We introduce a method to reconstruct the density matrix $\rho$ of a system of $n$ qubits and estimate its rank $d$ from data obtained by quantum-state-tomography measurements repeated $m$ times. The procedure consists of minimizing the risk of a linear estimator $\widehat{\rho }$ of $\rho$ penalized by a given rank (from 1 to $2^n$), where $\widehat{\rho }$ is previously obtained by the moment method. We obtain simultaneously an estimator of the rank and the resulting density matrix associated to this rank. We establish an upper bound for the error of the penalized estimator, evaluated with the Frobenius norm, which is of order $dn{(4/3)}^n/m$ and consistent for the estimator of the rank. The proposed methodology is computationally efficient and is illustrated with some example states and real experimental data sets.

We consider a general quantum system interacting with a bath and derive a master equation in the Lindblad form describing the evolution of the whole quantum system subjected to a measurement-based direct quantum feedback control (MDFC). As an example, we consider a qubit coupled with a dephasing environment under the MDFC. We show that for any given pure target state we can always find the corresponding MDFC scheme, which can effectively drive any initial state into this target state. By using an appropriate MDFC scheme with weak measurement we can stabilize a single qubit initially prepared in one of two nonorthogonal states against dephasing noise. Furthermore, we can effectively protect a kind of known mixed states composed of two nonorthogonal states by using the corresponding MDFC scheme.

We report the development of a direct variational method for calculating the first rotational excited state of diatomic molecules with $\sigma$ electrons where the Born-Oppenheimer approximation is not assumed. The method employs all-particle explicitly correlated Gaussian basis functions. The exponential parameters of the Gaussians are optimized with the aid of an analytically calculated energy gradient determined with respect to these parameters. The method is tested in calculations of the ortho-para spin isomerization of the hydrogen molecule in its all-bound vibrational states.

We present a rigorous method for accurate ab initio calculations of the electron electric-dipole-moment $\mathcal{P},\mathcal{T}$-odd interaction constant $W_d$. The approach uses configuration interaction wave functions and Dirac four-component spinors as one-particle basis functions, and the interaction constant $W_d$ is obtained as an expectation value over these correlated wave functions. We apply the method to the HfF${}^{+}$ molecular ion and determine spectroscopic constants for four low-lying electronic states. For one of these states ($\Omega =1$) we determine the effective electric field ($E_{\mathrm{eff}}=W_d\Omega$), which amounts to 23.3 GV/cm, correlating 34 valence and outer atomic core electrons and using wave-function expansions with nearly $5\times {10}^8$ coefficients.

We present a general analytical approach to investigate relativistic corrections in the dynamical Casimir effect (DCE). Particularly, we discuss the behavior of the additional frequency bands that appear in the spectral distribution of the created particles when the first relativistic corrections are taken into account. We do that in the context of circuit QED, by analyzing the setup used in the first measurement of the DCE, where a system with a time-dependent boundary condition simulates the mechanical motion of a single mirror in 1+1 dimensions. Our method is applicable to a large class of systems with oscillatory time-dependent parameters and can be generalized to higher dimensions and other fields.

Molecules in selected rotational states can remain well aligned for extended periods of time in the presence of an appropriate periodic train of nonresonant laser pulses. Here, we show that these states can be prepared by slowly switching the electromagnetic field amplitude during a sequence of laser pulses. For low-temperature ensembles, a high degree of alignment can be achieved by designing pulse trains that take into account the distribution of avoided crossings between quasienergy curves. Additionally, we present calculations that illustrate several effects causing misalignment. A discussion of subtleties concerning systems with unbounded rotational spectra is also included.

We study the first-order quantum phase transitions of Bose gases in optical lattices. A special emphasis is placed on an anomalous hysteresis behavior, in which the phase transition occurs in a unidirectional way and a hysteresis loop does not form. We first revisit the hardcore Bose-Hubbard model with dipole-dipole interactions on a triangular lattice to analyze accurately the ground-state phase diagram and the hysteresis using the cluster mean-field theory combined with cluster-size scaling. Details of the anomalous hysteresis are presented. We next consider the two-component and spin-1 Bose-Hubbard models on a hypercubic lattice and show that the anomalous hysteresis can emerge in these systems as well. In particular, for the former model, we discuss the experimental feasibility of the first-order transitions and the associated hysteresis. We also explain an underlying mechanism of the anomalous hysteresis by means of the Ginzburg-Landau theory. From the given cases, we conclude that the anomalous hysteresis is a ubiquitous phenomenon of systems with a phase region of lobe shape that is surrounded by the first-order boundary.

