INTRODUCTION TO SUPERCONDUCTING CIRCUITS
The goal of this first part is to give an up-to-date introduction to the field of super-conducting circuits from an experimentalist perspective while focusing on the devices that are used in this thesis work. The first chapter will focus on the transmon qubit, which is the most widespread qubit in the superconducting circuit community. A lot of other types of qubits have very promising performances but they are not the subject of this thesis. The second chapter is dedicated to the readout of superconducting qubits with a particular emphasis on dispersive and fluorescence readout that are crucial to the second part of this thesis. The third chapter is dedicated to three-level systems. Most applications of superconducting circuits rely on quantum operations on two-level systems while transmons oﬀer many more levels that are individually addressable. In-creasing the size of the Hilbert space for quantum operations or quantum algorithms opens up new perspectives for quantum physics and quantum information processing. The fourth chapter is dedicated to pumped microwave circuits and more specifically to linear microwave amplifiers. Three kinds of amplifiers were used in this work and a non-exhaustive review is given on the current state-of-the-art of microwave amplification at the quantum level.
In this first chapter, we will introduce the transmon as an elementary unit instrumen-tal to circuit QED. We will start by introducing quantum circuits from the solid-state physics perspective and then we combine it with the universal quantum mechanics description valid for a wide range of platforms dealing with single quantum systems.
circuit quantum electrodynamics
The goal of cavity quantum electrodynamics (CQED) is to study the properties of light (photons) coupled to matter (electrons, atoms, …). This field has led to numerous ground-breaking experiments [12, 65] well described in books and reviews such as Ex-ploring the quantum by Serge Haroche and Jean-Michel Raimond . Subsequently, a new branch of this field emerged in 1999 with the invention of the first superconduct-ing qubit  later followed by the demonstration of strong coupling regime between a transmon and a resonator  and was dubbed circuit QED. This first section is dedicated to the quantum optics of microwave circuits with superconducting artificial atoms.
A first striking property of these circuits is that they are macroscopic quantum sys-tems. While they contain a large number of microscopic particles, they host macroscopic degrees of freedom that behave quantum-mechanically. Secondly, their properties are not set by fundamental constants like the Rydberg energy. They are engineered at will by design with the technology of microelectronic chips. Lastly, a truly remarkable level of control was achieved in these systems thanks to the exponential growth of coherence times of these devices known as Schoelkopf ’s Law . The system used throughout this thesis is the transmon1, which is nowadays the most commonly used superconducting qubit. This qubit was originally envisioned to be coupled to a 2D resonator connected to a transmission line  but it can also be embedded into a 3D electromagnetic or into a lumped mode  in order to straightforwardly increase the quality factor of the LC resonator. Transmons are usually made of a Josephson junction shunted by a large capacitance  and the circuit is shielded and anchored to the base plate of a dilution refrigerator at about 20 mK so that the energy of the thermal fluctuations is much smaller than the energy quantum at a few GHz ~! kBT .
Another important thing to mention about superconducting circuits is that it is one of the many candidate platforms that could lead to the advent of a universal quantum computer thanks to quantum error correction [68, 69]. A lot of landmarks were achieved such as the implementation of multi qubit algorithms  or gate fidelities suﬃcient for error correction  and few small uncorrected quantum processors are already available online but the race is still ongoing.
