Using quantum optimal control to drive intramolecular vibrational redistribution and to perform quantum computing

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Using quantum optimal control to drive intramolecular

vibrational redistribution and to perform quantum computing

Ludovic Santos

Promotrice: Prof. Nathalie Vaeck

Co-directrice: Prof. Mich`ele Desouter-Lecomte

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Abstract

Using quantum optimal control to drive intramolecular vibrational redistribution and to perform quantum computing

December 2017 - Universit´e Libre de Bruxelles - Ludovic Santos

Quantum optimal control theory is applied to find optimal pulses for control-ling the motion of an ion and a molecule for two different applications. Those optimal pulses enable the control of the dynamics of the system by driving the atom or the molecule from an initial state to desired states.

The evolution equations obtained by means of the quantum optimal con-trol theory are resolved iteratively using a monotonic convergent algorithm. A number of simulation parameters are varied in order to get the optimal pulses including the duration of the pulses, the time step of the time grid, a penalty factor that limits the maximal intensity of the fields, and a guess pulse which is used to start the optimal control. The optimal pulses obtained for each ap-plication are analyzed by Fourier transform, and also by looking at the time evolution of the populations that they generate in the system.

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experi-tained have a high fidelity, have a spectrum with well-resolved peak frequencies, and their experimental feasibility seems achievable within the current abilities of experimental laboratories.

The second application is to propose an experimental realization of a mi-croscopic physical device able to simulate quantum dynamics. The idea is to use the motional states of a Cd+ ion trapped in an anharmonic potential to realize a quantum dynamics simulator of a single-particle Schr¨odinger equation. In this way, the motional states store the information and the optimal pulse manipulates this information to realize operations. In the present case, the simulated dynamics was the propagation of a wave packet in a harmonic po-tential. Starting from an initial quantum state, the pulse acts on the system to modify the motional states of the ion in such a way that the final superposition of motional states corresponds to the results of the dynamics. This simula-tion is performed with the Liouville–von Neumann equasimula-tion and also with the Lindblad equation as dissipation is included to test the robustness of the pulse against perturbations of the potential. The optimal pulses that are obtained have a high fidelity which shows that the ion trap system has correctly realized the quantum dynamics simulation. The optimal pulses are valid for any initial condition if the potential of the simulation or the mass of the propagated wave packet is unchanged.

Keywords

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Acknowledgements

When I started my Ph.D., I knew that I would enjoy the work which would keep me busy for four years. And indeed, doing research freely, discovering small things, going abroad for conferences and finally, ending by building a scientific manuscript were a great pleasure. This was only possible due to the environment in which I have evolved during those years. Thereby, I would like to thank all who contributed to this fulfilling environment.

My first thoughts go to my advisor, Nathalie Vaeck which, beginning with my memory thesis, was always available despite her full agenda. Her optimism was valuable in the more difficult moments. Michèle Desouter-Lecomte is the other person who significantly contributed to the completion of this thesis. I would like to thank her for her precious advice. During my thesis, I had the op-portunity to exchange with other professors including Michel Godefroid, Michel Herman and Jean Vander Auwera with whom I had great scientific discussions. Far from my office, I also had some interesting talks with Daniel Hurtmans about Fourier transforms or for informatics advice, and also with Nicolas Ia-cobellis during his master thesis. Jérôme Loreau also deserved some acknowl-edgments as he is an altruistic person who helped me in various problems met during my thesis. At the end of my four years, I had the chance to supervise a memorant of my lab, Antoine. It was a real pleasure to work with him, to help him in his research, to enhance our codes and to prepare the transition for his thesis. Otherwise, I am grateful to Robin for his help during the redaction of this work.

I also need to thank the people with whom I have collaborated which include Yves Justum in LCP (Paris) and David Perry (Ohio), the F.R.I.A./FRS-FNRS of Belgium for its research grant and my university, the Universit´e Libre de Bruxelles, for its friendly work environment.

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List of publications

L. Santos, N. Iacobellis, M. Herman, D. Perry, M. Desouter-Lecomte, and N. Vaeck. A test of optimal laser impulsion for controlling population within the Ns= 1, Nr =

5 polyad of 12C₂H₂. Mol. Phys., 113(24):4000–4006, 2015. doi: 10.1080/00268976. 2015.1102980. URL http://dx.doi.org/10.1080/00268976.2015.1102980.

L. Santos, Y. Justum, N. Vaeck, and M. Desouter-Lecomte. Simulation of the elementary evolution operator with the motional states of an ion in an anharmonic trap. J. Chem. Phys., 142:134304, April 2015. doi: 10.1063/1.4916355. URLhttp: //scitation.aip.org/content/aip/journal/jcp/142/13/10.1063/1.4916355.

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List of Figures

3.1 Axis system and Euler angles (reproduced from [4]) . . . 33

3.2 Overview of the Zhu–Rabitz algorithm . . . 47

4.1 The optimal test pulse (a) along with its Fourier transform (b), its Gabor transform (c) and the time evolution of populations generated by the pulse (d) . . . 55

5.1 Intramolecular vibrational redistribution in the Ns = 1, Nr = 5 polyad of acetylene . . . 66

5.2 (a) Optimal field +for preparation of the target state φ+ with a maximal amplitude of 8.40_{× 10}8 V/m and a performance index I of 0.99999. (b) Optimal field − for the target state φ− with a maximal amplitude of 4.11_{× 10}8 _{V/m and a performance index} I of 0.99999. The duration τf of the fields is 24.2 ps. . . 74 5.3 Squared modulus of the Fourier transform of the field + (a) and

of the field − (b) . . . 75 5.4 Gabor transform of the field + obtained with three exact

Black-man time windows. . . 76

5.5 Evolution of the populations of the eigenstates for the field + (a) and for the field − (b). . . 77 5.6 The dynamical free evolution of the zero-order state (01011₁1₎

(a) and (01011₁−1_{) (b). In both cases, the duration of the} dy-namical free evolution is equal to 121 ps, beginning immediately following the control pulse at t = 24.2 ps and ending at 145.2 ps on the horizontal axis. Note that the ground state curve is not distinguishable from the x-axis on the graphs. . . 78

5.7 Hamiltonian matrix in the rovibrational approximation. . . 80

5.8 Magnitude in a.u. of transition dipole moment matrix elements of the basis of the control. . . 85

5.9 Sketch of the eigenlevels of the basis where the odd subset is depicted in the gray zone. The energies of the levels are not scaled. 88

5.10 Optimal field for populating the dark vibrational states (01011₁1_). The maximal amplitude is 1.57 107 _{V/m and the duration equals} 2.721 ns. . . 91

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5.12 Evolution of the populations in the rovibrational zero-order levels during the application of the optimal field for populating the dark vibrational state (01011₁1_{) of acetylene. The dotted lines are the} rovibrational levels which are not targeted and the dashed lines are the levels of the ground state. . . 93

5.13 Evolution of the populations in the vibrational zero-order states of the acetylene polyad Ns = 1, Nr = 5 after tracing the rota-tional levels. . . 94

