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That's picking a pretty fine nit. Yes, obviously if an atom undergoes an energy transition then that energy has to come from (or go to) somewhere. That doesn't change the fundamental point, which is that the process doesn't happen instantaneously or discontinuously, and the continuity is provided by transition through a series of superposition states. Yes, to be strictly correct you'd have to say that the superposition is a joint state of the atom and something else and not the atom in isolation. But this post was targeted to a lay audience, and that sort of detail would be more of a distraction than a clarification IMHO.

You are right for spontaneous emission, there we need to connect the system to something else, or to formulate a detailed-enough relativistic model with retarded EM fields that is able to lose energy (ordinary Hamiltonian of charged particle variables only cannot describe loss of energy to radiation).

But for spontaneous absorption/emission, you are not right. Describing atoms by psi function is extremely successful, it is what Schroedinger did in his papers, what got him the right results for positions and intensities of spectral lines and what put his equation to the center of interest. It is true this successful description has some problems, such as the question of consistency of different world cuts (into system and environment), but the ordinary way to do the split for atom-radiation interaction (time-independent atom Hamiltonian and time-dependent external EM field) describes stimulated emission and absorption very well.

Stimulated absorption/emission does require presence of external field, but this field can be taken into account in the usual psi function formulation - this is called semi-classical theory of radiation, it is what Schroedinger proposed in 1920's for simple atoms and is still heavily used for description of interaction of radiation with atoms and molecules.

You’re right, I was thinking of a time-independent Hamiltonian. I may be mistaken but I thought that there are no stationary states otherwise (so it wouldn’t make sense either to say that the state is a superposition of the states 0 and 1).

> You are right for spontaneous emission

> But for spontaneous absorption/emission, you are not right

Was one of those instances of "spontaneous" supposed to be "stimulated"?

Yes, the second one should have been "stimulated". Thanks for pointing that out.

Not if we're really living in a simulation: e.g. if you initiate the state change of an atom ( the example was the addition of one quanta of energy ), then eventually the state will change from quanta 0 to quanta 1, with possibly many superpositions of both states during the described finite lifetime of the transition.

This, to me, sounds so much like eventual consistency ( for observers ), that it's a bit scary.

The whole business of superposition and measurement so reeks of lazy initialization that you might want to worry about what performance constraints the simulation is trying to beat too.

"An atom can be described by its quantum state only if it's isolated and in that case its energy is constant."

How do you figure? As a contradiction, take your atom+electromagnetic field system, describe the transition from excited atom to unexcited atom + photon state, and project out the E&M field. Voila, now you have a quantum description of an atom transitioning between different energy states. Its dynamics may look funny, i.e. they may appear nonlocal, they may not conserve energy, etc. but that's different from saying "there is not a quantum description of these dynamics" which is what you're claiming.

https://en.m.wikipedia.org/wiki/Quantum_state

> project out the E&M field. Voila, now you have a quantum description

There is no wavefunction describing the state of the subsystem because the system is not separable.

> Its dynamics may look funny, i.e. they may appear nonlocal, they may not conserve energy, etc. but that's different from saying "there is not a quantum description of these dynamics" which is what you're claiming.

What is the “quantum description of these dynamics”?

Quantum mechanics is usually based on something like the following postulate: “The state of a physical system is described by a well-behaved function of the coordinates and time, Ψ(q, t). The function contains all the information that can be known about the system.”

Not sure exactly what you're linking the Wikipedia article on quantum states, but if you like it as a reference, check out https://en.wikipedia.org/wiki/Quantum_state#Mixed_states. The operation of "projecting out" the E&M field in my previous comment would be realized on the density matrix as a partial trace over the electromagnetic field. You can also take the partial trace of this field of the Hamiltonian operator to see the effective dynamics of the atom when "ignoring" the E&M field. This is a fully quantum description of the state, so I stand by the statement that your claim "An atom can be described by its quantum state only if it's isolated and in that case its energy is constant." is incorrect.

Do you agree that “The state of a physical system is described by a well-behaved function of the coordinates and time, Ψ(q, t). The function contains all the information that can be known about the system.”?

When the atom is coupled with the electromagnetic field and the state of the system is not separable there is no complete description of the atom given by a wavefunction defining its quantum state. You can have an incomplete description by tracing out the rest of the system, I agree.

Let’s say then that "An atom can be completely described by its quantum state only if it's isolated and in that case its energy is constant."

Edit: in any case, my point was (and I think that we will agree) that it is misleading to say “Consider a system that transitions from energy state 0 to an adjacent energy state 1. [...] To go smoothly from 0 to 1, the system transitions through a series of superpositions of both states”.

The atom goes from the state 0 to the state 1 but during the transition it’s not described by a superposition of those states (that would be a pure state). If anything, it is described by an (improper) mixture of those states, obtained by tracing out the rest of the system.

Yes, but note that I deliberately used the word "system" rather than "atom". The system is the combination of the atom plus whatever it's absorbing energy from or emitting energy to. And that (entangled) system is in a superposition.

Ok, I was confused because if you say “Consider a system that transitions from energy state 0 to an adjacent energy state 1” it sounds as if the energy of the “system” is changing and when you say that “a particle [...] can be in two different energy states at the same time” it seems that you are referring to the atom being in a superposition of states with different energy.

