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The emerging technique of serial femtosecond X-ray crystallography (SFX) can be used to study the structure and dynamics of biological macromolecules to high spatial and temporal resolutions. An ongoing challenge for SFX is the damage caused by the ultrabright X-ray free electron laser pulse. Though it is often assumed that sufficiently femtosecond pulses `outrun' radiation damage, in reality electronic damage processes commence during exposure. We model the electronic damage to protein nanocrystals using a plasma model that tracks the continuous changes to the energy distribution of the unbound electrons. Tracking the continuous energy distribution is of particular importance for distinguishing the influence of differing elements on secondary damage processes. Heavy atoms have a ubiquitous but small presence in protein targets - typically as integral components of the macromolecule and as salts in the solvent. We find that these atoms considerably influence the simulated ionization and scattering behavior of realistic targets due to their rapid seeding of secondary ionization processes. In lysozyme, even the presence of native sulfur atoms significantly contributes to theoretical measures of damage-induced noise for >= 6 keV, 15 fs pulses. Contributing to the effect is that heavy atoms seed `intermediate' energy electron cascades that are particularly effective at ionizing the target on the femtosecond timescale. In addition, the disproportionate effect of heavy atoms means the damage to a protein crystal can be sensitive to their presence in the solvent. Outside of reducing the concentration of heavy atoms in the target, these results suggest the dose limits of SFX targets will be higher where the ionization of deep >~ 6 keV absorption edges is minimized, or, to a lesser extent, when such edges are only ionized with X-rays >> 2 keV above their binding energy.

Quantum chemistry is a promising application in the era of quantum computing since the unique effects of quantum mechanics that take exponential growing resources to simulate classically are controllable on quantum computers. Fermionic degrees of freedom can be encoded efficiently onto qubits and allow for algorithms such as the Quantum Equation-of-Motion method to find the entire energy spectrum of a quantum system. In this paper, we propose the Reduced Quantum Equation-of-Motion method by reducing the dimensionality of its generalized eigenvalue equation, which results in half the measurements required compared to the Quantum Equation-of-Motion method, leading to speed up the algorithm and less noise accumulation on real devices. In particular, we analyse the performance of our method on two noise models and calculate the excitation energies of a bulk Silicon and Gallium Arsenide using our method on an IBM quantum processor. Our method is fully robust to the uniform depolarizing error and we demonstrate that the selection of suitable atomic orbital complexity could increase the robustness of our algorithm under real noise. We also find that taking the average of multiple experiments tends towards the correct energies due to the fluctuations around the exact values. Such noise resilience of our approach could be used on current quantum devices to solve quantum chemistry problems.

The scattering of neutral particles by an atomic nucleus can lead to electronic ionisation and excitation through a process known as the Migdal effect. We revisit and improve upon previous calculations of the Migdal effect, using the Dirac-Hartree-Fock method to calculate the atomic wavefunctions. Our methods do not rely on the use of the dipole approximation, allowing us to present robust results for higher nuclear recoil velocities than was previously possible. Our calculations provide the theoretical foundations for future measurements of the Migdal effect using neutron sources, and searches for dark matter in direct detection experiments. We show that multiple ionisation must be taken into account in experiments with fast neutrons, and derive the semi-inclusive probability for processes that yield a hard electron above a defined energy threshold. We present results for the noble elements up to and including xenon, as well as carbon, fluorine, silicon and germanium. The transition probabilities from our calculations are publicly available., Comment: 19 pages, 12 figures; v2: journal version

A unified method for three-dimensional reconstruction of objects from transmission images collected at multiple illumination directions is described. The method may be applicable to experimental conditions relevant to absorption-based, phase-contrast or diffraction imaging using X-rays, electrons and other forms of penetrating radiation or matter waves. Both the phase retrieval (also known as contrast transfer function correction) and the effect of Ewald sphere curvature (in the cases with a shallow depth of field and significant in-object diffraction) are incorporated in the proposed algorithm and can be taken into account. Multiple scattering is not treated explicitly, but can be mitigated as a result of angular averaging that constitutes an essential feature of the method. The corresponding numerical algorithm is based on three-dimensional gridding which allows for fast computational implementation, including a straightforward parallelization. The algorithm can be used with any scanning geometry involving plane-wave illumination. A software code implementing the proposed algorithm has been developed, tested on simulated and experimental image data and made publicly available., Comment: 16 pages, 3 figures