5.14 Evolution of ¯J in the polyad during the application of the opti-mal field for populating the dark vibrational states (01011₁1_{) of} acetylene. . . 95

5.15 Field-free evolution of the vibrational zero-order states of the acetylene polyad Ns= 1, Nr= 5 : (a) after tracing the rotational levels of the basis; (b) with a zoom on a part of the graphic (a); (c) with only the rovibrational levels with J = 28; (d) with all the rovibrational levels. . . 96

6.1 Sketch of a linear Paul trap where the AC electrodes are depicted in orange, the Z-axis DC electrodes in blue and the circular DC electrodes in red (reproduced from [120]). . . 100

6.2 Potential (A) and pseudo-potential (B) produced by the AC elec-trodes (reproduced from [98]). . . 101

6.3 The anharmonic potential (Equation (6.1)) . . . 102

6.4 Motional wave functions of the ion in the anharmonic trap. . . . 103

6.5 Magnitude of the transition dipole moment matrix in a.u. . . 106

6.6 Sketch of the quantum dynamics simulator . . . 107

6.7 Links between the physical quantities of the simulated system and the physical system. . . 109

6.8 Mapping in the TDSE simulator. (a): harmonic (dashes) and anharmonic (full line) potentials of the Cd+ _{ion trap and the} superposed state Φ(Z, t = 0) with initial amplitudes cj(t = 0) = √∆xψ(xj, t = 0) in the j th motional eigenstate of the ion. The eigenenergies are indicated by arrows. (b): popula-tion _|cj(t = 0)|2 in the motional states|ji. (c): simulated system with a harmonic potential V (x) and initial localization probabil-ity _|ψ(xj, t = 0)|2. . . 111 6.9 Example of quantum dynamics simulator. . . 113

6.10 Evolution of the squared modulus of the motional wave func-tion _{|Φ(Z, t)|}2 after successive applications (l = 1, . . . , 10) of the rf -pulse driving the gate for the simulation of a coherent Gaus-sian wave packet in a harmonic potential (Figure 6.11a). The legend gives the pulse number. The wave function for t = 0 (red thick curve) is prepared by the initialization pulse so that cj(t = 0)/

√

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LIST OF FIGURES

6.11 Exact evolution of Gaussian wave packets (continuous lines) and results obtained from the mapping_|ψ(xj, t)|2 ↔ |cj(t)|2/∆xwith the populations of the ion eigenstates (markers) after application of the l th pulse. The legend gives the number l of the pulse. One observes the expected periodicity expressed in Equation (6.30). (a): coherent wave packet with σ = 1 a.u. ; (b): σ = 0.5 a.u. The applied field is EP(t) (Equation (3.47)). . . 118 6.12 Optimum fields and their spectrum _|S(ν)|2 in arbitrary units for

the preparation step and the simulation of the elementary trans-formation Us(∆t). (a) and (b): preparation of the initial wave packet _{|Φ(Z, t = 0)| for the simulation of the coherent Gaussian} (σ = 1 a.u. ); (c) and (d): field EF(t) (Equation (3.46)); (e) and (f): field EF(t) after filtering of the background and reoptimiza-tion; and (g) and (h): field EP(t) (Equation (3.47)). . . 119 6.13 Simulation in presence of dissipation due to fluctuating electric

fields. Exact evolution of the coherent wave packet (σ = 1 a.u.) (continuous lines) and results obtained from the populations of the ion eigenstates (markers) during the Lindblad dynamics with κ = 5_×10−18 a.u. (Equation (6.32)) (¯γ−1 = 55 ms) after applica-tion of the l th pulse. The legend gives the number l of the pulse. The applied field is EP(t) (Equation (3.47)). . . 121 6.14 Evolution of the fidelity (Equation (3.18)) during the simulation

of the coherent wave packet (σ = 1 a.u. ) for different values of the decoherence strength (Equation (6.32)) with the field calcu-lated by Equation (3.46) (dashed lines) or Equation (3.47) (full lines). . . 122

6.15 Evolution of the mean position of the ion_{hZ(t)i (a) (Section}2.1.4) and of the mean position of the coherent wave packet _{hx(t)i (b)} without decoherence (κ = 0) and for different values of the deco-herence strength (Equation (6.32)). The field is calculated using EP(t) (Equation (3.47)). . . 123 6.16 Squared modulus of the Fourier transform of the field ELP(t)

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List of Tables

1 Physical and mathematical constants [97] . . . 13

2 Atomic units [96] . . . 14

4.1 The main parameters and their limits . . . 54

5.1 The normal modes of vibration of acetylene. . . 59

5.2 Non-diagonal elements of the GAH [3, 112, 4]. Ri is the value of the effective parameter of the term which gives at same J an order of magnitude for the different resonances. . . 63

5.3 Vibronic transition dipole moment Dve for any vibrational tran-sitions with the zero-order states of the polyad Ns= 1, Nr = 5. 69 5.4 Energies of the states in the basis set of the control in _S_acev . . . 70

5.5 Summary of the parameters chosen for the simulation on the system in the vibrational approximation _S_acev . . . 73

5.6 Target populations of the optimal process for the target states φ+ and φ−. . . 76

5.7 Eigenlevels for J = 28 of the acetylene polyad Ns = 1, Nr = 5. . 81

5.8 Basis set of the control with the energies of each state. There are 52 rovibrational levels in the basis including 13 rovibrational ground levels and 39 rovibrational eigenlevels of the polyad Ns = 1, Nr = 5. . . 82

5.9 Truncated Boltzmann distribution of the rotational levels of the first three vibrational states of acetylene at 298K along the nor-malized distribution. . . 86

5.10 Coefficients of the linear distribution of the eigenlevels Γ1, Γ2, Γ3 for J = 28− 32 corresponding to the dark state (0101111) of acetylene. . . 87

5.11 Transition wavenumbers of peaks used in the guess field to start the control. Each transition is from or to the ground state Γ0. . 90

5.12 Summary of the parameters chosen for the simulation on the system in the rovibrational approximation _S_acev . . . 91

6.1 Energies of the motional states. . . 104

6.2 Assignation of the 4-qubit system. . . 112

6.3 Transition frequencies of peaks in the guess field. . . 115

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D.1 Eigenlevels of the basis . . . 144

D.2 Hamiltonian of the system in the rovibrational approximation_Srv ace145 E.1 Transition dipole moments matrix of the trapped ion in atomic

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List of definitions and

abbrevia-tions

Adaptive Feedback Control . . . 16

bra-ket notation . . . 23 superposition state . . . 23 pure state . . . 24 mixed state . . . 24 superoperator notation . . . 26 QOCT . . . 32 ZRA . . . 32

laboratory axis system . . . 33

molecular axis system . . . 33

MTOCT . . . 37

Off-resonant peaks . . . 55

exact Blackman window . . . 56

bright modes . . . 58 dark modes . . . 58 zero-order Hamiltonian . . . 59 zero-order state . . . 61 eigenstate . . . 65 IVR . . . 66

linear Paul trap . . . 100

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Units and constants

Physical and mathematical constants used in this work are shown in Table 1. Most of the calculations in this work are performed in atomic units. Those atomic units are shown in Table2. This system units is defined by setting the value of the four fundamental atomic units to unity,

~ = 1 (1)

me = 1 (2)

e = 1 (3)

ke = 1. (4)

Name Symbol Value Units

Boltzmann constant k 1.38064852(79)_{× 10}−23 J K−1 Speed of ligth in vacuum c 299792458 (exact) m s−1 Vacuum permittivity 0 8.854187817· · · × 10−12 F m−1

Pi π 3.141592653589793 . . .

