Comparison of density matrix and state vector approaches to dissipative evolution of hyperfine levels coupled to optical and radio-frequency fields in the two-level approximation
We consider a three-level atom interacting with two optical and one microwave fields in the adiabatic approximation resulting in a simplified description in the terms of a two-level system. Working within the reduced density operator framework, we assume this two-level system to be affected by two types of environment, described by some ad hoc non-Hermitian Hamiltonian and Gorini-Kossakowski-Sudarshan-Lindblad’s models. We compare the three types of dissipative evolution which can occur: driven by equations for a normalized density matrix, a non-normalized density matrix and a normalized state vector. Using the latter type, we derive an effective Hamiltonian, which encodes information about not only the Hamiltonian part of an original master equation but also its non-Hamiltonian (Liouvillian) part. The Hamiltonian turns out to be dependent on the wavefunction itself: the effects of above-mentioned environments induce, respectively, cubic and quintic nonlinear terms. For evolutions driven by density operators, we study various indicators of quantum purity. It is shown that if a trace of density operator is not conserved then conventional von Neumann entropy can no longer be used as a purity indicator; therefore we introduce the purity-normalized definition of quantum statistical entropy.
We consider a three-level atom interacting with two optical and one microwave fields in the adiabatic approximation resulting in a simplified description in the terms of a two-level system. Working within the reduced density operator framework, we assume this two-level system to be affected by two types of environment, described by some ad hoc non-Hermitian Hamiltonian and Gorini-Kossakowski-Sudarshan-Lindblad’s models. We compare the three types of dissipative evolution which can occur: driven by equations for a normalized density matrix, a non-normalized density matrix and a normalized state vector. Using the latter type, we derive an effective Hamiltonian, which encodes information about not only the Hamiltonian part of an original master equation but also its non-Hamiltonian (Liouvillian) part. The Hamiltonian turns out to be dependent on the wavefunction itself: the effects of above-mentioned environments induce, respectively, cubic and quintic nonlinear terms. For evolutions driven by density operators, we study various indicators of quantum purity. It is shown that if a trace of density operator is not conserved then conventional von Neumann entropy can no longer be used as a purity indicator; therefore we introduce the purity-normalized definition of quantum statistical entropy.
A unified quantum-statistical framework is proposed for open three-level atomic systems when two lower-energy states are coupled to a single upper state. We suggest a non-Hermitian generalization of the closed model’s Hamiltonian by the analytical continuation of real-valued parameters related to the energies into a complex domain, as well as by considering some terms previously neglected by the procedure of Hermitian conjugation. We show that such systems can have two types of density operator’s evolution, depending on whether they are expected to be sustainable, or in a process of irreversible decay. This approach not only explains previously heuristically introduced terms in equations of motion for the density matrix but also suggests new terms which predict new dissipative effects. We then consider a special case, namely the evolution of pure states described by a generalized Schrödinger equation, which turns out to be nonlinear with respect to the atomic wavefunctions. As a final step toward unification, in addition to the non-Hermitian Hamiltonian generalization, we also include Markovian quantum jumps by adding the Lindblad dissipator. The resulting equations not only expand the range of applicability of open three-level models, but also suggest that dissipative phenomena in quantum-optical systems in vivo are more diverse and significant than previously believed.
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
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.
Dissipation and decoherence, and the evolution from pure to mixed states in quantum physics, are handled through master equations for the density matrix. Master equations such as the Lindblad equation preserve the trace of this matrix. Viewing them as first-order time-dependent operator equations for the elements of the density matrix, a unitary integration procedure can be adapted to solve for these elements. A simple model for decoherence preserves the Hermiticity of the density matrix. A single, classical Riccati equation is the only one requiring numerical handling to obtain a full solution of the quantum evolution. The procedure is general, valid for any number of levels, but is illustrated here for a three-level system with two driving fields. For various choices of the initial state, we study the evolution of the system as a function of the amplitudes, relative frequencies, and phases of the driven fields and of the strength of the decoherence. The monotonic growth of the entropy is followed as the system evolves from a pure to a mixed state. An example is provided by the n=3 states of the hydrogen atom in a time-dependent electric field, such degenerate manifolds affording an analytical solution.
Abstract This chapter introduces the density operator and the Liouville equation and outlines how reduced dynamical description of open quantum systems naturally leads to the appearance of relaxation processes. (1) The density operator and the quantum Liouville equation (the density matrix for a pure system, statistical mixtures, representations, coherences, thermodynamic equilibrium). (2) An example: The time evolution of a two-level system in the density matrix formalism. (3) Reduced descriptions. (4) Time evolution equations for reduced density operators: The quantum master equation (projection operators, The Nakajima–Zwanzig equation, Derivation of the quantum master equation using the thermal projector, The quantum master equation in the interaction representation, The Markovian limit—the Redfield equation, The secular approximation and the Lindblad equation). (5) The Lindblad equation in the ‘Heisenberg representation’. (6) Relaxation of a two-level system and a harmonic oscillator revisited. (7) The optically driven two-level system in a thermal environment—the Bloch equations. (8) The oscillating dipole model of a two-level molecule. (9) Analogy of a coupled two-level system to a spin ½ system in a magnetic field.
In this work, we investigate the quantum coherence and purity in hydrogen atoms under dissipative dynamics, with a focus on the hyperfine structure states arising from the electron-proton spin interaction. Using the Lindblad master equation, we model the time evolution of the density matrix of the system, incorporating both the unitary dynamics driven by the hyperfine Hamiltonian and the dissipative effects due to environmental interactions. Quantum coherence is quantified using the L1 norm and relative entropy measures, while purity is assessed via von Neumann entropy, for initial states, including a maximally entangled Bell state and a separable state. Our results reveal distinct dynamics: for the Bell states, both coherence and purity decay exponentially with a rate proportional to the dissipation parameter, whereas for a kind of separable state, coherence exhibits oscillatory behavior modulated via the hyperfine coupling constant, superimposed on an exponential decay, and accompanied by a steady increase in entropy. Higher dissipation rates accelerate the loss of coherence and the growth of von Neumann entropy, underscoring the environment's role in suppressing quantum superposition and driving the system towards mixed states. These findings enhance our understanding of coherence and purity preservation in atomic systems and offer insights for quantum information applications where robustness against dissipation is critical.