although the mass of the mirror is negligible with respect to = + = 4 H H0 H1, H1 p , (17) that of the Earth, the effect originating from the mirror–particle m interaction should be taken into account since the mirror is placed where we neglect the higher order terms of β. Thus, if we denote at a distance much closer to the particle. The mirror can be seen as th the correction of δ coming from the GUP by , with a parallelepiped with density ρM , and located at −∞ <(x, y) < ∞ and −L < z < 0. With respect to the size of the particle, the mirror 1 0 0 0 0 can be considered as an infinite plane, so that the mirror behaves = ψ (z)|H1|ψ (z) − ψ (z)|H1|ψ (z) , (18) mg 2 2 1 1 as a disc of infinite radius. Thus, the total interacting potential in this case becomes ∗ then 0 R M rdr | |≤| + |≤4.5 μm. (19) V (h) =−2πmGρMkb dz , (24) b [r2 + (h − z)2](b+1)/2 −L 0 Let us now compute both of the corrections and in the fol- ∗ lowing section. where R is taken to be very large, but finite, in order to avoid the divergent integrals. Nevertheless, the integrals (24) are computed as follows 4. Corrections 3mgk h + L M = 1ρ + + V 1 (h) L ln(h L) h ln , The potential described in (15) corresponds to an interaction 2R h between two particles. However, we are dealing with a situation 3mgk2ρ h + L V M (h) =− ln , (25) where a point particle bounces on a plane mirror placed at the 2 2R h surface of the Earth. Therefore, we are required to derive the ef- 3mgkbρ 1 1 fective potentials between the test particle and the Earth, as well V M (h) =− + . b>2 − − b−2 b−2 as, between the particle and the mirror. 2R(b 1)(b 2) h (h + L) 328 A. Bhat et al. / Physics Letters B 770 (2017) 325–330

4.3. The perturbative correction from GUP deformation Table 1 Constrains on the power-law parameters. Now, let us consider the correction due to the perturbation b 123 term H (17) arising from the GUP deformation. The correction 1 | |(m) < 8.13 × 104 4.34 0.076 to the energy E at the lowest order in β is given by n β GUP = | 4| En ψ(x) p ψ(x) In order to obtain the bounds on the power-law parameters and m =− ˜ = b β kb, let us consider V b mgkb V b. Eq. (15) implies that kb , = [ 2 0 | 4| 0 =− b ˜ 4m ψn (z) pz ψn (z) so that V b mg V b. Therefore, we can write m + 0 | 2| 0 0 | 2 | 0 =− b ˜ =− b 0 | ˜ | 0 − 0 | ˜ | 0 2 ψn (z) pz ψn (z) ψ (y) p y ψ (y) b b ψ2 (z) V b ψ2 (z) ψ1 (z) V b ψ1 (z) , β 2 0 2 2 0 (35) = [4m [E − V 0(z)] + 8m Ec E − V 0(z) ] m n n which immediately leads to the constraint on the parameter as 0 0 0 2 = 4βm[E (E + 2Ec) − 2(E + Ec) V 0(z) + V (z) ], b ˜ n n n 0 | | = b/ b. According to (16), is bounded below 4.5 μm (26) and, so is bounded by the following relation where E = m ψ 0(y)|v2 |ψ 0(y) /2is the kinetic energy of the par- c y | |= b ˜ ticle along the horizontal direction. Note that a term proportional 4.5/ b. (36) to ψ 0(y)|p4 |ψ 0(y) has been omitted since it only leads to a y Using the results from (32)–(34), we can calculate ˜ correspond- global shift of the energy spectrum. Therefore, by computing the b ing to different values of b. If we turn off the GUP deformation following integrals (β = 0), we obtain the Table 1. In a similar way it is also possi- = 0 | | 0 V 0(z) ψn (z) V 0(z) ψn (z) ble to obtain the bounds for the cases when β = 0. On the other +∞ hand, if we turn off the power-law modification, we see that the 2 = 2 2 + = 0 GUP deformation parameter has the constraint mgNn zAi (θ z αn) En, 3 51 0 β<2.77167 × 10 , (37) (27) 2 = 0 | 2 | 0 V 0 (z) ψn (z) V 0 (z) ψn (z) which provides a tighter upper bound than those derived in the +∞ context of gravitational fields; see, for instance [55]. However, the 2 2 2 2 8 0 2 upper bound (37) is weaker in comparison to [7,56,57], which = (mg) N z Ai (θ z + αn) = (E ) , n 15 n were obtained in different contexts in the literature. Nevertheless, 0 what we notice is that we obtain positive contributions from both we obtain the final expression of the correction to the energy as SCQG and GUP deformation, which will make the theoretical val- follows ues of δ closer to the experiment. 4 10Ec EGUP = βm(E0)2(1 + ). (28) n n 0 6. A schematic proposal for experiment 5 3En

