I am reading the book "Decoherence and the Quantum-to-Classical Transition" by Maximilian A. Schlosshauer, and I have come to understand that for a two-level system with eigenstates $|a\rangle$, $|b\rangle$ for which the environmental respectively adopts eigenstates $|E_a\rangle$, $|E_b\rangle$ corresponding to the system states, the suppression of the coherent terms in the $a,b$ basis is proportional to $\langle E_a|E_b\rangle$. I also learned the general formula for how scattering of light environmental particles carries away information about a system, thereby reducing the environmental wavefunction overlap and decohering it. My question is: what about the kind of decoherence caused by photons either being absorbed or not being absorbed in a given molecule. For instance, if a molecule A absorbs at frequency $\omega$ in its ground state, and the excited molecule $A*$ absorbs at frequency $\omega_1$, it seems reasonable to me that irradiating the sample with $\omega$-frequency photons should suppress the ability of $A$ to exist in a coherent superposition of excited and ground states. However, I can't think of a mathematical theorem for what this decoherence would look like, and I can't find one in my book either. Does anyone know what theorem describes this specific type of decoherence?
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1$\begingroup$ Which book, author, chapter, page...? $\endgroup$– Tobias FünkeCommented Jan 29, 2023 at 9:08
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$\begingroup$ I added a reference to the specific book - I'm not referring to a specific page number, though! $\endgroup$– slithy_toveCommented Jan 29, 2023 at 14:56
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1$\begingroup$ The most detailed thing to do is explicitly write the interaction Hamiltonian between your system and all external systems, calculate the time evolution, then look at the reduced density operator of your system by "tracing out" the external degrees of freedom. This will work for any type of decoherence, including the one you have described with molecule-light interactions. $\endgroup$– Quantum MechanicCommented Feb 1, 2023 at 15:46
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$\begingroup$ Your premise is incorrect. If A can absorb $\omega$ to go to A* then by stimulated emission A* can go back to A in the presence of $\omega$ photon. $\endgroup$– Superfast JellyfishCommented Feb 5, 2023 at 6:30
2 Answers
Yes, shining light which is more/less likely to be absorbed by the excited vs unexcited states will certainly cause decoherence, as will any interaction which carries away information about the state of the system into the environment. As Schlosshauer discusses, this effect explains why it's so hard to keep atoms in coherent superpositions of energy eigenstates (see discussion in 2.8.2, although the assumption that $H_S >> H_{int}$ is not good in the case of laser light with enough energy to excite an atom).
Regarding a "mathematical theorem for what this decoherence would look like", the procedure would be very similar to that for photon scattering without absorbtion, but you'd have to use a Fock space to describe the system since the number of particles changes, and the S matrix would also encode information about photon creation/destruction. I recommend reading the section on Second Quantization in Sakurai (starts on page 461 of the second edition) or another good quantum mechanics text if you aren't familiar with Fock spaces. If a photon was definitely absorbed by an unexcited atom and definitely not absorbed by an excited atom, then the overlap between these states would be zero. A more realistic interaction would result in a final states which are some superposition of photon absorbed/scatter/unchanged and the overlap would be non-zero.
There are papers treating decoherence of relatively large molecules like fullerenes to test models of decoherence:
https://arxiv.org/abs/quant-ph/0402146
See also Section 4 of
https://arxiv.org/abs/1911.06282
which is a review of decoherence, which includes discussions of decoherence of atoms and molecules. The specific form of the decoherence depends on what regime is relevant, e.g.- the comparitive size of the wavelength of the molecule and the radiation it's interacting with, and it's unclear from your question what regime you're interested in.
