Publications by Christiane Koch

Sympathetic cooling of molecular ions through the Coulomb interaction with laser-cooled atomic ions is an efficient tool to prepare translationally cold molecules. Even at relatively high collisional energies of about 1eV (T 10000K), the nearest approach in the ion-ion collisions never gets closer than 1nm such that naively perturbations of the internal molecular state are not expected. The Coulomb field may, however, induce rotational transitions changing the purity of initially quantum state prepared molecules. Here, we use estimates of rotational state changes in collisions of diatomic ions with atomic ions (arXiv:1905.02130) and determine the overall rotational excitation accumulated over the sympathetic cooling. We also estimate the cooling time, considering both a single atomic ion and a Coulomb crystal of atomic ions.

We investigate rotational state changes in a single collision of diatomic molecular ions, both polar and apolar, with an atomic ion. These may occur when sympathetically cooling trapped molecular ions by co-trapped laser-cooled atomic ions, since the electric field due to the atom may excite molecular rotations. Thanks to the different time and energy scales of translational and rotational motion, we may treat the collision classically and describe rotations quantum mechanically. Based on numerical calculations for a number of examples, we derive closed-form estimates for the rotational excitation per collision, depending on the scattering energy and the molecular parameters.

Building on the insight that engineered noise, specifically, engineered dephasing can enhance the adiabaticity of controlled quantum dynamics, we investigate how a dephasing-generating coupling to an auxiliary quantum system affects quantum annealing protocols. By calculating the exact reduced system dynamics, we show how this coupling enhances the system’s adiabaticity solely through a coherent mechanism - an effective energy rescaling. We show that it can lead to an annealing speedup linearly proportional to the strength of the coupling. We discuss the experimental feasibility of the protocols, and investigate the trade-off between fidelity and implementability by examining two modified versions with fewer types of required physical couplings.

Quantum resonances in collisions and reactions are a sensitive probe of the intermolecular forces. They may dominate the final quantum state distribution, as recently observed for Feshbach resonances in a cold collision experiment (Science 380, 77 (2023)). This raises the question whether the sensitivity of such measurements is sufficient to assess the quality of theoretical models for the interaction. We here compare measured collision cross sections to those obtained with exact quantum coupled-channels scattering calculations for three different ab initio potential energy surfaces. We find that the ability to test the correct prediction of energy redistribution over molecular degrees of freedom is within reach, requiring only a modest improvement in energy resolution of current experiments. Such improvement will enable the separation of individual resonances and allow for an unambiguous experimental test of different theory approaches.

Cat states are a valuable resource for quantum metrology applications, promising to enable sensitivity down to the Heisenberg limit. Moreover, Schrödinger cat states, based on a coherent superposition of coherent states, show robustness against phase-flip errors making them a promising candidate for bosonic quantum codes. A pathway to realize cat states is via utilizing single Kerr-type anharmonicities as found in superconducting devices as well as in Rydberg atoms. Such platforms nevertheless utilize only the second order anharmonicity, which limits the time it takes for a cat state to be prepared. Here we show how proper tuning of multiple higher order nonlinear interactions leads to shorter cat state preparation time. We also discuss practical aspects including an optimal control scheme which allows us to start the state preparation from the vacuum state under standard single mode driving. Lastly, we propose an ensemble of Rydberg atoms that exhibits higher order nonlinearities as a platform to prepare cat states in the laboratory.

Transmon qubits arise from the quantization of nonlinear resonators, systems that are prone to the buildup of strong, possibly even chaotic, fluctuations. One may wonder to what extent fast gate operations, which involve the transient population of states outside the computational subspace, can be affected by such instabilities. We here consider the eigenphases and -states of the time evolution operators describing a universal gate set, and analyze them by methodology otherwise applied in the context of many-body physics. Specifically, we discuss their spectral statistic, the distribution of time dependent level curvatures, and state occupations in- and outside the computational subspace. We observe that fast entangling gates, operating at speeds close to the so-called quantum speed limit, contain transient regimes where the dynamics indeed becomes partially chaotic. We find that for these gates even small variations of Hamiltonian or control parameters lead to large gate errors and speculate on the consequences for the practical implementation of quantum control.

We develop photoelectron interferometry based on laser-assisted extreme ultraviolet ionization for randomly oriented chiral molecules. As in the well established ’reconstruction of attosecond beating by interference of two-photon transitions’, an infrared or visible laser pulse promotes interferences between components of the photoelectron wave packet ionized by a comb of XUV photons, applied here to a sample of chiral molecules. We show that the magnitude of the resulting chiral signal is simply controlled by the time delay between the XUV and laser pulses, the choice of the laser frequency determines the photoelectron energy at which the chiral signal is probed, and comparison of different polarization configurations in the two-photon process allows for disentangling the contributions of bound and continuum states to the chiral response. Our proposal provides a simple, experimentally feasible, robust and versatile tool for the control of photoelectron circular dichroism.

In molecular physics, it is often necessary to average over the orientation of molecules when calculating observables, in particular when modelling experiments in the liquid or gas phase. Evaluated in terms of Euler angles, this is closely related to integration over two- or three-dimensional unit spheres, a common problem discussed in numerical analysis. The computational cost of the integration depends significantly on the quadrature method, making the selection of an appropriate method crucial for the feasibility of simulations. After reviewing several classes of spherical quadrature methods in terms of their efficiency and error distribution, we derive guidelines for choosing the best quadrature method for orientation averages and illustrate these with three examples from chiral molecule physics. While Gauss quadratures allow for achieving numerically exact integration for a wide range of applications, other methods offer advantages in specific circumstances. Our guidelines can also by applied to higher-dimensional spherical domains and other geometries. We also present a Python package providing a flexible interface to a variety of quadrature methods.

Squeezing is a non-classical feature of quantum states that is a useful resource, for example in quantum sensing of mechanical forces. Here, we show how to use optimal control theory to maximize squeezing in an optomechanical setup with two external drives and determine how fast the mechanical mode can be squeezed. For the autonomous drives considered here, we find the inverse cavity decay to lower-bound the protocol duration. At and above this limit, we identify a family of protocols leveraging a two-stage control strategy, where the mechanical mode is cooled before it is squeezed. Identification of the control strategy allows for two important insights - to determine the factors that limit squeezing and to simplify the time-dependence of the external drives, making our protocol readily applicable in experiments.

Cooling a quantum system to its ground state is important for the characterization of non-trivial interacting systems, and in the context of a variety of quantum information platforms. In principle, this can be achieved by employing measurement-based passive steering protocols, where the steering steps are predetermined and are not based on measurement readouts. However, measurements, i.e., coupling the system to auxiliary quantum degrees of freedom, is rather costly, and protocols in which the number of measurements scales with system size will have limited practical applicability. Here, we identify conditions under which measurement-based cooling protocols can be taken to the dilute limit. For two examples of frustration-free one-dimensional spin chains, we show that steering on a single link is sufficient to cool these systems into their unique ground states. We corroborate our analytical arguments with finite-size numerical simulations and discuss further applications.

The structure and dynamics of a molecular system is governed by its potential energy surface (PES), representing the total energy as a function of the nuclear coordinates. Obtaining accurate potential energy surfaces is limited by the exponential scaling of Hilbert space, restricting quantitative predictions of experimental observables from first principles to small molecules with just a few electrons. Here, we present an explicitly physics-informed approach for improving and assessing the quality of families of PESs by modifying them through linear coordinate transformations based on experimental data. We demonstrate this "morphing" of the PES for the He-H⁺₂ complex for reference surfaces at three different levels of quantum chemistry and using recent comprehensive Feshbach resonance(FR) measurements. In all cases, the positions and intensities of peaks in the collision cross-section are improved. We find these observables to be mainly sensitive to the long-range part of the PES.

Achiral molecules can be made temporarily chiral by excitation with electric fields, in the sense that an average over molecular orientations displays a net chiral signal Tikhonov et al., Sci. Adv. 8, eade0311 (2022). Here, we go beyond the assumption of molecular orientations to remain fixed during the excitation process. Treating both rotations and vibrations quantum mechanically, we identify conditions for the creation of chiral vibrational wavepackets -- with net chiral signals -- in ensembles of achiral molecules which are initially randomly oriented. Based on the analysis of symmetry and controllability, we derive excitation schemes for the creation of chiral wavepackets using a combination of (a) microwave and IR pulses and (b) a static field and a sequence of IR pulses. These protocols leverage quantum rotational dynamics for pump-probe spectroscopy of chiral vibrational dynamics, extending the latter to regions of the electromagnetic spectrum other than the UV.

Operator controllability refers to the ability to implement an arbitrary unitary in SU(N) and is a prerequisite for universal quantum computing. Controllability tests can be used in the design of quantum devices to reduce the number of external controls. Their practical use is hampered, however, by the exponential scaling of their numerical effort with the number of qubits. Here, we devise a hybrid quantum-classical algorithm based on a parametrized quantum circuit. We show that controllability is linked to the number of independent parameters, which can be obtained by dimensional expressivity analysis. We exemplify the application of the algorithm to qubit arrays with nearest-neighbour couplings and local controls. Our work provides a systematic approach to the resource-efficient design of quantum chips.

We derive a set of functionals for optimization towards an arbitrary cat state and demonstrate their application by optimizing the dynamics of a Kerr-nonlinear Hamiltonian with two-photon driving. The versatility of our framework allows us to adapt our functional towards optimization of maximally entangled cat states, applying it to a Jaynes-Cummings model. We identify the strategy of the obtained control fields and determine the quantum speed limit as a function of the cat state’s excitation. Finally, we extend our optimization functionals to open quantum system dynamics and apply it to the Jaynes-Cummings model with decay on the oscillator. For strong dissipation and large cat radii, we find a change in the control strategy compared to the case without dissipation. Our results highlight the power of optimal control with functionals specifically crafted for complex physical tasks and the versatility of the quantum optimal control toolbox for practical applications in the quantum technologies.

The ability to implement any desired quantum logic gate on a quantum processing unit is equivalent to evolution-operator controllability of the qubits. Conversely, controllability analysis can be used to minimize the resources, i.e., the number of external controls and qubit-qubit couplings, required for universal quantum computing. Standard controllability analysis, consisting in the construction of the dynamical Lie algebra, is, however, impractical already for a comparatively small number of qubits. Here, we show how to leverage an alternative approach, based on a graph representation of the Hamiltonian, to determine controllability of arrays of coupled qubits. We provide a complete computational framework and exemplify it for arrays of five qubits, inspired by the ibmqquito architecture. We find that the number of controls can be reduced from five to one for complex qubit-qubit couplings and to two for standard qubit-qubit couplings.

Feshbach resonances are fundamental to interparticle interactions and become particularly important in cold collisions with atoms, ions, and molecules. In this work, we present the detection of Feshbach resonances in a benchmark system for strongly interacting and highly anisotropic collisions: molecular hydrogen ions colliding with noble gas atoms. The collisions are launched by cold Penning ionization, which exclusively populates Feshbach resonances that span both short- and long-range parts of the interaction potential. We resolved all final molecular channels in a tomographic manner using ion-electron coincidence detection. We demonstrate the nonstatistical nature of the final-state distribution. By performing quantum scattering calculations on ab initio potential energy surfaces, we show that the isolation of the Feshbach resonance pathways reveals their distinctive fingerprints in the collision outcome. Quantum scattering resonances, and Feshbach resonances in particular, are the key features of cold collisions that can fundamentally change collision outcomes. Numerous previous experiments have mainly focused on the initial channel, providing limited insight into the resonant scattering dynamics. Margulis et al. developed a method in which the energetics and decay channels of Feshbach resonances, formed by collisions induced by Penning ionization of metastable helium or neon atoms and the ground-state hydrogen molecule, were mapped out by ion-electron coincidence velocity map imaging with a precision of several Kelvin, enough to resolve all the final rovibrational quantum states in a single measurement. The proposed method offers a new approach to quantum state mapping of resonant collision dynamics. Feshbach resonances in reaction complexes of H₂⁺ ions and rare gas atoms are resolved in a tomographic manner.

