Schedule for: 17w5010 - Mathematical and Numerical Methods for Time-Dependent Quantum Mechanics - from Dynamics to Quantum Information

Quantum control is an important prerequisite for quantum devices [1]. A major obstacle is the fact that a quantum system can never completely be isolated from its environment. The interaction of a quantum system with its environment causes decoherence. Optimal control theory is a tool that can be used to identify control strategies in the presence of decoherence. I will show how to adapt optimal control theory to quantum information tasks for open quantum systems [2].

A key application of quantum control is to identify performance bounds, for tasks such as state preparation or quantum gate implementation, within a given architecture. One such bound is the quantum speed limit, which determines the shortest possible duration to carry out the task at hand. For open quantum systems, interaction with the environment may lead to a speed-up of the desired evolution. Here, I will show how initial correlations between system and environment may not only be exploited to speed up qubit reset but also to increase state preparation fidelities. Geometric control techniques provide an intuitive understanding of the underlying dynamics [3].

Control tasks such as state preparation or gate implementation are typically optimized for known, fixed parameters of the system. Showcasing the full capabilities of quantum optimal control, I will discuss how recent advances in quantum control techniques allow for going even further. Using a fully numerical quantum optimal control approach, it is possible to map out the entire parameter landscape for superconducting transmon qubits. This allows to determine the global quantum speed limit for a universal set of gates with gate errors limited solely by the qubit lifetimes. It thus provides the optimal working points for a given architecture [4].

While the interaction of qubits with their environment is typically regarded as detrimental, this does not need to be the case. I will show that the back-flow of amplitude and phase encountered in non-Markovian dynamics can be exploited to carry out quantum control tasks that could not be realized if the system was isolated [5]. The control is facilitated by a few strongly coupled, sufficiently isolated environmental modes. These can be found in a variety of solid-state devices including superconducting circuits.

[1] S. Glaser et al.: Training Schrödinger's cat: quantum optimal control, Eur. Phys. J. D 69, 279 (2015)
[2] C. P. Koch: Controlling open quantum systems: Tools, achievements, limitations, J. Phys. Cond. Mat. 28, 213001 (2016)
[3] D. Basilewitsch et al.: Beating the limits with initial correlations, arXiv:1703.04483
[4] M. H. Goerz et al.: Charting the circuit-QED Design Landscape Using Optimal Control Theory, arXiv:1606.08825
[5] D. M. Reich, N. Katz & C. P. Koch: Exploiting Non-Markovianity for Quantum Control, Sci. Rep. 5, 12430 (2015)

Controllable devices provide novel ways for the simulation of complex quantum open systems. In this talk, we will present different experimental platforms, developed in our group, where the dynamics of Born-Markov open quantum systems can be successfully simulated. In particular, we will discuss the observation of the so-called environment-assisted quantum transport in electrical oscillator networks, the survival of quantum coherence between indistinguishable particles propagating in quantum networks affected by noise, and the implementation of the first noise-enabled optical ratchet system.

We focus on quantum systems subject to external interactions (laser, magnetic fields) taken as controls, which are contaminated with non perturbative noise. The measured observables come thus in the form of probability laws; we ask the following question: is it possible, from the knowledge of these probability laws, to recover the free and interaction (dipole) Hamiltonians ? We see that the theoretical answer is positive (provided some assumptions on the controllability of the quantum system hold); then we explore numerical approaches which exploit particular metrics on the space of probability laws.

TBA

A numerical scheme that solves the time-dependent Dirac equation is presented in which the time evolution is performed by an operator-splitting decomposition technique combined with the method ofcharacteristics. On a classical computer, this numerical method has some nice features: it is very versatile and most notably, it can be parallellized efficiently. This makes for an interesting numerical tool for the simulation of quantum relativistic dynamical phenomena such as the electron dynamics in very high intensity lasers. Moreover, this numerical scheme can be implemented on a digital quantum computer due to its simple structure: the operator splitting is a sequence of streaming operators followed by rotations in spinor space. This structure is actually reminiscent of quantum walks, which can be implemented efficiently on quantum computers. We determine the resource requirements of the resulting quantum algorithm and show that under some conditions, it has an exponential speedup over the classical algorithm. Finally, an explicit decomposition of this algorithm into elementary gates from a universal set is carried out using the software Quipper. It is shown that a proof-of-principle calculation may be possible with actual quantum technologies.

