The bulk-boundary correspondence is among the central issues of non-Hermitian topological states. We show that a previously overlooked "non-Hermitian skin effect" necessitates redefinition of topological invariants in a generalized Brillouin zone. The resultant phase diagrams dramatically differ from the usual Bloch theory. Specifically, we obtain the phase diagram of the non-Hermitian Su-Schrieffer-Heeger model, whose topological zero modes are determined by the non-Bloch winding number instead of the Bloch-Hamiltonian-based topological number. Our work settles the issue of the breakdown of conventional bulk-boundary correspondence and introduces the non-Bloch bulk-boundary correspondence.

Edge States and Topological Invariants of Non-Hermitian Systems.

Microwave photonics with superconducting quantum circuits

A vision has formed in recent years of the components necessary for a large-scale quantum network. Single trapped atoms can serve as the nodes of this network, with the links established by flying photons that are coupled to the atoms using optical resonators. This review describes progress towards the goal of multinode networks using the current generation of experiments, which have achieved unprecedented levels of atomic qubit control and light-matter coupling efficiencies.

Cavity-based quantum networks with single atoms and optical photons

We report the creation of thermal, Fock, coherent, and squeezed states of motion of a harmonically bound {sup 9}Be{sup +} ion. The last three states are coherently prepared from an ion which has been initially laser cooled to the zero point of motion. The ion is trapped in the regime where the coupling between its motional and internal states, due to applied (classical) radiation, can be described by a Jaynes-Cummings-type interaction. With this coupling, the evolution of the internal atomic state provides a signature of the number state distribution of the motion. {copyright} {ital 1996 The American Physical Society.}

Generation of Non-classical Motional States of a Trapped Atom

In the past 20 years, impressive progress has been made both experimentally and theoretically in superconducting quantum circuits, which provide a platform for manipulating microwave photons. This emerging field of superconducting quantum microwave circuits has been driven by many new interesting phenomena in microwave photonics and quantum information processing. For instance, the interaction between superconducting quantum circuits and single microwave photons can reach the regimes of strong, ultra-strong, and even deep-strong coupling. Many higher-order effects, unusual and less familiar in traditional cavity quantum electrodynamics with natural atoms, have been experimentally observed, e.g., giant Kerr effects, multi-photon processes, and single-atom induced bistability of microwave photons. These developments may lead to improved understanding of the counterintuitive properties of quantum mechanics, and speed up applications ranging from microwave photonics to superconducting quantum information processing. In this article, we review experimental and theoretical progress in microwave photonics with superconducting quantum circuits. We hope that this global review can provide a useful roadmap for this rapidly developing field.

We propose a novel scheme for the preparation of a maximally entangled state of two atoms in an optical cavity. Starting from an arbitrary initial state, a singlet state is prepared as the unique fixed point of a dissipative quantum dynamical process. In our scheme, cavity decay is no longer undesirable, but plays an integral part in the dynamics. As a result, we get a qualitative improvement in the scaling of the fidelity with the cavity parameters. Our analysis indicates that dissipative state preparation is more than just a new conceptual approach, but can allow for significant improvement as compared to preparation protocols based on coherent unitary dynamics.

Dissipative Preparation of Entanglement in Optical Cavities

Engineered dissipation is used to deterministically produce and stabilize entanglement between two trapped-ion quantum bits, independently of their initial states; the entanglement is stabilized even in the presence of experimental noise and decoherence. Entangled states are a key resource in fundamental quantum physics, quantum cryptography and quantum computation. It has been generally assumed that the creation of such states requires the avoidance of contact with a dissipative environment, and minimization of decoherence. Some studies have shown, however, that dissipative interactions can be used to preserve coherence, and in this issue of Nature two groups demonstrate this principle for continuously driven physical systems. Lin et al. use engineered dissipation to deterministically produce and stabilize entanglement between two trapped-ion qubits, independent of their initial state. Shankar et al. use an autonomous feedback scheme to counteract decoherence and demonstrate the stabilization of an entangled Bell state of a quantum register of two superconducting qubits for an arbitrary time. This approach may be applied to a broad range of experimental systems to achieve desired quantum dynamics or steady states. Entangled states are a key resource in fundamental quantum physics, quantum cryptography and quantum computation1. Introduction of controlled unitary processes—quantum gates—to a quantum system has so far been the most widely used method to create entanglement deterministically2. These processes require high-fidelity state preparation and minimization of the decoherence that inevitably arises from coupling between the system and the environment, and imperfect control of the system parameters. Here we combine unitary processes with engineered dissipation to deterministically produce and stabilize an approximate Bell state of two trapped-ion quantum bits (qubits), independent of their initial states. Compared with previous studies that involved dissipative entanglement of atomic ensembles3 or the application of sequences of multiple time-dependent gates to trapped ions4, we implement our combined process using trapped-ion qubits in a continuous time-independent fashion (analogous to optical pumping of atomic states). By continuously driving the system towards the steady state, entanglement is stabilized even in the presence of experimental noise and decoherence. Our demonstration of an entangled steady state of two qubits represents a step towards dissipative state engineering, dissipative quantum computation and dissipative phase transitions5,6,7. Following this approach, engineered coupling to the environment may be applied to a broad range of experimental systems to achieve desired quantum dynamics or steady states. Indeed, concurrently with this work, an entangled steady state of two superconducting qubits was demonstrated using dissipation8.