The ability to characterise a Hamiltonian with high precision is crucial for the implementation of quantum technologies. In addition to the well-developed approaches utilising optimal probe states and optimal measurements, the method of optimal control can be used to identify time-dependent pulses applied to the system to achieve higher precision, especially in the presence of noise. Here, we extend optimally controlled estimation schemes for single qubits to non-commuting dynamics as well as two interacting qubits, demonstrating improvements in terms of maximal precision, time-stability, as well as robustness over uncontrolled protocols.

A planar molecule may become chiral upon excitation of an out-of-plane vibration, changing its handedness during half a vibrational period. When exciting such a vibration in an ensemble of randomly oriented molecules with an infrared laser, half of the molecules will undergo the vibration phase-shifted by π compared to the other half, and no net chiral signal is observed. This symmetry can be broken by exciting the vibrational motion with a Raman transition in the presence of a static electric field. Subsequent ionization of the vibrating molecules by an extreme ultraviolet pulse probes the time-dependent net handedness via the photoelectron circular dichroism. Our proposal for pump-probe spectroscopy of molecular chirality, based on quantum-chemical theory and discussed for the example of the carbonyl chlorofluoride molecule, is feasible with current experimental technology. Planar molecules can be made chiral with electric fields only, and a net chiral signature can be recorded upon ionization.

Non-radiative energy transfer between a Rydberg atom and a polar molecule can be controlled by a DC electric field. Here we show how to exploit this control for state-resolved, non-destructive detection and spectroscopy of the molecules where the lineshape reflects the type of molecular transition. Using the example of ammonia, we identify the conditions for collision-mediated spectroscopy in terms of the required electric field strengths, relative velocities, and molecular densities. Rydberg atom-enabled spectroscopy is feasible with current experimental technology, providing a versatile detection method as basic building block for applications of polar molecules in quantum technologies and chemical reaction studies.

Quantum optimal control, a toolbox for devising and implementing the shapes of external fields that accomplish given tasks in the operation of a quantum device in the best way possible, has evolved into one of the cornerstones for enabling quantum technologies. The last few years have seen a rapid evolution and expansion of the field. We review here recent progress in our understanding of the controllability of open quantum systems and in the development and application of quantum control techniques to quantum technologies. We also address key challenges and sketch a roadmap for future developments.

We propose a protocol that achieves fast adiabatic transfer between two orthogonal states of a qubit by coupling with an ancilla. The qubit undergoes Landau-Zener dynamics, whereas the coupling realizes a time-dependent Hamiltonian, which is diagonal in the spin’s instantaneous Landau-Zener eigenstates. The ancilla (or meter), in turn, couples to a thermal bath, such that the overall dynamics is incoherent. We analyse the protocol’s fidelity as a function of the strength of the coupling and of the relaxation rate of the meter. When the meter’s decay rate is the largest frequency scale of the dynamics, the spin dynamics is encompassed by a master equation describing dephasing of the spin in the instantaneous eigenbasis. In this regime the fidelity of adiabatic transfer improves as the bath temperature is increased. Surprisingly, the adiabatic transfer is significantly more efficient in the opposite regime, where the time scale of the ancilla dynamics is comparable to the characteristic spin time scale. Here, for low temperatures the coupling with the ancilla tends to suppress diabatic transitions via effective cooling. The protocol can be efficiently implemented by means of a pulsed, stroboscopic coupling with the ancilla and is robust against moderate fluctuations of the experimental parameters.

We present an analytical approach to construct the Lie algebra of finite-dimensional subsystems of the driven asymmetric top rotor. Each rotational level is degenerate due to the isotropy of space, and the degeneracy increases with rotational excitation. For a given rotational excitation, we determine the nested commutators between drift and drive Hamiltonians using a graph representation. We then generate the Lie algebra for subsystems with arbitrary rotational excitation using an inductive argument.

We prove complete controllability for rotational states of an asymmetric top molecule belonging to degenerate values of the orientational quantum number M. Based on this insight, we construct a pulse sequence that energetically separates population initially distributed over degenerate M-states, as a precursor for orientational purification. Introducing the concept of enantio-selective controllability, we determine the conditions for complete enantiomer-specific population transfer in chiral molecules and construct pulse sequences realizing this transfer for population initially distributed over degenerate M-states. This degeneracy presently limits enantiomer-selectivity for any initial state except the rotational ground state. Our work thus shows how to overcome an important obstacle towards separating, with electric fields only, left-handed from right-handed molecules in a racemic mixture.

Microwave three-wave mixing allows for enantiomer-selective excitation of randomly oriented chiral molecules into rotational states with different energy. The random orientation of molecules is reflected in the degeneracy of the rotational spectrum with respect to the orientational quantum number M and reduces, if not accounted for, enantiomer-selectivity. Here, we show how to design pulse sequences with maximal enantiomer-selectivity from an analysis of the M-dependence of the Rabi frequencies associated with rotational transitions induced by resonant microwave drives. We compare different excitations schemes for rotational transitions and show that maximal enantiomer-selectivity at a given rotational temperature is achieved for synchronized three-wave mixing with circularly polarized fields.

In continuous-variable quantum computing with qubits encoded in the infinite-dimensional Hilbert space of bosonic modes, it is a difficult task to realize strong and on-demand interactions between the qubits. One option is to engineer a beamsplitter interaction for photons in two superconducting cavities by driving an intermediate superconducting circuit with two continuous-wave drives, as demonstrated in a recent experiment. Here, we show how quantum optimal control theory (OCT) can be used in a systematic way to improve the beamsplitter interaction between the two cavities. We find that replacing the two-tone protocol by a three-tone protocol accelerates the effective beamsplitter rate between the two cavities. The third tone’s amplitude and frequency are determined by gradient-free optimization and make use of cavity-transmon sideband couplings. We show how to further improve the three-tone protocol via gradient-based optimization while keeping the optimized drives experimentally feasible. Our work exemplifies how to use OCT to systematically improve practical protocols in quantum information applications.

We investigate how optimal control theory can be used to improve Circular Dichroism (CD) signals for A--band of fenchone measured via the photoionization yield upon further excitation. These transitions are electric dipole forbidden to first order, which translates into low population transfer to the excited state (~8) but also allows for a clearer interplay between electric and magnetic transition dipole moments, which are of the same order of magnitude. Using a model including the electronic ground and excited A state as well as all permanent and transition multipole moments up to the electric quadrupole, we find that the absolute CD signal of randomly oriented molecules can be increased by a factor 3.5 when using shaped laser pulses, with the anisotropy parameter g increasing from 0.06 to 1. Our insights provide additional evidence on how optimal control can assist in amplifying chiral signatures via interactions of permanent and transition multipole moments.

We demonstrate a simplified method for dissipative generation of an entangled state of two trapped-ion qubits. Our implementation produces its target state faster and with higher fidelity than previous demonstrations of dissipative entanglement generation and eliminates the need for auxiliary ions. The entangled singlet state is generated in ∼7  ms with a fidelity of 0.949(4). The dominant source of infidelity is photon scattering. We discuss this error source and strategies for its mitigation.

Asymmetric spectral line shapes are a hallmark of interference of a quasi-bound state with a continuum of states. Such line shapes are well known for multichannel systems, for example, in photoionization or Feshbach resonances in molecular scattering. On the other hand, in resonant single channel scattering, the signature of such interference may disappear due to the orthogonality of partial waves. Here, we show that probing the angular dependence of the cross section allows us to unveil asymmetric Fano profiles also in a single channel shape resonance. We observe a shift in the peak of the resonance profile in the elastic collisions between metastable helium and deuterium molecules with detection angle, in excellent agreement with theoretical predictions from full quantum scattering calculations. Using a model description for the partial wave interference, we can disentangle the resonant and background contributions and extract the relative phase responsible for the characteristic Fano-like profiles from our experimental measurements.

Quantum coherent control of ultrafast bond making and the subsequent molecular dynamics is crucial for the realization of a new photochemistry, where a shaped laser field is actively driving the chemical system in a coherent way from the thermal initial state of the reactants to the final state of the desired products. We demonstrate here coherent control over the relative yields of Mg2 molecules that are generated via photoassociation and subsequently photodriven into different groups of final states. The strong-field process involves non-resonant multiphoton femtosecond photoassociation of a pair of thermally hot magnesium atoms into a bound Mg2 molecule and subsequent molecular dynamics on electronically excited states. The branching-ratio control is achieved with linearly chirped laser pulses, utilizing the different chirp dependence that various groups of final molecular states display for their post-pulse population. Our study establishes the feasibility of high degree coherent control over quantum molecular dynamics that is initiated by femtosecond photoassociation of thermal atoms.

Qubit reset is a key task in the operation of quantum devices which, for many quantum hardware platforms, presently limits device clock speed. While it is known that coupling the qubit to an ancilla on demand allows for the fastest qubit reset, the limits on reset accuracy and speed due to the choice of ancilla have not yet been identified—despite the great flexibility in device design for most quantum hardware platforms. Here, we derive bounds on qubit reset in terms of maximum fidelity and minimum time, assuming control over the qubit and no control over the ancilla. For two-level ancillas, we find a provably time-optimal protocol which consists of purity exchange between qubit and ancilla brought into resonance. The globally minimal time can only be realized for specific choices of coupling and control which we identify. When increasing the size of the ancilla Hilbert space, the maximally achievable fidelity increases, whereas the reset time remains constant. Our results translate into device design principles for realizing, in a given quantum architecture, the fastest and most accurate protocol for qubit reset.

Scattering resonances play a central role in collision processes in physics and chemistry. They help building an intuitive understanding of the collision dynamics due to the spatial localization of the scattering wavefunctions. For resonances that are localized in the reaction region, sharp peaks in the reaction rates are the characteristic signature, observed recently with state-of-the-art experiments in low energy collisions. If, however, the localization occurs outside of the reaction region, only the elastic scattering is modified. This may occur due to above barrier resonances, the quantum analogue of classical orbiting. By probing both elastic and inelastic scattering experimentally, we differentiate between the nature of quantum resonances - tunneling vs above barrier - and corroborate our findings by calculating the corresponding scattering wavefunctions.

Quantum discrimination and estimation are pivotal for many quantum technologies, and their performance depends on the optimal choice of probe state and measurement. Here we show that their performance can be further improved by suitably tailoring the pulses that make up the interferometer. Developing an optimal control framework and applying it to the discrimination and estimation of a magnetic field in the presence of noise, we find an increase in the overall achievable state distinguishability. Moreover, the maximum distinguishability can be stabilized for times that are more than an order of magnitude longer than the decoherence time.

The precise engineering of quantum states, a basic prerequisite for technologies such as quantum-enhanced sensing or quantum computing, becomes more challenging with increasing dimension of the system Hilbert space. Standard preparation techniques then require a large number of operations or slow adiabatic evolution and give access to only a limited set of states. Here, we use quantum optimal control theory to overcome this problem and derive shaped radio-frequency pulses to experimentally navigate the Stark manifold of a Rydberg atom. We demonstrate that optimal control, beyond improving the fidelity of an existing protocol, also enables us to accurately generate a nonclassical superposition state that cannot be prepared with reasonable fidelity using standard techniques. Optimal control thus substantially enlarges the range of accessible states. Our joint experimental and theoretical work establishes quantum optimal control as a key tool for quantum engineering in complex Hilbert spaces.

We suggest a quantum simulator that allows to study the role of memory effects in the dynamics of open quantum systems. A particular feature of our simulator is the ability to engineer both Markovian and non-Markovian dynamics by means of quantum measurements and the quantum Zeno dynamics induced by them. The simulator is realized by two subsystems of a bipartite quantum system or two subspaces of a single system which can be identified as system and meter. Exploiting the analogy between dissipation and quantum measurements, the interaction between system and meter gives rise to quantum Zeno dynamics, and the dissipation strength experienced by the system can be tuned by changing the parameters of the measurement, i.e., the interaction with the meter. Our proposal can readily be realized with existing experimental technology, such as cavity- or circuit-QED platforms or ultracold atoms.

