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Circuits Et Signaux Quantiques Quantum

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Circuits Et Signaux Quantiques Quantum Chaire de Physique Mésoscopique Michel Devoret Année 2008, 13 mai - 24 juin CIRCUITS ET SIGNAUX QUANTIQUES QUANTUM SIGNALS AND CIRCUITS Première leçon / First Lecture This College de France document is for consultation only. Reproduction rights are reserved. 08-I-1 VISIT THE WEBSITE OF THE CHAIR OF MESOSCOPIC PHYSICS http://www.college-de-france.fr then follow Enseignement > Sciences Physiques > Physique Mésoscopique > PDF FILES OF ALL LECTURES WILL BE POSTED ON THIS WEBSITE Questions, comments and corrections are welcome! write to "[email protected]" 08-I-2 1 CALENDAR OF SEMINARS May 13: Denis Vion, (Quantronics group, SPEC-CEA Saclay) Continuous dispersive quantum measurement of an electrical circuit May 20: Bertrand Reulet (LPS Orsay) Current fluctuations : beyond noise June 3: Gilles Montambaux (LPS Orsay) Quantum interferences in disordered systems June 10: Patrice Roche (SPEC-CEA Saclay) Determination of the coherence length in the Integer Quantum Hall Regime June 17: Olivier Buisson, (CRTBT-Grenoble) A quantum circuit with several energy levels June 24: Jérôme Lesueur (ESPCI) High Tc Josephson Nanojunctions: Physics and Applications NOTE THAT THERE IS NO LECTURE AND NO SEMINAR ON MAY 27 ! 08-I-3 PROGRAM OF THIS YEAR'S LECTURES Lecture I: Introduction and overview Lecture II: Modes of a circuit and propagation of signals Lecture III: The "atoms" of signal Lecture IV: Quantum fluctuations in transmission lines Lecture V: Introduction to non-linear active circuits Lecture VI: Amplifying quantum signals with dispersive circuits NEXT YEAR: STRONGLY NON-LINEAR AND/OR DISSIPATIVE CIRCUITS 08-I-4 2 LECTURE I : INTRODUCTION AND OVERVIEW 1. Review of classical radio-frequency circuits 2. Quantum information processing 3. Quantum-mechanical LC oscillator 4. A non-dissipative, non-linear element: the Josephson junction 5. Energy levels and transitions of the Cooper Pair Box 6. Summary of questions addressed by this course 08-I-5 OUTLINE 1. Review of classical radio-frequency circuits 2. Quantum information processing 3. Quantum-mechanical LC oscillator 4. A non-dissipative, non-linear element: the Josephson junction 5. Energy levels and transitions of the Cooper Pair Box 6. Summary of questions addressed by this course 08-I-5a 3 CLASSICAL RADIO-FREQUENCY CIRCUITS CommunicationCommunications Hergé, Moulinsart 1940's : MHz 2000's : GHz Computation Sony US Army Photo 08-I-6 SIMPLEST EXAMPLE : La radio? Mais c'est très simple! Eisberg, 1960 LC THE LC OSCILLATOR 08-I-7 4 WHAT IS A RADIO-FREQUENCY CIRCUIT? A RF CIRCUIT IS A NETWORK OF ELECTRICAL ELEMENTS node n element V branch np loop np Inp node p TWO DYNAMICAL VARIABLES CHARACTERIZE THE STATE OF EACH DIPOLE ELEMENT AT EVERY INSTANT: p JG J JG Signals: Voltage across the element: Vtnp ()= E ⋅dA any linear ∫n GJJG combination of these variables Current through the element: Itnp ()=⋅jdσ np ∫∫ 08-I-8 KIRCHHOFF’S LAWS c b ba b d c d ba ∑ Vλ = 0 ∑ Iν = 0 branches λ branchesν around loop tied to node 08-I-9 5 CONSTITUTIVE RELATIONS EACH ELEMENT IS TAKEN FROM A FINITE SET OF ELEMENT TYPES EACH ELEMENT TYPE IS CHARACTERIZED BY A RELATION BETWEEN VOLTAGE AND CURRENT LINEAR NON-LINEAR inductance: V = L dI/dt diode: I=G(V) capacitance: I = C dV/dt s d transistor: g Ids = G(Igs,Vds,) Vds resistance: V = R I Igs = 0 08-I-10 GENERALIZATIONS: MULTIPOLES, PORTS AN ELEMENT CAN BE CONNECTED TO MORE THAN TWO NODES dipole quadrupole hexapole etc.... n-1 currents n poles n-1 voltages ALSO, CONSTITUTIVE RELATION OF ELEMENT NEED NOT BE LOCAL IN TIME WILL DEAL WITH THESE HIGHER DEGREES OF COMPLEXITY LATER 08-I-11 6 COMPUTING THE DYNAMICAL STATE OF A CIRCUIT IN CLASSICAL PHYSICS specify choose constitutive solve for circuit set of relations currents with independent + and sources variables Kirchhoff’s voltages laws in circuit 08-I-12 COMPUTING THE DYNAMICAL STATE OF A CIRCUIT IN CLASSICAL PHYSICS specify choose constitutive solve for circuit set of relations currents with independent + and sources variables Kirchhoff’s voltages laws in circuit example : RCL circuit I t () φ (t) : flux thru inductance V φ(t) : voltage V across inductance φφ current thru inductance φ / L CItφ ++=() RL current thru resistance φ / R ETC... current thru capacitance Cφ 08-I-12a 7 OUTLINE 1. Review of classical radio-frequency circuits 2. Quantum information processing 3. Quantum-mechanical LC oscillator 4. A non-dissipative, non-linear element: the Josephson junction 5. Energy levels and transitions of the Cooper Pair Box 6. Summary of questions addressed by this course 08-I-5b THE POWER OF QUANTUM SUPERPOSITION REGISTER WITH N=10 BITS: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 2N = 1024 POSSIBLE CONFIGURATIONS classically, can store and work only on one number between 0 et 1023 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 08-I-13 8 THE POWER OF QUANTUM SUPERPOSITION REGISTER WITH N=10 BITS: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 2N = 1024 POSSIBLE CONFIGURATIONS classically, can store and work only on one number between 0 et 1023 1 1 1 1 1 1 1 1 1 0 “quantally”, can store and work on an 1 1 1 1 1 1 1 1 1 1 arbitrary superposition of these numbers! N 0 1 2 .....N 2 1 Ψ=ααα012 + + + + α21− − 08-I-13a QUANTUM PARALLELISM suppose a function f jnfj∈→=∈{0,1023} ( ) { 0,1023} Classically, need 1000 ×10-bit registers (10,000 bits) to store information about this function and to work on it. Quantum-mechanically, a 20-qubit register can suffice! 1 21N − j fj Ψ= N /2 ∑ () 2 j=0 Function encoded in a superposition of states of register 08-I-14 9 OUTLINE 1. Review of classical radio-frequency circuits 2. Quantum information processing 3. Quantum-mechanical LC oscillator 4. A non-dissipative, non-linear element: the Josephson junction 5. Energy levels and transitions of the Cooper Pair Box 6. Summary of questions addressed by this course 08-I-5c A CIRCUIT BEHAVING QUANTUM-MECHANICALLY AT THE LEVEL OF CURRENTS AND VOLTAGES ? SIMPLEST EXAMPLE: SUPERCONDUCTING LC OSCILLATOR CIRCUIT d ~ 0.5 mm 08-I-15 10 A CIRCUIT BEHAVING QUANTUM-MECHANICALLY AT THE LEVEL OF CURRENTS AND VOLTAGES ? SIMPLEST EXAMPLE: SUPERCONDUCTING LC OSCILLATOR CIRCUIT d ~ 0.5 mm MICROFABRICATION L ~ 3nH, C ~ 1pF, ωr /2π ~ 5GHz d << λ: LUMPED ELEMENT REGIME 08-I-15b A CIRCUIT BEHAVING QUANTUM-MECHANICALLY AT THE LEVEL OF CURRENTS AND VOLTAGES ? SIMPLEST EXAMPLE: SUPERCONDUCTING LC OSCILLATOR CIRCUIT d ~ 0.5 mm MICROFABRICATION L ~ 3nH, C ~ 1pF, ωr /2π ~ 5GHz d << λ: LUMPED ELEMENT REGIME SUPERCONDUCTIVITY ONLY ONE COLLECTIVE VARIABLE INCOMPRESSIBLE ELECTRONIC FLUID SLOSHES BETWEEN PLATES. NO INTERNAL DEGREES OF FREEDOM. 