when they are in thermal equilibrium. In fact, any method by which the atoms are directly and continuously excited from the ground state to the excited state (such as optical absorption) will eventually reach equilibrium with the de-exciting processes of spontaneous and stimulated emission. At best, an equal population of the two states, N1 = N2 = N/2, can be achieved, resulting in optical transparency but no net optical gain. Three-level lasers To achieve lasting non-equilibrium conditions, an indirect method of populating the excited state must be used. To understand how this is done, we may use a slightly more realistic model, that of a three-level laser. Again consider a group of N atoms, this time with each atom able to exist in any of three energy states, levels 1, 2 and 3, with energies E1, E2, and E3, and populations N1, N2, and N3, respectively. We assume that E1 < E2 < E3; that is, the energy of level 2 lies between that of the ground state and level 3. Initially, the system of atoms is at thermal equilibrium, and the majority of the atoms will be in the ground state, i.e., N1 ≈ N, N2 ≈ N3 ≈ 0. If we now subject the atoms to light of a frequency , the process of optical absorption will excite electrons from the ground state to level 3. This process is called pumping, and does not necessarily always directly involve light absorption; other methods of exciting the laser medium, such as electrical discharge or chemical reactions, may be used. The level 3 is sometimes referred to as the pump level or pump band, and the energy transition E1 → E3 as the pump transition, which is shown as the arrow marked P in the diagram on the right. Upon pumping the medium, an appreciable number of atoms will transition to level 3, such that N3 > 0. To have a medium suitable for laser operation, it is necessary that these excited atoms quickly decay to level 2. The energy released in this transition may be emitted as a photon (spontaneous emission), however in practice the 3→2 transition (labeled R in the diagram) is usually radiationless, with the energy being transferred to vibrational motion (heat) of the host material surrounding the atoms, without the generation of a photon. An electron in level 2 may decay by spontaneous emission to the ground