I shall present two pruned, nondirect product multi-configuration time dependent Hartree (MCTDH) methods for solving the Schr¨odinger equation. Both use a basis of products of natural orbitals. Standard MCTDH uses optimized 1D basis functions, called single particle functions, but the size of the basis scales exponentially with D, the number of coordinates.By replacing t → −iβ , β ∈ R>0 , we use the pruned methods to determine solutions of the time-independent Schroedinger equation. For a 12D Hamiltonian, we compare the pruned approach to standard MCTDH calculations for basis sizessmall enough that the latter are possible and demonstrate that pruning the basis reduces the CPU cost of computing vibrational energy levels of acetonitrile by more than two orders of magnitude. One of the pruned MCTDH methods uses an algebraic pruning constraint. The other uses a flexible basis that expands as the calculation proceeds. Results obtained with the expanded basis are compared to those obtained with the established multi-layer MCTDH (ML-MCTDH) scheme. Although ML-MCTDH is somewhat more efficient when low or intermediate accuracy is desired, pruned MCTDH is more efficient when high accuracy is required.

The minimum mode following method for finding first order saddle points on a potential energy surface is used, for example, in simulations of long time scale evolution of materials and surfaces of solids. Such simulations are increasingly being carried out in combination with computationally demanding electronic structure calculations of atomic interactions. Therefore, it becomes essential to reduce, as much as possible, the number of function evaluations needed to find the relevant saddle points. Several improvements to the method are presented here and tested on a benchmark system involving rearrangements of a heptamer island on a closed packed crystal surface. Instead of using a uniform or Gaussian random initial displacement of the atoms, as has typically been done previously, the starting points are arranged evenly on the surface of a hypersphere and its radius is adjusted during the sampling of the saddle points. This increases the diversity of saddle points found and reduces the chances of converging again to previously located saddle points. The minimum mode is estimated using the Davidson method, and it is shown that significant savings in the number of function evaluations can be obtained by assuming the minimum mode is unchanged until the atomic displacement exceeds a threshold value.

Electron transport in materials and mesoscopic is a very complex topic, not yet fully understood from a physical point of view. Some of the key mechanisms at the origin of electron transport have been identified though, and a wide variety of mathematical models have been proposed to account for these phenomena, among which the classical Drude model (1900), the Bloch model (1928), the BCS model (1957), the Anderson model (1958), the Hubbard model (1963), the Landauer-Büttiker formalism (1986), or the Haldane model (1988). Electron transport is described by a variety of models ranging from many-body quantum models, to semiclassical drift-diffusion models. Coherent transport inmesoscopic devices is usually using the Landauer-Büttiker formalism. For homogeneous periodic materials with or without defects, most of the available models are based on the quasiparticle picture of quasi Bloch electrons and holes scattered by phonons, impurities, and effective two-body interactions. Methods based on non-commutative geometry have also been introduced by Bellissard and collaborators to handle homogeneous aperiodic systems. In the last two decades, it has become possible to compute (more or less accurately) the various parameters of these models from first-principle calculations, using Density Functional Theory (DFT) and Green’s function methods (GW, Bethe-Salpeter equation). In this talk, I will review some of the recent progress and open questions in the mathematical understanding and numerical simulation of these models.

A large part of strong-field physics relies on trajectory-based semiclassical methods for the description of ionization and the subsequent dynamics, both for intuitive understanding and as quantitative models, including the workhorse Strong-Field Approximation and its various extensions. In this work I examine the underpinnings of this trajectory-based language in the saddle-point analysis of temporal integrals deformed over complex contours, and how this formalism can be extended to include interactions with the ion. I show that a first-principles approach requires the use of complex-valued positions as well as times, and that the evaluation of the ionic Coulomb potential at these complex-valued positions imposes new constraints on the allowed temporal integration contours. I show how the navigation of these constraints is infused with physical content and a rich geometry, and how the correct traversal of the resulting landscape has clear consequences on the photoelectron spectrum.

References:
1. E. Pisanty and M. Ivanov. Phys. Rev. A 93, 043408 (2016).
2. E. Pisanty and M. Ivanov. J. Phys. B: At. Mol. Opt. Phys. 49, 105601 (2016).

