Typically, optical microscopy uses the wavelike properties of light to image a scene. However, photon arrival times provide more information about emitter properties than the classical intensity alone. Here we show that the Hanbury Brown and Twiss experiment (second-order correlation function) measures the relative brightness of two single-photon emitters. By combining the total number of detected photons with the zero-lag value of the correlation function, the positions and relative brightness of two emitters in two dimensions can be resolved from only three measurement positions: trilateration. This result is impossible to achieve on the basis of classical intensity measurements alone and represents a minimal demonstration of the advantage gained using quantum measurements over conventional classical microscopy. The effective point-spread function for imaging scales approximately as the inverse square root of the acquisition time.

We present an extension of the Löwdin strategy to find arbitrary matrix elements of generic Slater determinants. Our method applies to an arbitrary number of fermionic operators, even in the case of a singular overlap matrix.

According to the Copenhagen interpretation of quantum mechanics a measurement leads to the collapse of a quantum system. A unified description of a quantum system before and after measurement of the position ${\mathit{r}}_1^a$ and the spin $s_1^a$ of one of its electrons is proposed based on the conditional amplitude and its density. Quantum collapse is described as reduction from continuum of the conditional amplitudes $\left\{\mathrm{\Phi }({\mathit{x}}_2,...,{\mathit{x}}_N|{\mathit{r}}_1^h,s_1^h)\right\}$ in an initial quantum state to the single actualized amplitude $\mathrm{\Phi }({\mathit{x}}_2,...,{\mathit{x}}_N|{\mathit{r}}_1^a,s_1^a)$ in a collapsed state. The latter state contains the information about quantum entanglement, the extraction of which with fragment entanglement measure is demonstrated for the realistic valence bond models of the dissociating dihydrogen and ethylene molecules.

We demonstrate the quantum-enhanced supervised classification of vectors in a nitrogen-vacancy ($NV$)–center system using the algorithm of Lloyd, Mohseni, and Rebentrost (arXiv:1307.0411). A ${}^{13}\mathrm{C}$ nuclear spin is employed to encode the vectors with the spin-triplet state of the $NV$ center as an ancilla. We design efficient methods to prepare the initial electron-nuclear entangled state within the coherence time $T_2^{*}$ of the electron spin, and then compute the distance between the test vector and the center of each class by measuring the level population through maximum likelihood estimation. Our experiment allows more than one reference vector in each class, thus forming an important enabling step toward quantum-enhanced machine learning.

Starting from explicitly constructing an entanglement witness for the mixture of a Dicke state with white noise (noisy Dicke state), we provide an entanglement criterion for any multiqubit state. The criterion is necessary and sufficient for noisy $W$ states of three, four, and five qubits. We demonstrate an entanglement criterion for any four-qubit state. The criterion is necessary and sufficient for the generalized noisy four-qubit Dicke states. We present the sufficient criterion of full separability for a noisy $2k$ qubit Dicke state with $k$ excitations. Then we generalize the entanglement witness and criterion to the mixture of white noise and generic $n$-qubit Dicke state with $k$ excitations based on strong numeric evidence. Apart from noisy Dicke states, we also consider the entanglement properties of simultaneously amplitude damped and dephased Dicke states, and depolarized multiqubit $W$ states.

We describe a method for creating small quantum processors in a crystal stoichiometric in an optically active rare-earth ion. The crystal is doped with another rare earth, creating an ensemble of identical clusters of surrounding ions, whose optical and hyperfine frequencies are uniquely determined by their spatial position in the cluster. Ensembles of ions in each unique position around the dopant serve as qubits, with strong local interactions between ions in different qubits. These ensemble qubits can each be used as a quantum memory for light, and we show how the interactions between qubits can be used to perform linear operations on the stored photonic state. We also describe how these ensemble qubits can be used to enact, and study, error correction.

We propose a two-qubit experiment for validating tunable antiferromagnetic XX interactions in quantum annealing. Such interactions allow the time-dependent Hamiltonian to be nonstoquastic, and the instantaneous ground state can have negative amplitudes in the computational basis. Our construction relies on how the degeneracy of the Ising Hamiltonian's ground states is broken away from the end point of the anneal: above a certain value of the antiferromagnetic XX interaction strength, the perturbative ground state at the end of the anneal changes from a symmetric to an antisymmetric state. This change is associated with a suppression of one of the Ising ground states, which can then be detected using solely computational basis measurements. We show that a semiclassical approximation of the annealing protocol fails to reproduce this feature, making it a candidate “quantum signature” of the evolution.

A recent realization of three-dimensional optical lattice clocks circumvents short-range collisional clock shifts which have been the bottle neck towards higher precision; the long-range electronic dipole-dipole interaction between the atoms becomes the primary source of clock shift due to interatomic interactions. We study the Rabi spectroscopy of three-dimensional optical lattice clocks with unity filling. From the Lindblad equation governing the time evolution of the density matrix of the atoms, we derive the Bloch equations in the presence of the external Rabi driving laser field, and solve the equations approximately to the first order of the coupling strength of the dipole-dipole interaction between the atoms. We find that the clock shift equals the product of the coupling strength, a factor determined by the parameters of the Rabi pulse, and another factor depending on the configuration of the three-dimensional optical lattice. Our result on the clock shift within the Rabi spectroscopy can be checked by measurements in a future experiment.

