With a coincidence counter, the dancing interference pattern would be stronger at times of increased intensity [common to both beams], and that pattern would be stronger than the background noise.

Almost all of optics had been concerned with first order coherence. The Hanbury-Brown and Twiss results prompted Glauber to look at higher order coherence, and he came up with a complete quantum-theoretic description of coherence to all orders in the electromagnetic field and a quantum-theoretic description of signal-plus-noise. He coined the term coherent state and showed that they are produced when a classical electric current interacts with the electromagnetic field.

Note that. The coherent state is an eigenstate of the annihilation operator in the Heisenberg picture. Each of these properties may lead to generalizations, in general different from each other see the article " Coherent states in mathematical physics " for some of these.

We emphasize that coherent states have mathematical features that are very different from those of a Fock state ; for instance, two different coherent states are not orthogonal,. However, since they obey a closure relation, any state can be decomposed on the set of coherent states. They hence form an overcomplete basis , in which one can diagonally decompose any state. This is the premise for the Sudarshan-Glauber P representation. This closure relation can be expressed by the resolution of the identity operator I in the vector space of quantum states,.

This resolution of the identity is intimately connected to the Segal—Bargmann transform.

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The following equality is the closest formal substitute, and turns out to be useful for technical computations,. On the left-hand side, the same is obtained by inserting. The resolution of the identity may also be expressed in terms of particle position and momentum. In addition to being an exact eigenstate of annihilation operators, a coherent state is an approximate common eigenstate of particle position and momentum.

Restricting to one dimension again,. The error in these approximations is measured by the uncertainties of position and momentum,. A single mode thermal coherent state [20] is produced by displacing a thermal mixed state in phase space , in direct analogy to the displacement of the vacuum state in view of generating a coherent state.

The density matrix of a coherent thermal state in operator representation reads. The partition function is equal to. We remark that if temperature goes to zero we have. The average number of photons in that state can be calculated as below. Here we have defined, for ease of notation,. Likewise, the photon number variance can be evaluated as. We deduce that the second moment cannot be uncoupled to the thermal and the quantum distribution moments, unlike the average value first moment.

In that sense, the photon statistics of the displaced thermal state is not described by the sum of the Poisson statistics and the Boltzmann statistics. The distribution of the initial thermal state in phase space broadens as a result of the coherent displacement. From Wikipedia, the free encyclopedia. Classical mechanics Old quantum theory Bra—ket notation Hamiltonian Interference. Advanced topics. Quantum annealing Quantum chaos Quantum computing Density matrix Quantum field theory Fractional quantum mechanics Quantum gravity Quantum information science Quantum machine learning Perturbation theory quantum mechanics Relativistic quantum mechanics Scattering theory Spontaneous parametric down-conversion Quantum statistical mechanics.

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Klauder and B. Klauder, The action option and a Feynman quantization of spinor fields in terms of ordinary c-numbers, Ann. Physics 11 — Breitenbach, S. Schiller, and J. Zhang, D.

### Coherent States and Bloch Functions

Feng, and R. Gilmore, Coherent states: Theory and some applications, Rev. Glauber, Coherent and incoherent states of radiation field, Phys. Sudarshan, Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams, Phys. Schwinger, Theory of quantized fields. III, Phys. Susskind and J.

- Coherent states - Wikipedia;
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Glogower, Quantum mechanical phase and time operator, Physics 1 Carruthers and M. Nieto, Phase and angle variables in quantum mechanics, Rev. Barnett and D. Pegg, On the Hermitian optical phase operator, J. Busch, M. Grabowski and P. Lahti, Who is afraid of POV measures? Unified approach to quantum phase observables, Ann.

## Quantum learning of coherent states

Dodonov, 'Nonclassical' states in quantum optics: a 'squeezed' review of the first 75 years, J. B: Quantum Semiclass. Dodonov and V. Vourdas, Analytic representations in quantum mechanics, J. A: Math.

Physical Review A. Bibcode : PhRvA..

Oz-Vogt, A. Mann and M. Revzen, Thermal coherent states and thermal squeezed states, J. Hyland, G. Rowlands, and F. Cummings, A proposal for an experimental determination of the equilibrium condensate fraction in superfluid helium, Phys. B 74 Olinto, Condensate fraction in superfluid He-4, Phys. B 35 Penrose and L.

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## Quantization physics

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