Spectroscopy using coherent light-matter interactions to probe quantum states.Coherent spectroscopy is a family of techniques that use light with a well-defined phase relationship — coherent light, typically from lasers — to probe the quantum states of matter. Unlike conventional spectroscopy, which simply measures how much light is absorbed or emitted at each frequency, coherent spectroscopy exploits the wave nature of light and matter to extract information about quantum superpositions, energy-level structures, and dynamical processes. It is the foundation of many modern quantum technologies, from atomic clocks to quantum computing.
The field emerged in the 1960s and 1970s with the development of lasers and the theoretical work on coherent transients by Richard Brewer and others. The technique of photon echoes, a cornerstone of coherent spectroscopy, was first predicted by Erwin Hahn in 1950 and demonstrated in optics by I. D. Abella, N. A. Kurnit, and S. R. Hartmann in 1966.
The key idea is that a coherent light pulse can prepare a quantum system in a superposition of states. A second pulse, delayed in time, can then interfere with the first, creating an echo or a beat signal that reveals the system's coherence properties. By varying the time delay and frequency of the pulses, researchers can map out the energy landscape of the system with high precision, even in the presence of noise or disorder.
Coherent spectroscopy is used in atomic clocks, quantum memory devices, and in the characterization of semiconductor nanostructures and molecular dynamics.
The photon echo, a key phenomenon in coherent spectroscopy, is the optical analogue of the spin echo used in MRI machines.
| C_{lm l'm'} | variable | Angular power spectrum coefficients — angular power spectrum |
| a_{lm} | variable | Spherical harmonic coefficients of temperature map — spherical harmonic coefficient kelvin |
| ⟨ ⟩ | operator | Ensemble average over realizations — ensemble average |
| * | operator | Complex conjugation — complex conjugate |
| C_{xy}(f) | variable | Magnitude squared coherence at frequency f — coherence |
| P_{xy}(f) | variable | Cross-spectral density between signals x and y at frequency f — cross spectral density watt per hertz |
| P_{xx}(f) | variable | Power spectral density of signal x at frequency f — power spectral density watt per hertz |
| P_{yy}(f) | variable | Power spectral density of signal y at frequency f — power spectral density watt per hertz |
| | \cdot | | operator | Magnitude (absolute value) of a complex number — absolute value |

