Recent investigations by Fuchs et al. [Matthias Fuchs, Mariano Trigo, Jian Chen, Shambhu Ghimire, Sharon Shwartz, Michael Kozina, Mason Jiang, Thomas Henighan, Crystal Bray, Georges Ndabashimiye et al., Anomalous nonlinear X-ray Compton scattering, Nat. Phys. 11, 964 (2015)] revealed an anomalous frequency shift in nonlinear Compton scattering of high-intensity x rays by electrons in solid beryllium. This frequency shift was at least 800 eV to the red of the values predicted by analytical free-electron models for the same process. In this paper, we describe a method for simulating nonlinear Compton scattering. The method is applied to the case of bound electrons in a local, spherical potential to explore the role of binding energy in the frequency shift of scattered x rays for different scattered angles. The results of the calculation do not exhibit an additional redshift for the scattered x rays beyond the nonlinear Compton shift predicted by the free-electron model. However, they do reveal a small blueshift relative to the free-electron prediction for nonlinear Compton scattering. The effect of electron-electron correlation effects is calculated and determined to be unlikely to be the source of the redshift. The case of linear Compton scattering from a photoionized electron followed by electron recapture is examined as a possible source of the redshift and ruled out.

In quantum theory, the description of radiation by a system of charged particles in stationary and nonstationary cases requires different approaches. While radiative processes in stationary systems are covered in textbooks, the emission of radiation by nonstationary systems driven by an external time-dependent force has not received due attention. In this paper, we develop a quantum theory describing the spectrum of photons emitted by a generic nonstationary atomic system and show how to implement it in practical calculations. We also discuss the approximation in which the exact quantum formula for the spectrum reduces to the well-known classical formula with the classical dipole moment replaced by the expectation value of the dipole moment calculated quantum mechanically. This approximation underlies the classical ansatz commonly used in the theory of high-order harmonic generation (HHG). To illustrate the theory, we apply it to calculating HHG spectrum for a one-dimensional zero-range-potential model. The spectra obtained from the quantum formula and from the ansatz are different and the difference grows as the probability of ionization grows. This difference is an observable effect predicted by the present theory.

To solve the time-dependent Schrödinger equation in spatially inhomogeneous pulses of electromagnetic radiation, we propose an iterative semiclassical complex trajectory approach. In numerical applications, we validate this method against ab initio numerical solutions by scrutinizing (a) electronic states in combined Coulomb and spatially homogeneous laser fields and (b) streaked photoemission from hydrogen atoms and plasmonic gold nanospheres. In comparison with streaked photoemission calculations performed in strong-field approximation, we demonstrate the improved reconstruction of the spatially inhomogeneous induced plasmonic infrared field near a nanoparticle surface from streaked photoemission spectra.

The above-threshold ionization (ATI) of argon atoms in intense circularly polarized laser fields has been investigated by numerically solving the three-dimensional time-dependent Schrödinger equation and the classical Newton-Lorentz equation. There are obvious angular shifts of the photoelectron momentum distribution corresponding to the peaks in the ATI spectrum. They have little dependence on the external field and are mainly determined by the final kinetic energy. It is found that the angular shift is monotonically increasing as a function of the photoelectron final kinetic energy. Based on the classical simulations, the initial kinetic energy of electrons can be obtained by separating the contribution of the external field and Coulomb potential from the final kinetic energy. We conclude that the stronger the laser field, the bigger the initial velocity of electrons for a definite final kinetic energy. Then, based on the final kinetic energy and the laser parameters, the initial velocity can be evaluated.

We propose a practical protocol to generate and observe a non-Abelian geometric phase using a periodically driven Raman process in the hyperfine ground-state manifold of atoms in a dilute ultracold gas. Our analysis is based upon recent developments and application of Floquet theory to periodically driven quantum systems. The simulation results show the non-Abelian gauge transformation and the noncommuting property of the $\mathrm{SU}\left(2\right)$ transformation operators. Based on these results, we propose a possible experimental implementation with an ultracold dilute Bose gas.

It was recently found that, under the action of the spin-orbit coupling (SOC) and Zeeman splitting (ZS), binary BEC with intrinsic cubic nonlinearity supports families of gap solitons, provided that the kinetic energy is negligible in comparison with the SOC and ZS terms. We demonstrate that, also under the action of SOC and ZS, a similar setting may be introduced for BEC with two components representing different atomic states, resonantly coupled by microwave radiation, while the Poisson's equation accounts for the feedback of the two-component atomic wave function onto the radiation. The microwave-mediated interaction induces an effective nonlinear trapping potential, which strongly affects the purport of the linear spectrum in this system. As a result, families of both gap and embedded solitons (those overlapping with the continuous spectrum) are found, being chiefly stable. The shape of the solitons features exact or broken skew symmetry. In addition to fundamental solitons (whose shape may or may not include a node), a family of dipole solitons is constructed too, which are even more stable than their fundamental counterparts. A nontrivial stability area is identified for moving solitons in the present system, which lacks Galilean invariance. Colliding solitons merge into a single one.

