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- I have some questions about spontaneous emission and if it is purely due to quantum fluctuations or if something called radiation reaction is involved as I encountered in some literature.

I am reading this chapter 3 from the book called The Quantum Vacuum by P.Milonni.(Attached in the pdf, look at chapter 3.2 Spontaneous emission)

There they say that spontaneous emission is due to both quantum fluctuations and radiation reaction. They say the transitions induced by the quantum fluctuations gives only the half of the Einstein A coefficient for spontaneous emission and in order to account for the full we need to consider radiation reaction.

„Consider now the effect of radiation reaction. As shown in Appendix A, the radiation reaction field is responsible for the rate 2 e 2 a 2 / 3 c 3 at which an oscillating charge loses energy to the electromagnetic field. For oscillation at frequency

What exactly do they mean with radiation reaction here? I mean the electron is bound to the nucleus and the position probability is calculated by the Schrödinger wavefunction, so I have difficulty understanding what is exactly oscillating. Of course the Schrödinger orbital resembles the wavefunticons for three dimensional oscillations but in the quantum case they represent a probability function where we can encounter the electron instead.

So what are they talking about for oscillation? From my understanding it seems as if they mean that the electron moving around in the orbital will have a oscillating motion and that when the charge accelerates in this oscillating motion it will according to classical electromagnetism emit electromagnetic radiation, but then my understanding from what they write, the half A coefficient due to radiation reaction would be emitting continuously all the time, which we know doesn’t happen, otherwise we are back at the level of why an electron orbiting the nucleus doesn’t emit radiation while spiraling down to the nucleus, just that it somehow just emitted half of the energy and the rest was due to the familiar quantum jumps.

So since we know the emission of photons from atoms are an all or none process, I personally can’t put the cause of spontaneous emission on anything else than the quantum fluctuations, and that a quantum fluctuation of the right frequency would induce a spontaneous emission to occur. Why does it in their calculating seem that the quantum fluctuations just is responsible for half of the emission rate?

They mention that their derivation is semiclassical can this be the cause for the inconsistencies in the reasoning?

In another paper by B. Fain Spontaneous Emission vs. Vacuum Fluctuations.(I also attached this paper) In this paper they conclude the following:

„Thus we come to the quite obvious conclusion that, in order to achieve noncontroversial results for the spontaneous emission, we must treat both the matter and the field according to the quantum theory. Another conclusion which stems from the example of the harmonic oscillator interacting with the radiation is that nonspontuneous emission is purely a quantum effect owing to the vacuum fluctuations, but the absence o/ this emission in the grou~,d state is purely a quantum el]cot which is due to the vacuum fluctuations.“

„From the above several conclusions follow.

1) Spontaneous emission exists both in classical and in quantum theories.

2) The semi-classical approach, when matter is treated quantum-me- chanically and radiation in the classical way, does not give correct results at lowlying levels of matter.

3) The crucial role of the vacuum fluctuations emerges in the ground state of matter. The stability o/ the ground state (i.e. the fact that it does not radiate) is purely a quantum eJ]evtwhivh is due to the vacuum ]luvtuations. „

They don't mention anything about radiation reaction.

So to summarize what I wonder: Is the correct answer that the only cause of spontaneous emission are quantum fluctuations?

In the chapter by Milonni why do they talk about radiation reaction and what do they actually mean with it, what does it refer to? If the radiation reaction idea is false why do they only come to half the A coefficient for quantum fluctuations? Is it because they treat it somewhat semiclassically? As they themselves say „On the basis of this simplistic semiclassical analysis, therefore, we have arrived at the conclusion that..“. And does this have to do with something they say in the other paper by B.Fain that one need to treat both the radiation and the matter quantum mechanically otherwise it will result in wrong results especially for the lower energy levels?

Another thing that makes me think Milonni reasoning is wrong comes from what I read about Cavity QED, which literally modifies the emission properties of atoms by the modification of the quantum vacuum. And in most literature I have read about Cavity QED I haven’t seen radiation reaction being mentioned.