In simple one-photon ionization, decoherence occurs due to entanglement between ion and photoelectron. Therefore, the preparation of coherent superpositions of electronic eigenstates of the hole in the photoion is extremely difficult. We demonstrate for the xenon atom that the degree of electronic coherence of the photoion in attosecond photoionization can be enhanced if the influence of many-body interactions is properly controlled. A mechanism at low photon energies involving multiphoton ionization is found, suppressing the loss of coherence through ionization into the same photoelectron partial waves. The degree of coherence found between the 4d0 and 5s hole states is, on the one hand, limited by Auger decay of the 4d0 hole. On the other hand, increasing the population ratio such that a significant portion of the state is in a true superposition of both states renders the maximization of the degree of coherence difficult.

Decay of bound states due to coupling with free particle states is a general phenomenon occurring at energy scales from MeV in nuclear physics to peV in ultracold atomic gases. Such a coupling gives rise to Fano-Feshbach resonances (FFR) that have become key to understanding and controlling interactions - in ultracold atomic gases, but also between quasiparticles, such as microcavity polaritons. Their energy positions were shown to follow quantum chaotic statistics. In contrast, their lifetimes have so far escaped a similarly comprehensive understanding. Here, we show that bound states, despite being resonantly coupled to a scattering state, become protected from decay whenever the relative phase is a multiple of π. We observe this phenomenon by measuring lifetimes spanning four orders of magnitude for FFR of spin-orbit excited molecular ions with merged beam and electrostatic trap experiments. Our results provide a blueprint for identifying naturally long-lived states in a decaying quantum system.

We present a new open-source Python package, krotov, implementing the quantum optimal control method of that name. It allows to determine time-dependent external fields for a wide range of quantum control problems, including state-to-state transfer, quantum gate implementation and optimization towards an arbitrary perfect entangler. Krotov’s method compares to other gradient-based optimization methods such as gradient-ascent and guarantees monotonic convergence for approximately time-continuous control fields. The user-friendly interface allows for combination with other Python packages, and thus high-level customization.

The measured multidimensional spectral response of different light harvesting complexes exhibits oscillatory features which suggest an underlying coherent energy transfer. However, making this inference rigorous is challenging due to the difficulty of isolating excited state coherences in highly congested spectra. In this work, we provide a coherent control scheme that suppresses ground state coherences, thus making rephasing spectra dominated by excited state coherences. We provide a benchmark for the scheme using a model dimeric system and numerically exact methods to analyze the spectral response. We argue that combining temporal and spectral control methods can facilitate a second generation of experiments that are tailored to extract desired information and thus significantly advance our understanding of complex open many-body structure and dynamics.

Quantum dynamical simulations of statistical ensembles pose a significant computational challenge due to the fact that mixed states need to be represented. If the underlying dynamics is fully unitary, for example, in ultrafast coherent control at finite temperatures, then one approach to approximate time-dependent observables is to sample the density operator by solving the Schrödinger equation for a set of wave functions with randomized phases. We show that, on average, random-phase wave functions perform well for ensembles with high mixedness, whereas at higher purities a deterministic sampling of the energetically lowest-lying eigenstates becomes superior. We prove that minimization of the worst-case error for computing arbitrary observables is uniquely attained by eigenstate-based sampling. We show that this error can be used to form a qualitative estimate of the set of ensemble purities for which the sampling performance of the eigenstate-based approach is superior to random-phase wave functions. Furthermore, we present refinements to both schemes which remove redundant information from the sampling procedure to accelerate their convergence. Finally, we point out how the structure of low-rank observables can be exploited to further improve eigenstate-based sampling schemes.

We determine how to optimally reset a superconducting qubit which interacts with a thermal environment in such a way that the coupling strength is tunable. Describing the system in terms of a time-local master equation with time-dependent decay rates and using quantum optimal control theory, we identify temporal shapes of tunable level splittings which maximize the efficiency of the reset protocol in terms of duration and error. Time-dependent level splittings imply a modification of the system-environment coupling, varying the decay rates as well as the Lindblad operators. Our approach thus demonstrates efficient reservoir engineering employing quantum optimal control. We find the optimized reset strategy to consist in maximizing the decay rate from one state and driving non-adiabatic population transfer into this strongly decaying state.

The angular momentum of molecules, or, equivalently, their rotation in three-dimensional space, is ideally suited for quantum control. Molecular angular momentum is naturally quantized, time evolution is governed by a well-known Hamiltonian with only a few accurately known parameters, and transitions between rotational levels can be driven by external fields from various parts of the electromagnetic spectrum. Control over the rotational motion can be exerted in one-, two-, and many-body scenarios, thereby allowing one to probe Anderson localization, target stereoselectivity of bimolecular reactions, or encode quantum information to name just a few examples. The corresponding approaches to quantum control are pursued within separate, and typically disjoint, subfields of physics, including ultrafast science, cold collisions, ultracold gases, quantum information science, and condensed-matter physics. It is the purpose of this review to present the various control phenomena, which all rely on the same underlying physics, within a unified framework. To this end, recall the Hamiltonian for free rotations, assuming the rigid rotor approximation to be valid, and summarize the different ways for a rotor to interact with external electromagnetic fields. These interactions can be exploited for control - from achieving alignment, orientation, or laser cooling in a one-body framework, steering bimolecular collisions, or realizing a quantum computer or quantum simulator in the many-body setting.

We report two schemes to generate perfect anisotropy in the photoelectron angular distribution of a randomly oriented ensemble of polyatomic molecules. In order to exert full control over the anisotropy of photoelectron emission, we exploit interferences between single-photon pathways and a manifold of resonantly enhanced two-photon pathways. These are shown to outperform nonsequential (w,2w) bichromatic phase control for the example of CHFClBr molecules. We are able to optimize pulses that yield anisotropic photoelectron emission thanks to a very efficient calculation of photoelectron momentum distributions. This is accomplished by combining elements of quantum chemistry, variational scattering theory, and time-dependent perturbation theory.

Three-wave mixing spectroscopy of chiral molecules, which exist in left-handed and right-handed conformations, allows for enantioselective population transfer despite random orientation of the molecules. This is based on constructive interference of the three-photon pathways for one enantiomer and the destructive one for the other. We prove here that three mutually orthogonal polarization directions are required to this end. Two different dynamical regimes exist to realize enantioselective population transfer, and we show that they correspond to different phase conditions in the three-wave mixing. We find the excitation scheme used in current rotational three-wave mixing experiments of chiral molecules with C₁ symmetry to be close to optimal and discuss the prospects for rovibrational three-wave mixing experiments of axially chiral molecules. Our comprehensive study allows us to clarify earlier misconceptions in the literature.

The definition of the scattering volume for p-wave collisions needs to be generalized in the presence of dipolar interactions for which the potential decreases with the interparticle separation as 1/R³. Here, we propose a way to define the scattering volume characterizing the short-range interactions in odd-parity waves, by analyzing the p-wave component of the two-body threshold wave function. Our approach uses an asymptotic model and introduces explicitly the anisotropic dipole-dipole interaction, which governs the ultracold collision dynamics at long range. The short-range interactions, which are essential to describe threshold resonances, are taken into account by a single parameter related to the nodal structure of the wave functions at short distances.

Interactions in a spin-polarized ultracold Fermi gas are governed by p-wave collisions and can be characterized by the p-wave scattering volume. Control of these collisions by Feshbach resonances is hampered by huge inelastic losses. Here, we suggest nonresonant light control of p-wave collisions, exploiting the anisotropic coupling of nonresonant light to the polarizability of the atoms. The p-wave scattering volume can be controlled by strong nonresonant light, in close analogy to the s-wave scattering length. For collision partners that are tightly trapped, the nonresonant light induces an energy shift directly related to the scattering volume (as defined by A. Crubellier et al., Phys. Rev. A 99, 032709 (2019)). This effect could be used to climb the ladder of the trap. We also show that controlling the scattering volume implies control, at least roughly, over the orientation at short interatomic distances of the interparticle axis relative to the polarization direction of the light. Our proposal is based on an asymptotic model that explicitly accounts for the anisotropic dipole-dipole interaction which governs the ultracold collision dynamics at long range.

In many chemical reactions with more than one possible outcome, the branching ratio is nearly constant over a wide range of collision energies. In barrierless systems governed by long-range interactions, however, the branching ratio is more sensitive to collision energy, and its dependence on it can be useful for better understanding the dynamics and reconstructing interaction potentials. Here we present the reaction rates of Penning and associative ionisation of metastable neon and helium with argon atoms. We obtain reaction rates in merge beam experiments, over a wide range of collision energies corresponding to that of room temperature, all the way down to a few millikelvins. We observe a change of two orders of magnitude in the branching ratio in the measured collision energy range and explain these changes using theoretical calculations.

We investigate the time-optimal control of the purification of a qubit interacting with a structured environment, consisting of a strongly coupled two-level defect in interaction with a thermal bath. On the basis of a geometric analysis, we show for weak and strong interaction strengths that the optimal control strategy corresponds to a qubit in resonance with the reservoir mode. We investigate under which conditions qubit coherence and correlation between the qubit and the environment can speed up the control process.

The performance of key tasks in quantum technology, such as accurate state preparation, can be maximized by utilizing external controls and deriving their shape with optimal control theory. For non-pure target states, the performance measure needs to match both the angle and the length of the generalized Bloch vector. A measure based on this simple geometric picture that separates angle and length mismatch into individual terms is introduced and the ensuing optimization framework is applied to maximize squeezing of an optomechanical oscillator at finite temperature. The results herein show that shaping the cavity drives can speed up squeezed state preparation by more than two orders of magnitude. Cooperativities and pulse shapes required to this end are fully compatible with the current experimental technology.

We demonstrate coherent control over the photoelectron circular dichroism in randomly oriented chiral molecules, based on quantum interference between multiple photoionization pathways. To significantly enhance the chiral signature, we use a finite manifold of indistinguishable (1+1’) resonantly enhanced multiphoton ionization pathways interfering at a common photoelectron energy but probing different intermediate states. We show that this coherent control mechanism maximizes the number of molecular states that constructively contribute to the dichroism at an optimal photoelectron energy and thus outperforms other schemes, including interference between opposite-parity pathways driven by bichromatic (w , 2w) fields as well as sequential pump-probe ionization.

Sympathetic cooling of molecular ions through the Coulomb interaction with laser-cooled atomic ions is an efficient tool to prepare translationally cold molecules. Even at relatively high collisional energies of about 1eV (T∼10000K), the nearest approach in the ion-ion collisions never gets closer than ∼1nm such that naively perturbations of the internal molecular state are not expected. The Coulomb field may, however, induce rotational transitions changing the purity of initially quantum state prepared molecules. Here, we investigate such rotational state changing collisions for both polar and apolar diatomic molecular ions and derive closed-form estimates for rotational excitation based on the initial scattering energy and the molecular parameters.

Entanglement generation can be robust against noise in approaches that deliberately incorporate dissipation into the system dynamics. The presence of additional dissipation channels may, however, limit fidelity and speed of the process. Here we show how quantum optimal control techniques can be used to both speed up the entanglement generation and increase the fidelity in a realistic setup, whilst respecting typical experimental limitations. For the example of entangling two trapped ion qubits Lin et al., Nature 504, 415 (2013), we find an improved fidelity by simply optimizing the polarization of the laser beams utilized in the experiment. More significantly, an alternate combination of transitions between internal states of the ions, when combined with optimized polarization, enables faster entanglement and decreases the error by an order of magnitude.

We theoretically analyze the efficiency of a quantum memory for single photons. The photons propagate along a transmission line and impinge on one of the mirrors of a high-finesse cavity. The quantum memory is constituted by a single atom within the optical resonator. Photon storage is realized by the controlled transfer of the photonic excitation into a metastable state of the atom and occurs via a Raman transition with a suitably tailored laser pulse, which drives the atom. Our study is supported by numerical simulations, in which we include the modes of the transmission line and we use the experimental parameters of existing experimental setups. It reproduces the results derived using input-output theory in the corresponding regimes and can be extended to compute dynamics where the input-output formalism cannot be straightforwardly applied. Our analysis determines the maximal storage efficiency, namely, the maximal probability to store the photon in a stable atomic excitation, in the presence of spontaneous decay and cavity parasitic losses. It further delivers the form of the laser pulse that achieves the maximal efficiency by partially compensating parasitic losses. We numerically assess the conditions under which storage based on adiabatic dynamics is preferable to non-adiabatic pulses. Moreover, we systematically determine the shortest photon pulse that can be efficiently stored as a function of the system parameters.

