Jaynes–Cummings model

In quantum optics, the Jaynes–Cummings model (sometimes abbreviated JCM) is a theoretical model that describes the system of a two-level atom interacting with a quantized mode of an optical cavity (or a bosonic field), with or without the presence of light (in the form of a bath of electromagnetic radiation that can cause spontaneous emission and absorption).

It was originally developed to study the interaction of atoms with the quantized electromagnetic field in order to investigate the phenomena of spontaneous emission and absorption of photons in a cavity.

The model was originally developed in a 1963 article by Edwin Jaynes and Fred Cummings to elucidate the effects of giving a fully quantum mechanical treatment to the behavior of atoms interacting with an electromagnetic field.

In order to simplify the math and allow for a tractable calculation, Jaynes and Cummings restricted their attention to the interaction of an atom with a single mode of quantum electromagnetic field.

This approach is in contrast to the earlier semi-classical method, in which only the dynamics of the atom are treated quantum mechanically, while the field with which it interacts is assumed to behave according to classical electromagnetic theory.

The quantum mechanical treatment of the field in the Jaynes–Cummings model reveals a number of novel features, including: To realize the dynamics predicted by the Jaynes–Cummings model experimentally requires a quantum mechanical resonator with a very high quality factor so that the transitions between the states in the two-level system (typically two energy sub-levels in an atom) are coupled very strongly by the interaction of the atom with the field mode.

In 1985, several groups using Rydberg atoms along with a maser in a microwave cavity demonstrated the predicted Rabi oscillations.

[5] It was not until 1987 that Gerhard Rempe, Herbert Walther, and Norbert Klein were finally able to use a single-atom maser to demonstrate the revivals of probabilities predicted by the model.

This successful demonstration of dynamics that could only be explained by a quantum mechanical model of the field spurred further development of high quality cavities for use in this research.

With the advent of one-atom masers it was possible to study the interaction of a single atom (usually a Rydberg atom) with a single resonant mode of the electromagnetic field in a cavity from an experimental point of view,[11][12] and study different aspects of the Jaynes–Cummings model.

[13] A quantum dot inside a photonic crystal nano-cavity is also a promising system for observing collapse and revival of Rabi cycles in the visible light frequencies.

Various experiments have demonstrated the dynamics of the Jaynes–Cummings model in the coupling of a quantum dot to the modes of a micro-cavity, potentially allowing it to be applied in a physical system of much smaller size.

[15][16][17][18] Other experiments have focused on demonstrating the non-linear nature of the Jaynes–Cummings ladder of energy levels by direct spectroscopic observation.

These experiments have found direct evidence for the non-linear behavior predicted from the quantum nature of the field in both superconducting circuits containing an artificial atom coupled to a very high quality oscillator in the form of a superconducting RLC circuit, and in a collection of Rydberg atoms coupled via their spins.

[19][20] In the latter case, the presence or absence of a collective Rydberg excitation in the ensemble serves the role of the two level system, while the role of the bosonic field mode is played by the total number of spin flips that take place.

[20] Theoretical work has extended the original model to include the effects of dissipation and damping, typically via a phenomenological approach.

For example, it was realized that during the periods of collapsed Rabi oscillations, the atom-cavity system exists in a quantum superposition state on a macroscopic scale.

Such a state is sometimes referred to as a Schrödinger cat, since it allows the exploration of the counter intuitive effects of how quantum entanglement manifests in macroscopic systems.

On the other hand, the two-level atom is equivalent to a spin-half whose state can be described using a three-dimensional Bloch vector.

It is possible, and often very helpful, to write the Hamiltonian of the full system as a sum of two commuting parts:

It is possible in the Heisenberg notation to directly determine the unitary evolution operator from the Hamiltonian:[29]

For ease of illustration, consider the interaction of two energy sub-levels of an atom with a quantized electromagnetic field.

Where we have made the following definitions: Next, the analysis may be simplified by performing a passive transformation into the so-called "co-rotating" frame.

So the probabilities to find the system in the ground or excited states after interacting with the cavity for a time

In this case, there was only a single quantum in the atom-field system, carried in by the initially excited atom.

As explained below, this discrete spectrum of frequencies is the underlying reason for the collapses and subsequent revivals probabilities in the model.

This non-linear splitting effect is purely quantum mechanical, and cannot be explained by any semi-classical model.

If the mean photon number is large, then since the statistics of the coherent state are Poissonian we have that the variance-to-mean ratio is

This result shows that the probability of occupation of the excited state oscillates with effective frequency

If the field were classical, the frequencies would have a continuous spectrum, and such re-phasing could never occur within a finite time.