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Chapter

1

Introduction

Nowadays, control of chemical reactions can be achieved using laser design with several techniques. Quantum optimal control theory is one of those and is applied in this work to find optimal pulses for controlling the motions an ion of cadmium and a molecule of acetylene. This means that the optimal pulse will drive the atom and the molecule from an initial state to desired final states which can be particular motional states, rotational states or vibrational states. In other words, control of the dynamics of the atom and the molecule is achieved. In the following lines, the origin of this method will be presented beginning from the birth of the laser control.

The first laser was realized in 1960 by T. Maiman [92]. Since then, chemists have wanted to use this new device to control reaction pathways. The idea was revolutionary as the experimenter was no longer limited to the observer role but could also become an actor of the chemical reactions. This goal was not easily achieved because the road to practical implementation was difficult [86,

16, 168, 17]. However, some early schemes made a breakthrough. Brumer and Shapiro proposed to use quantum interference to select between two reaction pathways. Their setup was composed of two monochromatic lasers of tunable intensities and phases, crossing each other to produce the quantum interference that leads to the coherent control of the system [21, 22, 122]. Some years later, Tannor, Kosloff and Rice proposed to use two femtosecond laser pulses with a time delay between them in a scheme called pump-dump [145, 146, 72, 122]. The role of the first laser was to populate an intermediate excited state until the second laser sends the population to the desired product channel. Other methods such as the stimulated Raman adiabatic passage [43,73,44,122] were developed but what greatly increased the control capabilities was the possibility to shape the laser pulses.

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eight-ies [133, 113, 129, 169, 72, 59]. This time, the quantum interference that pop-ulates the desired product was created by the shaping of the laser pulse, and especially the frequency and the phase of its components. Indeed, by proper shaping, a pulse could include different frequencies that could activate different transitions. The phase could also be tuned to induce the transitions at different times. This shaping has been encouraged by the significant advances in the experimental technology of femtosecond laser pulse shaping [66, 161, 162].

At the same time, a theoretical method to shape laser pulses appeared and became the leading one to achieve control over a quantum system [133,113,129,

72,59,130,131,31,132,47,141]. This method was different because in place of handling one control parameter (the phase difference between monochromatic fields or the time delay between pulses, for example) to reach the target, the pulse was built using multiple control parameters arising from the character-istics of the system and of the target of the control. This theoretical tool is called quantum optimal control theory (QOCT) and originates from the well-established optimal control used in engineering [115]. The tool is capable of shaping optimal pulses with a complex form both in time and frequency do-mains. The control objective is reached using the phase and amplitude of the frequency components of the field involving usually multiple quantum pathways. Numerical simulations of quantum optimal control requires knowledge of the Hamiltonian of the molecule to find optimal pulses but are interesting to suggest new experiments by obtaining insight on the feasibility of the control. It can also provide information on the quantum dynamics controlled as it simulates the re-action mechanisms. Today, the experimental realization of the control via laser pulse can be achieved using the Adaptive Feedback Control (AFC) [64] which is an empirical method of optimization. Indeed, the method relies on a feedback loop where the efficiency of the pulse is measured to improve the field through a learning algorithm (genetic algorithm [118]) during repeated experiments. AFC has proved its efficiency in numerous quantum applications [118, 46, 19, 105]. However, without a proper trial pulse, it is sometimes difficult to find the solu-tions in complex systems [117]. Moreover, the measurement of the efficiency of the pulse to realize the control objective is sometimes not possible because it destructs the control product as predicted by quantum mechanics, for example for the measurement of a quantum superposition. Therefore, the use of QOCT to suggest new experiments on complex systems is useful and offers an under-standing of the controlled mechanics as an asset. Much effort is also made to produce optimal pulses taking into account experimental implementation of the pulses [155,135].

In the recent years, different algorithms realizing QOCT have emerged to find the optimal pulse. The main ones are the gradient-ascent method [133,

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efficiency as the previous ones were sometimes leading to various computational efforts or were even unable to converge [173]. In the class of methods that constitutes the monotonic convergent algorithms, the central algorithm used in this work is the Zhu–Rabitz algorithm [176]. It will be presented in details.

Quantum optimal control theory using the Zhu–Rabitz algorithm is used in this work for two applications with distinct goals. In the first application, the aim is to find an optimal pulse to prepare vibrational dark states of the acetylene molecule by using a full dimension Hamiltonian calibrated by high-resolution spectroscopy.

Acetylene is a light and linear molecule, and resides at a border between the diatomic molecule and larger polyatomic molecules. It was an ideal tar-get for spectroscopic experimenters [53, 54, 109,160] who have tried to obtain high-quality spectroscopic data and to develop quantum models based on those data. Although relatively simple, it has also seven vibrational degrees of free-dom including those who are degenerate, allowing complex mechanism like the intramolecular vibrational redistribution (IVR) of populations between excited states. Consequently, the study of the molecule has been favored as the knowl-edge acquired could pave the way to the comprehension of larger molecules. Numerous studies have resulted in to the construction of a global Hamiltonian with an experimental precision typically between 10−4 and 10−3 cm−1 [5, 6].

To be realistic, theoretical simulations require a very precise molecular Hamil-tonian and are therefore restricted to atoms and small molecules [71, 70, 142,

170] or are performed in reduced dimensionality [75, 24]. Control in a real environment still remains a challenging topic. In the present case, this global acetylene Hamiltonian of full dimensionality (GAH) is used in the control giving very precise information of the controlled system and consequently, powering the QOCT with an uncommon precision.

This Hamiltonian is organized in polyads which are groups of states inter-acting together but isolated from the other states [52, 50, 51]. A particular polyad is the scope of the present work as this polyad contains a minimal set of vibrational states that exhibits non-trivial dynamical behavior. This polyad includes a bright state, which is reachable by transition from the ground vi-brational state, and two dark states, which are not reachable from the ground vibrational state. Those dark states are the target of the control and would remain unpopulated by usual spectroscopy.

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maximize the population in the vibrational dark states of acetylene will lead to an optimal pulse that could be produced experimentally.

Quantum optimal control theory using the Zhu–Rabitz algorithm is applied in a second application to propose the realization of a quantum dynamics sim-ulator using the motional states of a cadmium ion trapped in an anharmonic trap.

The proposal fits in the development of quantum computing and especially in the recent advances in quantum trap technologies. Quantum computing in-tends to encode information in atoms and molecules by making use of their quantum nature to realize logic operations [78]. These atoms and molecules then constitute a quantum computer and can be used to build quantum sim-ulators which are particular quantum computers design to simulate quantum systems. Today, those quantum simulators are believed to be able to improve the computational power for particular tasks such as the many-body problems. One of them is the simulation of quantum dynamics and it is the one targeted in this work.

