In this thesis, I present an experimental study of quantum control techniques for transferring population between vibrational states of atoms trapped in an optical lattice. Results from a range of techniques are compared, including techniques tested previously in the same system. In the study of the Adiabatic Rapid Passage (ARP) technique, control of population transfer is realized through varying the chirp rate and modulation amplitude of a frequency-chirped sinusoidal displacement of the lattice. Meanwhile, dependence of population transfer on the chirp direction is observed, which is explained by a model of ARP in a 3-level system. In the study of the coherent control technique, interference between a one-phonon transition at 2\omega and a two-phonon transition at omega is experimentally demonstrated. The omega and 2\omega transitions are realized by sinusoidally displacing the optical lattice at omega and sinusoidally modulating the lattice depth at 2\omega, respectively. The branching ratio of transitions to the first excited state and to higher excited states is controlled by varying the relative phase between these two pathways. The highest measured branching ratio of 17\pm2 is achieved among all the experiments using this coherent control scheme. In the study of the GRadient Ascent Pulse Engineering (GRAPE) technique, a "pulse" involving both displacement and depth-modulation of the lattice is used to transfer population. This pulse is theoretically engineered with the GRAPE algorithm to optimize the fidelity between the first excited state and the final state, when the lattice Hamiltonian without gravity for a specific lattice depth is considered. The experimental result shows that there is almost no excitation into higher excited states during population transfer from the ground to the first excited state, even when this process is affected by gravity and inhomogeneous broadening in reality. By comparing all the techniques, the GRAPE technique is found to be the best in terms of increasing population transfer into the first excited state while reducing excitation into higher excited states. On the other hand, the ARP technique creates the highest normalized population inversion, a ratio of the difference to the sum of the ground and the first excited state populations.

In this thesis, I present an experimental study of quantum control techniques for transferring population between vibrational states of atoms trapped in an optical lattice. Results from a range of techniques are compared, including techniques tested previously in the same system. In the study of the Adiabatic Rapid Passage (ARP) technique, control of population transfer is realized through varying the chirp rate and modulation amplitude of a frequency-chirped sinusoidal displacement of the lattice. Meanwhile, dependence of population transfer on the chirp direction is observed, which is explained by a model of ARP in a 3-level system. In the study of the coherent control technique, interference between a one-phonon transition at 2\omega and a two-phonon transition at omega is experimentally demonstrated. The omega and 2\omega transitions are realized by sinusoidally displacing the optical lattice at omega and sinusoidally modulating the lattice depth at 2\omega, respectively. The branching ratio of transitions to the first excited state and to higher excited states is controlled by varying the relative phase between these two pathways. The highest measured branching ratio of 17\pm2 is achieved among all the experiments using this coherent control scheme. In the study of the GRadient Ascent Pulse Engineering (GRAPE) technique, a "pulse" involving both displacement and depth-modulation of the lattice is used to transfer population. This pulse is theoretically engineered with the GRAPE algorithm to optimize the fidelity between the first excited state and the final state, when the lattice Hamiltonian without gravity for a specific lattice depth is considered. The experimental result shows that there is almost no excitation into higher excited states during population transfer from the ground to the first excited state, even when this process is affected by gravity and inhomogeneous broadening in reality. By comparing all the techniques, the GRAPE technique is found to be the best in terms of increasing population transfer into the first excited state while reducing excitation into higher excited states. On the other hand, the ARP technique creates the highest normalized population inversion, a ratio of the difference to the sum of the ground and the first excited state populations.

A major challenge in the field of condensed matter physics is to harness the quantum mechanical properties of atomic systems coupled to large environments. Thermal fluctuations destroy quantum information and obstruct the development of quantum technologies such as quantum computers and memory devices. Recent advances in quantum control enable the manipulation of complex quantum states, providing new paths to preserve quantum information and to employ the environment as a resource. In this dissertation, we develop practical quantum control protocols which quickly and efficiently transfer energy to/from an environment. A major contribution of this work is the design of powerful and efficient quantum engines and refrigerators, which use the environment either to generate useful work or to freeze a system to its ground state. In achieving its core objectives, this work has also expanded on several areas of condensed matter quantum physics, including (i) the characterization of special classes of entangled system-environment states, (ii) the discovery of novel quantum chaotic phases of matter, (iii) the design of control schemes which speed-up efficient adiabatic protocols, and (iv) the development of experimentally viable control schemes in trapped ion systems, semiconductors, and nano-diamonds.