Quantum LC oscillator
A quantum LC oscillator is the simplest circuit element that can be built with capac-itors and inductors [73, 74] as depicted in Fig. 2.1 a. In practice, the dynamics of a cavity mode is modeled by a harmonic oscillator. The most commonly used resonators in circuit QED are 3D rectangular cavities machined out of aluminum (Fig. 2.1 b), 3D coaxial / 4 cavities (Fig. 2.1 c) and / 2 coplanar-wave-guide (CPW) resonators (Fig. 2.1 d). There are no resistors in the circuit representation because the cavities are made out of superconducting materials that ensure that the supercurrent flows with negligible dissipation in the circuit. Losses will be introduced as a perturbation. In the case of a rectangular cavity of dimensions lx, ly, lz, the limit conditions impose that the resonant frequency of the TEmnl and TMmnl modes are given by  c s✓ m⇡ ◆ 2 + ✓ ⇡ ◆ 2 + ✓ l⇡ ◆ 2 fnml = n (2.1) 2⇡ lx ly lz where the indices n, m, l refer to the number of anti-nodes in standing wave pattern in the x, y, z directions. A cavity with dimensions (lx, ly, lz) = (26.5⇥26.5⇥9.6) mm3 has a first TE110 mode at f1,1,0 = 8 GHz. Similarly, a coaxial / 4 cavity has a fundamental resonance frequency f0 ‘ 4.25 GHz for l = 20 mm  well separated from its first har-monic frequency f1 = 3f0 = 12.75 GHz. Finally, / 2 CPW resonators are planar trans-mission lines terminated by two open circuits loads. The two terminations are suﬃcient to create a standing wave as in a Fabry-Pérot cavity and a l = 20 mm resonator will have a fundamental mode at f0 = 8.5 GHz. The first harmonic is f1 = 2f0 = 17 GHz.
circuit quantum electrodynamics
Figure 2.1: a. Electrical circuit of an LC oscillator. This system is analogous to a mass-spring system in mechanics with position coordinate taken to be Φ, the magnetic flux through the coil and the momentum variable is Q, the charge accumulated on the capacitor. The role of the spring constant is played by 1/L and the mass is C. The standard electrical variable V and I are obtained by Hamilton’s equation (2.5). b. Picture of two blocks of aluminum forming a microwave cavity resonating at 8 GHz . The cavity is created with a drilling machine from a raw block of aluminum. A qubit can be inserted in the cavity before closing it with an indium seal. c. Picture of a 3D coaxial / 4 cavity . The length of the central pin determines the resonant frequency of the cavity. The electromagnetic mode is confined at the bottom of the cavity and it is evanescent from the pin to the top opening. d. Schematics of a / 2 coplanar waveguide (CPW) resonator surrounded by its ground plane. A superconducting material (in our group Nb or TiN) in green is deposited on a silicon substrate in gray. The impedance of the resonator is determined by the width of the resonator w, the size of the gap g, the height of the superconducting material and the nature of the substrate. e. Energy levels of an harmonic oscillator. The energy levels in dashed line are evenly spaced by ~!r. The wavefunction amplitudes of the diﬀerent Fock states are represented in orange as a function of the flux Φ. The number of nodes of the wave function is equal to the number of photons in the cavity.
In order to derive the Lindblad Eq. (2.12) the following hypotheses are required 
• We usually deal with Eq. (2.12) by dividing the time into ‘slices’ of duration dt and every evolution ⇢t from time t to t + dt is incremental. This ‘coarse-grained’ description of the first order diﬀerential Eq. (2.12) screens out high frequency component of the dynamics with ! 1/dt. We thus perceive the dynamics of the studied system through a filter and this description will be accurate only for dt ⌧ TH where TH is the typical time scale of evolution of the observables of ⇢ due to unitary evolutions or damping processes.
• The environment must be a ‘sink’. We assume that this large system has a great number of degrees of freedom (represented by the collection of discrete electro-magnetic modes in Fig. 2.2 b) and that the dynamics of the bath does not impact the dynamics of the system. This amounts to neglect the memory eﬀects of the bath (also called reservoir in statistical physics). Mathematically we denote by ⌧E the time scale of the fluctuations and correlations of the environment. The inequality ⌧E ⌧ dt is required to ensure that the environment is amnesic at the scale dt. We thus renounce to the microscopic description of fluctuations much faster than dt.
• There is no notion of quantum measurement in the derivation of Eq. (2.12). Con-tinuous quantum measurement will be introduced in Chapter 3 and the dynamics of the quantum state will be predicted by the stochastic master equation in chap-ter 6.