The research on quantum computing gathers many fields because nowadays, different hardware are competing but none has emerged as the successful one. A lot of hope is placed in the trapped ions since the initial proposal by Cirac and Zoller [26] and the Nobel prize received by Wineland and his group for their achievements in trapping ions [95, 100, 84]. Since then, the research in this field continues to grow [57,39,80,81,99,49]. The trapped ions are an ideal system for quantum computing as they are isolated from their environment, and their quantum states can be modified by control fields. They also have a long coherence time which is an asset compared to the other supports of quantum computing [57]. Today, a lot of ions can be trapped like those with a single outer electron [57, 85].

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1.1. STRUCTURE OF THIS WORK

propagation of this simple wave packet. Otherwise, the simulation was tested against decoherence in order to reproduce experimental fields produced by the trap that could perturb the simulation.

This work was done in the Service de Chimique quantique et photophysique (CQP) of the Universit´e Libre de Bruxelles. The simulation were performed with QOCT codes originated from the Laboratoire de Chimie Physique (LCP) in Orsay with whom we were actively collaborating. Fruitful discussions occurred with the high-resolution infrared spectroscopy group of the CQP which provided the GAH data and insights about them. The research on the control of the acetylene dynamics begin to be rooted to the CQP as the project started with the master thesis of N. Iacobelis, before this work, and was continued during the master thesis of A. Aerts, at the end of this work. The project will go on as A. Aerts will start a Ph.D. research on this subject and as different research projects are funded all involving members of the CQP.

1.1 Structure of this work

This thesis is divided into seven chapters. After the first chapter that you will finish in a few lines, the Chapter2 will present some fundamentals of quantum dynamics which consists of the formalism and the mathematical representation used to perform the simulation. It will contain key explanations on the way the system is described. This chapter holds the mathematical ground for the optimal control theory that will be presented just after in Chapter 3.

To detail the method, subject of Chapter3, the way the efficiency of a pulse is measured, will be introduced. When this will be set into an expression it will be included in a functional. This functional is the starting point of the optimal control procedure as it contains the goal of the control and some constraints with in particular, a constraint on the way the system evolves. All the functionals of this work will be shown in this chapter. Then, the variational method will be applied on a functional to get evolution equations. Finally, the algorithm used to resolve those coupled evolution equations will be presented.

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In Chapter 5, the results of the first application, the preparation of the dark states, will be detailed after describing the acetylene molecule. In this description, the global Hamiltonian will be discussed. The specific polyad used in the control will be presented. The results for acetylene are divided into two parts. One for an approximation where the rotation was not yet included in the system. And one for the full calculation where the rotation was taken into account. At the end of this chapter, an overview will be given on the recent work realized with A. Aerts where dissipation processes are included.

The other chapter of the results (Chapter 6) will discuss the second appli-cation which use the motional states of the trapped ion. It will begin with the presentation of the cadmium ion along with the technology able to trap this ion. The structure of the Hamiltonian will also be shown. The results for this application are divided into two parts, one without dissipation and one with dissipation in the system.

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Chapter

2

Fundamentals of quantum dynamics

and the Liouville space _{L presented in Section} 2.2. In Section 2.3.3, another time evolution equation, the Lindblad equation will be introduced to deal with open quantum systems to take into account dissipative processes. At the end of this chapter (Section2.4), another type of evolution equations, which can speed up the calculation time of the simulations, will be presented.

2.1 Hilbert space

The complex Hilbert space_{H in which quantum mechanics formalism is usually} defined, is a linear vector space provided with a scalar product [48]. In this space, the state of a system is represented by a wave function or by a state vector. In this work, the Hilbert space _{H is used to describe the vibrational} polyad of acetylene in Section 5.1 and the trapped ion in Section6.2.

2.1.1 Wave function

The state of a system can be represented by a wave function ψ(r, t) which holds all the information on the system. This wave function measures the continuous probability amplitude to find a particle in r at time t. A probability density can also be defined as _{P(r) = |ψ(r, t)|}2.

In the present case, the wave function is described in the position space but other spaces such as the momentum space can be used. Therefore, this description need to be associated to an axis system. To simplify the formalism, another space has been introduced to described the quantum state of a particule without referring to the axis system of the position space. This abstract space is called the vector space. It is composed of state vectors which are a generalization of the wave function as, on the contrary to the wave function, they can describe the state of more systems like those including spin degrees of freedom.

2.1.2 State vector

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2.1. HILBERT SPACE

depicted by_{h·|·i. This notation is called the bra-ket notation and was proposed} by Dirac [37].

In this space, the state of the system is given by a state vector_{|ψ(t)i where} ψ(t) referred to the wave function ψ(r, t) without explicitly indicates the refer-ence to the position space. Indeed, if the state vector is decomposed in a basis of the eigenvectors of the position operator in 3D space ˆr,

|ψ(t)i = Z

d3r_{hr| ψ(t)i |ri ,} (2.1)

one finds the wave function as ψ(r, t) =_{hr| ψ(t)i.} The state vector can be expressed as

|ψ(t)i =X n

cn(t)|uni (2.2)

where_|uni are basis vectors which form an orthonormal basis in an n-dimensional Hilbert space and cn(t) are the coefficients of the basis vectors. The state vector needs to be normalized so the cn(t) follows the relation,

X n

|cn(t)|2 = 1. (2.3)

Those state vectors correspond to motional states for the trapped ion (Chap-ter 6) and to vibrational or rovibrational states for acetylene (Chapter5).

A state vector can also be expressed as a linear combination of state vectors. This linear combination is called a superposition state and is expressed as

|ψi =X k

ck|ψki (2.4)

where ck are the coefficients of each state vector |ψki.

2.1.3 Time-independent Schr¨

odinger equation

The time-independent Schr¨odinger equation is given by, ˆ

H_{|ψi = E |ψi .} (2.5)

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2.1.4 Time-dependent Schr¨

odinger equation

The time evolution of the state vector is given by the time-dependent Schr¨odinger equation,

i~∂

∂t|ψ(t)i = ˆH(t)|ψ(t)i (2.6)

driven by the Hamiltonian ˆH(t). This equation which is central in quantum mechanics, is an ordinary and linear first-order differential equation in t. Con-sequently, if the state vector _{|ψ(t)i is known at t = 0, it can be determined} for all other times [27]. Note that all linear combinations of the eigenstates are solution of this equation.

2.2 Liouville space

In Liouville space _{L [}8], state vectors are replaced by density operators to de-scribe the system. The_{L space is the Cartesian product of two Hilbert spaces} and is, therefore, a matrix space. The _{L space is used with the rovibrational} polyad of acetylene when the state of the system is represented by mixed states (Section 2.1). The computations in Liouville space are slower as the dimension is double in comparison with the Hilbert space.

2.2.1 Density operator

Up to now, a state vector, whether it was a superposition state or not, was the description of a system in a pure state in the sense that the quantum state was known. In addition to the description of a system in a pure state, the density operator can characterize the system in a statistical mixture of states called a mixed state. This description takes into account a classical probability for the system to be in a specific state. It differs from the probability amplitude for a system in a quantum superposition to be in one of the states of the superposition.

