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Open Systems & Information Dynamics

The GKLS Master Equation in High Energy Physics

Beatrix C. Hiesmayr

Published Sep 1, 2017• 3 citations
Publisher

Abstract

Utilizing the GKLS master equation we show that the decay property of a particle can be straightforwardly incorporated. In standard particle physics the decay is often described by an efficient non-hermitian Hamiltonian, in accord with the seminal Wigner-Weisskopf approximation. We show that by enlarging the Hilbert space and defining specific GKLS operators we have attained a formalism with a hermitian Hamiltonian and probability conserving states. This proves that the Wigner-Weisskopf approximation is Markovian and completely positive. In addition, this formalism allows a straightforward generalization to many-particle decays. Last, but not least, some impacts of the GKLS master equation onto systems at high energies are reported, such as for neutral meson, neutrino and hyperon systems.

DOI
10.1142/s123016121740008x

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Utilizing the GKLS master equation we show that the decay property of a particle can be straightforwardly incorporated. In standard particle physics the decay is often described by an efficient non-hermitian Hamiltonian, in accord with the seminal Wigner-Weisskopf approximation. We show that by enlarging the Hilbert space and defining specific GKLS operators we have attained a formalism with a hermitian Hamiltonian and probability conserving states. This proves that the Wigner-Weisskopf approximation is Markovian and completely positive. In addition, this formalism allows a straightforward generalization to many-particle decays. Last, but not least, some impacts of the GKLS master equation onto systems at high energies are reported, such as for neutral meson, neutrino and hyperon systems.
We consider an open quantum system which contains unstable states. The time evolution of the system can be described by an effective non-Hermitian Hamiltonian Heff{H}_{\mathrm{eff}}Heff​, in accord with the Wigner-Weisskopf approximation, and an additional term of the Lindblad form, the so-called dissipator. We show that, after enlarging the original Hilbert space by states which represent the decay products of the unstable states, the non-Hermitian part of Heff{H}_{\mathrm{eff}}Heff​---the ``particle decay''---can be incorporated into the dissipator of the enlarged space via a specific Lindblad operator. Thus the formulation of the time evolution on the enlarged space has a Hermitian Hamiltonian and is probability conserving. The equivalence of the formulation with the original one demonstrates that the time evolution which is governed by a non-Hermitian Hamiltonian and a dissipator of the Lindblad form is nevertheless completely positive, just as systems with Hermitian Hamiltonians.
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.
Derivation of a quantum master equation for a system weakly coupled to a bath which takes into account non-secular effects, but, nevertheless, has the mathematically correct Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) form (in particular, 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 non-secular 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.
The Lindblad (GKLS) master equation, which represents the mathematical form for the general evolution of a density matrix, is a versatile and widely-used tool in open quantum systems. In contrast with the typical approach of imposing additional conditions on the system, such as weak coupling or energy conservation, we explore the structure of the equation with no assumptions. We demonstrate that general quantum dynamics can be expressed through a combination of free evolution, exchanges of some physical quantities (generalised charges), not necessarily commuting with the Hamiltonian, between the system and the bath, and pure dephasing. This result comprises a novel perspective on quantum master equations, employing physical processes as elemental parts. We use it to explore the dynamics and stationary states of a two-level system and show that strong coupling, particle exchange, and non-Abelian effects all share the same physical origin. Moreover, we demonstrate that the generalised Gibbs state for all three cases contains a non-commutation term, which has not been previously considered.
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.