Coherent States - Coherent States in Quantum Optics

Coherent States in Quantum Optics

In quantum mechanics a coherent state is a specific kind of quantum state, applicable to the quantum harmonic oscillator, the electromagnetic field, etc. that describes a maximal kind of coherence and a classical kind of behavior. Erwin Schrödinger derived it as a minimum uncertainty Gaussian wavepacket in 1926 while searching for solutions of the Schrödinger equation that satisfy the correspondence principle. It is a minimum uncertainty state, with the single free parameter chosen to make the relative dispersion (standard deviation divided by the mean) equal for position and momentum, each being equally small at high energy. Further, contrary to the energy eigenstates of the system, the time evolution of a coherent state is concentrated along the classical trajectories. The quantum linear harmonic oscillator and hence, the coherent states, arise in the quantum theory of a wide range of physical systems. They are found in the quantum theory of light (quantum electrodynamics) and other bosonic quantum field theories.

While minimum uncertainty Gaussian wave-packets were well-known, they did not attract much attention until Roy J. Glauber, in 1963, provided a complete quantum-theoretic description of coherence in the electromagnetic field. In this respect, the concurrent contribution of E.C.G. Sudarshan should not be omitted, (there is, however, a note in Glauber's paper that reads: "Uses of these states as generating functions for the -quantum states have, however, been made by J. Schwinger ). Glauber was prompted to do this to provide a description of the Hanbury-Brown & Twiss experiment that generated very wide baseline (hundreds or thousands of miles) interference patterns that could be used to determine stellar diameters. This opened the door to a much more comprehensive understanding of coherence. (For more, see Quantum mechanical description.)

In classical optics light is thought of as electromagnetic waves radiating from a source. Often, coherent laser light is thought of as light that is emitted by many such sources that are in phase. Actually, the picture of one photon being in-phase with another is not valid in quantum theory. Laser radiation is produced in a resonant cavity where the resonant frequency of the cavity is the same as the frequency associated with the atomic transitions providing energy flow into the field. As energy in the resonant mode builds up, the probability for stimulated emission, in that mode only, increases. That is a positive feedback loop in which the amplitude in the resonant mode increases exponentially until some non-linear effects limit it. As a counter-example, a light bulb radiates light into a continuum of modes, and there is nothing that selects any one mode over the other. The emission process is highly random in space and time (see thermal light). In a laser, however, light is emitted into a resonant mode, and that mode is highly coherent. Thus, laser light is idealized as a coherent state. (Classically we describe such a state by an electric field oscillating as a stable wave. See Fig.1)

The energy eigenstates of the linear harmonic oscillator (e.g., masses on springs, lattice vibrations in a solid, vibrational motions of nuclei in molecules, or oscillations in the electromagnetic field) are fixed-number quantum states. The Fock state (e.g. a single photon) is the most particle-like state; it has a fixed number of particles, and phase is indeterminate. A coherent state distributes its quantum-mechanical uncertainty equally between the canonically conjugate coordinates, position and momentum, and the relative uncertainty in phase and amplitude are roughly equal—and small at high amplitude.