Like the state vector, the density operator contains all the information on the system and is given by

ˆ

ρ(t) =_{|ψ(t)i hψ(t)|} (2.7)

for a system in a pure state. In this case, it corresponds to the projection operator onto the state _{|ψ(t)i. If the state vectors are decomposed into the} basis vectors_|uni, one obtains

ˆ

ρ(t) =X n,m

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2.2. LIOUVILLE SPACE

where * indicates the conjugate. The density operator can also be described by a matrix in the basis _{|uni}, called the density matrix where the elements of the matrix are written as

ρnm(t) =hum| ˆρ(t) |uni = c∗n(t)cm(t). (2.9)

The general form of the density operator, which holds for a system in a mixed state and also for a system in a pure state (all pk equal zero except one) is given by,

ˆ

ρ(t) =X k

pkρˆk(t) (2.10)

where k is the number of pure states in the statistical mixture, pk is their classical probability and ˆρk(t) is their density operator. Note that the sum of the probabilities is equal to one P

kpk = 1. This density operator can also be represented by a matrix in the basis_{|uni}, where the elements with n = m are the populations of the states and are defined as

ρnn(t) = X

k

pk|cnk(t)|2. (2.11)

They represent the probability of the system to be in the basis state_|uni if the state of the system is _|ψki. In a density matrix, the elements with n 6= m are the coherences between the states and are given by,

ρnm(t) = X

k

pkc∗nk(t)cmk(t). (2.12)

Those coherences represent the quantum interference between the basis states and are responsible for various quantum properties of the system such as quan-tum superposition or also quanquan-tum entanglement.

The trace of an operator, noted Tr, is equal to the sum of its diagonal elements. For the density matrix, the trace is always Tr [ ˆρ] = 1 by definition. The trace of the squared density matrix, called the purity, is Tr [ ˆρ2] = 1 for a pure state and Tr [ ˆρ2_] _{≤ 1 for a mixed state. Note that dissipative processes} that will be presented in Section 2.3 lead to a non unitary evolution of the system and will affect the purity of the density matrix.

2.2.2 Liouville–von Neumann equation

In Liouville space, the evolution of the system can be obtained by the Liouville– von Neumann equation [89, 175, 134],

i~∂

∂tρ(t) =ˆ

h ˆ_{H, ˆ}_ρ(t)i

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where [_{·, ·] corresponds to a commutator.}

Note that the factor ₋i ~

h ˆ_H,_·i

can be represented equivalently by a super-operatorLˆˆnd 1 acting on the density operator |ρ(t)ii,

∂

∂t|ρ(t)ii = ˆ ˆ

Lnd|ρ(t)ii . (2.14)

by using the superoperator notation [30, 90, 103, 8]. In this notation, the op-erators are represented by _{|·ii and by hh·|, whereas superoperators which act} on other operators are depicted with a double hat O. Moreover, the doubleˆˆ bracket scalar product_{hhA|Bii is given by the trace of the product of the} ele-ments Tr [A_{× B].}

In this way, superoperators are considered as operators and operators are considered as vectors in a super Hilbert space formed by the direct product of the original Hilbert space and its dual. To express the Liouville–von Neumann equation in the superoperator notation shows that this equation for matrices (density operators) is similar to the Schr¨odinger equation (Equation (2.6)) for vectors (state vectors).

2.3 Liouville space with dissipation

Microscopic systems always interact with their environment by causing some-times great changes in the system. One of the effect of this interaction is the appearance of decoherence which is purely a quantum effect. The decoherence alters the coherence of the system and can especially destroy quantum superpos-tion of states [125]. Indeed, it affects the non-diagonal elements of the density matrices. The interaction with the environment can also affect consequently the dynamics of the system by generating dissipation or creating fluctuations in the system. The dissipation processes cause a non-unitary evolution of the system by losing energy to the environment, whereas the fluctuations arise from random processes in the environment causing an infinitesimal variation of the observ-ables of the system. Unlike the dissipation, the variation averaged over time of the fluctuations is null. Note that those interactions lead also to decoherence in the system.

For some systems studied in this work, the influence of the environment through dissipative processes has been analyzed. Those systems are treated as open quantum systems interacting with their environment. The dynamics of the system-environment combination is hard to analyze because of the complexity of the environment. That is why master equations offer a convenient way to treat

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2.3. LIOUVILLE SPACE WITH DISSIPATION

with open quantum systems as they express the time evolution of the reduced density matrix of the system [8, 125]. The advantage is that one only needs to know how the environment is acting on the system without describing it. The knowledge of the initial density matrix and the ability to solve the master equation are sufficient to determine all the dynamics of the system at any time t. Compared to the evolution of closed systems of the previous Sections 2.1.4

and 2.2.2, the evolution of a master equation will be non-unitary.

In this work, dissipative processes are taken into account thanks to the Lind-blad master equation for the trapped ion when dissipation is included in the system. This equation follows the Markovian dynamics and to use it, two ap-proximations need to be done. It is the Born and Markov apap-proximations [125] which are presented in Sections2.3.1and2.3.2. Other more complex Markovian master equations can be used like the Redfield equation [119, 18] but they are not investigated in this work.

2.3.1 Born approximation

The Born approximation, which is also called the weak coupling assumption, concerns the interaction of the system with the environment [104, 8, 125]. Firstly, the approximation states that the coupling between the system and the environment needs to remain weak. Secondly, the environment needs to be large enough so that changes of the density matrix ρE are negligible. This approximation implies that

ˆ

ρ(t)_{≈ ˆρ}S(t)⊗ ˆρE (2.15)

where ˆρS(t) and ˆρE are the density operators describing the state of the system and the environment, respectively. ˆρEhas no time-dependency as it is considered constant over time.

2.3.2 Markov approximation

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the system created by the interaction with the environment. In this case, the information transferred to the environment can never come back to the system.

2.3.3 Lindblad equation

The Lindblad master equation gives the evolution of a system in the Born and Markov approximations including dissipation processes [89, 175, 134]. It is the most general master equation that ensures positivity of the reduced density matrix ˆρS(t) [125]. This equation was chosen to take into account dissipation and is given, in the superoperator notation, by

∂ ∂t|ρ(t)ii = _ˆ ˆ Lnd+Lˆˆd |ρ(t)ii . (2.16)

The action of the Lindblad superoperatorLˆˆ=Lˆˆnd+Lˆˆd

on the density opera-tor is composed of two parts that are expressed, without using the superoperaopera-tor notation, by:

• one unitary part (Lˆˆnd|ρ(t)ii) which is the right part of the Liouville–von Neumann equation,

− i ~

h ˆ_{H(t), ˆ}_ρ(t)i

. (2.17)

• one non-unitary part (Lˆˆd|ρ(t)ii) which represents the decoherence and will contain the dissipation processes,

X jk ˆ Ljkρ(t) ˆˆ L†_jk − 1 2 h ˆ ρ(t), ˆL†_jkLˆjk i + (2.18)

where ˆLjk are the Lindblad operators that describe the dissipative pro-cesses and [_{·, ·]}₊ is an anti-commutator.