In the case of a cavity losing photons ‘one-by-one’ via a coupling to a transmission line, only one jump operator is non zero L = p aˆ where is the coupling rate to the transmission line. We will see the jump operators associated to a qubit in a following section and for any given system that satisfies the above-listed conditions, there exists a set of jump operators describing the decoherence of the system.
Cavity coupled to two transmission lines
In this thesis, we used two-port cavities with jump operators Lˆ1 = p 1aˆ and Lˆ2 = p 2aˆ. The input-output relation (see appendix A) p iaˆ = aˆini + aˆouti (2.13) relates the input and output propagating modes of the transmission line to the station-ary mode aˆ of the device. In the case of 3D resonators, the value of the coupling is determined by the length of the pins of the SMA connectors mounted on the cavity, while the coupling is given by a planar capacitance in the case of 2D resonators. In our case, we have ⇠ 2⇡ ⇥ 2.3 MHz for the most coupled port that will be used to collect the outgoing signals.
From Eq. (2.17), we can see that the transmission profile is always a Lorentzian in amplitude accompanied with a ⇡ phase shift (see Fig 2.3). The width of Lorentzian and the phase shift is always given by tot. In reflection on port 1, three regimes are observed.
• The over-coupled regime is defined by 1 L + 2. In this regime the losses are negligible so in reflection |S11(!)| = 1 and a 2⇡ phase shift is observed (blue curve in Fig 2.3).
• The critical coupling corresponds to 1 = L + 2. In this regime a ⇡ phase shift is observed in reflection while the amplitude vanishes at resonance (green curve in Fig 2.3).
• The under-coupled regime corresponds to L+ 2 1. A bigger dip in amplitude is observed in reflection due to the important losses with a phase shift ⇡ (red curve in Fig 2.3).
Now that we know how to deal with open quantum systems, we need to add a non linear element to complete the cQED toolbox. The transmon is an artificial atom made of a Josephson junction connected to two superconducting islands. In our case the aluminum/alumina/aluminum junction is typically 250 ⇥ 200 nm and the associated tunnel resistance at room temperature of the order of 2 to 8 k ⌦(see appendix B). The coupling Hamiltonian associated to the coherent tunneling of a Cooper pair through the barrier reads ˆ EJ +1 Htunneling = N X (2.18) 2 (|Ni hN + 1| + |N + 1i hN|) where EJ is a macroscopic parameter, which is proportional to the DC conductance Gn of the junction in the normal state, which can be adjusted during the fabrication process and to the superconducting gap Δ of the material EJ = ΔGn 8he2 . The state |Ni corresponds to exactly N Cooper pairs having passed through the junction.
Black-box quantization of a transmon embedded in a cavity
General theory with a single junction
Typical circuits used in our experiments have more degrees of freedom than a simple harmonic oscillator and all the complexity of the system can be captured in a ‘black box’. The black box quantization method was originally introduced by Nigg et al. . Gener-alizations of this method have been proposed by Solgun et. al.  and Malekakhlagh et. al.  to model more complex environment but we restrict the discussion to the Foster decomposition of the environment. Its principle relies on solving the linearized problem and treat the non linearity perturbatively. In the case of a linear circuit, knowing the impedance Z(!) or admittance Y (!) = Z 1(!) of a dipole black box connected as a function of frequency completely characterizes its quantum properties.