We consider a spin-imbalanced Fermi gas at zero temperature in the normal phase on the BCS side of the BCS-BEC crossover and around unitarity. We compute the critical polarization for pairing, the correlated occupation numbers, and the contact in an extension of particle-particle random-phase approximation (RPA) (also called non-self-consistent $T$-matrix approach or ladder approximation). The so-called renormalized RPA consists in computing the $T$ matrix with self-consistently determined occupation numbers. The occupation numbers are determined either by keeping the self-energy only to first order or by resumming the Dyson equation. In this way, the result for the critical polarization, strongly overestimated in standard RPA, is clearly improved. We also discuss some problems of this approach.

Compressive sensing is used to perform high-dimensional quantum channel estimation with classical light. As an example, we perform a numerical simulation for the case of a three-dimensional classically nonseparable state that is propagated through atmospheric turbulence. Using a singular-value-thresholding-algorithm-based compressive sensing, we determine the channel matrix, which we subsequently use to correct for the atmospheric-turbulence-induced distortions. As a measure of the success of the procedure, we calculate the fidelity and the trace distance of the corrected density matrix with respect to the input state, and compare the results to those of the density matrix for the uncorrected state. Furthermore, we quantify the amount of classical nonseparability in the density matrix of the corrected state by calculating its negativity. The results show that compressive sensing could contribute to the development and implementation of free-space quantum and optical communication systems.

The reduced dynamics of an open quantum system obtained from an underlying microscopic Hamiltonian can in general only approximately be described by a time-local master equation. The quality of that approximation depends primarily on the coupling strength and the structure of the environment. Various such master equations have been proposed with different aims. Choosing the most suitable one for a specific system is not straightforward. By focusing on the accuracy of the reduced dynamics we provide a thorough assessment for a selection of methods (Redfield equation, quantum optical master equation, coarse-grained master equation, a related dynamical map approach, and a partial-secular approximation). We consider secondary, here, whether or not an approach guarantees positivity. We use two qubits coupled to a Lorentzian environment in a spin-boson-like fashion modeling a generic situation with various system and bath time scales. We see that, independent of the initial state, the simple Redfield equation with time-dependent coefficients is significantly more accurate than all other methods under consideration. We emphasize that positivity violation in the Redfield equation formalism becomes relevant only in a regime where any of the perturbative master equations considered here are rendered invalid anyway. This implies that the loss of positivity should in fact be welcomed as an important feature: it indicates the breakdown of the weak-coupling assumption. In addition we present the various approaches in a self-contained way and use the behavior of their errors to provide further insight into the range of validity of each method.

This paper presents a framework for quantum causal modeling based on the interpretation of causality as a relation between an observer's probability assignments to hypothetical or counterfactual experiments. The framework is based on the principle of “causal sufficiency”: that it should be possible to make inferences about interventions using only the probabilities from a single “reference experiment” plus causal structure in the form of a directed acyclic graph. This leads to several interesting results: we find that quantum measurements deserve a special status distinct from interventions, and that a special rule is needed for making inferences about what would happen if they are not performed (“unmeasurements”). One natural candidate for this rule is found to be an equation of importance to the quantum Bayesianism interpretation of quantum mechanics. We find that the causal structure of quantum systems must have a “layered” structure, and that the model can naturally be made symmetric under reversal of the causal arrows.

Common flux-qubit readout schemes are qubit dominated, meaning they measure in the energy eigenbasis of the qubit. For various applications, measurements in a basis different from the energy eigenbasis are required. Here we present an indirect measurement protocol, which is detector dominated instead of qubit dominated, yielding projective measurements in the persistent-current basis for arbitrary bias points. We show that with our setup it is possible to perform a quantum nondemolition measurement in the persistent-current basis at all flux bias points with fidelities reaching almost $100\%$.

Nonadiabatic holonomic gates provide a promising way to realize high-fidelity quantum gates because of their robustness and high-speed implementation. Compared with quantum computation that consists of many nonadiabatic holonomic one-qubit and two-qubit gates, the nonadiabatic holonomic multiqubit gate, which can reduce the number of quantum gates, is more efficient to construct simpler and faster complicated quantum computation. We propose the simplest scheme to realize nonadiabatic holonomic three-qubit controlled gates. On the one hand, the parameters we set to realize the three-qubit controlled gates are all time independent, which makes it easy to control and operate the quantum systems. On the other hand, our scheme avoids the combination of the elementary gates and realizes the three-qubit controlled gates only by one-shot implementation.