We combine a quantum dynamical propagator that explicitly accounts for quantum mechanical time ordering with optimal control theory. After analyzing its performance with a simple model, we apply it to a superconducting circuit under so-called Pythagorean control. Breakdown of the rotating-wave approximation is the main source of the very strong time-dependence in this example. While the propagator that accounts for the time ordering in an iterative fashion proves its numerical efficiency for the dynamics of the superconducting circuit, its performance when combined with optimal control turns out to be rather sensitive to the strength of the time-dependence. We discuss the kind of quantum gate operations that the superconducting circuit can implement including their performance bounds in terms of fidelity and speed.

Preparation of a so-called circular state in a Rydberg atom where the projection of the electron angular momentum takes its maximum value is challenging due to the required amount of angular momentum transfer. Currently available protocols for circular state preparation are either accurate but slow or fast but error prone. Here we show how to use quantum optimal control theory to derive pulse shapes that realize fast and accurate circularization of a Rydberg atom. In particular, we present a theoretical proposal for optimized radio-frequency pulses that achieve high fidelity in the shortest possible time, given current experimental limitations on peak amplitudes and spectral bandwidth. We also discuss the fundamental quantum speed limit for circularization of a Rydberg atom when lifting these constraints.

Fast and reliable reset of a qubit is a key prerequisite for any quantum technology. For real world open quantum systems undergoing non-Markovian dynamics, reset implies not only purification, but in particular erasure of initial correlations between qubit and environment. Here, we derive optimal reset protocols using a combination of geometric and numerical control theory. For factorizing initial states, we find a lower limit for the entropy reduction of the qubit as well as a speed limit. The time-optimal solution is determined by the maximum coupling strength. Initial correlations, remarkably, allow for faster reset and smaller errors. Entanglement is not necessary.

With recent improvements in coherence times, superconducting transmon qubits have become a promising platform for quantum computing. They can be flexibly engineered over a wide range of parameters, but also require us to identify an efficient operating regime. Using state-of-the-art quantum optimal control techniques, we exhaustively explore the landscape for creation and removal of entanglement over a wide range of design parameters. We identify an optimal operating region outside of the usually considered strongly dispersive regime, where multiple sources of entanglement interfere simultaneously, which we name the quasi-dispersive straddling qutrits regime. At a chosen point in this region, a universal gate set is realized by applying microwave fields for gate durations of 50 ns, with errors approaching the limit of intrinsic transmon coherence. Our systematic quantum optimal control approach is easily adapted to explore the parameter landscape of other quantum technology platforms.

We present a protocol for interleaved randomized benchmarking of arbitrary quantum gates using Monte Carlo sampling of quantum states. It is generally applicable, including non-Clifford gates while preserving key advantages of randomized benchmarking such as error amplification as well as independence from state preparation and measurement errors. This property is crucial for implementations in many contemporary systems. Although the protocol scales exponentially in the number of qubits, it is superior to direct Monte Carlo sampling of the average gate fidelity in both the total number of experiments by orders of magnitude and savings in classical preprocessing, that are exponential.

A pair of atoms interacts with nonresonant light via its anisotropic polarizability. This effect can be used to tune the scattering properties of the atoms. Although the light-atom interaction varies with interatomic separation as 1/R³, the effective s-wave potential decreases more rapidly as 1/R⁴ such that the field-dressed scattering length can be determined without any formal difficulty. The scattering dynamics are essentially governed by the long-range part of the interatomic interaction and can thus be accurately described by an asymptotic model A. Crubellier et al., New J. Phys. 17, 045020 (2015). Here we use the asymptotic model to determine the field-dressed scattering length from the s-wave radial component of a particular threshold wave function. Applying our theory to the scattering of two strontium isotopes, we calculate the variation of the scattering length with the intensity of the nonresonant light. Moreover, we predict the intensities at which the scattering length becomes infinite for any pair of atoms.

Photoelectron circular dichroism refers to the forward/backward asymmetry in the photoelectron angular distribution with respect to the propagation axis of circularly polarized light. It has recently been demonstrated in femtosecond multi-photon photoionization experiments with randomly oriented camphor and fenchone molecules C. Lux et al., Angew. Chem. Int. Ed. 51, 5001 (2012); C. S. Lehmann et al., J. Chem. Phys. 139, 234307 (2013). A theoretical framework describing this process as (2+1) resonantly enhanced multi-photon ionization is constructed, which consists of two-photon photoselection from randomly oriented molecules and successive one-photon ionisation of the photoselected molecules. It combines perturbation theory for the light-matter interaction with ab initio calculations for the two-photon absorption and a single-center expansion of the photoelectron wavefunction in terms of hydrogenic continuum functions. It is verified that the model correctly reproduces the basic symmetry behavior expected under exchange of handedness and light helicity. When applied it to fenchone and camphor, semi-quantitative agreement with the experimental data is found, for which a sufficient d wave character of the electronically excited intermediate state is crucial.

Anisotropy is a fundamental property of particle interactions. It occupies a central role in cold and ultracold molecular processes, where orientation-dependent long-range forces have been studied in ultracold polar molecule collisions. In the cold collisions regime, quantization of the intermolecular degrees of freedom leads to quantum scattering resonances. Although these states have been shown to be sensitive to details of the interaction potential, the effect of anisotropy on quantum resonances has so far eluded experimental observation. Here, we directly measure the anisotropy in atom-molecule interactions via quantum resonances by changing the quantum state of the internal molecular rotor. We observe that a quantum scattering resonance at a collision energy of kB x 270mK appears in the Penning ionization of molecular hydrogen with metastable helium only if the molecule is rotationally excited. We use state-of-the-art ab initio theory to show that control over the rotational state effectively switches the anisotropy on or off, disentangling the isotropic and anisotropic parts of the interaction.

Photoionization with attosecond pulses populates hole states in the photoion. Superpositions of hole states represent ideal candidates for time-dependent spectroscopy, for example via pump-probe studies. The challenge consists in identifying pulses that create coherent superpositions of hole states while satisfying practical constraints. Here, we employ quantum optimal control to maximize the degree of coherence between these hole states. To this end, we introduce a derivative-free optimization method with sequential parametrization update (SPA optimization). We demonstrate the versatility and computational efficiency of SPA optimization for photoionization in argon by maximizing the coherence between the 3s and 3p₀ hole states using shaped attosecond pulses. We show that it is possible to maximize the hole coherence while simultaneously prescribing the ratio of the final hole state populations.

The advent of quantum devices, which exploit the two essential elements of quantum physics, coherence and entanglement, has sparked renewed interest in the control of open quantum systems. Successful implementations face the challenge of preserving relevant nonclassical features at the level of device operation. A major obstacle is decoherence, which is caused by interaction with the environment. Optimal control theory is a tool that can be used to identify control strategies in the presence of decoherence. Here we review recent advances in optimal control methodology that allow typical tasks in device operation for open quantum systems to be tackled and discuss examples of relaxation-optimized dynamics. Optimal control theory is also a useful tool to exploit the environment for control. We discuss examples and point out possible future extensions.

Photoelectron spectra and photoelectron angular distributions obtained in photoionization reveal important information on, e.g., charge transfer or hole coherence in the parent ion. Here we show that optimal control of the underlying quantum dynamics can be used to enhance desired features in the photoelectron spectra and angular distributions. To this end, we combine Krotov’s method for optimal control theory with the time-dependent configuration interaction singles formalism and a splitting approach to calculate photoelectron spectra and angular distributions. The optimization target can account for specific desired properties in the photoelectron angular distribution alone, in the photoelectron spectrum, or in both. We demonstrate the method for hydrogen and then apply it to argon under strong XUV radiation, maximizing the difference of emission into the upper and lower hemispheres, in order to realize directed electron emission in the XUV regime.

Penning ionization reactions in merged beams with precisely controlled collision energies have been shown to accurately probe quantum mechanical effects in reactive collisions. A complete microscopic understanding of the reaction is, however, faced with two major challenges - the highly excited character of the reaction’s entrance channel and the limited precision of even the best state-of-the-art ab initio potential energy surfaces. Here, we suggest photoassociation spectroscopy as a tool to identify the character of orbiting resonances in the entrance channel and probe the ionization width as a function of interparticle separation. We introduce the basic concept, using the example of metastable helium and argon, and discuss the general conditions under which this type of spectroscopy will be successful.

We show that a pseudospectral representation of the wavefunction using multiple spatial domains of variable size yields a highly accurate, yet efficient method to solve the time-dependent Schrödinger equation. The overall spatial domain is split into non-overlapping intervals whose size is chosen according to the local de Broglie wavelength. A multi-domain weak formulation of the Schrödinger equation is obtained by representing the wavefunction by Lagrange polynomials with compact support in each domain, discretized at the Legendre-Gauss-Lobatto points. The resulting Hamiltonian is sparse, allowing for efficient diagonalization and storage. Accurate time evolution is carried out by the Chebychev propagator, involving only sparse matrix-vector multiplications. Our approach combines the efficiency of mapped grid methods with the accuracy of spectral representations based on Gaussian quadrature rules and the stability and convergence properties of polynomial propagators. We apply this method to high-harmonic generation and examine the role of the initial state for the harmonic yield near the cutoff.

It is control that turns scientific knowledge into useful technology: in physics and engineering it provides a systematic way for driving a dynamical system from a given initial state into a desired target state with minimized expenditure of energy and resources. As one of the cornerstones for enabling quantum technologies, optimal quantum control keeps evolving and expanding into areas as diverse as quantumenhanced sensing, manipulation of single spins, photons, or atoms, optical spectroscopy, photochemistry, magnetic resonance (spectroscopy as well as medical imaging), quantum information processing and quantum simulation. In this communication, state-of-the-art quantum control techniques are reviewed and put into perspective by a consortium of experts in optimal control theory and applications to spectroscopy, imaging, as well as quantum dynamics of closed and open systems. We address key challenges and sketch a roadmap for future developments.

The role of internal molecular degrees of freedom, such as rotation, has scarcely been explored experimentally in low-energy collisions despite their significance to cold and ultracold chemistry. Particularly important to astrochemistry is the case of the most abundant molecule in interstellar space, hydrogen, for which two spin isomers have been detected, one of which exists in its rotational ground state whereas the other is rotationally excited. Here we demonstrate that quantization of molecular rotation plays a key role in cold reaction dynamics, where rotationally excited ortho-hydrogen reacts faster due to a stronger long-range attraction. We observe rotational state-dependent non-Arrhenius universal scaling laws in chemi-ionization reactions of para-H₂ and ortho-H₂ by He(2³ P₂), spanning three orders of magnitude in temperature. Different scaling laws serve as a sensitive gauge that enables us to directly determine the exact nature of the long-range intermolecular interactions. Our results show that the quantum state of the molecular rotor determines whether or not anisotropic long-range interactions dominate cold collisions.

Optimal control theory is a powerful tool for solving control problems in quantum mechanics, ranging from the control of chemical reactions to the implementation of gates in a quantum computer. Gradient-based optimization methods are able to find high fidelity controls, but require considerable numerical effort and often yield highly complex solutions. We propose here to employ a two-stage optimization scheme to significantly speed up convergence and achieve simpler controls. The control is initially parametrized using only a few free parameters, such that optimization in this pruned search space can be performed with a simplex method. The result, considered now simply as an arbitrary function on a time grid, is the starting point for further optimization with a gradient-based method that can quickly converge to high fidelities. We illustrate the success of this hybrid technique by optimizing a holonomic phasegate for two superconducting transmon qubits coupled with a shared transmission line resonator, showing that a combination of Nelder-Mead simplex and Krotov’s method yields considerably better results than either one of the two methods alone.

