This page is still in progress.
«»
Quantum States
One popular characterization of quantum states is through Glauber's correlators $g^{(n)}$ (the most famous one being $g^{(2)}$). We provided a nice way to explore the Hilbert space of all quantum states using those as flashlights (see Wading through the Hilbert space).
Fock states
Coherent states
Because quantum theory is more fundamental and general than classical theory, it should be able to encompass it. The classical field $\alpha\in\mathbb{C}$ described by an amplitude $|\alpha|$ and a phase $\operatorname{Arg}(\alpha)$, is described in quantum theory by the coherent state $\ket{\alpha}$. This state is best defined in the Fock basis: $$\ket{\alpha}=\exp\Big(-{|\alpha|^2\over2}\Big)\sum_{n=0}^\infty{\alpha^n\over\sqrt{n!}}\ket{n}\,.$$ It has the remarkable property that it is the eigenstate of the annihilation operator: $$a\ket{\alpha}=\alpha\ket{\alpha}\,.$$ This is as expected for a classical state: removing (detecting) a particle leaves the state itself unchanged. The coherent state has no notion of particles, just as a wave.
The coherent state was first discovered by Schrödinger as a solution to his equation that does not diffuse in space, and thus provides a tenable interpretation for a particle, which, however, was not the good one.
The concept was more usefully brought back to the fore by Sudarshan and was quickly took over by Glauber as the basis for his theory of quantum optical coherence. This caused bitter controversies between the two, especially as Glauber would earn the Nobel Prize in Physics 2005 for this work.
A champion of coherent states is Klauder, who extended and formalized them. [ISBN: 9971966522] Notable papers include his definition of coherent states for complex systems, such as the Coulomb potential (coherent states for the Hydrogen atom[1]), or the quantum-to-classical correspondence.[2]
In contrast to Klauder who relaxed the group-theoretic constraints of coherent states, Perelomov[3] provided a general definition of coherent states for arbitrary Lie groups, while Gilmore[4] also gave them a geometric interpretation.
- 🕮Coherent States and Their Applications. J.-P. Antoine and F. Bagarello and J.-P. Gazeau. Springer Proceedings in Physics, 2019. [ISBN: 978-3-319-76731-4] [DOI: 10.1007/978-3-319-76732-1]
Thermal states
Cothermal states
Interpolate between thermal and coherent states.
Theoretical Aspects of Mixtures of Thermal and Coherent Radiation. G. Lachs in Phys. Rev. 138:B1012 (1965).
Negative binomial states
Interpolate between thermal and coherent states.
Glauber-Sudarshan P representation of negative binomial states. K. Matsuo in Phys. Rev. A 41:519 (1990). Negative Binomial States of Quantized Radiation Fields. H. Fu and R. Sasaki in J. Phys. Soc. Jpn. 66:1989 (1997).
Binomial states
Interpolate between Fock and coherent states.
Phase states
Number-phase states
Intermediate number-phase states of the quantized radiation field. B. Baseia, A. d. Lima and G. Marques in Phys. Lett. A 204:1 (1995).
Gaussian states
Gaussian states are those which can be created only with displacement operators and squeezing. See Ref. Quantum optics in the phase space. S. Olivares in Eur. Phys. J. Spec. Top. 203:3 (2012). for a tutorial. The most Gaussian state of all is the Coherent state previously introduced.
Another one-mode definition is:[5]
A Gaussian state is pure iff the determinant of the coherence variance matrix = 1.[6][7]
Squeezing
Squeezed states have been first introduced by Stoler,[8] and then by Henry Yuen as the two-photon coherent state[9][10]. See also Walls.[11]
Beyond the diagonal
Randomly phased coherent states
Pure thermal distribution
Pure states having thermal photon distribution revisited: generation and phase-optimization. B. Baseia, C. M. Dantas and M. Moussa in Physica A 258:203 (1998).
References
- ↑ Coherent states for the hydrogen atom. J. R. Klauder in J. Phys. A.: Math. Gen. 29:L293 (1996).
- ↑ Coherent states for the hydrogen atom. J. R. Klauder in J. Phys. A.: Math. Gen. 29:L293 (1996).
- ↑ Coherent states for arbitrary Lie group. A. M. Perelomov in Commun. Math. Phys. 26:222 (1972).
- ↑ Geometry of symmetrized states. R. Gilmore in Annals of Physics 74:391 (1972).
- ↑ Quantifying coherence of Gaussian states. J. Xu in Phys. Rev. A 93:032111 (2016).
- ↑ Quantum information with Gaussian states. X.-B. Wang, T. Hiroshima, A. Tomita and M. Hayashi in Phys. Rep. 448:1 (2007).
- ↑ Quantifying coherence in infinite-dimensional systems. Y. Zhang, L. Shao, Y. Li and H. Fan in Phys. Rev. A 93:012334 (2016).
- ↑ Equivalence Classes of Minimum Uncertainty Packets. D. Stoler in Phys. Rev. D 1:3217 (1970).
- ↑ Generalized coherent states and the statistics of two-photon lasers. H. P. Yuen in Phys. Lett. A 51:1 (1975).
- ↑ Two-photon coherent states of the radiation field. H. P. Yuen in Phys. Rev. A 13:2226 (1976).
- ↑ Squeezed states of light. D. F. Walls in Nature 306:141 (1983).