Electromagnetic radiation creates a polarization in a sample and that polarization will re-radiate light. This process forms the basis of spectroscopy. The polarization on an individual chromophore takes the form of a **coherence**. A coherence arises from the quantum mechanical entanglement of two states induced by an electromagnetic field perturbation, and must be understood quantum mechanically. A quantum state $n$ has its own time-independent wavefunction, $\psi(x)_n$, and it's own time-dependent amplitude, $c_n(t)$. If *a* and *b* are the two entangled quantum states, then the system can be observed as the square of the total wavefunction:

$$\left| \Psi(x,t)\right|^2 = \left| c_a(t) \psi_a(x) + c_b(t) \psi_b(x) \right|^2$$

which, when expanded, yields:

$$\left| \Psi(x,t)\right|^2 = c_a(t)c_b^*(t) \psi_a(x)\psi_b(x) + c_b(t)c_a^*(t) \psi_b(x)\psi_a(x) + \left|c_a(t)\right|^2\left|\psi_a(x)\right|^2 + \left|c_b(t)\right|^2\left|\psi_b(x)\right|^2$$

The mixed-amplitude products $c_a c_b^*$ and $c_b c_a^*$ are the coherences (non-mixed amplitude products, such as $c_a c_a^*$ and $c_b c_b^*$, are special cases of a coherence known as **populations**). Often we use *density matrix notation *to identify these amplitude products: $\rho_{ij} \equiv c_i c_j^*$ (shorthand will also refer to the coherence as the quantum state indices, i.e. $\rho_{ij} = ij$). These coherences launch the electromagnetic fields that are responsible for molecular spectroscopy. The coherence’s imaginary term is responsible for absorption (i.e. the part of the polarization that is out-of-phase with the electromagnetic field) and the real term is responsible for refraction (i.e. the in-phase part of the polarization).

Time dependent quantum mechanics describes the evolution of a quantum state as it becomes entangled with another state (in our case, the means of entanglement is the applied electromagnetic field). By controlling the incident light fields, the coherences and populations can be controlled.

The properties of a coherent nonlinear process can be defined by the sequence of coherences and populations that are created by successive interactions with the excitation fields. Let’s first start with the simplest case and follow the sequence of coherences and populations when a system starting in the ground state interacts with a resonant electromagnetic field that promotes transitions between states *g* and *a*. The figure to the left diagrams the transitions between the energy levels of a molecule undergoing absorption or refraction. Absorption and refraction are described by a single interaction, *gg→ag* (the upwards arrow in the figure), i.e. the field interacts with the ground state population, *gg*, to create an quantum entanglement of states *a* and *g*. The *ag* coherence creates an output field at the same frequency as the excitation field (the downwards striped arrow) so the new field and the exciting field are superimposed and interfere so the net field forming the output is smaller than the input field (absorption) and phase shifted (refraction).

There are many types of coherences that are important for nonlinear spectroscopy. In analogy to NMR, an *ag* coherence is an example of a **single quantum coherence**. A second excitation can create an excitation on the ket resulting in an *(a+b),g* coherence or on the bra resulting in an *ab* coherence. These coherences are called a **double quantum** or a **zero quantum coherence**, respectively.