Many aspects of the well-known mapping between the partition function of a classical spin model and the quantum entangled state have been studied in recent years. However, the consequences of the existence of a classical (critical) phase transition on the corresponding quantum state have been mostly ignored. In this paper, we consider this problem for an important example of the Kitaev toric code model which has been shown to correspond to the two-dimensional (2D) Ising model though a duality transformation. We show that the temperature on the classical side is mapped to bit-flip noise on the quantum side. It is then shown that a transition from a coherent superposition of a given quantum state to a noncoherent mixture corresponds exactly to paramagnetic-ferromagnetic phase transition in the Ising model. To identify such a transition further, we define an order parameter to characterize the decoherence of such a mixture and show that it behaves similar to the order parameter (magnetization) of the 2D Ising model, a behavior that is interpreted as a robust coherence in the toric code model. Furthermore, we consider other properties of the noisy toric code model exactly at the critical point. We show that there is a relative stability to noise for the toric code state at the critical noise which is revealed by a relative reduction in susceptibility to noise.

$n\to m$ random access code (RAC) is an information task to encode an $n$-bit random message into a shorter $m$-bit string. Resulting from the compression loss from $n$ to $m$ bits, there exists success probabilities for the string receiver to recover any bits of the initial message. It has been shown that success probabilities of bit recovery can be enhanced by employing quantum resources. Here we present experimental realizations of $2\to 1$ and $3\to 1$ distributed entanglement-assisted random access codes (EARACs) with a two-photon entangled source. The average success probabilities obtained in our experiment are $0.8491\left(41\right)$ for a $2\to 1$ EARAC and $0.7724\left(34\right)$ for a $3\to 1$ EARAC, exceeding the classical counterparts, respectively. Moreover, through the existing experimental data, success probabilities for $n$ varying from 4 to 10 ($m=1$) are estimated.

We derive an explicit expression for the functional derivative of the subleading term in the strong interaction limit expansion of the generalized Levy-Lieb functional for the special case of two electrons in one dimension. The expression is derived from the zero-point-energy (ZPE) functional, which is valid if the quantum state reduces to strongly correlated electrons in the strong coupling limit. The explicit expression is confirmed numerically and respects the relevant sum rule. We also show that the ZPE potential is able to generate a bond midpoint peak for homonuclear dissociation and is properly of purely kinetic origin. Unfortunately, the ZPE diverges for Coulomb systems, whereas the exact peaks should be finite.

The transition between the linear phase and zigzag phase in an ion trap is widely used to study the mechanism of the second-order continuous phase transition. For a linear ion chain, during the quenching across the critical point, kinks are formed. The relation between the density of kinks and quenching rate can be described by the Kibble-Zurek mechanism. In this work, we consider a one-dimensional trapped-ion chain with the motion on the phase-transition plane. Using the interacting Langevin equation, we show that when one ion is kicked out from the chain, the off-axis action caused by this ion will propagate to others one by one with a defined velocity. This velocity will be the same as the sound velocity in some extreme cases but has obviously different meaning. In the presence of finite temperature, which is modeled using a many-body stochastic Langevin equation to unveil a microscopic mechanism for kink formations in this model, we find that the kink can be formed when the diffusion radius is larger than the mean displacement of the ions along the radial direction. This criterion provides an alternative mode for the formation of kinks in realistic experiments. The predictions based on this mechanism can be qualitatively consistent with those from the Kibble-Zurek mechanism in homogeneous structures; however, the microscopic details, such as kink formation and kink disappearance can be seen much clearer in our simplified model.

We propose an experimental realization of a time crystal using an atomic Bose-Einstein condensate in a high finesse optical cavity pumped with laser light detuned to the blue side of the relevant atomic resonance. By mapping out the dynamical phase diagram, we identify regions in parameter space showing stable limit cycle dynamics. Since the model describing the system is time independent, the emergence of a limit cycle phase indicates the breaking of continuous time translation symmetry. Employing a semiclassical analysis to demonstrate the robustness of the limit cycles against perturbations and quantum fluctuations, we establish the emergence of a time crystal.

