RÉSONATEURS NANOMÉCANIQUES DANS LE RÉGIME QUANTIQUE NANOMECHANICAL RESONATORS IN QUANTUM REGIME

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1 Chaire de Physique Mésoscopique Michel Devoret Année 1, 15 mai - 19 juin RÉSONATEURS NANOMÉCANIQUES DANS LE RÉGIME QUANTIQUE NANOMECHANICAL RESONATORS IN QUANTUM REGIME Deuxième leçon / Second lecture This College de France document is for consultation only. Reproduction rights are reserved. Last update: II-1

2 PROGRAM OF THIS YEAR'S LECTURES Lecture I: Introduction to nanomechanical systems Lecture II: How do we model the coupling between electromagnetic modes and mechanical motion? Lecture III: Is zero-point motion of the detector variables a limitation in the measurement of the position of a mechanical oscillator? Lecture IV: In the active cooling of a nanomechanical resonator, is measurement-based feedback inferior to autonomous feedback? Lecture V: How close to the ground state can we bring a nanoresonator? Lecture VI: What oscillator characteristics must we choose to convert quantum information from the microwave domain to the optical domain? 1-II-

3 CALENDAR OF 1 SEMINARS May 15: Rob Schoelkopf (Yale University, USA) Quantum optics and quantum computation with superconducting circuits. May : Konrad Lehnert (JILA, Boulder, USA) Micro-electromechanics: a new quantum technology. May 9: Olivier Arcizet (Institut Néel, Grenoble) A single NV defect coupled to a nanomechanical oscillator: hybrid nanomechanics. June 5: Ivan Favero (MPQ, Université Paris Diderot) From micro to nano-optomechanical systems: photons interacting with mechanical resonators. June 1: A. Douglas Stone (Yale University, USA) Lasers and anti-lasers: a mesoscopic physicist s perspective on scattering from active and passive media. June 19: Tobias J. Kippenberg (EPFL, Lausanne, Suisse) Cavity optomechanics: exploring the coupling of light and micro- and nanomechanical oscillators. 1-II-3

4 LECTURE II : INTERACTION BETWEEN LIGHT AND MOVING MIRROR IN THE QUANTUM REGIME OUTLINE 1. A simple hamiltonian, whose parameters span many orders of magnitude. Radiation pressure and the Doppler effect in a 1D deformable cavity: a fundamentally non-linear effect 3. Full coupling hamiltonian between radiation and moving mirror 4. Adiabatic approximation 1-II-4

5 QUANTUM ELECTRO/OPTO-MECHANICS IS GOVERNED BY A SIMPLE HAMILTONIAN Hˆ = ω ˆˆ ( ) ˆ ˆ ˆ ˆ ˆ ˆ eaa + ω mb b + g3 b + b aa K ω m = Zm M = KM X ZPF = Z m ( ) X ˆ = X b+ b ZPF g = X ZPF 3 ω e ω e JILA/NIST ω e ω m M K ω m X ω, ω LKB-Paris, U. Vienna EPFL, Caltech U. Vienna 1-II-5 e m

6 QUANTUM ELECTROMECHANICS courtesy of K. Lenhert T~4mK (8MHz) Teufel et al., Nature 471, 8 (11) JILA/NIST 1-II-6

7 QUANTUM OPTOMECHANICS (1) ω m ω e γ T=.6K (1GHz) Verhagen, Deleglise, Weis, Schliesser and Kippenberg, Nature 48, 66 (1) EPFL 1-II-7

8 QUANTUM OPTOMECHANICS() T~K (4GHz) Chan et al., Nature 478, 9 (11) CALTECH, U. VIENNA 1-II-8

9 ORDERS OF MAGNITUDE Group ω m Q m =ω m /γ X ZPF ω e Q e =ω e /κ g 3 ω m /κ JILA/NIST K. Lehnert, R. Simmonds 1.56 MHz fm 7.54 GHz khz 5 EPFL T. Kippenberg 78 MHz 34 ~1fm 385 THz khz 11 Caltech/Vienna O. Painter, M. Aspelmeyer 3.68 GHz 1 5 3fm 195 THz khz 7 1-II-9

10 HOW IS THE HAMILTONIAN ABOVE, ALLOWING TO MEASURE fm DISPLACEMENTS, CONNECTED TO THE ONE RESPONSIBLE FOR THE DOPPLER MEASUREMENT OF VELOCITY AND THE ATTRACTION BETWEEN TWO PLATES SUBMITTED TO AN AC VOLTAGE? WHAT APPROXIMATIONS ARE MADE IN THIS SIMPLE HAMILTONIAN? THE CONNECTION BETWEEN RADIATION PRESSURE AND THE DOPPLER EFFECT CAN BE EXPLORED USING A SIMPLE MODEL, WHICH WILL ANSWER THESE QUESTIONS. Reference: C. K. Law, Phys. Rev. A 51, 537 (1995) 1-II-1