The variational determination of the two-fermion reduced density matrix is described for harmonically trapped, ultracold few-fermion systems in one dimension with equal spin populations. This is accomplished by formulating the problem as a semi-definite program, with the two-fermion reduced density matrix being subject to well-known $N$-representability conditions. The ground-state energies, as well as the density, pair-correlation function, and lower-order eigenvalues of the two-fermion reduced density matrix of various fermionic systems are found by utilising an augmented Lagrangian method for semi-definite programming. The ground-state energies are found to match well to those determined by full-configuration interaction and coupled-cluster calculations and the density, pair-correlation function, and eigenvalue results demonstrate that the salient features of these systems are well-described by this method. These results collectively demonstrate the utility of the reduced density matrix method firstly in describing strong correlation arising from short-range interactions, suggesting that the well-known N-representability conditions are sufficient to model ultracold fermionic systems, and secondly in illustrating the prospect of treating larger systems currently out of the reach of established methods., Comment: 18 pages, 6 figures

In recent years, x-ray speckle tracking techniques have emerged as viable tools for wavefront metrology and sample imaging applications. These methods are based on the measurement of near-field images. Thanks to the simple experimental set-up, high angular sensitivity and compatibility with low coherence sources these methods have been actively developed for use with synchrotron and laboratory light sources. Not only do speckle-tracking techniques give the potential for high resolution imaging, but they also provide rapid and robust characterisation of aberrations of x-ray optical elements, focal spot profiles and the sample position and transmission properties. In order to realise these capabilities, we require software implementations that are equally rapid and robust. To address this need, a software suite has been developed for the "ptychographic x-ray speckle tracking technique" -- an x-ray speckle based method suitable for highly divergent wavefields. The software suite is written in Python 3, with an OpenCL back end for GPU and multi-CPU core processing. It is accessible as a Python module, through the command line or through a graphical user interface and is available as source code under version 3 or later of the GNU General Public License.

We present a method for the measurement of the phase gradient of a wavefront by tracking the relative motion of speckles in projection holograms as a sample is scanned across the wavefront. By removing the need to obtain an un-distorted reference image of the sample, this method is suitable for the metrology of highly divergent wavefields. Such wavefields allow for large magnification factors, that, according to current imaging capabilities, will allow for nano-radian angular sensitivity and nano-scale sample projection imaging. Both the reconstruction algorithm and the imaging geometry are nearly identical to that of ptychography, except that the sample is placed downstream of the beam focus and that no coherent propagation is explicitly accounted for. Like other x-ray speckle tracking methods, it is robust to low-coherence x-ray sources making is suitable for lab based x-ray sources. Likewise it is robust to errors in the registered sample positions making it suitable for x-ray free-electron laser facilities, where beam pointing fluctuations can be problematic for wavefront metrology. We also present a modified form of the speckle tracking approximation, based on a second-order local expansion of the Fresnel integral. This result extends the validity of the speckle tracking approximation and may be useful for similar approaches in the field.

It is proved that the width of a function and the width of the distribution of its values cannot be made arbitrarily small simultaneously. In the case of ergodic stochastic processes, an ensuing uncertainty relationship is demonstrated for the product of correlation length and variance. A closely related uncertainty principle is also established for the average degree of fourth-order coherence and the spatial width of modes of bosonic quantum fields. However, it is shown that, in the case of stochastic and quantum observables, certain non-classical states with sub-Poissonian statistics, such as for example photon number squeezed states in quantum optics, can overcome the "classical" noise-resolution uncertainty limit. The described uncertainty relationship, which is fundamentally different from the Heisenberg and related uncertainty principles, can define an upper limit for the information capacity of communication and imaging systems. It is expected to be useful in a variety of problems in classical and quantum physics., Comment: 12 pages, 1 figure, to be submitted to a journal

Simulations of radiation damage in single molecule imaging using a X-ray free electron laser use atomic rates calculated in the lowest order. We investigate the difference in ion yield predictions using Hartree-Fock and Hartree-Fock-Slater pproximations for light and heavy elements of biological significance. The results show that for the biologically abundant elements of the second and third rows of the periodic table both approximations agree to about 6%. For the heavier elements beyond the fourth row the discrepancy rices to 11% for the range of the pulse parameters covered in this work. Presented analysis can be used for an error estimation in a wide range of ab initio simulations of the X-ray pulse interaction with biological molecules. We also discuss other atomic structure effects and show that their account has considerably smaller effect on the ion yields of respective elements compared to the choice of the approximation.

A reciprocal relationship between the autocovariance of the light intensity in the source plane and in the far-field detector plane is presented in a form analogous to the classical van Cittert - Zernike theorem, but involving intensity correlation functions. A "classical" version of the reciprocity relationship is considered first, based on the assumption of circular Gaussian statistics of the complex amplitudes in the source plane. The result is consistent with the theory of Hanbury Brown - Twiss interferometry, but it is shown to be also applicable to estimation of the source size or the spatial resolution of the detector from the noise power spectrum of flat-field images. An alternative version of the van Cittert - Zernike theorem for intensity correlations is then derived for a quantized electromagnetic beam in a coherent state, which leads to Poisson statistics for the intrinsic intensity of the beam.

