Strong quantum violation of the gravitational weak equivalence principle by a nonGaussian wavepacket
Abstract
The weak equivalence principle of gravity is examined at the quantum level in two ways. First, the position detection probabilities of particles described by a nonGaussian wavepacket projected upwards against gravity around the classical turning point and also around the point of initial projection are calculated. These probabilities exhibit massdependence at both these points, thereby reflecting the quantum violation of the weak equivalence principle. Secondly, the mean arrival time of freely falling particles is calculated using the quantum probability current, which also turns out to be mass dependent. Such a massdependence is shown to be enhanced by increasing the nonGaussianity parameter of the wave packet, thus signifying a stronger violation of the weak equivalence principle through a greater departure from Gaussianity of the initial wave packet. The massdependence of both the position detection probabilities and the mean arrival time vanish in the limit of large mass. Thus, compatibility between the weak equivalence principle and quantum mechanics is recovered in the macroscopic limit of the latter. A selection of Bohm trajectories is exhibited to illustrate these features in the free fall case.
pacs:
03.65.Ta, 04.20.CvKeywords: nonGaussian wavepacket, Weak equivalence principle, Bohm trajectory
I Introduction
In the famous gedanken experiment conceived by Galileo, the universality of the ratio between the gravitational and inertial masses has been studied with test bodies in free fall from the leaning tower of Pisa Gabook1638 (). Since then several tests of weak equivalence principle have been performed with classical test bodies such as very sensitive pendula or torsion balances establishing extremely accurate conformation of the equality of gravitational and inertial masses. The most familiar tests of the weak equivalence principle are experiments of the Eötvöstype EoMath1890 (), which measure the gravitational acceleration of macroscopic objects.Thus Einstein’s weak equivalence principle is established in the Newtonian picture of the gravitational force which is exactly proportional to the inertial mass. The validity of the principle even for quantum mechanical particles is also proved using gravityinduced interference experiments CoOvWePRL1975 (); PeChChNature1999 ().
The traditional equivalence principle is fundamentally both classical and local, and it is interesting to enquire how it is to be understood in quantum mechanics. Principle of equivalence states that inertial mass is equal to gravitational mass; . Another statement is that when all sufficiently small test bodies fall freely, they acquire an equal acceleration independent of their mass or constituent in a gravitational field. Quantum mechanically, it Hobook1993 () can be said that ” The results of experiments in an external potential comprising just a sufficiently weak, homogeneous gravitational field, as determined by the wave function, are independent of the mass of the system”. The last statement is also known as weak equivalence principle of quantum mechanics (WEQ).
The possibility of violation of weak equivalence principle in quantum mechanics is discussed in a number of papers, for instance using neutrino mass oscillations in a gravitational potential GaPRD1988 (). The evidence of violation of WEQ can be shown both experimentally and theoretically. Experimental evidence exists in the interference phenomenon associated with the gravitational potential in neutron and atomic interferometry experiments CoOvWePRL1975 (); PeChChNature1999 (). Theoretically, for a particle bound in an external gravitational potential, it is seen that the radii, frequencies and binding energy depend on the mass of the bound particle GrRMP1979 (); GrAP1968 (); GrRMP1983 ().
The free fall of test particles in a uniform gravitational field in the case of quantum states with and without a classical analogue was reexamined by Viola and OnofrioViPRD1997 (). Another quantum mechanical approach of the violation of WEQ was given by Davies DaCQG2004 () for a quantum particle in a uniform gravitational field using a model quantum clock PeAJP1980 (). A quantum particle moving in a gravitational field can penetrate the classically forbidden region beyond the classical turning point of the gravitational potential and the tunnelling depth depends on the mass of the particle. So, there is a small massdependent quantum delay in the return time representing a violation of weak equivalence principle. This violation is found within a distance of roughly one de Broglie wavelength from the turning point of the classical trajectory.
