This reply is in response to the comment by Shea and Gauthier on our paper [Phys. Rev. A 87, 063408 (2013)].

It has recently been shown that it is possible to represent the complete quantum state of any system as a phase-space quasiprobability distribution (Wigner function) [Phys. Rev. Lett. 117, 180401 (2016)]. Such functions take the form of expectation values of an observable that has a direct analogy to displaced parity operators. In this work we give a procedure for the measurement of the Wigner function that should be applicable to any quantum system. We have applied our procedure to IBM's Quantum Experience five-qubit quantum processor to demonstrate that we can measure and generate the Wigner functions of two different Bell states as well as the five-qubit Greenberger–Horne–Zeilinger state. Because Wigner functions for spin systems are not unique, we define, compare, and contrast two distinct examples. We show how the use of these Wigner functions leads to an optimal method for quantum state analysis especially in the situation where specific characteristic features are of particular interest (such as for spin Schrödinger cat states). Furthermore we show that this analysis leads to straightforward, and potentially very efficient, entanglement test and state characterization methods.

We derive a versatile protocol to implement a fast and robust not quantum gate, driven by single-shot-shaped pulses with appropriate time-dependent phase and intensity profiles. The pulses are derived by combining analytic computations and numerical optimizations. We experimentally demonstrate the applicability, efficiency, and robustness of the derived pulses to rephase atomic coherences in a ${\text{Pr}}^{3+}\text{:}{\text{Y}}_2{\text{SiO}}_5$ crystal.

We report on the alteration of photon-emission properties of a single trapped ion coupled to a high-finesse optical fiber cavity. We show that the vacuum field of the cavity can simultaneously affect the emissions in both the infrared (IR) and ultraviolet (UV) branches of the $\mathrm{\Lambda }$-type level system of ${}^{40}\mathrm{Ca}^{+}$ despite the cavity coupling only to the IR transition. The cavity induces strong emission in the IR transition through the Purcell effect resulting in a simultaneous suppression of the UV fluorescence. The measured suppression of this fluorescence is as large as 66% compared with the case without the cavity. Through analysis of the measurement results, we have obtained an ion-cavity coupling of ${\overline{g}}_0=2\pi \times (5.3\pm 0.1)$ MHz, the largest reported so far for a single ion in the IR domain.

Periodic structures may grow in both conservative and dissipative systems. A multiplicity of examples can be found in nature and in the laboratory. However, periodic structures may grow and decay. We show that the effects of dissipation are essential for these structures to remain. Using the nonlinear Schrödinger equation and its extensions as basic examples of conservative and dissipative systems we show that there are two ways of adiabatic transformations of a continuous-wave solution into a train of pulses.

Optomagnonics and optomechanics have various applications, ranging from tunable light sources to optical manipulation for quantum information science. We propose a hybrid system based on the interaction between a phonon and a magnon that can be tuned by an electromagnetic field based on the radiation pressure and magneto-optical effects. The self-energies of the magnon and phonon induced by the electromagnetic field and the influence of the thermal noise are studied. Moreover, the topological features of the hybrid system are illustrated considering the dynamical encircling with the exceptional points, and the chiral characteristics under these encirclements are found.

Hybrid spin-mechanical systems have great potential in sensing, macroscopic quantum mechanics, and quantum information science. In order to induce strong coupling between an electron spin and the center-of-mass motion of a mechanical oscillator, a large magnetic gradient usually is required, which is difficult to achieve. Here we show that strong coupling between the electron spin of a nitrogen-vacancy (NV) center and the torsional vibration of an optically levitated nanodiamond can be achieved in a uniform magnetic field. Thanks to the uniform magnetic field, multiple spins can strongly couple to the torsional vibration at the same time. We propose utilizing this coupling mechanism to realize the Lipkin-Meshkov-Glick (LMG) model by an ensemble of NV centers in a levitated nanodiamond. The quantum phase transition in the LMG model and finite number effects can be observed with this system. We also propose generating torsional superposition states and realizing torsional matter-wave interferometry with spin-torsional coupling.

We show that the paradox published by Saldanha and Vedral [Phys. Rev. A 87, 042102 (2013)] is based on an improper interpretation of the localization probability density and a miscalculation of the wave functions. This Comment shows that there is no paradox as stated.