There they say that spontaneous emission is due to both quantum fluctuations and radiation reaction. They say the transitions induced by the quantum fluctuations gives only the half of the Einstein A coefficient for spontaneous emission and in order to account for the full we need to consider radiation reaction.

„Consider now the effect of radiation reaction. As shown in Appendix A, the radiation reaction field is responsible for the rate 2 e 2 a 2 / 3 c 3 at which an oscillating charge loses energy to the electromagnetic field. For oscillation at frequency

*ωσ*with amplitude*x0*this rate is…“What exactly do they mean with radiation reaction here? I mean the electron is bound to the nucleus and the position probability is calculated by the Schrödinger wavefunction, so I have difficulty understanding what is exactly oscillating. Of course the Schrödinger orbital resembles the wavefunticons for three dimensional oscillations but in the quantum case they represent a probability function where we can encounter the electron instead.

So what are they talking about for oscillation? From my understanding it seems as if they mean that the electron moving around in the orbital will have a oscillating motion and that when the charge accelerates in this oscillating motion it will according to classical electromagnetism emit electromagnetic radiation, but then my understanding from what they write, the half A coefficient due to radiation reaction would be emitting continuously all the time, which we know doesn’t happen, otherwise we are back at the level of why an electron orbiting the nucleus doesn’t emit radiation while spiraling down to the nucleus, just that it somehow just emitted half of the energy and the rest was due to the familiar quantum jumps.

So since we know the emission of photons from atoms are an all or none process, I personally can’t put the cause of spontaneous emission on anything else than the quantum fluctuations, and that a quantum fluctuation of the right frequency would induce a spontaneous emission to occur. Why does it in their calculating seem that the quantum fluctuations just is responsible for half of the emission rate?

They mention that their derivation is semiclassical can this be the cause for the inconsistencies in the reasoning?

In another paper by B. Fain Spontaneous Emission vs. Vacuum Fluctuations.(I also attached this paper) In this paper they conclude the following:

„Thus we come to the quite obvious conclusion that, in order to achieve noncontroversial results for the spontaneous emission, we must treat both the matter and the field according to the quantum theory. Another conclusion which stems from the example of the harmonic oscillator interacting with the radiation is that nonspontuneous emission is purely a quantum effect owing to the vacuum fluctuations, but the absence o/ this emission in the grou~,d state is purely a quantum el]cot which is due to the vacuum fluctuations.“

„From the above several conclusions follow.

1) Spontaneous emission exists both in classical and in quantum theories.

2) The semi-classical approach, when matter is treated quantum-me- chanically and radiation in the classical way, does not give correct results at lowlying levels of matter.

3) The crucial role of the vacuum fluctuations emerges in the ground state of matter. The stability o/ the ground state (i.e. the fact that it does not radiate) is purely a quantum eJ]evtwhivh is due to the vacuum ]luvtuations. „

They don't mention anything about radiation reaction.

So to summarize what I wonder: Is the correct answer that the only cause of spontaneous emission are quantum fluctuations?

In the chapter by Milonni why do they talk about radiation reaction and what do they actually mean with it, what does it refer to? If the radiation reaction idea is false why do they only come to half the A coefficient for quantum fluctuations? Is it because they treat it somewhat semiclassically? As they themselves say „On the basis of this simplistic semiclassical analysis, therefore, we have arrived at the conclusion that..“. And does this have to do with something they say in the other paper by B.Fain that one need to treat both the radiation and the matter quantum mechanically otherwise it will result in wrong results especially for the lower energy levels?

Another thing that makes me think Milonni reasoning is wrong comes from what I read about Cavity QED, which literally modifies the emission properties of atoms by the modification of the quantum vacuum. And in most literature I have read about Cavity QED I haven’t seen radiation reaction being mentioned.