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arXiv:2209.14209 · Open Systems & Information Dynamics

Nature and Origin of Operators Entering the Master Equation of an Open Quantum System

Giovanni Spaventa, Paola Verrucchi

Published Jun 1, 2022• 2 citations
arXivHTMLPDF

Abstract

By exploiting the peculiarities of a recently introduced formalism for describing open quantum systems (the parametric representation with environmental coherent states) we derive an equation of motion for the reduced density operator of an open quantum system that has the same structure of the celebrated Gorini–Kossakowski–Sudarshan–Lindblad equation, but holds regardless of Markovianity being assumed. The operators in our result have explicit expressions in terms of the Hamiltonian describing the interactions with the environment, and can be computed once a specific model is considered. We find that, instead of a single set of Lindblad operators, in the general (non-Markovian) case there one set of Lindblad-like operators for each and every point of a symplectic manifold associated to the environment. This intricacy disappears under some assumptions (which are related to Markovianity and the classical limit of the environment), under which it is possible to recover the usual master-equation formalism. Finally, we find such Lindblad-like operators for two different models of a qubit in a bosonic environment, and show that in the classical limit of the environment their renown master equations are recovered.

DOI
10.1142/s123016122250010x

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By exploiting the peculiarities of a recently introduced formalism for describing open quantum systems (the parametric representation with environmental coherent states) we derive an equation of motion for the reduced density operator of an open quantum system that has the same structure of the celebrated Gorini–Kossakowski–Sudarshan–Lindblad equation, but holds regardless of Markovianity being assumed. The operators in our result have explicit expressions in terms of the Hamiltonian describing the interactions with the environment, and can be computed once a specific model is considered. We find that, instead of a single set of Lindblad operators, in the general (non-Markovian) case there one set of Lindblad-like operators for each and every point of a symplectic manifold associated to the environment. This intricacy disappears under some assumptions (which are related to Markovianity and the classical limit of the environment), under which it is possible to recover the usual master-equation formalism. Finally, we find such Lindblad-like operators for two different models of a qubit in a bosonic environment, and show that in the classical limit of the environment their renown master equations are recovered.
We develop a framework that allows us to describe the dynamics of the total state of an open quantum system and its bosonic environment in the usual Born (weak coupling) and Markov approximation. By shifting the whole time-dependence into an unnormalized s-operator of the open system, the full dynamics is captured by an s-master equation of similar structure than the well-known Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation for the reduced dynamics. By varying the ordering parameter s (0 ≤ s ≤ 1) we obtain the partial Husimi representation (s = 0) and the partial Glauber-Sudarshan representation (s = 1) for the dynamics of the total state. For the reduced density operator the GKSL master equation can be derived easily. The case of s = 1/2, leading to a partial Wigner representation, is helpful to study the overlap of states in the total Hilbert space of system and environment.
Abstract Chapter 22 investigates the dynamics of open quantum systems that interact with their environment. We discuss the time-evolution of the joint system in terms of a time-ordered exponential of an integral over the Hamiltonian and determine a similar formulation for a Liouvillian super-operator. Based on these ideas we develop the notion of dynamical maps and apply the Born-Markov approximation to arrive at a dynamical semi-group and general Markovian master equations. We then turn to a specific example of such an equation of motion for the density operator: the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) equation which captures both dissipative dynamics and decoherence using Lindblad operators. We employ the GKLS equation to model the absorption, and spontaneous as well as stimulated emission of radiation by a two-level atom. We then discuss the Jaynes-Cummings model that describes the interaction of such an atom with a single mode of the electromagnetic field in a cavity
Derivation of a quantum master equation for a system weakly coupled to a bath which takes into account nonsecular effects, but nevertheless has the mathematically correct Gorini-Kossakowski-Lindblad-Sudarshan form (in particular, it preserves positivity of the density operator) and also satisfies the standard thermodynamic properties is a known long-standing problem in theory of open quantum systems. The nonsecular terms are important when some energy levels of the system or their differences (Bohr frequencies) are nearly degenerate. We provide a fully rigorous derivation of such equation based on a formalization of the weak-coupling limit for the general case.
A longstanding tool to characterize the evolution of open Markovian quantum systems is the GKSL (Gorini-Kossakowski-Sudarshan-Lindblad) master equation. However, in some cases, open quantum systems can be effectively described with non-Hermitian Hamiltonians, which have attracted great interest in the last twenty years due to a number of unconventional properties, such as the appearance of exceptional points. Here, we present a short review of these two different approaches aiming in particular to highlight their relation and illustrate different ways of connecting non-Hermitian Hamiltonian to a GKSL master equation for the full density matrix.
The Schrödinger equation describes a quantum system that does not interact with anything outside itself. The participation measure evolves unitarily; the norm is preserved; coherent superpositions are maintained indefinitely. This is a useful idealization but it is not what real quantum systems do. Every real quantum system is in contact with its environment — thermal photons, lattice vibrations, stray fields, gas‑phase collisions, observation by experimenters. Coherent superpositions decay; populations relax toward thermal equilibrium; entanglement with the environment proliferates. The closed‑system Schrödinger equation cannot describe these phenomena because, by construction, it has no environment. In 1976, Lindblad — and independently Gorini, Kossakowski, and Sudarshan — proved that the most general Markovian dynamics for a quantum density operator that preserves Hermiticity, trace, and complete positivity has a specific functional form: ∂ₜ ρ̂ = −(i/ℏ)[Ĥ, ρ̂] + ∑_α ( L̂_α ρ̂ L̂_α† − ½{ L̂_α† L̂_α , ρ̂ } ) The first term is the von Neumann equation. The sum is the “dissipator” — jump operators L̂_α mediating environmental coupling. This is the master equation of decoherence theory: qubits coupled to reservoirs, atoms in optical cavities, mesoscopic conductors with electron‑phonon coupling, open‑system everything. The standard derivation traces out the environment and applies the Born–Markov approximation. It works empirically but leaves open: why is Markov valid? Why is Born valid? What is the substrate‑level meaning of the jump operators? This walkthrough shows how the Event Density framework forces the Lindblad form at substrate level rather than deriving it from approximations. The density‑operator description is forced by environmental phase‑independence, derived from substrate primitives, rather than postulated as the natural object after partial‑tracing. Between commitment events, the density operator evolves unitarily — U3 in density‑operator form. The jump operators are the substrate‑level representation of Primitive 11 commitment events: discrete, channel‑selective updates triggered when environmental coupling drives the chain’s individuation. The Markov property is forced by P11’s locality (commitment events do not carry long‑range memory). The anti‑commutator normalization is forced by P04 (bandwidth conservation) plus trace preservation. Complete positivity is automatic from the projector structure of commitment events. The structural payoff: decoherence is not approximate physics derived from a Born–Markov simplification of a more fundamental closed‑system theory. It is what U3 plus substrate primitives produces directly when the chain is non‑isolated. The Lindblad form is forced rather than approximate. Complete positivity, trace preservation, and the Markovian property — usually justified by appeals to physical reasonableness — are forced consequences of bandwidth conservation, environmental phase‑independence, and commitment‑event locality respectively.
The theory of open quantum systems is one of the most essential tools for the development of quantum technologies. Furthermore, the Lindblad (or Gorini-Kossakowski-Sudarshan-Lindblad) master equation plays a key role as it is the most general generator of Markovian dynamics in quantum systems. In this paper, we present this equation together with its derivation and methods of resolution. The presentation tries to be as self-contained and straightforward as possible to be useful to readers with no previous knowledge of this field.