In the recent past, Ali et al. AlCQG2006 () have shown an explicit mass dependence of the position probabilities for quantum particles projected upwards against gravity around both the classical turning point and the point of initial projection using Gaussian wave packet. They also have shown an explicit mass dependence of the mean arrival time at an arbitrary detector location for a Gaussian wave packet under free fall. Thus both the position probabilities and the mean arrival time show the violation of WEQ for quantum particles represented by Gaussian wave packets. In the present paper, using a class of nonGaussian wave packets involving a tunable parameter, we shall compute position detection probabilities around the classical turning point and the point of initial projection and also compute the mean arrival time for atomic and molecular mass particles represented by nonGaussian wave packets to observe stronger violation of WEQ for quantum particles. To compute the mean arrival time, we consider the case when the particles are dropped from a height with zero initial velocity.
Violation of WEQ arises as a consequence of the spread of wave packets the magnitude of which itself depends on the mass. In order to illustrate this effect we present an analysis in terms of Bohmian trajectories. In the Bohmian model BoPR1952 () each individual particle is assumed to have a definite position, irrespective of any measurement. The premeasured value of position is revealed when an actual measurement is done. Over an ensemble of particles having the same wave function , these ontological positions are distributed according to the probability density where the wave function evolves with time according to the Schrödinger equation and the equation of motion of any individual particle is determined by the guidance equation , where is the Bohmian velocity of the particle and is the probability current density. Solving the guidance equation one gets the trajectory of the particle. In the present work we describe the evolution of Bohmian trajectories in terms of the mass and the tunable parameter of the nonGaussian wave packet.
Ii Position Detection Probability
Let us consider an ensemble of quantum particles projected upwards against gravity with a given initial mean position and mean velocity. Our motivation is to look for greater violation of the equivalence principle compared to that achieved by a Gaussian wave packet. Hence, we formulate a onedimensional nonGaussian wave function which represents the initial state of each particle, given by
(1) 
Here, is the tuneable parameter varying from to . The normalisation factor is given by
(2) 
The salient features corresponding to the above wave packet are its asymmetry, its infinite tail, and its reduciblity to a Gaussian wave packet upon a continuous decrement of the parameter to zero. The property of an infinite tail of the above wave function is not associated with nonGaussian forms that are generated by truncating the Gaussian distribution. The asymmetry of the wave packet (due to the sine function) entails a difference between its mean and peak, as will be further evident in our following analysis.
The initial group velocity is given by , where is the mass of the particle and is the wave number. For , we get the Gaussian wave function. If and represent the initial wave functions for particles 1 and 2 in the Schrödinger picture, then the mean initial conditions are
(3) 
where and represent the expectation values for position and momentum operators respectively. Here, we have restricted ourselves to a onedimensional representation along the vertical direction for simplicity. With the propagator
(4) 
of a particle in the linear gravitational potential and with the relation
(5) 
one can get the Schrödinger time evolved wave function at any subsequent time as
(6)  
where, and . The probability density is given by
(7) 
where
and
(8) 
is the spreading of the Gaussian wave packet. The probability density is explicitly mass dependent. Here, the spreading of the wave packet is given by
(9) 
where,
(10) 
which is also mass dependent.
Fig. 1 gives the plot of vs .
From this figure, we can see that for large masses, becomes almost constant with . For a particular value of , decreases with increasing value of mass .
The expression for expectation value of z is given by
(11) 
Ehrenfest’s theorem states that expectation value of quantum mechanical operators obey classical laws of motion, the instances of which are given below
(12) 
It is possible to check that Ehrenfest’s theorem is satisfied for our nonGaussian wave packet. In the Gaussian limit (i.e., ), and the dynamics of is like a classical point particle, as expected.
In Eq. (11) the first term is dependent. There is no in this expression. So, is dependent but mass independent. With the solutions obtained from the equation , we can get the value of for which is maximum. But the equation can’t be solved analytically. So, we have computed numerically. depends both on and mass. In Table 1, it is shown numerically how and vary with , for . In Table 2, it is shown numerically how those two vary with mass. It is clear that and increases with and mass but increases with , and remains constant for all masses.