We propose a scheme to characterize topological phases with integer invariants through detecting topological charges via quench dynamics. The essential advantage of the current scheme is that only a single-(pseudo)spin component needs to be measured to extract the complete information of topological charges which determine the bulk topology of the phase. A topological charge depicts the chirality of a monopole at Dirac or Weyl point captured by the node of the (pseudo)spin-orbit coupling, and the topological phases are determined by total charges enclosed by the so-called band-inversion surfaces (BISs). We show a duality that measuring the $i\mathrm{th}$ spin component in quenching $j\mathrm{th}$ spin axis precisely equals to measuring the $j\mathrm{th}$ spin component in quenching the $i\mathrm{th}$ spin axis. With this we find that characterizing the complete information of topological charges is precisely mapped to measuring the quench (pseudo)spin dynamics of a single spin component caused by quenches along all spin axes. Therefore, the detection of topological physics involves measuring only a single spin component and has great experimental advantages. We numerically examine two concrete models, and propose a cold-atom experimental setup with high feasibility for measurement. This work paves the way to dynamically classify topological phases.

We present a framework for the construction of solvable models of optical settings with genuinely two-dimensional landscapes of refractive index. Solutions of the associated nonseparable Maxwell equations in paraxial approximation are found using the time-dependent supersymmetry. We discuss peculiar theoretical aspects of the construction. In particular, we focus on the existence of localized solutions specific for the new systems. Sufficient conditions for their existence are discussed. Localized solutions vanishing for large $|\vec{x}|$, which we call light dots, as well as the guided modes that vanish exponentially outside the wave guides, are constructed. We consider different definitions of the parity operator and analyze general properties of the $\mathcal{PT}$-symmetric systems, e.g., presence of localized states or existence of symmetry operators. Despite the models with parity-time symmetry are of the main concern, the proposed framework can serve for construction of non-$\mathcal{PT}$-symmetric systems as well. We explicitly illustrate the general results on a number of physically interesting examples, e.g., wave guides with periodic fluctuation of refractive index or with a localized defect, curved wave guides, two coupled wave guides, or a uniform refractive index system with a localized defect.

The development of new techniques to improve measurements is crucial for all sciences. By employing quantum systems as sensors to probe some physical property of interest allows the application of quantum resources, such as coherent superpositions and quantum correlations, to increase measurement precision. Here we experimentally investigate a scheme for quantum target detection based on linear optical measurement devices, when the object is immersed in unpolarized background light. By comparing the quantum (polarization-entangled photon pairs) and the classical (separable polarization states) strategies, we found that the quantum strategy provides us an improvement over the classical one in our experiment when the signal-to-noise ratio is greater than 1/40, or about 16 dB of noise. This is in constrast to quantum target detection considering nonlinear optical detection schemes, which have shown resilience to extreme amounts of noise. A theoretical model is developed which shows that, in this linear-optics context, the quantum strategy suffers from the contribution of multiple background photons. This effect does not appear in our classical scheme. By improving the two-photon detection electronics, it should be possible to achieve a polarization-based quantum advantage for a signal-to-noise ratio that is close to 1/400 for current technology.

Under the action of a weak constant force a wave packet in periodic potential undergoes periodic oscillations in space, returning to the initial position after one oscillation cycle. This wave phenomenon, known as Bloch oscillations (BOs), pertains to many physical systems. Can BOs also occur in topological insulators with topologically protected edge states? This question is highly nontrivial, because in topological insulators with broken time-reversal symmetry, the edge states propagate unidirectionally without backscattering, hence BOs that typically involve stages, where a wave packet moves along and against the direction of the force, seem to be impossible in such systems when force acts parallel to the edge of the insulator. Here we reveal that BOs still occur with topological edge states, but in a nonconventional way: they are accompanied not only by oscillations along the edge in the direction of force, but also by oscillations in the direction transverse to that of the force. A full BO cycle involves switching between edge states at the opposite edges through delocalized bulk modes. Bloch oscillations of the topological edge states require to scan the first Brillouin zone twice to complete one cycle, thus they have a period that is two times larger than the period of usual BOs. All these unusual properties are in contrast to BOs in nontopological systems.

