Cavity optomechanics. Mishkat Bhattacharya Omjyoti Dutta Swati Singh

Transcription

1 Cavity optomechanics Pierre Meystre B Institute, Dept. of Physics and College of Optical Sciences The University of Arizona Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR

2 quantum metrology How can quantum resources be exploited with maximal robustness and minimal technical overhead? What are the inherent sensitivity-reliability tradeoffs in quantum sensors? What is the role of particle statistics in quantum metrology? How do decoherence mechanisms set fundamental limits to measurement precision? This talk: Cooled nanoscale cantilevers for quantum control of AMO systems

3 outline Brief review of optomechanical mirror cooling Two- and three-mirror cavities Cantilever coupling to dipolar molecules Single molecule Phonon squeezing in molecular lattice Coherent control Outlook (J. Sankey, Yale University)

4 micromechanical cantilevers single-particle detectors Micromechanical cantilever with a single E. Coli cell A micromechanical cantilever with a single E. Coli cell attached. The cantilever is used to detect changes in mass due to selectively attached biological agents present in small quantities. Craighead Research Group, Cornell University Protein detection Three cantilevers coated with antibodies to PSA, a prostate cancer marker found in the blood. The left cantilever bends as the protein PSA binds to the antibody. Photo courtesy of Kenneth Hsu/UC Berkeley & the Protein Data Bank

5 micromirrors D. Kleckner & D. Bouwmeester, Nature 444, 75 (006) SEM photograph of a vertical thermal actuator with integrated micromirror. Application of a current to the actuator arm produces vertical motion of the mirror, which can either reflect an optical beam or allow it to be transmitted. (Southwest Research Institute,

6 radiation pressure mirror cooling Basic idea: change frequency and damping of a mirror mounted on a spring using radiation pressure Number of quanta of vibrational motion: n k T kbt env Γ M hωeff hωef f Γeff B eff = [ H. Metzger and K. Karrai, Nature 43, 100 (004) ] Cold damping: laser radiation increases mirror damping. Requires laser tuned below cavity resonance Little change in mirror frequency Increase effective mirror frequency Requires laser tuned above cavity resonance Use two lasers! [ T. Corbitt et al, PRL 98, (007) ]

7 basic optomechanics cold damping Goal: increasing Ω eff Trapping due to `optical spring : k opt = df rp dx ( x) B. S. Sheard et al. PRA 69, (R) (004) Need blue-detuned cavity

8 basic optomechanics mirror cooling Goal: decreasing T = Γ Γ eff Tenv ( M / eff ) Cooling due to asymmetric pumping of sidebands ω 0 Ω F ω 0 + Ω ω 0 ω0 Ω M. Lucamarini et al., PRA 74, (006) A. Schliesser et al. PRL 97, (006) Need red-detuned cavity

9 competing requirements Blue detuned cavity Restoring force (optical spring) Antidamping Red detuned cavity Anti-restoring force Damping

10 two-wavelengths approach T. Corbitt et al, PRL 98, (007) Spring constant damping k>0 Γ>0 Dynamically unstable (subcarrier) k>0 Γ<0 Stable k<0 Γ>0 anti-stable k<0 Γ<0 Statically unstable (carrier)

11 Prehistory V. Braginsky, Measurement of weak forces in physics experiments (Univ. of Chicago Press, Chicago, 1977). V.B. Braginsky and F.Y. Khalili, "Quantum measurement'' Cambridge University Press, Cambridge, 199 C. M. Caves, Quantum mechanical noise in an interferometer, Phys. Rev. D3, 1693 (1981).

12 prehistory

13 early history Cavity cooling of a microlever, C. Höhberger-Metzger & K. Karrai, Nature 43, 100 (004) 18K

14 recent history S. Gigan et al. (Zeilinger group) Self-cooling of a micromirror by radiation pressure, Nature 444, 67 (006) < 10K O. Arcizet et al. (Pinard/Heidman group) Radiation pressure cooling and optomechanical instability of a micromirror, Nature 444, 71 (006) 10K D. Kleckner et al. (Bouwmeester group) Sub-Kelvin optical cooling of a micromechanical resonator, Nature 444, 75 (006) 135 mk A. Schliesser et al. (Vahala/Kippenberg groups) Radiation pressure cooling of a micromechanical oscillator using dynamical back-action PRL 97, (006) T. Corbitt et al. (LIGO) An all-optical trap for a gram-scale mirror, PRL 98, (007), Optical dilution and feedback cooling of a gram-scale oscillator, PRL 99, (007) 11K 6.9 mk A. Vinante et al (AURIGA gravitational wave detector mirror), M eff = 1,100 Kg, PRL 101, (008) 0.17 mk

