Farhi et al. recently proposed a class of quantum algorithms, the quantum approximate optimization algorithm (QAOA), for approximately solving combinatorial optimization problems (E. Farhi et al., arXiv:1411.4028; arXiv:1412.6062; arXiv:1602.07674). A level-$p$ QAOA circuit consists of $p$ steps; in each step a classical Hamiltonian, derived from the cost function, is applied followed by a mixing Hamiltonian. The $2p$ times for which these two Hamiltonians are applied are the parameters of the algorithm, which are to be optimized classically for the best performance. As $p$ increases, parameter optimization becomes inefficient due to the curse of dimensionality. The success of the QAOA approach will depend, in part, on finding effective parameter-setting strategies. Here we analytically and numerically study parameter setting for the QAOA applied to MaxCut. For the level-1 QAOA, we derive an analytical expression for a general graph. In principle, expressions for higher $p$ could be derived, but the number of terms quickly becomes prohibitive. For a special case of MaxCut, the “ring of disagrees,” or the one-dimensional antiferromagnetic ring, we provide an analysis for an arbitrarily high level. Using a fermionic representation, the evolution of the system under the QAOA translates into quantum control of an ensemble of independent spins. This treatment enables us to obtain analytical expressions for the performance of the QAOA for any $p$. It also greatly simplifies the numerical search for the optimal values of the parameters. By exploring symmetries, we identify a lower-dimensional submanifold of interest; the search effort can be accordingly reduced. This analysis also explains an observed symmetry in the optimal parameter values. Further, we numerically investigate the parameter landscape and show that it is a simple one in the sense of having no local optima.

Measurement-based quantum computation describes a scheme where entanglement of resource states is utilized to simulate arbitrary quantum gates via local measurements. Recent works suggest that symmetry-protected topologically nontrivial, short-ranged entangled states are promising candidates for such a resource. Miller and Miyake [npj Quantum Inf. 2, 16036 (2016)] recently constructed a particular ${\mathbb{Z}}_2\times {\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetry-protected topological state on the Union Jack lattice and established its quantum-computational universality. However, they suggested that the same construction on the triangular lattice might not lead to a universal resource. Instead of qubits, we generalize the construction to qudits and show that the resulting $(d-1)$ qudit nontrivial ${\mathbb{Z}}_d\times {\mathbb{Z}}_d\times {\mathbb{Z}}_d$ symmetry-protected topological states are universal on the triangular lattice, for $d$ being a prime number greater than 2. The same construction also holds for other 3-colorable lattices, including the Union Jack lattice.

We propose a superconducting circuit platform for simulating spin-1 models. To this purpose we consider a chain of $N$ ultrastrongly coupled qubit-resonator systems interacting through a grounded superconducting quantum interference device (SQUID). The anharmonic spectrum of the qubit-resonator system and the selection rules imposed by the global parity symmetry allow us to activate well controlled two-body quantum gates via ac pulses applied to the SQUID. We show that our proposal has the same simulation time for any number of spin-1 interacting particles. This scheme may be implemented within the state-of-the-art circuit QED in the ultrastrong coupling regime.

Owing to its ability to provide unique information on electron dynamics, time-resolved electron momentum spectroscopy (EMS) is used to study theoretically a laser-driven electronic motion in atoms. Specifically, a chirped laser pulse is used to adiabatically transfer the populations of lithium atoms from the ground state to the first excited state. During this process, impact ionization near the Bethe ridge by time-delayed ultrashort, high-energy electron pulses is used to image the instantaneous momentum density of this electronic population transfer. Simulations with 100 fs and 1 fs pulse durations demonstrate the capability of EMS to image the time-varying momentum density, including its change of symmetry as the population transfer progresses. Moreover, the spectra corresponding to different pulse durations reveal different kinds of electronic motion. We discuss how to properly interpret these time-resolved EMS spectra, which represent a generalization of time-independent EMS.

