Symmetry protected topological orders of 1D spin systems with D2+T symmetry

In [Z.-X. Liu, M. Liu, X.-G. Wen, arXiv:1101.5680], we studied 8 gapped symmetric quantum phases in S=1 spin chains %/ladders which respect a discrete spin rotation $D_2 \subset SO(3)$ and time reversal $T$ symmetries. In this paper, using a generalized approach, we study all the 16 possible gapped symmetric quantum phases of 1D integer spin systems with only $D_2+T$ symmetry. Those phases are beyond Landau symmetry breaking theory and cannot be characterized by local order parameters, since they do not break any symmetry. They correspond to 16 symmetry protected topological (SPT) orders. We show that all the 16 SPT orders can be fully characterized by the physical properties of the symmetry protected degenerate boundary states (end `spins') at the ends of a chain segment. So we can measure and distinguish all the 16 SPT orders experimentally. We also show that all these SPT orders can be realized in S=1 spin ladder models. The gapped symmetric phases protected by subgroups of $D_2+T$ are also studied. Again, all these phases can be distinguished by physically measuring their end `spins'.Comment: 10+page

Coupled-resonator-induced transparency with a squeezed vacuum

We present the first experimental observation of quantum fluctuation spectra in two coupled optical cavities with an injected squeezed vacuum light. The quadrature components of the reflected squeezed vacuum spectra are measured by phase sensitive homodyne detector. The experimental results demonstrate coupled-resonator-induced transparency in the quantum regime, in which electromagnetically-induced-transparency-like characteristic of the absorption and dispersion properties of the coupled optical cavities determines the line-shape of the reflected quantum noise spectra.Comment: 4 pages, 4 figures, appear in Phys. Rev. Let

Complete classification of 1D gapped quantum phases in interacting spin systems

Quantum phases with different orders exist with or without breaking the symmetry of the system. Recently, a classification of gapped quantum phases which do not break time reversal, parity or on-site unitary symmetry has been given for 1D spin systems in [X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B \textbf{83}, 035107 (2011); arXiv:1008.3745]. It was found that, such symmetry protected topological (SPT) phases are labeled by the projective representations of the symmetry group which can be viewed as a symmetry fractionalization. In this paper, we extend the classification of 1D gapped phases by considering SPT phases with combined time reversal, parity, and/or on-site unitary symmetries and also the possibility of symmetry breaking. We clarify how symmetry fractionalizes with combined symmetries and also how symmetry fractionalization coexists with symmetry breaking. In this way, we obtain a complete classification of gapped quantum phases in 1D spin systems. We find that in general, symmetry fractionalization, symmetry breaking and long range entanglement(present in 2 or higher dimensions) represent three main mechanisms to generate a very rich set of gapped quantum phases. As an application of our classification, we study the possible SPT phases in 1D fermionic systems, which can be mapped to spin systems by Jordan-Wigner transformation.Comment: 15 pages, 3 figure

Tensor product representation of topological ordered phase: necessary symmetry conditions

The tensor product representation of quantum states leads to a promising variational approach to study quantum phase and quantum phase transitions, especially topological ordered phases which are impossible to handle with conventional methods due to their long range entanglement. However, an important issue arises when we use tensor product states (TPS) as variational states to find the ground state of a Hamiltonian: can arbitrary variations in the tensors that represent ground state of a Hamiltonian be induced by local perturbations to the Hamiltonian? Starting from a tensor product state which is the exact ground state of a Hamiltonian with $\mathbb{Z}_2$ topological order, we show that, surprisingly, not all variations of the tensors correspond to the variation of the ground state caused by local perturbations of the Hamiltonian. Even in the absence of any symmetry requirement of the perturbed Hamiltonian, one necessary condition for the variations of the tensors to be physical is that they respect certain $\mathbb{Z}_2$ symmetry. We support this claim by calculating explicitly the change in topological entanglement entropy with different variations in the tensors. This finding will provide important guidance to numerical variational study of topological phase and phase transitions. It is also a crucial step in using TPS to study universal properties of a quantum phase and its topological order.Comment: 10 pages, 6 figure

Nonperturbative signatures in pair production for general elliptic polarization fields

The momentum signatures in nonperturbative multiphoton pair production for general elliptic polarization electric fields are investigated by employing the real-time Dirac-Heisenberg-Wigner formalism. For a linearly polarized electric field we find that the positions of the nodes in momenta spectra of created pairs depend only on the electric field frequency. The polarization of external fields could not only change the node structures or even make the nodes disappear but also change the thresholds of pair production. The momentum signatures associated to the node positions in which the even-number-photon pair creation process is forbid could be used to distinguish the orbital angular momentum of created pairs on the momenta spectra. These distinguishable momentum signatures could be relevant for providing the output information of created particles and also the input information of ultrashort laser pulses.Comment: 8 pages, 4 figures, submitted to Europhysics Letter