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Quantum networks provide a prospective paradigm to connect separated quantum nodes, which relies on the distribution of long-distance entanglement and active feedforward control of qubits between remote nodes. Such approaches can be utilized to construct nonlocal quantum gates, forming building blocks for distributed quantum computing and other novel quantum applications. However, these gates have only been realized within single nodes or between nodes separated by a few tens of meters, limiting the ability to harness computing resources in large-scale quantum networks. Here, we demonstrate nonlocal photonic quantum gates between two nodes spatially separated by 7.0 km using stationary qubits based on multiplexed quantum memories, flying qubits at telecom wavelengths, and active feedforward control based on field-deployed fibers. Furthermore, we illustrate quantum parallelism by implementing the Deutsch-Jozsa algorithm and the quantum phase estimation algorithm between the two remote nodes. These results represent a proof-of-principle demonstration of quantum gates over metropolitan-scale distances and lay the foundation for the construction of large-scale distributed quantum networks relying on existing fiber channels., Comment: 26 pages, 18 figures

The non-Hermitian skin effect (NHSE) undermines the conventional bulk-boundary correspondence (BBC) since it results in a distinct bulk spectrum in open-boundary systems compared to the periodic counterpart. Using the non-Hermitian (NH) Su-Schrieffer-Heeger (SSH) model as an example, we propose an intuitive approach, termed ``doubling and swapping" method, to restore the BBC. Explicitly, we construct a modified system by swapping the asymmetric intracell hoppings in every second primitive unit cell, such that it has double-sized unit cells compared to the NH SSH model and is free of NHSE. Importantly, the modified system and the NH SSH chain exhibit identical spectra under open boundary conditions (OBC). As a result, the modified system can serve as the valid bulk for defining topological invariants that correctly predicts edge states and topological phase transitions. The basic principle is applicable to many other systems such as the non-Hermitian Creutz ladder model. Furthermore, we extend the study to disordered systems in which the asymmetric hoppings are randomly swapped. We show that two types of winding numbers can be defined to account for the NHSE and topological edge states, respectively., Comment: 4 figures

The non-Hermitian skin effect can arise in materials that have asymmetric hoppings between atoms or resonating units, which makes the bulk eigenspectrum sensitive to boundary conditions. When skin effect emerges, eigenstates in the bulk continuum can become localized on the edges, making the distinction between edge and bulk states challenging. We establish the bulk-boundary correspondence for a Chern insulator model with non-Hermitian skin effect by combining two approaches ("non-Bloch" approach and "biorthogonal" approach). Both approaches can suppress the skin effect but they are based on different mathematical tools. A biorthogonal inverse participation ratio is used as a measure to distinguish between bulk states and edge states, and a non-Bloch Chern number is used to characterize the topology and predict the number of topological edge bands. In addition to tangential degeneracies, crossing degeneracies are found to occur between the bulk and edge bands. Their presence enriches the (de)localization behavior of the edge states but does not affect the Chern number. The phase diagram of the system has interesting features that are not found in Hermitian systems. For example, one topological transition and two non-Hermitian phase transitions can be induced by tuning a single parameter. The gapless phase is topologically protected due to the stable existence of the non-Hermitian band degeneracies guaranteed by nonzero discriminant numbers., Comment: 8 pages, 4 figures

An astroid-shaped loop of exceptional points (EPs), comprising four cusps, is found to spawn from the triple degeneracy point in the Brillouin zone (BZ) of a Lieb lattice with nearest-neighbor hoppings when non-Hermiticity is introduced. The occurrence of the EP loop is due to the realness of the discriminant which is guaranteed by the non-Hermitian chiral symmetry. The EPs at the four cusps involve the coalescence of three eigenstates, which is the combined result of the non-Hermitian chiral symmetry and mirror-T symmetry. The EP loop is exactly an astroid in the limit of an infinitesimal non-Hermiticity. The EP loop expands from the $M$ point with increasing non-Hermiticity and splits into two EP loops at a critical non-Hermiticity. The further increase of non-Hermiticity contracts the two EP loops towards and finally to two EPs at the $X$ and $Y$ points in the BZ, accompanied by the emergence of Dirac-like cones. The two EPs vanish at a larger non-Hermiticity. The EP loop disappears and several discrete EPs are found to survive when next-nearest hoppings are introduced to break the non-Hermitian chiral symmetry. A topological invariant called the discriminant number is used to characterize their robustness against perturbations. Both discrete EPs and those on the EP loop(s) are found to show anisotropic asymptotic behaviors. Finally, the experimental realization of the Lieb lattice using a coupled waveguide array is discussed.

Praseodymium (Pr$^{3+}$) ions doped in the site 1 of yttrium orthosilicate (Y$_2$SiO$_5$) has been widely employed as the photonic quantum memory due to their excellent optical coherence and spin coherence. While praseodymium ions occupying the site 2 in Y$_2$SiO$_5$ crystal have better optical coherence as compared with those at site 1, which may enable better performance in quantum memory. Here we experimentally characterize the hyperfine interactions of the ground state $^3$H$_4$ and excited state $^1$D$_2$ of Pr$^{3+}$ at site 2 in Y$_2$SiO$_5$ using Raman heterodyne detected nuclear magnetic resonance. The Hamiltonians for the hyperfine interaction are reconstructed for both ground state $^3$H$_4$ and excited state $^1$D$_2$ based on the Raman heterodyne spectra in 201 magnetic fields. The two-pulse spin-echo coherence lifetime for the ground state is measured to be 2.6$\pm$0.1 ms at site 2 with zero magnetic field, which is more than five times longer than that at site 1. The magnetic fields with zero first order Zeeman shift in the hyperfine transition for Pr$^{3+}$ at site 2 in Y$_2$SiO$_5$ are identified.