5. The modified GQW spectrum To this end, we make a proposal for an experimental quantum bouncer through an opto-mechanical set-up, which would be able Combining the corrections coming from the SCQG, we obtain to provide an understanding of the combined effect of SCQG and GUP. Opto-mechanical devices yield a promising avenue for prepar- V = V E (z) + V M (z), (29) b b b ing and investigating quantum states of massive objects ranging which when combined with the correction to the energy arising from a few picograms up to several kilograms [58]. Significant from the GUP deformation, we obtain the total energy shift experimental progress has been achieved by using such devices in different contexts, including coherent interactions [59], laser E = V + EGUP. (30) n b n cooling of nano and micro-mechanical devices into their quantum Therefore, we have ground state [60]. Recently, such types of systems have been uti- 1 lized for the understanding of more exciting features like the SNE + = 0 | | 0 + GUP b ψ2 (z) V b ψ2 (z) E2 in SCQG [39,40] and GUP [41]. Here, our goal is to understand mg the combined effect of SCQG and GUP through the given opto- − 0 | | 0 − GUP ψ1 (z) V b ψ1 (z) E1 . (31) mechanical system. The underlying principle behind the experiment is that the ul- Let us now compute the numerical values of the above expressions, tra cold neutrons (UCN) move freely in the gravitational field above = = 2 = 2 = with R 6378 km, m 939 MeV/c , g 9.81 m/s , L 10 cm, a mirror and make a total reflection from the surface of the mirror = 2 μ ρ 1 and Ec m v y /2 0.221 eV when the corresponding de Broglie wavelength is bigger than the −11 −57 interatomic distances of the matter. Thus, the set-up gives rise to a 1 + = k1 × 5.59203 × 10 + β × 1.62357 × 10 m GQW, where the UCN form bound quantum states in the Earth’s < 4.5 μm, (32) gravitational field. The eigenvalues are non-equidistant and, are −7 −1 −57 in the range of pico-eV. These type of scenarios offer fascinating 2 + = k2 × 2.38493 × 10 m + β × 1.62357 × 10 m possibilities to combine the effects of Newton’s gravity at short < 4.5 μm, (33) distances with the high precision resonance spectroscopy methods − − of quantum mechanics. The schematic diagram of the experimental + = k × 0.0101239 m 2 + β × 1.62357 × 10 57 m 3 3 set-up is given in Fig. 1, which consists of an anti-vibrator granite < 4.5 μm. (34) table, a convex magnification system (mirror), an inclinometer, a A. Bhat et al. / Physics Letters B 770 (2017) 325–330 329

and both of the theories provide positive contributions in the correction of the energy spectrum, it raises a natural possibility that the combined theory may guide us towards a theory beyond the SCQG. Thus our proposal may yield an intermediate theory beyond the SCQG and the complete theory of quantum gravity. Moreover, we have provided a schematic experimental set-up which would help the laboratory experts to explore our theory further in the lab. There are many natural directions that may follow up our in- vestigation. First, it will be interesting to study the similar kind of effects of GUP in the context of other theories of gravity. There exist many other type of GUP deformation, which may be useful to study the similar theories to confirm our findings. However, the most interesting future problem lies on the understanding of the