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arXiv:2308.05160 · Journal of Physics A Mathematical and Theoretical

Extending third quantization with commuting observables: a dissipative spin-boson model

Luka Medic, A. Ramšak, Tomaž Prosen

Published Jan 25, 2024• 0 citations
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Abstract

Abstract We consider the spectral and initial value problem for the Lindblad–Gorini–Kossakowski–Sudarshan master equation describing an open quantum system of bosons and spins, where the bosonic parts of the Hamiltonian and Lindblad jump operators are quadratic and linear respectively, while the spins couple to bosons via mutually commuting spin operators. Needless to say, the spin degrees of freedom can be replaced by any set of finite-level quantum systems. A simple, yet non-trivial example of a single open spin-boson model is worked out in some detail.

DOI
10.1088/1751-8121/ad2297

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•Mar 17, 2026•0 citations
Abstract We consider the spectral and initial value problem for the Lindblad–Gorini–Kossakowski–Sudarshan master equation describing an open quantum system of bosons and spins, where the bosonic parts of the Hamiltonian and Lindblad jump operators are quadratic and linear respectively, while the spins couple to bosons via mutually commuting spin operators. Needless to say, the spin degrees of freedom can be replaced by any set of finite-level quantum systems. A simple, yet non-trivial example of a single open spin-boson model is worked out in some detail.
Abstract The aims of this paper are twofold. The first aim is to give a brief review of the most relevant theoretical results concerning the uniqueness of the steady-state solution of the Lindblad–Gorini–Kossakowski–Sudarshan master equation and the criteria which guarantee relaxingness and irreducibility of dynamical semigroups. In particular, we test and discuss their physical meaning by considering their applicability to the characterisation of the simplest open quantum system i.e. a two-level system coupled to a bath of harmonic oscillators at zero temperature. The second aim is to provide a set of sufficient conditions which guarantees the uniqueness of the steady-state solution and its attractivity. Starting from simple assumptions, we derive simple criteria that can be efficiently exploited to characterise the behavior of dissipative systems of spins and bosons (with truncated Fock space), and a wide variety of other open quantum systems recently studied.
An open quantum system can be described by a master equation, of which one of the most popular is the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation. We revisit description of driven-dissipative quantum systems focusing on the appropriate choice of the system's basis and the respective transformations. We consider the GKSL equation in different bases and calculate the dynamics for a qubit and for a qudit. An appropriate choice of the basis is a fundamental problem for theoretical consideration of open quantum systems and provides an opportunity to obtain the desired evolution in practice.
Abstract The dynamics of Markovian open quantum systems are described by Lindblad master equations. For fermionic and bosonic systems that are quasi-free, i.e. with Hamiltonians that are quadratic in the ladder operators and Lindblad operators that are linear in the ladder operators, we derive the equation of motion for the covariance matrix. This determines the evolution of Gaussian initial states and the steady states, which are also Gaussian. Using ladder super-operators (a.k.a. third quantization), we show how the Liouvillian can be transformed to a many-body Jordan normal form which also reveals the full many-body spectrum. Extending previous work by Prosen and Seligman, we treat fermionic and bosonic systems on equal footing with Majorana operators, shorten and complete some derivations, also address the odd-parity sector for fermions, give a criterion for the existence of bosonic steady states, cover non-diagonalizable Liouvillians also for bosons, and include quadratic systems. In extension of the quasi-free open systems, quadratic open systems comprise additional Hermitian Lindblad operators that are quadratic in the ladder operators. While Gaussian states may then evolve into non-Gaussian states, the Liouvillian can still be transformed to a useful block-triangular form, and the equations of motion for k -point Green’s functions form a closed hierarchy. Based on this formalism, results on criticality and dissipative phase transitions in such models are discussed in a companion paper.
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.
Abstract In the conventional quantum mechanics of conserved systems, Hamiltonian is assumed to be a Hermitian operator. However, when it comes to quantum systems in presence of dissipation and/or noise, including open quantum optical systems, the strict hermiticity requirement is nor longer necessary. In fact, it can be substantially relaxed: the non-Hermitian part of a Hamiltonian is allowed, in order to account for effects of dissipative environment, whereas its Hermitian part would be describing subsystem’s energy. Within the framework of the standard approach to dissipative phenomena based on a master equation for the reduced density operator, we propose a replacement of the hermiticity condition by a more general condition of commutativity between Hermitian and anti-Hermitian parts of a Hamiltonian. As an example, we consider a dissipative two-mode quantum system coupled to a single-mode electromagnetic wave, where we demonstrate that the adjoint-commutativity condition does simplify the parametric space of the model.
This text is a short introduction to the physics of driven-dissipative many-body systems, focusing on a few selected topics. Beyond its more historical interest in the study of atomic physics and quantum optics, presently the modeling and studying of dissipative phenomena in open quantum systems are pivotal to understanding quantum hardware platforms. Although the lack of a thermodynamic potential for these out-of-equilibrium open systems makes it theoretically challenging to investigate their physics, at the same time it allows one to go beyond the thermodynamic paradigms and investigate new and exotic phenomena. We will focus on one of the simplest, yet most effective, descrip-tions of open quantum systems, namely the (Gorini-Kossakowski-Sudarshan-) Lindblad master equation. This phenomenological approach has been devel-oped to describe quantum systems that weakly interact (Born approximation) with a memoryless (Markovian) surrounding environment. However, it is often a remarkably simple and predictive tool to characterize the essential features of an open quantum system. Although many of the results derived below will ap-ply to any quantum system, we will focus in particular on bosonic/spin systems.