Workshop on Nano-Opto-Electro-Mechanical Systems Approaching the Quantum Regime September 2010

Transcription

1 Workshop on Nano-Opto-Electro-Mechanical Systems Approachg the Quantum Regime 6-10 September 010 Quantum Signatures of the Dynamics of a Vibrational Mode of a Th Membrane with an Optical Cavity David VITALI School of Science & Tech. University of Camero via Madonna delle Carceri, 9b, Camero 603 MC ITALY

2 Quantum signatures of the dynamics of a vibrational mode of a th membrane with an optical cavity David Vitali M. Karuza, C. Biancofiore, G. Di Giuseppe, R. Natali, M. Galassi, P. Tombesi School of Science and Technology, Physics Division, University of Camero, Italy ICTP Workshop on Nano-Opto-Electro-Mechanical Systems Approachg the Quantum Regime, Sept. 6-10, 010, Trieste 1

3 Outle of the talk 1. Optomechanical systems: the case of a th membrane with a Fabry-Perot cavity (also with some experimental results). Theory predictions on quantum phenomena: entanglement, ground-state coolg (with one or two mechanical modes), ponderomotive squeezg of the light mode

4 Why enterg the quantum regime for opto- and electro-mechanical systems? quantum-limited sensors, i.e., workg at the sensitivity limits imposed by Heisenberg uncertaty prciple explorg the boundary between the classical macroscopic world and the quantum microworld (how far can we go the demostration of macroscopic quantum phenomena?) quantum formation applications (optomechanical and electromechanichal devices as light-matter terfaces and quantum memories), or transducers for quantum computg architectures 3

5 We focus on cavity optomechanics 1. Fabry-Perot cavity with a movg micromirror micropillar mirror (LKB, Paris) Monocrystalle Si cantilever, (Vienna). Silica toroidal optical microcavities spokesupported microresonator (Munich, Lausanne) With electronic actuation, (Brisbane) 4

6 Evanescent couplg of a SiN nanowire to a toroidal microcavity (Munich, Lausanne) microdisk and a vibratg nanomechanical beam waveguide (Yale) Photonic crystal zipper cavity (Caltech) membrane the middle scheme: Fabry-Perot cavity with a th SiN membrane side (Yale, and more recently Caltech, Camero) 5

7 We focus here on the cavity-membrane system Many cavity modes (still Gaussian TEM mn for an aligned membrane close to the waist) Hcav k a k k a k Many vibrational modes u mn (x,y) of the membrane u mn x, y s nx s d my d nm T td m n Vibrational frequencies T = surface tension = SiN density, t = membrane thickness d = membrane side length m,n = 1, 6

8 z( x, y) M n, m H M n, m nm nm q nm p u nm nm ( x, y) Membrane axial deformation field q nm M td 4 qnm, plk l mk Mode mass Mechanical Hamiltonian Dimensionless position and momentum of vibrational modes Optomechanical teraction due to radiation pressure Ht dxdy P rad ( x, y) z( x, y) (at first order z) Prad ( x, y) 0 t / nm 1 dze ( x, y, z) B( x, y, z) z t / Radiation pressure field 7

9 Hˆ t c nmlk l, k, n, m a l a k q nm Trilear couplg describg photon scatterg between cavity modes mediated by the vibratg membrane c nmlk l L k M nm nmlk nmlk = dimensionless couplg constants dependg upon membrane position, thickness, transverse spatial overlap between optical and vibrational modes.. Some first experimental data Camero We have observed scatterg between modes: simultaneous presence of a TEM00 mode (driven by the laser) and TEM0n (n 6) mode (scattered by the membrane) CCD camera picture of the transverse patterns of the tracavity mode, showg the simultaneous presence of a TEM00 and TEM0n (n 6) mode 8

10 Mode couplg and the correspondg frequency shifts can be tuned by adjustg the position and orientation of the membrane Avoided crossg Relative frequency of the two modes TEM00 and TEM0n versus the membrane displacement. The data are consistent with a splittg of about 1 MHz (see also J. Sankey et al., Nat. Phys, July 010, for a much more detailed study of mode couplg) Couplg quadratic q Spectrum of the transmitted signal Excitation spectrum of the vibrational modes of the SiN membranes, both the presence and absence of electromechanical drivg (room temperature, low mechanical Q -> well the classical regime) 9

11 Let us now focus on a simpler situation: sgle mechanical oscillator, nonlearly coupled by radiation pressure, to a sgle optical oscillator This is possible when: The external laser (with frequency L a ) drives only a sgle cavity mode a and scatterg to the other cavity modes is negligible (no frequency close mode) a bandpass filter the detection scheme can be used, isolatg a sgle mechanical resonance Hˆ t 0 G a aq Hˆ drive i Ee a E i t * e L L i t a E P L L amplitude of the drivg laser with put power P L detection bandwidth 10

12 The membrane is contact with an ohmic environment at temperature T; Fluctuation-dissipation theorem presence of a quantum Langev force with correlation functions Also dampg and noise act on the system.. m itt' t t' e m d coth kt 1 Gaussian, generally non-markovian The cavity mode is damped by two dependent processes: 1. photon leakage through the mirrors, with decay rate 1. absorption by the membrane, with decay rate (q), non-standard because of membrane position dependence --> further nonlearity Each decay is associated with a vacuum put Langev noise a j (t) with correlation functions a j k j k j k t a t' a t a t' 0 a t a t' t t' jk Gaussian, Markovian 11