11 1-II-11 AN ELEMENTARY, DEFORMABLE 1D RESONATOR z y fixed end wall (fixed mirror) parallel strip transmission line (want fields uniform in y and z) movable end wall (movable mirror). v(t) = X(t) E ( xt, ) X(t) electric field vanishes at mirror l l x

12 A SIMPLE, DEFORMABLE 1D RESONATOR (CNT'D). v(t) = X(t) B( x, t) X(t) Magnetic field is maximum at mirrors l l x A Treat vector potential in radiation gauge (a.k.a. Coulomb or transverse gauge) A x, t A x, zˆ ( ) = ( t) The mirrors are perfectly conducting and inside the cavity, no sources are present. Thus: ( () t t) A( t) 1-II-1, =, = ( xt, ) The vector potential obeys: A( x t) ( xt, ) A A E( x, t) = zˆ B( x, t) = yˆ t x 1, = x c t speed of light

13 ENERGY INSIDE RESONATOR Instantaneous energy density inside resonator: E ε 1 μ ( xt, ) = E( x, t) + B( x, t) ε 1 = A x t + (, ) A( x, t) t μ x Instantaneous total energy inside resonator: ε ( ) ( ) ( ) 1 U = S E x, t dx= S A x, t A x, t dx + t μ x mirror area Crucial details are that l is an implicit function of time through the dynamics of X and that at all times A ( ) ( t t) A( t), =, = 1-II-13

14 RADIATION PRESSURE Pressure = derivative of energy inside an enclosure with respect to volume of enclosure Thus, pressure on mirror is: p U = = ( x, t) dx E V = E (, t) 1 = μ x A ( xt, ) 1 = B ( x, t) μ x= A x= A =, = ( = t) ( t), = This remains correct even if l varies with time. Only space derivative contributes. Time derivative of A always zero at mirror. Pressure is given by square of magnetic field at mirror. Coupling between a macroscopic scatterer and light is fundamentally non-linear. 1-II-14

15 PHOTON APPROACH TO RADIATION PRESSURE Photons each have energy E momentum E/c Photons bounce on mirror Momentum transfer to mirror: E/c per photon (when perfect back-scattering) Pressure: momentum transfer per unit area per unit time Radiation pressure: incoming energy flux / c TEM mode in cavity p S E B = = c μc Bin Bmirror = = μ μ in in in Poynting vector Confirms wave approach! 1-II-15

16 NORMAL MODES OF A CAVITY WHOSE SHAPE DEPENDS ON TIME W k ( x, t ) k=3 W (, ) k xt l(t') l(t) x W k π kx, = sn i ( xt) () t () t Both wavelength and amplitude are time dependent. Symbol k : integer. () t (, ) (, ) W xt W xt = δ k k' Mode πck frequency: ω k () t = t () kk, ' Also varies with time! Orthonormality condition is independent of time. (scheme is not relativistic, OK here) 1-II-16

17 GENERALIZED NORMAL COORDINATES OF FIELD INSIDE CAVITY Normal coordinates k ( ) t ( ) = (, ) W (, ) q t A x t x t dx k Countable infinity of degrees of freedom. In turn, the field can be expressed in terms of normal coordinates: (, ) ( ) (, ) A xt = q t W xt Completeness k = 1 k k of mode functions Dimensional analysis: [ q ] = [ Φ] [ ] 1/ flux 1-II-17

18 CLASSICAL EQUATIONS OF MOTION FOR FIELD NORMAL COORDINATES AND MIRROR POSITION Field equation: c x (, ) = A( xt, ) A x t t ω kqk = qk + gkjq j gkjq j + g jkg jlql j j j, l Newton's Law for mirror: M ( ) V 1 = + μ c mech al force k, j ( 1) k+ j ωωq q k j k j radiation pressure force coupled equations: light presses on mirror, mirror pumps energy into or out of light. where: k ( ) ω = πck and: g kj k+ j kj ( 1 ), = j k, k = j k j 1-II-18

19 DOPPLER EFFECT fixed mirror far away v =constant v c wave packet, velocity c & frequency f After reflection: v frequency is now f 1+ c energy in pulse has increased by same ratio 1-II-19

20 DOPPLER EFFECT FROM PHOTON POINT OF VIEW Process is so far completely classical but can be interpreted in particle language: before collision with mirror v =constant P = mv v c after collision with mirror P = mv 1+ v V Particle bouncing elastically on a moving heavy wall sees its momentum P increased by mv, where m is particle mass and v the wall velocity. Relative increase in P is thus (1+ v/v), where V is the velocity of the particle. We can apply this idea to the photon: v P ω = P = ω 1+ c c c 1-II-

21 FULL HAMILTONIAN OF CAVITY WITH MOVING MIRROR IN THE NON-RELATIVISTIC APPROX TION From the classical equations, it is possible to construct a Lagrangian. One obtains the conjugate momenta and the quantum Hamiltonian by imposing standard commutation relation between coordinates and momenta. where ( pˆ + Πˆ ) Hˆ = + V a a M Πˆ ( ) ( ) + ω ˆ ˆ k k k k = 1 1/ i k = + k, j j and πc ( ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ) gkj akaj akaj akaj ajak ( ), ˆ ( ) aˆ a δ j k = jk Casimir interaction (1-D) Hamiltonian contains Doppler shift, radiation pressure and Casimir effect. First term resembles usual gauge coupling of charged particle. But here, mirror is neutral. Motion is non-relativistic and bounded by condition : l >. 1-II-1