An uncertainty inequality is presented that establishes a lower limit for the product of the variance of the time-averaged intensity of a mode of a quantized electromagnetic field and the degree of its spatial localization. The lower limit is determined by the vacuum fluctuations within the volume corresponding to the width of the mode. This result also leads to a generalized form of the Heisenberg uncertainty principle for boson fields in which the lower limit for the product of uncertainties in the spatial and momentum localization of a mode is equal to the product of Planck's constant and a dimensionless functional which reflects the joint signal-to-noise ratio of the position and momentum of vacuum fluctuations in the region of the phase space occupied by the mode. Experimental X-ray synchrotron measurements provide an initial verification of the proposed theory in the case of Poisson statistics.

During the last five years, serial femtosecond crystallography using x-ray laser pulses has developed into a powerful technique for determining the atomic structures of protein molecules from micrometer and sub-micrometer sized crystals. One of the key reasons for this success is the "self-gating" pulse effect, whereby the x-ray laser pulses do not need to outrun all radiation damage processes. Instead, x-ray induced damage terminates the Bragg diffraction prior to the pulse completing its passage through the sample, as if the Bragg diffraction was generated by a shorter pulse of equal intensity. As a result, serial femtosecond crystallography does not need to be performed with pulses as short as 5--10 fs, as once thought, but can succeed for pulses 50--100 fs in duration. We show here that a similar gating effect applies to single molecule diffraction with respect to spatially uncorrelated damage processes like ionization and ion diffusion. The effect is clearly seen in calculations of the diffraction contrast, by calculating the diffraction of average structure separately to the diffraction from statistical fluctuations of the structure due to damage ("damage noise"). Our results suggest that sub-nanometer single molecule imaging with 30--50 fs pulses, like those produced at currently operating facilities, should not yet be ruled out. The theory we present opens up new experimental avenues to measure the impact of damage on single particle diffraction, which is needed to test damage models and to identify optimal imaging conditions., Comment: 23 pages; 5 figures

X-ray Free-Electron Lasers (XFELs) deliver X-ray pulses with a coherent flux that is approximately eight orders of magnitude greater than that available from a modern third generation synchrotron source. The power density in an XFEL pulse may be so high that it can modify the electronic properties of a sample on a femtosecond timescale. Exploration of the interaction of intense coherent X-ray pulses and matter is of both intrinsic scientific interest, and of critical importance to the interpretation of experiments that probe the structures of materials using high-brightness femtosecond XFEL pulses. In this letter, we report observations of the diffraction of an extremely intense 32 fs nanofocused X-ray pulses by a powder sample of crystalline C60. We find that the diffraction pattern at the highest available incident power exhibits significant structural signatures that are absent in data obtained at both third-generation synchrotron sources or from XFEL sources operating at low output power. These signatures are consistent with a highly ordered structure that does not correspond with any previously known phase of crystalline C60. We argue that these data indicate the formation of a transient phase that is formed by a dynamic electronic distortion induced by inner-shell ionisation of at least one carbon atom in each C60 molecule., Comment: 9 pages, 3 figures

The application of powder diffraction methods in two-dimensional crystallography is regarded as intractable because of the uncertainties associated with overlapping reflections. Here, we report an approach that resolves these ambiguities and provides reliable low-resolution phase information directly from powder diffraction data. We apply our method to the recovery of the structure of the bacteriorhodopsin (bR) molecule to a resolution of 7 angstroms using only powder diffraction data obtained from two-dimensional purple membrane (PM) crystals., Comment: 14 pages, 4 figures, 3 tables

Proposals to determine biomolecular structures from diffraction experiments using femtosecond X-ray free-electron laser (XFEL) pulses involve a conflict between the incident brightness required to achieve diffraction-limited atomic resolution and the electronic and structural damage induced by the illumination. Here we show that previous estimates of the conditions under which biomolecular structures may be obtained in this manner are unduly restrictive, because they are based on a coherent diffraction model that is not appropriate to the proposed interaction conditions. A more detailed imaging model derived from optical coherence theory and quantum electrodynamics is shown to be far more tolerant of electronic damage. The nuclear density is employed as the principal descriptor of molecular structure. The foundations of the approach may also be used to characterize electrodynamical processes by performing scattering experiments on complex molecules of known structure., Comment: 16 pages, 2 figures