Fig. 2 shows the variation of and with time simultaneously. Dotted curve shows the motion of whereas the continuous curve shows the motion of . As we take cm, the dependence of and becomes vanishingly small. So, this dependence can not be shown graphically. As a result, the curves for and coincide totally in this graph.
in cm  in cm  

0  38.59999999999815  38.599999999999910 
0.1  38.61540633397262  38.611498366598200 
0.2  38.62925626009014  38.622755653868810 
0.3  38.64090000000000  38.633500000000000 
0.4  38.65026513575232  38.643679695976970 
0.5  38.65786585282045  38.652999876917650 
0.6  38.66400000000000  38.661400000000000 
0.7  38.66910000000000  38.668800000000000 
0.8  38.67337906376510  38.675246198243485 
0.9  38.67696171641449  38.680689435211890 
1.0  38.68002216630916  38.685197660661515 
Mass  

in a.m.u.  in cm  in cm 
30  38.657865898827980  38.65299987691765 
60  38.657865903136155  38.65299987691765 
90  38.657865903934166  38.65299987691765 
120  38.657865904213070  38.65299987691765 
150  38.657865904342295  38.65299987691765 
Now, we want to observe the expression for in the classical limit. If a full classical description is to emerge from quantum mechanics, we must be able to describe quantum systems in phasespace. The Wigner distribution function WiPR1932 (); FoCoAJP2002 () which is one of the phasespace distributions, can provide a reexpression of quantum mechanics in terms of classical concepts.
The Wigner distribution function which is calculated from the time evolved wave function , is given by
(13) 
and the marginals of which yield the correct quantum probabilities for position and momentum separately.
By substituting the value of and using Eq.(5) we obtain the Wigner distribution function for nonGaussian wave packet. It satisfies the classical Liouville’s equation DaSeCS2002 () given by
(14) 
Here, and are obtained from Hamilton’s equation of motion given by
where, is the Hamiltonian of the system.
The corresponding position distribution function is given by
(16) 
So, the expression for the average value of using Wigner distribution function is given by
(17)  
which is exactly equal to the quantum mechanical expectation value of .
The particles are projected upwards against gravity. Cassicaly, the refrence point reaches the maximum height at the time . So, the probability of finding the particles within a very narrow detector region around is given by
(18) 
At a later time , the classical peak of the wave packet returns to it’s initial projection point at . Similarly, the probability of finding the particles within a very narrow detector region around is given by
(19) 
From Fig. 3 and Fig. 4, it is clear that both the probabilities are mass dependent for an initial nonGaussian position distribution for smaller masses. This effect of mass dependence of the probabilities occur essentially as the spreading of the wave packet under linear gravitational potential depends on the mass of the particles. These probabilities become saturated in the limit of large mass. As increases, both the probabilities decrease for a particular value of . The departure from the result for the Gaussian () case increases with the value of the nonGaussian parameter, signifying a stronger violation of the weak equivalence principle.
Iii Mean Arrival Time
Now, Let us consider the quantum particles subjected to free fall from the initial position at under gravity with the initial state given by Eq.(1) and with . Using the probability current approach prob () the mean arrival time of the quantum particles to reach a detector location at is given by
(20) 
where, is the quantum probability current density. However, we emphasize that the definition of the mean arrival time used in the above equation is not a uniquely derivable result within standard quantum mechanics. It should also be noted that can be negative, so we take the modulus sign in order to use the above definition. From Eq.(20), it can be seen that the integral of the denominator converges whereas the integral of the numerator diverges formally. To avoid this problem, several techniques have been employed DaEgMuquant2001 (). In this paper, to obtain convergent results of the numerator we use the simple technique of getting a cutoff at in the upper limit of the time integral with . is the width of the wave packet at time . Thus, our calculations of the arrival time are valid upto the level of spread in the wave function.
Here, the expression for the Schrödinger probability current density with the initial nonGaussian position distribution is given by
(21) 
where
and where the ’s are defined below Eq.(7).