Continuous-variable systems in quantum theory can be fully described through any one of the $s$-ordered family of quasiprobabilities ${\mathrm{\Lambda }}_s\left(\alpha \right), s\in [-1,1]$. We ask for what values of $(s,a)$ is the scaling map ${\mathrm{\Lambda }}_s\left(\alpha \right)\to a^{-2}{\mathrm{\Lambda }}_s\left(a^{-1}\alpha \right)$ a positive map? Our analysis based on a duality we establish settles this issue: (i) the scaling map generically fails to be positive, showing that there is no useful entanglement witness of the scaling type beyond the transpose map, and (ii) in the two particular cases $(s=1,|a|\le 1)$ and $(s=-1,|a|\ge 1)$, and only in these two nontrivial cases, the map is not only positive but also completely positive as seen through the noiseless attenuator and amplifier channels. We also present a “phase diagram” for the behavior of the scaling maps in the $s-a$ parameter space with regard to its positivity, obtained from the viewpoint of symmetric-ordered characteristic functions. This also sheds light on similar diagrams for the practically relevant attenuation and amplification maps with respect to the noise parameter, especially in the range below the complete-positivity (or quantum-limited) threshold.

Recent years have witnessed a rapidly growing interest in high-dimensional quantum entanglement for fundamental studies as well as towards novel applications. Therefore, the ability to verify entanglement between physical qudits, $d$-dimensional quantum systems, is of crucial importance. To show nonclassicality, Hardy's paradox represents “the best version of Bell's theorem” without using inequalities. However, so far it has only been tested experimentally for bidimensional vector spaces. Here, we formulate a theoretical framework to demonstrate the ladder proof of Hardy's paradox for arbitrary high-dimensional systems. Furthermore, we experimentally demonstrate the ladder proof by taking advantage of the orbital angular momentum of high-dimensionally entangled photon pairs. We perform the ladder proof of Hardy's paradox for dimensions 3 and 4, both with the ladder up to the third step. Our paper paves the way towards a deeper understanding of the nature of high-dimensionally entangled quantum states and may find applications in quantum information science.

In the spirit of the thin-layer quantization approach, we give the formula of the geometric influences of a particle confined to a curved surface embedded in three-dimensional Euclidean space. The geometric contributions can result from the reduced commutation relation between the acted function depending on normal variable and the normal derivative. According to the formula, we obtain the geometric potential, geometric momentum, geometric orbital angular momentum, geometric linear Rashba, and cubic Dresselhaus spin-orbit couplings. As an example, a truncated cone surface is considered. We find that the geometric orbital angular momentum can provide an azimuthal polarization for spin, and the sign of the geometric Dresselhaus spin-orbit coupling can be flipped through the inclination angle of generatrix.

Currently, epsilon-near-zero (ENZ) materials have become important for controlling the propagation of light and enhancing by several orders of magnitude the Kerr and other nonlinearities. Given this advance it is important to examine the quantum electrodynamic processes and information tasks near ENZ materials. We study the entanglement between two two-level systems near ENZ materials and compare our results with the case where the ENZ material is replaced by a metal. It is shown that with ENZ materials substantial entanglement can be achieved over larger distances than for metal films. We show that this entanglement over large distances is due to the fact that one can not only have large emission rates but also large energy transmission rates at the epsilon-near-zero wavelength. This establishes the superiority of ENZ materials for studying processes specifically important for quantum information tasks.

We study the question of existence, i.e., stability with respect to dissociation of the spin-quartet permutation- and reflection-symmetric ${}^4{(-3)}^{+}{}_g$ ($S_z=-3/2,M=-3$) state of the $\left(\alpha \alpha eee\right)$ Coulomb system: the ${{\mathrm{He}}_2}^{+}$ molecular ion, placed in a magnetic field $0\le B\le 10000$ a.u. We assume that the $\alpha$ particles are infinitely massive (Born-Oppenheimer approximation of zero order) and adopt the parallel configuration, when the molecular axis and the magnetic field direction coincide, as the optimal configuration. The study of the stability is performed variationally with a physically adequate trial function. To achieve this goal, we explore several helium-containing compounds in strong magnetic fields, in particular; we study the spin-quartet ground state of the ${\mathrm{He}}^{-}$ ion and the ground (spin-triplet) state of the helium atom, both for a magnetic field in $100\le B\le 10000$  a.u. The main result is that the ${{\mathrm{He}}_2}^{+}$ molecular ion in the state ${}^4{(-3)}^{+}{}_g$ is stable towards all possible decay modes for magnetic fields $B\gtrsim 120\mathrm{a}.\mathrm{u}.$ and with the magnetic field increase the ion becomes more tightly bound and compact with a cigar-type form of electronic cloud. At $B=1000\mathrm{a}.\mathrm{u}.$, the dissociation energy of ${{\mathrm{He}}_2}^{+}$ into ${\mathrm{He}}^{-}+\alpha$ is $\sim 702\mathrm{eV}$ and the dissociation energy for the decay channel to $\mathrm{He}+\alpha +e$ is $\sim 729\mathrm{eV}$, and both energies are in the energy window for one of the observed absorption features of the isolated neutron star 1E1207.4-5209.

