Small Quantum Systems Research Articles

Controlled molecule injector for cold, dense, and pure molecular beams at the European x-ray free-electron laser.

A permanently available molecular-beam injection setup for controlled molecules (COMO) was installed and commissioned at the small quantum systems (SQS) instrument at the European x-ray free-electron laser (EuXFEL). A b-type electrostatic deflector allows for pure state-, size-, and isomer-selected samples of polar molecules and clusters. The source provides a rotationally cold (T ≈ 1 K) and dense (ρ ≈ 108 cm-3) molecular beam with pulse durations up to 100 µs generated by a new version of the Even-Lavie valve. Here, a performance overview of the COMO setup is presented along with characterization experiments performed both with an optical laser at the Center for Free-Electron-Laser Science and with x rays at EuXFEL under burst-mode operation. COMO was designed to be attached to different instruments at the EuXFEL, in particular, the SQS and single particles, clusters, and biomolecules (SPB) instruments. This advanced controlled-molecules injection setup enables x-ray free-electron laser studies using highly defined samples with soft and hard x-ray FEL radiation for applications ranging from atomic, molecular, and cluster physics to elementary processes in chemistry and biology.

Dynamics of a small quantum system open to a bath with thermostat

Dynamics of a small quantum system open to a bath with thermostat

General theory for discrete symmetry-breaking equilibrium states in quantum systems.

Spontaneous symmetrybreaking in phase transitions occurs when the system Hamiltonian is symmetric under a certain transformation, but the equilibrium states observed in nature are not. Here we build two noncommuting quantities from the order parameter of the transition and the symmetry operator that are constants of motion when such equilibrium states exist. Then, we derive a general equilibrium ensemble for the ordered phase and show that equilibrium states consisting of superpositions of different symmetry-breaking states, like positive and negative magnetized states, may exist. We propose an experimental realization of such equilibrium states with the state-of-the-art quantum technologies, and test it by means of numerical calculations. Finally, we show that a small symmetry-breaking perturbation in the Hamiltonian stabilizes the conservation of one of the two former quantities, implying that symmetry-breaking equilibrium states become stable even in small quantum systems.

A 1D imaging soft X-ray spectrometer for the small quantum systems instrument at the European XFEL.

A 1D imaging soft X-ray spectrometer installed on the small quantum systems (SQS) scientific instrument of the European XFEL is described. It uses movable cylindrical constant-line-spacing gratings in the Rowland configuration for energy dispersion in the vertical plane, and Wolter optics for simultaneous 1D imaging of the source in the horizontal plane. The soft X-ray fluorescence spectro-imaging capability will be exploited in pump-probe measurements and in investigations of propagation effects and other nonlinear phenomena.

Sketching phase diagrams using low-depth variational quantum algorithms

Mapping out phase diagrams of quantum systems using classical simulations can be challenging or intractable due to the computational resources required to simulate even small quantum systems far away from the thermodynamic limit. We investigate using quantum computers and the variational quantum eigensolver (VQE) for this task. In contrast to the task of preparing the exact ground state using VQE, sketching phase diagrams might require less quantum resources and accuracy, because low fidelity approximations to the ground state may be enough to correctly identify different phases. We used classical numerical simulations of low-depth VQE circuits to compute order parameters for four well-studied spin and fermion models which represent a mix of 1D and 2D, and exactly-solvable and classically hard systems. We find that it is possible to predict the location of phase transitions up to reasonable accuracy using states produced by VQE even when their overlap with the true ground state is small. Further, we introduce a model-agnostic predictor of phase transitions based on the speed with which the VQE energy improves with respect to the circuit depth, and find that in some cases this is also able to predict phase transitions.

Qudit machine learning

We present a comprehensive investigation into the learning capabilities of a simple d-level system (qudit). Our study is specialized for classification tasks using real-world databases, specifically the Iris, breast cancer, and MNIST datasets. We explore various learning models in the metric learning framework, along with different encoding strategies. In particular, we employ data re-uploading techniques and maximally orthogonal states to accommodate input data within low-dimensional systems. Our findings reveal optimal strategies, indicating that when the dimension of input feature data and the number of classes are not significantly larger than the qudit’s dimension, our results show favorable comparisons against the best classical models. This trend holds true even for small quantum systems, with dimensions d < 5 and utilizing algorithms with a few layers ( L=1,2 ). However, for high-dimensional data such as MNIST, we adopt a hybrid approach involving dimensional reduction through a convolutional neural network. In this context, we observe that small quantum systems often act as bottlenecks, resulting in lower accuracy compared to their classical counterparts.

