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Luigi Frunzio THESE DE DOCTORAT DE L’UNIVERSITE PARIS-SUD XI spécialité: Physique des solides présentée par: Luigi Frunzio pour obtenir le grade de DOCTEUR de l’UNIVERSITE PARIS-SUD XI Sujet de la thèse: Conception et fabrication de circuits supraconducteurs pour l'amplification et le traitement de signaux quantiques soutenue le 18 mai 2006, devant le jury composé de MM.: Michel Devoret Daniel Esteve, President Marc Gabay Robert Schoelkopf Rapporteurs scientifiques MM.: Antonio Barone Carlo Cosmelli ii Table of content List of Figures vii List of Symbols ix Acknowledgements xvii 1. Outline of this work 1 2. Motivation: two breakthroughs of quantum physics 5 2.1. Quantum computation is possible 5 2.1.1. Classical information 6 2.1.2. Quantum information unit: the qubit 7 2.1.3. Two new properties of qubits 8 2.1.4. Unique property of quantum information 10 2.1.5. Quantum algorithms 10 2.1.6. Quantum gates 11 2.1.7. Basic requirements for a quantum computer 12 2.1.8. Qubit decoherence 15 2.1.9. Quantum error correction codes 16 2.2. Macroscopic quantum mechanics: a quantum theory of electrical circuits 17 2.2.1. A natural test bed: superconducting electronics 18 2.2.2. Operational criteria for quantum circuits 19 iii 2.2.3. Quantum harmonic LC oscillator 19 2.2.4. Limits of circuits with lumped elements: need for transmission line resonators 22 2.2.5. Hamiltonian of a classical electrical circuit 24 2.2.6. Quantum description of an electric circuit 27 2.2.7. Caldeira-Leggett model for dissipative elements 27 2.2.8. Beyond the harmonic oscillator: need for non-linearity 29 3. A non-linear circuit element: the Josephson tunnel junction 31 3.1. A model for the Josephson tunnel junction 32 3.2. Josephson equations 33 3.3. Quantum Hamiltonian of an isolated Josephson tunnel junction 34 4. Superconducting circuits based on the Cooper Pair Box 37 4.1. Different competing implementations of quantum computers 37 4.2. The Cooper Pair box 38 4.3. Manipulation of the state of a Cooper Pair box 41 4.4. Brief summary on other superconducting quantum circuits 42 5. Read-out strategy with the Cooper Pair Box 47 5.1. Read-out requirements 47 iv 5.2. Weakly coupled read-out of a Cooper Pair Box: charge strategy 49 5.3. Weakly coupled read-out of a Cooper Pair Box: loop current strategy 50 5.4. Dispersive read-out of Cooper Pair Box: non-linear inductive read-out in the RF-Quantronium architecture 53 5.5. Dispersive read-out of Cooper Pair Box: non-linear capacitive read-out in the Circuit QED architecture 57 5.6. Non-linear amplification of quantum signals: the Josephson Bifurcation Amplifier 60 6. Reprints of experimental and theoretical results on the Quantronium architecture with the JBA 63 7. Reprints of experimental and theoretical results on the Circuit QED architecture 89 8. Fabrication techniques used for this work 111 8.1. Substrates 111 8.2. Optical photolithography for resonator patterning 112 8.3. Capacitor fabrication 116 8.4. Dolan bridge technique to fabricate JTJ-based circuits 119 9. Conclusions and perspectives 125 Appendix: text and slides from the defense presentation 127 Bibliography 185 v vi List of Figures Figure 2.1.1. The Bloch sphere for a single qubit. Figure 2.1.2. The collapse of the Bloch sphere at measurement. Figure 2.2.1. Electrical circuit version of a harmonic oscillator. Figure 2.2.2. Convention on the branch variables. Figure 2.2.3. Caldeira-Leggett model for a dissipative element. Figure 3.1. Josephson tunnel junction. Figure 3.2. Josephson tunnel junction circuit model. Figure 4.1. The Cooper Pair Box. Figure 4.2. The split Cooper Pair Box. Figure 5.1. The “Quantronium” architecture. Figure 5.2. Dispersive read-out in the RF-Quantronium architecture. Figure 5.3. The Circuit Quantum Electrodynamics architecture. Figure 5.4. Dispersive read-out. Figure 5.5. The Josephson Bifurcation Amplifier. Figure 8.1. The double layer resist “mushroom”. Figure 8.2. Aluminum profile after lift-off. Figure 8.3. Edges of a Niobium resonator. Figure 8.4. Josephson Bifurcation Amplifier capacitor fabrication. Figure 8.5. Capacitors for a Quantronium circuit. Figure 8.6. Resist bilayer patterned for a double angle deposition. Figure 8.7. Undercut double layer resist. Figure 8.8. Josephson tunnel junction fabricated by double angle evaporation. vii viii List of Symbols α, β Superposition coefficients γ Rotation angle between frames in a Bloch sphere γ Qubit depolarization rate in the cQED reprints γˆ Gauge invariant phase difference operator Γ Decay rate of the excited state in a damped harmonic oscillator Γ1 Depolarization rate Γ2 Decoherence rate ON Γ1 Depolarization rate in the ON state OFF Γ2 Decoherence rate in the OFF state Γϕ Dephasing rate δ Gauge invariant phase difference ∆ Superconducting energy gap ∆ Detuning