# Quantum solitonic wave-packet of a meso-scopic system in singularity free gravity

###### Abstract

In this paper we will discuss how to localise a quantum wave-packet due to self gravitating macroscopic object by modifying the Schrödinger equation beyond General Relativity. In particular, we will study soliton-like solutions in infinite derivative ghost free theories gravity, which resolves the gravitational singularity in the potential. We will show a unique feature that a quantum spread of such a gravitational system is larger than that of the Newtonian’s gravity, therefore enabling us a window of opportunity to test such theories of gravity in near future at a table-top experiment.

Einstein’s general relativity (GR) has been widely accepted theory of gravity on large scales and late times, i.e. in the infrared (IR) regime, where it has been tested to a very high precision -C.-M. . The recent detection of gravitational waves from binary Blackholes has matched its predictions extremely well with numerical simulations -B.-P. . Despite the great success of GR, there are problems at short distances and small time scales, the theory allows Blackhole and Cosmological singularities in the ultraviolet (UV) regime. In addition, there is an open question - to what extent Einstein’s GR is valid in the UV? In fact, the inverse-square law of Newton’s potential has been tested only up to -D.-J. . This means that any modification from the Newtonian potential can occur in the vast desert of scales spanning about more than 30 orders of magnitude, i.e from to the Planck scale in spacetime dimensions.

This provides us an ample motivation to explore gravitation beyond GR, and put such theories on test in a laboratory, even at table-top experiment. A very well-known interesting observation is that any macroscopic object naturally provides a scale for localising its own quantum wave packet in a semi-classical approach, where gravity is still treated classically, and the matter component is treated like quantum. For instance, in the Newtonian gravitational potential, a quantum spread of a wave packet has been computed by Diósi, which is solely given by the Newton’s constant, , and the mass, , of a macroscopic object by solving the Schrödinger’s equation. A solitonic solution has been found due to self-gravitating potential which tries to confine the wave packet, while quantum property will try to de-localize the wave packet. An optimum quantum spread, , of the wave packet can be found by minimising the energy, and it is given by Diosi:1988uy :

(1) |

Indeed, this quantum spread is very tiny for a massive object, i.e. the value of m for kg. Therefore, measuring such a spread of a wave packet provides a simple way to constrain theory of gravitation beyond GR in a table-top experiment.

The aim of this paper will be to show explicitly that in theories beyond GR, it is possible to constrain the new scale of gravitation by quantifying the solitonic wave packet for a massive object and compare the results with that of the Newtonian case. In particular, we will be studying a particular modification of GR, where the gravitational force vanishes in the UV limit, i.e. as , where stands for the force between particles separated by the distance . Such theory of gravity has recently been advocated to ameliorate the UV aspects of gravitation Biswas:2011ar .

It has been known for a while that a quadratic curvature gravity ( derivatives in metric) is a renormalizable theory of gravity, but contains massive spin-2 ghost, signaling instability in the vacuum and therefore lacking predictions -K.-S. . In order to resolve this ghost problem, we require infinite derivative theories of gravity (IDG) as pointed out in Ref. Biswas:2011ar . The most general quadratic curvature covariant action with infinite derivatives has been constructed in dimensions, around constant curvature backgrounds, see Biswas:2011ar ; Biswas:2016etb :

(2) |

where is a dimensionful coupling, , where is the new scale of gravitation, i.e. GeV. The d’Alembertian operator is given by: , where , and we take mostly metric signature. The three form factors are very similar to pion form factors in strong interaction, and they are reminiscence to derivative nature of interaction depicted by the massless nature of gravity, i.e. they contain infinite order covariant derivatives and are analytic functions of , . These form factors have already been constrained by the general covariance, in the IR the form factors should be such that they match the predictions of GR, and through-out from IR to UV, they should not introduce any new dynamical degrees of freedom, i.e. the graviton remains massless and transverse-traceless, therefore the coefficients are all fixed. Around Minkowski background, they follow a simple relationship given by Biswas:2011ar . In fact, around a constant curvature background, we can treat , without loss of generality.

