Cavity optomechanics: manipulating mechanical resonators with light

Transcription

1 Cavity optomechanics: manipulating mechanical resonators with light David Vitali School of Science and Technology, Physics Division, University of Camerino, Italy in collaboration with M. Abdi, Sh. Barzanjeh, C. Biancofiore, P. Tombesi, M. Karuza, G. Di Giuseppe, R. Natali, M. Galassi Congresso SIF 011, L Aquila, September

2 Outline of the talk 1. Introduction to cavity optomechanics. The special case of a thin membrane within a Fabry- Perot cavity (membrane-in-the-middle setup): (theory and experimental results in Camerino) 3. Theory predictions on steady state continuous variable entanglement 4. Quantum information applications

3 Why entering the quantum regime for opto- and electro-mechanical systems? 1. quantum-limited sensors, i.e., working at the sensitivity limits imposed by Heisenberg uncertainty principle Nano-scale: Single-spin MRFM D. Rugar group, IBM Almaden Macro-scale-scale: gravitational wave interferometers (VIRGO, LIGO) 3

4 Toward single molecule nanomechanical sensors (tiny small mass detectable frequency shift) M. Roukes group, Caltech 4

5 . exploring the boundary between the classical macroscopic world and the quantum microworld (how far can we go in the demostration of macroscopic quantum phenomena?) Optical detection of Schrodinger cat states of a cantilever? 5

6 3. quantum information applications (optomechanical and electromechanichal devices as light-matter interfaces and quantum memories), or transducers for quantum computing architectures 6

7 We focus on cavity optomechanics: many possible schemes 1. Fabry-Perot cavity with a moving micromirror micropillar mirror (LKB, Paris) Monocrystalline Si cantilever, (Vienna). Silica toroidal optical microcavities spokesupported microresonator (Munich, Lausanne) With electronic actuation, (Brisbane) 7

8 Evanescent coupling of a SiN nanowire to a toroidal microcavity (Munich, Lausanne) microdisk and a vibrating nanomechanical beam waveguide (Yale, H. Tang group) Photonic crystal zipper cavity (Caltech) membrane in the middle scheme: Fabry-Perot cavity with a thin SiN membrane inside (J. Harris group Yale, Kimble group Caltech, Camerino) 8

9 Example: the membrane-in-the-middle setup Many cavity modes (still Gaussian TEM mn for an aligned membrane close to the waist) Hcav k z0 k a k a k Many vibrational modes u mn (x,y) of the membrane u mn x, y sin nx sin d my d nm T L d d m n Vibrational frequencies T = surface tension = SiN density, Ld = membrane thickness d = membrane side length m,n = 1, 9

10 Optomechanical interaction due to radiation pressure H int dxdy P rad ( x, y) z( x, y) (at first order in z) P rad ( x, y) 0 z L nm 1 dz E( x, y, z) B( x, y, z) z z 0 0 L d d / / Radiation pressure field z( x, y) M n, m nm q nm u nm ( x, y) Membrane axial deformation field Hˆ int L l, k, n, m lk M nm nmlk lk q nm a l a k C. Biancofiore et al., Phys. Rev. A 84, (011) Trilinear coupling describing photon scattering between cavity modes caused by the vibrating membrane 10

11 Let us now simplify the system: single mechanical oscillator, nonlinearly coupled to a single optical oscillator Possible when: The external laser (with frequency L c ) drives only a single cavity mode a and scattering into the other cavity modes is negligible (no frequency close mode) a bandpass filter in the detection scheme can be used, isolating a single mechanical resonance Hˆ m p q qa a Hdrive q c G0q Cavity optomechanics Hamiltonian applicable in a wide variety of systems E Hˆ drive detection bandwidth P L L i Ee a E i t * e L L i t a amplitude of the driving laser with input power P L 11

12 SOME EXPERIMENTAL RESULTS IN THE MEMBRANE-IN-THE-MIDDLE SETUP The first order expansion based on the radiation pressure force is a poor approximation at nodes and antinodes; it is better to adopt the general dependence p q 1 arcsin R cos k z q q 0 G0q R = membrane reflectivity Theory curves for a perfectly aligned membrane close to the waist 1

13 Misalignment and shift of the membrane from the waist couple via scattering the TEM mn cavity modes: linear combinations of nearby TEM mn modes become the new cavity modes: (q) is changed significantly: tunable optomechanical interaction experimental data in Camerino: 50 nm membrane, misaligned by an angle of 0.77 mrad, and 1. mm shifted from waist 13

14 Results well fitted by a perturbative solution of the wave equation in the cavity with the membrane inside: the various (q) are well reproduced: See also J. Sankey et al., Nat. Phys. 010 Avoided crossings between TEM 00 and the TEM 0 triplet q G 0 ~ 10 Hz MHz / nm Linear coupling q 4.46MHz / nm Strong quadratic coupling 14