It is clear that is explicitly mass dependent. So, is also mass dependent. From Fig. 5, we can observe that for smaller masses is mass dependent and for larger masses it becomes mass independent. For a particular value of mass, increases with .
From Fig. 6, we can see that for large masses, becomes almost constant with . For a particular value of , decreases with increasing value of mass .
It is clear that the massdependence of the mean arrival time gradually vanishes as the mass is increased. Thus, compatibility with the equivalence principle emerges in the limit of large mass, or classical limit. Note that with the increase of the parameter , one needs to go to higher values of mass in order for the massdependence to become insignificant. Thus, a nonGaussian wave packet can exhibit nonclassical features in a mass range where its Gaussian counterpart behaves classically.
Iv Bohm Trajectory Approach
Within the Bohmian interpretation of quantum mechanics BoPR1952 () each individual particle is assumed to have a definite position, irrespective of any measurement. The equation of motion of the particle is,
(22) 
where is called ’quantum potential energy’ and it is explicitly massdependent and determines the path of the particle. In the gravitational potential the quantum version of Newton’s second law is given by
(23) 
To quote from Holland Hobook1993 (), ”It is immediately clear that the assumption of the equality of the inertial and passive gravitational masses does not imply that all bodies fall with an equal acceleration in a given gravitational field, due to the intervention of the massdependent quantum force term.” Thus, WEQ is violated. The arrivaltime problem is unambiguously solved in the Bohmian mechanics, where for an arbitrary scattering potential , one finds Lebook2002 () that for those particles that actually reach , the arrivaltime distribution is given by the modulus of the probability current density, i.e., .
The quantum potential is nonlocal, and thus it is expected that its effect on the violation of WEQ is manifested in a way similar to the violation observed as a consequence of the spread of the wave packet. We compute a set of Bohmian trajectories corresponding to free fall of our nonGaussian wave packet with the tunable parameter . Fig. 7 shows a selection of Bohmian trajectories exhibiting their spread depending upon the values of and . Note that for small mass, the trajectories with initial positions on the left and right of the center of the wave packet (mean ) spread out with evolution, indicating violation of WEQ. The magnitude of spread increases with , signifying stronger violation of WEQ with increased departure from Gaussianity. The spread of the trajectories decreases as mass is increased, leading to the emergence of WEQ in the classical limit.
V Conclusions
To summarize, in this work we have studied the violation of the gravitational weak equivalence principle in quantum mechanics using a nonGaussian wave packet. The wave packet is constructed in such a way, that its departure from Gaussianity is represented by a continuous and tunable parameter . The asymmetry of the wave packet entails a difference between its mean and peak, and causes the peak to evolve differently from a classical point particle, whereas the mean evolves like a classical point particle modulo a constant depending upon the parameter . Such a result is consistent with the Ehrenfest therorem, as we have shown, and reconfirmed using the Wigner distribution. The correspondence with the results following from a Gaussian wave packet is achieved in the limit of .
The weak equivalence principle is shown to be violated first through the
dependence on
mass of the position detection probabilities of freely falling particles.
The magnitude of violation increases with the increase of the nonGaussian
parameter , signifying stronger viotation of WEQ with larger
deviation from Gaussianity. We next compute the arrival time of freely
falling wave packets through the probability current distribution
corresponding to the nonGaussian wave packet. The violation of WEQ is
again exhibited with the dependence on mass of the mean arrival time.
It is observed that in the limit of large mass the classical value for the
mean arrival time is approached, thereby indicating the emergence of the
WEQ in the classical limit. In this
context it is worthwhile to mention that though our work follows as a
natural consequence of quantum mechanics or quantum theory in the
nonrelativistic limit, it does not imply a violation of general covariance
in the relativistic domain. An interesting connection between the
nonrelativistic limit of quantum theory and the principle of equivalence
has recently been discussed PaPaPRD2011 (). Finally, a computation of Bohmian trajectories further
establishes the stronger violation of WEQ by a nonGaussian wave packet.