A pair of anisotropic exceptional points (EPs) of arbitrary order are found in a class of non-Hermitian random systems with asymmetric hoppings. Both eigenvalues and eigenvectors exhibit distinct behaviors when these anisotropic EPs are approached from two orthogonal directions in the parameter space. For an order-$N$ anisotropic EP, the critical exponents $\nu$ of phase rigidity are $(N-1)/2$ and $N-1$, respectively. These exponents are universal within the class. The order-$N$ anisotropic EPs split and trace out multiple ellipses of EPs of order $2$ in the parameter space. For some particular configurations, all the EP ellipses coalesce and form a ring of EPs of order $N$. Crossover to the conventional order-$N$ EPs with $\nu=(N-1)/N$ is discussed., Comment: Main text contains 14 pages and 4 figures

The faithful storage and coherent manipulation of quantum states with matter-systems enable the construction of large-scale quantum networks based on quantum repeater. To achieve useful communication rates, highly multimode quantum memories will be required to construct a multiplexed quantum repeater. Here, we present the first demonstration of the on-demand storage of orbital-angular-momentum states with weak coherent pulses at the single-photon-level in a rare-earth-ion doped crystal. Through the combination of this spatial degree-of-freedom with temporal and spectral degrees of freedom, we create a multiple-degree-of-freedom memory with high multimode capacity. This device can serve as a quantum mode converter with high fidelity, which is the fundamental requirements for the construction of a multiplexed quantum repeater. This device further enables essentially arbitrary spectral and temporal manipulations of spatial-qutrit-encoded photonic pulses in real-time. Therefore, the developed quantum memory can serve as a building block for scalable photonic quantum information processing architectures.

Probing collective spin dynamics is a current challenge in the field of magnetic resonance spectroscopy and has important applications in material analysis and quantum information protocols. Recently, the rare-earth ion doped crystals are an attractive candidate for making long-lived quantum memory. Further enhancement of its performance would benefit from the direct knowledge on the dynamics of nuclear-spin bath in the crystal. Here we detect the collective dynamics of nuclear-spin bath located around the rare-earth ions in a crystal using dynamical decoupling spectroscopy method. From the measured spectrum, we analyze the configuration of the spin bath and characterize the flip-flop time between two correlated nuclear spins in a long time scale ($\sim $1s). Furthermore, we experimentally demonstrate that the rare-earth ions can serve as a magnetic quantum sensor for external magnetic field. These results suggest that the rare-earth ion is a useful probe for complex spin dynamics in solids and enable quantum sensing in the low-frequency regime, revealing promising possibilities for applications in diverse fields., Comment: The unit in Fig.4 has been corrected

The spin coherence time of $^{151}$Eu$^{3+}$ which substitutes the yttrium at site 1 in Y$_2$SiO$_5$ crystal has been extended to 6 hours in a recent work [\textit{Nature} \textbf{517}, 177 (2015)]. To make this long-lived spin coherence useful for optical quantum memory applications, we experimentally characterize the hyperfine interaction of the optically-excited state $^5$D$_0$ using Raman-heterodyne-detected nuclear magnetic resonance. The effective spin Hamiltonians for excited and ground state are fitted based on the experimental spectra obtained in 200 magnetic fields with various orientations. To show the correctness of the fitted parameters and potential application in quantum memory protocols, we also characterize the ground-state hyperfine interaction and predict the critical magnetic field which produces the 6-hour-long coherence time. The complete energy level structure for both the $^7$F$_0$ ground state and $^5$D$_0$ excited state at the critical magnetic field are obtained. These results enable the design of quantum memory protocols and the optimization of optical pumping strategy for realization of photonic quantum memory with hour-long lifetime.

It is commonly believed that Anderson localized states and extended states do not coexist at the same energy. Here we propose a simple mechanism to achieve the coexistence of localized and extended states in a band in a class of disordered quasi-1D and quasi-2D systems. The systems are partially disordered in a way that a band of extended states always exists, not affected by the randomness, whereas the states in all other bands become localized. The extended states can overlap with the localized states both in energy and in space, achieving the aforementioned coexistence. We demonstrate such coexistence in disordered multi-chain and multi-layer systems., Comment: 5 pages, 3 figures

It is well known that inversion symmetry in one-dimensional (1D) systems leads to the quantization of the geometric Zak phase to values of either 0 or {\pi}. When the system has particle-hole symmetry, this topological property ensures the existence of zero-energy interface states at the interface of two bulk systems carrying different Zak phases. In the absence of inversion symmetry, the Zak phase can take any value and the existence of interface states is not ensured. We show here that the situation is different when the unit cell contains multiple degrees of freedom and a hidden inversion symmetry exists in a subspace of the system. As an example, we consider a system of two Su-Schrieffer-Heeger (SSH) chains coupled by a coupler chain. Although the introduction of coupler chain breaks the inversion symmetry of the system, a certain hidden inversion symmetry ensures the existence of a decoupled $2\times2$ SSH Hamiltonian in the subspace of the entire system and the two bands associated with this subspace have quantized Zak phases. These "quantized" bands in turn can provide topological boundary or interface states in such systems. Since the entire system has no inversion symmetry, the bulk-boundary correspondence may not hold exactly. The above is also true when next-nearest-neighbor hoppings are included. Our systems can be realized straightforwardly in systems such as coupled single-mode optical waveguides or coupled acoustic cavities., Comment: 40 pages, 5 figures