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arXiv:quant-ph/0607220 · Journal of Physics A Mathematical and General

Entanglement by linearSU(2) transformations: generation and evolution of quantum vortex states

G. S. Agarwal, J Banerji

Published Aug 29, 2006• 32 citations
arXivHTMLPDF

Abstract

We consider the evolution of a two-mode system of bosons under the action of a Hamiltonian that generates linear SU (2) transformations. The Hamiltonian is generic in that it represents a host of entanglement mechanisms, which can thus be treated in a unified way. We start by solving the quantum dynamics analytically when the system is initially in a Fock state. We show how the two modes get entangled by evolution to produce a coherent superposition of vortex states in general, and a single vortex state under certain conditions. The degree of entanglement between the modes is measured by finding the explicit analytical dependence of the Von Neumann entropy on the system parameters. The reduced state of each mode is analysed by means of its correlation function and spatial coherence function. Remarkably, our analysis is shown to be equally as valid for a variety of initial states that can be prepared from a two-mode Fock state via a unitary transformation and for which the results can be obtained by mere inspection of the corresponding results for an initial Fock state. As an example, we consider a quantum vortex as the initial state and also find conditions for its revival and charge conjugation. While studying the evolution of the initial vortex state, we have encountered and explained an interesting situation in which the entropy of the system does not evolve whereas its wavefunction does. Although the modal concept has been used throughout the paper, it is important to note that the theory is equally applicable for a two-particle system in which each particle is represented by its bosonic creation and annihilation operators.

DOI
10.1088/0305-4470/39/37/011

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Entanglement by linear SU(2) transformations: generation and evolution of quantum vortex states