Recent HHG and ionization experiments in solids have given birth to the field of attosecond condensed matter physics. Potential applications range from solid state coherent xuv radiation sources, to resolving ultrafast processes in the condensed matter phase, to PHz (petahertz) opto-electronic elements. Theoretical analysis of these processes has so far been mainly confined to the single-active electron (SAE) limit. In the first part of the talk theoretical progress in understanding HHG and ionization in the SAE limit will be reviewed. Both, bulk and nano-confined systems will be explored. While SAE analysis can reasonably describe many features of strong field processes in solids on a qualitative level, there is no doubt that many-body effects play an important role quantitatively. Beyond that, we expect them to add additional signatures to the one-body results which will make them identifiable experimentally. In the second part of the talk many-body features of strong field processes in solids will be explored. We use a multi-configuration time-dependent Hartree-Fock (MCTDHF) method developed for atomic and molecular attosecond science. MCTDHF is used to model 1D model quantum wires in strong fields, which consist of a string of atoms with one electron per lattice site. Chains of up to more than 20 atoms are investigated. Our results exhibit clear multi-electron signatures in HHG spectra.

Introduction

After the formulation of multi-configuration time-dependent Hartree-Fock (MCTDHF) method to treat electronic dynamics in atoms and molecules induced by the interaction with intense laser pulses from first principles [1], the theoretical efforts exerted on the developments of the method has been changed their aspects from the basic formulations and the proof-of-principle type calculations to practical calculations in order to elucidate the many electron dynamics by comparisons with experimental results [2].

Recently, efforts have been made to improve the numerical performance of the MCTDHF method aiming to reduce the size of the configuration space, i.e., Slater determinantal expansion length, by restricting the orbital excitation schemes [3,4].

A different approximation of factorized configuration interaction coefficients [5] as well as the multi-layer formulation of MCTDHF [6] have also been introduced recently.

In the present study, we propose a new formulation for the time propagation of a time-dependent multi-configuration wave function in which the spin-orbitals follow a single-particle time-dependent Schrödinger equation (TDSE) specified by a multiplicative time-dependent local effective potential $v_{\rm eff}(\mathbf{r},t)$.

Theory

We consider an $N$-electron time-dependent wave function $\Psi(x_1,x_2,\cdots,x_N,t)$ perturbed by a time-dependent external field.

The wave function is assumed to be represented by $$\Psi(x_1,x_2,\cdots,x_N,t) = \sum_{K=1}^{\mathcal{L}} C_K(t) \Phi_K(x_1,x_2,\cdots,x_N,t),$$ where ${\{C}_{K}(t)\}$ represent time-dependent configuration interaction coefficients and $\Phi_{K}(t)$ time-dependent Slater determinants.

The time-dependence of each Slater determinant is due to the time dependence of the constituent spin-orbitals.

The total Hamiltonian of the system is represented by $$\hat{H}(t) = \hat{T} + \hat{V}_{\rm ext}(t) + \hat{V}_{\rm ee}$$ , where $\hat{T}$, $\hat{V}_{\rm ext}(t) = \sum_{j=1}^N v_{\rm ext}(\mathbf{r}_j,t)$, and $\hat{V}_{\rm ee}$ represent the kinetic energy operator, the sum of nuclear attraction potential and the time-dependent external perturbation, and the electron-electron repulsion potential, respectively.

The spin-orbitals are assumed to obey a single-particle TDSE expressed by $$\left[ i \hbar \frac{\partial }{\partial t} -\left( - \frac{\hbar^2}{2m_{\rm e}} \frac{\partial^2 }{\partial {\mathbf{r}}^2} + v_{\rm eff}(\mathbf{r},t) \right) \right] \phi_k(x,t) = 0.$$ where $x=(\mathbf{r},\sigma)$ denotes the spatial and spin-coordinates of an electron, and $v_{\rm eff}(\mathbf{r},t)$ is the effective potential to be calculated.

We define an effective Hamiltonian for the relevant system as $$\hat{H}_{\rm eff}(t) = \hat{T} + \sum_{j=1}^N v_{\rm eff}(\mathbf{r}_j,t) = \hat{T} + \hat{V}_{\rm eff}.$$ The effective potential is formulated by using McLachlan's minimization principle in which the difference of the time-evolution of the wave function $\Psi(x_1,x_2,\cdots,x_N,t)$ is minimized between the TDSEs specified by $\hat{H}(t)$ and $\hat{H}_{\rm eff}(t)$.