2.4 Interaction picture

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2.4. INTERACTION PICTURE

operators. Therefore, there is an evolution equation for the states but also for the operators. These two pictures differ in the way of looking at the time evolution, but the choice of the picture does not affect the physical quantities measured for each observable. In the interaction picture, the vectors will be depicted with the symbol _{∼ above them, the operators will be represented by}

˜

O in place of ˆO and the superoperator by O in place of˜˜ O. In this work, theˆˆ interaction picture is used to speed up the numerical calculation in comparison with the Schr¨odinger picture.

In the interaction picture, it is useful to decompose the Hamiltonian in a part independent of time and another interaction part with time dependency,

ˆ

H(t) = ˆH0+ ˆH1(t). (2.19)

Therefore, the state vector in interaction picture

˜

ψ(t)E can be defined as,

˜ ψ(t) E ≡ e~i ˆ H0t |ψ(t)i . (2.20)

Accordingly, the density operator in interaction picture ˜ρ(t) [8] is given by

˜

ρ(t) = e~iH0tˆ ρ(t)eˆ − i

~H0tˆ (2.21)

and the operators by

˜ O(t) = e~i ˆ H0t_O(t)e_ˆ −i ~ ˆ H0t_. _(2.22)

One consequence of the last equation is that the Hamiltonian ˆH0 is invariant under the transformation to the interaction picture.

The evolution equations presented in the previous Sections2.1.4,2.2.2and2.3.3, can also be expressed in interaction picture.

2.4.1 Time-dependent Schr¨

odinger equation in

interac-tion picture

The time-dependent Schr¨odinger equation in interaction picture is defined as

i~∂ ∂t ˜ ψ(t)E= ˜H1(t) ˜ ψ(t)E (2.23)

where the interaction Hamiltonian in interaction picture is equal to ˜ H1(t) = e i ~H0ˆ tHˆ₁(t)e− i ~H0tˆ . (2.24)

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2.4.2 Liouville–von Neumann equation in interaction

pic-ture

In interaction picture, the Liouville–von Neumann equation looks like ∂ ∂t| ˜ρ(t)ii = ˜ ˜ L1| ˜ρ(t)ii . (2.25) where L˜˜1 is equivalent to −i ~ h ˜_H₁_,_·i

when the superoperator notation is not used. The development to obtain this equation is shown in AppendixB.2.

2.4.3 Linblad equation in interaction picture

In interaction picture, the Lindblad equation is defined as, ∂

∂t| ˜ρ(t)ii = ˜ ˜

L1| ˜ρ(t)ii +L˜˜d| ˜ρ(t)ii (2.26)

whereL˜˜d =Lˆˆdas shown in AppendixB.3along with the development to obtain this equation. If there is no time-dependency in the Hamiltonian as in a field-free evolution, the equation reduces to,

∂

∂t| ˜ρ(t)ii = ˆ ˆ

Ld| ˜ρ(t)ii . (2.27)

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Chapter

3

Quantum optimal control theory

-QOCT

In this chapter, the theory of QOCT will be described to show how this optimization method can obtain solutions for the control of quantum dynam-ics. First, the interaction Hamiltonian used to take into account the laser field along with the approximation set to use this Hamiltonian will be presented in Section3.1. Then, all the knowledge to build a functional which is the start-ing point of the QOCT, will be shown. After that, the different functionals used in this work will be described. In the Sections 3.9 and 3.10, the evolution equations will be obtained from the functional using the variational method. Finally, the Zhu–Rabitz algorithm (ZRA) used to get the optimal pulse, will be discussed in details in Section 3.11. This algorithm has been used in place of the common gradient-ascent method [133, 72, 59, 28, 116] or the Krotov method [72, 139, 147] as it offers a fast quadratic monotonic convergence with a high accuracy in only a few iterations. Another difference with those algo-rithms is the way to incorporate the feedback of the action of the pulse in a cycle, which consists of a forward and backward propagation of the evolution equations. The gradient-ascent method includes the feedback every cycle and calculates one new version of the field by cycle. The Krotov method includes the feedback at each time step and determines one new version of the field by cycle. Finally, the ZRA includes the feedback at each time step and calculates two new versions of the field by cycle. For a deeper analysis, the reader can refer to the following references [122,74].

3.1 Perturbed Hamiltonian and electric dipole

approximation

The control of the dynamics, in the QOCT approach, is carried out by coupling the system with a linearly polarized field (t) with a finite duration. In con-sequence, the system undergoes a perturbation. The total Hamiltonian taking into account this perturbation is,

ˆ

H(t) = ˆH0+ ˆV (t) (3.1)

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3.1. PERTURBED HAMILTONIAN AND ELECTRIC DIPOLE APPROXIMATION

The interaction of the field with the system is considered in the electric dipole approximation [27, 62] as some higher interaction terms are not take into account. This mean that the spatial variation of the radiation field is neglected across the atom or the molecule. Note also that the magnetic part of the electromagnetic field is considered to be negligible.

The system of coordinates used to describe this interaction is the laboratory axis system (LAS) composed of the d = X, Y , Z axis. This system of coor-dinates is fixed in space, on the contrary of the molecular axis system (MAS) which is fixed to the molecule and is composed of the α = x, y, z axis. The link between the two axis systems is defined using the Euler angles θ, φ, χ as shown in Figure 3.1. Indeed, the MAS axis system is transformed to the LAS axis system through a rotation given by the direction cosines matrix λd,α [167].

Figure 3.1: Axis system and Euler angles (reproduced from [4])

As the field vector is oriented in the Z direction of the LAS, the projection of the dipole moment is considered along the Z axis. Therefore, the perturbation is given by,

ˆ

V (t) =_−ˆµZ(t) (3.2)

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Finally, the total Hamiltonian is then the following, ˆ

H(t) = ˆH0− ˆµZ(t). (3.3)

3.2 Path and target state

The optimized dynamics is,

|ψ(t = 0)i (t)→ |ψ(t = τf)i (3.4)

where τf is the duration of the pulse that drives the evolution. This equation defines a path. At the end of the pulse, the goal is to obtained,

|ψ(τf)i = |φfi (3.5)

where_|φfi is the target state which can be an eigenstate but also a normalized superposition of eigenstates. In Section 3.6, the multi-target optimal control theory is presented where multiple paths are optimized with one pulse.

3.3 Type of optimization

There are two classes of optimization. The first one is the optimization of the expectation value of an observable Dψ_{| ˆ}O_|ψE and the second one is the optimization of an unitary transformation ˆO_{|ψi. The choice of the operator ˆ}O defines what will be optimized and therefore, the goal of the control.