The new scale of physics, , also signifies the scale of non-locality -Yu.-V. ; Tomboulis ; Tseytlin ; Siegel ; Modesto ; Biswas:2005qr , where the gravitational interaction in this class of theory becomes non-local, see Talaganis:2014ida . Furthermore, it has been argued that the above action becomes UV finite beyond 1-loop -Yu.-V. ; Tomboulis ; Modesto ; Talaganis:2014ida . In this regard, we will be exploring for the first time quantum localization of a wave-packet in such non-local theories of gravity. Classically, such theories can resolve cosmological singularity as pointed out in Biswas:2005qr ; Biswas:2011ar ; Koshelev:2012qn , and possibly even blackhole singularity Koshelev:2017bxd , due to the fact that the physical effect of non-locality can be spread out on macroscopic scale in spacetime.

We now wish to consider a semi-classical approach to IDG described by the action in Eq. (2), and work in the non-relativistic and in the weak-field regime. The starting point is the field equations with a semi-classical source term:

(3) |

where arises from the equations of motion of IDG action, see Ref. Biswas:2013cha . The right hand side of the above expression contains the expectation value of the quantised energy-momentum operator in the quantum state
. We are interested in the linear regime of Eq. (3), which can be obtained by expanding the spacetime metric around the Minkowski background, , and neglecting higher order terms in the perturbation . Moreover, by imposing the DeDonder gauge, we can now show that the semi-classical linearized equations are given by ^{1}^{1}1Note that the perturbation is a classical field, i.e.
it is not quantized in the semi-classical approach. Biswas:2011ar ; Biswas:2013cha :

(4) |

where and The coefficient is defined in terms of Biswas:2011ar :

(5) |

and . Note, that in the linear approximation we are working with, acts as the quantized energy-momentum tensor operator in a flat spacetime.

The coefficient is not an arbitrary function as we had discussed above. By demanding that the gravity remains massless and does not introduce any new dynamical degrees of freedom, the function should be exponential of an entire function Tomboulis ; Biswas:2011ar . Such functions do not introduce any new poles in the propagator, and therefore no new dynamical degrees of freedom other than the massless graviton. One simple choice is
^{2}^{2}2One way to show that the choice
(in the momentum space) does not introduce any additional gravitational
degrees of freedom is to consider the poles in the propagator. As
shown in Ref. Biswas:2011ar ; Biswas:2013kla ; Buoninfante the propagator corresponding
to the action around Minkowski spacetime is given by ,
where and are the so called
spin projection operators along the spin- and spin- components,
respectively. For the choice , there are no additional poles in the propagator,
where the GR propagator is given by: .:

(6) |

In fact other choices of entire function can be made without any loss of generality, as they provide similar universal UV and IR behaviour as pointed out in Ref. Edholm:2016hbt . In the UV the gravitational potential, for , therefore the force vanishes in this regime, while in the IR for , the gravitational potential yields , as expected in the case of GR and in Newtonian gravity Biswas:2011ar ; Koshelev:2017bxd .

We now wish to solve Eq. (4) in the weak-gravitational field and static spacetime limit, and , for the metric:

(7) |

where , and the gravitational potential, , satisfies the differential equation:

(8) |

whose solution is given by:

(9) |

where we have used .

With the help of gravitational potential, Eq. (9), we can now compute the semi-classical interaction Hamiltonian:

(10) |

where we have used the fact that provides the dominant contribution, and .

We can now obtain a one-particle Schrödinger equation, with a non-linear term that takes into account the IDG self-interaction ^{3}^{3}3 In
Bahrami a similar derivation has been presented for the Newtonian gravity.:

(11) |

where is the one particle wave-function and we have assumed that only one kind of particle is present, . Thus, Eq. (11) describes the dynamics of a self-gravitating wave-function in a non-relativistic, weak-field regime. Note that in the limit when , we recover the well-known expression of Schrödinger-Newton equation, see Diosi:1988uy ; Penrose ; Bahrami , as expected.

Let us now study the stationary solutions of Eq. (11), namely seeking solutions which balance quantum-mechanical spreading and contraction due to the attractiveness of the gravitational interaction. Following Ref. Diosi:1988uy , we can show that Eq. (11) admits soliton-like solutions for the ground-state of the form , where is a real function depending only on the space-coordinates , while the time-dependence only appears in the phase, and is the Lagrange multiplier arising from minimising the ground-state. By assuming that is a Gaussian wave-packet, with a normalized , and with a characteristic width , i.e.