We present high-precision calculations of local Lorentz invariance (LLI) violating parameters of the $3d{}^2{D}_{3/2;5/2}$ states in ${\mathrm{Ca}}^{+}$. We have employed three variants of the relativistic coupled-cluster (RCC) theory to determine these coefficients by gradually including electron correlation effects through the single, double, and triple excitation determinants from the Dirac-Hartree-Fock wave function. A precise estimate of the energy shift due to LLI violation depends on accurate evaluations of the expectation values of the square of the momentum operator ($\langle p^2\rangle$) and a second rank tensor ($\langle T_0^{\left(2\right)}\rangle$). It is found that the $\langle T_0^{\left(2\right)}\rangle$ values converge smoothly with the systematic inclusion of higher-order correlation effects in an expectation value evaluation approach, however, that is not the case for $\langle p^2\rangle$. Similar trends were also observed in the finite-field approach. To circumvent these problems, we determine $\langle p^2\rangle$ values very precisely by developing and applying an analytic gradient approach in the RCC framework. Corrections due to the Breit and quantum electrodynamics interactions are also estimated. Further, these calculations are validated by evaluating the energies and the quadrupole moments of the above states at different levels of approximations.

Previous schemes of nonadiabatic holonomic quantum computation were focused mainly on realizing a universal set of elementary gates. Multiqubit controlled gates could be built by decomposing them into a series of the universal gates. In this article, we propose an approach for realizing nonadiabatic holonomic multiqubit controlled gates in which a $(n+1)$-qubit controlled-$\left(\mathrm{n}·\mathbf{\sigma }\right)$ gate is realized by $(2n-1)$-basic operations instead of decomposing it into the universal gates, whereas a $(n+1)$-qubit controlled arbitrary rotation gate can be obtained by combining only two such controlled-$\left(\mathrm{n}·\mathbf{\sigma }\right)$ gates. Our scheme greatly reduces the operations of nonadiabatic holonomic quantum computation.

We study quantum anomaly detection with density estimation and multivariate Gaussian distribution. Both algorithms are constructed using the standard gate-based model of quantum computing. Compared with the corresponding classical algorithms, the resource complexities of our quantum algorithm are logarithmic in the dimensionality of quantum states and the number of training quantum states. We also present a quantum procedure for efficiently estimating the determinant of any Hermitian operators $\mathcal{A}\in {\mathcal{R}}^{N\times N}$ with time complexity $O(polylogN)$, which forms an important subroutine in our quantum anomaly detection with multivariate Gaussian distribution. Finally, our results also include the modified quantum kernel principal component analysis (PCA) and the quantum one-class support vector machine (SVM) for detecting classical data.

Current techniques in quantum process tomography typically return a single point estimate of an unknown process based on a finite albeit large amount of measurement data. Due to statistical fluctuations, however, other processes close to the point estimate can also produce the observed data with near certainty. Unless appropriate error bars can be constructed, the point estimate does not carry any sound operational interpretation. Here, we provide a solution to this problem by constructing a confidence region estimator for quantum processes. Our method enables reliable estimation of essentially any figure of merit for quantum processes on few qubits, including the diamond distance to a specific noise model, the entanglement fidelity, and the worst-case entanglement fidelity, by identifying error regions which contain the true state with high probability. We also provide a software package, QPtomographer, implementing our estimator for the diamond norm and the worst-case entanglement fidelity. We illustrate its usage and performance with several simulated examples. Our tools can be used to reliably certify the performance of, e.g., error correction codes, implementations of unitary gates, or more generally any noise process affecting a quantum system.

We determine the frequency of the ultranarrow ${}^{87}\mathrm{Sr}{}^{1}S_0-{}^{3}P_2$ transition by spectroscopy of an ultracold gas. This transition is referenced to four molecular iodine lines that are observed by Doppler-free saturation spectroscopy of hot iodine vapor. The frequency differences between the Sr and the ${\mathrm{I}}_2$ transitions are measured with an uncertainty of $0.5\mathrm{MHz}$. The absolute frequency of the ${}^{87}\mathrm{Sr}{}^{1}S_0-{}^{3}P_2$ (${\text{F}}^{\prime }=7/2$) transition is $446648775\left(30\right)\mathrm{MHz}$ and limited in accuracy by the iodine reference. This work prepares the use of the Sr ${}^{1}S_0-{}^{3}P_2$ transition for quantum simulation and computation.