Enhancing the performance of quantum reservoir computing and solving the time-complexity problem by artificial memory restriction

We propose a scheme that can enhance the performance and reduce the computational cost of quantum reservoir computing. Quantum reservoir computing is a computing approach which aims at utilizing the complexity and high dimensionality of small quantum systems, together with the fast trainability of reservoir computing, in order to solve complex tasks. The suitability of quantum reservoir computing for solving temporal tasks is hindered by the collapse of the quantum system when measurements are made. This leads to the erasure of the memory of the reservoir. Hence, for every output, the entire input signal is needed to reinitialize the reservoir, leading to quadratic time complexity. Another critical issue for the hardware implementation of quantum reservoir computing is the need for an experimentally accessible means of tuning the nonlinearity of the quantum reservoir. We present an approach which addresses both of these issues. We propose artificially restricting the memory of the quantum reservoir by only using a small number inputs to reinitialize the reservoir after measurements are performed. This strongly influences the nonlinearity of the reservoir response due to the influence of the initial reservoir state, while also substantially reducing the number of quantum operations needed to perform time-series prediction tasks due to the linear rather than quadratic time complexity. The reinitialization length therefore provides an experimental accessible means of tuning the nonlinearity of the response of the reservoir, which can lead to significant task-specific performance improvement. We numerically study the linear and quadratic algorithms for a fully connected transverse Ising model and a quantum processor model. Published by the American Physical Society 2024

PyBEST: Improved functionality and enhanced performance

PyBEST: Improved functionality and enhanced performance

Identifying an Environment-Induced Localization Transition from Entropy and Conductance.

Environment-induced localization transitions (LT) occur when a small quantum system interacts with a bath of harmonic oscillators. At equilibrium, LTs are accompanied by an entropy change, signaling the loss of coherence. Despite extensive efforts, equilibrium LTs have yet to be observed. Here, we demonstrate that ongoing experiments on double quantum dots that measure entropy using a nearby quantum point contact realize the celebrated spin-boson model and allow to measure the entropy change of its LT. We find a Kosterlitz-Thouless flow diagram, leading to a universal jump in the spin-bath interaction, reflected in a discontinuity in the zero temperature QPC conductance.

Einstein-Podolsky-Rosen Experiment with Two Bose-Einstein Condensates

In 1935, Einstein, Podolsky, and Rosen (EPR) conceived a gedanken experiment which became a cornerstone of quantum technology and still challenges our understanding of reality and locality today. While the experiment has been realized with small quantum systems, a demonstration of the EPR paradox with massive many-particle systems remains an important challenge, as such systems are particularly closely tied to the concept of local realism in our everyday experience and may serve as probes for new physics at the quantum-to-classical transition. In this work we report an EPR experiment with two spatially separated Bose-Einstein condensates, each containing about 700 rubidium atoms. Entanglement between the condensates results in strong correlations of their collective spins, allowing us to demonstrate the EPR paradox between them. Our results represent the first observation of the EPR paradox with spatially separated, massive many-particle systems. They show that the conflict between quantum mechanics and local realism does not disappear as the system size increases to more than a thousand massive particles. Furthermore, EPR entanglement in conjunction with individual manipulation of the two condensates on the quantum level, as we demonstrate here, constitutes an important resource for quantum metrology and information processing with many-particle systems.Received 9 November 2022Revised 20 February 2023Accepted 3 April 2023DOI:https://doi.org/10.1103/PhysRevX.13.021031Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasBose-Einstein condensatesEntanglement detectionEntanglement in quantum gasesNonlocalityQuantum fluctuations & noiseQuantum metrologyQuantum InformationAtomic, Molecular & Optical

Harmonic radiation contribution and X-ray transmission at the Small Quantum Systems instrument of European XFEL.

