Physique quantique: de la théorie à la technologie

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Layout of this dissertation

The content of this thesis has been organized as follows:
Chapter 3: We start with a Hamiltonian identification problem. We prove that identifying the Hamiltonian from population measurements using a family of dis-criminating control fields is a well posed problem. These results are based on [Leghtas 2012d].
Chapter 4: We then move on to a state preparation problem. We propose a control field which could perform any state permutation on a multi-level system. The power of this scheme is that this one field may act on an ensemble of systems with different parameter values, and permute every single one of them. This result appeared in [Leghtas 2011].
Chapter 5: Here we tackle the problem of quantum state stabilization. Report-ing the results of [Sarlette 2012], we describe a scheme for stabilizing superposition states and entangled states in a cavity. In this aim, we suggest tailoring the inter-action of the cavity with a controlled environment: a stream of two level atoms. This proposal doesn’t require any real time data processing and feedback, making it very appealing for experimental implementation. This controlled dissipation forces

Layout of this dissertation

any initial state in the cavity to converge towards the desired target state. The convergence is proved where the cavity state space is an infinite dimensional Hilbert space, having to overcome technical problems such as the non compactness of the unit ball.
Chapter 6: In this final chapter, we start with a state preparation scheme and then see how this can be used to stabilize a whole manifold of states, thus performing quantum error correction. We propose a sequence of pulses which prepares any superposition of non overlapping coherent states in a microwave resonator. This preparation scheme can also be applied to generate entangled states of two modes and entangled states of an arbitrary large number of qubits. This was the subject of a recent paper [Leghtas 2012b]. We then show that these coherent state superpositions can be used for efficient quantum error correction [Leghtas 2012c]. This brings us one step closer to a readily realizable quantum memory.
Each chapter can be read independently. A conclusion in chapter 7 gives an insight on future work.

Enhanced observability utilizing quantum control

Ce chapitre traite l’identification d’Hamiltonien pour un système commandable avec des transitions non dégénérées et un état initial connu. On considère un seul contrôle scalaire et une mesure de population à instant T arbitrairement grand.
On démontre que la matrice de moment dipolaire est localement observable: pour tout couple de matrices différentes mais suffisamment proches, il existe un con-trôle qui permet d’obtenir deux mesures différentes. Ce résultat suggère qu’une sim-ple mesure de population à un seul instant peut être transformée en une source d’information très riche permettant l’identification unique du moment dipolaire, lorsque cette mesure est précédée d’une commande bien choisie. Ce chapitre est basé sur [Leghtas 2012d], qui paraîtra comme une note technique dans IEEE Trans-actions of Automatic Control. Ce résultat a été obtenu en collaboration avec Gabriel Turinici, Herschel Rabitz et Pierre Rouchon.
This chapter considers Hamiltonian identification for a controllable quantum system with non-degenerate transitions and a known initial state. We assume to have at our disposal a single scalar control input and the population measure of only one state at an (arbitrarily large) final time T. We prove that the quantum dipole moment matrix is locally observable in the following sense: for any two close

Enhanced observability utilizing quantum control

but distinct dipole moment matrices, we construct discriminating controls giving two different measurements. This result suggests that what may appear at first to be very restrictive measurements are actually rich for identification, when combined with well designed discriminating controls, to uniquely identify the complete dipole moment of such systems. This chapter is based on [Leghtas 2012d], which is about to appear as a technical note in IEEE transactions of automatic control. This result was obtained in collaboration with Gabriel Turinici, Herschel Rabitz and Pierre Rouchon
Quantum control has been receiving increasing attention [Brif 2010] and one of its promising applications is to Hamiltonian identification [Warren 1993] by using the ability to actively control a quantum system as a means to gain information about the underlying Hamiltonian governing its dynamics. The underlying premise is that controls may be found which make the measurements not only robust to noise but also highly sensitive to the unknown parameters in the Hamiltonian. Hence, although the performance of laboratory measurements may be constrained, the abil-ity to control a quantum system has the prospect of turning this data into a rich source of information on the system’s Hamiltonian.
In this chapter, we consider the problem of identifying the dipole moment (which is assumed to be real) of an N −level quantum system, initialized to a known state (ground state), from a single population measurement at one arbitrarily large time T . We suppose an ability to freely control the system with a time dependent electric field u(t). The measurements are obtained by (i) initializing at time t = 0 the system’s state at a known state |i , (ii) controlling in open loop and without measurement the system with an electric field u_k (t) for t ∈ [0, T ] where T > 0, and (iii) measuring at final time T the population of one state |f . This may be repeated for many controls (u_k )_k . We prove the existence of controls which make the identification from one population measurement a well posed problem (theorem 3.1). These controls have a simple physical interpretation in analogy with Ramsey interferometry (see Fig. 3.1).
The perspective above combined with control theory is motivated by three prac-tical arguments. First, measuring a state population at one time T is a technique which can have a very high signal to noise ratio (∼ 100). Second, technological progress with spatial light modulators (SLM) permits generating a broad variety of controls in the laboratory. Third, ultra short pulsed fields can be well measured in the laboratory [Iaconis 1998]. Hence, we are able to design a variety of precisely known control inputs.
Le Bris et al [LeBris 2007] prove the observability of the dipole moment when the population of all states are measured over an arbitrarily large interval of time. Algorithms to reconstruct the dipole from the measured data were proposed using nonlinear observers [Leghtas 2009, Bonnabel 2009]. A different setting is considered in [Schirmer 2010a, Schirmer 2010b] where it is supposed that one can prepare and measure the system in a set of orthogonal states at various times, and the available data is the probability to measure the system in a certain state when it was prepared in another; Bayesian estimation is used to reconstruct the energy levels, the damping constants and the dipole moment from the measured data. We consider here the less demanding case where the only available measurement is the population of one state at one arbitrarily large time, and the initial state is known and coincides with the ground state. We may summarize the scheme to identify an arbitrary matrix element l|µ|k of the dipole moment operator µ by the following:

1. We use the controllability of the system to steer it from the ground state to state |l .
2. “Gently” Rabi flop the transition |k → | l using the assumed unique transition frequency in a way that does not affect nearby transitions.
3. Finally use the controllability of the system to steer the system to a state which is detectable by the measurement apparatus: population of a state |f .
4. From the measured population, deduce the dipole matrix element.