/pdf/dissipative-production-of-a-maximally-entangled-steady-state-4fzr211l28.pdf

Dissipative production of a maximally entangled steady state of two quantum bits

We present an effective operator formalism for open quantum systems. Employing perturbation theory and adiabatic elimination of excited states for a weakly driven system, we derive an effective master equation which reduces the evolution to the ground-state dynamics. The effective evolution involves a single effective Hamiltonian and one effective Lindblad operator for each naturally occurring decay process. Simple expressions are derived for the effective operators which can be directly applied to reach effective equations of motion for the ground states. We compare our method with the hitherto existing concepts for effective interactions and present physical examples for the application of our formalism, including dissipative state preparation by engineered decay processes.

Effective operator formalism for open quantum systems

Entangled states are a key resource in fundamental quantum physics, quantum cryptography and quantum computation. Introduction of controlled unitary processes—quantum gates—to a quantum system has so far been the most widely used method to create entanglement deterministically. These processes require high-fidelity state preparation and minimization of the decoherence that inevitably arises from coupling between the system and the environment, and imperfect control of the system parameters. Here we combine unitary processes with engineered dissipation to deterministically produce and stabilize an approximate Bell state of two trapped-ion quantum bits (qubits), independent of their initial states. Compared with previous studies that involved dissipative entanglement of atomic ensembles or the application of sequences of multiple time-dependent gates to trapped ions, we implement our combined process using trapped-ion qubits in a continuous time-independent fashion (analogous to optical pumping of atomic states). By continuously driving the system towards the steady state, entanglement is stabilized even in the presence of experimental noise and decoherence. Our demonstration of an entangled steady state of two qubits represents a step towards dissipative state engineering, dissipative quantum computation and dissipative phase transitions. Following this approach, engineered coupling to the environment may be applied to a broad range of experimental systems to achieve desired quantum dynamics or steady states. Indeed, concurrently with this work, an entangled steady state of two superconducting qubits was demonstrated using dissipation.

/pdf/dissipative-production-of-a-maximally-entangled-steady-state-4vpys4e7i2.pdf

Dissipative production of a maximally entangled steady state

Quantum-enhanced measurements hold the promise to improve high-precision sensing ranging from the definition of time standards to the determination of fundamental constants of nature. However, quantum sensors lose their sensitivity in the presence of noise. To protect them, the use of quantum error-correcting codes has been proposed. Trapped ions are an excellent technological platform for both quantum sensing and quantum error correction. Here we present a quantum error correction scheme that harnesses dissipation to stabilize a trapped-ion qubit. In our approach, always-on couplings to an engineered environment protect the qubit against spin-flips or phase-flips. Our dissipative error correction scheme operates in a continuous manner without the need to perform measurements or feedback operations. We show that the resulting enhanced coherence time translates into a significantly enhanced precision for quantum measurements. Our work constitutes a stepping stone towards the paradigm of self-correcting quantum information processing. Quantum error correction plays a key role in quantum information and metrology, but generally requires complex gates and measurements sequences. Here, the authors use trapped ions to implement a scheme in which always-on coupling to an engineered environment protects the qubit against errors.

/pdf/dissipative-quantum-error-correction-and-application-to-nyr1y92ktw.pdf

Florentin Reiter

Papers

Dissipative Preparation of Entanglement in Optical Cavities

Dissipative production of a maximally entangled steady state of two quantum bits

Effective operator formalism for open quantum systems

Dissipative production of a maximally entangled steady state

Dissipative quantum error correction and application to quantum sensing with trapped ions.