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The Importance of Gravitational Force in Coherence

In the HOM interference (Section A), two photons will bunch together if they arrive at the beam splitter simultaneously. Its underlying mechanism is discussed in this section. The mechanism also suggests the importance of gravitational force in coherence that leads to quantum entanglement and Bose–Einstein condensation.

Binding Energy

The interaction between two photons may create a "binding energy" which refers to the released energy when two or more particles are combined into a larger system. For example, the combination of an electron and a proton into a hydrogen atom will release 13.6 eV of energy. To separate the two, 13.6 eV or more must be supplied, usually by a photon. The rocket-Earth interaction also creates a binding energy. In order for the rocket to go to space and completely free from the binding of the Earth, energy above the binding energy must be added, usually by the rocket fuel. The combination of an electron and a proton is mediated by the electromagnetic (EM) force between them, whereas the binding between the rocket and the Earth is mediated by the gravitational (GR) force. Then, how do photons bunch together?

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Figure 37f. The interference of two waves. When in phase, the two lower waves produce constructive interference, resulting in a wave of larger amplitude. When 180° out of phase, they cause destructive interference.
[Source: Wikipedia]

"The energy of a wave is proportional to the square of its amplitude." This statement applies to EM waves (photons) and various other waves (Lumenlearning; Stack Exchange), including GR waves (Kokkotas, 2002, eq. 2.25). If two identical waves are generated at the same place simultaneously, their superposition will produce constructive interference, resulting in a wave of larger amplitude. This appears to violate the law of energy conservation.

Consider two EM waves (photons) with the same frequency, amplitude, and velocity: wave 1 and wave 2. Let E1 = the energy of wave 1; E2 = the energy of wave 2. Because these two waves have the same amplitude, E1 = E2. If they do not interact, their total energy ET should be equal to 2E1. Constructive interference doubles the amplitude, leading to a superposed wave with energy of 4E1. According to the law of energy conservation, the total energy of a closed system cannot be changed. Then, where does the extra energy in the superposed system come from?

In fact, the total energy of the system has not changed. The total energy must include the potential energy created by the interaction between two waves, denoted by Ep. If Ep is negative, it means that the two subsystems are bound together. Its absolute value is equal to the binding energy. Take the hydrogen atom as an example, an electron and a proton can be bound in the atom because their electric interaction produces negative potential energy (equals to -13.6 eV). Hence, the total energy of the superposed wave should be

ET = 4E1 + Ep

Due to the conservation of energy, the total energy of the superposed wave must equal to the total energy of separated waves, that is, 2E1. Therefore,

Ep = - 2E1

This explains why constructive interference will create negative potential energy, causing two waves to bind together.

Importance of Gravitational Force

The gravitational force has been shown to induce quantum entanglement and coherence between massive particles (Marletto and Vedral, 2017; Krisnanda et al., 2019; Ghoshal et al., 2020). Photons are massless. However, a photon has an energy of hν, where h is the Plank's constant and ν is its frequency. Through Einstein's famous equation, E = mc2, the photon also has an effective mass which can exhibit gravitational interaction. Such GR force is very weak, unable to induce the entanglement of a large number of photons. To date, less than 20-photon entanglement has been observed (Wang et al, 2016; Zhong et al., 2018).

As discussed earlier, the ions passing through ion channels may emit both EM and GR waves. GR waves are the carrier of gravitational force. The attractive force between GR waves, or between GR and EM waves, is strong. It has been estimated that the GR-GR attractive force is more than 80 orders of magnitude stronger than the EM-EM attraction (Faraoni and Dumse, 1998, Eq. 6.1). Therefore, the GR waves radiated from the ions passing through ion channels could play an important role in quantum entanglement (Section B) as well as Bose-Einstein condensation (Section E).

 

Author: Frank Lee
Last updated: 2020-09-25