(12) |

we obtain the expectation value of the energy in the solitonic ground-state:

(13) |

Note that in the limit , we recover the energy in the case of Newtonian self-interaction (see Ref. Diosi:1988uy ), as expected.

Further note, that the presence of increases the energy of the ground state, in fact for any values of and , we always find:

(14) |

Now, we can also find the spread of the soliton by minimising the energy in Eq. (13) with respect to and we obtain:

(15) |

It is not instructive to write down the full physical solution here, we will provide its full behaviour in Fig. 1. Note that in the limit , we recover the Newtonian case already, seen in Eq. (1), obtained in Ref. Diosi:1988uy . In the opposite limit, when , by expanding the energy in Eq. (13), up to the order , and simplifying we find

(16) |

The above equation suggests that the quantum state of a self-gravitating system is localized within a region of size , which is governed by the scale of non-locality, this will become evident by looking at the plot below in Fig. 1, and at the table in which several values of the spread of the wave-function have been shown.

mass | eV) | eV) | GeV) | GeV) | |
---|---|---|---|---|---|

kg | m | m | m | m | m |

kg | m | m | m | m | m |

kg | m | m | m | m | m |

kg | m | m | m | m | m |

In Fig. 1, the physical solution of Eq. (15) has been plotted, i.e. the spread of the solitonic wave-packet with respect to the mass for different values of the parameter . The quantum spread and are plotted as functions of the mass . Note that in the case of IDG, the wave-function spread turns out to be much larger than that of the Newtonian gravitational potential,

(17) |

This is an effect induced by IDG, and the presence of non-locality in the gravitational interaction. Such theories possess mass-gap Frolov in the gravitational potential, governed by the scale , and therefore within , effectively the gravitational force vanishes, but for , the gravitational attractive force balances the quantum spread of the wave-function. In this regard, the quantum wave-function of the meso-scopic system provides a solitonic solution, as depicted in Fig. 1. In the plot-region the behaviour of the IDG spread is well approximated by Eq. (16). Moreover, the larger is the value of , smaller is the value of , and in the limit in which , we have . Generally, we obtain: .

In the table , let us consider some numerical values of the spread of the wave-function obtained by evaluating the physical solution of Eq. (15) for different values of the mass, , and the parameter . Note that the larger the mass is, smaller is the spread. For a mass kg, in the Newtonian potential, meters, while the IDG spread is always larger, and for eV, , which is much larger than .

This indeed provides a new scenario for testing the classical properties of IDG from the quantum localisation of the wave-packet in a molecule interferometry gerlich , and in optomechanical tests, see yang ; andregro . A larger spread in the wave-function might provide us a smoking gun signature of the nature of a gravitational potential. We should be able to study the free expansion and contraction of the wave-function and place limits on the new scale of physics, . It is worthwhile to mention that these are very sensitive experiments, and there are several sources of decoherence effects, for a review see Bassi , which pose enormous experimental challenges for observing some interesting quantum phenomena in a ghost free and singularity free theory of gravitation. One promising experiment along these lines will be to test the spread of a quantum wave-packet in a free-fall experiment performed in microgravity, see Muntinga:2013pta , where the authors have put forward an interesting experimental proposal to test quantum mechanics of weakly coupled Bose-Einstein condensate in a freely falling system.

Before we conclude, it is worthwhile to mention that it is also possible to study modified gravitational potentials, such as Yukawa-like potential capozziello of the form , arising in theories, where similar computations suggest an opposite scenario compared to that of IDG. Indeed, the energy and the spread of the ground-state turn out to be smaller compared to that of the Newtonian case: .

This proof-of concept paper provides us, for the first time, how we can put theories of gravity on to test-bed by studying the solitonic wave-function of a self-gravitating quantum system in theories beyond Einstein’s GR. In particular, singularity free theories of gravity provides an intriguing observation that the minimum energy for the IDG is always larger compared to that of the Newtonian case, and the spread of the wave-function is always larger than that of the Newtonian potential for meso-scopic systems. These predictions are indeed testable in a table-top experiment in near future, and might allow us a deeper understanding of the gravitational interaction at short distances.

## Acknowledgments.

The authors would like to thank Sougato Bose for discussions.

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