# Quantum defect

The term quantum defect is ambiguous. Various meanings are discussed below. Characteristic is that the defect deals with the loss on the smallest energy scale of light: that of the quantum.

## In laser science

In laser science, the term "quantum defect" refers to the fact that the energy of a pump photon is generally higher than that of a signal photon (photon of the output radiation). The difference of energies goes to the heat; this heat may carry away the excess of entropy delivered with the multimode incoherent pump.

The quantum defect of a laser can be defined as part of the energy of the pumping photon, which is lost (not turned into photons at the lasing wavelength) in the gain medium at the lasing. At given frequency $~\omega _{\rm {p}}~$ of pump and given frequency $~\omega _{\rm {s}}~$ of lasing, the quantum defect $~q=\hbar \omega _{\rm {p}}-\hbar \omega _{\rm {s}}~$ . Such quantum defect has dimension of energy; for the efficient operation, the temperature of the gain medium (measured in units of energy) should be small compared to the quantum defect.

At a fixed pump frequency, the higher the quantum defect, the lower is the upper bound for the power efficiency.

## In Rydberg atoms

The quantum defect of a Rydberg atom refers to a correction applied to the equations governing Rydberg atom behavior to take into account the fact that the inner electrons do not entirely screen their associated charge in the nucleus. It is used particularly for the alkalis that contain a single electron in their outer shell.

The perfect 1/r potential in the hydrogen atom leads to an electron binding energy given by

$E_{\text{B}}=-{\dfrac {Rhc}{n^{2}}}$ ,

where R is the Rydberg constant, h is Planck's constant, c is the speed of light and n is the principal quantum number.

For multi-electron atoms in Rydberg states with a low value of the orbital angular momentum, there is a high probability of finding the excited electron near the nucleus where it can polarize or even penetrate the ion core, modifying the potential; the resulting shift of the energy levels is represented mathematically as an angular momentum dependent quantum defect, δl:

$E_{\text{B}}=-{\dfrac {Rhc}{(n-\delta _{l})^{2}}}$ .

The largest shifts occur when the orbital angular momentum is equal to 0 (normally labelled 's') and these are shown in the table for the alkali metals:

Element Configuration n-δs δs
Li 2s 1.59 0.41
Na 3s 1.63 1.37
K 4s 1.77 2.23
Rb 5s 1.81 3.19
Cs 6s 1.87 4.13