13 Description terms of Heisenberg-Langev equations ( the frame rotatg at L ) a i p 1 G qa qa E a q a q m L 0 G a 0 1 a m q p m p q q q 1 aa a a a Nonlear cavity decay Nonlear noise Additional non-standard terms due to membrane absorption; how much do they affect quantum effects? 1

14 Classical steady state and learization around it Strong drivg and high-fesse cavity steady-state with an tense tracavity field (amplitude s ) and deformed membrane. We focus on the learized dynamics of the quantum fluctuations around this steady state (only cavity mode is learized exact for s >> 1) a aa s q q s qq 1 s q s E i s s c L G 0 m s steady-state radiation pressure shift Nonlear eqn. for the tracavity steady-state amplitude Effective cavity detung Radiation pressure optical bistability (Dorsel et al., 1983, more recently cavity-bec systems, (see Esslger talk) 13

15 Optical bistability by radiation pressure observed also our cavity-membrane system E s s Experimental data Dynamical transition to the new steady state at mechanical frequencies 14

16 Back to theory: Quantum dynamics of the fluctuations: Learized quantum Langev equations XX pp qq m pp GXX XX YY sq YY YY XX Gqq m qq pp m q q s 1 s q q 1 X q X 1 s Y 1 q s s q Y Y X Y a a a a i Amplitude quadrature Phase quadrature Additional terms due to membrane absorption X Y j j a a j j G a i G0 0 a j j P L L Amplitude noise Phase noise Effective radiation pressure couplg 15

17 1. STEADY STATE ENTANGLEMENT When the system is stable, it reaches for t a Gaussian steady state, due to: 1. Learized dynamics. Gaussian quantum noises Gaussian Gaussian characteristic function T i V T Tr exp id e T T q, p, X, Y V ij i i j j j i i jj correlation matrix (CM) fully characterizg the steady state and its entanglement properties (we use log-negativity) Review paper: C. Genes, A. Mari, D. Vitali and P. Tombesi, Quantum Effects Optomechanical Systems, Advances Atomic, Molecular, and Optical Physics, Vol , Academic Press, 009, pp

18 . GROUND STATE COOLING OF THE MEMBRANE MODES The steady state CM, V, contas also the fo about the stationary energy of the membrane mode, U V11 qq V pp Is it possible to get simultaneous optomechanical steady-state entanglement and ground state coolg (q = p =½)ofa membrane mode with state of the art parameters, despite membrane absorption (Im n ~ 10-4 ) 17

19 For parameters similar to those of our current experiment: M = 35 ng m KHz, Q m, P L = 650 W, L = 7 cm, F 0 = 0000, T = 4 K, t = 50 nm, ~ m, n M =. + i 10-4 Blue: n eff = ground state occupancy Red: E N, Log-negativity 18

20 Cavity resonant with the laser blue sideband t (50 nm) membrane thickness 19

21 Relaxg the sgle mechanical mode description: What if a nearby mechanical mode is present? Everythg depends upon the frequency mismatch between the two modes 1 = 1 Coolg is not disturbed if the two modes are not too close: the two modes are even simultaneously cooled = F = , 0. m F = , m 0

22 Coolg is hibited when the frequencies are close! It happens when the modes are separated by less than the effective mechanical width, 1 < (net laser coolg rate) = = 1 one mode only C. Genes et al., New J. Phys. 10 (008)

23 This hibition is due to a classical destructive terference phenomenon, similar to a classical analogue of electromagnetically duced transparency (EIT) Two modes when 1

24 Alternative explanation: when 1 =, radiation pressure couples the cavity mode only with the effective center-ofmass of the two mechanical modes When 1 =, the relative motion is decoupled from the center-of-mass and the cavity mode is uncooled and therefore also the two modes are uncooled. 3

25 EFFECT OF NEARBY MODE ON ENTANGLEMENT Similar to coolg: the two modes are simultaneously entangled with the cavity mode if the are not too close 1 > = one mode only Entanglement is more fragile and more affected than coolg 4

26 EFFECT OF NEARBY MODE ON ENTANGLEMENT The situation is more volved when the modes are close 1 < T = 0 one mode only Entanglement at T = 0 creases at resonance because the center-of-mass is strongly entangled with the cavity T = 0.4 K But entanglement at resonance is soon destroyed by temperature due to the uncooled relative motion 5

27 FURTHER POSSIBLE QUANTUM EFFECT: GENERATION OF SQUEEZED LIGHT AT THE CAVITY OUTPUT Predicted by Manci-Tombesi, and Fabre et al Squeezed light Feedback-assisted generation of squeezg? 6

28 Shot noise Optimized homodyne spectrum of the output light, cavity-membrane system; feedback (full) yields little improvement over no feedback (dashed) Feedback does not help, but squeezg is possible with stateof-the art devices (ma problem: low-frequency phase noise) D. Vitali & P. Tombesi, CR Physique, to appear 7

29 CONCLUSIONS 1. Some prelimary experimental results with a cavity-membrane-the-middle system. Membrane absorption does not seriously affects ground state coolg and entanglement 3. Simultaneous coolg and entanglement of two mechanical modes is possible only if they are not too close frequency 4. Quadrature squeezg of the cavity output is feasible with state-ofthe art systems 8