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We consider the evolution of a two-mode system of bosons under the action of a Hamiltonian that generates linear SU (2) transformations. The Hamiltonian is generic in that it represents a host of entanglement mechanisms, which can thus be treated in a unified way. We start by solving the quantum dynamics analytically when the system is initially in a Fock state. We show how the two modes get entangled by evolution to produce a coherent superposition of vortex states in general, and a single vortex state under certain conditions. The degree of entanglement between the modes is measured by finding the explicit analytical dependence of the Von Neumann entropy on the system parameters. The reduced state of each mode is analysed by means of its correlation function and spatial coherence function. Remarkably, our analysis is shown to be equally as valid for a variety of initial states that can be prepared from a two-mode Fock state via a unitary transformation and for which the results can be obtained by mere inspection of the corresponding results for an initial Fock state. As an example, we consider a quantum vortex as the initial state and also find conditions for its revival and charge conjugation. While studying the evolution of the initial vortex state, we have encountered and explained an interesting situation in which the entropy of the system does not evolve whereas its wavefunction does. Although the modal concept has been used throughout the paper, it is important to note that the theory is equally applicable for a two-particle system in which each particle is represented by its bosonic creation and annihilation operators.
Abstract. We consider the evolution of a two-mode system of bosons under the action of a Hamiltonian that generates linear SU(2) transformations. The Hamiltonian is generic in that it represents a host of entanglement mechanisms, which can thus be treated in a unified way. We start by solving the quantum dynamics analytically when the system is initially in a Fock state. We show how the two modes get entangled by evolution to produce a coherent superposition of vortex states in general, and a single vortex state under certain conditions. The degree of entanglement between the modes is measured by finding the explicit analytical dependence of the Von Neumann entropy on the system parameters. The reduced state of each mode is analyzed by means of its correlation function and spatial coherence function. Remarkably, our analysis is shown to be equally as valid for a variety of initial states that can be prepared from a two-mode Fock state via a unitary transformation and for which the results can be obtained by mere inspection of the corresponding results for an initial Fock state. As an example, we consider a quantum vortex as the initial state and also find conditions for its revival and charge conjugation. While studying the evolution of the initial vortex state, we have encountered and explained an interesting situation in which the entropy of the system does not evolve whereas its wave function does. Although the modal concept has been used throughout the paper, it is important to note that the theory is equally applicable for a two-particle system in which each particle is represented by its bosonic creation and annihilation operators. (Figures in this article are in colour only in the electronic version) PACS numbers: 03.67.-a, 03.75.Gg, 03.75.Lm, 03.67.Mn, 05.30.Jp, 42.50.Dv 1.
Utilizing the general theory of open quantum systems to investigate the exact dynamical evolution of simple bilinear systems, we discover a mechanism of the dynamical genesis of quantum entanglement. We focus in detail on the exact quantum evolution dynamics of two photonic modes (or any two bosonic modes) coupled to each other through a linear interaction, as the simplest system of open quantum systems that we have investigated in the past two decades. Such a linear coupling alone fails to produce two-mode entanglement. We also start with an initially separable pure state of the two modes. By solving exactly the quantum equation of motion without relying on the probabilistic interpretation, we find that when the initial state of one mode is different from a coherent state (a minimum uncertainty wave packet with equal variance in the conjugate quadratures that corresponds to a well-defined classically "particle"), the causality in the time-evolution of each mode is internally violated. It also leads to the emergence of quantum entanglement between the two modes. The lack of causality is the nature of statistics. We discover that it is the internal violation of causality in the reduced (subsystem) dynamical evolution that results in the emergence of entanglement and statistic probability in quantum mechanics, even though the dynamical evolution of the whole system completely obeys the deterministic Schrödinger equation. This conclusion is valid for the quantum dynamics of more complicated composite systems. It may provide the fundamental mechanism of the dynamical genesis for both the entanglement and the statistical probability within the deterministic framework of quantum mechanics, which is the longest-standing problem that has not been fully understood since the birth of quantum mechanics.
The dynamical generation of entanglement in closed bipartite systems is investigated in the semiclassical regime. We consider a model of two particles, initially prepared in a product of coherent states, evolving in time according to a generic Hamiltonian, and derive a formula for the linear entropy of the reduced density matrix using the semiclassical propagator in the coherent-state representation. The formula is explicitly written in terms of quantities that define the stability of classical trajectories of the underlying classical system. The formalism is then applied to the problem of two nonlinearly coupled harmonic oscillators, and the result is shown to be in remarkable agreement with the exact quantum measure of entanglement in the short-time regime. An important by-product of our approach is a unified semiclassical formula, which contemplates both the coherent-state propagator and its complex conjugate.
An exactly solvable case of an interacting Hamiltonian of two bosonic modes is considered to study fundamental properties of the entanglement dynamics for coupled nonlinear oscillators. Such an interaction is of physical importance, either in a two-species Bose–Einstein condensate or in the case of two modes of electromagnetic fields interacting in Kerr media. The time-evolved state is obtained analytically for initial products of two Fock and two coherent states, and the purification times of the subsystems are determined. The possibility of dynamical generation of a quantum superposition state is discussed at such purification times. We also identify the existence of two regimes: the short time, phase spread regime where subsystem entropy rises monotonically and the self-interference regime where it oscillates and a purification phenomenon can be observed. Our results also show that the break time from the first regime to the second one becomes longer, as well as the purification and reversibility times, as the Planck constant becomes much smaller than a typical action in phase space.
Based on our recently published work [Front. Phys. 21, 093201 (2026)], we extend the analysis to the density-operator evolution of a two-mode entangled quantum system subject to amplitude decay and show that the output state retains the same functional structure as the initial density operator throughout the decay. We further examine the evolutions of several quantum statistical distributions and the von Neumann entropy used to quantify entanglement during the process. This work offers a systematic approach to studying two-mode entangled open systems and yields results on the von Neumann entropy that may help facilitate entanglement measurement.
Following the approach of Solano et al (2003 Phys. Rev. Lett. 90 027903) we propose a scheme for a generation of a few classes of entangled (nonlinear) coherent states. To achieve this purpose, the interaction of a spatially narrow collection of two-level atoms with a quantized field in a high-Q factor cavity in the presence of a strong-driving classical field is studied. We perform appropriate Hamiltonians describing the atom-field interaction by focusing on two particular forms of intensity-dependent functions which are directly related to su (1, 1) and su (2) Lie algebras. It is shown that the dynamical evolution of the considered systems can generate bipartite, tripartite (nonlinear) and more complicated entangled states corresponding to the mentioned groups depending on the number of atoms in the cavity. In the processes of the abovementioned generation schemes, even and odd nonlinear coherent states are produced. In the end, in a particular circumstance with the two-mode quantized field we can turn easily from Jaynes–Cummings to anti-Jaynes–Cummings interactions which brings us to the maximally entangled number state . Finally, to quantify the degree of entanglement of the produced states, the measures of von Neumann and linear entropies are applied.