INTRODUCTION À LA PHYSIQUE MÉSOSCOPIQUE: ÉLECTRONS ET PHOTONS INTRODUCTION TO MESOSCOPIC PHYSICS: ELECTRONS AND PHOTONS

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1 Chaire de Physique Mésoscopique Michel Devoret Année 2007, Cours des 7 et 14 juin INTRODUCTION À LA PHYSIQUE MÉSOSCOPIQUE: ÉLECTRONS ET PHOTONS INTRODUCTION TO MESOSCOPIC PHYSICS: ELECTRONS AND PHOTONS Deuxième leçon / Second Lecture This College de France document is for consultation only. Reproduction rights are reserved. 07-II-1

2 What do "electron" and "photon" mean in mesoscopic physics? Purpose: provide groundwork for Landauer's approach of transport phenomena and quantum circuit theory 07-II-1

3 THE MESOSCOPIC RESISTOR + V _ quantum coherent wire I source reservoir L L ϕ drain reservoir The Landauer reservoir is to Fermi waves what a black-body is to Bose waves. 07-II-2

4 Mesoscopic wire: a collection of independent channels reservoir t r t r mode m µ + µ E ev right-moving quasiparticles f + f left-moving quasiparticles 07-II-3

5 Mesoscopic wire: a collection of independent channels t r t r mode m reservoir µ + µ E ev right-moving quasiparticles f + f left-moving quasiparticles 07-II-3bis

6 Mesoscopic wire: a collection of independent channels t r t r mode m reservoir µ + µ E ev right-moving quasiparticles f + f left-moving quasiparticles 07-II-3ter

7 THE LANDAUER-BÜTTIKER FORMULA FOR THE AVERAGE CURRENT I = I I + e + I = f ( E) t ( E) 2 d E ± ± m h m f ± E ( ) = 1 E µ ± 1+ exp kt B µ µ = + Electrons interact with the voltage source but not between themselves ev 07-II-4

8 THE USUAL ELECTRON OF ATOMIC AND HIGH ENERGY PHYSICS PARTICLE IDENTIFICATION CARD Last Name: Electron First name: Bare Address: Vacuum Genre: Fermion Occupation: Wave packet Lifetime: infinite Average energy: ω Average momentum: k Velocity: v=dω/dk Mass: dk/dv=m e Charge: -e Spin: 1/2 Magnetic moment: µ B 07-II-5

9 An example of a Feynman diagram involving the usual electron and photon of atomic physics propagating in vacuum γ γ e - e - 07-II-6

10 THE "ELECTRON" OF MESOSCOPICS PARTICLE IDENTIFICATION CARD Last Name: Electron First name: Quasi Address: Metal Genre: Fermion Occupation: Wave packet Lifetime: finite, k F Average energy: ω Average momentum: k Velocity: v=dω/dk Mass: dk/dv=m eff (k) Transverse charge: -e Longitudinal charge: 0 (q 0) Spin: 1/2 Magnetic moment: g µ B 07-II-7

11 Definition of the longitudinal and transverse part of a field: F = F + F l F = 0 t F = 0 l t The longitudinal and transverse charges are the sources of the longitudinal and transverse parts of the electrical field, respectively. 07-II-8

12 A METAL AT LOW ENERGY: FERMI QUASIPARTICLES + BOSONIC PLASMONS cannot solve the full many body problem, but. low-lying excitations of strongly interacting bare electrons nearly free quasielectrons and holes bosonic plasma modes photons 07-II-9

13 PLASMA PHYSICS APPLIED TO METALS positive background negatively charged fluid current density charge density constitutive equation approximation ne 0 ne = ( n ) 0 + δ n e j = env ρ ρ = eδn; + i j = 0 t nv ( ) + = ( m + ) v nv en E v B P t δ n vb small; small n E 0 (jellium) Quantum Mechanics enter in internal pressure internal pressure 07-II-10

14 FERMI PRESSURE Box volume V Nb of fermions N k y kf a Total energy E K Length scale a 0 Energy scale Ry 2 4 me e = 4 πε ; Ry = 2 4πε me e 0 k x N k a n r k V = = ( π ) = = a a = r a 3 1 r 2 4π F ; 2 s ; F 3 2 4π 3 0 s 0 ( k ) F F g m E e C s 2 E K 3 F = = E F = 2 N 5 2m e 5 rs v = k = v E N = Ry F Ry c 0 E K V = Fermi pressure E V m n N = = 1 3 v F 07-II-11