08-I-15c 11 DEGREE OF FREEDOM IN ATOM vs CIRCUIT: SEMI-CLASSICAL DESCRIPTION Superconducting Rydberg atom LC oscillator L C 08-I-16 DEGREE OF FREEDOM IN ATOM vs CIRCUIT: SEMI-CLASSICAL DESCRIPTION Superconducting Rydberg atom LC oscillator L C 2 Possible correspondences: charge dynamics velocity of electron → current through inductor force on electron → voltage across capacitor field dynamics velocity of electron → voltage across capacitor force on electron → current through inductor 08-I-16a 12 DEGREE OF FREEDOM IN ATOM vs CIRCUIT semiclassical picture Superconducting Rydberg atom LC oscillator L φ C 2 Possible correspondences: charge dynamics velocity of electron → current through inductor force on electron → voltage across capacitor field dynamics velocity of electron → voltage across capacitor force on electron → current through inductor 08-I-16b FLUX AND CHARGE IN LC OSCILLATOR electrical world mechanical world φ I X +Q f k V M -Q 1 φ = LI -I QV= C X = f P = MV k 08-I-17 13 FLUX AND CHARGE IN LC OSCILLATOR electrical world mechanical world φ I X +Q f k V M -Q 1 φ = LI -I QV= C X = f P = MV k position variable: φ X momentum variable: Q P generalized force : Ι f generalized velocity: V V generalized mass : CM generalized spring constant: 1/L k 08-I-17a HAMILTONIAN FORMALISM electrical world mechanical world φ I X +Q f k V M -Q time-integral position of voltage on of mass capacitor charge on capacitor w/r wall momentum of mass 22 Q φ PkX22 HQ()φ, =+ HXP(), =+ 22CL 22M conjugate variables conjugate variables 08-I-18 14 HAMILTON'S EQUATION OF MOTION electrical world mechanical world φ I X +Q f k V M -Q ∂HQ ∂HP φ == X == ∂QC 1 ∂P M k ω = ωr = r ∂H M ∂H φ LC P =− =kX Q =− = ∂X ∂φ L 08-I-19 FROM CLASSICAL TO QUANTUM PHYSICS: CORRESPONDENCE PRINCIPLE XX→ ˆ position operator P → Pˆ momentum operator ˆˆˆ HXP( ,,) → HXP( ) hamiltonian operator ∂∂AB ∂∂ AB −={}A,,/BABi →⎡ ˆ ˆ ⎤ = commutator ∂∂XP ∂∂ PX PB ⎣ ⎦ Poisson bracket position and X ,1,PXPi= →=⎡⎤ˆˆ = momentum {}PB ⎣⎦ operators do not commute 08-I-20 15 FLUX AND CHARGE DO NOT COMMUTE For every branch in the circuit: φ I +Q ⎡⎤φˆˆ,Q = i= V ⎣ ⎦ -Q We have obtained this result thru correspondance principle after having recognized flux and charge as canonical conjugate coordinates. "Cannot quantize the equations of the stock market!" A.J. Leggett We could also obtain this result from quantum field theory thru the commutation relations of the electric and magnetic field. Tedious. 08-I-21 LC CIRCUIT AS QUANTUM HARMONIC OSCILLATOR E φ +Q -Q hωr φ Haaˆ =+=ω ˆˆ† 1 r ( 2) φφˆˆQQ ˆˆ aiaˆˆ=+; † =− i annihilation and creation operators φφrrQQ rr φωrr= 2= L QCrr= 2=ω 08-I-22 16 THERMAL EXCITATION OF LC CIRCUIT E φ I hωr φ Can place the circuit =ω kT in its ground state r B 5 GHz 10mK 08-I-22a WAVEFUNCTIONS OF LC CIRCUIT E φ I hωr φ 2φr Ψ(φ) In every energy eigenstate, Ψ (photon state) 0 current flows in opposite directions simultaneously! 0 Ψ φ 1 08-I-23 17 EFFECT OF DAMPING E φ φ important: negligible dissipation dissipation broadens energy levels ⎡ ⎛⎞i 1 ⎤ see Ennr=++=ω ⎢ ⎜⎟1 ⎥ ⎣ ⎝⎠22Q ⎦ why later Q = RCωr 08-I-24 ALL TRANSITIONS ARE DEGENERATE E φ hωr φ CANNOT STEER THE SYSTEM TO AN ARBITRARY STATE IF PERFECTLY LINEAR 08-I-25 18 NEED NON-LINEARITY TO FULLY REVEAL QUANTUM MECHANICS Potential energy Position coordinate 08-I-26 OUTLINE 1.
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