Discussion

We report the detailed theoretical analysis of the properties of the effective potential associated with an exact wave function.

Furthermore, as an elementary application of the present formalism, we propose a direct method to calculate the so-called Brueckner orbitals [7] as a special solution of a set of spin-orbitals calculated as eigenfunctions for a single-particle Schrödinger equation specified by a time-independent effective potential $v_{\rm eff}(\mathbf{r})$ that is associated with an exact ground-state wave function [8].

Also, the relationship between the present effective potential and the Slater's effective potential will be clarified [9].

References

1. For example, T. Kato and H. Kono, Chem. Phys. Lett. 392 (2004) 533-540.

2. K.L. Ishikawa and T. Sato, IEEE J. Sel. Topics Quantum Electron. 21 (2015) 8700916-1-16.

3. H. Miyagi and L.B. Madsen, Phys. Rev. A 87 (2013) 062511-1-12.

4. T. Sato and K. L. Ishikawa, Phys. Rev. A 91 (2015) 023417-1-15.

5. E. Lötstedt, T. Kato, and Y. Yamanouchi, J. Chem. Phys. 144 (2016) 154116-1-13.

6. H. Wang and M. Thoss, J. Chem. Phys. 131 (2009) 024114-1-14.

7. R.K. Nesbet, Phys. Rev. 109 (1958) 1632-1638.

8. P.O. Löwdin, J. Math. Phys. 3 (1962) 1171-1184.

9DJ.C. Slater, Phys. Rev. 91 (1953) 528-530.

In most of the past studies of processes involving interaction of lasers with atoms and molecules the tiny photon momentum has not been taken into account nor the issue of momentum sharing between a photoelectron and an ion has not been addressed despite the fact than when intense lasers are used a huge amount of infrared photons are absorbed. This situation has been related to the fact that in most theoretical investigations the dipole approximation has been used for description of the photoionization processes. In this talk I emphasize the importance of using the non-dipole approaches in description of the interaction of intense lasers with atoms and molecules. I will review some surprising results obtained by us using numerical solutions of the time-dependent Schroedinger equation in [1-3] and present new results related to the photon-momentum effect using counter-propagating pulses and the specific non-dipole effects in diatomic molecules.

[1] S. Chelkowski, A.D. Bandrauk, and P.B. Corkum, Phys.Rev.Let. 113, 263005 (2014).
[2] S. Chelkowski, A.D. Bandrauk, and P.B. Corkum, Phys.Rev. A 92, 051401 (R) (2015).
[3] S. Chelkowski, A.D. Bandrauk, and P.B. Corkum, Phys.Rev. A 95, 053402 (2017).

Use of elliptically-polarized light opens the possibility of investigating effects that are not accessible with linearly-polarized pulses. This talk presents new physical effects that are predicted for ionization of the helium atom by few-cycle, elliptically-polarized attosecond pulses. For double ionization of He by an intense elliptically-polarized attosecond pulse, we predict a nonlinear dichroic effect (i.e., the difference of the two-electron angular distributions in the polarization plane for opposite helicities of the ionizing pulse) that is sensitive to the carrier-envelope phase, ellipticity, peak intensity I, and temporal duration of the pulse [1]. For single [2,3] and double ionization [4] of He by two oppositely circularly-polarized, time-delayed attosecond pulses, we predict that the photoelectron momentum distributions in the polarization plane have helical vortex structures that are exquisitely sensitive to the time-delay between the pulses, their relative phase, and their handedness [2-4]. These effects manifest the ability to control the angular distributions of the ionized electrons by means of the attosecond pulse parameters. Our predictions are obtained numerically by solving the two-electron time-dependent Schrödinger equation for the six-dimensional case of elliptically-polarized attosecond pulses. They are interpreted analytically by means of perturbation theory analyses of the two ionization processes.

*This work is supported in part by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), Award No. DE-FG03-96ER14646.

