The Jaynes-Cummings model (JCM), a soluble fully quantum mechanical model of an atom in a field, was first used (in 1963) to examine the classical aspects of spontaneous emission and to reveal the existence of Rabi oscillations in atomic excitation probabilities for fields with sharply defined energy (or photon number). For fields having a statistical distributions of photon numbers the oscillations collapse to an expected steady value. In 1980 it was discovered that with appropriate initial conditions (e.g. a near-classical field), the Rabi oscillations would eventually revive, only to collapse and revive repeatedly in a complicated pattern. The existence of these revivals, present in the analytic solutions of the JCM, provided direct evidence for discreteness of field excitation (photons) and hence for the truly quantum nature of radiation. Subsequent study revealed further non-classical properties of the JCM field, such as a tendency of the photons to antibunch. Within the last two years it ha...

The Jaynes-Cummings Model

Seventy five years ago, three remarkable papers by Schr¨ odinger, Kennard and Darwin were published. They were devoted to the evolution of Gaussian wave packets for an oscillator, a free particle and a particle moving in uniform constant electric and magnetic fields. From the contemporary point of view, these packets can be considered as prototypes of the coherent and squeezed states, which are, in a sense, the cornerstones of modern quantum optics. Moreover, these states are frequently used in many other areas, from solid state physics to cosmology. This paper gives a review of studies performed in the field of so-called ‘nonclassical states’ (squeezed states are their simplest representatives) over the past seventy five years, both in quantum optics and in other branches of quantum physics. My starting point is to elucidate who introduced different concepts, notions and terms, when, and what were the initial motivations of the authors. Many new references have been found which enlarge the ‘standard citation package’ used by some authors, recovering many undeservedly forgotten (or unnoticed) papers and names. Since it is practically impossible to cite several thousand publications, I have tried to include mainly references to papers introducing new types of quantum states and studying their properties, omitting many publications devoted to applications and to the methods of generation and experimental schemes, which can be found in other well known reviews. I also mainly concentrate on the initial period, which terminated approximately at the border between the end of the 1980s and the beginning of the 1990s, when several fundamental experiments on the generation of squeezed states were performed and the first conferences devoted to squeezed and ‘nonclassical’ states commenced. The 1990s are described in a more ‘squeezed’ manner: I have confined myself to references to papers where some new concepts have been introduced, and to the most recent reviews or papers with extensive bibliographical lists.

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'Nonclassical' states in quantum optics: a 'squeezed' review of the first 75 years

The time-dependent treatment of a three-level atom in the V configuration confined in an optical cavity as well as the dynamics of the cavity field in two different regimes, in the presence and absence of the nonlinear mirror, has been discussed. The time-dependent infinite coupled differential equations of motion are solved by the matrix continued fraction method. In deriving the numerical inverse Laplace transform, the quotient difference algorithm and the fast Fourier transform have been applied. The nonlinear effects of the system over the time evolution of the population inversion, the mean photon number and the second-order correlation function for some dimensionless parameters are delineated. Ultimately the reduction of the three-level atom to an effective two-level one under specific conditions and in a weak-driving limit has been performed.

Temporal behavior of an atom-cavity system in two distinct regimes

Publisher Summary Atoms commonly do not act as isolated objects, but rather are open quantum systems, as they interact with the environment Typically, this environment is formed by the electromagnetic vacuum field The interaction of atoms with the environment modifies the atomic dynamics, with spontaneous emission as the most obvious example Spontaneous emission is generally recognized as incoherent process, which leads to decoherence and, therefore, forms a major limitation for many schemes of current theoretical and experimental interest But vacuum-induced processes can also generate coherent time evolution These coherences can be interpreted as arising from vacuum-induced transitions between different atomic states The situation becomes even more interesting if different atoms can exchange energy via the vacuum Such dipole–dipole interactions induce both coherent and incoherent atomic dynamics, leading to significant deviations from the single-atom properties Finally, a complex interplay of vacuum-induced interatomic and intraatomic dynamics may arise if several multilevel atoms are considered These vacuum-induced processes form the basis for a large number of applications, for which the vacuum-induced dynamics can be favorable, perturbing, or even both Most applications can be improved, if the vacuum-induced processes can be modified or even controlled Thus, a profound understanding of vacuum-induced processes is desirable Motivated by this, this chapter discusses vacuum-induced processes in multilevel atoms

