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Journal of Physics Conference Series

Exact solution of rate equations for a two-spin-qubit system

Le Thi Ha Linh, Bich Ha Nguyen

Published Sep 1, 2009• 1 citation
Publisher

Abstract

The quantum dynamics of a system of two interacting spin-qubits is studied for elaborating the physical mechanism of the quantum information transfer between them. A simple model with their Heisenberg XY exchange interaction is investigated. The rate equations are established. The analytical expressions of their solution are exactly derived. They explicitly demonstrate, how the quantum information encoded into a spin-qubit at the initial time t = 0 is transfered to other one at any time t > 0.

DOI
10.1088/1742-6596/187/1/012031

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The quantum dynamics of a system of two interacting spin-qubits is studied for elaborating the physical mechanism of the quantum information transfer between them. A simple model with their Heisenberg XY exchange interaction is investigated. The rate equations are established. The analytical expressions of their solution are exactly derived. They explicitly demonstrate, how the quantum information encoded into a spin-qubit at the initial time t = 0 is transfered to other one at any time t > 0.
The aim of this topical review is a systematic and concise presentation of the results of a series of theoretical works on the quantum dynamics of two-spin-qubit systems towards the elaboration of a physical mechanism of the quantum information transfer between two spin-qubits. For this purpose the main attention is paid to exactly solvable models of two-spin-qubit systems, since the analytical expressions of the elements of their reduced density matrices explicitly exhibit the mutual dependence of the quantum information encoded into the spin-qubits. The treatment of their decoherence due to the interaction with the environment is performed in the Markovian approximation. Rate equations for axially symmetric systems of two coupled spin-qubits non-interacting, as well as interacting, with the environment are exactly solved. It is shown how the solutions of rate equations demonstrate the physical mechanism of the quantum information exchange between the spin-qubits. This mechanism holds also in all two-spin-qubit systems whose rate equations can be solved only by means of numerical calculations. Exact solutions of rate equations for two uncoupled spin-qubits interacting with two separate environments reveal an interesting physical phenomenon in the time evolution of the qubit-qubit entanglement generated by their interaction with the environments: the entanglement sudden death and revival. A two-spin-qubit system with an asymptotically decoherence free subspace was also explicitly constructed. The presented calculations and reasonings can be extended for application to the study of spin-qubit chains or networks.
Abstract The quantum dynamics of a system of three interacting identical spin-qubits with the XY Heisenberg exchange interaction between two nearest-neighbour spin-qubits is studied in the Markovian approximation with respect to the interaction between this system and the environment. A Lindblad formula with two physical mechanisms of the decoherence is used. Two different cases are considered simultaneously: three spin-qubits interact with three district baths of excitations or with one and the same common bath of excitations. The systems of rate equations are solved in the first-order approximation with respect to the small constants of decoherence.
A physical model of the quantum information exchange between two qubits is studied theoretically. The qubits are two identical two-level atoms, the physical mechanism of the quantum information exchange is the mutual dependence of the reduced density matrices of two qubits generated by their couplings with a multimode radiation field. The Lehmberg-Agarwal master equation is exactly solved. The explicit form of the mutual dependence of two reduced density matrices is established. The application to study the entanglement of two qubits is discussed.
We study the dynamics evolution of a two-qubit Heisenberg XXX spin chain under a time-dependent rotating magnetic field.Based on the algebraic structure of the non-autonomouse system,the exact solution of the Schrdinger equation is obtained by using the method of algebraic dynamics.Based on the time-dependent analytical solution,we further study the entanglement evolution between the two coupled spins for different initial states,and find that the entanglement is determined by the coefficients of the initial state and the coupling constant J of the system.
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.