[1] J.M. Ngoko Djiokap, N.L. Manakov, A.V. Meremianin, S.X. Hu, L.B. Madsen, and A.F. Starace, “Nonlinear dichroism in back-to-back double ionization of He by an intense elliptically-polarized few-cycle XUV pulse,” Phys. Rev. Lett. 113, 223002 (2014).
[2] J.M. Ngoko Djiokap, S.X. Hu, L.B. Madsen, N.L. Manakov, A.V. Meremianin, and A.F. Starace, “Electron Vortices in Photoionization by Circularly Polarized Attosecond Pulses,” Phys. Rev. Lett. 115, 113004 (2015).
[3] J.M. Ngoko Djiokap, A.V. Meremianin, N.L. Manakov, S.X. Hu, L.B. Madsen, and A.F. Starace, “Multistart Spiral Electron Vortices in Ionization by Circularly Polarized UV Pulses,” Phys. Rev. A 94, 013408 (2016).
[4] J.M. Ngoko Djiokap, A.V. Meremianin, N.L. Manakov, S.X. Hu, L.B. Madsen, and A.F. Starace, “Kinematical Vortices in Double Photoionization of Helium by Attosecond Pulses,” Phys. Rev. A (accepted 19 June 2017, in press).

The nuclear dynamics of molecules following photoexcitation are fundamentally quantum dynamical processes owing to the breakdown of the Born-Oppenheimer approximation in regions close to the intersection of potential energy surfaces. To describe such processes, the solution of the time-dependent Schrödinger equation is necessitated. If large amplitude nuclear motion is involved, then this is a particularly challenging task for conventional gridbased quantum dynamics methods, owing to the need to construct model Hamiltonians that accurately describe multiple potential energy surfaces and the couplings between them over a large subvolume of nuclear configuration space. A powerful alternative is to expand the nuclear wavefunction in terms of localised time-dependent parameterised basis functions, and to exploit this locality to calculate Hamiltonian matrix elements using information at a small number of nuclear geometries. Such methods not only promise to break the curse of exponential scaling suffered by conventional grid-based methods, but also allow for quantum dynamics calculations to be performed ‘on-the-fly’ using information from ab initio electronic structure calculations calculated as and when it is needed. In this talk, I will discuss one such method: the ab initio multiple spawning (AIMS) method. In the AIMS method, the solution of the time-dependent Schrödinger equation is achieved via the expansion of the nuclear wavefunction in an adaptive set of Gaussian basis functions, which is increased in size when needed in order to efficiently describe the transfer of population between electronic states. In the first part of the talk, the theoretical foundations and details of the AIMS method will be discussed. In the second part, representative examples of the study of excited state molecular dynamics will be given, illustrating the power and success of the AIMS method in studying photochemical processes.

Networks of interacting spin-1/2 particles form the basis for a wide range of quantum technologies including quantum communication, simulation and computation devices. Optimal control provides methods to steer their dynamics to implement specific quantum operations. It is usually implemented to find optimal time-dependent control fields to implement quantum gates or transformations of quantum states or observables in the context of open-loop quantum control. We recently proposed an alternative approach of static controls, based on shaping the energy landscape of quantum systems. For coupled spin systems this type of control could be realized in terms of spatially distributed gates that introduce energy level shifts using quasi-static local electric or magnetic fields. Although there are insufficient control degrees of freedom for the system to be completely controllable, many practically interesting operations can be implemented using these controls, including efficient transfer of excitations in spin networks. Furthermore, the resulting controllers combine high fidelity and strong robustness properties under device uncertainties, surpassing traditional limits in classical control. In particular, we observe positive correlations between the logarithmic sensitivity and the control error in many cases, i.e., the highest fidelity controllers are also the most robust. Structured singular value analysis shows the same trend for large structured variation using $\mu$-analysis tools.
One way to understand the surprising robustness of the controllers is in terms of feedback control laws. The energy biases create direct feedback loops. Similar to feedback loops in electronic circuits such as operational amplifiers, this feedback is highly effective and does not require measurements.
I will discuss results on energy landscape control for quantum spin devices with a focus on robustness at high fidelities of the operations and the interpretation of the controllers in terms of feedback control laws.

