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Jaynes–Cummings model

Jaynes–Cummings model is a physics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Jaynes–Cummings model rather than just read about it. In short: 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). The model assumes the rotating-wave approximation, neglects dissipation initially, and treats only a single field mode and a two-level atom, with or without the presence of light (in the form of a bat…

Jaynes–Cummings model — main illustration
Jaynes–Cummings model — illustration

Key takeaways

  • Jaynes–Cummings model belongs to physics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Jaynes–Cummings model to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Jaynes–Cummings model from memory before moving on to harder problems.

Reference excerpt

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). The model assumes the rotating-wave approximation, neglects dissipation initially, and treats only a single field mode and a two-level atom, 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. It is named after Edwin Thompson Jaynes and Fred Cummings in the 1960s and was confirmed experimentally in 1987. The Jaynes–Cummings model is of great interest to atomic physics, quantum optics, solid-state physics and quantum information circuits, both experimentally and theoretically. Journal special issues have commemorated the 50th anniversary, (which contains numerous relevant articles, including two interesting editorials, one by Cummings), and 60th anniversary. It also has applications in coherent control and quantum information processing.

History

1963: Jaynes and Cummings 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 mathematics and allow for a tractable calculation, Jaynes and Cummings restricted their attention to the interaction of an atom with a single mode of a quantised electromagnetic field. (See below for further mathematical details.) 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:

The existence of Rabi oscillations between the states of the two-level system as it interacts with the quantum field. This was originally believed to be a purely quantum mechanical effect, although a semi-classical explanation for it was later provided in terms of linear dispersion and absorption A ladder of quantized energy levels, called the Jaynes–Cummings ladder, that scales in energy non-linearly as n {\displaystyle {\sqrt {n}}} where n {\displaystyle n} is the total number of quanta in the coupled system. This quantization of energies and non-linear scaling is purely quantum mechanical in nature. The collapse and subsequent revivals of the probability to detect the two-level system in a given state when the field is initially in a coherent state. While the collapse has a simple classical explanation, the revivals can only be explained by the discreteness of the energy spectrum due to the quantum nature of the field. 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. This simultaneously suppresses any coupling between other sub-levels in atom and coupling to other modes of the field, and thus makes any losses small enough to observe the dynamics predicted by the Jaynes–Cummings model. Because of the difficulty in realizing such an apparatus, the model remained a mathematical curiosity for quite some time. In 1985, several groups using Rydberg atoms along with a maser in a microwave cavity demonstrated the predicted Rabi oscillations. However, as noted before, this effect was later found to have a semi-classical explanation.

1987: Rempe, Walther and Klein 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. Before that time, research groups were unable to build experimental setups capable of enhancing the coupling of an atom with a single field mode, simultaneously suppressing other modes. Experimentally, the quality factor of the cavity must be high enough to consider the dynamics of the system as equivalent to the dynamics of a single mode field. 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, and study different aspects of the Jaynes–Cummings model. It was found that an hourglass geometry could be used to maximize the volume occupied by the mode, while simultaneously maintaining a high quality factor in order to maximize coupling strength, and thus better approximate the parameters of the model. To observe strong atom-field coupling in visible light frequencies, hour-glass-type optical modes can be helpful because of their large mode volume that eventually coincides with a strong field inside the cavity. 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.

… excerpt ends here. Continue reading the full article.

Illustrations

Jaynes–Cummings model: Illustration of the Jaynes–Cummings model. An atom in an optical cavity is shown as a red dot on the top left. The energy levels of the atom that couple to the field mode within the cavity are shown in the circle on the bottom right. Transfer between the two states causes photon emission (absorption) by the atom into (out of) the cavity mode.
Illustration of the Jaynes–Cummings model. An atom in an optical cavity is shown as a red dot on the top left. The energy levels of the atom that couple to the field mode within the cavity are shown in the circle on the bottom right. Transfer between the two states causes photon emission (absorption) by the atom into (out of) the cavity mode.
Jaynes–Cummings model: A plot of the probability to find the system in the excited state as a function of the unit-less parameter g t {\displaystyle gt} for a system with mean photon number ⟨ n ⟩ = 25 {\displaystyle \langle n\rangle =25} . Note the initial collapse over short times, followed by revival at longer times. This behavior is attributable to the discrete spectrum of frequencies caused by quantization of the field.
A plot of the probability to find the system in the excited state as a function of the unit-less parameter g t {\displaystyle gt} for a system with mean photon number ⟨ n ⟩ = 25 {\displaystyle \langle n\rangle =25} . Note the initial collapse over short times, followed by revival at longer times. This behavior is attributable to the discrete spectrum of frequencies caused by quantization of the field.
Jaynes–Cummings model illustration

Worked examples

Example 1 — a first encounter with Jaynes–Cummings model

Start with the simplest possible case. Write down what Jaynes–Cummings model claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Jaynes–Cummings model before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Jaynes–Cummings model ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Jaynes–Cummings model

In research
Jaynes–Cummings model appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Jaynes–Cummings model in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Jaynes–Cummings model is common in secondary-school and first-year university syllabi. It links to neighbouring topics Quantum optics, so understanding it makes those chapters shorter.
In everyday life
Look for Jaynes–Cummings model outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.
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How to study Jaynes–Cummings model in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Jaynes–Cummings model means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Jaynes–Cummings model out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Jaynes–Cummings model in simple terms?

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). The model assumes the rotating-wave approximation, neglects dissipation initiall…

Why does Jaynes–Cummings model matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Jaynes–Cummings model?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Jaynes–Cummings model.

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  • Quantum optics

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