The first step in the coherent control of a photoinduced binary reaction is bond making or photoassociation. We have recently demonstrated coherent control of bond making in multi-photon femtosecond photoassociation of hot magnesium atoms, using linearly chirped pulses Levin et al., arXiv:1411.1542. The detected yield of photoassociated magnesium dimers was enhanced by positively chirped pulses which is explained theoretically by a combination of purification and chirp-dependent Raman transitions. The yield could be further enhanced by pulse optimization resulting in pulses with an effective linear chirp and a sub-pulse structure, where the latter allows for exploiting vibrational coherences. Here, we systematically explore the efficiency of phase-shaped pulses for the coherent control of bond making, employing a parametrization of the spectral phases in the form of cosine functions. We find up to an order of magnitude enhancement of the yield compared to the unshaped transform-limited pulse. The highly performing pulses all display an overall temporally increasing instantaneous frequency and are composed of several overlapping sub-pulses. The time delay between the first two sub-pulses almost perfectly fits the vibrational frequency of the generated intermediate wavepacket.These findings are in agreement with chirp-dependent Raman transitions and exploitation of vibrational dynamics as underlying control mechanisms.

When the environment of an open quantum system is non-Markovian, amplitude and phase flow not only from the system into the environment but also back. Here we show that this feature can be exploited to carry out quantum control tasks that could not be realized if the system was isolated. Inspired by recent experiments on superconducting phase circuits, we consider an anharmonic ladder with resonant amplitude control only. This restricts realizable operations to SO(N). The ladder is immersed in an environment of two-level systems. Strongly coupled two-level systems lead to non-Markovian effects, whereas the weakly coupled ones result in single-exponential decay. Presence of the environment allows for implementing diagonal unitaries that, together with SO(N), yield the full group SU(N). Using optimal control theory, we obtain errors that are solely T₁-limited.

We combine multi-channel electronic structure theory with quantum optimal control to derive Raman pulse sequences that coherently populate a valence excited state. For a neon atom, Raman target populations of up to 13 percent are obtained. Superpositions of the ground and valence Raman states with a controllable relative phase are found to be reachable with up to 4.5 percent population and phase control facilitated by the pump pulse carrier envelope phase. Our results open a route to creating core-hole excitations in molecules and aggregates that locally address specific atoms and represent the first step towards realization of multidimensional spectroscopy in the xuv and x-ray regimes.

We demonstrate for the first time coherent control of bond making, a milestone on the way to coherent control of photo-induced bimolecular chemical reactions. In strong-field multiphoton femtosecond photoassociation experiments, we find the yield of detected magnesium dimer molecules to be enhanced for positively chirped pulses and suppressed for negatively chirped pulses. Our ab initio model shows that control is achieved by purification via Franck-Condon filtering combined with chirp-dependent Raman transitions. Experimental closed-loop phase optimization using a learning algorithm yields an improved pulse that utilizes vibrational coherent dynamics in addition to chirp-dependent Raman transitions. Our results show that coherent control of binary photo-reactions is feasible even under thermal conditions.

Optimal control theory is a powerful tool for improving figures of merit in quantum information tasks. Finding the solution to any optimal control problem via numerical optimization depends crucially on the choice of the optimization functional. Here, we derive a functional that targets the full set of two-qubit perfect entanglers, gates capable of creating a maximally-entangled state out of some initial product state. The functional depends on easily-computable local invariants and uniquely determines when a gate evolves into a perfect entangler. Optimization with our functional is most useful if the two-qubit dynamics allows for the implementation of more than one perfect entangler. We discuss the reachable set of perfect entanglers for a generic Hamiltonian that corresponds to several quantum information platforms of current interest.

The difficulty of an optimization task in quantum information science depends on the proper mathematical expression of the physical target. Here we demonstrate the power of optimization functionals targeting an arbitrary perfect two-qubit entangler, creating a maximally-entangled state out of some initial product state. For two quantum information platforms of current interest, nitrogen vacancy centers in diamond and superconducting Josephson junctions, we show that an arbitrary perfect entangler can be reached faster and with higher fidelity than specific two-qubit gates or local equivalence classes of two-qubit gates. Our results are obtained with two independent optimization approaches, underlining the crucial role of the optimization target.

We derive a universal model for atom pairs interacting with non-resonant light via the polarizability anisotropy, based on the long range properties of the scattering. The corresponding dynamics can be obtained using a nodal line technique to solve the asymptotic Schrödinger equation. It consists in imposing physical boundary conditions at long range and vanishing of the wavefunction at a position separating inner zone and asymptotic region. We show that nodal lines which depend on the intensity of the non-resonant light can satisfactorily account for the effect of the polarizability at short range. The approach allows to determine the resonance structure, energy, width, channel mixing and hybridization even for narrow resonances.

Non-resonant light interacting with diatomics via the polarizability anisotropy couples different rotational states and may lead to strong hybridization of the motion. The modification of shape resonances and low-energy scattering states due to this interaction can be fully captured by an asymptotic model, based on the long-range properties of the scattering (Crubellier et al., arXiv:1412.0569). Remarkably, the properties of the field-dressed shape resonances in this asymptotic multi-channel description are found to be approximately linear in the field intensity up to fairly large intensity. This suggests a perturbative single-channel approach to be sufficient to study the control of such resonances by the non-resonant field. The multi-channel results furthermore indicate the dependence on field intensity to present, at least approximately, universal characteristics. Here we combine the nodal line technique to solve the asymptotic Schrödinger equation with perturbation theory. Comparing our single channel results to those obtained with the full interaction potential, we find nodal lines depending only on the field-free scattering length of the diatom to yield an approximate but universal description of the field-dressed molecule, confirming universal behavior.

The electronic structure of the (LiYb)⁺ molecular ion is investigated with two variants of the coupled cluster method restricted to single, double, and noniterative or linear triple excitations. Potential energy curves for the ground and excited states, permanent and transition electric dipole movements, and long-range interaction coefficients C₄ and C₆ are reported. The data is subsequently employed in scattering calculations and photoassociation studies. Feshbach resonances are shown to be measurable despite the ion’s micromotion in the Paul trap. Molecular ions can be formed in their singlet electronic ground state by one-photon photoassociation and in triplet states by two-photon photoassociation; and control of cold atom-ion chemistry based on Feshbach resonances should be feasible. Conditions for sympathetic cooling of an Yb⁺ ion by an ultracold gas of Li atoms are found to be favorable in the temperature range of 10 mK to 10 nK; and further improvements using Feshbach resonances should be possible. Overall, these results suggest excellent prospects for building a quantum simulator with ultracold Yb⁺ ions and Li atoms.

The predissociation dynamics of lithium iodide (LiI) in the first excited A-state is investigated for molecules in the gas phase and embedded in helium nanodroplets, using femtosecond pump-probe photoionization spectroscopy. In the gas phase, the transient Li+ and LiI+ ion signals feature damped oscillations due to the excitation and decay of a vibrational wave packet. Based on high-level ab initio calculations of the electronic structure of LiI and simulations of the wave packet dynamics, the exponential signal decay is found to result from predissociation predominantly at the lowest avoided X-A potential curve crossing, for which we infer a coupling constant V=650(20) reciprocal cm. The lack of a pump-probe delay dependence for the case of LiI embedded in helium nanodroplets indicates fast droplet-induced relaxation of the vibrational excitation.

A Ramsey-type interferometer is suggested, employing a cold trapped ion and two time-delayed off-resonant femtosecond laser pulses. The laser light couples to the molecular polarization anisotropy, inducing rotational wavepacket dynamics. An interferogram is obtained from the delay dependent populations of the final field-free rotational states. Current experimental capabilities for cooling and preparation of the initial state are found to yield an interferogram visibility of more than 80 percent. The interferograms can be used to determine the polarizability anisotropy with an accuracy of about ± 2 percent, respectively ± 5 percent, provided the uncertainty in the initial populations and measurement errors are confined to within the same limits.

Controlled phase (CPHASE) gates can in principle be realized with trapped neutral atoms by making use of the Rydberg blockade. Achieving the ultra-high fidelities required for quantum computation with such Rydberg gates is however compromised by experimental inaccuracies in pulse amplitudes and timings, as well as by stray fields that cause fluctuations of the Rydberg levels. We report here a comparative study of analytic and numerical pulse sequences for the Rydberg CPHASE gate that specifically examines the robustness of the gate fidelity with respect to such experimental perturbations. Analytical pulse sequences of both simultaneous and stimulated Raman adiabatic passage (STIRAP) are found to be at best moderately robust under these perturbations. In contrast, optimal control theory is seen to allow generation of numerical pulses that are inherently robust within a predefined tolerance window. The resulting numerical pulse shapes display simple modulation patterns and their spectra contain only one additional frequency beyond the basic resonant Rydberg gate frequencies. Pulses of such low complexity should be experimentally feasible, allowing gate fidelities of order 99.90 - 99.99 to be achievable under realistic experimental conditions.

We study optimal quantum control of the dynamics of trapped Bose-Einstein condensates: The targets are to split a condensate, residing initially in a single well, into a double well, without inducing excitation; and to excite a condensate from the ground to the first excited state of a single well. The condensate is described in the mean-field approximation of the Gross-Pitaevskii equation. We compare two optimization approaches in terms of their performance and ease of use, namely gradient ascent pulse engineering (GRAPE) and Krotov’s method. Both approaches are derived from the variational principle but differ in the way the control is updated, additional costs are accounted for, and second order derivative information can be included. We find that GRAPE produces smoother control fields and works in a black-box manner, whereas Krotov with a suitably chosen step size parameter converges faster but can produce sharp features in the control fields.

For qubits, Monte Carlo estimation of the average fidelity of Clifford unitaries is efficient — it requires a number of experiments that is independent of the number n of qubits and classical computational resources that scale only polynomially in n. Here, we identify the requirements for efficient Monte Carlo estimation and the corresponding properties of the measurement operator basis when replacing two-level qubits by p-level qudits. Our analysis illuminates the intimate connection between mutually unbiased measurements and the existence of unitaries that can be characterized efficiently. It allows us to propose a ’hierarchy’ of generalizations of the standard Pauli basis from qubits to qudits according to the associated scaling of resources required in Monte Carlo estimation of the average fidelity.

For certain quantum operations acting on qubits, there exist bases of measurement operators such that estimating the average fidelity becomes efficient. The number of experiments required is then independent of system size and the classical computational resources scale only polynomially in the number of qubits. Here we address the question of how to optimally choose the measurement basis for efficient gate characterization when replacing two-level qubits by d-level qudits. We define optimality in terms of the maximal number of unitaries that can be efficiently characterized. Our definition allows us to construct the optimal measurement basis in terms of their spectra and eigenbases: The measurement operators are unitaries with d-nary spectrum and partition into d+1 Abelian groups whose eigenbases are mutually unbiased.

The efficient initialization of a quantum system is a prerequisite for quantum technological applications. Here we show that several classes of quantum states of a harmonic oscillator can be efficiently prepared by means of a Jaynes-Cummings interaction with a single two-level system. This is achieved by suitably tailoring external fields which drive the dipole and/or the oscillator. The time-dependent dynamics that leads to the target state is identified by means of Optimal Control Theory (OCT) based on Krotov’s method. Infidelities below 10⁻⁴ can be reached for the parameters of the experiment of the ENS group in Paris, where the oscillator is a mode of a high-Q microwave cavity and the dipole is a Rydberg transition of an atom. For this specific situation we analyze the limitations on the fidelity due to parameter fluctuations and identify robust dynamics based on pulses found using ensemble OCT. Our analysis can be extended to quantum-state preparation of continuous-variable systems in other platforms, such as trapped ions and circuit QED.

We investigate the performance of different control techniques for ion transport in state-of-the-art segmented miniaturized ion traps. We employ numerical optimization of classical trajectories and quantum wavepacket propagation as well as analytical solutions derived from invariant based inverse engineering and geometric optimal control. We find that accurate shuttling can be performed with operation times below the trap oscillation period. The maximum speed is limited by the maximum acceleration that can be exerted on the ion. When using controls obtained from classical dynamics for wavepacket propagation, wavepacket squeezing is the only quantum effect that comes into play for a large range of trapping parameters. We show that this can be corrected by a compensating force derived from invariant based inverse engineering, without a significant increase in the operation time.

We show that optimizing a quantum gate for an open quantum system requires the time evolution of only three states irrespective of the dimension of Hilbert space. This represents a significant reduction in computational resources compared to the complete basis of Liouville space that is commonly believed necessary for this task. The reduction is based on two observations: The target is not a general dynamical map but a unitary operation; and the time evolution of two properly chosen states is sufficient to distinguish any two unitaries. We illustrate gate optimization employing a reduced set of states for a controlled phasegate with trapped atoms as qubit carriers and a √iSWAP gate with superconducting qubits.