Implementing high-fidelity quantum control and suppressing the unwanted environmental noise has been one of the essential challenges in developing quantum information technologies. In the past, driving pulse sequences based on Dirac delta functions or square wave functions, such as Hahn spin echo or CPMG, have been developed to dynamically correcting the noise effects. However, implementing these ideal pulses with high fidelity is a challenging task in real experiments. In this thesis, we provide a new and simple method to explore the entire solution space of driving pulse shapes that suppress environmental noise in the evolution of the system. In this method, any single-qubit phase gate that is first-order robust against quasi-static transversal noise corresponds to a closed curve on a two-dimensional plane, and more general first-order robust single-qubit gates correspond to closed three-dimensional space curves. Second-order robust gates correspond to closed curves having the property that their projection onto any two-dimensional planes shall enclose a zero net area. The driving pulse shapes that implement the gates can be determined by the curvature, torsion, and the length of the curve. By utilizing the framework it is possible to obtain globally optimal solutions in pulse shaping in respect of experimental constraints by mapping them into geometrical optimization problems. One such problem we solved is to prove that the fastest possible single-qubit phase gates that are second-order noise-resistant shall be implemented using sign-flipping square functions. Since square waves are not experimentally feasible, we provide a method to smooth these pulses with minimal loss in gate speed while maintaining the robustness, based on the geometrical framework. This framework can also be useful in diagnosing the noise-cancellation properties of pulse shapes generated from numerical methods such as GRAPE. We show that this method for pulse shaping can significantly improve the fidelity of single-qubit gates through numerical simulation. / Doctor of Philosophy / Controlling a quantum system with high-fidelity is one of the main challenges in developing quantum information technologies, and it is an essential task to reduce the error caused by unwanted environmental noise. In this thesis, we developed a new geometrical formalism that enables us to explore all possible driving fields and provides a simple recipe to generate an infinite number of experimentally feasible driving pulse shapes for implementing quantum gates. We show that single-qubit operations that could suppress quasi-static noise to first-order correspond to closed three-dimensional space curves, and single-qubit gates that are second-order robust correspond to closed curves with zero enclosed net area. This simple geometrical framework can be utilized to obtain optimal solutions in quantum control problems, and can also be used as a method to diagnose driving pulse shapes generated from other means. We show that this method for pulse shaping can significantly improve the fidelity of single-qubit gates through numerical simulation.

Η θεωρία κβαντικού ελέγχου είναι μια νέα αναπτυσσόμενη μαθηματική θεωρία, η οποία ανζητά μεθόδους για τον καθορισμό της συμπεριφοράς των κβαντικών συστημάτων. Ο πρώτος στόχος της παρούσης εργασίας είναι η επισήμανση και η διευκρίνηση των θεωρητικών ερωτημάτων που προκύπτουν από την παρουσίαση της θεωρίας του κβαντικού ελέγχου ως προέκτασης του κλασικού ελέγχου. Ο δεύτερος στόχος της εργασίας είναι η παρουσίαση ορισμένων απλών παραδειγμάτων και μεθόδων κβαντικού ελέγχου, έτσι ώστε ο αναγνώστης να αποκτήσει μιά πρώτη εικόνα σχετικά με τα προβλήματα που ανακύπτουν στην προσπάθεια των ερευνητών να τιθασεύσουν τον μικρόκοσμο. Καθώς ο κβαντικός έλεγχος είναι κοινό ερευνητικό πεδίο για την επιστήμη των ηλεκτρολόγων μηχανικών και των φυσικών, η εργασία αυτή είναι φτιαγμένη έτσι ώστε να παρέχει σε ένα μηχανικό και ένα φυσικό τις γνώσεις που είναι απαραίτητες για τη μελέτη του κβαντικού ελέγχου. Για το σκοπό αυτό το πρώτο κεφάλαιο απευθύνεται κυρίως σε φυσικούς και αποτελεί μια παρουσίαση των βασικών αρχών της κλασικής θεωρίας ελέγχου. Αντίστοιχα, το δεύτερο κεφάλαιο απευθύνεται σε μηχανικούς και παρέχει τις απαραίτητες γνώσεις κβαντομηχανικής. Τέλος, το τρίτο κεφάλαιο αποτελεί τον κύριο κορμό της εργασίας και απευθύνεται σε αναγνώστες που προέρχονται και από τους δύο επιστημονικούς κλάδους. / The theory of quantum control is a new mathematical theory, which investigates methods for the description of the behaviour of quantum systems. The first target of this work is the identification and discussion of the theoretical questions of the quantum control theory as an extension of the classical control theory. The second target is to present some simple examples and methos of of quantum control in order to give to the reader a first insight to the related problems. The work contains three chapters. The first one gives an overview of the principales of the classic control theory. The second chapter provides the basic principals of quantum mechanics. Finally, the third chapter, which is the main part of the work, contains the examples and their discussion.