Expectation value of an observable

In the case of the expectation value of an observable, the goal can be very diverse like, for example, the projection of populations to a state with the projection operator, the maximization of the density at one point of the space with the local operator or also the inversion of populations. In this work, the preparation of a specific target state _|φfi of acetylene (Chapter 5) is made using the projection operator,

ˆ

P =_|φfi hφf| (3.6)

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3.4. PERFORMANCE INDEX

Unitary transformation

In the case of the unitary transformation, the target state is determined by applying a Hermitian operator on the initial state. This operator can also be very diverse as the unitary transformation can correspond, for example, to a quantum gate like the CNOT, Hadamard or also the QFT gate, in a qubit system [104]. In this work, the unitary transformation, denoted Us(∆t), is presented in details in Section6.3.2. Note already that the multi-target optimal control explained in Section3.6 will be used for this unitary transformation as the operator Us(∆t) acts simultaneously on multiple paths.

3.4 Performance index

When the target of the control is defined, the performance index can give the accuracy of the optimal field to realize the operation. The performance index is built from the operator ˆO chosen for the optimization and a quantum state at the final time of the simulation τf. For the optimization of the expectation value of an operator, the performance index is given by,

I =_hψ(τf)| ˆO|ψ(τf)i . (3.7) In the case of the projection operator, ˆO = ˆP with a projection onto the state |φfi, the performance index reduces to,

IP r=|hφf |ψ(τf)i|2. (3.8)

This integral quantifies the overlap between the target state_|φfi and the prop-agated state at the end of the control_|ψ(τf)i. Note that IP r is invariant under the transformation

|φfi → eiγ|φfi (3.9)

where γ is a phase varying between 0 and 2π. For the optimization of an unitary transformation, the performance index is also the overlap between the target state and the propagated state at the end of the control.

3.5 Basic objective functional in Hilbert space

To get evolution equations of the system under optimization of the pulse, an objective functional F needs to be defined. This functional is the sum of an objective and some constraints Cn on the dynamics of the system

F = I +X n

Cn . (3.10)

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3.5.1 Constraints

Constraint on the field intensity

The intensity of the field needs to be limited to stay in a perturbative regime as seen in Section3.1. Another reason is to ensure the feasibility of the optimal pulse to be reproduced experimentally. For this purpose, the total integrated intensity of the pulse is limited by the constraint CI which is included in the functional in the following way,

CI =− Z τf

0

α(t) _|(t)|2 dt (3.11)

with α(t) = _s(t)α0 where α0 is the penalty factor, a parameter that controls the strength of the constraint, and s(t) is a temporal sine function envelope given by sin2(πt/τf) [143]. The use of a temporal envelope ensures that the pulse switches smoothly on and off which is another requirement for experimenters. Other temporal envelopes could have been used to get better Fourier transform spectrum of the optimal pulses which are presented in Section 4.4.

Constraint to follow an evolution equation

In the functional, a constraint is included to force the system to follow an evo-lution equation which will always be satisfied during the entire process [176]. In Hilbert space, the evolution equation is the time-dependent Schr¨odinger equa-tion shown in Secequa-tion 2.1.4. The constraint in the Schr¨odinger case, looks like

CS =−2 Re hφf| ψ(τf)i Z τf 0 λ(t) ∂ ∂t+ i ~ ˆ_H₀_{− ˆµ}_Z_(t) ψ(t) dt (3.12)

where Re [_{·] is the real part of the content in the square brackets and λ is a} Lagrange multiplier introduced for the variational method. The λ corresponds to the wave function that will be obtained iteratively backward in time which is therefore called the backward wave function. The constraint is multiplied by two because the complex conjugate of ψ(t) needs also to follow the evolution equation. Furthermore, Zhu, Botina and Rabitz [176] have introduced a factor, the decoupled factor, which sits in front of the integral, to decouple the bound-ary conditions between the evolution equations presented in Equations (3.43) and (3.44); i.e. to set the final condition for the backward propagation to φf.

Other constraints

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3.6. MULTI-TARGET QOCT

on the spectral range has for goal to limit to a range of frequencies of the optimal field obtained after optimization. This spectral limitation can be needed for experimental reproduction of the pulse. The constraint on the area of the pulse ensures that the total time-integrated area of the field tends to nearly zero in order to produce only optical fields [142]. In the present case, those constraints were not applied. Indeed, for the first one, the spectral range produced was admissible without the constraint, and for the second one, the area of the pulse obtained without the constraint was nearly zero. More constraints or filtering techniques can be found in the work of Werschnik and Gross [163].

3.5.2 Basic functional

Given the simplest performance index IP r and the two primary constraints, the functional can be written as,

F =_|hφf |ψ(τf)i|2− Z τf 0 α(t) _|(t)|2 dt − 2 Re hφf|ψ(τt)i Z τf 0 λ(t) ∂ ∂t + i ~ ˆ_H₀_{− ˆµ}_Z_(t) ψ(t) dt . (3.13) It corresponds to the functional proposed in the article which presents the Zhu– Rabitz algorithm for the control of population [176].

3.6 Multi-target QOCT

To realize an unitary transformation, multiple paths need to be optimized at the same time. Each one of these paths corresponds to the evolution of a basis state in which the unitary evolution is defined, and the field obtained after optimization is common for all paths. Moreover, regardless of the initial state, the optimal field needs to drive the system to the target state of each path. To do so, the multi-target optimal control theory (MTOCT) is used [163,148,60]. New objectives need to be defined and new functionals need to be chosen.

In the optimal control, the objective does not always take into account the phase as shown in Section3.4. However, it is a requirement when multiple paths are optimized and when the phase of the final state needs to be controlled. Therefore, special functionals are built where the relative phase of the final states:

|φi,1i → |φf,1i eiγ1 |φi,2i → |φf,2i eiγ2

. . . _{→ . . .}

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are taken into account in the performance index. In this work, two different performance indexes for the MTOCT case have been used based on the projec-tion operator. One is the sum of transiprojec-tion probabilities and the other is called the fidelity. Each one takes into account the phase of each path as required in a MTOCT objective functional.

3.6.1 Sum of transition probabilities

The performance index based on the sum of the transition probabilities [149,

171] is a sum of the different paths and can be expressed as,

IP = N +1 X k=1 |hφf,k| ψk(τf)i| 2 N (3.14)

where_hφf,k| are the target states and |ψk(τf)i are the propagated states at the final time for each path k. The sum is over N + 1, where N is the number of paths, because a supplemental path (N + 1) is added to ensure a good phase control such that the unitary transformation is valid for any superposition,

|φi,si → |φf,si (3.15)

to take into account the phase of the different final states [171]. Without this supplemental path, the performance index would be insensitive to the phase as the square modulus of the overlap of the states is taken [157]. This path is defined thanks to _|φi,si which holds for the normalized sum of the initial states and_|φf,si which is calculated from the normalized sum of the final states including a phase factor,

|φi,si = 1 √ N N X k=1 |φi,ki (3.16) |φf,si = 1 √ N N X k=1 |φf,ki eiγk (3.17)

where γk are phases varying between 0 and 2π.