Nanoparticle position detection is prevented in highly symmetric whispering-gallery-mode optical microcavities due to the redistributable electric field of resonant light therein. In asymmetric tubular microcavities formed by a rolled-up slab waveguide, the optical resonant modes are split and locked to the spiral-shaped geometry which provides a reference point for the nodes and antinodes of the electric field within the microcavity. The discriminative responses of neighboring resonant modes to a local disturbance provide a method for angular position detection of a single nanoparticle on a rolled-up optical microcavity. These findings add functionality to microcavity applications and a deeper understanding of cavity electrodynamics.

We prove that all functions obeying the Kramers-Kronig relations can be approximated as superpositions of Lorentzian functions, to any precision. As a result, the typical textbook analysis of dielectric dispersion response functions in terms of Lorentzians may be viewed as encompassing the whole class of causal functions. A further consequence is that Lorentzian resonances may be viewed as possible building blocks for engineering any desired metamaterial response, for example, by use of split-ring resonators of different parameters. Two example functions, far from typical Lorentzian resonance behavior, are expressed in terms of Lorentzian superpositions: a steep dispersion medium that achieves large negative susceptibility with arbitrarily low loss or gain and an optimal realization of a perfect lens over a bandwidth. Error bounds are derived for the approximation.

Taking the non-Markovian effect into account, we study how to store a single photon of arbitrary temporal shape in a single atom coupled to an optical cavity. Our model applies to Raman transitions in three-level atoms, with one branch of the transition controlled by a driving pulse, and the other coupled to the cavity. For any couplings of input field to the optical cavity and detunings of the atom from the driving pulse and cavity, we extend the input-output relation from Markovian to non-Markovian dynamics. For most possible photon shapes, we derive an analytic expression for the driving pulse in order to completely map the input photon into the atom. We find that the amplitude of the driving pulse depends only on the detuning of the atom from the frequency of the cavity, i.e., the detuning of the atom to the driving pulse has no effect on the strength of the driving pulse.

Slow highly charged ions penetrating surfaces quickly capture many electrons in highly excited states, leaving empty the innermost shells, forming hollow atoms. These hollow atoms then fill their innermost shells in a stepwise manner through a long cascade of Auger and x-ray transitions. We have measured the mean emission depths of the series of x rays emitted during the decay cascade of Ar hollow atoms formed below the surface of metal and dielectric materials. It has been found that the decay times of these hollow atoms are much longer in dielectrics than in metals, and at keV/$q$ kinetic energies, at depths of the order of 10–20 nm, considerably deeper than any expected value. These findings have been tentatively explained by the different responses of metals and dielectrics to the slow penetration of a highly charged ion.

The quantum state $\psi$ is a mathematical object used to determine the outcome probabilities of measurements on physical systems. Its fundamental nature has been the subject of discussions since the origin of the theory: Is it ontic, that is, does it correspond to a real property of the physical system? Or is it epistemic, that is, does it merely represent our knowledge about the system? Recent advances in the foundations of quantum theory show that epistemic models that obey a simple continuity condition are in conflict with quantum theory already at the level of a single system. Here we report an experimental test of continuous epistemic models using high-dimensional attenuated coherent states of light traveling in an optical fiber. Due to nonideal state preparation (of coherent states with imperfectly known phase) and nonideal measurements (arising from losses and inefficient detection), this experiment tests only epistemic models that satisfy additional constraints which we discuss in detail. Our experimental results are in agreement with the predictions of quantum theory and provide constraints on a class of $\psi$-epistemic models.

We calculate analytically the Bogoliubov excitation spectrum of a toroidal spin-1 Bose-Einstein condensate that is subjected to a homogeneous magnetic field and contains vortices with arbitrary winding numbers in the $m_F=\pm 1$ components of the hyperfine spin. We show that a rotonlike spectrum can be obtained, or an initially stable condensate can be made unstable by adjusting the magnitude of the magnetic field or the trapping frequencies. The structure of the instabilities can be analyzed by measuring the particle densities of the spin components. We confirm the validity of the analytical calculations by numerical simulations.