Magnetically tunable Feshbach resonances for polar paramagnetic ground-state diatomics are too narrow to allow for magnetoassociation starting from trapped, ultracold atoms. We show that non-resonant light can be used to engineer the Feshbach resonances in their position and width. For non-resonant field strengths of the order of 10⁹ W/cm², we find the width to be increased by three orders of magnitude, reaching a few Gauss. This opens the way for producing ultracold molecules with sizeable electric and magnetic dipole moments and thus for many-body quantum simulations with such particles.

We apply two recent generalizations of monotonically convergent optimization algorithms to the control of molecular orientation by laser fields. We show how to minimize the control duration by a step-wise optimization and maximize the field-free molecular orientation using state-dependent constraints. We discuss the physical relevance of the different results.

Shaped pulses obtained by optimal control theory often possess unphysically broad spectra. In principle, the spectral width of a pulse can be restricted by an additional constraint in the optimization functional. However, it has so far been impossible to impose spectral constraints while strictly guaranteeing monotonic convergence. Here, we show that Krotov’s method allows for simultaneously imposing temporal and spectral constraints without perturbing monotonic convergence, provided the constraints can be expressed as positive semi-definite quadratic forms. The optimized field is given by an integral equation which can be solved efficiently using the method of degenerate kernels. We demonstrate that Gaussian filters suppress undesired frequency components in the control of non-resonant two-photon absorption.

Laser cooling of molecules employing broadband optical pumping involves a timescale separation between laser excitation and spontaneous emission. Here, we optimize the optical pumping step using shaped laser pulses. We derive two optimization functionals to drive population into those excited state levels that have the largest spontaneous emission rates to the target state. We show that, when using optimal control, laser cooling of molecules works even if the Franck-Condon map governing the transitions is preferential to heating rather than cooling. Our optimization functional is also applicable to the laser cooling of other degrees of freedom provided the cooling cycle consists of coherent excitation and dissipative deexcitation steps whose timescales are separated.

We show that the minimum experimental effort to characterize the proper functioning of a quantum device scales as 2**n for n qubits and requires classical computational resources ~ n**2 2**3n. This represents an exponential reduction compared to the best currently available protocol, Monte Carlo characterization. The reduction comes at the price of either having to prepare entangled input states or obtaining bounds rather than the average fidelity itself. It is achieved by applying Monte Carlo sampling to so-called two-designs or two classical fidelities. For the specific case of Clifford gates, the original version of Monte Carlo characterization based on the channel-state isomorphism remains an optimal choice. We provide a classification of the available efficient strategies for device characterization in terms of the number of required experimental settings, average number of actual measurements and classical computational resources.

We show how additional constraints, restricting the spectrum of the optimized pulse or confining the system dynamics, can be used to steer optimization in quantum control towards distinct solutions. Our examples are multi-photon excitation in atoms and vibrational population transfer in molecules. We show that a spectral constraint is most effective in enforcing non-resonant two-photon absorption pathways in atoms and avoiding unnecessarily broad spectra in Raman transitions in molecules. While a constraint restricting the system to stay in an allowed subspace is also capable of identifying non-resonant excitation pathways, it does not avoid spurious peaks in the pulse spectrum. Both constraints are compatible with monotonic convergence but imply different additional numerical costs.

Two-photon photoassociation of hot magnesium atoms by femtosecond laser pulses, creating electronically excited magnesium dimer molecules, is studied from first principles, combining ab initio quantum chemistry and molecular quantum dynamics. This theoretical framework allows for rationalizing the generation of molecular rovibrational coherence from thermally hot atoms L. Rybak, S. Amaran, L. Levin, M. Tomza, R. Moszynski, R. Kosloff, C. P. Koch, and Z. Amitay, Phys. Rev. Lett.107, 273001 (2011). Random phase thermal wavefunctions are employed to model the thermal ensemble of hot colliding atoms. Comparing two different choices of basis functions, random phase wavefunctions built from eigenstates are found to have the fastest convergence for the photoassociation yield. The interaction of the colliding atoms with a femtosecond laser pulse is modeled non-perturbatively to account for strong-field effects.

We derive an algebraic framework which identifies the minimal information required to assess how well a quantum device implements a desired quantum operation. Our approach is based on characterizing only the unitary part of an open system’s evolution. We show that a reduced set of input states is sufficient to estimate the average fidelity of a quantum gate, avoiding a sampling over the full Liouville space. Surprisingly, the minimal set consists of only two input states, independent of the Hilbert space dimension. The minimal set is, however, impractical for device characterization since one of the states is a totally mixed thermal state and extracting bounds for the average fidelity is impossible. We therefore present two further reduced sets of input states that allow, respectively, for numerical and analytical bounds on the average fidelity.

We show that time induces a dynamical renormalization of the system-environment coupling in open-quantum-system dynamics. The renormalizability condition, of the interactions being either local, or, alternatively, defined on a finite continuum support, is generally fulfilled for both discrete and continuous environments. As a consequence, we find a generalized Lieb-Robinson bound to hold for local and, surprisingly, also for nonlocal interactions. This unified picture allows us to devise a controllable approximation for arbitrary non-Markovian dynamics with an a priori estimate of the worst-case computational cost.

In this paper we formulate the theory of the interaction of a diatomic linear molecule in a spatially degenerate state with the non-resonant laser field and of the rovibrational dynamics in the presence of the field. We report on ab initio calculations employing the double electron attachment intermediate Hamiltonian Fock space coupled cluster method restricted to single and double excitations for all electronic states of the Rb₂ molecule up to 5s+5d dissociation limit of about 26.000 cm⁻¹. In order to correctly predict the spectroscopic behavior of Rb₂, we have also calculated the electric transition dipole moments, non-adiabatic coupling and spin-orbit coupling matrix elements, and static dipole polarizabilities, using the multireference configuration interaction method. When a molecule is exposed to a strong non-resonant light, its rovibrational levels get hybridized. We study the spectroscopic signatures of this effect for transitions between the X¹Σg⁺ electronic ground state and the A¹Σu⁺ and b³Πu excited state manifold. The latter is characterized by strong perturbations due to the spin-orbit interaction. We find that for non-resonant field strengths of the order 10⁹ W/cm², the spin-orbit interaction and coupling to the non-resonant field become comparable. The non-resonant field can then be used to control the singlet-triplet character of a rovibrational level.

We apply two different monotonically convergent optimization algorithms to the control of molecular rotational dynamics by laser pulses. This example represents a quantum control problem where the interaction of the system with the external field is nonlinear. We test the validity and accuracy of the two methods on the key control targets of producing molecular orientation and planar delocalization at zero temperature, and maximizing permanent alignment at nonzero temperature.

Photoassociation, assembling molecules from atoms using laser light, is limited by the low density of atom pairs at sufficiently short interatomic separations. Here we show that nonresonant light with intensities of the order of 10¹⁰ W/cm2 modifies the thermal cloud of atoms, enhancing the Boltzmann weight of shape resonances and pushing scattering states below the dissociation limit. This leads to an enhancement of photoassociation rates by several orders of magnitude and opens the way to significantly larger numbers of ground-state molecules in a thermal ensemble than achieved so far.

We discuss the production of ultracold molecules in their electronic ground state by photoassociation employing electronically excited states with ion-pair character and strong spin-orbit interaction. A short photoassociation laser pulse drives a non-resonant three-photon transition for alkali atoms colliding in their lowest triplet state. The excited state wave packet is transferred to the ground electronic state by a second laser pulse, driving a resonant two-photon transition. After analyzing the transition matrix elements governing the stabilization step, we discuss the efficiency of population transfer using transform-limited and linearly chirped laser pulses. Finally, we employ optimal control theory to find the most efficient stabilization pathways. We find that the stabilization efficiency can be increased by one and two orders of magnitude for linearly chirped and optimally shaped laser pulses, respectively.

State-of-the-art ab initio techniques have been applied to compute the potential energy curves for the electronic states in the A¹Σu+, c³Πu, and a³Σu⁺ manifold of the strontium dimer, the spin-orbit and nonadiabatic coupling matrix elements between the states in the manifold, and the electric transition dipole moment from the ground X¹Σg⁺ to the nonrelativistic and relativistic states in the A+c+a manifold. The potential energy curves and transition moments were obtained with the linear response (equation of motion) coupled cluster method limited to single, double, and linear triple excitations for the potentials and limited to single and double excitations for the transition moments. The spin-orbit and nonadiabatic coupling matrix elements were computed with the multireference configuration interaction method limited to single and double excitations. Our results for the nonrelativistic and relativistic (spin-orbit coupled) potentials deviate substantially from recent ab initio calculations. The potential energy curve for the spectroscopically active (1)0u⁺ state is in quantitative agreement with the empirical potential fitted to high-resolution Fourier transform spectra A. Stein, H. Knoeckel, and E. Tiemann, Eur. Phys. J. D 64, 227 (2011). The computed ab initio points were fitted to physically sound analytical expressions, and used in converged coupled channel calculations of the rovibrational energy levels in the A+c+a manifold and line strengths for the A^1\Sigma_u^+ <-- X^1\Sigma_g^+ transitions. Positions and lifetimes of quasi-bound Feshbach resonances lying above the ^1S + ^3P_1 dissociation limit were also obtained. Our results reproduce (semi)quantitatively the experimental data observed thus far. Predictions for on-going and future experiments are also reported.

We predict feasibility of the photoassociative formation of Sr₂ molecules in arbitrary vibrational levels of the electronic ground state based on state-of-the-art ab initio calculations. Key is the strong spin-orbit interaction between the c³Πu, A¹Σu⁺ and B¹Σu⁺ states. It creates not only an effective dipole moment allowing free-to-bound transitions near the 1S + 3P₁ intercombination line but also facilitates bound-to-bound transitions via resonantly coupled excited state rovibrational levels to deeply bound rovibrational levels of the ground X¹Σg⁺ potential, with v" as low as v"=6. The spin-orbit interaction is responsible for both optical pathways. Therefore, those excited state levels that have the largest bound-to-bound transition moments to deeply bound ground state levels also exhibit a sufficient photoassociation probability, comparable to that of the lowest weakly bound excited state level previously observed by Zelevinsky et al. Phys. Rev. Lett. 96, 203201 (2006). Our study paves the way for an efficient photoassociative production of Sr_2 molecules in ground state levels suitable for experiments testing the electron-to-proton mass ratio.

We consider the optimal control problem of transferring population between states of a quantum system where the coupling proceeds only via intermediate states that are subject to decay. We pose the question whether it is generally possible to carry out this transfer. For a single intermediate decaying state, we recover the Stimulated Raman Adiabatic Passage (STIRAP) process for which we present analytic solutions in the finite time case. The solutions yield perfect state transfer only in the limit of infinite time. We also present analytic solutions for the case of transfer that has to proceed via two consecutive intermediate decaying states. We show that in this case, for finite power the optimal control does not approach perfect state transfer even in the infinite time limit. Our four-level results agree with those of Khaneja et al. J. Magnet. Reson. 162, 311 (2003) derived in a different way. We generalize our findings to characterize the topologies of paths that can be achieved by coherent control.

We apply the optimization algorithm developed by Konnov and Krotov Automation and Remote Control 60, 1427 (1999) to quantum control problems. Using a second order construction, we derive a class of monotonically convergent optimization algorithms. We show that for most quantum control problems, the second order contribution can be straightforwardly estimated since optimization is performed over compact sets of candidate states. Generally, quantum control problems can be classified according to the optimization functionals, equations of motion and dependency of the Hamiltonian on the control. For each problem class, we outline the resulting monotonically convergent algorithm. While a second order construction is necessary to ensure monotonic convergence in general, for the ’standard’ quantum control problem of a convex final-time functional, linear equations of motion and linear dependency of the Hamiltonian on the field, both first and second order algorithms converge monotonically. We compare convergence behavior and performance of first and second order algorithms for two generic optimization examples.

