The Heisenberg energy-time uncertainty tells us that we can have so-called virtual states between eigenstates as long as the lifetime of these states is at most:

$\tau = (\frac{h}{4 \pi E_v})$

Where $h$ is the Planck constant ($6.62607 * 10^{-34} J*s$, or $4.135668 * 10^{-15} eV*s$), and $E_v$ is energy difference between our virtual state and the nearest eigenstate. If we assume $E_v \approx 1 eV$ for a virtual state, we can estimate that this state can only exist for at most $\tau_2 \approx 3.29106 * 10^{-16}$ seconds.

Now, in the literature and wikipedia, I've repeatedly seen the claim that two-photon absorption (http://en.wikipedia.org/wiki/Two-photon_absorption) involves a single photon event promoting a molecule to a virtual state, and because the energy-time uncertainty allows this state to exist for some time $\tau$ (as estimated above for a $\approx 1 eV$ transition), if another photon strikes within this time window the energy of the two photons can be summed to allow for a transition with double the energy of a single photon. For example, two $\lambda \approx 2.48 \mu m$ photons with energies of $\approx 0.5 eV$ each can, through a virtual state, sum to allow for a $\approx 1 eV$ transition correspond to the absorbance of a single $\lambda \approx 1.24 \mu m$ photon.

However, I'm having trouble understanding where the "memory" for the first photon absorbance is stored, since a virtual state is not a real eigenstate with distinct observables. What mechanism allows for the energies of the two photons to be summed? And are these energies being summed, or is there some energy loss associated with transitioning through a virtual state? Please note that I fully understand there may not be an intuitive "memory mechanism", I just mean "memory" in the sense of being able to write down something like a descriptive state vector for a virtual state that accounts for the state's energy.

Finally, what does it mean to say that a virtual state can exist for "at most a time $\tau$" in the context of the energy-time uncertainty principle? Is this just the expected lifetime for a state decaying exponentially with decay parameter $\lambda = \frac{1}{\tau}$?

A good soundbite for this question would be: how can we write down a quantum mechanical expression for a virtual state (say, in the context of raman spectroscopy) in Hilbert space? And if it's ultimately correct to describe the virtual state using an interaction Hamiltonian between some particle's dipole and the EM field of an incoming photon (of some wavelength), how do we write down a description of this Hamiltonian in a rigorous manner? Does this Hamiltonian give us any insight into the lifetime of the virtual state?

Also, why do I often see references to a Rigged Hilbert space when doing literature searches on this topic? Take for example these two papers by Gadella, M.:

Gadella, M. A description of virtual scattering states in the rigged Hilbert space formulation of quantum mechanics. J. Math. Phys. 24, 2142 (1983). http://scitation.aip.org/content/aip/journal/jmp/24/8/10.1063/1.525966

Gadella, M. Construction of rigged Hilbert spaces to describe resonances and virtual states. Physica A: Statistical Mechanics and its Applications 124(1–3), March 1984, 317 (1984). http://www.sciencedirect.com/science/article/pii/0378437184902498

From (http://arxiv.org/pdf/1209.5620.pdf): Leon-Montiel, R. J. et. al. Role of the spectral shape of quantum correlations in two-photon virtual-state spectroscopy. (2013).

"The quantum mechanical calculation of the TPA [Two-Photon Absorption] transition probability shows that its value can be understood as a weighted sum of many energy non-conserving atomic transitions (virtual-state transitions) [22, 23] between energy levels."

  • $\begingroup$ Can you give a reference for some of the papers you speak of (especially the rigged Hilbert space)? Do you have Göppert‐Mayer's original paper? "Über Elementarakte mit zwei Quantensprüngen", Annalen der Physik vol 401 issue 3 1931? There is a translation also, both behind paywalls. $\endgroup$ – WetSavannaAnimal Dec 7 '13 at 9:02
  • $\begingroup$ @WetSavannaAnimalakaRodVance Thanks for your comment. I updated my question with two papers by Gadella, M. that I was looking at - I've had trouble finding other relevant papers. Also, here's Goppert-Mayer's original paper (onlinelibrary.wiley.com/doi/10.1002/andp.19314010303/pdf) for free, but I don't have a translation. $\endgroup$ – RGrey Dec 8 '13 at 19:55
  • $\begingroup$ Fantastic! The original is enough! I shall have a look through the papers you cite: your question is an excellent one and I shall think about it some more. A trite answer is that the "memory" can be in a phenomenological model: we could add terms to the Hamiltonian of the form $\kappa(\nu_1,j_1,\nu_2,j_2)\, a_0^\dagger \,a(\nu_1,j_1)\, a(\nu_2,j_2) + H.c.$ where $a_0^\dagger$ raises the molecule state and $a(\nu,j)$ the annihilation operator for EM mode $j$ at frequency $\nu$ but I suspect this doesn't help much. You've likely already thought of the obvious that the "memory" could be .... $\endgroup$ – WetSavannaAnimal Dec 8 '13 at 22:14
  • $\begingroup$ ... in the coupling of the molecule with the EM field and these are the lines I'll try thinking along. $\endgroup$ – WetSavannaAnimal Dec 8 '13 at 22:16
  • $\begingroup$ @WetSavannaAnimalakaRodVance I look forward to seeing what you can come up with! I've been reading about the electric dipole approximation, and I did think of using a coupling Hamiltonian to describe the virtual state, but it's unclear to me how to proceed in a rigorous way w.r.t. the virtual state. $\endgroup$ – RGrey Dec 9 '13 at 22:14

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