The chapter is organized as follows. In subsection 3.2 we state the main result in Theorem 3.1, and subsection 3.3 gives the proof of the Theorem and an important lemma on which the main result is based. Finally concluding remarks are presented in subsection 3.4.

Observability of the quantum dipole moment

Problem setting

We consider a quantum system in a pure state described by the wave function |ψ ∈ S . Here S is the set of N dimensional complex vectors of unit norm. The system interacts with an electric field (the real control input) u ∈ U_T for some T > 0 with U_T ≡ {f : [0, T ] → R |f piecewise continuous }. For a given control u we measure the population of the state |f at time T denoted as P_if (u). We denote by H₀ the free Hamiltonian, due to the kinetic and potential energy of the system (Hermitian matrix) and by µ the dipole moment operator, also a Hermitian matrix. In the notation of (2.1), we have µ = H₁ and u(t) = u₁(t). This choice was made to adapt to the notation used in the mathematical physics community. The initial state |i and the measured state |f are eigenvectors of H₀. We consider a semi-classical model for the light-matter interaction
For all T > 0, we suppose that we can create any field u ∈ U_T and that we can measure P_if (u). For M different fields {u₁, .., u_M } we can collect the measurements {P_if (u₁), .., P_if (u_M )}. Through (3.1), P_if is a function of µ and a functional of u, and when necessary this explicit dependence will be written as P_if (u, µ). The aim of this chapter is to explore the feasibility of estimating the dipole moment µ from the measured data {P_if (u₁), .., P_if (u_M )} using well chosen controls {u₁, .., u_M }. Below, P_if (u, µ) refers to the measurement achieved on the real system using a control u, and for any µˆ, P_if (u, µˆ) is the estimated measurement which is obtained by simulating system (3.1) with the control u and coupling µˆ.

Table of contents :

1 Introduction (version française) 
1.1 Physique quantique: de la théorie à la technologie
1.2 Le contrôle quantique
1.2.1 Formulation générale
1.2.2 Vue d’ensemble
1.3 Contributions
1.4 Plan de la thèse
2 Introduction 
2.1 Quantum physics: from theory to technology
2.2 Quantum control theory
2.2.1 General formulation
2.2.2 Brief Overview
2.3 Contributions
2.4 Layout of this dissertation
3 Enhanced observability utilizing quantum control 
3.1 Introduction
3.2 Observability of the quantum dipole moment
3.2.1 Problem setting
3.2.2 Main result
3.3 Proofs
3.3.1 Existence of discriminating controls
3.3.2 Proof of Theorem 3.1
3.3.3 Proof of lemma 3.1
3.3.4 Proof of lemma 3.2
3.4 Conclusion
3.5 Definitions and computation
4 Adiabatic passage and ensemble control 
4.1 Introduction
4.2 Problem setting
4.2.1 Standard formulation
4.2.2 Change of frame
4.3 Robust ensemble transfer from |ki to |N − k − 1i
4.3.1 Transfer Theorem
4.3.2 Simulations
4.4 Robust ensemble transfer from |li to |pi
4.4.1 From |0i to any |pi
4.4.2 From any |li to any |pi
4.4.3 Simulations
4.5 Robust ensemble permutation of populations
4.5.1 Permutation theorem
4.5.2 Example and simulations
4.6 Proofs
4.6.1 Proof of Theorem 4.1
4.6.2 Proof of Theorem 4.2 and corollary 4.1
4.7 Summary and discussion
5 Stabilizing non-classical states in cavity QED 
5.1 Proposal to stabilize non-classical states of one- and two-mode radiation fields by reservoir engineering
5.1.1 General description
5.1.2 Engineered reservoir for coherent state stabilization
5.1.3 Kerr Hamiltonian simulation in the dispersive regime
5.1.4 Regime of arbitrary detunings
5.1.5 Decoherence and experimental imperfections
5.1.6 A reservoir for two-mode entangled states
5.1.7 Summary and discussion
5.1.8 Detailed computations
5.2 A first convergence proof
5.2.1 Problem setting
5.2.2 Main result
5.2.3 Proofs
6 Schrödinger cats and quantum error correction 
6.1 From cavity QED to circuit QED
6.2 Deterministic protocol for mapping a qubit to coherent state super- positions in a cavity
6.2.1 Introduction
6.2.2 The qcMAP gate
6.2.3 Preparing arbitrary superpositions of coherent states in one and two cavity modes
6.2.4 Using the qcMAP gate to entangle qubits
6.2.5 Summary and discussion
6.2.6 Detailed sequence for conditional displacements and preparation of superpositions of coherent states
6.3 Hardware-efficient autonomous quantum error correction
6.3.1 Introduction
6.3.2 Cavity logical 1 and logical 0, and MBQEC
6.3.3 Autonomous QEC
6.3.4 Encoding, decoding and correcting operations
6.3.5 Simulations
6.3.6 Conclusion
7 Conclusion 
Bibliography

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