Vacuum-Induced Processes in Multilevel Atoms

We review the concepts and the present state of theoretical studies of spin-imbalanced superfluidity, in particular the elusive Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, in the context of ultracold quantum gases. The comprehensive presentation of the theoretical basis for the FFLO state that we provide is useful also for research on the interplay between magnetism and superconductivity in other physical systems. We focus on settings that have been predicted to be favourable for the FFLO state, such as optical lattices in various dimensions and spin-orbit coupled systems. These are also the most likely systems for near-future experimental observation of the FFLO state. Theoretical bounds, such as Bloch's and Luttinger's theorems, and experimentally important limitations, such as finite-size effects and trapping potentials, are considered. In addition, we provide a comprehensive review of the various ideas presented for the observation of the FFLO state. We conclude our review with an analysis of the open questions related to the FFLO state, such as its stability, superfluid density, collective modes and extending the FFLO superfluid concept to new types of lattice systems.

https://iopscience.iop.org/article/10.1088/1361-6633/aaa4ad/pdf

The Fulde-Ferrell-Larkin-Ovchinnikov state for ultracold fermions in lattice and harmonic potentials: a review.

First-order correlation functions of the electric field and photon-number probabilities in the case of a three-level atom interacting with two-mode radiation field are obtained to investigate the coherent properties of the stimulated emission under different initial conditions. It is found that double stimulation will cause the field to approach its initial coherent state and the oscillation of the photon-number probabilities to collapse and revive.

Coherent properties of the stimulated emission from a three-level atom.

Atomic inversion is studied in the Jaynes-Cummings model with a Kerr nonlinearity when the interacting field is initially prepared in a binomial state (BS), a thermal state (TS), or a squeezed vacuum (SV). The inversion can, under suitable conditions, show a very steady beat phenomenon with a time evolution very similar to that found in classical physics for the BS, and even for the TS and SV. This means that in this system any field can reveal its ``granularity.''

"Classical" beat phenomena from the Jaynes-Cummings model with a Kerr nonlinearity.

Kth-power amplitude squeezing (KPAS) is introduced and examined in the nonlinear-harmonic-oscillator model and the Jaynes-Cummings model with an intensity coupling. We discover that, in these two cases, the long-time behavior of KPAS (k\ensuremath{\le}4) for the strong initial field periodically displays a collapse-revival-valley and squeezed collapse-revival phenomenon, respectively. For a nonclassical state, the relation between KPAS and ordinary squeezing, Hong-Mandel squeezing, and antibunching is investigated. The results show that KPAS is another intrinsic and independent nonclassical effect inequivalent to the last three.

Higher-order squeezing for the quantized light field : Kth-power amplitude squeezing

By using the reservoir theory and Dekker's quantization procedure for a dissipation system, a set of new surface-dressed optical Bloch equations including the surface-induced frequency shift as well as the decay rate is derived to calculate the resonance fluorescence spectrum of an adatom near a bulk solid surface. A mediate layer composed of nonabsorbing dielectric is placed to separate the adatom and the bulk solid. The size and the dielectric effects of this layer are found to be as important as those effects from the bulk solid. These effects will be directly exhibited in the decay rates and the frequency shifts. The resonance fluorescence spectrum is discussed in different cases and is found to be strongly affected by the surfaces.

Resonance fluorescence of an adatom near solid surfaces.

The interaction between electromagnetic radiation and two three-level atoms is investigated. The atom-atom separation is assumed to be smaller than the corresponding mean resonance wavelength. The total radiance rate of the atomic system is calculated as a function of time. Explicit results are given for several different initial states of the atomic system. Some of them exhibit superradiance, and some act as photon-trapping states. Instead of exponential decay, the oscillations appear in the time evolution of the radiation rate due to the correlation between atoms. The decay constants and coupling decay constants show strong effects on the emission rates. Completely differing from two identical double-level atoms, the cooperative effects always exist in the three-level atomic system except for the case in which the coupling decay constants ${\ensuremath{\gamma}}_{12}$ and ${\ensuremath{\gamma}}_{21}$ are zero.

Chang-de Gong

Papers

Coherent properties of the stimulated emission from a three-level atom.

"Classical" beat phenomena from the Jaynes-Cummings model with a Kerr nonlinearity.

Higher-order squeezing for the quantized light field : Kth-power amplitude squeezing

Resonance fluorescence of an adatom near solid surfaces.

Spontaneous emission by two three-level atoms