15 At the membrane positions where there is an avoided crossing the optomechanical interaction becomes quadratic in the membrane position q 4.46MHz / nm H int q q a a Dispersive interaction useful for the nondestructive measurement of mechanical energy: possible detection of phonon quantum jumps 15

16 Also damping and noise act on the system The mechanical element is in contact with its environment at temperature T Fluctuation-dissipation theorem presence of a quantum Langevin force x with correlation functions x m itt' t x t' e m d coth kt 1 Gaussian, generally non-markovian (nondelta-correlated) The cavity mode is damped by photon leakage with decay rate presence of a vacuum input Langevin noise a in (t) with correlation functions a in t a t' a t a t' 0 a t a t' t t' in in in in in Gaussian, Markovian Vacuum electromagnetic field outside the cavity reservoir at T 0 at optical frequencies, because 1 a a kt 1 N e 1 0 kt 16

17 Here we consider also an important noise of technical origin, which is usually neglected: phase noise of the driving laser It is practically unavoidable and it could be extremely destructive for continuous variable (CV) entanglement = quantum correlations between field and mechanical quadratures at a given phase Modification of the driving term E t (t) laser amplitude fluctuations (t) (zero-mean) laser phase fluctuations Noisy laser is also phase reference for any quadrature measurement the dynamics must be described in the frame rotating at the fluctuating laser frequency L t e i t M. Abdi et al., PRA 84, 0335 (011) 17

18 Heisenberg-Langevin equations in such a fluctuating frame Laser phase noise Laser amplitude noise coupling CV entanglement = strong quantum correlations between quadratures better achievable with strong driving steady-state with an intense intracavity field (amplitude a s ) and deformed/displaced resonator. Linearized dynamics of the quantum fluctuations around this steady state exact for a s >> 1) a a a s q q s q 18

19 Linearized Quantum Langevin equations for the quantum fluctuations amplitude noise negligible Large phase noise ( a s» 1) Nonlinear terms have been neglected Effective cavity detuning 19

20 Phase noise statistics and spectrum (t) = zero-mean Gaussian noise, related to laser spectrum S L () S i de t t frequency noise spectrum; we take a bandpass noise spectrum G l = laser linewidth = center of the frequency noise band = frequency noise bandwidth 0

21 STATIONARY OPTOMECHANICAL ENTANGLEMENT When the system is stable, it reaches for t a Gaussian steady state S, due to: 1. Linearized dynamics. Gaussian quantum noises V ij S Gaussian Gaussian characteristic function T i x V T Tr exp id x x x x i j e T j i x i xj T x q, p, X, Y correlation matrix (CM) fully characterizing the steady state and its entanglement properties (we use log-negativity E N ) Review paper: C. Genes, A. Mari, D. Vitali and P. Tombesi, Quantum Effects in Optomechanical Systems, Advances in Atomic, Molecular, and Optical Physics, Vol. 1 57, Academic Press, 009, pp

22 (no phase noise) EFFECT OF PHASE NOISE ON log-neg E N G l / = 0 G l / = 0.1 khz G l / = 1 khz D l = m = m State-of-art experimental parameters: m=10 ng, m / 10 MHz, m / 5 Hz, G 0 = 1 khz, L = 1 mm, T = 400 mk, / 50 khz, /

23 EFFECT OF PHASE NOISE SPECTRUM ON log-neg E N / = 30 khz / = 80 khz / = 140 khz D l = m = m G l / = 0.1 khz M. Abdi et al., arxiv: v1 [quant-ph], in press on PRA 3

24 Relevant quantity: eff m m APPROXIMATE ANALYTICAL RESULTS G D D m m D D m m Frequency noise spectrum at the effective mechanical resonance m eff Frequency modified by the optomechanical coupling Without phase noise: E N is maximum at the bistability threshold: close to it one has When Maximizing over D ln(5/3) ~ 0.51 With phase noise, one easily has - > 0.5 E N 0 close to bistability, and E N becomes maximum FAR FROM threshold (maximum achievable E N ) 4

25 . GROUND STATE COOLING OF THE MEMBRANE MODES The steady state CM, V, contains also the info about the stationary energy of the membrane mode, U V 11 q V p Cavity laser-cooling, realized by the radiation pressure of the cavity mode, which behaves as an effective additional zero-temperature reservoir for the oscillator Phase noise is much less detrimental on laser cooling to ground state of the mechanical resonator (recently achieved at JILA and in Caltech) 5

26 EFFECT OF PHASE NOISE ON COOLING G l / = 0 G l / = 0.1 khz G l / = 1 khz D l = m = m State-of-art experimental parameters: m=10 ng, m / 10 MHz, m / 5 Hz, G 0 = 1 6 khz, L = 1 mm, T = 400 mk, / 50 khz, /

