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Repulsive Gravitational Effect of a Quantum Wave Packet Front. Phys. 10, 100401 (2015) DOI 10.1007/s11467-015-0478-9 RESEARCH ARTICLE Repulsive gravitational effect of a quantum wave packet and experimental scheme with superfluid helium Hongwei Xiong1,2 1Wilczek Quantum Center, Zhejiang University of Technology, Hangzhou 310023, China 2College of Science, Zhejiang University of Technology, Hangzhou 310023, China Corresponding author. E-mail: [email protected] Received April 21, 2015; accepted May 14, 2015 We consider the gravitational effect of quantum wave packets when quantum mechanics, gravity, and thermodynamics are simultaneously considered. Under the assumption of a thermodynamic origin of gravity, we propose a general equation to describe the gravitational effect of quantum wave packets. In the classical limit, this equation agrees with Newton’s law of gravitation. For quantum wave packets, however, it predicts a repulsive gravitational effect. We propose an experimental scheme using superfluid helium to test this repulsive gravitational effect. Our studies show that, with present technology such as superconducting gravimetry and cold atom interferometry, tests of the repulsive gravitational effect for superfluid helium are within experimental reach. Keywords gravitational effect of quantum wave packet, precision measurement, cold atoms PACS numb ers 04.60.Bc, 04.80.Cc, 05.70.-a itational effect for quantum wave packets. It is clear that, 1 Introduction without a well-defined solution to the quantum gravita- tional problem at the Planck length, this phenomenologi- Although the unification of quantum mechanics and gen- cal theory requires experimental testing. Fortunately, our eral relativity is elusive, considerable theoretical studies studies show that current techniques for measuring the to reveal possible macroscopic quantum gravitational ef- gravitational force such as superconducting gravimetry fect have been presented. A classic example is Hawking and cold atom interferometry are able to test repulsive radiation [1], predicted by combining general relativity, gravitational effects in superfluid helium. quantum mechanics, and thermodynamics. Although the The paper is organized as follows. In Section 2, we pro- quantum gravitational problem is far from solved, vari- vide an explanation of the attractive gravitational force ous approaches are being used to find evidence of quan- in Newton’s law of gravitation. Section 3 is devoted to the tum gravitational effects in high-energy scattering exper- consideration of the gravitational effects for a quantum iments and astronomical observations [2, 3]. Many novel wave packet when both quantum mechanics and thermo- ideas are also proposed to test the quantum gravitational dynamics are considered. We present a general equation effects, e.g., probing the Planck-scale physics with a me- for the gravitational force for a quantum wave packet. chanical oscillator [4], the search for dark energy using In Section 4, using the general equation presented in the atom interferometry [5], and the measurement of homo- third section, we propose an experimental scheme to test logical noise [6, 7]. At present, no evidence of quantum the repulsive gravitational effect for superfluid helium. A gravitational effects have been observed in experiments brief summary and discussion are presented in the final on Earth. section. In this study, we consider the gravitational effects for a quantum wave packet based on the thermodynamic origin of gravity [8]. By considering the fact that the di- 2 Thermodynamic understanding of the rection of force can be naturally interpreted in thermo- direction of gravitational force in Newton’s dynamics, we find some justification for the attractive law of gravitation characteristics of Newton’s law of gravitation. With the same considerations, we discuss a possible repulsive grav- It is well known that the law of gravity closely resem- c The Author(s) 2015. This article is published with open access at www.springer.com/11467 RESEARCH ARTICLE bles the laws of thermodynamics and hydrodynamics [9– the ordinary temperature for an ensemble of particles in 14]. This has led to intensive studies [15–26] into the thermal equilibrium. In this acceleration process, there thermodynamic origin of gravitation [8]. When the ther- is no entropy increase for the particle itself. In the ac- modynamics of a system are considered, the direction celeration process of the particle, the only possibility for of the force (e.g., pressure) can be determined from the entropy increase comes from the vacuum, if Eq. (1) is thermodynamic properties of the system. This leads to correct. Thus, TV refers to the vacuum temperature at a question about whether there is a physical mechanism the location of the particle. This shows that the accel- determining the direction of the gravitational force, if the eration of a particle will