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.
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.
We compare two approaches to open quantum systems, namely, the non-Hermitian dynamics and the Lindblad master equation. In order to deal with more general dissipative phenomena, we propose the unified master equation that combines the characteristics of both of these approaches. This allows us to assess the differences between them as well as to clarify which observed features come from the Lindblad or the non-Hermitian part, when it comes to experiment. Using a generic two-mode single-atom laser system as a practical example, we analytically solve the dynamics of the normalized density matrix operator. We study the two-level model in a number of cases (depending on parameters and types of dynamics), compute different observables and study their physical properties. It turns out that one can not only able to describe the different types of damping in dissipative quantum optical systems but also mimic the undamped anharmonic oscillatory phenomena which happen in quantum systems with more than two levels (while staying within the framework of the analytically simple two-mode approximation).
Abstract If an open quantum system is initially uncorrelated from its environment, then its dynamics can be written in terms of a Lindblad-form master equation. The master equation is divided into a unitary piece, represented by an effective Hamiltonian, and a dissipative piece, represented by a hermiticity-preserving superoperator; however, the division of open system dynamics into unitary and dissipative pieces is non-unique. For finite-dimensional quantum systems, we resolve this non-uniqueness by specifying a norm on the space of dissipative superoperators and defining the canonical Hamiltonian to be the one whose dissipator is minimal. We show that the canonical Hamiltonian thus defined is equivalent to the Hamiltonian initially defined by Lindblad, and that it is uniquely specified by requiring the dissipator’s jump operators to be traceless, extending a uniqueness result known previously in the special case of Markovian master equations. For a system weakly coupled to its environment, we give a recursive formula for computing the canonical effective Hamiltonian to arbitrary orders in perturbation theory, which we can think of as a perturbative scheme for renormalizing the system’s bare Hamiltonian.
Non-Hermitian systems have observed numerous novel phenomena and might lead to various applications. Unlike standard quantum physics, the conservation of energy guaranteed by the closed system is broken in the non-Hermitian system, and the energy can be exchanged between the system and the environment. Here we present a scheme for simulating the dissipative phase transition with an open quantum optical system. The competition between the coherent interaction and dissipation leads to the second-order phase transition. Furthermore, the quantum correlation in terms of squeezing is studied around the critical point. Our work may provide a new route to explore the non-Hermitian quantum physics with feasible techniques in experiments.
If an open quantum system is initially uncorrelated from its environment, then its dynamics can be written in terms of a Lindblad-form master equation. The master equation is divided into a unitary piece, represented by an effective Hamiltonian, and a dissipative piece, represented by a hermiticity-preserving superoperator; however, the division of open system dynamics into unitary and dissipative pieces is non-unique. For finite-dimensional quantum systems, we resolve this non-uniqueness by specifying a norm on the space of dissipative superoperators and defining the canonical Hamiltonian to be the one whose dissipator is minimal. We show that the canonical Hamiltonian thus defined is equivalent to the Hamiltonian initially defined by Lindblad, and that it is uniqueley specified by requiring the dissipator's jump operators to be traceless. For a system weakly coupled to its environment, we give a recursive formula for computing the canonical effective Hamiltonian to arbitrary orders in perturbation theory, which we can think of as a perturbative scheme for renormalizing the system's bare Hamiltonian.
Within the framework of the quantum-statistical approach, utilizing both non-Hermitian Hamiltonian and Lindblad’s jump operators, one can derive various generalizations of the von Neumann equation for reduced density operators, also known as hybrid master equations. If one considers the evolution of pure states only, i.e., disregarding the coherence between states and spontaneous transitions from pure to mixed states, then one can resort to quantum-mechanical equations of the Schrödinger type. We derive them from the hybrid master equations and study their main properties, which indicate that our equations have a larger range of applicability compared to other generalized Schrödinger equations proposed hitherto. Among other features, they can describe not only systems which remain in the stationary eigenstates of the Hamiltonian as time passes, but also those which evolve from those eigenstates. As an example, we consider a simple but important model, a quantum harmonic oscillator driven by both Hamiltonian and non-Hamiltonian terms, and derive its classical limit, which turns out to be the damped harmonic oscillator. Using this model, we demonstrate that the effects of dissipative environments of different types can cancel each other, thus resulting in an effectively dissipation-free classical system. Another discussed phenomenon is whether a non-trivial quantum system can reduce to a classical system in free motion, i.e., without experiencing any classical Newtonian forces. This uncovers a large class of quantum-mechanical non-Hamiltonian systems whose dynamics are not determined by conventional mechanics’ potentials and forces, but rather come about through quantum statistical effects caused by the system’s environment.
Traditional thermodynamics, built upon the foundation of Hermitian quantum mechanics, primarily describes closed or weakly open systems where energy is conserved. However, the ubiquitous presence of dissipation and gain in realistic physical and biological systems necessitates a more generalized framework. Non-Hermitian quantum mechanics, characterized by Hamiltonians with complex eigenvalues, naturally accounts for such open systems and their interaction with the environment. This paper explores the burgeoning field of non-Hermitian thermodynamics, which aims to extend classical and quantum thermodynamic principles to systems exhibiting energy exchange with their surroundings. We delve into how non-Hermiticity provides a natural language for describing processes involving dissipation, gain, and decoherence, thereby offering a unified perspective on these phenomena. Key concepts such as PT-symmetry, exceptional points, and generalized definitions of work, heat, and entropy in non-Hermitian settings are reviewed. The paper discusses the implications for fundamental thermodynamic laws, the performance of non-Hermitian heat engines, and the role of coherence in energy transport. By embracing the mathematical elegance and physical insights of non-Hermitian physics, we propose a comprehensive framework that unifies the seemingly disparate concepts of dissipation and coherence within a consistent thermodynamic theory, paving the way for advanced applications in quantum technologies, materials science, and biological energy transfer.