Editing Coherent state (section)

== Coherent states in quantum optics ==
[[Image:Coherent noise compare3.png|thumb|Figure 1:  The electric field, measured by optical [[homodyne detection]], as a function of phase for three coherent states emitted by a Nd:YAG laser. The amount of quantum noise in the electric field is completely independent of the phase. As the field strength, i.e. the oscillation amplitude α of the coherent state is increased, the quantum noise or uncertainty is constant at 1/2, and so becomes less and less significant. In the limit of large field the state becomes a good approximation of a noiseless stable classical wave. The average photon numbers of the three states from top to bottom are {{angbr|n}}=4.2, 25.2, 924.5<ref>{{cite journal | last1=Breitenbach | first1=G. | last2=Schiller | first2=S. | last3=Mlynek | first3=J. | title=Measurement of the quantum states of squeezed light | journal=Nature | publisher=Springer Nature | volume=387 | issue=6632 | year=1997 | issn=0028-0836 | doi=10.1038/387471a0 | pages=471–475| bibcode=1997Natur.387..471B | s2cid=4259166 |url=http://gerdbreitenbach.de/publications/nature1997.pdf}}</ref>]]

[[Image:coherent state wavepacket.jpg|thumb|300px|Figure 2: The oscillating [[wave packet]] corresponding to the second coherent state depicted in Figure 1. At each phase of the light field, the distribution is a [[Normal distribution|Gaussian]] of constant width.]]

[[Image:Wigner function coherent state.png|thumb|300px|Figure 3: [[Wigner quasiprobability distribution|Wigner function]] of the coherent state depicted in Figure 2. The distribution is centered on state's amplitude α and is [[phase space formulation|symmetric around this point]]. The ripples are due to experimental errors.]]

In [[quantum optics]] the coherent state refers to a state of the quantized [[electromagnetic field]], etc.<ref name="klau-ska"/><ref>{{cite journal | last1=Zhang | first1=Wei-Min | last2=Feng | first2=Da Hsuan | last3=Gilmore | first3=Robert | title=Coherent states: Theory and some applications | journal=Reviews of Modern Physics | publisher=American Physical Society (APS) | volume=62 | issue=4 | date=1990-10-01 | issn=0034-6861 | doi=10.1103/revmodphys.62.867 | pages=867–927| bibcode=1990RvMP...62..867Z }}</ref><ref name="gazeau">[[J-P. Gazeau]], ''Coherent States in Quantum Physics'', Wiley-VCH, Berlin, 2009.</ref> that describes a maximal kind of [[Coherence (physics)|coherence]] and a classical kind of behavior.  [[Erwin Schrödinger]] derived it as a "minimum [[Uncertainty principle|uncertainty]]" [[Wave packet#Gaussian wavepackets in quantum mechanics|Gaussian wavepacket]] in 1926,  searching for solutions of the [[Schrödinger equation]] that satisfy the [[correspondence principle]].<ref name="schrod"/>  It is a '''minimum uncertainty state''', with the single free parameter chosen to make the relative dispersion (standard deviation in natural dimensionless units) equal for position and momentum, each being equally small at high energy.

Further, in contrast to the [[Stationary state|energy eigenstates]] of the system, the time evolution of a coherent state is concentrated along the classical [[trajectory|trajectories]]. The quantum linear harmonic oscillator, and hence coherent states, arise in the quantum theory of a wide range of physical systems.  They occur in the quantum theory of light ([[quantum electrodynamics]]) and other [[bosonic]] [[quantum field theories]].

While minimum uncertainty Gaussian wave-packets had been well-known, they did not attract full attention until [[Roy J. Glauber]], in 1963, provided a complete quantum-theoretic description of coherence in the electromagnetic field.<ref>{{cite journal | last=Glauber | first=Roy J. | title=Coherent and Incoherent States of the Radiation Field | journal=Physical Review | publisher=American Physical Society (APS) | volume=131 | issue=6 | date=1963-09-15 | issn=0031-899X | doi=10.1103/physrev.131.2766 | pages=2766–2788| bibcode=1963PhRv..131.2766G }}</ref> In this respect, the concurrent contribution of [[E.C.G. Sudarshan]] should not be omitted,<ref>{{cite journal | last=Sudarshan | first=E. C. G. | title=Equivalence of Semiclassical and Quantum Mechanical Descriptions of Statistical Light Beams | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=10 | issue=7 | date=1963-04-01 | issn=0031-9007 | doi=10.1103/physrevlett.10.277 | pages=277–279| bibcode=1963PhRvL..10..277S }}</ref> (there is, however, a note in Glauber's paper that reads: "Uses of these states as [[generating function]]s for the <math>n</math>-quantum states have, however, been made by J. Schwinger<ref>{{cite journal | last=Schwinger | first=Julian | title=The Theory of Quantized Fields. III | journal=Physical Review | publisher=American Physical Society (APS) | volume=91 | issue=3 | date=1953-08-01 | issn=0031-899X | doi=10.1103/physrev.91.728 | pages=728–740| bibcode=1953PhRv...91..728S }}</ref>).
Glauber was prompted to do this to provide a description of the [[Hanbury Brown and Twiss effect|Hanbury-Brown & Twiss experiment]], which generated very wide baseline (hundreds or thousands of miles) [[Interference (wave propagation)|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 definition|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 (waves)|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 electron transition]]s 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 [[exponential growth|increases exponentially]] until some [[nonlinear optics|nonlinear 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 [[Coherence (physics)|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)

Besides describing lasers, coherent states also behave in a convenient manner when describing the quantum action of [[beam splitter]]s: two coherent-state input beams will simply convert to two coherent-state beams at the output with new amplitudes given by classical electromagnetic wave formulas;<ref name=Leonhardt>{{cite book |last1=Leonhardt |first1=Ulf |title=Measuring the Quantum State of Light |date=1997 |publisher=Cambridge University Press |isbn=9780521497305}}</ref> such a simple behaviour does not occur for other input states, including number states. Likewise if a coherent-state light beam is partially absorbed, then the remainder is a pure coherent state with a smaller amplitude, whereas partial absorption of non-coherent-state light produces a more complicated statistical [[Mixed quantum state|mixed state]].<ref name=Leonhardt/> Thermal light can be described as a statistical mixture of coherent states, and the typical way of defining [[nonclassical light]] is that it cannot be described as a simple statistical mixture of coherent states.<ref name=Leonhardt/>

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 [defined [[heuristic]]ally] and amplitude are roughly equal—and small at high amplitude.