👀Quantum Optics Unit 6 – Jaynes–Cummings Model

Key Concepts and Foundations

• Jaynes–Cummings Model (JCM) describes the interaction between a two-level atom and a single mode of the quantized electromagnetic field
• Assumes a single atom interacting with a single mode of the cavity field, neglecting any dissipative processes
• Two-level atom characterized by ground state $|g\rangle$ and excited state $|e\rangle$, separated by energy $\hbar\omega_a$
  • Atom modeled as a simple dipole, interacting with the electric field of the cavity mode
• Single mode of the quantized electromagnetic field described by the creation and annihilation operators, $\hat{a}^\dagger$ and $\hat{a}$, respectively
  • Field mode has a frequency $\omega_f$, close to the atomic transition frequency $\omega_a$
• Interaction between the atom and the field leads to the phenomena of Rabi oscillations and collapses and revivals of the atomic population
• JCM serves as a fundamental model for understanding light-matter interactions at the quantum level, with applications in quantum optics, quantum information processing, and cavity quantum electrodynamics (cavity QED)

Mathematical Framework

• Hamiltonian of the Jaynes–Cummings Model consists of three parts: atomic Hamiltonian $\hat{H}_a$, field Hamiltonian $\hat{H}_f$, and interaction Hamiltonian $\hat{H}_I$
  • $\hat{H} = \hat{H}_a + \hat{H}_f + \hat{H}_I$
• Atomic Hamiltonian: $\hat{H}_a = \frac{1}{2}\hbar\omega_a\hat{\sigma}_z$, where $\hat{\sigma}_z$ is the Pauli Z operator
• Field Hamiltonian: $\hat{H}_f = \hbar\omega_f(\hat{a}^\dagger\hat{a} + \frac{1}{2})$, describing a quantum harmonic oscillator
• Interaction Hamiltonian: $\hat{H}_I = \hbar g(\hat{a}^\dagger\hat{\sigma}_- + \hat{a}\hat{\sigma}_+)$, where $g$ is the coupling strength, and $\hat{\sigma}_+$ and $\hat{\sigma}_-$ are the atomic raising and lowering operators
• Rotating Wave Approximation (RWA) applied to simplify the interaction Hamiltonian by neglecting the rapidly oscillating terms $\hat{a}^\dagger\hat{\sigma}_+$ and $\hat{a}\hat{\sigma}_-$
• Jaynes–Cummings Hamiltonian in the RWA: $\hat{H}_{JC} = \frac{1}{2}\hbar\omega_a\hat{\sigma}_z + \hbar\omega_f\hat{a}^\dagger\hat{a} + \hbar g(\hat{a}^\dagger\hat{\sigma}_- + \hat{a}\hat{\sigma}_+)$
• Eigenstates of the Jaynes–Cummings Hamiltonian are the dressed states, denoted as $|n,\pm\rangle$, which are superpositions of the bare atom-field states $|g,n\rangle$ and $|e,n-1\rangle$

Quantum Dynamics

• Time evolution of the system governed by the Schrödinger equation: $i\hbar\frac{d}{dt}|\psi(t)\rangle = \hat{H}_{JC}|\psi(t)\rangle$
• Rabi oscillations: periodic exchange of energy between the atom and the field, characterized by the Rabi frequency $\Omega_n = 2g\sqrt{n}$, where $n$ is the number of photons in the field
  • Rabi frequency depends on the square root of the photon number, demonstrating the quantum nature of the field
• Collapses and revivals: periodic disappearance and reappearance of Rabi oscillations due to the discrete nature of the photon number distribution
  • Collapse time: $t_c \approx \frac{1}{2g\sqrt{\bar{n}}}$, where $\bar{n}$ is the average photon number
  • Revival time: $t_r \approx \frac{2\pi\sqrt{\bar{n}}}{g}$
• Jaynes–Cummings ladder: energy level structure of the dressed states, with energy splitting between adjacent levels given by the generalized Rabi frequency $\Omega_n$
• Vacuum Rabi splitting: energy splitting between the dressed states $|0,\pm\rangle$ in the absence of photons, equal to $2g$

Atom-Field Interactions

• Dipole interaction between the atom and the electric field of the cavity mode, described by the interaction Hamiltonian $\hat{H}_I$
• Coupling strength $g$ depends on the atomic dipole moment $\vec{\mu}$ and the electric field per photon $\vec{E}_0$: $g = -\vec{\mu} \cdot \vec{E}_0$
  • Coupling strength determines the strength of the interaction and the timescale of the quantum dynamics
• Resonant interaction: when the atomic transition frequency $\omega_a$ is close to the field frequency $\omega_f$, leading to efficient energy exchange between the atom and the field
• Off-resonant interaction: when there is a detuning $\Delta = \omega_a - \omega_f$ between the atomic transition frequency and the field frequency, resulting in modified Rabi frequency and reduced energy exchange
• Lamb shift: shift in the atomic energy levels due to the interaction with the vacuum field fluctuations, which can be incorporated into the Jaynes–Cummings Model as a renormalization of the atomic transition frequency