experimental realization of the combined theory of SCQG and GUP Fig. 1. The schematic of the opto-mechanical setup for the proposed experiment. by using the opto-mechanical set-up, while it has already been used to understand each of the individual frameworks of SCQG and ceiling scatterer and a position sensitive detector. The UCN shall GUP. be formed by passing the neutrons of the proposed wavelength 4 between 0.75–0.90 nm through the superfluid of He of a vol- Acknowledgements ume of about 4–5 liters. The UCN are brought to the experimental set-up by a guided system composed of a slit, a neutron collima- AB is supported by MHRD, Government of India and would like tor and an aperture. The neutrons fly in the slit and are screened to thank Department of Physics and MMED at NIT Srinagar for in the range of 4–10 m/s. The upper wall of the slit is an effi- carrying out her research pursuit. SD is supported by a CARMIN cient absorber which only lets the surviving neutrons to pass and, postdoctoral fellowship by IHES and IHP. The work of QZ is sup- hence forth controlling first quantum states of the neutron flux. ported by NUS Tier 1 FRC Grant R-144-000-360-112. The mirror consists of a magnification system, which is made of a bi-metal (NiP) coated glass rod of radius between 3–3.5 nm with References the roughness of about 1.5 nm. However the mirror system shall be modified to elliptical shape in order to harness the neutrons [1] H.S. Snyder, Quantized space–time, Phys. Rev. 71 (1947) 38. of much smaller wavelength as well. The distribution of 100 μm [2] N. Seiberg, E. Witten, String theory and noncommutative geometry, J. High En- in height is magnified to ∼ 2.5mm at the position on the detec- ergy Phys. 1999 (1999) 032. tor. This gives the average magnification power of about 25 at the [3] A. Kempf, G. Mangano, R.B. Mann, Hilbert space representation of the minimal length uncertainty relation, Phys. Rev. D 52 (1995) 1108. glancing angle at 200 to make the reflection high. [4] S. Das, E.C. Vagenas, Universality of quantum gravity corrections, Phys. Rev. The most important aspect of the position sensitive neutron Lett. 101 (2008) 221301. detector is that the neutrons must be changed into a charged par- [5] M. Gomes, V.G. Kupriyanov, Position-dependent noncommutativity in quantum ticles by a nuclear reaction. The converter materials are available mechanics, Phys. Rev. D 79 (2009) 125011. as 3He, 6Li, 10B, and 157Gd, etc. A black thin CCD (charge cou- [6] B. Bagchi, A. Fring, Minimal length in quantum mechanics and non-Hermitian Hamiltonian systems, Phys. Lett. A 373 (2009) 4307–4310. pled detector) could be used along with a thin layer of 3He, 10B [7] C. Quesne, V.M. Tkachuk, Composite system in deformed space with minimal 10 10 or Ti– B–Ti. The ideal thickness of B should be about 200 nm, length, Phys. Rev. A 81 (2010) 012106. and must be evaporated directly on the CCD surface. The standard .[8] C.A Mead, Possible connection between gravitation and fundamental length, pixel size is 24 μm × 24 μm and the thickness of the active volume Phys. Rev. 135 (1964) B849. should be about 20 μm. The kinetic energy is deposited on the ac- [9] G. Veneziano, A stringy nature needs just two constants, Europhys. Lett. 2 (1986) 199. tive area and a charge cluster is created, which typically spreads [10] D.J. Gross, P.F. Mende, String theory beyond the Planck scale, Nucl. Phys. B 303 into nine pixels. The weighted center of the charge cluster is a (1988) 407–454. good estimation of the incident neutron position. [11] D. Amati, M. Ciafaloni, G. Veneziano, Can spacetime be probed below the string At the exit of the UCN system the wavefucntion changes due to size?, Phys. Lett. B 216 (1989) 41–47. the absence of suppressing slit and as such the neutrons turn as [12] T. Yoneya, On the interpretation of minimal length in string theories, Mod. Phys. Lett. A 4 (1989) 1587–1595. bouncers on the mirror surface and their trajectories are measured [13] K. Konishi, G. Paffuti, P. Provero, Minimum physical length and the generalized by the Time of flight (TOF) method. They are characterized by a uncertainty principle in string theory, Phys. Lett. B 234 (1990) 276–284. Gaussian distribution with a mean of 9.4 m/s and the standard [14] C. Rovelli, Loop quantum gravity, Living Rev. Relativ. 