Jaynes–Cummings model
The Jaynes–Cummings model is a seminal theoretical construct in quantum optics that models the coherent interaction between a two-level quantum system, typically representing an atom or qubit with states separated by transition frequency \omega_0, and a single quantized mode of an electromagnetic field in a cavity with frequency \omega, under the rotating-wave approximation that neglects rapidly oscillating counter-rotating terms.[1] This exactly solvable model, introduced in 1963, captures the quantum mechanical dynamics of light-matter coupling through its Hamiltonian \hat{H} = \hbar \omega \hat{a}^\dagger \hat{a} + \frac{\hbar \omega_0}{2} \hat{\sigma}_z + \hbar g (\hat{a}^\dagger \hat{\sigma}_- + \hat{a} \hat{\sigma}_+ ), where \hat{a}^\dagger and \hat{a} are the field's creation and annihilation operators, \hat{\sigma}_z, \hat{\sigma}_-, and \hat{\sigma}_+ are the atomic Pauli operators, and g denotes the coupling strength.[2] It provides a minimal yet rich description of cavity quantum electrodynamics (QED), highlighting purely quantum effects absent in semiclassical treatments.[1] Originally developed to compare quantum and semiclassical radiation theories in the context of maser operation, the model reveals key phenomena such as Rabi oscillations, where energy oscillates between the atom and field at a frequency $2g \sqrt{n+1} (with n the initial photon number) in the resonant case (\omega = \omega_0), manifesting as vacuum Rabi splitting in the strong-coupling regime where g exceeds dissipation rates.[2] For coherent field states with mean photon number \bar{n} \gg 1, the model predicts collapse and revival of these oscillations: an initial dephasing collapse over timescale t_c \sim 1/(g \sqrt{\bar{n}}) due to the superposition of Rabi frequencies, followed by revivals at t_r \sim 2\pi \sqrt{\bar{n}}/g owing to the discrete photon spectrum, a signature of field quantization first experimentally observed in 1987 using Rydberg atoms in a superconducting cavity.[2] These dynamics underscore the model's role in distinguishing quantum from classical behavior, as validated in early beam maser analyses.[1] Beyond its foundational predictions, the Jaynes–Cummings model has enduring relevance in modern quantum technologies, serving as the paradigmatic description for circuit QED systems where superconducting qubits couple to microwave cavities with coupling ratios exceeding g/\kappa = 200 ( \kappa the cavity decay rate), enabling quantum computing gates, entanglement generation, and simulation of many-body physics like the Dicke model phase transition.[2] Extensions without the rotating-wave approximation yield the quantum Rabi model, incorporating ultrastrong coupling regimes (g \sim 0.1 \omega) observed in solid-state platforms, while lattice variants model photon transport in coupled cavities.[3] Its simplicity and solvability continue to drive advances in quantum information science, with the original paper highly cited, garnering over 5,000 citations as of 2024, reflecting its profound impact on understanding non-classical light-matter interfaces.[4]History
Origins in 1963
The semiclassical Rabi model, introduced by I. I. Rabi in 1936, provided an early framework for understanding the interaction between a two-level atomic system and a classical electromagnetic field, predicting periodic energy exchange known as Rabi oscillations. This approach treated the field as a classical wave, which sufficed for strong-field regimes but failed to account for quantum fluctuations and photon statistics in low-intensity scenarios, such as those involving single atoms or few photons. As interest grew in quantum optical devices like masers during the early 1960s, there arose a need for a fully quantum mechanical treatment that quantized both the atomic dipole and the electromagnetic field to reveal nonclassical effects inherent to the radiation-matter coupling. In response to this need, Edwin T. Jaynes and Fred W. Cummings published their seminal work in 1963, titled "Comparison of quantum and semiclassical radiation theories with application to the beam maser," in the Proceedings of the IEEE. The paper's primary motivation was to rigorously compare the predictions of quantum electrodynamics—where field amplitudes are operators—with the semiclassical