Haldane predicted that the isotropic quantum Heisenberg spin chain is in a "massive" phase if the spin is integral. The first rigorous example of an isotropic model in such a phase is presented. The Hamiltonian has an exact SO(3) symmetry and is translationally invariant, but we prove the model has a unique ground state, a gap in the spectrum of the Hamiltonian immediately above the ground state and exponential decay of the correlation functions in the ground state. Models in two and higher dimension which are expected to have the same properties are also presented. For these models we construct an exact ground state, and for some of them we prove that the two-point function decays exponentially in this ground state. In all these models exact ground states are constructed by using valence bonds. Table of Contents 1. Introduction 478 2. The One-Dimensional Model (1.1) 2.1. The Ground State 481 2.2. The Ground State Two-Point Correlation Function 485 2.3. The Energy Gap 487 2.4. The Infinite Chain 492 3. The Spin 3/2 Model on the Hexagonal Lattice 3.1. The Ground State 496 3.2. Properties of the Ground State 499 3.3. Proof of Exponential Decay 500 4. VBS States on an Arbitrary Lattice 4.1. The Ground States 504 4.2. The VBS State on the Cayley Tree 506 4.3. SU{n) VBS State and Random Loop Representation 510 4.4. SU(n) Quantum "Spin" System 514
An elementary quantum network operation involves storing a qubit state in an atomic quantum memory node, and then retrieving and transporting the information through a single photon excitation to a remote quantum memory node for further storage or analysis. Implementations of quantum network operations are thus conditioned on the ability to realize such matter-to-light and/or light-tomatter quantum state mappings. Here, we report generation, transmission, storage and retrieval of single quanta using two remote atomic ensembles. A single photon is generated from a cold atomic ensemble at Site A via the protocol of Duan, Lukin, Cirac, and Zoller (DLCZ) [1] and is directed to Site B through a 100 meter long optical fiber. The photon is converted into a single collective excitation via the dark-state polariton approach of Fleischhauer and Lukin [2]. After a programmable storage time the atomic excitation is converted back into a single photon. This is demonstrated experimentally, for a storage time of 500 nanoseconds, by measurement of an anticorrelation parameter α. Storage times exceeding ten microseconds are observed by intensity cross-correlation measurements. The length of the storage period is two orders of magnitude longer than the time to achieve conversion between photonic and atomic quanta. The controlled transfer of single quanta between remote quantum memories constitutes an important step towards distributed quantum networks.A quantum network, consisting of quantum nodes and interconnecting channels, is an outstanding goal of quantum information science. Such a network could be used for distributed computing or for the secure sharing of information between spatially remote parties [1,3,4,5,6,7]. While it is natural that the network's fixed nodes (quantum memory elements) could be implemented by using matter in the form of individual atoms or atomic ensembles, it is equally natural that light fields be used as carriers of quantum information (flying qubits) using optical fiber interconnects.The matter-light interface seems inevitable since the local storage capability of ground state atomic matter cannot be easily recreated with light fields. Interfacing material quanta and single photons is therefore a basic primitive of a quantum network.
Haldane predicted that the isotropic quantum Heisenberg spin chain is in a "massive" phase if the spin is integral. The first rigorous example of an isotropic model in such a phase is presented. The Hamiltonian has an exact SO(3) symmetry and is translationally invariant, but we prove the model has a unique ground state, a gap in the spectrum of the Hamiltonian immediately above the ground state and exponential decay of the correlation functions in the ground state. Models in two and higher dimension which are expected to have the same properties are also presented. For these models we construct an exact ground state, and for some of them we prove that the two-point function decays exponentially in this ground state. In all these models exact ground states are constructed by using valence bonds. Table of Contents 1. Introduction 478 2. The One-Dimensional Model (1.1) 2.1. The Ground State 481 2.2. The Ground State Two-Point Correlation Function 485 2.3. The Energy Gap 487 2.4. The Infinite Chain 492 3. The Spin 3/2 Model on the Hexagonal Lattice 3.1. The Ground State 496 3.2. Properties of the Ground State 499 3.3. Proof of Exponential Decay 500 4. VBS States on an Arbitrary Lattice 4.1. The Ground States 504 4.2. The VBS State on the Cayley Tree 506 4.3. SU{n) VBS State and Random Loop Representation 510 4.4. SU(n) Quantum "Spin" System 514
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