The three-body fragmentation dynamics of ${{\mathrm{CH}}_4}^{2+}$ dications induced by single-electron capture of slow (3-keV/u) $\mathrm{A}{\mathrm{r}}^{8+}$ ions is investigated. The experiment is performed on a newly built, highly charged ion collision platform which consists of an electron cyclotron resonance ion source and a cold target recoil ion momentum spectroscopy (COLTRIMS) setup. Using the COLTRIMS methodology, the complete kinematical information is determined for two three-body breakup channels, ${{\mathrm{CH}}_4}^{2+}\to {\mathrm{H}}^{+}+{{\mathrm{CH}}_2}^{+}+\mathrm{H}$ and ${{\mathrm{CH}}_4}^{2+}\to {{\mathrm{H}}_2}^{+}+\mathrm{C}{\mathrm{H}}^{+}+\mathrm{H}$. Then analyzing the complete kinematics with the Dalitz plot, very different fragmentation mechanisms (e.g., sequential and/or concerted pathway) are clearly identified for the two channels. To confirm the existence of some possible fragmentation pathways, we also simulate corresponding Dalitz plots employing a simple classical mechanical model. For the ${\mathrm{H}}^{+}+{{\mathrm{CH}}_2}^{+}+\mathrm{H}$ channel, the dependence of the fragmentation pathway on its kinetic energy release is studied, which reflects the different nature of the corresponding states of ${{\mathrm{CH}}_4}^{2+}$ dications. Furthermore, the kinetic energy ratio of two ionic fragments is analyzed to infer the three-body fragmentation mechanism of ${{\mathrm{CH}}_4}^{2+}$ dications.

The single, double, and triple Auger decay processes from the $1s^{-1}2p^{-1}np$ shake-up states of Ne have been studied with a multielectron coincidence method. It is revealed that the Rydberg electron promoted in the initial photoionization step generally acts as a spectator both in the single and double Auger decays. The single Auger decay predominantly produces three-hole one-particle final states while leaving the Rydberg electron virtually unchanged. In the cascade double Auger paths, the Rydberg electron is a spectator in the first step of the cascade process but is subsequently released in the second step, leading to the formation of three-hole final states. The direct paths of double Auger decays, which are much weaker than the cascade paths, produce preferably four-hole one-particle states due to the spectator behavior of the Rydberg electron. The distributions of the $\mathrm{N}{\mathrm{e}}^{4+}$ states populated by the triple Auger decays are also presented.

Tightly focused optical dipole traps induce vector light shifts (“fictitious magnetic fields”) which complicate their use for single-atom trapping and manipulation. The problem can be mitigated by adding a larger, real magnetic field, but this solution is not always applicable; in particular, it precludes fast switching to a field-free configuration. Here we show that this issue can be addressed elegantly by deliberately adding a small elliptical polarization component to the dipole trap beam. In our experiments with single ${}^{87}\mathrm{Rb}$ atoms laser-cooled in a chopped trap, we observe improvements up to a factor of 11 of the trap lifetime compared to the standard, seemingly ideal linear polarization. This effect results from a modification of heating processes via spin-state diffusion in state-dependent trapping potentials. We develop Monte Carlo simulations of the evolution of the atom's internal and motional states and find that they agree quantitatively with the experimental data. The method is general and can be applied in all experiments where the longitudinal polarization component is non-negligible.

The transport of excitations governs fundamental properties of matter. Particularly rich physics emerges in the interplay between disorder and environmental noise, even in small systems such as photosynthetic biomolecules. Counterintuitively, noise can enhance coherent quantum transport, which has been proposed as a mechanism behind the high transport efficiencies observed in photosynthetic complexes. This effect has been called “environment-assisted quantum transport”. Here, we propose a quantum simulation of the excitation transport in an open quantum network, taking advantage of the high controllability of current trapped-ion experiments. Our scheme allows for the controlled study of various different aspects of the excitation transfer, ranging from the influence of static disorder and interaction range, over the effect of Markovian and non-Markovian dephasing, to the impact of a continuous insertion of excitations. Our paper discusses experimental error sources and realistic parameters, showing that it can be implemented in state-of-the-art ion-chain experiments.

We show that the Kapitza stabilization can occur in the context of nonlinear quantum fields. Through this phenomenon, an amplitude-modulated lattice can stabilize a Bose–Einstein condensate with repulsive interactions and prevent the spreading for long times. We present a classical and quantum analysis in the framework of Gross-Pitaevskii equation, specifying the parameter region where stabilization occurs. Effects of nonlinearity lead to a significant increase of the stability domain compared with the classical case. Our proposal can be experimentally implemented with current cold atom settings.

