Publications

Feshbach resonances are fundamental to interparticle interactions and have been detected in collisions with atoms, ions, and molecules. However, extracting the resonance contribution to the collision dynamics and identifying their effect on the collision outcome has remained elusive. Here we present the detection of Feshbach resonance states with complete final quantum state resolution in a benchmark system for strongly interacting and highly anisotropic collisions — molecular hydrogen ions colliding with noble gas atoms. We launch the collisions by cold Penning ionization which allows us to exclusively populate Feshbach resonances that span both short- and long-range parts of the interaction potential. We simultaneously resolve all final molecular channels of the Feshbach resonances in a tomographic manner using ion-electron coincidence velocity map imaging. Despite the strong mixing of all the degrees of freedom during the collision, we do not observe a statistical distribution of the final quantum states. We demonstrate that isolation of the Feshbach resonance pathway enables the detection of unique fingerprints of the resonant dynamics as confirmed by full \textitab initio quantum scattering calculations.

We extend here the optimization functional targeting arbitrary cat states, derived in the companion paper, to open quantum system dynamics. Applying it to a Jaynes-Cummings model with decay on the oscillator, we find, for strong dissipation and large cat radii, a change in the control strategy for preparing an entangled cat state. Our results illustrate the versatility of the quantum optimal control toolbox for practical applications in the quantum technologies.

A key task in the field of quantum optimal control is to encode physical targets into figures of merit to be used as optimization functionals. Here we derive a set of functionals for optimization towards an arbitrary cat state. We demonstrate the application of the functionals by optimizing the dynamics of a Kerr-nonlinear Hamiltonian with two-photon driving. Furthermore, we show the versatility of the framework by adapting the functionals towards optimization of maximally entangled cat states, applying it to a Jaynes-Cummings model. Finally, we identify the strategy of the obtained control fields and determine the quantum speed limit as a function of the cat state’s excitation. Our results highlight the power of optimal control with functionals specifically crafted for complex physical tasks. They allow for optimizing the preparation of entangled cat states in more realistic settings, including, e.g., dissipative effects which we investigate in the companion paper.

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.

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.

We investigate how optimal control theory can be used to improve Circular Dichroism (CD) signals for the 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 but 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 of 2.5 when using shaped laser pulses, with the anisotropy parameter g increasing from 0.06 to 1. We find that this effect is caused by the interference between the excitation pathways prompted by the different multipole moments of the molecule.

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.

Qubit reset is a basic prerequisite for operating quantum devices, requiring the export of entropy. The fastest and most accurate way to reset a qubit is obtained by coupling the qubit to an ancilla on demand. Here, we derive fundamental bounds on qubit reset in terms of maximum fidelity and minimum time, assuming control over the qubit and no control over the ancilla. Using the Cartan decomposition of the Lie algebra of qubit plus ancilla, we identify the types of interaction and controls for which the qubit can be purified. For these cases, we show that a time-optimal protocol is resonant purity exchange between qubit and ancilla, where the maximum fidelity is identical for all cases but the minimum time strongly depends on the type of interaction and control.

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.

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.

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.

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.

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.

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 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.

The wave-function Monte-Carlo method, also referred to as the use of "quantum-jump trajectories", allows efficient simulation of open systems by independently tracking the evolution of many pure-state "trajectories". This method is ideally suited to simulation by modern, highly parallel computers. Here we show that Krotov’s method of numerical optimal control, unlike others, can be modified in a simple way, so that it becomes fully parallel in the pure states without losing its effectiveness. This provides a highly efficient method for finding optimal control protocols for open quantum systems and networks. We apply this method to the problem of generating entangled states in a network consisting of systems coupled in a unidirectional chain. We show that due to the existence of a dark-state subspace in the network, nearly-optimal control protocols can be found for this problem by using only a single pure-state trajectory in the optimization, further increasing the efficiency.

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 both purification and erasure of 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 and a speed limit. The time-optimal solution is determined by the maximum coupling strength. Initial correlations, remarkably, allow for faster reset and smaller errors.

We use quantum optimal control theory to systematically map out the experimentally reachable parameter landscape of superconducting transmon qubits. With recent improvements in decoherence times, transmons have become a promising platform for quantum computing. They can be engineered over a wide range of parameters, giving them great flexibility, but also requiring us to identify good regimes to operate at. Using state-of-the-art control techniques, we exhaustively explore the landscape for the potential creation and distribution of entanglement, for a wide range of system parameters and applied microwave fields. We find the greatest success outside the usually considered dispersive regime. A universal set of gates is realized for gate durations of 50 ns, with gate errors approaching the theoretical limit. Our quantum optimal control approach is easily adapted to other platforms for quantum technology.

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.

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.

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 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.

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.

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.

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 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 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.

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.

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.