approximation, using the beam maser as a concrete test case involving a stream of two-level atoms passing through a resonant cavity. By focusing on the simplest nontrivial system—a single two-level atom interacting with a single quantized cavity mode—they derived an exactly solvable model under the rotating-wave approximation, highlighting how quantum treatment of the field alters the dynamics compared to classical predictions. A central theoretical prediction of the model was the structure of the quantized energy levels, forming a ladder of dressed states where each manifold labeled by the total excitation number consists of a doublet split by an amount proportional to the vacuum Rabi frequency times \sqrt{n+1}, with n denoting the photon number in the initial field state. This \sqrt{n} dependence revealed the quantum scaling of the interaction strength, contrasting with the field-amplitude-linear scaling in semiclassical theory. Initially applied to maser-like systems, the model illuminated nonclassical effects, such as the coherent Rabi oscillations driven by discrete photon exchanges, which underscore the quantized nature of the field and enable phenomena like photon blockade absent in classical descriptions.Experimental confirmation in 1987
The first experimental realization of the Jaynes–Cummings model was achieved in 1987 by Gerhard Rempe, Herbert Walther, and Norbert Klein using a one-atom maser configuration at the Max Planck Institute for Quantum Optics.[5] In this setup, highly excited Rydberg atoms of rubidium were prepared in the 63P_{3/2} state via laser excitation and injected at low flux rates—ensuring on average fewer than one atom present in the interaction region at any time—into a superconducting niobium cavity tuned to the microwave transition frequency of 21.456 GHz between the 63P_{3/2} and 61D_{5/2} states.[5][6] The cavity, a cylindrical resonator with a quality factor Q ≈ 5 × 10^{10}, was cooled to 2 K to suppress thermal excitations and operated in the single-photon regime, where the mean photon number remained low (typically 2.5–5, including about 2 thermal photons).[5][6] A Fizeau velocity selector ensured well-defined atomic transit times through the cavity (around 140–200 μs), allowing precise control over the atom-field interaction duration.[6] The atoms' state after interaction was detected via field ionization, enabling measurement of the population inversion as a function of time. This configuration achieved the strong-coupling regime, where the vacuum Rabi frequency (characterizing the atom-cavity coupling strength g) exceeded both the cavity decay rate κ and the atomic decay rate γ, with g/2π ≈ 25 kHz, κ/2π ≈ 2 kHz, and γ/2π ≈ 0.3 kHz, permitting multiple Rabi cycles during the atomic transit.[5][6] The key observation was the time evolution of the atomic inversion, revealing vacuum Rabi oscillations that collapsed due to dephasing over initial field states and subsequently revived at later times, directly confirming the quantized nature of the field-atom interaction as predicted by the Jaynes–Cummings model.[5] These dynamics, absent in semiclassical treatments, highlighted the discrete photon number effects, with collapse occurring after 50–80 μs and revivals beyond 140 μs for mean photon numbers around 2.5–5.[5][6] The experiment overcame significant challenges, including maintaining ultralow temperatures to minimize thermal photon noise, fabricating a high-Q cavity for long photon lifetimes (τ_c ≈ 0.25 ms), and selecting atomic velocities to isolate coherent dynamics from Doppler broadening.[5][6] This work marked the inaugural demonstration of quantum collapse and revival, validating the model's predictions for a single atom interacting with a quantized cavity mode.[5]Developments since the 1990s