Experimental Realizations

• Cavity Quantum Electrodynamics (cavity QED): experimental platform for realizing the Jaynes–Cummings Model, using atoms or quantum dots coupled to high-finesse optical or microwave cavities
  • Examples: Fabry-Pérot cavities, whispering gallery mode resonators, photonic crystal cavities
• Superconducting Circuit QED: implementation of the Jaynes–Cummings Model using superconducting qubits (artificial atoms) coupled to microwave resonators
  • Advantages: strong coupling, tunability, scalability, and long coherence times
• Trapped ions: realization of the Jaynes–Cummings Model using the interaction between the internal states of a trapped ion and the quantized motion of the ion in the trap
• Optomechanical systems: extension of the Jaynes–Cummings Model to the interaction between a cavity field and a mechanical oscillator, enabling the study of quantum effects in macroscopic systems

Applications and Implications

• Quantum information processing: Jaynes–Cummings Model provides a platform for implementing quantum gates, quantum state transfer, and quantum entanglement between atoms and photons
  • Examples: quantum computing, quantum communication, quantum cryptography
• Quantum simulation: Jaynes–Cummings Model can be used to simulate other quantum systems, such as spin systems or many-body physics, by mapping the system onto the atom-field interaction
• Quantum metrology: Jaynes–Cummings Model enables high-precision measurements of physical quantities, such as time, frequency, or electric fields, by exploiting the sensitivity of the atom-field interaction to these parameters
• Quantum optics: Jaynes–Cummings Model provides a foundation for understanding and controlling the interaction between light and matter at the quantum level, with applications in nonlinear optics, quantum light sources, and quantum memories
• Fundamental tests of quantum mechanics: Jaynes–Cummings Model allows for the investigation of fundamental concepts in quantum mechanics, such as entanglement, decoherence, and the quantum-classical boundary

Limitations and Extensions

• Validity of the Rotating Wave Approximation (RWA): RWA breaks down in the ultrastrong coupling regime, where the coupling strength becomes comparable to the atomic transition frequency or the field frequency
  • Extensions: Jaynes–Cummings Model without RWA, leading to new physical effects such as the Bloch-Siegert shift and the breakdown of the conservation of excitation number
• Dissipation and decoherence: Jaynes–Cummings Model in its basic form does not include the effects of dissipation and decoherence, which can limit the coherence time and the fidelity of quantum operations
  • Extensions: Master equation approach, incorporating dissipation and decoherence through the coupling to external reservoirs (e.g., atomic spontaneous emission, cavity decay)
• Multi-atom and multi-mode extensions: Jaynes–Cummings Model can be extended to include multiple atoms interacting with a single field mode (Tavis-Cummings Model) or a single atom interacting with multiple field modes (multi-mode Jaynes–Cummings Model)
  • Applications: Dicke superradiance, quantum phase transitions, quantum many-body physics
• Ultrastrong and deep strong coupling regimes: When the coupling strength becomes comparable to or larger than the atomic transition frequency or the field frequency, new physical phenomena emerge, requiring more advanced theoretical treatments beyond the Jaynes–Cummings Model
  • Examples: Quantum Rabi Model, Dicke Model, Hopfield Model

Problem-Solving Techniques

• Dressed state basis: Transforming the Hamiltonian into the basis of the dressed states $|n,\pm\rangle$, which diagonalizes the Jaynes–Cummings Hamiltonian and simplifies the calculation of the system dynamics
• Rotating frame: Moving to a rotating frame that eliminates the time-dependent factors in the Hamiltonian, simplifying the equations of motion and the interpretation of the results
• Quantum master equations: Incorporating dissipation and decoherence effects by deriving and solving quantum master equations for the reduced density matrix of the system
  • Examples: Lindblad master equation, Bloch-Redfield master equation
• Numerical simulations: Using numerical methods to solve the time-dependent Schrödinger equation or the master equation, especially in cases where analytical solutions are not available or the system is too complex
  • Techniques: Runge-Kutta methods, split-operator methods, quantum Monte Carlo methods
• Perturbation theory: Applying perturbative techniques to solve the Jaynes–Cummings Model in the presence of additional interactions or detunings, such as the dispersive regime or the ultrastrong coupling regime
  • Examples: Rotating Wave Approximation, adiabatic elimination, effective Hamiltonian approach
• Quantum trajectory simulations: Simulating the stochastic evolution of the system under continuous measurements, such as homodyne or heterodyne detection, using quantum trajectory methods
  • Examples: Quantum jump approach, quantum state diffusion, stochastic master equations