15 systems From T. Kippenberg and K. Vahala Science 31, 117 (008)

16 three-mirror geometry [ A. Dorsel et al., PRL 51, 1550 (1983); PM et al, JOSA B11, 1830 (1985); J. D. McCullen et al., Optics Lett. 9, 193 (1984) ]

17 three-mirror cavity Nature 45, 7 (008) See also: M. Bhattacharya and PM, PRL 99, (007) M. Bhattacharya et al, PRA 77, (008) Early discussion: PM et al., JOSA B, 1830 (1985)

18 basic optomechanics quantum approach p 1 H = hωn( q) a a + mω q m + M But: nπ c 1 q ωn( q) = = ωn ωn 1 L + q 1 + q / L L So: H p 1 hωna a + + mω M q hξa aq m

19 three-mirror cavity, R=1 ω ω n, l n, r ω (1 q / L) n ω (1 + q / L) n L 0 q L H p 1 = hωn ( a a + b b) + + mω M q hξ ( a a b b) q m

20 three-mirror cavity, R<1 q0 0 ω ω δ ξ n, e n e L( q q 0) ω ω + δ + ξ n, o n o L( q q 0) q / L W. J. Fader, IEEE J. Quant. Electron. 1, 1838 (1985) p 1 H = h( ωn δe) a a + h( ωn + δo) b b + + mω M q hξl( a a b b) q m

21 three-mirror cavity, R< 1 q0 0, ξq ( 0) ωn e ωn q q, ω + + ξq ( 0) ωn o n o q q q / L H p 1 = h( ω δ ) ω δ + ξ ( m n e a a + h( n + o) b b + mω M q h Q a a b b) q allows preparation of energy eigenstates & observation of quantum jumps

22 linear vs. quadratic coupling Linear coupling Quadratic coupling back-action: mirror motion changed cavity frequency changed intracavity power changed radiation pressure Important when storage time of light comparable to inverse mirror oscillation frequency Quadratic opto-mechanical coupling simply modifies the effective frequency of oscillating mirror Purely dispersive Requires asymmetry in frequency change for two directions of mirror motion

23 a flavor of the theory (R=1) p 1 ( ) Ω M q hξ ( ) H = hωc a a + b b + + m a a b b q m Quantum Langevin equations of motion, input-output formalism: ( ξ ) a& = i Q + γ / a + b& = i ( ξ Q ) + γ / b + q& = p / m p& γ a γ b in in ( ) ( M / ) mωm q + hξ a a b b Γ m p = + ε in Noise operators: ( ) ( ) ( ) ( ) ain t = ain, s + δ ain t δ ain t, δ ain t ' = δ ( t t ') ΓM dω iω ( t t ') hω εin ( t) = 0 εin ( t) εin ( t ') = e ω 1 coth m h + π kbt

24 steady-state linear coupling Effective frequency: in Ωeff ΩM ML δ 4ξγ P 1 [( γ / ) + δ ] Cooling: 4ξγ P 1 δ ML [( γ / ) + δ ] in Γeff ΓM + 3 stiffening field: δ δ > 0 P s in P s cooling field: δ δ < 0 P c in P c M. Bhattacharya and PM, PRL 99, (007) M. Bhattacharya, H. Uys and PM, PRA 77, (008)

25 steady-state quadratic coupling Effective frequency: ξ γ P 1 Ω Ω + + Q in eff M Mωn ( γ / ) δ Effective stiffness: Γ eff Γ M maximized on resonance

26 cooling of rotational motion H Lz 1 = hωca a + + Iωφφ hξφa aφ I [ L, φ ] = ih I z = MR / Spiral phase elements with opposite windings on both sides ω D ξ γ P φ in eff ωφ Iωc [ + ( γ / ) ] ξ γ P φ in eff Dφ + 3 Iωc [ + ( γ / ) ] as = ωc ωφ hξφ Iω φ M. Bhattacharya & PM, PRL 99, (007)

27 rovibrational entanglement scaling: z : h / Mω p : hmω z z z φ : h / Iω L : hiω φ z φ Hamiltonian (dimensionless form) hω hω z φ H = hωca a + ( pz + z ) + ( Lz + φ ) hg za az + hgφa aφ g z ω h cl h L M L Iω c = gφ = ωz φ

28 rovibrational entanglement Entanglement measure: Ru ( ω) Rv ( ω) E( ω) [ R ( ω), R ( ω)] z p Rovibrational entanglement for E( ω ) < 1 z δu = δ z δφ δ v = δ p + δ L R u z z = [ δu( ω) + δu( ω)] / M. Bhattacharya P.-L Giscard and PM, PRA 77, (R) (008)