We conclude by noting that realization of exactly Gaussian wave packets is
rather difficult in real experiments. One of the consequences of our results
is to enable relaxation of the wave packet to be Gaussian, and therefore it
can facilitate the experimental observation of the violation of WEQ, as
well as enable a quantitative verification of the way the violation of WEQ
depends on the departure from the Gaussian nature of the wave packet. Other
ramifications of this result call for further study.
Acknowledgements: PC, DH, ASM and SS acknowledge support from DST India Project no. SR/S2/PU16/2007. MRM and SVM acknowledge finantial support of the University of Qom.
References
 (1) Galilei G 1638 Discorsi Intorno a due Nuove Scienze(Leiden: Elsevier) (English translation edited by H Crew and A de Salvio (New York: Macmillan 1914) pp 2123)

(2)
Eötvös R V Math. Nat. Ber. Ungarn. 8 65 (1890);
R. V. Eötvös, V. Pekar, and E. Fekete, Ann. Phys. (N.Y.) 68 11 (1922).  (3) Colella R, Overhauser A W and Werner S A Phys. Rev. Lett. 34 1472 (1975).
 (4) Peters A, Chung K Y and Chu S, Nature 400 849 (1999).
 (5) Holland P 1993 The Quantum Theory of Motion (London: Cambridge University Press) pp 25966.

(6)
Gasperini M, Phys. Rev. D 38 2635 (1988);
Krauss L M and Tremaine S Phys. Rev. Lett. 60 176 (1988);
Halprin A and Leung C N Phys. Rev. D 53 5365 (1996);
Mureika J R Phys. Rev. D 56 2408 (1997);
Gago A M, Nunokawa H and Funchal R Z Phys. Rev. Lett. 84 4035 (2000);
Adunas G Z, Milla E R and Ahluwalia D V Gen. Rel. Grav. 33 183 (2001).  (7) Greenberger D M and Overhauser A W 1979 Rev. Mod. Phys. 51 43
 (8) Greenberger D M 1968 Ann. Phys. 47 116
 (9) Greenberger D M 1983 Rev. Mod. Phys. 55 875
 (10) Viola L and Onofrio R 1997 Phys. Rev. D 55 455
 (11) Davies P C W 2004 Class. Quantum Grav. 21 2761
 (12) Peres A 1980 Am. J. Phys. 48 552
 (13) Ali M A, Majumdar A S, Home D and Pan A K (2006) Class. Quantum Grav. 23 6493

(14)
Bohm D 1952 Phys. Rev. 85 166;
Bohm D 1952 Phys. Rev. 85 180;
Bohm D 1953 Phys. Rev. 89 458  (15) Wigner E 1932 Phys. Rev. 40 749

(16)
Ford G W and O’Connell R F 2002 Am. J. Phys. 70 319;
Robinett R W, Doncheski M A and Bassett L C 2005 Preprint quantph/0502097  (17) Das S K and Sengupta S 2002 Current Science 83 1234
 (18) Dumont R S and Marchioro T L II 1993 Phys. Rev. A 47 85; Leavens C R 1993 Phys Lett A 178 27

(19)
Damborenea J A, Egusquiza I L and Muga J G 2001 Preprint quantph/0109151;
Hahne G E 2003 J. Phys. A: Math. Gen. 36 7149 
(20)
Leavens C R 2002 Bohm trajectory approach to timing electrons Time in Quantum Mechanics ed J G Muga, R Sala and I L Egusquiza (Berlin: Springer);
Leavens C R 1993 Phys. Lett. A 178 27;
Leavens C R 1996 The âtunnelingtime problemâ for electrons Bohmian Mechanics and Quantum Theory: An Appraisal ed J Cushing, A Fine and S Goldstein (Dordrecht: Kluwer Academic) p 111;
Leavens C R 1998 Phys. Rev. A 58 840  (21) Padmanabhan H and Padmanabhan T 2011 Phys. Rev. D 84 085018