We investigate the large-distance asymptotic behavior of the static density-density and field-field correlation functions in the one-dimensional Bose gas at finite temperature. The asymptotic expansions of the Bose gas correlators are obtained performing a specific continuum limit in the similar low-temperature expansions of the longitudinal and transversal correlation functions of the $XXZ$ spin chain. In the lattice system the correlation lengths are computed as ratios of the largest and next-largest eigenvalues of the $XXZ$ spin chain quantum transfer matrix. In both cases, lattice and continuum, the correlation lengths are expressed in terms of solutions of Yang-Yang type [C. N. Yang and C. P. Yang, J. Math. Phys. 10, 1151 (1969)] nonlinear integral equations which are easily implementable numerically.

We calculate the optical force and torque applied to an electric dipole by a spinning light field. We find that the dissipative part of the force depends on the orbital energy flow of the field only, because the latter is related to the phase gradient generalized for such a light field. As for the remaining spin energy flow, it gives rise to an optical torque. The resulting change in the optical force is detailed for different experimentally relevant configurations, and we show in particular how this change is critical when surface plasmon modes are involved.

Rosenblum et al. showed recently that a three-level atom in a cavity can function as a perfect “photon turnstile.” Here we explore how this device reshapes the incident pulses, generalize it to the case of an input coherent state, and consider its use in reverse as a “photon adder.” We find that for initially unentangled pulses, adding one photon to an $|n\rangle$ number state is possible with a success probability equal to $1/(n+1)$.

Chiral electron-vortex beams, carrying a well-defined orbital angular momentum (OAM) about the propagation axis, are potentially useful as probes of magnetic and other chiral materials. We present an effective operator, expressible in a multipolar form, describing the inelastic processes in which electron-vortex beams interact with atoms, including those present in Bose-Einstein condensates, involving exchange of OAM. We show clearly that the key properties of the processes are dependent on the dynamical state and location of the atoms involved as well as the vortex-beam characteristics. Our results can be used to identify scenarios in which chiral-specific electron-vortex spectroscopy can probe magnetic sublevel transitions normally studied using circularly polarized photon beams with the advantage of atomic-scale spatial resolution.

Formalizing an old desire of Einstein, “$\psi$-epistemic theories” try to reproduce the predictions of quantum mechanics, while viewing quantum states as ordinary probability distributions over underlying objects called “ontic states.” Regardless of one's philosophical views about such theories, the question arises of whether one can cleanly rule them out by proving no-go theorems analogous to the Bell inequality. In the 1960s, Kochen and Specker (who first studied these theories) constructed an elegant $\psi$-epistemic theory for Hilbert space dimension $d=2$, but also showed that any deterministic $\psi$-epistemic theory must be “measurement contextual” in dimensions 3 and higher. Last year, the topic attracted renewed attention, when Pusey, Barrett, and Rudolph (PBR) showed that any $\psi$-epistemic theory must “behave badly under tensor product.” In this paper, we prove that even without the Kochen-Specker or PBR assumptions, there are no $\psi$-epistemic theories in dimensions $d\ge 3$ that satisfy two reasonable conditions: (1) symmetry under unitary transformations and (2) “maximum nontriviality” (meaning that the probability distributions corresponding to any two nonorthogonal states overlap). This no-go theorem holds if the ontic space is either the set of quantum states or the set of unitaries. The proof of this result, in the general case, uses some measure theory and differential geometry. On the other hand, we also show the surprising result that without the symmetry restriction, one can construct maximally nontrivial $\psi$-epistemic theories in every finite dimension $d$.

In this paper we investigate properties of several randomness generation protocols in the device-independent framework. Using Bell-type inequalities it is possible to certify that the numbers generated by an untrusted device are indeed random. We present a selection of certificates which guarantee two bits of randomness for each run of the experiment in the noiseless case and require the parties to share a maximally entangled state. To compare them we study their efficiency in the presence of white noise. We find that for different amounts of noise different operators are optimal for certifying most randomness. Therefore, the vendor of the device should use different protocols depending on the amount of noise expected to occur. Another of our results that we find particularly interesting is that using a single Bell operator as a figure of merit is rarely optimal.

The rates of convergence for the partial-wave expansion with odd-power $r_{12}$ terms for the ground-state energy of the helium atom are derived. For both the second-order $1/Z$ expansion and the Rayleigh-Ritz variational method, the energy increments of the partial-wave expansion converge as $O\left(L^{-N-7}\right)$, where $N$ is the highest odd-power $r_{12}$ function. The derivations require assumptions of the regularities for the ground-state helium wave function, which have not been established. Numerical results are presented for supporting the theoretical rates of convergence.