induce vacuum excitations, and thermodynamic origin of gravitation is assumed. In this thus, lead to finite vacuum temperature at the location section, we provide possible answers to this question. of the particle. Because there is a decay of these vac- Recently, the thermodynamic understanding of gravi- uum excitations in the propagation process, TV should tation has been significantly advanced by Verlinde’s work be a spatially dependent temperature field distribution [27], in which the change of entropy S after a displace- with a maximum value given by |a|/(2πkBc). As we do ment x is given by the following formula: not have exact knowledge at the Planck length, it is not mc within the scope of the present work to provide the ex- S =2πkB x. (1) act spatial scale of the decay. However, at least in the Here, m is the mass of a fundamental particle, c is the present work, the maximum value of TV is the physical speed of light, and kB is the Boltzmann constant. In quantity we need. the original work by Verlinde, this postulation (moti- It is noteworthy that the above equation is the same vated by Bekenstein’s work [9] regarding black holes and as the Unruh temperature [13]. This indicates the self- entropy) plays a key role in deriving Newton’s law of consistency of the above derivation. In Verlinde’s work, gravitation. The proportional relation between S and x both Eqs. (1) and (5) are used to obtain Newton’s law can be partially explained by an entropy increase with of gravitation. Here, we show the possibility of further information loss. For a particle’s motion in the vacuum simplification because Eq. (5) can be derived from Eq. background, when the particle arrives at a location after (1). a displacement x, the information regarding its path is The meaning of Eq. (5) is further explained by Fig. 1. lost, which leads to an entropy increase when the vacuum For a particle at location B with acceleration aB (shown background is also included. by the red arrow), the coupling between the particle and Although there is significant controversy regarding the the vacuum modes establishes the vacuum temperature meaning and validity of this thermodynamic formula, it field distribution (shown by the dashed blue line) with a deserves further study. Considering a particle with accel- peak value of TV (aB)=|aB|/(2πkBc). The strong cou- eration a,wehave pling between the particle and the vacuum modes leads 2 to a “dressed” state that includes the local vacuum ex- ajt xj = . (2) citations and the particle itself. If the particle has no 2 size, roughly speaking, the width of the local vacuum j , , Here, =1 2 3. In this paper, all bold symbols represent excitations is of the order of the Planck length (using vectors. From Eq. (1), we have the same treatment for the derivation of Newton’s law of πk mc 2 B 2 gravitation used in the latter part of this section). dS = (ajt) dt. (3) Eq. (5) shows the vacuum temperature field due to an j accelerating particle. It is natural to consider the oppo- In addition, from E = m2c4 + p2c2,usingthenonrel- site problem: What is the acceleration of a particle in ativistic approximation, we have the presence of a finite vacuum temperature field distri- bution? In Fig. 1, we assume that there is a vacuum dE = m a2tdt. (4) j temperature field distribution (shown by the red line) j that is due to a system denoted by A, e.g., a celestial Using the fundamental thermodynamic relation dE = body. At location B, there is a particle (denoted by a TV dS,wehave red sphere). To establish local thermal equilibrium, the 2 red sphere will accelerate such that the peak vacuum j aj |a| TV = = . (5) temperature of the dressed state is equal to the temper- 2πkBc a2 2πkBc j j ature of the vacuum temperature field at location B, i.e., TV (aB)=TV (B). In this situation, we have It is clear that this temperature TV is different from 100401-2 Hongwei Xiong, Front. Phys. 10, 100401 (2015) RESEARCH ARTICLE show that the above equation provides an explanation of why the gravitational force is attractive between two spatially separated objects. Now, we consider Newton’s law of gravitation under the assumption of the thermodynamic origin of gravita- tion. In Fig. 2, we consider a space with a Planck length 3 lp ≡ G/c , which shows the structure of the phase space due to the quantum gravitational effect. Here, G is the gravitational constant. Although the microscopic mechanisms of lp are not completely clear, it is not unrea- Fig. 1 The relation between the vacuum temperature field dis- sonable to assume the existence of lp in the phase space. tribution and acceleration. The red line shows the vacuum temper- ature field distribution for a system denoted by A, e.g., a celestial We assume that the energy of a particle (indicated by body. The red sphere represents an initially at rest classical par- a red sphere in Fig. 2) is ε. We further assume that the ticle at a location B.
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