Measuring topological invariants in photonic systems

Motivated by the recent theoretical and experimental progress in implementing topological orders with photons, we analyze photonic systems with different topologies and present a scheme to probe their topological features. Specifically, we propose a scheme to modify the boundary phases to manipulate edge state dynamics. Such a scheme allows one to measure the winding number of the edge states. Furthermore, we discuss the effect of loss and disorder on the validity of our approach.Comment: 5 pages, 5 figure

Stability of Fractional Quantum Hall States in Disordered Photonic Systems

The possibility of realizing fractional quantum Hall liquids in photonic systems has attracted a great deal of interest of late. Unlike electronic systems, interactions in photonic systems must be engineered from non-linear elements and are thus subject to positional disorder. The stability of the topological liquid relies on repulsive interactions. In this paper we investigate the stability of fractional quantum Hall liquids to impurities which host attractive interactions. We find that for sufficiently strong attractive interactions these impurities can destroy the topological liquid. However, we find that the liquid is quite robust to these defects, a fact which bodes well for the realization of topological quantum Hall liquids in photonic systems.Comment: 11 pages, 5 figure

Measurement of many-body chaos using a quantum clock

There has been recent progress in understanding chaotic features in many-body quantum systems. Motivated by the scrambling of information in black holes, it has been suggested that the time dependence of out-of-time-ordered (OTO) correlation functions such as $\langle O_2(t) O_1(0) O_2(t) O_1(0) \rangle$ is a faithful measure of quantum chaos. Experimentally, these correlators are challenging to access since they apparently require access to both forward and backward time evolution with the system Hamiltonian. Here, we propose a protocol to measure such OTO correlators using an ancilla which controls the direction of time. Specifically, by coupling the state of ancilla to the system Hamiltonian of interest, we can emulate the forward and backward time propagation, where the ancilla plays the role of a 'quantum clock'. Within this scheme, the continuous evolution of the entire system (the system of interest and the ancilla) is governed by a time-independent Hamiltonian. Our protocol is immune to errors that could occur when the direction of time evolution is externally controlled by a classical switch.Comment: 11 pages + Appendices, 9 figure

Topologically robust transport of entangled photons in a 2D photonic system

We theoretically study the transport of time-bin entangled photon pairs in a two-dimensional topological photonic system of coupled ring resonators. This system implements the integer quantum Hall model using a synthetic gauge field and exhibits topologically robust edge states. We show that the transport through edge states preserves temporal correlations of entangled photons whereas bulk transport does not preserve these correlations and can lead to significant unwanted temporal bunching or anti-bunching of photons. We study the effect of disorder on the quantum transport properties; while the edge transport remains robust, bulk transport is very susceptible, and in the limit of strong disorder, bulk states become localized. We show that this localization is manifested as an enhanced bunching/anti-bunching of photons. This topologically robust transport of correlations through edge states could enable robust on-chip quantum communication channels and delay lines for information encoded in temporal correlations of photons.Comment: Revised versio

Minimal Injective Resolutions and Auslander-Gorenstein Property for Path Algebras

Let $R$ be a ring and $\mathcal{Q}$ be a finite and acyclic quiver. We present an explicit formula for the injective envelopes and projective precovers in the category $\rm{Rep} (\mathcal{Q} ,R)$ of representations of $\mathcal{Q}$ by left $R$-modules. We also extend our formula to all terms of the minimal injective resolution of $R\mathcal{Q}$. Using such descriptions, we study the Auslander-Gorenstein property of path algebras. In particular, we prove that the path algebra $R\mathcal{Q}$ is $k$-Gorenstein if and only if $\mathcal{Q}=\overrightarrow{A_{n}}$ and $R$ is a $k$-Gorenstein ring, where $n$ is the number of vertices of $\mathcal{Q}$.Comment: Accepted for publication in Comm. Algebr