Master of Science / Department of Physics / Itzhak Ben-Itzhak / Ultrafast lasers allow us to study molecular dynamics on their natural timescale. The electronic dynamics can be studied using attosecond pulses, while the vibrational and rotational dynamics can be probed using tens of femtosecond and picosecond laser pulses, respectively. This capability has led to a broad understanding of the electronic dynamics in atoms and molecules as well as vibrational and rotational dynamics of molecules, which is one of the important goals in basic science. Moreover, it is possible to control quantum mechanical processes using ultrafast intense lasers. In this thesis, we focus on a couple of experiments. The first involves quantum control of the formation of neutral molecular fragments while the second focuses on three-body fragmentation of molecules employing the native-frames analysis method, which was recently introduced by our group [J. Rajput et al., Phys. Rev. Lett. 120, 103001 (2018)]. Experimental studies focused on the formation of excited neutral D fragments from D2 molecules are presented. We show that by manipulating the chirp of the intense laser pulses, i.e. the “time order” of the frequency components within the pulse, the formation of these fragments is controlled. To achieve this control we implement a single-prism compressor to manipulate the chirp of the laser pulses. Three-body fragmentation of CO₂ resulting in C+ + O+ + O+ is also studied. We show that if the two bonds break in a two-step process, i.e. a sequential breakup, the pathways from which the two identical O+ fragments originate can be separated using the native-frames analysis method. In contrast, the two O+ fragments cannot be distinguished if the two C-O bonds break simultaneously.

Quantum information is a growing field showing exciting possibilities for computational speed-up and communications. For the successful implementation of quantum computers, high-precision control is required to reach fault-tolerant thresholds. Control of quantum systems pertains to the manipulation of states and their evolution. In order to minimize the effects of the environment on the control operations of the qubits, control pulses should be made time-optimal. In addition, control pulses should be made robust to noise in the system, dispersion in energies and coupling elements, and uncertain parameters. In this thesis, we examine a robust time-optimal gradient ascent technique which is used to develop controls of the motional degrees of freedom for an ensemble of neutral atoms in a one-dimensional optical lattice in the high dispersion regime with shallow trapping potentials. As such, the system is analyzed in the delocalized basis. The system is treated as an ensemble of atoms with a range of possible quasimomenta across the first Brillouin zone. This gives the ensemble of Hamiltonians, indexed by the quasimomenta, a distinct spectra in their motional states and highly inhomogeneous control Hamiltonians. Thus, the optical lattice is seen as a model system for robust control. We find optimized control pulses designed using an ensemble modification of gradient-ascent pulse engineering robust to any range of quasimomentum. We show that it is possible to produce rotation controls with fidelities above 90\% for half of the first Brillouin zone with gate times in the order of several free oscillations. This is possible for a spectrum that shows upwards of 75\% dispersion in the energies of the band structure. We also show that NOT controls for qubit rotations on the entire Brillouin zone fidelities above 99\% were possible for 0.6\% dispersion in energies. The gate times were also in the order of several free oscillations. It is shown that these solutions are palindromic in time due to phase differences in some of the energy couplings when comparing one half of the Brillouin zone to another. We explore the limits of discretized sampling of a continuous ensemble for control.

This dissertation focuses on two connected areas: quantum computation and quantum control. Two proposals to construct a quantum computer, using nuclear magnetic resonance (NMR) and superconductivity, are introduced. We give details about the modeling, qubit realization, one and two qubit gates and measurement in the language that mathematicians can understand and fill gaps in the original literatures. Two experimental examples using liquid NMR are also presented. Then we proceed to investigate an example of quantum control, that of a magnetometer using quantum feedback. Previous research has shown that feedback makes the measurement robust to an unknown parameter, the number of atoms involved, with the assumption that the feedback is noise free. To evaluate the effect of the feedback noise, we extend the original model by an input noise term. We then compute the steady state performance of the Kalman filter for both the closed-loop and open-loop cases and retrieve the estimation error variances. The results are compared and criteria for evaluating the effects of input noise are obtained. Computations and simulations show that the level of input noise affects the measurement by changing the region where closed loop feedback is beneficial.