The formation of diatomic molecules with rotational and vibrational coherence is demonstrated experimentally in free-to-bound two-photon femtosecond photoassociation of hot atoms. In a thermal gas at a temperature of 1000 K, pairs of magnesium atoms, colliding in their electronic ground state, are excited into coherent superpositions of bound rovibrational levels in an electronically excited state. The rovibrational coherence is probed by a time-delayed third photon, resulting in quantum beats in the UV fluorescence. A comprehensive theoretical model based on ab initio calculations rationalizes the generation of coherence by Franck-Condon filtering of collision energies and partial waves, quantifying it in terms of an increase in quantum purity of the thermal ensemble. Our results open the way to coherent control of a binary reaction.

We investigate femtosecond photoassociation of thermally hot atoms in the gas phase and its coherent control. In the photoassociation process, formation of a chemical bond is facilitated by light in a free-to-bound optical transition. Here, we study free-to-bound photoassociation of a diatomic molecule induced by femtosecond pulses exciting a pair of scattering atoms interacting via the van-der-Waals-type electronic ground state potential into bound levels of an electronically excited state. The thermal gas of reactants is at temperatures in the range of hundreds of degrees. Despite this incoherent initial state, rotational and vibrational coherences are observed in the probing of the created Mg2 molecules.

Atom chips are a promising candidate for a scalable architecture for quantum information processing provided a universal set of gates can be implemented with high fidelity. The difficult part in achieving universality is the entangling two-qubit gate. We consider a Rydberg phase gate for two atoms trapped on a chip and employ optimal control theory to find the shortest gate that still yields a reasonable gate error. Our parameters correspond to a situation where the Rydberg blockade regime is not yet reached. We discuss the role of spontaneous emission and the effect of noise from the chip surface on the atoms in the Rydberg state.

State-of-the-art ab initio techniques have been applied to compute the potential energy curves for the SrYb molecule in the Born-Oppenheimer approximation for the ground state and first fifteen excited singlet and triplet states within the coupled-cluster framework. The leading long-range coefficients describing the dispersion interactions at large interatomic distances are also reported. The electric transition dipole moments have been obtained as the first residue of the polarization propagator computed with the linear response coupled-cluster method restricted to single and double excitations. Spin-orbit coupling matrix elements have been evaluated using the multireference configuration interaction method restricted to single and double excitations with a large active space. The electronic structure data was employed to investigate the possibility of forming deeply bound ultracold SrYb molecules in an optical lattice in a photoassociation experiment using continuous-wave lasers. Photoassociation near the intercombination line transition of atomic strontium into the vibrational levels of the strongly spin-orbit mixed b³Σ⁺, a³Π, A¹Π, and C¹Π states with subsequent efficient stabilization into the v’’=1 vibrational level of the electronic ground state is proposed. Ground state SrYb molecules can be accumulated by making use of collisional decay from v’’=1 to v’’=0. Alternatively, photoassociation and stabilization to v’’=0 can proceed via stimulated Raman adiabatic passage provided that the trapping frequency of the optical lattice is large enough and phase coherence between the pulses can be maintained over at least tens of microseconds.

Optimal control theory is a versatile tool that presents a route to significantly improving figures of merit for quantum information tasks. We combine it here with the geometric theory for local equivalence classes of two-qubit operations to derive an optimization algorithm that determines the best entangling two-qubit gate for a given physical setting. We demonstrate the power of this approach for trapped polar molecules and neutral atoms.

We study controlled phasegates for ultracold atoms in an optical potential. A shaped laser pulse drives transitions between the ground and electronically excited states where the atoms are subject to a long-range 1/R³ interaction. We fully account for this interaction and use optimal control theory to calculate the pulse shapes. This allows us to determine the minimum pulse duration, respectively, gate time T that is required to obtain high fidelity. We accurately analyze the speed limiting factors, and we find the gate time to be limited either by the interaction strength in the excited state or by the ground state vibrational motion in the trap. The latter needs to be resolved by the pulses in order to fully restore the motional state of the atoms at the end of the gate.

A (diatomic) shape resonance is a metastable state of a pair of colliding atoms quasi-bound by the centrifugal barrier imposed by the angular momentum involved in the collision. The temporary trapping of the atoms’ scattering wavefunction corresponds to an enhanced atom pair density at low interatomic separations. This leads to larger overlap of the wavefunctions involved in a molecule formation process such as photoassociation, rendering the process more efficient. However, for an ensemble of atoms, the atom pair density will only be enhanced if the energy of the resonance comes close to the temperature of the atomic ensemble. Herein we explore the possibility of controlling the energy of a shape resonance by shifting it toward the temperature of atoms confined in a trap. The shifts are imparted by the interaction of non-resonant light with the anisotropic polarizability of the atom pair, which affects both the centrifugal barrier and the pair’s rotational and vibrational levels. We find that at laser intensities of up to 5× 10⁹ W/cm² the pair density is increased by one order of magnitude for ⁸⁷Rb atoms at 100 μK and by two orders of magnitude for ⁸⁸Sr atoms at 20 μK.

The transfer of weakly bound KRb molecules from levels just below the dissociation threshold into the vibrational ground state with shaped laser pulses is studied. Optimal control theory is employed to calculate the pulses. The complexity of modelling the molecular structure is successively increased in order to study the effects of the long-range behavior of the excited state potential, resonant spin-orbit coupling and singlet-triplet mixing.

Two atoms in an ultracold gas are correlated at short inter-atomic distances due to threshold effects where the potential energy of their interaction dominates the kinetic energy. The correlations manifest themselves in a distinct nodal structure of the density matrix at short inter-atomic distances. Pump-probe spectroscopy has recently been suggested Phys. Rev. Lett. 103, 260401 (2009) to probe these pair correlations: A suitably chosen, short photoassociation laser pulse depletes the ground state pair density within the photoassociation window, creating a non-stationary wave packet in the electronic ground state. The dynamics of this non-stationary wave packet is monitored by time-delayed probe and ionization pulses. Here, we discuss how the choice of the pulse parameters affects experimental feasibility of this pump-probe spectroscopy of two-body correlations.

A propagation method for time-dependent Schrödinger equations with an explicitly time-dependent Hamiltonian is developed where time ordering is achieved iteratively. The explicit time-dependence of the time-dependent Schrödinger equation is rewritten as an inhomogeneous term. At each step of the iteration, the resulting inhomogeneous Schrödinger equation is solved with the Chebychev propagation scheme presented in J. Chem. Phys. 130, 124108 (2009). The iteratively time-ordering Chebychev propagator is shown to be robust, efficient and accurate and compares very favorably to all other available propagation schemes.

We suggest pump-probe spectroscopy to study pair correlations that determine the many-body dynamics in weakly interacting, dilute ultracold gases. A suitably chosen, short laser pulse depletes the pair density locally, creating a ’hole’ in the electronic ground state. The dynamics of this non-stationary pair density is monitored by a time-delayed probe pulse. The resulting transient signal allows to spectrally decompose the ’hole’ and to map out the pair correlation functions.

Photoassociation of ultracold rubidium atoms with femtosecond laser pulses is studied theoretically. The spectrum of the pulses is cut off in order to suppress pulse amplitude at and close to the atomic resonance frequency. This leads to long tails of the laser pulse as a function of time giving rise to coherent transients in the photoassociation dynamics. They are studied as a function of cut-off position and chirp of the pulse. Molecule formation in the electronically excited state is attributed to off-resonant excitation in the strong-field regime.

We investigate the interaction of femtosecond laser pulses with an ensemble of ultracold rubidium atoms by applying shaped excitation pulses with two different types of spectral filtering. Although the pulses, which are frequency filtered with a high pass, have no spectral overlap with molecular states, we observe coherent molecular transients. Similar transients obtained with nearly transform-limited pulses, where only the atomic resonance is removed, reveal two differing oscillatory components. The resulting transients are compared among themselves and supported with quantum dynamical simulations which indicate a photoassociation process. The effect is due to the strong field interaction of the pulse with the colliding atom pair.

We present spectrally resolved pump-probe experiments on the photoassociation of ultracold rubidium atoms with shaped ultrashort laser pulses. The pump pulse causes a free-bound transition leading to a coherent transient signal of rubidium molecules in the first excited state. In order to achieve a high frequency resolution the bandwidth of the pump pulse is reduced to a few wavenumbers. The frequency dependence of the transient signal close to the D1 atomic resonance is investigated for characteristic pump-probe delay times. The observed spectra, which show a pronounced dip for pump-probe coincidence, are interpreted using quantum dynamical calculations.

We investigate the possibility of forming deeply bound ultracold RbCs molecules by a two-color photoassociation experiment. We compare the results with those for Rb₂ in order to understand the characteristic differences between heteronuclear and homonuclear molecules. The major differences arise from the different long-range potential for excited states. Ultracold 85Rb and 133Cs atoms colliding on the X1Sigma+ potential curve are initially photoassociated to form excited RbCs molecules in the region below the Rb(5S) + Cs(6P1/2) asymptote. We explore the nature of the Omega=0⁺ levels in this region, which have mixed A1Sigma⁺ and b3Pi character. We then study the quantum dynamics of RbCs by a time-dependent wavepacket (TDWP) approach. A wavepacket is formed by exciting a few vibronic levels and is allowed to propagate on the coupled electronic potential energy curves. For a detuning of 7.5 cm-1, the wavepacket for RbCs reaches the short-range region in about 13 ps, which is significantly faster than for the homonuclear Rb₂ system; this is mostly because of the absence of an R⁻³ long-range tail in the excited-state potential curves for heteronuclear systems. We give a simple semiclassical formula that relates the time taken to the long-range potential parameters. For RbCs, in contrast to Rb2, the excited-state wavepacket shows a substantial peak in singlet density near the inner turning point, and this produces a significant probability of deexcitation to form ground-state molecules bound by up to 1500 cm-1. Our analysis of the role of spin-orbit coupling concerns the character of the mixed states in general and is important for both photoassociation and stimulated Raman deexcitation.

Photoassociation with short laser pulses has been proposed as a technique to create ultracold ground state molecules. A broad-band excitation seems the natural choice to drive the series of excitation and deexcitation steps required to form a molecule in its vibronic ground state from two scattering atoms. First attempts at femtosecond photoassociation were, however, hampered by the requirement to eliminate the atomic excitation leading to trap depletion. On the other hand, molecular levels very close to the atomic transition are to be excited. The broad bandwidth of a femtosecond laser then appears to be rather an obstacle. To overcome the ostensible conflict of driving a narrow transition by a broad-band laser, we suggest a two-photon photoassociation scheme. In the weak-field regime, a spectral phase pattern can be employed to eliminate the atomic line. When the excitation is carried out by more than one photon, different pathways in the field can be interfered constructively or destructively. In the strong-field regime, a temporal phase can be applied to control dynamic Stark shifts. The atomic transition is suppressed by choosing a phase which keeps the levels out of resonance. We derive analytical solutions for atomic two-photon dark states in both the weak-field and strong-field regime. Two-photon excitation may thus pave the way toward coherent control of photoassociation. Ultimately, the success of such a scheme will depend on the details of the excited electronic states and transition dipole moments. We explore the possibility of two-photon femtosecond photoassociation for alkali and alkaline-earth metal dimers and present a detailed study for the example of calcium.

A propagation scheme for time-dependent inhomogeneous Schrödinger equations is presented. Such equations occur in time dependent optimal control theory and in reactive scattering. A formal solution based on a polynomial expansion of the inhomogeneous term is derived. It is subjected to an approximation in terms of Chebychev polynomials. Different variants for the inhomogeneous propagator are demonstrated and applied to two examples from optimal control theory. Convergence behavior and numerical efficiency are analyzed.

Optical Feshbach resonances Phys. Rev. Lett. 94, 193001 (2005) and pump-dump photoassociation with short laser pulses Phys. Rev. A 73, 033408 (2006) have been proposed as means to coherently form stable ultracold alkali dimer molecules. In an optical Feshbach resonance, the intensity and possibly frequency of a cw laser are ramped up linearly followed by a sudden switch-off of the laser. This is applicable to tightly trapped atom pairs. In short-pulse photoassociation, the pump pulse forms a wave-packet in an electronically excited state. The ensuing dynamics carry the wave-packet to shorter internuclear distances where, after half a vibrational period, it can be deexcited to the electronic ground state by the dump pulse. Short-pulse photoassociation is suited for both shallow and tight traps. The applicability of these two means to produce ultracold molecules is investigated here for $^88$Sr. Dipole-allowed transitions proceeding via the $B^1\Sigma_u^+$ excited state as well as transitions near the intercombination line are studied.