3.6.2 Fidelity

The normalized fidelity IF is a performance index that takes directly into ac-count the phase [107, 110, 111, 151],

IF = N X k=1 hφf,k| ψk(τf)i N 2 (3.18)

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3.7. QOCT IN LIOUVILLE SPACE

3.6.3 Functionals

Based on those two performance indexes, two functionals FP and FF were de-fined in Hilbert space,

FP = N +1 X k=1 |hφf,k| ψk(τf)i|2− Z τf 0 α(t) _|(t)|2 dt − 2 Re "_{N +1} X k=1 hφf,k| ψk(τf)i Z τf 0 hλk(t)| ∂ ∂t + i ~ ˆ_H₀_{− ˆµ}_Z_(t) |ψk(t)i dt # (3.19) FF = N X k=1 hφf,k| ψk(τf)i 2 − Z τf 0 α(t) _|(t)|2 dt − 2 Re " _N X j=1 hφf,j| ψj(τf)i N X k=1 Z τf 0 hλk(t)| ∂ ∂t + i ~ ˆ_H₀_{− ˆµ}_Z_(t) |ψk(t)i dt # (3.20) with the second term and last term being the constraint on the intensity of the field and the constraint to follow the Schr¨odinger equation, respectively. The difference between the two functionals is the position of the decoupled factor that is in or outside the sum of the third part of the equation.

Notice that the coefficients of normalization of the performance indexes are not included in the functionals as they will not affect the variational method and the resulting evolution equations obtained with it.

3.7 QOCT in Liouville space

In Liouville space, Ohtsuki and coworkers [108] have shown that the evolution equations obtained after applying the variational method on functionals are similar to those obtained for the Hilbert space. The difference being that in Liouville space, density operators are used in place of the state vectors and that the evolution equations are based on functionals where the Schr¨odinger equation is replaced by a master equation.

3.7.1 Functional

For the Liouville space, the performance index is defined as IL=|hhρf| ρ(τf)ii|

2

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where ρf is the target density matrix and the double bra-ket in the superoper-ator notation is given by Trhρ†_tρ(τf)

i .

The constraint CI is still the same than in Equation (3.11). But the con-straint CS is different as the Schr¨odinger equation is replaced by the Liouville– von Neumann equation presented in Section 2.2.2.

Therefore, the functional for the QOCT in Liouville space is defined as

where_{|η(t)ii is a Lagrange multiplier that corresponds to the density operator} |ρ(t)ii resolved backward in time.

3.7.2 Uhlmann Fidelity

For density operators that represent mixed states, the performance index IL is less appropriate as it does not reflect the likeness of two mixed states accurately. Therefore, the Uhlmann fidelity [63] as defined by Nielsen and Chuang [104] is used to measure the efficiency of the pulse,

IU = T r " q√ ρfρ(τf)√ρf # . (3.23)

for density operator of mixed states. This performance index is not used in the functional because the conservation of the monotonic convergence of the algorithm with this performance index have not been investigated in this work since it requires the completion of a long and complicated mathematical proof.

3.7.3 Dissipation

Once in Liouville space, dissipation can be included in the Hamiltonian. The transformation does not affect the general form of the functional or the evolution equations obtained as the only modification for both is

ˆ ˆ

Lnd(t)→Lˆˆnd(t) +Lˆˆd (3.24)

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3.8. MULTI-TARGET QOCT IN LIOUVILLE SPACE WITH DISSIPATION

3.8 Multi-target QOCT in Liouville space with

dissipation

MTOCT can also be performed in Liouville space with dissipation. In this case, the functionals are based at the same time on those from Sections 3.6 and 3.7

where evolution equations are replaced by the Lindblad equation presented in Section 2.3.3. They are given by:

FLP = N +1 X k=1 |hhρf,k| ρk(τf)ii| 2 − Z τf 0 α(t) _|(t)|2 dt_{− 2 Re} "_{N +1} X k=1 hhρf,k| ρk(τf)ii Z τf 0 ηk(t) ∂ ∂t+ _ˆ ˆ Lnd(t) +Lˆˆd ρk(t) dt # (3.25) FLF = N X k=1 hhρf,k| ρk(τf)ii 2 − Z τf 0 α(t) _|(t)|2 dt_{− 2 Re} " _N X j=1 hhρf,j| ρj(τf)ii N X k=1 Z τf 0 ηk(t) ∂ ∂t+ _ˆ ˆ Lnd(t) +Lˆˆd ρk(t) dt # . (3.26)

Those functionals are used for the trapped ion when some dissipation is included in the system (Section 6.6).

3.9 Variational method

When the functional is defined, the next step is to maximize this functional to get evolution equations using the variational method [143],

δF (λ, ψ, ) = 0 (3.27)

where δF are small variations against each dependency of the functional. There-fore, three equations need to be resolved, one for each dependency of the func-tional (δλF = 0, δψF = 0, δF = 0) [9, 143]. For the basic functional (Sec-tion 3.5.2), the development is shown in the following lines. For the other functionals, the development is explained in AppendixA.

The first equation to be resolved is

δλF = F [ψ(t), λ(t)+δλ(t), (t)]− F [ψ(t), λ(t), (t)] = 0 (3.28) which can be expressed after development of the F functional as follows,

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The integral is null only if ψ(t) is solution of the time-dependent Schr¨odinger equation with_{|ψ(t = 0)i = |φ}ii as initial condition,

i~∂

∂t|ψ(t)i = ˆH0 − ˆµZ(t)

|ψ(t)i , |ψ(0)i = |φii . (3.30)

The second equation is the variation of F in function of the state ψ(t), δψF = F [ψ(t)+δψ(t), λ(t), (t)]− F [ψ(t), λ(t), (t)] = 0 (3.31) which can be developed in,

δψF = 2 Re [hφf|ψ(τf)i hφf|δψ(τf)i] − 2 Re hφf|δψ(τf)i Z τf 0 λ(t) ∂ ∂t + i ~ ˆ_H₀_{− ˆµ}_Z_(t) ψ dt − 2 Re hφf|ψ(τf)i Z τf 0 λ(t) ∂ ∂t + i ~ ˆ_H 0− ˆµZ(t) δψ dt = 0. (3.32)

Following Equation (3.30), it is known that the second term is null. Moreover, as δψ is arbitrary, it is chosen independent of time. Therefore, the partial derivative ∂/∂t in the third term disappears. The equation reduces to,

δψlF = 2 Re [hφf|ψ(τf)i hφf|δψi] − 2 Re hφf|ψ(τf)i Z τf 0 λ(t) i ~ ˆ_H₀_{− ˆµ}_Z_(t) δψ dt = 0 (3.33) and i ~ ˆ_H₀_{− ˆµ}_Z_(t)

can be placed in the bra vector_{hλ(t)| in the second term} to get, δψlF = 2 Re " hφf|ψ(τf)i hφf|δψi + Z τf 0 i ~ ˆ_H₀ _{− ˆµ}_Z_(t) λ(t) δψ dt !# = 0. (3.34)

This equation can only be true if λ(t) follows the time-dependent Schr¨odinger equation with the target state_{|λ(t = τ}f)i = |φfi as final condition,

i~∂

∂t|λ(t)i =

ˆ_H₀_{− ˆµ}_Z_(t)

|λ(t)i , |λ(τf)i = |φfi . (3.35)

In practice, this equation is resolved backward in time since it obeys a final condition. In consequence, λ(t) is called the backward wave function.

Using quantum optimal control to drive intramolecular vibrational redistribution and to perform quantum computing