The Tavis–Cummings model, which extends the Jaynes–Cummings model to multi-atom regimes and describes collective interactions of multiple two-level atoms with a single quantized field mode, was introduced in 1968.[7] In the 1990s, theoretical advancements included exact solutions for generalized multi-atom cases, facilitating analysis of symmetric and asymmetric atomic ensembles in cavity quantum electrodynamics (QED), and enabling studies of superradiance and subradiance phenomena. Concurrently, dissipative extensions incorporated open-system dynamics via Lindblad master equations, accounting for cavity decay and atomic dephasing, which revealed collapse-revival patterns modified by environmental noise.[8] The 2000s marked the integration of the model with solid-state systems, particularly semiconductor quantum dots coupled to optical microcavities, where excitonic two-level systems exhibited strong light-matter coupling akin to atomic implementations.[9] Seminal experiments demonstrated vacuum Rabi splitting in quantum dot-cavity hybrids, validating the model's predictions for few-photon interactions and paving the way for scalable quantum emitters.[9] Parallel developments in circuit QED utilized superconducting qubits as artificial atoms interacting with microwave resonators, achieving ultrastrong coupling regimes and enabling precise control of Jaynes–Cummings dynamics at the single-photon level. The 50th anniversary in 2013 prompted a special issue in the Journal of Physics B: Atomic, Molecular and Optical Physics, which highlighted the model's enduring impact, particularly through circuit QED implementations that realized dispersive regimes for quantum information processing. This collection underscored how superconducting circuits extended the model to multi-qubit scenarios, observing collective effects beyond single-atom limits. A 60th anniversary review in 2024, published in the Journal of the Optical Society of America B, titled "The Jaynes–Cummings model: 60 years and still counting," emphasized the model's ongoing relevance in diverse platforms, from trapped ions to nanophotonic structures, while addressing challenges like decoherence mitigation.[10] Recent advances have incorporated the model into hybrid quantum systems, combining disparate platforms such as optomechanical cavities and spin ensembles to explore multimode interactions and entanglement generation.[3] Additionally, investigations beyond the rotating-wave approximation have revealed counter-rotating terms' effects in ultrastrong coupling, altering energy spectra and dynamics in circuit QED setups.[11]Physical system
The two-level atom
In the Jaynes–Cummings model, the atomic component is approximated as a two-level system, consisting of a ground state denoted as |g\rangle and an excited state |e\rangle. This simplification captures the essential quantum behavior relevant to resonant interactions with a quantized electromagnetic field, neglecting higher energy levels that are far detuned from the transition frequency of interest. The approximation holds under conditions where the field-atom coupling is dominated by the near-resonant transition between these two states, such as in cavity quantum electrodynamics experiments with atoms or artificial qubits. The quantum mechanical description of the atom employs the computational basis \{ |e\rangle, |g\rangle \}. The Pauli spin operators are defined in this basis as\sigma_z = |e\rangle\langle e| - |g\rangle\langle g|,
which acts as the population inversion operator,
\sigma_+ = |e\rangle\langle g|,
the raising operator that excites the atom from |g\rangle to |e\rangle, and
\sigma_- = |g\rangle\langle e|,
the lowering operator for de-excitation. These operators satisfy the commutation relations [\sigma_z, \sigma_\pm] = \pm 2 \sigma_\pm and [\sigma_+, \sigma_-] = \sigma_z, analogous to spin-1/2 operators. The free Hamiltonian of the atom, describing its internal energy without field interaction, is given by
H_a = \frac{\hbar \omega_a}{2} \sigma_z,
where \omega_a is the angular frequency of the transition between |e\rangle and |g\rangle, with eigenvalues +\frac{\hbar \omega_a}{2} for |e\rangle and -\frac{\hbar \omega_a}{2} for |g\rangle. This form sets the zero of energy at the midpoint between the levels. Under the electric dipole approximation, the interaction of the atom with the electromagnetic field is mediated by its dipole moment operator, expressed as
\mathbf{d} = \mathbf{d}_0 (\sigma_+ + \sigma_-),
where \mathbf{d}_0 is the transition dipole matrix element \langle e| \mathbf{d} |g \rangle, assumed real and aligned along a principal direction. This form neglects any permanent dipole moments in the bare states |e\rangle and |g\rangle, which vanish due to parity selection rules in centrosymmetric atoms, ensuring the dipole operator only connects the two levels.