Ground state

The ground state is the lowest-energy state of a quantum system: the electronic configuration of an atom or ion with nothing excited, the lowest vibrational level of the lowest electronic state of a molecule, or, in a semiconductor, the state with the valence band full and the conduction band empty. In thermal equilibrium nearly all of the population sits there whenever the next level lies many $k_B T$ above it. At 300 K, $k_B T = 25.85$ meV (208.5 cm⁻¹), so for a transition of 1 eV the relative population of the upper level is

$$e^{-1/0.02585} \approx 1.6 \times 10^{-17},$$

and for a 0.80 eV transition (1550 nm) about $3.6 \times 10^{-14}$, with equal degeneracies assumed. For optical transitions the excited states are therefore empty at room temperature unless something pumps them, and an unpumped medium absorbs at its transition wavelengths.

Ground state in three-level and four-level lasers

Whether the lower laser level is the ground state decides how hard a medium is to invert. In a three-level system such as ruby, the laser transition ends on the ground state, so more than half the ions must be removed from it before there is any gain (see population inversion). In a four-level laser such as Nd:YAG, the lower level lies about 2110 cm⁻¹ above the ground state; its Boltzmann factor at 300 K is about $4 \times 10^{-5}$, so it is nearly empty and inversion begins with the first excited ions.

Erbium and ytterbium in silica fall between these cases. In Er³⁺ the 1530–1565 nm transition runs from the $^4I_{13/2}$ manifold to the $^4I_{15/2}$ ground manifold, so unpumped erbium fiber absorbs strongly at the signal wavelengths; this ground-state absorption is why an EDFA must be pumped along its whole length and why its gain spectrum depends on the fraction of ions inverted. In Yb³⁺ the lower level for emission near 1030 nm is an upper Stark component of the ground manifold a few hundred cm⁻¹ up (612 cm⁻¹ in Yb:YAG, with a Boltzmann factor of about 0.05 at 300 K), so the medium is quasi-three-level: it reabsorbs at the laser wavelength unless pumped hard, and cooling it lowers the threshold. The same ground-state absorption, made saturable, is the operating principle of a saturable absorber.

Excited-state absorption

An ion already in an excited state can absorb a second photon and move higher still. This excited-state absorption competes with the ground-state absorption that pumping relies on, or removes signal photons. Erbium pumped near 800 nm loses efficiency this way, which is one reason 980 nm and 1480 nm pumps are used. Ytterbium-doped fiber has only two manifolds within reach of the pump and is free of it.

Ground states in quantum wells and dots

In a quantum well, confinement splits the conduction and valence bands into subbands, and the lowest of each is called the ground-state subband. For an ideal infinitely deep 10 nm GaAs well with an electron effective mass of 0.067 $m_e$, the electron ground state lies 56 meV above the bulk band edge; a real well with Al₀.₃Ga₀.₇As barriers gives about 30 meV. Quantum-well lasers normally emit on the transition between the electron and heavy-hole ground subbands.

In a quantum-dot laser the dot states are discrete, and "ground-state lasing" means emission from the lowest electron–hole transition, around 1.3 µm in InAs/GaAs dots. The ground state of each dot holds only two electrons of opposite spin, so its gain saturates at a modest value. When a short or lossy cavity demands more gain than the ground state can supply, the laser switches to, or adds, emission from the excited state, which lies typically several tens of meV higher: 70 meV above a 1300 nm ground-state line corresponds to about 1211 nm.

Common questions

What is the difference between the ground state and an excited state?

The ground state is the lowest energy level a system can occupy; any higher level is an excited state. At thermal equilibrium the population of an excited state relative to the ground state is $e^{-\Delta E/k_BT}$ times the ratio of their degeneracies.

Why is a laser transition that ends on the ground state harder to invert?

A transition ending on the ground state starts with that level full, so more than half of the population has to be pumped out before stimulated emission exceeds absorption. In a four-level medium the lower laser level starts nearly empty.

References: A. E. Siegman, Lasers (University Science Books, 1986); O. Svelto, Principles of Lasers, 5th ed. (Springer, 2010); E. Desurvire, Erbium-Doped Fiber Amplifiers: Principles and Applications (Wiley, 1994); L. A. Coldren, S. W. Corzine and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012).