Engineering three-body interaction and Pfaffian states in circuit QED systems

We demonstrate a scheme to engineer the three-body interaction in circuit-QED systems by tuning a fluxonium qubit. Connecting such qubits in a square lattice and controlling the tunneling dynamics, in the form of a synthesized magnetic field, for the photon-like excitations of the system, allows the implementation of a parent Hamiltonian whose ground state is the Pfaffian wave function. Furthermore, we show that the addition of the next-nearest neighbor tunneling stabilizes the ground state, recovering the expected topological degeneracy even for small lattices. Finally, we discuss the implementation of these ideas with the current technology.Comment: 5 pages, 4 figure

Thermal management and non-reciprocal control of phonon flow via optomechanics

Engineering phonon transport in physical systems is a subject of interest in the study of materials and plays a crucial role in controlling energy and heat transfer. Of particular interest are non-reciprocal phononic systems, which in direct analogy to electric diodes, provide a directional flow of energy. Here, we propose an engineered nanostructured material, in which tunable non-reciprocal phonon transport is achieved through optomechanical coupling. Our scheme relies on breaking time-reversal symmetry by a spatially varying laser drive, which manipulates low-energy acoustic phonons. Furthermore, we take advantage of recent developments in the manipulation of high-energy phonons through controlled scattering mechanisms, such as using alloys and introducing disorder. These combined approaches allow us to design an acoustic isolator and a thermal diode. Our proposed device will have potential impact in phonon-based information processing, and heat management in low temperatures.Comment: 7 pages, 3 figure, and supplementary materia

Optical Lattice with Torus Topology

We propose an experimental scheme to construct an optical lattice where the atoms are confined to the surface of a torus. This construction can be realized with spatially shaped laser beams which could be realized with recently developed high resolution imaging techniques. We numerically study the feasibility of this proposal by calculating the tunneling strengths for atoms in the torus lattice. To illustrate the non-trivial role of topology in atomic dynamics on the torus, we study the quantized superfluid currents and fractional quantum Hall (FQH) states on such a structure. For FQH states, we numerically investigate the robustness of the topological degeneracy and propose an experimental way to detect such a degeneracy. Our scheme for torus construction can be generalized to Riemann surfaces with higher genus for exploration of richer topological physics.Comment: 5+8 pages, 4+5 figure

Hardware-efficient fermionic simulation with a cavity-QED system

In digital quantum simulation of fermionic models with qubits, non-local maps for encoding are often encountered. Such maps require linear or logarithmic overhead in circuit depth which could render the simulation useless, for a given decoherence time. Here we show how one can use a cavity-QED system to perform digital quantum simulation of fermionic models. In particular, we show that highly nonlocal Jordan-Wigner or Bravyi-Kitaev transformations can be efficiently implemented through a hardware approach. The key idea is using ancilla cavity modes, which are dispersively coupled to a qubit string, to collectively manipulate and measure qubit states. Our scheme reduces the circuit depth in each Trotter step of the Jordan-Wigner encoding by a factor of $N^2$, comparing to the scheme for a device with only local connectivity, where $N$ is the number of orbitals for a generic two-body Hamiltonian. Additional analysis for the Fermi-Hubbard model on an $N\times N$ square lattice results in a similar reduction. We also discuss a detailed implementation of our scheme with superconducting qubits and cavities.Comment: 10 pages + Appendices, 5 figures, 1 tabl

Measurement of topological invariants in a 2D photonic system

A hallmark feature of topological physics is the presence of one-way propagating chiral modes at the system boundary. The chirality of edge modes is a consequence of the topological character of the bulk. For example, in a non-interacting quantum Hall model, edge modes manifest as mid-gap states between two topologically distinct bulk bands. The bulk-boundary correspondence dictates that the number of chiral edge modes, a topological invariant called the winding number, is completely determined by the bulk topological invariant, the Chern number. Here, for the first time, we measure the winding number in a 2D photonic system. By inserting a unit flux quantum at the edge, we show that the edge spectrum resonances shift by the winding number. This experiment provides a new approach for unambiguous measurement of topological invariants, independent of the microscopic details, and could possibly be extended to probe strongly correlated topological orders.Comment: Revised explanation of experimental results as anomalous spectral flow of edge state resonance