Κατά τη διάρκεια των τελευταίων ετών, το ενδιαφέρον πολλών επιστημόνων έχει στραφεί στη μελέτη της δυναμικής ηλεκτρονίων τα οποία είναι τοποθετημένα σε συζεύξεις κβαντικών τελειών. Στις μέρες μας, τέτοιου είδους κβαντικά συστήματα, όπου έχουν παγιδευτεί ηλεκτρόνια, μελετώνται με εντατικό ρυθμό. Ειδικότερα, στο πρώτο μέρος της εργασίας, μελετάμε τη δυναμική δύο αλληλεπι- δρώντων ηλεκτρονίων, τα οποία είναι παγιδευμένα σε μία δομή ζεύγους κβαντικών τελειών, κάτω από την επίδραση διχρωματικών ηλεκτρικών πεδίων. Η θεωρητική ανάλυση βασίζεται στην προσέγγιση του συστήματος δύο ενεργειακών επιπέδων και καταλήγουμε στις αναλυτικές συνθήκες εντοπισμού των δύο ηλεκτρονίων στην ίδια κβαντική τελεία. Τα αναλυτικά αποτελέσματα συγκρίνονται με τα αριθμητικά, τα οποία προκύπτουν από την επίλυση της χρονοεξαρτώμενης εξίσωσης Schrödinger. Στο δεύτερο μέρος της εργασίας αυτής, μελετάμε τον βέλτιστο έλεγχο για δυναμικό διπλής συμμετρικής κβαντικής τελείας, όπου έχει παγιδευτεί ένα ηλεκτρόνιο, κατά τη διάρκεια του οποίου το σύστημα αλληλεπιδρά με έναν παλμό ηλεκτρομαγνητικού πεδίου. Αρχικά χρησιμοποιούμε τις προσεγγίσεις του περιστρεφόμενου κύματος και του ακριβούς συντονισμού και προσεγγίζουμε το σύστημα με ένα σύστημα τριών ενεργειακών καταστάσεων. Στη συνέχεια, περιγράφουμε το σύστημα μέσω διαφορι- κών εξισώσεων, οι οποίες πρέπει να ικανοποιούνται από το βέλτιστο ηλεκτρομαγνη- τικό πεδίο. Τέλος, καταλήγουμε σε αναλυτικές εκφράσεις για το σχήμα του παλμού βέλτιστου ελέγχου, ο οποίος οδηγεί σε χρονικά μέση, αλλά και ολική μεγιστοποίηση του πληθυσμού μιας ενεργειακής στάθμης που έχουμε επιλέξει. / During the last years, the study of dynamics of electrons trapped by systems of quantum dots has attracted the interest of many scientists. Nowadays, such quantum systems, where electrons have been trapped, are being studied intensively. More specifically, in the first part of this thesis, we investigate the dynamics of two interacting electrons confined in a symmetric double quantum dot structure, under the influence of bichromatic electric fields. The theoretical analysis is based on an effective two-level system approach and the conditions for two-electron localization in the same quantum dot are analytically derived. The analytical results are compared to numerical results obtained from the solution of the time-dependent Schrödinger equation. In the second part of this thesis, we study the potential for optimal control of a symmetric double quantum dot structure, where a single electron has been trapped, interacting with a single pulsed electromagnetic field. We first use the rotating wave and resonant approximations and reduce the dynamics of the system to that of a degenerate three-level-type system. We also formulate the optimal control problem in terms of differential equations that have to be fulfilled by the optimal electromagnetic fields. We then obtain general analytical expressions for the optimal pulse shapes that lead to global maximization of the final population of the target state and of the timeaveraged population of the target state in the quantum dot structure.

Quantum information is a growing field showing exciting possibilities for computational speed-up and communications. For the successful implementation of quantum computers, high-precision control is required to reach fault-tolerant thresholds. Control of quantum systems pertains to the manipulation of states and their evolution. In order to minimize the effects of the environment on the control operations of the qubits, control pulses should be made time-optimal. In addition, control pulses should be made robust to noise in the system, dispersion in energies and coupling elements, and uncertain parameters. In this thesis, we examine a robust time-optimal gradient ascent technique which is used to develop controls of the motional degrees of freedom for an ensemble of neutral atoms in a one-dimensional optical lattice in the high dispersion regime with shallow trapping potentials. As such, the system is analyzed in the delocalized basis. The system is treated as an ensemble of atoms with a range of possible quasimomenta across the first Brillouin zone. This gives the ensemble of Hamiltonians, indexed by the quasimomenta, a distinct spectra in their motional states and highly inhomogeneous control Hamiltonians. Thus, the optical lattice is seen as a model system for robust control. We find optimized control pulses designed using an ensemble modification of gradient-ascent pulse engineering robust to any range of quasimomentum. We show that it is possible to produce rotation controls with fidelities above 90\% for half of the first Brillouin zone with gate times in the order of several free oscillations. This is possible for a spectrum that shows upwards of 75\% dispersion in the energies of the band structure. We also show that NOT controls for qubit rotations on the entire Brillouin zone fidelities above 99\% were possible for 0.6\% dispersion in energies. The gate times were also in the order of several free oscillations. It is shown that these solutions are palindromic in time due to phase differences in some of the energy couplings when comparing one half of the Brillouin zone to another. We explore the limits of discretized sampling of a continuous ensemble for control.