Transmission measurements of the soft X-ray beamline to the Small Quantum Systems (SQS) scientific instrument at the SASE3 undulator of European XFEL are presented. Measurements are reported for a wide range of photon energies (650 eV to 2400 eV), using X-ray gas monitors as well as a bolometric radiometer. The results are in good agreement with simulations for the beam transport and show a transmission of up to 80% over the whole photon energy range. The contribution of second- and third-harmonic radiation of the soft X-ray undulator is determined at selected photon energies by performing transmission measurements using a gas absorber to provide variable attenuation of the incoming photon flux. A comparison of the results with semi-analytic calculations for the generation of free-electron laser pulses in the SASE3 undulator reveals an influence of apertures along the beam transport on the exact harmonic content to be accounted for at the experiment. The second-harmonic content is measured to be in the range of 0.1% to 0.3%, while the third-harmonic contributed a few percent to the SASE3 emission. For experiments at the SQS instrument, these numbers can be reduced through specific selections of the mirror reflection angles.

Truncated Wigner approximation for the bosonic model of large spin baths

The central spin model has a wide applicability. It is ideally suited to describe a small quantum system, for instance a quantum bit, in contact to a bath of spins, e.g., nuclear spins or other small quantum systems in general. According to previous work [R. R\"ohrig et al., Phys. Rev. B 97, 165431 (2018)], a large bath of quantum spins can be described as a bath of quantum harmonic oscillators. But the resulting quantum model is still far from being straightforwardly solvable. Hence we consider a chain representation for the bosonic degrees of freedom to study how well a semi-classical truncated Wigner approximation of the effective model of harmonic oscillators (bTWA) works in comparison with other approximate and exact methods. We find that the bTWA works well for short times, but deviates from the results of other methods for long times. The general message is that the applicability of semi-classical approaches strongly depends on the variables in terms of which the model is formulated. Numerically, we examine the effect of the number of bath spins and of the truncation level, i.e., the chain length.

Exploring quantum thermodynamics with NMR

Quantum thermodynamics seeks to extend non-equilibrium stochastic thermodynamics to small quantum systems where non-classical features are essential to its description. Such a research area has recently provided meaningful theoretical and experimental advances by exploring the wealth and the power of quantum features along with informational aspects of a system’s thermodynamics. The relevance of such investigations is related to the fact that quantum technological devices are currently at the forefront of science and engineering applications. This short review article provides an overview of some concepts in quantum thermodynamics highlighting test-of-principles experiments using nuclear magnetic resonance techniques.

The beam transport system for the Small Quantum Systems instrument at the European XFEL: optical layout and first commissioning results.

The Small Quantum Systems instrument is one of the six operating instruments of the European XFEL, dedicated to the atomic, molecular and cluster physics communities. The instrument started its user operation at the end of 2018 after a commissioning phase. The design and characterization of the beam transport system are described here. The X-ray optical components of the beamline are detailed, and the beamline performances, transmission and focusing capabilities are reported. It is shown that the X-ray beam can be effectively focused as predicted by ray-tracing simulations. The impact of non-ideal X-ray source conditions on the focusing performances is discussed.

Molecular photodissociation dynamics revealed by Coulomb explosion imaging.

Coulomb explosion imaging (CEI) methods are finding ever-growing use as a means of exploring and distinguishing the static stereo-configurations of small quantum systems (molecules, clusters, etc). CEI experiments initiated by ultrafast (femtosecond-duration) laser pulses also allow opportunities to track the time-evolution of molecular structures, and thereby advance understanding of molecular fragmentation processes. This Perspective illustrates two emerging families of dynamical studies. 'One-colour' studies (employing strong field ionisation driven by intense near infrared or single X-ray or extreme ultraviolet laser pulses) afford routes to preparing multiply charged molecular cations and exploring how their fragmentation progresses from valence-dominated to Coulomb-dominated dynamics with increasing charge and how this evolution varies with molecular size and composition. 'Two-colour' studies use one ultrashort laser pulse to create electronically excited neutral molecules (or monocations), whose structural evolution is then probed as a function of pump-probe delay using an ultrafast ionisation pulse along with time and position-sensitive detection methods. This latter type of experiment has the potential to return new insights into not just molecular fragmentation processes but also charge transfer processes between moieties separating with much better defined stereochemical control than in contemporary ion-atom and ion-molecule charge transfer studies.

