Two Photon Absorption in Si nanowire waveguide.
Si wire waveguides
The two-photon absorption is a quantum process in which two photons are simultaneously absorbed, giving a valence-band electron enough energy to transition into the conduction band. This transition proceeds through a virtual state within the bandgap.
page created in Dec. 2025
Two-electron absorption. The electron path |
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Two-electron absorption. The electron path |
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Middle gap level
There are two possible mechanisms, which forms a middle gap levels in semiconductor:
(middle-gap level type 1): bulk type
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(middle-gap level type 2): surface type
The surface states exist due to breaking of periodicity of semiconductor lattice and a boundary of the semiconductor.
Other paths for the loss of photons due to TPA
(path 1) :
emitting a second-harmonic photon.
emmiting of 2nd harmonic photon |
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| After an electron undergoes a transition from the valence band to the conduction band via two-photon absorption, it can subsequently relax back to the valence band, emitting a second-harmonic photon. The energy of the emitted photon equals the sum of the energies of the two absorbed photons. |
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(path 2) : emitting photons at energies different from that of absorbed photons
emitting photons at energies different from that of absorbed photons |
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Questions. Answers. Discussion
(about lifetime on middle-bandgap level)
(Mark's thoughts)
(instantaneous lifetime due to Heisenberg uncertainty principle )
the lifetime of the virtual level in two photon absorption (TPA) is essentially instantaneous and limited by the Heisenberg uncertainty principle. Measurement evidence was also found in this paper 
describing the autocorrelation measurement of ~7fs pulses by TPA, which if its virtual state lifetime was longer then 7fs, then that would have limited the autocorrelation measurement resolution to > 7 fs so it seems cased closed on that question.
(Heisenberg uncertainty principle vs. separation between two photons)
My interpretation was that the virtual energy level in two photon absorption (TPA) is just that-virtual only and the two photons need to be at the same place at the same time, and the precision of defining the position of the electron that can absorb those two photons is limited by the Heisenberg uncertainty principle, which sounds reasonable. On the other hand, if considering their wavefunctions, then I agree, TPA is still possible for even widely separated photons in space however, the probability would be far less. The autocorrelation measurement of ultra short pulses using TPA is a convincing way of showing that the lifetime of the virtual energy level is mostly instantaneous (by determining the minimum temporal resolution of the material), at least in that material, and the low probability of TPA for more widely separated photons is background noise.
As you pointed out, Heisenberg’s uncertainty principle was referring to the uncertainty of the electron’s spatial position (distance) in space, not time, and the lifetime of the virtual energy level in TPA is interpreted as the time it takes for the photon to traverse that distance (spatial location where the electron exists), limited by the Heisenberg uncertainty principle for defining the electron’s position in space. Nevertheless, I agree much longer lifetimes are still possible as determined by the overlapping wavefunctions, although with much less probability.
(my thoughts)
(two-photon absorption used for autocorrelation)
Yes, two-photon absorption (TPA) is often used for autocorrelation of short optical pulses, which is commonly interpreted as evidence that the lifetime of the mid-gap (virtual) state is short. However, reported values for this lifetime are quite controversial, ranging from a few femtoseconds up to nearly 1000 fs (~1 ps).
I do not think that this lifetime is fundamentally fixed by the Heisenberg uncertainty principle. In my view, the uncertainty principle does not play a key role in TPA and may not influence it at all.
(Role of the Heisenberg uncertainty principle)
The Heisenberg uncertainty principle applies to processes in which energy is not conserved. This is not the case for TPA: in two-photon absorption, energy is fully conserved—the combined energy of the two absorbed photons equals the energy gained by the excited electron.
(example 1): (where the Heisenberg uncertainty principle works): dark matter / vacuum fluctuations.:
An electron–positron pair can be created from the vacuum state even though the energy of the electron- positron pair is larger than the vacuum energy.. This breaking of the energy conservation is allowed only for a very short time due to the Heisenberg uncertainty principle.
(example 2): (where the Heisenberg uncertainty principle works): magnetic moment of the ground state:
If the ground state has no spin or orbital magnetic moment, but an excited state does, the ground state can still acquire a small magnetic moment. This occurs because the electron briefly occupies the excited state without an external energy source. Again, energy conservation is violated for a very short time due to the uncertainty principle.
(Lifetime of the mid-gap state: bulk and interface contributions)
I believe there are two fundamentally different contributions to the lifetime of the mid-gap state: a bulk contribution, characterized by a very short lifetime, and an interface contribution, which can have a longer lifetime.
(Bulk contribution)
It is incorrect to assume that there are no energy levels inside the bandgap of a semiconductor. Such states do exist, but their energies are complex numbers.
This situation is analogous to harmonic and evanescent waves in a waveguide. When the refractive index is real, the spatial field distribution is harmonic; when the refractive index is complex, the field exhibits exponential decay.
Similarly, inside the bandgap, the electron energy becomes complex due to resonant Bragg reflection from the periodic atomic lattice. While electrons in band states have a harmonic temporal evolution, electrons in bandgap states exhibit exponential temporal decay.
The decay time (i.e., the lifetime) is inversely proportional to the imaginary part of the complex energy. According to the linear k⋅p theory, the farther the energy is from the band edge, the larger the imaginary component of the energy and, therefore, the shorter the lifetime. For states near the middle of the gap, linear perturbation theory is no longer valid, and the dependence should be more complex.
(Interface contribution)
At an interface, the periodicity of the atomic lattice—responsible for the bandgap—is broken. As a result, interface states often appear. The energies of these interface states are real rather than complex, therefore the interface states have a long lifetime.
Even when the interface is not sharp enough to form well-defined surface states, the breaking of lattice periodicity should still reduce the imaginary part of the electron energy and thus increase the lifetime. This occurs because the conditions for Bragg reflection are partially broken.
An important point is that surface and interface states can often be controlled externally (for example, by applying a voltage). Therefore, it may be possible to control the strength of TPA externally via interface engineering or a gate voltage.
Clearly, the contribution with the longer lifetime will dominate the TPA process.
