Model Index
Use this index to find a model by the physical question it answers. A Hamiltonian model specifies joint interactions; a channel specifies a finite-time transformation; a master equation adds an approximate dynamical description. They can describe different aspects of the same experiment.
Required background. Closed vs Open Quantum Systems explains how retaining or discarding degrees of freedom changes the description.
Measurement and qubit noise models
Section titled “Measurement and qubit noise models”| Model and reading | Physical question |
|---|---|
| von Neumann measurement | How does a system variable become correlated with a pointer position? |
| Quantum instrument | What are the outcome probabilities and the state after each recorded outcome? |
| Dephasing channel and pure-dephasing generator | How does coherence change while populations in a specified basis remain fixed? |
| Amplitude-damping channel and damping generator | How does an excitation relax, and what happens to coherence at the same time? |
| Depolarizing channel | What changes under isotropic qubit noise used as a benchmark? |
Dephasing and damping cannot reliably be distinguished from one initial state alone. A thermal damping process leaves equilibrium populations unchanged, even though other inputs relax. Compare several preparations or a physically justified microscopic model; Common Channels gives an example.
Baths, dissipation, and light–matter coupling
Section titled “Baths, dissipation, and light–matter coupling”| Model and reading | Retained system and interaction |
|---|---|
| Spin–boson model | A two-level system coupled to oscillator modes; tunneling and the coupling axis determine whether the interaction produces pure dephasing or relaxation |
| Caldeira–Leggett model | A coordinate coupled linearly to oscillator coordinates, with an explicit counterterm convention |
| Thermal oscillator | A harmonic mode exchanging quanta with a thermal reservoir in a weak-coupling limit |
| Central-spin example | One retained spin interacting with environmental spins, which need not have Gaussian oscillator statistics |
| Jaynes–Cummings model | Coherent excitation exchange between a two-level system and one mode under a rotating-wave approximation |
| Quantum optical master equation | Atoms and modes with emission, loss, pumping, or dephasing from specified reservoirs |
| Input–output theory | Cavity or waveguide dynamics connected to the fields entering and leaving a port |
| Circuit QED | Microwave qubit–resonator coupling and dispersive readout |
| Quantum optics and stochastic master equations | Optical states conditioned on counting or quadrature records |
The closed Jaynes–Cummings Hamiltonian does not contain cavity leakage. A damped model combines a coherent Hamiltonian with reservoir terms, but the appropriate jumps depend on coupling and frequency resolution. Local bare-mode damping is not automatically accurate in every strong-coupling regime.
Likewise, the Caldeira–Leggett Hamiltonian and its high-temperature master equation are different levels of description. The Hamiltonian does not require the high-temperature or Markov approximations used in that particular reduced equation. See Common Master Equations.
Explicit environments and methods for memory
Section titled “Explicit environments and methods for memory”| Model or method | Useful structure |
|---|---|
| Collision models | Sequential interactions with auxiliary systems; fresh, reused, or correlated auxiliaries give different reduced processes |
| Pseudomode methods | Structured reservoir correlations represented by suitable damped auxiliary modes |
| Reaction-coordinate mapping | A collective bath coordinate is retained explicitly in an enlarged system |
| Hierarchical equations of motion | Gaussian bath correlations expanded into a hierarchy of auxiliary operators |
The last three are representations or computational methods as well as modeling strategies. They do not identify one universal kind of bath. Check the correlation expansion, residual-bath approximation, and numerical truncation relevant to the chosen method.
Independent fresh auxiliaries give a sequence of channels at collision boundaries. Identical collisions produce a discrete semigroup under repetition, but a continuous-time GKLS limit requires an appropriate scaling of interaction strength and collision duration. Reused auxiliaries can carry correlations back to the system; reuse permits memory but does not prove a particular memory signature for every initial state.
Exercises
Section titled “Exercises”Choose the system boundary. A qubit exchanges energy coherently with a narrow cavity resonance, which weakly leaks into a broad external field. What is a useful enlarged system?
Solution
Retain the qubit and cavity mode together, and model leakage to the external field separately. Coherent qubit–cavity exchange is then explicit. Tracing out the cavity too early can hide a long correlation time and produce an inaccurate qubit-only Markov model.
A formula with the same name. Does solving a spin–boson Hamiltonian exactly require its weak-coupling master equation to be valid?
Solution
No. The microscopic Hamiltonian can be studied beyond the regime of that reduced approximation. The question is whether the Hamiltonian represents the experiment and whether the chosen analytical or numerical solution is controlled.
References
Section titled “References”- Breuer, H.-P., and F. Petruccione. The Theory of Open Quantum Systems. Oxford University Press, 2002. Microscopic models and reduced descriptions.
- Carmichael, H. J. Statistical Methods in Quantum Optics 1: Master Equations and Fokker–Planck Equations. Springer, 1999. Light–matter and damping models.
- Leggett, A. J., et al. “Dynamics of the dissipative two-state system.” Reviews of Modern Physics 59, 1–85 (1987). doi:10.1103/RevModPhys.59.1.
- Weiss, U. Quantum Dissipative Systems. 4th edition, World Scientific, 2012. Oscillator baths and dissipative models.