What determines the time between atomic absorption and emission of photons?

[dsaun777]

What determines the time between atomic absorption and emission of photons? Is there a correlation to blackbody radiation?

 

[Orodruin]

Electromagnetic interaction strength, wave function overlap between the initial and final states and the interaction operator, and available phase space.

No. It is not related to blackbody mechanics.

 

[dsaun777]

Since physics is the business of making predictions, what is our ability to predict the time interval using all of these mathematical tools?

 

[Vanadium 50]

what is our ability to predict the time interval using all of these mathematical tools?

Pretty good.

 

[Orodruin]

Pretty good.

Pretty damn good!

 

[dsaun777]

Pretty damn good!

Can I see some general equations or references to sources?

 

[Orodruin]

This should be discussed in most intermediate level textbooks on quantum mechanics.

 

[Delta2]

Maybe @vanhees71 would have something to share with us on this topic. He has a vast knowledge and likes to write posts that are mini treatises on various subjects.

 

[vanhees71]

We don't need to speculate about this. My atomic colleagues recently measured it:

https://www.goethe-university-frank..._time_it_takes_for_an_electron_to_be_released

 

[Delta2]

We don't need to speculate about this. My atomic colleagues recently measured it:

https://www.goethe-university-frank..._time_it_takes_for_an_electron_to_be_released

I know I might have a bad name as a speculator but I didn't speculate here all I asked is if you can give us your lights, regarding the main theory points around this problem in a post/mini treatise of yours which I know you like to write.

 

[Orodruin]

We don't need to speculate about this. My atomic colleagues recently measured it:

https://www.goethe-university-frank..._time_it_takes_for_an_electron_to_be_released

This seems to be about measuring the time of release of an electron in the photoelectric effect. If I understand the OP correctly, the question is about the time between an atom absorbing a photon (thus becoming excited) and relaxing to the ground state again through photon emission.

 

[DrClaude]

What determines the time between atomic absorption and emission of photons?

For one particular event, it is completely random.

Since physics is the business of making predictions, what is our ability to predict the time interval using all of these mathematical tools?

The theory allows us to know what the times will be on average, but not for a single event. Such is the nature of quantum mechanics.

 

[Orodruin]

For one particular event, it is completely random.

Just to qualify this a bit because people tend to take statements like this as ”we have no clue” even when followed by more precise statements:

Even something ”completely random” has a probability distribution, ie, a function that tells you how likely each decay time is in this case. Even if any positive time could result for a single decay, each time is not equally likely. We can model the distribution very well and that includes, as mentioned, descriptive information about it such as the mean and standard deviation.

 

[Twigg]

What determines the time between atomic absorption and emission of photons? Is there a correlation to blackbody radiation?

To give a more technical answer, the lifetime of spontaneous emission is determined by the vacuum state E-field and the transition multipole moment of the atomic states.

The simplest example is a dipole-allowed transition in a two-level atom. In hydrogen, the lifetime of spontaneous emission is determined by the transition dipole moments $\langle nlm| -er |n' l' m' \rangle$, where r is the radial coordinate of the electron, n, l and m are the usual quantum numbers for the ground state and n', l', and m' are the usual quantum numbers for the excited state (of course, you have to add spin quantum numbers and LS coupling to get the full picture, but it's the same idea). Here's some notes outlining a derivation of the lifetime from the vacuum field and dipole matrix from Fermi's golden rule. There is a more verbose derivation in chapter 2 or 3 of Metcalf's book.

 

[vanhees71]

That's of course also an interesting question, but I'm not aware whether there's any time-resolved measurement of a spontaneous atomic transition. It's also not clear to me, in which way this "transition time" is to be defined, let alone how it can be meausured, because the interpretation of transient states of interacting relativistic quantum fields is not defined. What's usually predicted are transition-probability rates between asymptotic free states. In this example the asymptotic free initial state is an "excited atom and no photons" and the final state is "unexcited atom and one photon". What's shown in the quoted calculation using Fermi's golden rule (1st-order perturbation theory) is the mean life time of the excited state (which is just the inverse width of the corresponding spectral line), and here the question what happens in the "transient" is not addressed at all.

There is of course also a relation with black-body radiation. Black-body radiation is thermal radiation, i.e., you have, e.g., a cavity with walls held at a constant temperature. Due to the thermal motion of the charged constituents of the walls some electromagnetic radiation is present in the cavity, and if the cavity walls are held on constant temperature for some time, this radiation is in thermal equilibrium with the walls. Microscopically you have all the time spontaneous and induced emission as well as absorption of photons from/by the walls, and equilibrium is reached when the rates of emission and absorption are the same and thus the mean energy of the radiation within the cavity does not change with time anymore. The fact that you get a Planck spectrum (i.e., a Bose-Einstein distribution of massless non-interacting bosons) for the thermal radiation is indeed due to the unitarity of the S-matrix, implying (weak) detailed balance between emission and absorption processes, which necessarily leads to Boltzmann's H-theorem and thus determines the equilibrium phasespace distribution. The spontaneous emission contribution here is specifically of quantum nature, leading to a Bose-Einstein rather than a Maxwell-Boltzmann (or Wien in the context of radiation) distribution. This "kinetic approach" to the black-body radiation law has been worked out by Einstein already in 1917, i.e., about 10 years before modern quantum theory and particularly relativistic quantum field theory and the quantization of the em. field was discovered.