29 Schrödinger drum Relative motion: q = q m ω eff eff q 1 = m / = 3ω m Center-of-mass motion: Q = ( q + q ) / M eff eff 1 = m Ω = ω m Cavity spectrum (allowed values of k) sin ( θ + 3 kl) + sin θ sin k(3 L q) = sinθ cos( θ + kq) cos kq sinθ = R M. Bhattacharya and PM, PRA78, (R) (008)

30 cavity spectrum ω ( q, Q) = + B ( q q ) + M ( q q ) n, i n, i n, i 0 n, i + B ' n, i ( Q Q ) + M ( Q Q n, i 0 n, i 0 + P ( q q )( Q Q ) ) o Can cool normal modes separately can perform QND energy measurements on normal modes separately Can use mode coupling to control one mode with the other Can generate quantum entanglement of modes (Hartmann and Plenio, PRL to be published)

31 cantilevers for quantum detection and control Pushing quantum mechanics to truly macroscopic systems Quantum superpositions and entanglement in macroscopic systems Novel detectors Coherent control Single-domain ferromagnet with oscillatory component B(t) couples to atomic spin F Described by Tavis-Cummings Hamiltonian [P. Treutlein et al., PRL 99, (007)]

32 Tavis-Cummings Hamiltonian Magnetic field at center of microtrap: B ( t) = G a( t) e $ x r m Cantilever-condensate interaction: H = µ B ( t) = µ g F a( t) r b F x Detuning: µ B gf B0 δ = ω L ω c = ω c h Hamiltonian: H = H + H + V = ω S + ω a a + g S a + S a + BEC r h L z h c h ( ) spin N/ g = µ BGm h h mω r [P. Treutlein et al., PRL 99, (007)]

33 cantilever coupling to dipolar molecules

34 single molecule Used in Magnetic Force Microscopy Frequency shift squeezing G dd = 1 4πε Electric dipoles 0 ω t = ωt + π 3d m d c 5 ε0mr 1/ G dd = µ 0 4π Magnetic dipoles r = ( R + xc ) + x m 1/

35 motional squeezing Classical cantilever motion, = I ( ) V = hc b + b ω ω c t C 15dmd c h = N 6 4πεo mωt R mcωc 1/ N = k B T c / hω c u = Ct S. Singh, M. Bhattacharya, O. Dutta, PM, arxiv:

36 lattice of molecular dipoles E-field One-dimensional molecular chain: H p N N pi dm 1 = + + V 3 t i m 4π ε o i< j x x i j l Small displacements: ω = ω sin( kl / ) k ω = d 0 (3 / πε ml ) 0 m 0 5 1/ Acoustic phonons H ω 1 p = h k bk bk + k P. Rabl and P. Zoller PRA (007)

37 coupling to micromechanical oscillator Hc = hω c a a + k 1 V I dmd c 3( R + xc ) = 1 3 i ri ri Lengths hierarchy: qi l Nl R Squeezing! Frequency shift: ω ω + k k 1 4πε 0 6dmd mω R k c 5 Phonons-oscillator interaction ( )( ) V = hc a + a b b + b b + b b + b b I k k k k k k k k k k C k 3dmd c h = 6 πεo mωk R mcωc 1/

38 phonon squeezing Classical cantilever motion, RWA, VI = h NCk ( bk b k + b kbk ) Two-mode squeezing: 1 ( k k k k ) s1 = b + b + b + b 1 ( k k k k ) s = b b b + b i ω = MHz SrO molecule 16 m = 10 Kg c c N = 100 d c = 10 1 R = µ m l = 00 nm C = k u= C k Cm Nt

39 coherent control V = m c [ R x 7 com + xrel + xc ε0r 4 ( ) 48R x x 48R x x 140lx + 60lx x 60lx d d 8π 3 c com c com rel rel com x + 10lx x x rel c c rel com ] Center-of-mass and relative modes of motion, classical cantilever V exp[ ( ω ω ω )] +.. Lc ab i rel com c ( t) h c a, a b, b L c = N h mcωc 1/ Varied for coherent control

40 simple case ψ (0) = 1,0 com rel ψ ( t) = α 1,0 + β 0,1 com rel com rel Cantilever frequency dependence last generation

41 More interesting start from a thermal state Average occupation Cantilever frequency generation time (Preliminary results) Mean energy generation

42 outlook Limits of cooling Preparation of exotic states Quantized cantilever motion Quantum phase transitions Wenzhou Chen & PM (under construction) Condensates in high-q cavities Esslinger et al, Science 3, 35 (008) Stamper-Kurn et al, Nature Physics 4, 561 (008)