Quantum work statistics in regular and classical-chaotic dynamical billiard systems.

In the thermodynamics of nanoscopic systems the relation between classical and quantum mechanical description is of particular importance. To scrutinize this correspondence we have chosen two two-dimensional billiard systems. Both systems are studied in the classical and the quantum mechanical settings. The classical conditional probability density p(E,L|E_{0},L_{0}) as well as the quantum mechanical transition probability P(n,l|n_{0},l_{0}) are calculated, which build the basis for the statistical analysis. We calculate the work distribution for one particle. The results in the quantum case in particular are of special interest since a suitable definition of mechanical work in small quantum systems is already controversial. Furthermore, we analyze the probability of both zero work and zero angular momentum difference. Using connections to an exactly solvable system analytical formulas are given for both systems. In the quantum case we get numerical results with some interesting relations to the classical case.

High-resolution electron time-of-flight spectrometers for angle-resolved measurements at the SQS Instrument at the European XFEL.

A set of electron time-of-flight spectrometers for high-resolution angle-resolved spectroscopy was developed for the Small Quantum Systems (SQS) instrument at the SASE3 soft X-ray branch of the European XFEL. The resolving power of this spectrometer design is demonstrated to exceed 10 000 (E/ΔE), using the well known Ne 1s-13p resonant Auger spectrum measured at a photon energy of 867.11 eV at a third-generation synchrotron radiation source. At the European XFEL, a width of ∼0.5 eV full width at half-maximum for a kinetic energy of 800 eV was demonstrated. It is expected that this linewidth can be reached over a broad range of kinetic energies. An array of these spectrometers, with different angular orientations, is tailored for the Atomic-like Quantum Systems endstation for high-resolution angle-resolved spectroscopy of gaseous samples.

Adiabatic Lindbladian Evolution with Small Dissipators

We consider a time-dependent small quantum system weakly coupled to an environment, whose effective dynamics we address by means of a Lindblad equation. We assume the Hamiltonian part of the Lindbladian is slowly varying in time and the dissipator part has small amplitude. We study the properties of the evolved state of the small system as the adiabatic parameter and coupling constant both go to zero, in various asymptotic regimes. In particular, we analyse the deviations of the transition probabilities of the small system between the instantaneous eigenspaces of the Hamiltonian with respect to their values in the purely Hamiltonian adiabatic setup, as a function of both parameters.

Thermalization of small quantum systems: from the zeroth law of thermodynamics

Thermalization of isolated quantum systems has been studied intensively in recent years and significant progresses have been achieved. Here, we study thermalization of small quantum systems that interact with large chaotic environments under the consideration of Schrödinger evolution of composite systems, from the perspective of the zeroth law of thermodynamics. Namely, we consider a small quantum system that is brought into contact with a large environmental system; after they have relaxed, they are separated and their temperatures are studied. Our question is under what conditions the small system may have a detectable temperature that is identical with the environmental temperature. This should be a necessary condition for the small quantum system to be thermalized and to have a well-defined temperature. By using a two-level probe quantum system that plays the role of a thermometer, we find that the zeroth law is applicable to quantum chaotic systems, but not to integrable systems.

Timing and X-ray pulse characterization at the Small Quantum Systems instrument of the European X-ray Free Electron Laser.

This contribution presents the initial characterization of the pump-probe performance at the Small Quantum Systems (SQS) instrument of the European X-ray Free Electron Laser. It is demonstrated that time-resolved experiments can be performed by measuring the X-ray/optical cross-correlation exploiting the laser-assisted Auger decay in neon. Applying time-of-arrival corrections based on simultaneous spectral encoding measurements allow us to significantly improve the temporal resolution of this experiment. These results pave the way for ultrafast pump-probe investigations of gaseous media at the SQS instrument combining intense and tunable soft X-rays with versatile optical laser capabilities.