We investigate hyperfine induced electron spin and entanglement dynamics in a system of two quantum dot spin qubits. We focus on the situation of zero external magnetic field and concentrate on approximation-free theoretical methods. We give an exact solution of the model for homogeneous hyperfine coupling constants ͑with all coupling coefficients being equal͒ and varying exchange coupling, and we derive the dynamics therefrom. After describing and explaining the basic dynamical properties, the decoherence time is calculated from the results of a detailed investigation of the short-time electron-spin dynamics. The result turns out to be in good agreement with experimental data.

We investigate the relation between integrability and decoherence in central spin models with more than one central spin. We show that there is a transition between integrability ensured by the Bethe ansatz and integrability ensured by complete sets of commuting operators. This has a significant impact on the decoherence properties of the system, suggesting that it is not necessarily integrability or nonintegrability which is related to decoherence, but rather its type or a change from integrability to nonintegrability. DOI: 10.1103/PhysRevLett.105.177602 PACS numbers: 76.20.+q, 02.30.Ik, 03.65.Fd, 76.30.Àv The Liouville-Arnol'd theorem states that if a system with n degrees of freedom has n involutive integrals of motion, which are functionally independent, its Hamiltonian equations of motion are solvable via quadratures [1]. Such a system is called integrable. Despite a huge effort, so far it has not been achieved to adapt the concept of integrability to the quantum mechanical framework satisfactorily. At the present time there are two commonly accepted definitions: A quantum mechanical system is called integrable (i) if there is a Bethe ansatz [2] or (ii) if the system has a complete set of commuting operators (CSCO) [3] sharing ''suitable'' properties (to be further explained below). Note that the notion of integrability in classical mechanics does not require the solvability of the quadratures. In this sense both of the aforementioned approaches are in direct analogy with classical mechanics.In investigations mainly focused on the first type of integrability, evidence has been found that it is related to transport properties [4], to quantum phase transitions [5], and to decoherence [6,7]. Here systems of the formhave been considered, where H c denotes a central system and H c$b a coupling term between the central system and a bath. Mainly two roads have been followed. On the one hand, the influence of chaotic or regular baths on the decoherence of the central system has been investigated [6]. On the other hand, the decoherence properties of the central systems of models which are integrable or nonintegrable have been studied [7]. The usual procedure within such considerations is to evaluate numerically the level statistics of the respective system and to relate a possible change in the statistics to a change of other properties of the system happening at the same point. Motivated by their important role in the context of solid state quantum information processing [8], we investigate in the present letter integrability and its relation to decoherence in central spin models. Here we define a quantum system to be integrable if it is possible to compute all eigenstates and eigenvalues of the respective Hamiltonian using operations with less complexity than the direct diagonalization of the Hamiltonian matrix [9]. Here we refer to the computional complexity. The exact diagonalization of a Hamiltonian matrix, for example, grows exponentially with the system size. This very strict notion of integrability conta...

We provide a unified framework for the treatment of special integrable systems which we propose to call 'generalized mean-field systems'. Thereby previous results on integrable classical and quantum systems are generalized. Following Ballesteros and Ragnisco, the framework consists of a unital algebra with brackets, a Casimir element and a coproduct which can be lifted to higher tensor products. The coupling scheme of the iterated tensor product is encoded in a binary tree. The theory is exemplified by the case of a spin octahedron. The relation to other generalizations of the coalgebra approach is discussed.

We numerically study the hyperfine induced nuclear spin dynamics in a system of two coupled quantum dots in zero magnetic field. Each of the electron spins is considered to interact with an individual bath of nuclear spins via homogeneous coupling constants (all coupling coefficients being equal). In order to lower the dimension of the problem, the two baths are approximated by two single long spins. We demonstrate that the hyperfine interaction enables to utilize the nuclear baths for quantum information purposes. In particular, we show that it is possible to swap the nuclear ensembles on time scales of seconds and indicate that it might even be possible to fully entangle them. As a key result, it turns out that the larger the baths are, the more useful they become as a resource of quantum information. Interestingly, the nuclear spin dynamics strongly benefits from combining two quantum dots of different geometry to a double dot set up. Introduction.-Electron spins confined in semiconductor quantum dots with an s-type conduction band, like for example GaAs quantum dots, experience decoherence through the spin-orbit interaction, and by the hyperfine interaction with surrounding nuclear spins. With respect to possible future solid state quantum computation systems utilizing the electron spin as the qubit 1,2 , these interactions act as a source of decoherence. Due to the spatial confinement of the electron spin in a quantum dot, the relaxation time T 1 induced by the spin-orbit interaction is enhanced for low temperatures 3,4 . As the dephasing time T 2 due to the spin orbit interaction turns out to be as long as the T 1 time under realistic conditions 5 , the major source of decoherence in semiconductor quantum dots results from the hyperfine interaction 6-10 . For related reviews the reader is referred to Refs.11-15 . Similar situations arise in carbon nanotube quantum dots 16 , phosphorus donors in silicon 17 and nitrogen vacancies in diamond [18][19][20] .

Central spin models describe several types of solid state nanostructures which are presently considered as possible building blocks of future quantum information processing hardware. From a theoretical point of view, a key issue remains the treatment of the flip-flop terms in the Hamiltonian in the presence of a magnetic field. We systematically study the influence of these terms, both as a function of the field strength and the size of the spin baths. We find crucial differences between initial states with central spin configurations of high and such of low polarizations. This has strong implications with respect to the influence of a magnetic field on the flip-flop terms in central spin models of a single and more than one central spin. Furthermore, the dependencies on bath size and field differ from those anticipated so far. Our results might open the route for the systematic search for more efficient perturbative treatments of central spin problems.Comment: 7 pages, 3 figure