 

[Cthugha]

That's of course also an interesting question, but I'm not aware whether there's any time-resolved measurement of a spontaneous atomic transition. It's also not clear to me, in which way this "transition time" is to be defined, let alone how it can be meausured, because the interpretation of transient states of interacting relativistic quantum fields is not defined. What's usually predicted are transition-probability rates between asymptotic free states. In this example the asymptotic free initial state is an "excited atom and no photons" and the final state is "unexcited atom and one photon". What's shown in the quoted calculation using Fermi's golden rule (1st-order perturbation theory) is the mean life time of the excited state (which is just the inverse width of the corresponding spectral line), and here the question what happens in the "transient" is not addressed at all.

I am a bit puzzled here because - in the ensemble average over many realizations - this is a standard experiment that is performed in a huge number of labs in chemistry and physics routinely. Leaving aside the simplest versions using time-correlated single photon counting, this is for example routinely realized in optical quantum computing. One uses a short pulsed light field resonant with the transition of interest and adjusts the power such that exactly performs half a Rabi cycle, so the atom will be found in the excited state with a probability of 100% percent. One may then, e.g., perform pump probe spectroscopy and investigate the transmission of a weak probe beam also resonant with that transition. If the probability to find the atom in the ground state is high, the transmission of the probe beam will be low as it can be absorbed. If the probability to find the atom in the excited state is high, the probe beam cannot be absorbed, so its transmission increases and the "transmission" of the probe beam may even be above one as it can induce stimulated emission. One may use femtosecond or picosecond pulses and deterministically vary the delay between pump and probe pulse using a mechanical delay line. As light is fast one can easily reach few femtosecond resolution this way. In that way, one can easily monitor the probabilities with which the atom will be found in the excited state or ground state on average. This is also how Rabi oscillations are typically investigated experimentally.

More thorough approaches such as 2D Fourier transform spectroscopy exist which yield direct access to the coherent polarization (the off-diagonal matrix elements in the light matter coupling Hamiltonian), but they are rather sophisticated.

 

[vanhees71]

Well, yes, but that's not resolving a transient state!

 

[Cthugha]

Why? You get the probability amplitudes for the system being in the excited/ground state for each and every delay tau after the excitation process. In principle one can get also the off-diagonal elements (typically called coherences within the optics-related communities). Which deeper description than the full density matrix of the atom (or possibly the atom field system if you want a closed system) would a theorist desire?
At high energies I can see that things are way more complicated, but for typical atomic transitions where non-relativistic approximations and standard QED work well, I am not sure what deeper characterization of a spontaneous decay process one might want.
Of course this is not a description of an individual decay process. In some limits one get something similar using weak measurements for individual decays as well, but this would be a huge technical challenge. However, it works, e.g., for spontaneous photon decay from a cavity. That can be resolved for an individual "emission" process.

 

[Delta2]

This is an unsolved mystery in QED right? I mean if there is some sort of transient process as the atom emits the photon and drops to the ground state, or if that happens instantaneously, i.e. atom switch states instantaneously WITHOUT transient process.

 

[Cthugha]

This is an unsolved mystery in QED right? I mean if there is some sort of transient process as the atom emits the photon and drops to the ground state, or if that happens instantaneously, i.e. atom switch states instantaneously WITHOUT transient process.

I would argue that if you accept that the quantum state is a fundamental description of what is going on, then the question is settled.

Classically, it is clear what you need to have radiation: the simple "accelerated charges radiate" is something everybody has heard of. In QM things are of course more complicated. We are talking about probability amplitudes for events here and that atomic states are nominally stable and electrons are not really revolving around the core of the atom in any classical manner. Otherwise atoms would not be stable as a whole.

However, an excited atomic state is fully stationary only in the absence of external fields. Any Stark shift by an external field mixes states. This is also true for the vacuum field. It acts as a perturbation and puts the atom into a superposition of the excited and the ground state. Now, there are still no accelerating electrons here, but in such a superposition state of two states with different parity at different energies the spatial distribution of the electron probability density varies with time as well. Here, the spatial change of the probability amplitude may mimic accelerated motion which in turn will couple to the probability amplitude for photon emission. Basically, you get an oscillating probability amplitude for finding the excitation in the atomic subsystem and for finding it in the field. It looks a bit like a typical Rabi oscillation.

A nice experiment where these effects were demonstrated is PRL 108, 093602 (2012) ArXiv link here

They took single photons from the resonance fluorescence of a quantum dot and put them into a Michelson interferometer. They still saw interference when changing the difference between the two arms to a delay of more than 2 nanoseconds which means that the single photon is still delocalized in a sense in the cm-range. While one can get esoteric in explaining this interference of the photon with itself, the commonly accepted explanation is quite simple. In the interferometer you have a long and a short path. The probability amplitudes for the photon being emitted at an earlier time and taking the longer path and the photon being emitted at a later time and taking the shorter path interfere. The system is in a superposition of having the excitation in the atomic subsystem and in the field subsystem as long a sit can be isolated from the environment. Any decohering interaction then destroys the superposition.

Of course the "mystery" how to get from the information known about the ensemble average to the single individual measurement result when the system is perturbed remains. But this is always the case in QM, even in less complicated scenarios. But the behaviour of the probability amplitudes is well-defined and non-instantaneous.

Edit: In the single-shot case, the best example of monitoring an emission process I know is Haroche's landmark Nature paper ArXiv Link - that is "just" photon emission from a cavity, but it is ingenious how they are able to observe this in real time in the single shot case via QND measurements.