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Entropy

Pseudo-Qutrit Formed by Two Interacting Identical Spins (s = 1/2) in a Variable External Magnetic Field

Yury M. Belousov, I. V. Chernousov, Vladimir I. Man’ko

Published Apr 26, 2023• 2 citations
Publisher

Abstract

An analytical solution is obtained for the problem of two interacting, identical but separated spin 1/2 particles in a time-dependent external magnetic field, in a general case. The solution involves isolating the pseudo-qutrit subsystem from a two-qubit system. It is shown that the quantum dynamics of a pseudo-qutrit system with a magnetic dipole-dipole interaction can be described clearly and accurately in an adiabatic representation, using a time-dependent basis set. The transition probabilities between the energy levels for an adiabatically varying magnetic field, which follows the Landau-Majorana-Stuckelberg-Zener (LMSZ) model within a short time interval, are illustrated in the appropriate graphs. It is shown that for close energy levels and entangled states, the transition probabilities are not small and strongly depend on the time. These results provide insight into the degree of entanglement of two spins (qubits) over time. Furthermore, the results are applicable to more complex systems with a time-dependent Hamiltonian.

DOI
10.3390/e25050716

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An analytical solution is obtained for the problem of two interacting, identical but separated spin 1/2 particles in a time-dependent external magnetic field, in a general case. The solution involves isolating the pseudo-qutrit subsystem from a two-qubit system. It is shown that the quantum dynamics of a pseudo-qutrit system with a magnetic dipole-dipole interaction can be described clearly and accurately in an adiabatic representation, using a time-dependent basis set. The transition probabilities between the energy levels for an adiabatically varying magnetic field, which follows the Landau-Majorana-Stuckelberg-Zener (LMSZ) model within a short time interval, are illustrated in the appropriate graphs. It is shown that for close energy levels and entangled states, the transition probabilities are not small and strongly depend on the time. These results provide insight into the degree of entanglement of two spins (qubits) over time. Furthermore, the results are applicable to more complex systems with a time-dependent Hamiltonian.
We derive the density matrix of a nonstationary system in the adiabatic representation and obtain the corresponding Liouville equation. This representation is particularly suitable when the standard interaction representation is inapplicable. We illustrate the approach with two identical, spatially separated spin-1/2 particles coupled by a magnetic dipole-dipole interaction in a slowly varying external magnetic field. Specifically, we analyze the spin polarization of pseudo-qutrit states in regimes where the states become entangled. The application of the adiabatic representation to calculate the dependence of observed physical quantities on time for quantum systems is illustrated. Our results show that entangled states can significantly modify the behavior of these observables and provide additional opportunities for their control.
A time-dependent model of two interacting spin qutrits is analyzed is analyzed and solved. The two interacting qutrits are subjected to a longitudinal field linearly varying over time as in the Landau-Majorana-St\"uckelberg- Zener (LMSZ) scenario. Although a transverse field is absent, we show the occurrence of LMSZ transitions assisted by the coupling between the two spin-qutrits. Such a physical effect permits us to estimate experimentally the coupling strength between the spins and allows the generation of entangled states of the two qutrits by appropriately setting the slope of the ramp. Furthermore, the possibility of local and nonlocal control as well as the existence of dark states of the two qutrits have been brought to light. Effects stemming from a noisy surrounding environment are also taken into account by introducing a random fluctuating field component as well as non-Hermitian terms in the Hamiltonian model.
We investigate a system of two identical and distinguishable spins 1/2, with a direct magnetic dipole-dipole interaction, in an external magnetic field. Constraining the hyperfine tensor to exhibit axial symmetry generates the notable symmetry properties of the corresponding Hamiltonian model. In fact, we show that the reduction of the anisotropy induces the invariance of the Hamiltonian in the 3×3 subspace of the Hilbert space of the two spins in which S^2 invariably assumes its highest eigenvalue of 2. By means of appropriate mapping, it is then possible to choose initial density matrices of the two-spin system that evolve in such a way as to exactly simulate the time evolution of a pseudo-qutrit, in the sense that the the actual two-spin system nests the subdynamics of a qutrit regardless of the strength of the magnetic field. The occurrence of this dynamic similitude is investigated using two types of representation for the initial density matrix of the two spins. We show that the qutrit state emerges when the initial polarizations and probability vectors of the two spins are equal to each other. Further restrictions on the components of the probability vectors are reported and discussed.
We introduce a new approach for describing nonstationary quantum systems with a discrete energy spectrum. The essence of this approach is that we describe the evolution of a quantum system in a time-dependent basis. In a sense, this approach is similar to the description of the system in the interaction representation. However, the time dependence of the basic states of the representation is determined not by the evolution operator with a time-independent Hamiltonian but by the eigenstates of the time-dependent Hamiltonian defined at the current time. The time dependence of the basic states of the representation leads to the appearance of an additional term in the Schrödinger equation, which in the case of slowly changing parameters of the Hamiltonian can be considered as a small perturbation. The adiabatic representation is suitable in cases where it is impossible to apply the standard interaction representation. The application of the adiabatic representation is illustrated by the example of two spins connected by a magnetic dipole–dipole interaction in a slowly varying external magnetic field.
Abstract In this study, we investigate the entanglement dynamics of two qutrits under Dzyloshinsky–Moriya (DM) interaction within a spin-qubit environment, influenced by two identical time-dependent external magnetic fields described by B ( 1 − e − γ t ) . To quantify the entanglement between the two spin qutrits, we employ the Holstein–Primakoff transformation to compute the negativity measure. Our analysis reveals that the entanglement dynamics are affected by various factors, including the initial states of the spin environment and the qutrits, the system-environment coupling strength, the strength of the DM interaction, and the intensity of the external magnetic field. Notably, we find that increasing the spin coherence amplitude and the intensity of the magnetic field contributes to the stabilization of entanglement. In the weak coupling regime, an increase in the DM interaction enhances entanglement, irrespective of whether a magnetic field is present. Conversely, in the strong coupling regime, an increase in the DM interaction negatively impacts entanglement, regardless of the presence or absence of a magnetic field.
We propose a method to describe the evolution of two spins coupled by hyperfine i nteraction in an external time- dependent magnetic field. We apply the approach to the case of hyperfine interaction with axial symmetry, which can be solved exactly in a constant, appropriately oriented magnetic field. In order to t reat t he n onstationary d ynamical p roblem, we modify the time-dependent Schrödinger equation through a change of representation that, by exploiting an instantaneous (adiabatic) basis makes the time-dependent Hamiltonian diagonal at any time instant. The solution of the transformed time-dependent Schrödinger FRVBUJPO in the form of chronologically ordered exponents with transparent pre-exponential coefficients is reported. This solution is highly simplified when an adiabatically varying magnetic field perturbs the system. The approach here proposed may be used for the perturbative treatment of other dynamical problems with no exact solution.