We present a three-dimensional theoretical comparison between the radiation-pressure forces exerted on an atom in an isotropic light cooling scheme and in a six-beam molasses. We demonstrate that, in the case of a background vapor where all the space directions of the atomic motion have to be considered, the mean cooling rate is equal in both configurations. Nevertheless, we also point out what mainly differentiates the two cooling techniques: the force component orthogonal to the atomic motion. If this transverse force is always null in the isotropic light case, it can exceed the radiation-pressure-force longitudinal component in the six-beam molasses configuration for high atomic velocities, hence reducing the velocity capture range.

We study the attractive Fermi mixture of a ${}^6\mathrm{Li}\text{−}{}^{40}\mathrm{K}$ gas in one dimension using the continuous matrix product states variational ansatz and obtain the $T=0$ phase diagram. We predict an axial density profile that contains four distinct phases trapped induced along one-dimensional (1D) tubes, which is more intricate than those observed in 1D mass-balanced systems or in higher-dimensional gas clouds. The parameter regimes explored are realistic in view of possible future experiments. This is an application of continuous matrix product states to a nonintegrable fermionic system.

It is known that, in the thermodynamic limit, the Chern number of a translationally invariant system cannot change under unitary time evolutions that are smooth in momentum space. Yet a real-space counterpart of the Chern number, the Bott index, has been shown to change in periodically driven systems with open boundary conditions. Here we prove that the Bott index and the Chern number are identical in translationally invariant systems in the thermodynamic limit. Using the Bott index, we show that, in finite-size translationally invariant systems, a Fermi sea under a periodic drive that is turned on slowly can acquire a different topology from that of the initial state. This can happen provided that the gap-closing points in the thermodynamic limit are absent in the discrete Brillouin zone of the finite system. Hence, in such systems, a periodic drive can be used to dynamically prepare topologically nontrivial states starting from topologically trivial ones.

We study the nonequilibrium quench dynamics of a one-dimensional anyonic gas. We focus on the integrable anyonic Lieb-Liniger model and consider the quench from noninteracting to hard-core anyons. We study the dynamics of the local properties of the system. By means of integrability-based methods, we compute analytically the one-body density matrix and the density-density correlation function at all times after the quench and in particular at infinite time. Our results show that the system evolves from an initial state where the local momentum distribution function is nonsymmetric to a steady state where it becomes symmetric. Furthermore, while the initial momentum distribution functions (and the equilibrium ones) explicitly depend on the anyonic parameter, the final ones do not. This is reminiscent of the dynamical fermionization observed in the context of free expansions after release from a confining trap.

Spontaneous parametric down-conversion is a well-known process in quantum nonlinear optics in which a photon incident on a nonlinear crystal spontaneously splits into two photons. Here we propose an analogous physical process where one excited atom directly transfers its excitation to a pair of spatially separated atoms with probability approaching 1. The interaction is mediated by the exchange of virtual rather than real photons. This nonlinear atomic process is coherent and reversible, so the pair of excited atoms can transfer the excitation back to the first one: the atomic analog of sum-frequency generation of light. The parameters used to investigate this process correspond to experimentally demonstrated values in ultrastrong circuit quantum electrodynamics. This approach can be extended to realize other nonlinear interatomic processes, such as four-atom mixing, and is an attractive architecture for the realization of quantum devices on a chip. We show that four-qubit mixing can efficiently implement quantum repetition codes and, thus, can be used for error-correction codes.

We derive quantum trajectories (also known as stochastic master equations) that describe an arbitrary quantum system probed by a propagating wave packet of light prepared in a continuous-mode Fock state. We consider three detection schemes of the output light: photon counting, homodyne detection, and heterodyne detection. We generalize to input field states in superpositions and mixtures of Fock states and illustrate our formalism with several examples.

In this work we first examine transverse and longitudinal fluxes in a $\mathcal{PT}$-symmetric photonic dimer using a coupled-mode theory. Several surprising understandings are obtained from this perspective: The longitudinal flux shows that the $\mathcal{PT}$ transition in a dimer can be regarded as a classical effect, despite its analogy to $\mathcal{PT}$-symmetric quantum mechanics. The longitudinal flux also indicates that the so-called giant amplification in the $\mathcal{PT}$-symmetric phase is a subexponential behavior and does not outperform a single gain waveguide. The transverse flux, on the other hand, reveals that the apparent power oscillations between the gain and loss waveguides in the $\mathcal{PT}$-symmetric phase can be deceiving in certain cases, where the transverse power transfer is in fact unidirectional. We also show that this power transfer cannot be arbitrarily fast even when the exceptional point is approached. Finally, we go beyond the coupled-mode theory by using the paraxial wave equation and also extend our discussions to a $\mathcal{PT}$ diamond and a one-dimensional periodic lattice.