A system of neutral atoms trapped in an optical lattice and dispersively coupled to the field of an optical cavity can realize a variation of the Bose-Hubbard model with infinite-range interactions. This model exhibits a first-order quantum phase transition between a Mott insulator and a charge density wave, with spontaneous symmetry breaking between even and odd sites, as was recently observed experimentally [Landig et al., Nature (London) 532, 476 (2016)]. In the present paper, we approach the analysis of this transition using a variational model which allows us to establish the notion of an energy barrier separating the two phases. Using a discrete WKB method, we then show that the local tunneling of atoms between adjacent sites lowers this energy barrier and hence facilitates the transition. Within our simplified description, we are thus able to augment the phase diagram of the model with information concerning the height of the barrier separating the metastable minima from the global minimum in each phase, which is an essential aspect for the understanding of the reconfiguration dynamics induced by a quench across a quantum critical point.

We investigate the high-order-sideband generation (HSG) in a hybrid cavity electro-photomechanical system in which an optical cavity is driven by two optical fields (a monochromatic pump field and a nanosecond Gaussian probe pulse with huge numbers of wave cycles), and at the same time a microwave cavity is driven by a monochromatic ac voltage bias. We show that even if the input powers of two driven optical fields are comparatively low the HSG spectra can be induced and enhanced, and the sideband plateau is extended remarkably with the power of the ac voltage bias increasing. It is also shown that the driven ac voltage bias has profound effects on the carrier-envelope-phase–dependent effects of the HSG in the hybrid cavity electro-photomechanical system. Our research may provide an effective way to control the HSG of optical fields by using microwave fields in cavity optomechanics systems.

Spectroscopy is a unique experimental tool for measuring the fundamental Casimir-Polder interaction between excited-state atoms, or other polarizable quantum objects, and a macroscopic surface. Spectroscopic measurements probe atoms at nanometric distances away from the surface where QED retardation is usually negligible and the atom-surface interaction is proportional to the inverse cube of the separation distance, otherwise known as the van der Waals regime. Here we focus on selective reflection, one of the main spectroscopic probes of Casimir-Polder interactions. We calculate selective reflection spectra using the full, distance dependent, Casimir-Polder energy shift and linewidth. We demonstrate that retardation can have significant effects, in particular for experiments with low-lying energy states. We also show that the effective probing depth of selective reflection spectroscopy depends on the transition linewidth. Our analysis allows us to calculate selective reflection spectra with composite surfaces, such as metasurfaces, dielectric stacks, or even bidimensional materials.

The Dicke model describes the coherent interaction of a laser-driven ensemble of two-level atoms with a quantized light field. It is realized within cavity QED experiments, which in addition to the coherent Dicke dynamics feature dissipation due to, e.g., atomic spontaneous emission and cavity photon loss. Spontaneous emission supports the uncorrelated decay of individual atomic excitations as well as the enhanced collective decay of an excitation that is shared by $N$ atoms and whose strength is determined by the cavity geometry. We derive a many-body master equation for the dissipative Dicke model including both spontaneous emission channels and analyze its dynamics on the basis of Heisenberg-Langevin and stochastic Bloch equations. We find that the collective loss channel leads to a region of bistability between the empty and the superradiant state. Transitions between these states are driven by nonthermal Markovian noise. The interplay between dissipative and coherent elements leads to a genuine nonequilibrium dynamics in the bistable regime, which is expressed via a nonconservative force and a multiplicative noise kernel appearing in the stochastic Bloch equations. We present a semiclassical approach, based on stochastic nonlinear optical Bloch equations, which for the infinite-range Dicke model become exact in the large-$N$-limit. The absence of an effective free-energy functional, however, necessitates the inclusion of fluctuation corrections with $O(1/N)$ for finite $N<\infty$ to locate the nonthermal first-order phase transition between the superradiant and the empty cavity.

