# Exactly solvable model for the dynamics of two spin- particles embedded in separate spin star environments

###### Abstract

Exact analytical results for the dynamics of two interacting qubits each of which is embedded in its own spin star bath are presented. The time evolution of the concurrence and the purity of the two-qubit system is investigated for finite and infinite numbers of environmental spins. The effect of qubit-qubit interactions on the steady state of the central system is investigated.

## 1 Introduction

Exactly solvable models play a very useful role in various fields of physics. They help improving our understanding of physical processes and allow us gain more insight into complicated phenomena that take place in nature [1]. Needless to recall the usefulness of exactly solvable models such as the harmonic oscillator, the nuclear shell model and the Ising model, to mention but a few. From a practical point of view, exactly solvable models serve as a very convenient tool for testing the accuracy of numerical algorithms, often used in the study of problems that cannot be analytically solved due to the complexity of the systems under investigation.

In nature, quantum systems are influenced by their surrounding environment through, in general, complicated coupling interactions, leading them to lose their coherence [2]. This refers to as the decoherence process [3, 4, 5]. Moreover, quantum systems exhibit properties that do not have classical analogous [6]. Of great interest is entanglement, the main ingredient for quantum teleportation and quantum computation [7, 8, 9, 10, 11, 12]. Over the last years, many proposals have been made for the implementation of quantum information processing. Solid state systems are very promising [13, 14] and have been the subject of many investigations. In particular, decoherence and entanglement of qubits coupled to spin environments [15] attracted much attention [16, 17]. Thus new exactly solvable models describing the dynamics of qubits in spin baths are highly welcome. Recently, the spin star configuration, initially proposed by Bose, has been extensively investigated [18, 19, 20, 21, 22]. An exact treatment of the dynamics of two qubits coupled to common spin star bath via interactions is presented in [23, 24]. In this paper we propose to investigate analytically the dynamics when the two qubits interact with separate spin star baths.

The paper is organized as follows. In section 2 the model Hamiltonian is introduced. In section 3 we present a detailed derivation of the time evolution operator and we investigate the dynamics of the qubits at finite for some particular initial conditions. In section 4 we study the thermodynamic limit, in which the sizes of the spin environments become infinite. Section 5 is devoted to the second-order master equation. We end the paper with a short summary.

## 2 Model

The system under study consists of two two-level systems ( e.g., spin- particles) each of which is embedded in its own spin star environment composed of spins-. The central particles interact with each other through a Ising interaction; the corresponding coupling constant is equal to , where the factor 4 is introduced for later convenience. We shall assume that each qubit couples to its environment via Heisenberg interaction whose coupling constant is , which is, in turn, scaled by in order to ensure good thermodynamic behavior. The spin baths will be denoted by and . The Hamiltonian for the composite system has the form

(1) |

where

(2) |

and

(3) |

Here and denote the spin operators corresponding to the central qubits, whereas denotes the spin operator corresponding to the particle within the environment. Introducing the total spin operators and of the environments and , respectively, one can rewrite the full Hamiltonian as

(4) |

The dynamics of the two-qubit system is fully described by its density matrix obtained, as usual, by tracing the time-dependent total density matrix , describing the composite system, with respect to the environmental degrees of freedom, namely,

(5) | |||||

where and designate the time evolution operator and the initial total density matrix, respectively.

At the central qubits are assumed to be uncoupled with the environments; the latter are assumed to be at infinite temperature. This means that the initial total density density matrix can be written as

(6) |

Here is the initial density matrix of the two-qubit system, and is the unit matrix on the space . The former can be written as , with for . Similarly, we introduce the basis state vectors of , such that ( for even and for odd), and . The time-dependent reduced density matrix can be expressed as

(7) |

where , and [25]. Hence, our task reduces to finding the exact form of the matrix elements of the time evolution operator (). This will be the subject of the next section.

## 3 Derivation of the exact form of the time evolution operator

The time evolution operator can be expanded as

(8) |

In order to derive analytical expressions for even and odd powers of the total Hamiltonian let us notice that anticommutes with , that is,

(9) |

This can easily be shown using the following properties for spin- operators: , and . Moreover, it is easily seen that , which simply implies that for ,

(10) |

In the standard basis of , it can be shown that powers of and are given by

(11) | |||||

(12) | |||||

(13) | |||||

(14) |

It follows that

(15) | |||||

where

(16) |

and

(17) |

Using the fact that

(18) | |||||

(19) |

one obtains

(20) |

where

(21) | |||

(22) | |||

(23) | |||

(24) |

Inserting equation (20) into equation (10), yields

(25) |

where

(26) | |||

(27) | |||

(28) | |||

(29) |

The above operators satisfy

(30) | |||

(31) |

Furthermore, one can show that the matrix elements of are given by

(32) | ||||

(33) | ||||

(34) | ||||

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(37) |

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(49) |

(50) | |||||

(51) | |||||

(52) | |||||

(53) | |||||

(54) | |||||

(55) |

Having in hand the explicit expressions of powers of the total Hamiltonian, it can easily be verified that the elements of the time evolution operator, obtained by inserting equations (25) and (32)-(55) into equation (8), are given by

(56) | ||||

(57) | ||||

(58) |

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