15 ARRIVE AT LINEARIZED EQUATIONS FOR FIELDS AND ELECTRON FLUID j vs ρ = ωpε0e t ρ El = ε 0 ρ jl + = 0 t 1 E t B = µ 2 0 j c t B Et + = 0 t B = 0 t ω v s P = 2 en0 mε P = mn n E = El + E j = jl + jt El; jl; ρ E ; j ; B { } { } t t t 0 plasma frequency sound velocity longitudinal part transverse part 07-II-12

16 BOUNDARY CONDITIONS: WIRE ABOVE A GROUND PLANE h x 07-II-13

17 BOUNDARY CONDITIONS: WIRE ABOVE A GROUND PLANE h x Field lines from wire end on ground plane h << λ 07-II-13bis

18 LONGITUDINAL MODE CURRENTS TRANSVERSE MODE CURRENTS λ 07-II-14

19 LONGITUDINAL MODE CHARGES TRANSVERSE MODE CHARGES λ II-14bis

20 DISPERSION RELATION OF ELECTRODYNAMIC MODES OF WIRE ω PLASMA MODE ω p TRANS- MISSION LINE MODE SLOPE ~ c eff PLASMA MODE : LONGITUDINAL = "SOUND" T.L.M. : TRANSVERSE = GUIDED "LIGHT" k s ω v = P s k 07-II-15

21 RESPONSE : SCREENING 1 ω 1 ρ ρ s = P t ext s = v ω s P ~ a 0 for 3D metals plane wave dispersion relation: ω = ω + vk P s dielectric response function: ( ) r 2 screened potential! V eff r, 0 = ( ω ) ε k, ω = 1+ e ε 0 r e r s k 2 2 s 1 ω ω 2 P 07-II-16

22 NEUTRAL METALLIC SPHERE density of ions density of electrons x x x s jellium total charge density electroneutrality x 07-II-17

23 CHARGED METALLIC SPHERE _ density of ions density of electrons x x x s _ total charge density electroneutrality x 07-II-17bis

24 SELF-CONSISTENT PICTURE OF ELECTRON STATES ρ total x E x U W x E F ρ elec 07-II-18 x

25 07-II-19 WHAT IS µ? U W x E F add charge U W 0 +δw(µ) stays same to a very good approximation µ E F x

26 IDEAL METALLIC TOROIDAL WIRE ions electrons 07-II-20

27 LOW ENERGY ELECTRODYNAMIC EXCITATIONS 07-II-21

28 LOW ENERGY ELECTRODYNAMIC EXCITATIONS 07-II-21

29 LOW ENERGY ELECTRODYNAMIC EXCITATIONS 07-II-21

30 LOW ENERGY ELECTRODYNAMIC EXCITATIONS 07-II-21

31 LOW ENERGY ELECTRODYNAMIC EXCITATIONS BOSONIC EXCITATIONS " PHOTONS" 07-II-21

32 ELECTROSTATIC EXCITATION + _ + _ + _ + _ + _ + _ Example: torus in parallel plate capacitor 07-II-22

33 OTHER QUASI-STATIC MACROSCOPIC EXCITATION OF ELECTRONS IN TORUS: ELECTRICAL CURRENT Electrons move bodily with respect to ions. No surface charge. Example: flux through torus increases linearly with time 07-II-23

34 QUASIPARTICLE EXCITATIONS electron hole In other words, just heat! 07-II-24

35 E QUASIPARTICLE EXCITATIONS GROUND STATE ONE "ELECTRON" ONE "HOLE" 07-II-25

36 E FINITE LIFETIME OF QUASIPARTICLES ONE "ELECTRON" TWO "ELECTRONS" + ONE "HOLE" 07-II-26

37 DISPERSION RELATION OF ELEMENTARY EXCITATIONS IN A METAL energy ω p electrodynamic modes holes velocity = v F electrons velocity ~ c 0 k F momentum 07-II-27

