The Jaynes-Cummings model reveals the fascinating interplay between atoms and light in cavities. Dressed states, formed by this interaction, create a unique energy structure called the Jaynes-Cummings ladder.

These dressed states exhibit intriguing spectral properties, like vacuum Rabi splitting. Understanding their dynamics and dissipation helps us grasp how quantum systems behave when atoms and light strongly interact.

Dressed states in Jaynes-Cummings model

Concept and formation of dressed states

• Dressed states are the eigenstates of the Jaynes-Cummings Hamiltonian describing the interaction between a two-level atom and a single mode of the electromagnetic field in a cavity
• Result from the strong coupling between the atom and the cavity field leading to the formation of entangled states that are a combination of the atomic and photonic states
• Labeled as |n, ±⟩ where n represents the number of photons in the cavity and ± indicates the superposition of the atomic excited and ground states (|e⟩ and |g⟩)

Energy level shifts and Jaynes-Cummings ladder

• Energy levels of the dressed states are shifted from the bare states (uncoupled atom and field states) due to the atom-field interaction
• Shifts result in the formation of the Jaynes-Cummings ladder, a unique energy level structure
• Ground state of the ladder is the |0, -⟩ state, a superposition of the atomic ground state and the cavity vacuum state
• As the number of photons in the cavity increases, the energy splitting between the dressed states within each rung also increases leading to a nonlinear energy level structure

Eigenstates and eigenvalues of the Jaynes-Cummings Hamiltonian

Jaynes-Cummings Hamiltonian

• Hamiltonian given by $H = ħω_a σ_z/2 + ħω_c a^†a + ħg(a^†σ_- + aσ_+)$
  • $ω_a$ is the atomic transition frequency
  • $ω_c$ is the cavity frequency
  • $g$ is the coupling strength
  • $σ_z, σ_+, and σ_-$ are the Pauli operators
  • $a^†$ and $a$ are the creation and annihilation operators for the cavity field
• Diagonalized in the basis of the uncoupled states $|e, n⟩$ and $|g, n+1⟩$ to find eigenstates and eigenvalues
  • $e$ and $g$ represent the excited and ground states of the atom
  • $n$ is the number of photons in the cavity

Eigenstates and eigenvalues

• Eigenstates of the Jaynes-Cummings Hamiltonian are the dressed states $|n, ±⟩ = c_n± |e, n⟩ ± s_n± |g, n+1⟩$
  • $c_n±$ and $s_n±$ are the coefficients determined by the diagonalization procedure
• Eigenvalues of the dressed states are $E_n± = ħ(n+1/2)ω_c ± ħΩ_n/2$
  • $Ω_n = √(Δ^2 + 4g^2(n+1))$ is the generalized Rabi frequency
  • $Δ = ω_a - ω_c$ is the detuning between the atomic transition and the cavity frequency

Energy level structure and Jaynes-Cummings ladder

Ladder-like energy level structure

• Energy level structure of the Jaynes-Cummings model forms a ladder-like pattern known as the Jaynes-Cummings ladder
• Each rung of the ladder consists of a pair of dressed states $|n, ±⟩$, separated by the generalized Rabi frequency $Ω_n$
• Ladder structure arises from the coupling between the atom and the cavity field causing a splitting of the energy levels compared to the uncoupled system

Nonlinear energy level spacing

• As the number of photons in the cavity increases, the energy splitting between the dressed states within each rung also increases
• Leads to a nonlinear energy level structure
• Spacing between adjacent rungs is not constant and depends on the photon number $n$
• Nonlinearity is a consequence of the square root dependence of the generalized Rabi frequency $Ω_n$ on the photon number

Spectral properties of the Jaynes-Cummings model

Vacuum Rabi splitting

• One of the key spectral features is the vacuum Rabi splitting occurring when the atom-cavity coupling strength $g$ exceeds the decay rates of the atom and the cavity
• Manifests as a doublet in the absorption and emission spectra with two peaks separated by the generalized Rabi frequency $Ω_0 = 2g$ at the resonance condition ($Δ = 0$)
• Magnitude of the vacuum Rabi splitting provides a direct measure of the strength of the atom-cavity coupling and is a signature of the strong coupling regime

Spectral linewidths and decoherence

• Spectral linewidths of the dressed states are determined by the decay rates of the atom and the cavity
• Linewidths can be influenced by various decoherence mechanisms such as atomic spontaneous emission, cavity losses, and dephasing
• Decoherence leads to the broadening of the spectral lines and the reduction of the visibility of the vacuum Rabi splitting
• Study of the spectral properties allows for the characterization of the coherence and dissipation in the Jaynes-Cummings system

Dressed-state formalism for Jaynes-Cummings dynamics

Time evolution and Rabi oscillations

• Dressed-state formalism provides a powerful tool for describing the dynamics of the Jaynes-Cummings system, including the evolution of the atomic and photonic states
• Time evolution of the system can be expressed in terms of the dressed states rotating at their respective eigenfrequencies determined by the generalized Rabi frequency $Ω_n$
• Rabi oscillations between the dressed states within each rung of the Jaynes-Cummings ladder can be observed with the oscillation frequency given by $Ω_n$

Dynamics and initial state preparation

• Dynamics of the system can be influenced by the initial state preparation such as the coherent excitation of the atom or the cavity field
• Different initial states lead to different dynamical behaviors and allow for the study of various phenomena
  • Collapses and revivals of the atomic population
  • Entanglement dynamics between the atom and the cavity field
  • Generation of non-classical states of light (Fock states, squeezed states)

Dissipative dynamics and decoherence

• Dissipative dynamics of the Jaynes-Cummings system can be incorporated into the dressed-state formalism by including the decay rates of the atom and the cavity
• Leads to the description of damped Rabi oscillations and the decay of coherences
• Decoherence mechanisms such as atomic spontaneous emission and cavity losses can be modeled using master equation approaches
• Study of the dissipative dynamics allows for the understanding of the interplay between coherent interactions and dissipation in the Jaynes-Cummings system