Table of contents :
1.2 Individual quantum systems
1.3 Decoherence and readout of a superconducting qubit
1.4 Quantum trajectories
1.5 Post-selected evolution
i measurement and control of superconducting circuits
2 introduction to superconducting circuits
2.1 Circuit quantum electrodynamics
2.1.2 Quantum LC oscillator
2.1.3 Open quantum systems
2.1.4 Cavity coupled to two transmission lines
2.2 Transmon qubit
2.2.1 Black-box quantization of a transmon embedded in a cavity
2.2.2 Finite element simulation – Energy participation ratios
2.3 Open system dynamics of a qubit
2.3.2 Entropy of a qubit
2.3.3 Decoherence mechanisms
3 readout of a superconducting qubit
3.1 Measuring a quantum system
3.1.1 Generalized measurement
3.1.2 Continuous measurement
3.2 Dispersive readout
3.2.1 Homodyne detection of the cavity field
3.2.2 AC stark shift and measurement induced dephasing
3.3 Measurement of fluorescence
3.3.1 Heterodyne detection of the fluorescence of a qubit
3.3.2 Destructive and QND measurements
3.4 Full quantum tomography
3.4.1 Direct access to the Bloch vector
3.4.2 Tomography of a qubit undergoing Rabi oscillations
3.4.3 Comparing the fidelities of a weak and projective quantum tomography
4.2 Preparation of an arbitrary quantum superposition of three levels
4.3 Projective tomography of a three-level system
4.3.1 From the measurement output to the density matrix
4.3.2 Representation of the density matrix of a qutrit
4.4 Quadrature plane calibration and temperature measurement
4.4.1 Calibration of the IQ plane
4.4.2 Direct temperature measurement
4.5 Open-system dynamics of a three-level atom
4.5.1 Lindblad equation
4.5.2 Energy relaxation
4.5.3 Ramsey experiments
4.6 Continuous measurement of a three-level system
4.6.1 Dispersive measurement of a qutrit
4.6.2 Quantum jumps between three levels
5 microwave amplifiers
5.2 Quantum parametric amplification
5.2.1 Phase preserving amplification
5.2.2 Phase sensitive amplification
5.3 Josephson parametric amplifier
5.3.1 Different JPAs and different pumping schemes
5.3.3 Frequency tunability
5.4 Josephson parametric converter
5.4.1 Josephson ring modulator
5.4.2 Amplification mode
5.4.3 Flux tunability
5.5 Travelling wave parametric amplifier
5.5.1 Phase matching condition
5.5.2 Amplification performance
5.6 Figures of merit of amplifiers
5.6.1 Amplifying setup
5.6.3 Quantum efficiencies
5.6.4 Dynamical bandwidth
5.6.5 Static bandwidth
5.6.6 Dynamical range
5.6.7 Comparison between two detection chains and a JTWPA
ii measurement back-action
6 quantum trajectories
6.1 Quantum back-action of measurement
6.1.1 Kraus operators formulation
6.1.2 Dispersive interaction
6.1.3 Measurement along the orthogonal quadrature
6.1.4 Fluorescence signal
6.2 Quantum trajectories
6.2.1 Repeated Kraus map and Markov chain
6.2.2 The stochastic master equation
6.2.3 From measurement records to quantum trajectories
6.2.4 Validation by an independent tomography
6.2.5 Parameter estimation
6.3 Trajectories statistics
6.3.1 Different regimes
6.3.2 Zeno dynamics – Interplay between detectors
6.3.3 Rabi oscillations
6.3.4 Exploration of several regimes
6.4 Diffusion of quantum trajectories
6.4.2 Fokker-Planck equation
6.4.3 Impact of the efficiencies on the statistics
6.4.4 Convection velocity
6.4.5 Diffusion tensor
6.4.6 Dimensionality of the diffusion
6.4.8 An Heisenberg-like inequality for pure states
7 post-selected quantum trajectories
7.1 Past quantum state
7.1.1 Prediction and retrodiction
7.1.2 Continuous time dynamics
7.2 Pre and post-selected trajectories
7.2.1 Time symmetric Rabi evolution
7.2.2 « Anomalous » weak values
7.2.3 Influence of the post-selection
a transmon coupled to a transmission line
a.1 Classical equation of motions
a.1.1 Dynamics of the system
a.1.2 Asymptotic expansion in ✏
a.2 Quantum description
b experimental techniques
b.1.1 Fabrication of JPA and JPC
b.1.2 3D transmons
b.1.3 2D CPW chips
b.2 Measurement setup