38 DISSIPATION CORRESPONDS TO CREATION OF ELECTRON-HOLE PAIRS FROM ELECTRODYNAMIC EXCITATIONS energy IMPURITIES MOMENTUM NO LONGER CONSERVED ω p electrodynamic modes holes electrons ω RF ω = ε + ε RF h e 0 k F momentum 07-II-28

39 REVERSE PROCESS CORRESPONDS TO JOHNSON NOISE energy ω p electrodynamic modes holes electrons ω RF ω = ε + ε RF h e 0 k F momentum 07-II-28bis

40 JOHNSON NOISE IS EQUIVALENT TO BLACK-BODY RADIATION T R power per unit frequency: P (, T) ν = e 2hν hν kt B 1 1-D version of Planck's radiation law I(, T) ν = 2 hν / c emission of "photons" by excited quasielectron-hole pairs analogous to emission of photons by black-body atoms e hν kt B II-28ter

41 energy DISPERSION RELATION OF ELEMENTARY EXCITATIONS IN A SUPERCONDUCTING METAL (caveat: S-wave, with gap) ω p electrodynamic modes SUPERCONDUCTING QUASIPARTICLES 0 NO DISSIPATION? k F ENERGY GAP momentum 07-II-29

42 CONCLUSIONS "Electrons" and "photons" in mesoscopic physics are "dressed" particles with properties which can greatly differ from their counterparts in free space. These properties can be designed. We can construct a quantum Lego set, explore its various combinations and "invent" new quantum effects. 07-II-30

43 COMPARISON BETWEEN QUANTUM OPTICS AND QUANTUM TRANSPORT EXP MENTS sources atom detectors DC mesoscopic device V I 07-II-31

44 COMPARISON BETWEEN QUANTUM OPTICS AND RF QUANTUM TRANSPORT EXP MENTS sources atom detectors RF mesoscopic device generator amplifier 07-II-31bis

45 QUANTUM OPTICS atoms, molecules light beams, fibers mirrors, beam splitters, etc light sources : lasers photodetectors, photomultipliers T background = 300K cavity weak atom-field coupling QUANTUM CRYOLECTRONICS tunnel devices, semic. dots coax. transmission lines filters, couplers, circulators microwave generators cryogenic amplifiers T background = 30mK resonator, oscillator strong artificial atom field coupling photon loss and dispersion resistance and reactance 07-II-32

SOME KEY IDEAS Rolf Landauer David Thouless Joe

quasi-electrons) what is important is loss of

46 SOME KEY IDEAS Rolf Landauer David Thouless Joe Imry Tony Leggett "think conductance, not conductivity!" 07-II-33 E Thouless D = 2 L "(for quasi-electrons) what is important is loss of quantum information: decoherence" dissipative non-linear circuits: to what extent do they obey quantum mechanics?

47 NEXT YEAR: "QUANTUM CIRCUITS AND SIGNALS" How do we treat a macroscopic circuit quantum-mechanically? How do we describe non-linear elements like tunnel junctions, both normal and superconducting? What are the properties of quantum noise? How does it limit the processing of signals? 07-II-34

48 LE COURS DE L'AN PROCHAIN: "CIRCUITS ET SIGNAUX QUANTIQUES" Début: 13 mai 2008 Comment traiter quantiquement un circuit électronique macroscopique? Comment décrire les composants non-linéaires comme les jonctions tunnel? Quels sont les propriétés du buit quantique? Quel est son influence sur le traitement du signal? 07-II-34bis

49 Acknowledgements : D. Esteve, H. Pothier, D. Stone and C. Urbina W.M. KECK

INTRODUCTION À LA PHYSIQUE MÉSOSCOPIQUE: ÉLECTRONS ET PHOTONS INTRODUCTION TO MESOSCOPIC PHYSICS: ELECTRONS AND PHOTONS

Chaire de Physique Mésoscopique Michel Devoret Année 2007, Cours des 7 et 14 juin INTRODUCTION À LA PHYSIQUE MÉSOSCOPIQUE: ÉLECTRONS ET PHOTONS INTRODUCTION TO MESOSCOPIC PHYSICS: ELECTRONS AND PHOTONS