References:
[1] Emergence of Classicality under decoherence in robust quantum transport, S. Schirmer, E. Jonckheere, S. O’Neil, and F. Langbein, in preparation.
[2] Design of Feedback Control Laws for Information Transfer in Spintronics Networks, S Schirmer, E Jonckheere, F Langbein, arXiv:1607.05294, 2016.
[3] Time optimal information transfer in spintronics networks, FC Langbein, S Schirmer, E Jonckheere, 2015 IEEE 54th Annual Conference on Decision and Control (CDC), 6454-58, 5, 2015.
[4] Structured singular value analysis for spintronics network information transfer control, E Jonckheere, S Schirmer, F Langbein, IEEE Transactions on Automatic Control, DOI: 10.109/TAC.2017.2714623, arXiv:1706.03247, in press, 2017.
[5] Jonckheere-Terpstra test for nonclassical error versus log-sensitivity relationship of quantum spin network controllers, E Jonckheere, S Schirmer, F Langbein, arXiv:1612.02784, 2016.
[6] Information transfer fidelity in spin networks and ring-based quantum routers, E Jonckheere, F Langbein, S Schirmer, Quantum Information Processing 14 (12), 4751-4785.
[7] Multi-fractal Geometry of Finite Networks of Spins, P Bogdan, E Jonckheere, S Schirmer, arXiv:1608.08192, 2016.

It will be shown that the Rabi model and the ion laser interaction Hamiltonians may be related via a simmilarity transformation which allows much faster regimes than the ones reached with the rotating wave approximation. This is because the speed of the interaction is dictated the Rabi (intensity of the laser) and our approach does not impose a condition on it.
It will also be shown that the unitary transformation may be done in the time dependent case, and the Rabi model Hamiltonian parameters will depend also on time.

The well-know pseudo-spectral second-order splitting method has been the work-horse algorithm for solving the time-dependent Schrodinger equation for decades. However, it has been only ~30 years that one learns how to generalize this algorithm to fourth and higher orders. During that period, one also learned about the crucial distinction between solving time-reversible and time-irreversible quantum evolution equations. In this work, I show that there is only one class of fourth-order, forward time step algorithm that can solve both time-reversible and time-irreversible equations with equal efficiency. Other higher order algorithms based on multi-product expansion and complex coefficient will also be mentioned.

The electronic system is driven far from its ground state in many applications today: attosecond control and manipulation of electron dynamics and the consequent ion dynamics, photovoltaic design, photoinduced processes in general. Time-dependent density functional theory is a good candidate by which to computationally study such problems. Although it has had much success in the linear response regime for calculations of excitation spectra and response, its reliability in the fully non-perturbative regime is less clear even though it is increasingly used. In the first part of the talk, I will show some of our recent work exploring exact features of the time-dependent exchange-correlation potential that are necessary to yield accurate dynamics and new approaches to develop functionals going beyond the adiabatic approximation. In the second part of the talk, I broaden the focus to the description of coupled electron-ion motion. When the coupling to quantum nuclear dynamics is accounted for, we find additional terms in the potential acting on the electronic subsystem, that fully account for electron-nuclear correlation, and that can yield significant differences to the traditional potentials used when computing coupled electron-ion dynamics.

To solve the equation of motion for a quantum-mechanical wavefunction $\psi$ the Schrödinger equation, for a many-particle system is a major problem in many branches of physics. Based on the idea that the joint probability density $|\psi|^2$ can be written as the product of a marginal probability density (that depends only on some of the particle variables) and a conditional probability density, it can be shown that there exists a marginal wavefunction which also obeys a Schrödinger equation, but with an effective potential that encodes the interaction with all parti- cles. In this talk, I present an application of this idea to a many-electron system: By taking only the variables of one electron as the marginal coordinates, an exact single-electron picture that describes the correlated electron dynamics of many electrons is obtained. All many-electron interactions are then encoded in the structure and time-dependence of an effective single-electron potential. This approach is applied to the description of strong field phenomena, because they are often interpreted in a single-electron picture, while at the same time the understanding and measurement of many-electronic interactions is a main topic in this field. First results for 2- and 3-electron model systems in strong laser fields are presented and used to illustrate how an exact single electron picture of ionization or high-harmonic generation looks like. Additionally, I show first steps towards a feasible method for the calculation of the many-electron dynamics in complex molecules.

[1] Axel Schild, E.K.U. Gross, Phys. Rev. Lett. 118, 163202 (2017).