We propose an improved photoassociation scheme to produce ultracold molecules in their vibronic ground state for the generic case where non-adiabatic effects facilitating transfer to deeply bound levels are absent. Formation of molecules is achieved by short laser pulses in a Raman-like pump-dump process where an additional near-infrared laser field couples the excited state to an auxiliary state. The coupling due to the additional field effectively changes the shape of the excited state potential and allows for efficient population transfer to low-lying vibrational levels of the electronic ground state. Repetition of many pump-dump sequences together with collisional relaxation allows for accumulation of molecules in v=0.

Optimal control theory is developed for the task of obtaining a primary objective in a subspace of the Hilbert space while avoiding other subspaces of the Hilbert space. The primary objective can be a state-to-state transition or a unitary transformation. A new optimization functional is introduced which leads to monotonic convergence of the algorithm. This approach becomes necessary for molecular systems subject to processes implying loss of coherence such as predissociation or ionization. In these subspaces controllability is hampered or even completely lost. Avoiding the lossy channels is achieved via a functional constraint which depends on the state of the system at each instant in time. We outline the resulting new algorithm, discuss its convergence properties and demonstrate its functionality for the example of a state-to-state transition and of a unitary transformation for a model of cold Rb2.

We have studied the effect of resonant electronic state coupling on the formation of ultracold ground-state ⁸⁵Rb₂. Ultracold Rb₂ molecules are formed by photoassociation (PA) to a coupled pair of 0u⁺ states, 0u+(P1/2) and 0u+(P3/2), in the region below the 5S+5P1/2 limit. Subsequent radiative decay produces high vibrational levels of the ground state, X ¹Σg⁺. The population distribution of these X state vibrational levels is monitored by resonance-enhanced two-photon ionization through the 2 ¹Σu⁺ state. We find that the populations of vibrational levels v‘=112-116 are far larger than can be accounted for by the Franck-Condon factors for 0u+(P1/2) \to X ¹Σg⁺ transitions with the 0u+(P1/2) state treated as a single channel. Further, the ground-state molecule population exhibits oscillatory behavior as the PA laser is tuned through a succession of 0u⁺ state vibrational levels. Both of these effects are explained by a new calculation of transition amplitudes that includes the resonant character of the spin-orbit coupling of the two 0u⁺ states. The resulting enhancement of more deeply bound ground-state molecule formation will be useful for future experiments on ultracold molecules.

The total number of molecules produced in a pulsed photoassociation of ultracold atoms is a crucial link between theory and experiment. A calculation based on first principles can determine the experimental feasibility of a pulsed photoassociation scheme. The calculation method considers an initial thermal ensemble of atoms. This ensemble is first decomposed into a representation of partial spherical waves. The photoassociation dynamics is calculated by solving the multichannel time-dependent Schrödinger equation on a mapped grid. The molecules are primarily assembled in a finite region of internuclear distances, the ’photoassociation window’. The ensemble average was calculated by adding the contributions from initial scattering states confined to a finite volume. These states are Boltzmann averaged where the partition function is summed numerically. Convergence is obtained for a sufficiently large volume. The results are compared to a thermal averaging procedure based on scaling laws which leads to a single representative initial partial wave which is sufficient to represent the density in the ’photoassociation window’. For completeness a third high-temperature thermal averaging procedure is described which is based on random phase thermal Gaussian initial states. The absolute number of molecules in the two first calculation methods agree to within experimental error for photoassociation with picosecond pulses for a thermal ensemble of rubidium or caesium atoms in ultracold conditions.

We investigate pump-dump photoassociation of ultracold molecules with amplitude- and phase-modulated femtosecond laser pulses. For this purpose a perturbative model for the light-matter interaction is developed and combined with a genetic algorithm for adaptive feedback control of the laser pulse shapes. The model is applied to the formation of 85Rb2 molecules in a magneto-optical trap. We find for optimized pulse shapes an improvement for the formation of ground state molecules by more than a factor of 10 compared to unshaped pulses at the same pump-dump delay time, and by 40 compared to unshaped pulses at the respective optimal pump-dump delay time. Since our model yields directly the spectral amplitudes and phases of the optimized pulses, the results are directly applicable in pulse shaping experiments.

Photoassociation of two ultracold rubidium atoms and the subsequent formation of stable molecules in the singlet ground and lowest triplet states is investigated theoretically. The method employs laser pulses inducing transitions via excited states correlated to the 5S+5P1/2 asymptote. Weakly bound molecules in the singlet ground or lowest triplet state can be created by a single pulse while the formation of more deeply bound molecules requires a two-color pump-dump scenario. More deeply bound molecules in the singlet ground or lowest triplet state can be produced only if efficient mechanisms for both pump and dump steps exist. While long-range 1/R³-potentials allow for efficient photoassociation, stabilization is facilitated by the resonant spin-orbit coupling of the 0u⁺ states. Molecules in the singlet ground state bound by a few wavenumbers can thus be formed. This provides a promising first step toward ground state molecules which are ultracold in both translational and vibrational degrees of freedom.

This theoretical paper investigates the formation of ground state molecules from ultracold cesium atoms in a two-color scheme. Following previous work on photoassociation with chirped picosecond pulses Luc-Koenig et al., Phys. Rev. A 70, 033414 (2004), we investigate stabilization by a second (dump) pulse. By appropriately choosing the dump pulse parameters and time delay with respect to the photoassociation pulse, we show that a large number of deeply bound molecules are created in the ground triplet state. We discuss (i) broad-bandwidth dump pulses which maximize the probability to form molecules while creating a broad vibrational distribution as well as (ii) narrow-bandwidth pulses populating a single vibrational ground state level, bound by 113 cm$^{-1}$. The use of chirped pulses makes the two-color scheme robust, simple and efficient.

The photodesorption of NO from NiO(100) is studied from first principles, with electronic relaxation treated by the use of the surrogate Hamiltonian approach. Two nuclear degrees of freedom of the adsorbate-substrate system are taken into account. To perform the quantum dynamical wave-packet calculations, a massively parallel implementation with a one-dimensional data decomposition had to be introduced. The calculated desorption probabilities and velocity distributions are in qualitative agreement with experimental data. The results are compared to those of stochastic wave-packet calculations where a sufficiently large number of quantum trajectories is propagated within a jumping wave-packet scenario.

We propose to create ultracold ground state molecules in an atomic Bose-Einstein condensate by adiabatic crossing of an optical Feshbach resonance. We envision a scheme where the laser intensity and possibly also frequency are linearly ramped over the resonance. Our calculations for ⁸⁷Rb show that for sufficiently tight traps it is possible to avoid spontaneous emission while retaining adiabaticity, and conversion efficiencies of up to 50 can be expected.

The dissipative quantum dynamics of an anharmonic oscillator coupled to a bath is studied with the purpose of elucidating the differences between the relaxation to a spin bath and to a harmonic bath. Converged results are obtained for the spin bath by the Surrogate Hamiltonian approach. This method is based on constructing a system-bath Hamiltonian, with a finite but large number of spin bath modes, that mimics exactly a bath with an infinite number of modes for a finite time interval. Convergence with respect to the number of simultaneous excitations of bath modes can be checked. The results are compared to calculations that include a finite number of harmonic modes carried out by using the multi-configuration time-dependent Hartree method of Nest and Meyer, J. Chem. Phys. 119, 24 (2003). In the weak coupling regime, at zero temperature and for small excitations of the primary system, both methods converge to the Markovian limit. When initially the primary system is significantly excited, the spin bath can saturate restricting the energy acceptance. An interaction term between bath modes that spreads the excitation eliminates the saturation. The loss of phase between two cat states has been analyzed and the results for the spin and harmonic baths are almost identical. For stronger couplings, the dynamics induced by the two types of baths deviate. The accumulation and degree of entanglement between the bath modes have been characterized. Only in the spin bath the dynamics generate entanglement between the bath modes.

In recent experiments on ultracold matter, molecules have been produced from ultracold atoms by photoassociation, Feshbach resonances, and three-body recombination. The created molecules are translationally cold, but vibrationally highly excited. This will eventually lead them to be lost from the trap due to collisions. We propose shaped laser pulses to transfer these highly excited molecules to their ground vibrational level. Optimal control theory is employed to find the light field that will carry out this task with minimum intensity. We present results for the sodium dimer. The final target can be reached to within 99 if the initial guess field is physically motivated. We find that the optimal fields contain the transition frequencies required by a good Franck-Condon pumping scheme. The analysis is able to identify the ranges of intensity and pulse duration which are able to achieve this task before other competing process take place. Such a scheme could produce stable ultracold molecular samples or even stable molecular Bose-Einstein condensates.

A microscopic model for electronic quenching in the photodesorption of NO from NiO(100) is developed. The quenching is caused by the interaction of the excited adsorbate–substrate complex with electron hole pairs (O 2p to Ni 3d states) in the surface. The electron hole pairs are described as a bath of two level systems which are characterized by an excitation energy and a dipole charge. The parameters are connected to estimates from photoemission spectroscopy and configuration interaction calculations. Due to the localized electronic structure of NiO a direct optical excitation mechanism can be assumed, and a reliable potential energy surface for the excited state is available. Thus a treatment of all steps in the photodesorption event from first principles becomes possible for the first time. The surrogate Hamiltonian method, which allows one to monitor convergence, is employed to calculate the desorption dynamics. Desorption probabilities of the right order of magnitude and velocities in the experimentally observed range are obtained

A microscopic model for the excitation and relaxation processes in photochemistry at surfaces is developed. Our study is based on ab initio calculations and the surrogate Hamiltonian method treating surface electron-hole pairs as a bath of two-level systems. Desorption probabilities and velocities in the experimentally observed range are obtained. The excited state lifetime is calculated, and a dependence of observables on pulse length is predicted.

An ultrafast photoinduced charge transfer event in condensed phase is simulated. The interaction with the field is treated explicitly within a time-dependent framework. The description of the interaction of the system with its environment is based on the surrogate Hamiltonian method where the infinite number of degrees of freedom of the environment is approximated by a finite set of two-level modes for a limited time. This method is well suited to ultrafast events, since it is not limited by weak coupling between system and environment. Moreover, the influence of the external field on the system-bath coupling is included naturally. The surrogate Hamiltonian method is generalized to incorporate two electronic states including all possible system-bath interactions. The method is applied to a description of a pump-probe experiment where every step of the cycle is treated consistently. Dynamical variables are considered which go beyond rates of charge transfer such as the transient absorption spectrum. The parameters of the model are chosen to mimic the mixed valence system (NH3)5RuNCRu(CN)5

An analysis of the eigenstates of a symmetry-broken spin-boson Hamiltonian is performed by computing Bloch and Husimi projections. The eigenstate analysis is combined with the calculation of absorption bands of asymmetric dimer configurations constituted by monomers with nonidentical excitation energies and optical transition matrix elements. Absorption bands with regular and irregular fine structures are obtained and related to the transition from the coexistence to a mixing of adiabatic branches in the spectrum. It is shown that correlations between spin states allow for an interpolation between absorption bands for different optical asymmetries.

The fine structure of the complex quantum spectrum of a dimer constituted by monomers with a finite lifetime in the excited states and a strong excitonic–vibronic coupling has been investigated in detail. Lifetime distributions of the spectrum are analysed for different system parameter sets. It is shown that in case of an asymmetric configuration the spectrum may be characterised by a broad distribution of the lifetimes of the eigenstates. This can give rise to a strongly varying relaxation behaviour, which is due to the mixing of the monomer spectra with two different excitonic lifetimes in the dimer spectrum.

Coherent control exploits quantum pathway interference to steer the dynamics of a quantum system in the desired way. The dynamical interferences can be introduced in various ways. In this chapter the overall concept of coherent control together with different realizations is introduced, and the difficulty of coherently controlling cold collisions is analyzed in depth. As an example, photoassociation, i.e., the formation of a chemical bond by an external field, is considered. Prospects for controlling cold collisions beyond photoassociation are discussed.