The light propagating in a waveguide array or photonic lattice has become an ideal platform to control light and to mimic quantum behaviors in a classical system. We here investigate the propagation of light in a coupled waveguide array with one of the waveguides periodically modulated in its geometric structure or refractive index. Within the framework of Floquet theory, it is interesting to find that the light shows the localized propagation in the modulated waveguide as long as bound quasistationary modes are formed in the band-gap area of the Floquet eigenvalue spectrum. This mechanism gives a useful instruction to confine light via engineering the periodic structure to form the bound modes. It also serves as a classical simulation of decoherence control via temporally periodic driving in open quantum systems.

Within the framework of exact quantum electrodynamics in dispersing and absorbing media, we have studied the emission from an initially in the upper state prepared emitter in a high-quality cavity in the case when there are two cavity modes that resonantly interact with the emitter. In the case when one of the modes is in resonance with the emitter transition and the other mode is tuned to the frequency equal to the Rabi splitting of the first mode, the effect of Rabi resonance is observed. In particular, the second mode exhibits a substantial increase of the one-photon Fock state efficiency and enhancement of the emission spectrum in comparison to the exact resonant case.

Quantization processes generally assume the Hamiltonian formulation of classical mechanics. The notion of dissipation does not fit easily into the Hamiltonian structure. There appeared in the literature a quantization process that circumvents the use of the Hamiltonian approach and derives the Schrödinger equation from first principles. Thus, the usual approach of assuming a dissipative force of the type $f_d=k\left(t\right)v$, where $k\left(t\right)$ is a scalar depending only on the time and $v$ is the velocity, can be approached using this quantization process to mathematically derive a Schrödinger equation. The derivation for this simple case was already performed by the authors elsewhere. The full generalization of a dissipative force in the context of a linear response theory, as is usual in classical mechanics, would be to write this force as $f_i(\mathbf{x},t)={\mathcal{K}}_{ij}(\mathbf{x},t)v_j$, where $\mathcal{K}(\mathbf{x},t)$ is a tensor giving the nonhomogeneity and nonisotropy of the process. In this paper, we present this generalization and connect our results with nonlinear Schrödinger-like equations already known from the literature.

The interaction of a quantum system with its environment causes decoherence, setting a fundamental limit on its suitability for quantum information processing. However, we show that if the system consists of coupled parts with different internal energy scales then the interaction of one part with a thermal bath need not lead to loss of coherence from the other. Remarkably, we find that the protected part can remain coherent for longer when the coupling to the bath becomes stronger or the temperature is raised. Our theory will enable the design of decoherence-resistant hybrid quantum computers.

We report an experimental implementation of automated state transformations on spatial photonic qutrits following the theoretical proposal made by Baldijão et al. [Phys. Rev. A 96, 032329 (2017)]. A qutrit state is simulated by using three Gaussian beams, and after some state operations, the transformed state is available in the end in terms of the basis state. The state transformation setup uses a spatial light modulator and a calcite-based interferometer. The results reveal the usefulness of the operation method. The experimental data show a good agreement with theoretical predictions, opening possibilities for explorations in higher dimensions and in a wide range of applications. This is a necessary step in qualifying spatial photonic qudits as a competitive setup for experimental research in the implementation of quantum algorithms which demand a large number of steps.

We embarked upon the task of experimental protection of different classes of tripartite entangled states, namely, the maximally entangled Greenberger-Horne-Zeilinger (GHZ) and W states and the tripartite entangled state called the $W\overline{W}$ state, using dynamical decoupling. The states were created on a three-qubit NMR quantum information processor and allowed to evolve in the naturally noisy NMR environment. Tripartite entanglement was monitored at each time instant during state evolution, using negativity as an entanglement measure. It was found that the W state is most robust while the GHZ-type states are most fragile against the natural decoherence present in the NMR system. The $W\overline{W}$ state, which is in the GHZ class yet stores entanglement in a manner akin to the W state, surprisingly turned out to be more robust than the GHZ state. The experimental data were best modeled by considering the main noise channel to be an uncorrelated phase damping channel acting independently on each qubit, along with a generalized amplitude damping channel. Using dynamical decoupling, we were able to achieve a significant protection of entanglement for GHZ states. There was a marginal improvement in the state fidelity for the W state (which is already robust against natural system decoherence), while the $W\overline{W}$ state showed a significant improvement in fidelity and protection against decoherence.