Dynamics of a quantum system coupled to an environment (bath) — decoherence, dissipation, relaxation, and loss of purity. Master-equation formalism: Redfield (non-secular Born-Markov), Lindblad / GKSL (1976 Gorini-Kossakowski-Sudarshan,…
open-quantum-systems
Environment-induced decoherence: off-diagonals decay exp(−t/T₂)
Zeh 1970 / Zurek 1981-1991 / Joos-Zeh 1985: coupling of a quantum system to a macroscopic environment causes the off-diagonal coherences of…
Born-Markov: weak coupling + memoryless bath → Redfield / Lindblad master equation
The Nakajima-Zwanzig projection-operator formalism yields the exact non-local-in-time integro-differential equation for ρ_S(t); the Born +…
Redfield equation: non-secular Born-Markov master equation
Redfield 1957 nuclear-magnetic-resonance relaxation theory, repurposed for general Born-Markov open dynamics: non-secular…
Lindblad trace preservation: Tr(dρ/dt) = 0 from canonical form
Gorini-Kossakowski-Sudarshan 1976 / Lindblad 1976: the most general generator of a completely-positive trace-preserving (CPTP) quantum…
Markov validity: τ_B ≪ τ_S; error ≈ (τ_B/τ_S)²
Breuer-Petruccione §3 / Rivas-Huelga 2011: the Markov approximation in Born-Markov master-equation theory replaces the non-local-in-time…
Fermi golden rule decoherence: Γ = (2π/ℏ) |⟨f|V|i⟩|² ρ(E_f)
Dirac 1927 / Fermi 1950 golden rule: transition rate from initial eigenstate |i⟩ of H₀ to a continuum / quasi-continuum of final states…
Caldeira–Leggett dissipative bath: Ohmic spectral density J(ω) = η·ω; Legendre-transform path integral
Caldeira–Leggett model of quantum dissipation. Setup: a system coordinate q coupled linearly to an infinite bath of harmonic oscillators,…
Jaynes–Cummings dressed states: H = ℏω(a†a + σ_z/2) + ℏg(aσ₊ + a†σ₋); harmonic Rabi oscillations
Jaynes–Cummings model: a two-level atom + single cavity mode at resonance under the rotating-wave approximation. Setup: atomic Hamiltonian…
Quantum Zeno effect: P^N_survive → 1 as measurement rate N → ∞; trace-class projection limit
Quantum Zeno effect: frequent measurement freezes a quantum state's evolution. Setup: an initial state |ψ⟩, a Hamiltonian H, a projector P…
Caldeira–Leggett classical damping: η = π·J_c²·ρ/2; FD theorem ⟨f(t)f(t')⟩ = 2η·kB·T·δ(t−t')
Caldeira–Leggett damping coefficient and fluctuation-dissipation. From framework F7, integrating out the Ohmic bath gives a Langevin…
Jaynes–Cummings vacuum Rabi splitting: Ω_n = g·√(n+1); dressed-state gap = 2ℏg·√(n+1)
Jaynes–Cummings vacuum Rabi splitting. From framework F8, the dressed-state energies for |n,±⟩ are E± = ℏω·(n+1/2) ± ℏg·√(n+1). The…
Zeno suppression: P_survive → 1 via N measurements; effective decay Γ_eff = Γ²·t/N → 0
Quantum Zeno suppression of decay. From framework F9, the N-fold Zeno probability P_surv(t, N) = (1 − (Γt/N)²/2)^N tends to 1 as N → ∞ for…
Lindblad (1976)
G Lindblad 1976 most-general Markovian master-equation preserving complete-positivity + trace; basis of decoherence + quantum-noise…
Quantum trajectories (Carmichael 1993)
H J Carmichael 1993 quantum-jump unraveling; numerical-efficiency over density-matrix; modern stochastic-Schrodinger-equation simulations.
Decoherence (Zurek 1981)
W Zurek 1981 environment-induced superselection; pointer-states; modern Schlosshauer 2007 'Decoherence and the Quantum-to-Classical…
Non-Markovian (BLP 2009)
Breuer-Laine-Piilo 2009 non-Markovianity measure via trace-distance; modern memory-effects characterization in quantum-information.
Born-Markov-secular
Born + Markov + secular-approximations standard derivation of master-eq; modern numerically-exact alternatives HEOM Tanimura 1989.
Quantum Zeno (1977)
Misra-Sudarshan 1977 frequent-measurement freezes evolution; experimental Itano 1990 trapped-ion; modern Zeno-protected logical-qubits.
Lindblad (1976)
G Lindblad 1976 + Gorini-Kossakowski-Sudarshan 1976 GKLS; modern modern foundational text + post-Markovian + Davies-microscopic-derivation.
Nakajima-Zwanzig (1958)
S Nakajima 1958 + R Zwanzig 1960 projection-operator; modern modern foundational text + non-Markovian + memory-kernel + dissipation.
Davies derivation (1974)
E Davies 1974 weak-coupling-limit; modern modern foundational text + secular + RWA + microscopic-derivation thermal-bath.
Breuer-Petruccione (2002)
H-P Breuer-F Petruccione 2002 'Theory of Open Quantum Systems'; modern modern foundational text + non-Markovian-quantification BLP-measure.
Decoherence (Zurek 1981)
W Zurek 1981 + Joos-Zeh 1985; modern modern foundational text + einselection + quantum-Darwinism Zurek 2009.
Collisional model (Rau 1963)
J Rau 1963 + Scarani 2002 + Ciccarello 2017 collisional-models; modern modern foundational + repeated-interaction quantum-thermo bath.