We demonstrate the production of large ${}^7\mathrm{Li}$ Bose-Einstein condensates in an optical dipole trap using $D_1$ gray molasses. The sub-Doppler cooling technique reduces the temperature of $4\times {10}^9$ atoms to $25\mu \mathrm{K}$ in 3 ms. After microwave evaporation cooling in a magnetic quadrupole trap, we transfer the atoms to a crossed optical dipole trap, where we employ a magnetic Feshbach resonance on the $|F=1,m_F=1\rangle$ state. Fast evaporation cooling is achieved by tilting the optical potential using a magnetic field gradient on the top of the Feshbach field. Our setup produces pure condensates with $2.7\times {10}^6$ atoms in the optical potential for every 11 s. The trap tilt evaporation allows rapid thermal quench, and spontaneous vortices are observed in the condensates as a result of the Kibble-Zurek mechanism.

Excitation of surface plasmon-polariton (SPP) waves was investigated at an interface of a metal and a one-dimensional photonic crystal (1DPC) along the direction of periodicity of the photonic crystal in a prism-coupled configuration. The rigorous coupled-wave approach was used to formulate the problem and the absorptance spectra were obtained as a function of the incidence angle, the period, and the thickness of the photonic crystal. The correlation of the absorptance bands with the underlying canonical problem and the spatial power profiles confirmed the excitation of SPP waves. The excitation of SPP waves at the interface of the 1DPC along the direction of the periodicity opens up new avenues for applications.

We show that arrays of ${\chi }^{\left(2\right)}$ nonlinear waveguides in the second-harmonic generation regime are a promising source of continuous-variable entanglement. We indeed demonstrate analytically that optical arrays with an odd number of waveguides injected with the zero-eigenvalue fundamental supermode entangle this fundamental supermode with a collective harmonic field. Moreover, the fundamental individual modes are multipartite entangled and their entanglement grows with propagation length. The device is scalable, robust to losses, does not rely on specific values of nonlinearity and coupling, and is easily realized with current technology. It thus stands as an unprecedented candidate for generation of multipartite continuous-variable entanglement for optical quantum information processing.

Do the prior history and the wave-packet size and form of a free electron have a physical effect in its interaction with light? Here we answer these fundamental questions on the interpretation of the electron quantum wave function by analyzing spontaneous and stimulated emissions of a quantum electron wave packet, interacting with a general, quantized radiation field. For coherent radiation (Glauber state), we confirm that stimulated emission and absorption of photons depends on the preinteraction-history-dependent size, exhibiting spectral cutoff when it exceeds the interacting radiation wavelength. Furthermore, stimulated emission of an optically modulated electron wave packet has a characteristic harmonic emission spectrum beyond the cutoff, which depends on the modulation features. In either case, there is no wave-packet-dependent radiation of the Fock state, and particularly the vacuum state spontaneous emission is wave-packet independent. The classical-to-quantum transition of radiation from the point-particle to the plane-wave limits, and the effects of wave-packet modulation indicate a way of measuring the wave-packet size of a single electron wave function, and suggest an alternative direction for exploring light-matter interaction.

Linear optical computing (LOC) with thermal light has recently gained attention because the problem is connected to the permanent of a Hermitian positive semidefinite matrix (HPSM), which is of importance in the computational complexity theory. Despite the several theoretical analyses on the computational structure of an HPSM in connection to LOC, the experimental demonstration and the computational complexity analysis via the linear optical system have not been performed yet. We present, herein, experimental LOC for estimating the permanent of an HPSM. From the linear optical experiments and theoretical analysis, we find that the LOC efficiency for a multiplicative error is dependent on the value of the permanent and that the lower bound of the computation time scales exponentially. Furthermore, our results are generalized and applied to LOC of permanents of unitary matrices, which can be implemented with a multiport quantum interferometer involving single photons at the input ports. We find that LOC with single photons, for the permanent estimation, is on average less efficient than the most efficient classical algorithm known to date, even in ideal conditions.

Proton transfer is a common process occurring in biologic tissues and chemical reactions involving water which evolves on the femtosecond timescale. We induce and investigate proton transfer through strong-field ionization of isolated water dimers using ultrashort intense laser pulses. Ionization gives rise to ionic molecular dynamics which terminates in proton transfer and dissociation of the singly charged ion, but also in undissociated ${\left({\mathrm{H}}_2\mathrm{O}\right)}_2{}^{+}$ ions. Double ionization allows us to experimentally determine the proton transfer rate in ${\left({\mathrm{H}}_2\mathrm{O}\right)}_2{}^{+}$ in the electronic ground state.