1 (1998) 1. deviation of 2.8 m/s which is subject to change due to various [15] P. Dzierzak, J. Jezierski, P. Malkiewicz, W. Piechocki, The minimum length prob- modifications of the design. lem of loop quantum cosmology, Acta Phys. Pol. A 41 (2010) 717–726. [16] M. Maggiore, A generalized uncertainty principle in quantum gravity, Phys. Lett. B 304 (1993) 65–69. 7. Conclusions [17] M. Park, The generalized uncertainty principle in (A)dS space and the modification of Hawking temperature from the minimal length, Phys. Lett. B 659 We have studied a two-fold correction in the energy spectrum (2008) 698–702. [18] T. Padmanabhan, Physical significance of Planck length, Ann. Phys. 165 (1985) of a GQW. The first correction arises from the scheme of the SCQG 38–58. itself, where we considered the SNE in a particular framework [19] J. Greensite, Is there a minimum length in d = 4 lattice quantum gravity?, Phys. of braneworld model with infinite extra dimensions. The second Lett. B 255 (1991) 375–380. correction emerges from a GUP deformation of the given SCQG [20] G. Amelino-Camelia, Testable scenario for relativity with minimum length, framework. The combined effect of these two interesting theories Phys. Lett. B 510 (2001) 255–263. [21] J. Magueijo, L. Smolin, Lorentz invariance with an invariant energy scale, Phys. provide a new bound on the energy spectrum of the GQW, which Rev. Lett. 88 (2002) 190403. is closer to the observed values in the GRANIT experiment. Since, [22] F. Girelli, E.R. Livine, D. Oriti, Deformed special relativity as an effective flat both of the effects of SCQG and GUP occur at small length scales, limit of quantum gravity, Nucl. Phys. B 708 (2005) 411–433. 330 A. Bhat et al. / Physics Letters B 770 (2017) 325–330

[23] J.L. Cortes, J. Gamboa, Quantum uncertainty in doubly special relativity, Phys. [41] I. Pikovski, M.R. Vanner, M. Aspelmeyer, M. Kim, C. Brukner, Probing Planck- Rev. D 71 (2005) 065015. scale physics with quantum optics, Nat. Phys. 8 (2012) 393–397. [24] F. Scardigli, Generalized uncertainty principle in quantum gravity from micro- [42] P. Pedram, K. Nozari, S.H. Taheri, The effects of minimal length and maximal black hole gedanken experiment, Phys. Lett. B 452 (1999) 39–44. momentum on the transition rate of ultra cold neutrons in gravitational field, J.[25] Y. Ng, H. Van Dam, Spacetime foam, holographic principle, and black hole J. High Energy Phys. 2011 (2011) 1. quantum computers, Int. J. Mod. Phys. A 20 (2005) 1328–1335. [43] S. Dey, A. Fring, L. Gouba, PT -symmetric non-commutative spaces with mini- [26] L.J. Garay, Quantum gravity and minimum length, Int. J. Mod. Phys. A 10 (1995) mal volume uncertainty relations, J. Phys. A, Math. Theor. 45 (2012) 385302. 145–165. [44] S. Dey, V. Hussin, Entangled squeezed states in noncommutative spaces with [27]J. Y. Ng, Selected topics in Planck-scale physics, Mod. Phys. Lett. A 18 (2003) minimal length uncertainty relations, Phys. Rev. D 91 (2015) 124017. 1073–1097. [45] S. Dey, A. Fring, B. Khantoul, Hermitian versus non-Hermitian representations [28] S. Hossenfelder, Minimal length scale scenarios for quantum gravity, Living Rev. for minimal length uncertainty relations, J. Phys. A, Math. Theor. 