We investigate the ability of time-dependent density functional theory (TDDFT) to capture attosecond valence electron dynamics resulting from sudden X-ray ionization of a core electron. In this special case the initial state can be constructed unambiguously, allowing for a simple test of the accuracy of the dynamics. The response following nitrogen K-edge ionization in nitrosobenzene shows excellent agreement with fourth order algebraic diagrammatic construction (ADC(4)) results, suggesting that a properly chosen initial state allows TDDFT to adequately capture attosecond charge migration. Visualizing hole motion using an electron localization picture (ELF), we provide an intuitive chemical interpretation of the charge migration as a time-dependent superposition of Lewis-dot resonance structures. Coupled with the initial state solution to obtain such dynamics with TDDFT, this chemical picture facilitates interpretation of electron .

Ending up with a lot of dynamics data ( be it trajectories or wavefunctions) is a common situation we face whenever we are working with real (and sometimes, evenmodel) systems. How can we make sense of these massive data in terms of concepts we understand? In the past two decades I and my team have been using modern dynamical systems theory to uncover simple structures buried under massive trajectory calculations. The power of this tool is due to a very simple property: It allows you to focus on collective behavior of families of trajectories rather than individual ones, thereby helping you to isolate generic behavior. I will give you specific examples from our research which illustrate the use of this powerful tool.

We numerically study the interaction of a terahertz pulse with monolayer graphene. We use a numerical solution of the two-dimensional Dirac equation in Fourier space with time-evolution based on split-operator method to describe the dynamics of electron-hole pair creation in graphene. We notice that the electron momentum density is affected by the carrier-envelope phase (CEP) of the few-cycle terahertz laser pulse that induces the electron dynamics. Two main features are observed: (1) interference pattern for any values of the CEP and (2) asymmetry, for non-zero values of the CEP. We explain the origin of the quantum interferences and the asymmetry within the adiabatic-impulse model by finding conditions to reach minimal adiabatic gap between the valence band and the conduction band in graphene. The quantum interferences emanate from successive non-adiabatic transitions at this minimal gap. We discuss how these conditions and the interference pattern are modified by the CEP. This opens the door to control fundamental time-dependent electron dynamics in the tunneling regime in Dirac materials. Also, this suggests a way to measure the CEP of a terahertz laser pulse when it interacts with condensed matter systems.

Joint work C. Lefebvre, F. Fillion-Gourdeau, D. Gagnon and S. MacLean

A discrete-time Quantum Walk (QW) is essentially a unitary operator driving the evolution of a single particle on the lattice. Some QWs admit a continuum limit, leading to familiar PDEs (e.g. the Dirac equation). In this paper, we study the continuum limit of a wide class of QWs, and show that it leads to an entire class of PDEs, encompassing the Hamiltonian form of the massive Dirac equation in (1+1) curved spacetime. Therefore a certain QW, which we make explicit, provides us with a unitary discrete toy model of a test particle in curved spacetime, in spite of the fixed background lattice. Mathematically we have introduced two novel ingredients for taking the continuum limit of a QW, but which apply to any quantum cellular automata: encoding and grouping.

Quantum feedback control is challenging to implement as a measurement on a quantum state only reveals partial information of the state. A feedback procedure can be developed based on a trusted model of the system dynamic, which is typically not available in practical applications. We aim to devise tractable methods to generate effective feedback procedures that do not depend on trusted models. As an application, we construct a reinforcement-learning algorithm to generate adaptive for quantum-enhanced phase estimation in the presence of arbitrary phase noise. Our algorithm exploits noise-resistant differential evolution and introducesan accept-reject criterion. Our robust method shows a path forward to realizing adaptive quantum metrology with unknown noise properties.

Dirac equation, proposed by Paul Dirac in 1928, is a relativistic version of the Schroedinger equation for quantum mechanics. It describes the evolution of spin-1/2 massive particles, e.g. electrons. Due to its applications in graphene and 2D materials, Dirac equations has drawn considerable interests recently. We are concerned with the numerical methods for solving the Dirac equation in the non-relativistic limit regime, involving a small parameter inversely proportional to the speed of light. We begin with commonly used numerical methods in literature, including finite difference time domain and time splitting spectral, which need very small time steps to solve the Dirac equation in the non-relativistic limit regime. We then propose and analyze a multi-scale time integrator pseudospectral method for the Dirac equation, and prove its uniform convergence in the non-relativistic limit regime. We will extend the study to the nonlinear Dirac equation case.