22 If ˆX = TWO APPROXIMATIONS Very well obeyed for nanoresonators in quantum regime! (small displacement condition) ( ) Then, after a gauge transformation: Hˆ ˆ ˆ ˆ ˆ ˆ 1 = T HT; T = exp iπˆ X ˆ ˆ P H + U Xˆ + a a XˆF ˆ ( ) 1 ω ˆ ˆ k k k M k = 1 ˆ F ˆ ˆ ˆ = k+ j ( ) ( 1 ω ˆ ˆ ˆ ˆ ˆ ) kωj akaj akaj akaj ajak k, j Harmonic restoring force for mirror: ( ˆ ) K M If ω = ω ω ( k j) m k j we finally arrive at:,, Hˆ 1 U X = Adiabatic condition ˆ ˆ ˆ P + KX + ω e 1+ X M KXˆ includes Casimir force ˆ ˆ a a radiation pressure For a strictly 1D resonator, it corresponds to period of mirror motion much slower than roundtrip of photon. QED! 1-II-

23 M JILA/NIST L ω e ω m ELECTROSTATIC PRESSURE C ( ) K 1 ω e = ; Ze = LC Q ZPF = Z Since ωe ωm can neglect terms a, e a in Hˆ L C X = S S X C = ε ε 1 ˆ ˆ ˆ ˆ ˆ ˆ Φ Q X P KX H = L C M Qˆ = Q a+ a Xˆ = X b+ b ZPF ( ) ( ; ) ZPF ω e X ZPF = ( a a ) + ( a+ a ) 1 ( b+ b ) + ωmb b 4 Hˆ X ZPF ω 1 ( ) eaa b+ b + ωmbb 1-II-3

24 4-WAVE NON-LINEARITY Sankey,Yang, Zwickl, Jayich, and Harris, Nature Physics 6, 77 (1) fixed mirrors movable piece of dielectric Symmetry excludes from coupling hamiltonian terms odd in mechanical position Hˆ ( bˆ bˆ ) a ω ˆˆ+ g ˆ 4 a = ω ˆ ˆ bb ea + m + Can be used to read phonon number directly. aˆ e 4 = ( ) g X ZPF ω 1-II-4

25 SELECTED BIBLIOGRAPHY FOR THE COURSE Books Braginsky, V. B., and F. Y. Khalili, "Quantum Measurements" (Cambridge University Press, Cambridge, 199). Clarke, J. and Braginsky, A. I., eds., "The SQUID Handbook" (Wiley-VCH, Weinheim, Germany, 6). Cleland, A. N., "Foundations of Nanomechanics", (Springer, Berlin, 1) Cohen-Tannoudji, C., Dupont-Roc, J., Grynberg, G., "Photons and Atoms", (Wiley, New York, 1989) Gardiner, C.W. and Zoller, P. "Quantum Noise" (Springer, Berlin, 4). Haroche, S. and Raimond, J-M., "Exploring the Quantum" (Oxford University Press, 6). Pozar, D. M., "Microwave Engineering" 3rd edn (Wiley, New York, 4 ) Walls, D.F., and Milburn, G.J. "Quantum Optics" (Springer, Berlin, 8) Wiseman, H.M. and Milburn, G.J., "Quantum Measurement and Control" (Cambridge, 11) Articles O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, A. Heidmann, Nature 444, 71 (6) Chan J., et al., "Laser cooling of a nanomechanical oscillator into its quantum ground state", Nature 478, 89 (11) P. F. Cohadon, A. Heidmann, M. Pinard, Phys. Rev. Lett. 83, 3174 (1999) O Connell A. D. et al., "Quantum ground state and single-phonon control of a mechanical resonator", Nature 464, 697 (1) Clerk A. A., Devoret M. H., Girvin S. M., Marquardt F., and Schoelkopf R. J., "Introduction to Quantum Noise, Measurement and Amplification", Rev. Mod. Phys. 8, 1155 (1) Devoret M. H., and Schoelkopf R.J., Nature 46, () Favero, I., and Karrai, K., "Optomechanics of deformable optical cavities", Nature Photonics 3, 1 (9) Marquardt F., Chen J. P., Clerk A. A. and Girvin S. M., "Quantum theory of cavity-assisted sideband cooling of mechanical motion", Phys. Rev. Lett. 99, 939 (7) Kippenberg, T. J. and Vahala, K. J., "Cavity optomechanics: back-action at the mesoscale", Science 31, (8) Safavi-Naeini A. H., et al., "Observation of Quantum Motion of a Nanomechanical Resonator", Phys. Rev. Lett. 18, 336 (1) Teufel J.D. et al., "Sideband cooling of micromechanical motion to the quantum ground state", Nature 475, 359 (11) 1-II-5

26 END OF LECTURE