46 (2013) Relativ. 16 (2013) 2. 335304. [29] S.-Q. Yang, B.-F. Zhan, Q.-L. Wang, C.-G. Shao, L.-C. Tu, W.-H. Tan, J. Luo, Test of [46] V.V. Nesvizhevsky, et al., Quantum states of neutrons in the Earth’s gravita- the gravitational inverse square law at millimeter ranges, Phys. Rev. Lett. 108 tional field, Nature 415 (2002) 297–299. (2012) 081101. [47] S. Flügge, Practical Quantum Mechanics, Springer Verlag, Berlin, 1999. [30] R. Ruffini, S. Bonazzola, Systems of self-gravitating particles in general relativity [48] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions: With For- and the concept of an equation of state, Phys. Rev. 187 (1969) 1767. mulas, Graphs, and Mathematical Tables, Dover Publications, New York, 1964. [31] R. Penrose, Quantum computation, entanglement and state reduction, Philos. [49] V.V. Nesvizhevsky, et al., Study of the neutron quantum states in the gravity Trans. R. Soc. Lond. Ser. A 356 (1998) 1927–1939. field, Eur. Phys. J. C 40 (2005) 479–491. [50] O. Bertolami, J.G. Rosa, C.M.L. de Aragão, P. Castorina, D. Zappalà, Noncommu- [32] P. Tod, I.M. Moroz, An analytical approach to the Schrödinger–Newton equa- tative gravitational quantum well, Phys. Rev. D 72 (2005) 025010. tions, Nonlinearity 12 (1999) 201. [51] F. Brau, F. Buisseret, Minimal length uncertainty relation and gravitational [33] J.R. van Meter, Schrödinger–Newton ‘collapse’ of the wavefunction, Class. Quan- quantum well, Phys. Rev. D 74 (2006) 036002. tum Gravity 28 (2011) 215013. [52] R. Banerjee, B.D. Roy, S. Samanta, Remarks on the noncommutative gravita- [34] H. Yang, H. Miao, D.-S. Lee, B. Helou, Y. Chen, Macroscopic quantum mechanics tional quantum well, Phys. Rev. D 74 (2006) 045015. in a classical spacetime, Phys. Rev. Lett. 110 (2013) 170401. [53] F. Buisseret, B. Silvestre-Brac, V. Mathieu, Modified Newton’s law, braneworlds, [35] D. Giulini, A. Großardt, The Schrödinger–Newton equation as a non-relativistic and the gravitational quantum well, Class. Quantum Gravity 24 (2007) 855. limit of self-gravitating Klein–Gordon and Dirac fields, Class. Quantum Gravity [54] L. Randall, R. Sundrum, An alternative to compactification, Phys. Rev. Lett. 83 29 (2012) 215010. (1999) 4690. [36] D. Giulini, A. Großardt, Gravitationally induced inhibitions of dispersion accord- [55] F. Scardigli, R. Casadio, Gravitational tests of the generalized uncertainty prin- ing to the Schrödinger–Newton equation, Class. Quantum Gravity 28 (2011) ciple, Eur. Phys. J. C 75 (2015) 425. 195026. [56] M. Bawaj, et al., Probing deformed commutators with macroscopic harmonic [37] G. Manfredi, P.-A. Hervieux, F. Haas, Variational approach to the time- oscillators, Nat. Commun. 6 (2015) 7503. dependent Schrödinger–Newton equations, Class. Quantum Gravity 30 (2013) [57] F. Scardigli, G. Lambiase, E.C. Vagenas, GUP parameter from quantum correc- 075006. tions to the Newtonian potential, Phys. Lett. B 767 (2017) 242–246. [38] S. Bera, R. Mohan, T.P. Singh, Stochastic modification of the Schrödinger– J.[58] T. Kippenberg, K.J. Vahala, Cavity optomechanics: back-action at the Newton equation, Phys. Rev. D 92 (2015) 025054. mesoscale, Science 321 (2008) 1172–1176. [39] A. Großardt, J. Bateman, H. Ulbricht, A. Bassi, Optomechanical test of the [59] A.D. O’Connell, et al., Quantum ground state and single-phonon control of a Schrödinger–Newton equation, Phys. Rev. D 93 (2016) 096003. mechanical resonator, Nature 464 (2010) 697–703. [40] C.C. Gan, C.M. Savage, S.Z. Scully, Optomechanical tests of a Schrödinger– [60] J.D. Teufel, et al., Sideband cooling of micromechanical motion to the quantum Newton equation for gravitational quantum mechanics, Phys. Rev. D 93 (2016) ground state, Nature 475 (2011) 359–363. 124049.