between the qubit and the resonator frequency δˆ Gauge invariant phase difference operator for the split Cooper pair box ˆ δi Gauge invariant phase difference operator for a single junction ∆δˆ Flux fluctuations ∆I01 Current difference between the qubit states ∆Ng Effective gate charge number change ∆U Barrier height in a JTJ ∆U01 Charge difference between the qubit states ε o Vacuum dielectric permeability ε r Relative dielectric permeability η, ξ, ζ Numerically determined constants η o Electromagnetic wave impedance in vacuum θ Latitude on the Bloch sphere ix θ Phase difference between the two superconducting electrodes θˆ Phase difference operator κ Photon decay rate in a 1D transmission line resonator λ Wavelength λcQED Resonator-qubit coupling in circuit QED λRF− Q Resonator-qubit coupling in RF-Quantronium µ o Vacuum magnetic permeability µ r Relative magnetic permeability ν01 Qubit transition frequency νr Resonator frequency νRabi Rabi frequency νRF Pulse frequency νs Spectroscopy tone frequency ρo Normal state resistivity of a metal ρ(ω,Τ) Residual resistivity of a BCS superconductor σˆi Pauli operators σ + Spin creation operator σ − Spin annihilation operator ϕ Longitude on the Bloch sphere Φ Magnetic field flux Φˆ Magnetic field flux operator Φ0 Superconducting magnetic flux quantum Φb (t) Branch flux Φext External magnetic flux Φl ,Φl External flux bias through a loop Φn (t) Node flux ˆ Φni, Node flux operators x Φm Intermediate node flux for the m-th oscillator ΦJ(t) Josephson element branch flux χ, χ(t) Bias current normalized to the critical current in a JTJ ψ , ψ Arbitrary normalized superposition ψˆ Unit norm state vector ω01 Transition angular frequency between the qubit states ωo Resonant angular frequency of the oscillator ωd Drive angular frequency ωp Bare plasma angular frequency ωr Resonator angular frequency ωRabi Rabi angular frequency ωRamsey Ramsey angular frequency ωRF Angular frequency of an RF voltage signal ωs Shifted resonant angular frequency of the oscillator “0” Classical zero state of bit “1” Classical one state of bit 0 Ground state of the qubit 1 Excited state of the qubit Aˆ Generic operator G a Representative vector of the operator Aˆ aˆ Photon annihilation operator aˆ† Photon creation operator G ai Components of a in a reference frame b Branch c Field propagation velocity in a medium C Capacitance CΣ Cooper Pair Box total capacitance xi Cg Cooper Pair Box gate capacitance Ci Cooper Pair Box island capacitance CJ Josephson tunnel junction capacitance Cm Capacitive component of the m-th oscillator CTL Transmission line total capacitance d Junction asymmetry coefficient D Number of dimensions G dl Line path element dt Infinitesimal time interval e Electron charge G E Electric field EB Barrier Energy EC Coulomb charging energy for one Cooper pair in a Cooper Pair Box d EC Detector junction charging energy ECJ Coulomb charging energy for one Cooper pair in an isolated Josephson tunnel junction EJ Josephson energy ∗ EJ Effective Josephson energy of a symmetric split Cooper pair box d EJ Detector junction Josephson energy EL Inductive energy Ez Minimum energy splitting of the qubit states F Read-out fidelity or discriminating power g Photon-qubit coupling in the circuit QED Hamiltonian G h Representative vector of the Hamiltonian operator = Planck constant divided by 2π G H Magnetic field ˆ Hc Noise coupling Hamiltonian xii ˆ HcQED Circuit QED Hamiltonian ˆ * HcQED Effective circuit QED Hamiltonian ˆ int HcQED QND probe operator in circuit QED G hi Components of h in a reference frame hJ(t) Energy stored in the Josephson element ˆ hJ ,sCPB Split Cooper Pair Box Hamiltonian operator ˆ * H JC Jaynes-Cummings Hamiltonian operator ˆ hJTJ Isolated Josephson tunnel junction Hamiltonian operator ˆ Hqubit Qubit Hamiltonian operator G hR,TOT Representative vector of the total Hamiltonian operator in the rotating frame ˆ HtRF ( ) Time dependent Hamiltonian operator ˆ * H RFQ− Effective RF-Quantronium Hamiltonian ˆ int H RF− Q QND probe operator in RF-Quantronium G ˆ hRFR, Representative vector of HtRF ( ) in the rotating frame ˆ H RFSQUID− RF-SQUID Hamiltonian h(X) Energy stored in a component hY Hamiltonian of the admittance Iˆ Identity operator Iˆ Current operator itb ( ) Branch current Ic Josephson tunnel junction critical current in Node current Irf(t) Current drive amplitude xiii i(t) Current I(t) Current drive Kˆ Tunneled Cooper pair number operator kB Boltzmann constant L Inductance l Circuit component size l Loop LJ Phase-dependent Josephson inductance LJ0 Josephson inductance Lm Inductive component of the m-th oscillator n Node n Number of photon in a resonator nˆ Number operator for photons in a harmonic oscillator Nˆ Cooper pair number operator NΦ Flux frustration Noff Residual offset
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