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The time evolution for Qubit – spin coherent state interaction system using majorana sphere representation

Muhammad Nadzim Zamri, Ainul Zaim Zakaria, Bahari Iskandar

Published Nov 10, 2023• 0 citations
OpenAlex

Abstract

We studied the interacting system of qubit with a collection of N qubits or known as spin coherent state. From the interaction of the qubit – spin coherent state system, we can observe many interesting phenomena such as ‘collapse and revival’ in the time evolution of the qubit state, the qubit of attractor state, the qubit – spin coherent state entanglement, and the occurrence of the Schrӧdinger’s cat state. The event of ‘collapse and revival’ will be mainly focused in this study. Prior studies had shown the Bloch Sphere and Majorana Sphere representations for the interaction system between qubit and a coherent state. Therefore, in this research, we will analyze the time evolution of the qubit state in the qubit – spin coherent state interacting system for the one qubit case, and represent the dynamics of the qubit state using the Majorana Sphere representation.

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•Sep 30, 2014•0 citations
We studied the interacting system of qubit with a collection of N qubits or known as spin coherent state. From the interaction of the qubit – spin coherent state system, we can observe many interesting phenomena such as ‘collapse and revival’ in the time evolution of the qubit state, the qubit of attractor state, the qubit – spin coherent state entanglement, and the occurrence of the Schrӧdinger’s cat state. The event of ‘collapse and revival’ will be mainly focused in this study. Prior studies had shown the Bloch Sphere and Majorana Sphere representations for the interaction system between qubit and a coherent state. Therefore, in this research, we will analyze the time evolution of the qubit state in the qubit – spin coherent state interacting system for the one qubit case, and represent the dynamics of the qubit state using the Majorana Sphere representation.
We study the dynamics of a system comprised of a single qubit interacting equally with NNN qubits (a ``spin star'' system). Although this model can be solved exactly, the exact solution does not give much intuition for the dynamics of the model. Here, we find an approximation that gives some insight into the dynamics for a particular class of initial spin-coherent states of the NNN qubits. We find an effective Hamiltonian for the system that is a finite Kerr (one-axis twisting) Hamiltonian for the N+1N+1N+1 qubits. The initial spin-coherent state evolves to spin-squeezed states on short time scales, and to ``multiple-Schr\"odinger-cat'' states (superpositions of many spin-coherent states) on longer time scales, a manifestation of the phenomenon of fractional revivals of the initial state. The evolution of the system is visualized with phase-space plots (QQQ functions) that, when plotted against time, reveal a ``quantum carpet'' pattern. Of particular interest is the fact that our approximation captures the qualitative features of the model even for small values of NNN. This suggests the possibility of observing the phenomenon of fractional revival in this model for systems of few qubits.
Abstract By representing a quantum state and its evolution with the Majorana stars on the Bloch sphere, the Majorana representation provides us an intuitive way to study a physical system with SU(2) symmetry. In this work, based on coherent states, we propose a method to establish the generalization of Majorana representation for a general symmetry. By choosing a generalized coherent state as a reference state, we give a more general Majorana representation for both finite and infinite systems and the corresponding star equations are given. Using this method, we study the squeezed vacuum states for three different symmetries, Heisenberg–Weyl, SU(2) and SU(1,1), and express the effect of squeezing parameter on the distribution of stars. Furthermore, we also study the dynamical evolution of stars for an initial coherent state driven by a nonlinear Hamiltonian, and find that at a special time point, the stars are distributed on two orthogonal large circles.
We obtain exact dynamics of a two-qubit central spin model (CSM) consisting of two interacting qubits homogeneously coupled to a spin bath via the XXZXXZXXZ-type coupling, with the bath initially prepared in linear superpositions of the symmetric Dicke states. Using the interaction picture Hamiltonian with respect to the non-spin-flipping part of the model, we derive a sequence of equations of motion within each magnetization sector satisfied by the probability amplitudes of the time-evolved state. These equations of motion admit analytical solutions for the single-qubit CSM in which one of the two central qubits decouples from the rest of the system. Based on this, we provide a quantitative interpretation to the observed collapse-revival phenomena in the single-qubit Rabi oscillations when the bath is prepared in the spin coherent state. We then study the disentanglement and coherence dynamics of two initially entangled noninteracting qubits when the two qubits interact with individual baths or with a common bath. For individual baths, the coherent dynamics is found to be positively correlated to the single-qubit purity dynamics, and entanglement sudden disappearance and revivals are observed in both cases. The entanglement creation of two initially separable qubits coupled to a common bath is also studied and collapse and revival behaviors in the entanglement dynamics are observed. Choosing the equally weighted state and the WWW-class states as the bath initial states, we finally study the dynamics of entanglement between two individual bath spins and demonstrate the entanglement sharing mechanism in such a system.
We extend the study of the Jayne–Cummings (JC) model involving a pair of identical two-level atoms (or qubits) interacting with a single mode quantized field. We investigate the effects of replacing the radiation field mode with a composite spin, comprising [Formula: see text] qubits, or spin-1/2 particles. This model is relevant for physical implementations in superconducting circuit QED, ion trap and molecular systems. For the case of the composite spin prepared in a spin coherent state, we demonstrate the similarities of this set-up to the qubits-field model in terms of the time evolution, attractor states and in particular the collapse and revival of the entanglement between the two qubits. We extend our analysis by taking into account an effect due to qubit imperfections. We consider a difference (or “mismatch”) in the dipole interaction strengths of the two qubits, for both the field mode and composite spin cases. To address decoherence due to this mismatch, we then average over this coupling strength difference with distributions of varying width. We demonstrate in both the field mode and the composite spin scenarios that increasing the width of the “error” distribution increases suppression of the coherent dynamics of the coupled system, including the collapse and revival of the entanglement between the qubits.
Coherent control of quantum dynamics by phase-manipulation of the driving fields, has long been established as an essential tool for state-selective preparation of systems. On the other hand, the basic manipulations of qubits in most physical realizations of quantum-computation devices use Rabi pulses that operate on the Bloch sphere, particularly for weakly-coupled, slow-decoherent systems. In this work we analyze the role of phase-control and phase-dependence of Rabi pulses that prepare Bell states in a system of distinguishable qubits interacting in a harmonic trap. We show that the population dynamics and the properties of the entanglement exhibit a strong dependence to the relative phase. For coherent phonon distributions, collapse and full revival of entanglement occur, while for thermal distributions, except for a few "protected" phases, decoherence with partial revivals are observed.
The generation of non-classical states of large quantum systems is an important topic of study. It is of fundamental interest because the generation of larger and larger non-classical states extends quantum theory further and further into the classical domain, and it is also of practical interest because such states are an important resource for quantum technologies. \nThe focus of this thesis is the “spin star model” for the interaction of a single spin-1/2 particle with N other spin-1/2 particles. Although this is a simple model, we show that its dynamics include many interesting quantum phenomena, including fractional revival, and Jaynes- Cummings-like collapse and revival. Starting with a spin coherent state of the N spin system – an easily prepared state, in principle – we show that these dynamics can be used to generate a wide variety of non-classical states of the N spin system, including Schrodinger cat states, GHZ states, multiple-Schrodinger cat states and spin squeezed states.