27 HOW TO EXPLOIT OPTOMECHANICAL ENTANGLEMENT? For quantum communication applications one manipulates traveling rather than intracavity photons it is more important to analyze the entanglement of the mechanical mode with the optical cavity output By considering the output, one can manipulate a multipartite system: in fact, by means of spectral filters, one can select many different traveling output modes originating from a single intracavity mode 7

28 PROPER DEFINITION OF OUTPUT MODES input-output relation From the continuous cavity output a out (t), one can extract N independent output modes by selecting appropriate time (or frequency) intervals Independent modes 0 dtg * j t g k t dg j g k jk ~ * ~ g k (t) causal filter function C. Genes et al., PRA 78, (008) 8

29 SELECTING ONE OUTPUT MODE By appropriately choosing the center and the bandwidth of the detected output mode, the entanglement with the mechanical resonator can be distilled and increased = output center frequency = m, inverse bandwidth Optomechanical entanglement with the output mode centered at Entanglement with the mechanical mode is optimally carried by the output mode centered at the Stokes sideband of the laser; optimal width G effective width of the mechanical resonance 9

30 ENTANGLEMENT WITH OUTPUT FIELD TELEPORTATION ONTO MECHANICAL MODES One can use this optomechanical entanglement with the cavity output to teleport the state of an optical mode (Victor) onto the mechanical mode entangled Braunstein-Kimble teleportation protocol 1. Homodyne measurements by Alice. Communication of the results to Bob 3. Conditional actuation (phase-space displacement) by Bob 30

31 The Braunstein-Kimble protocol is designed for optical twomode squeezed states (high correlation between amplitude and phase quadrature of the two modes). The present optomechanical entanglement implies fieldmechanical correlations between generic quadratures, which are not easy to find PROBLEMS 1. One has to make local transformations (local quadrature rotations, squeezing, ) in order to optimize teleportation. Which ones?. For a given optomechanical entanglement, what is the maximum achievable fidelity after the local optimization? 31

32 We have given some answers in A. Mari, D. Vitali, Phys. Rev. A 78, (008). We consider a generic bipartite Gaussian state with CM V and entanglement E N as shared resource, and local tracepreserving Gaussian CP (TGCP) maps (unitary Gaussian operations, eventually with additional ancillas in Gaussian states which are then traced out) as operations for improving the teleportation fidelity Under a TGCP map V V ' SVS G i iss T T G 0 sympletic matrix we proved that 3

33 33 1. The optimizing local transformation is always a local Gaussian UNITARY at Alice and Bob site, supplemented at most by an attenuation either on Bob or Alice mode (i.e., mixing with vacuum at a beam splitter). This optimal local TGCP map leads to a final CM, which can always be written in the following normal form N N N E E E e F e e For a given log negativity E N, the locally optimized fidelity F (for teleporting coherent states) always satisfies m d m d d n d n The upper bound is achieved if and only if V is symmetric

34 CONCLUSIONS 1. One can create stationary (virtually infinite lifetime) optomechanical entanglement between the cavity mode and a nanomechanical resonator. Laser phase noise can seriously affect optomechanical entanglement (more than ground state cooling), but can be still achieved by appropriately filtering phase noise (it depends upon ) 3. Optomechanical entanglement can be used in quantum communication protocols, as for example, to teleport the state of an optical field onto a nanomechanical resonator 34

35 Classical-like description of cavity cooling frequency-dependent effective damping >> m suppression of susceptibility at resonance negligible sensitivity to thermal noise cooling Laser-cooling description of cavity cooling (I. Wilson-Rae et al., 007) Rates at which photons are scattered by the moving oscillator, simultaneously with the absorption (Stokes, A + ) or emission (anti- Stokes, A - ) of vibrational phonons G opt A A net laser cooling rate 36

36 Connection between the two descriptions eff m G A A m m opt m In the limits n eff G mn0 8 1qs A A m A Ground state cooling achieved when A n, G A, q s m not too large If these conditions are satisfied, absorption by the membrane does not hinder reaching the quantum regime 37

37 For parameters similar to those of our current experiment: M = 35 ng, m / 50 KHz, Q m 10 6, P L = 650 W, L = 9 cm, F 0 = 0000, T = 4 K, L d = 50 nm, D ~ m, n M =.0 + i 10-4 Blue: n eff = ground state occupancy Red: E N, Log-negativity Quantum regime achievable! C. Biancofiore et al., arxiv:110.10v1 [quant-ph] 38

38 Cavity resonant with the laser blue sideband t (50 nm) membrane thickness 39

39 Example: Equivalence with capacitively-coupled nanoresonator-superconducting microwave cavity system equivalent circuit (flux, charge Q) Annihilation op. of cavity photons Hamiltonian in the frame rotating at the driving frequency is equivalent to the Fabry-Perot case 40