One can speak meaningfully of "an atom in a superposition of energy states" despite the fact that, strictly speaking, such a thing is not possible, just as one can speak meaningfully of "the force of gravity" despite the fact that, strictly speaking, there is no such force. The latter is understood as the force-like effect of curved spacetime, and the former is understood as "an atom being a component of a system in a superposition of states with different distributions of energy" (or something like that). Communications between humans becomes more productive when we cut each other a little terminological slack.

The system is separable after you measure out your ancilla. Imagine a stream of qubits B_1, B_2, ... prepared in +X who interact weakly with qubit A prepared in +X. and are then strongly measured projectively. After the brief interaction, A and a certain B will be entangled. Projectively measuring this B updates our knowledge of A, but due to the weak entanglement, the adjustment is small and incremental. The state of such B qubit is known and thus there is no remaining entanglement between A and B, and thus A and B are separable. After many such Bs, an outside observer privy to all measurement outcomes will understand that A is doing an absorbing random walk to the poles of the Bloch sphere. Throwing away this information and taking an ensemble average, you'd see an exponential decay of coherence.

This is a toy model of a qubit being weakly measured by an incident flying field.

> Throwing away this information and taking an ensemble average, you'd see an exponential decay of coherence.

That is described by a density matrix but is not a pure state. The true state may be a pure state (because as you said the systems are separable after the measurement) but our description is a mixture reflecting our imperfect information (and not a superposition of the pure states corresponding to the potential outcomes).

What puzzles me is where the photon goes when it bumps up the energy of the electron, and what does it look like when the electron shell drops back down. I realize too much "realism" in QM is just a waste of effort, but I keep seeing the field sort of pinching off a photon.

The photon goes into the energy of the electron. Because mass and energy are the same thing the mass of the photon is converted into the energy of the electrons new state.

This is incorrect. See the stochastic schroedinger equation. A quantum system will stay pure under measurement as long as the measurement process is completely efficient.

I’m not familiar with this extension of QM. What is the “system” here? Would the atom stay in a pure state or would a larger system stay in a pure state?

The whole system does. The trick is you never trace anything out. You needn't. A spin interacting with a flying field is itself a pure system. It's just when you trace out the flying field that the system may become impure. If the spin interacts with the field, and the field is measured with measurements local to the field subsystem, the entire system will stay pure, and you'll see the qubit's wavefunction "collapse."

Thanks. In that case I don’t think there is a contradiction with what I said, if there is no wave function corresponding to the atom (alone). I understood the original comment as saying that the quantum state of the atom during the transition between energy levels was a superposition of the corresponding pure states (i.e. a pure state) and I was objecting to that.

Yeah I mean that’s still fine. The entanglement between system and field never persists since the field is taken to be measured immediately. The systems are only non-separable as long as they’re entangled. See the notes by Steck on the derivation of the SME and SSE.

I agree that if the field is taken to be a degree of freedom entangled with the system under measurement, things are funnier. This experiment isn’t really like spontaneous or stimulated emission... where yeah, DURING the process, the photon pooped out is entangled with your subsystem. In the case of the present experiment there’s a much more subtle thing going on with a three level system where the bright-zero manifold is used as a witness to the dark-zero manifold, where the presence or lack thereof of fast jumps in the bright-zero manifold betray information about the dark-zero manifold, necessarily.

What "is still fine"?

I've been reading a bit about quantum state diffusion, continuous measurements and quantum trajectories. My superficial understanding of the subject is the following. Let's say that you have an open system described by a reduced density operator evolving according to some master equation. The system will be in general described by a mixed state.

You can have an alternative representation with the quantum state changing in a non-deterministic way according to a stochastic equation. In this representation the quantum state remains pure for one "trajectory" but to describe the system you need to consider the ensemble of realizations. So you still have a mixture and the same density matrix as before.

I think it's a bit more subtle than this. Taking ensemble averages is itself a gesture of throwing information away, so yes, you do now have classical uncertainty that now may be represented by a density matrix. I think what I take issue with is "to describe the system." The master equations are deep down, kalman filters. They take some inputs and through bayes rule produce their best estimate of what state the system is in. Given perfect information, in this situation, they'll produce, somewhat miraculously, a complete description of the state of the system. If you had a person that was being bombarded by soccer balls from random directions you wouldn't say that an average of their muscle movements described the human-soccerball system. You'd wind up with something not particularly informative. Yes, you do need to take averages at some point... but you might want to average over a solid angle of impinging soccer balls... then you get somewhat deterministic, informative behavior.

I want to get across that this experiment is in some sense probing the way a quantum system processes quantum fluctuations. An ensemble average of traces therefore throws away/averages out the very thing that the system is responding to.

> Given perfect information, in this situation, they'll produce, somewhat miraculously, a complete description of the state of the system.

I don't think that the complete description of the system when the atom is going from a pure excited state to a pure ground state will include pure states of the atom which are superpositions of the excited state and the ground state. If we have a complete description of the system, the atom may be part of a larger system which is in a pure state and in that case the quantum state of the atom may be an improper mixture of the excited state and the ground state. The atom may also be in a pure state on its own, but then it will be either in the excited state or in the ground state.

That's all I said. I may be wrong but I fail to see in your comments a reason to think so.

Edit: By the way, I know an atom can in principle be in a state which is a superposition of states with different energies (I said so in my first comment). It's just that I don't think that happens during the spontaneous transition from one state to another. If it does happen, I would be glad to learn about it.

Edit2: Looking again at some quantum optics papers I see people argue that stochastic equations have a physical meaning and are not just a calculation device. Anyway, those interaction models are quite complex and full of approximations so it's not clear what "pure" means anymore...