Introduction to Optomechanics

Transcription

1 Centre for Quantum Engineering and Space-Time Research Introduction to Optomechanics Klemens Hammerer Leibniz University Hannover Institute for Theoretical Physics Institute for Gravitational Physics (AEI) IMPRS Lecture Week Mardorf -- 19

2 Optomechanics Cavity Optomechanics: Mechanical degree of freedom coupled to a cavity mode thermal force Mean energy in thermal equilibrium: radiation pressure force Can be used to cool the mirror: Can it be used to observe quantum effects with macroscopic systems? Quantum Optomechanics

3 Optomechanical Systems D. Bowmeester, Santa Barbara/Leiden LIGO Laser Interferometer Gravitational Wave Observatory

4 Optomechanical Systems Micromirrors Micromembranes Microtoroids Aspelmeyer (Vienna) Heidmann (Paris) Bouwmeester (St Barbara, Leiden) Harris (Yale) Kimble (Caltech) Treutlein (Basel) Kippenberg (MPQ) Weig (LMU) Vahala (Caltech) Bowen (UQ)

5 Optomechanical Systems Optomechanical Crystals Painter (Caltech) Tang (Yale) Optomechanics with BEC Esslinger (Caltech) Stamper-Kurn (Berkeley) Levitated Nanoobjects Theory: Chang (Caltech) Romero-Isart (MPQ, ICFO) Experiment: Raizen (Austin)

6 Mechanical Systems Coupled to Light Quantum Optomechanics M. Aspelmeyer, F. Marquardt, T. Kippenberg, RMP, arxiv: Macroscopic Quantum Mechanics: Theory and Experimental Concepts of Optomechanics Y. Chen arxiv:

7 Quantum MECHANICAL Tests of the Superposition Principle Marshall Bouwmeester Penrose PRL 2003

8 Quantum Mechanical Tests of the Superposition Principle Ramsey Approach Create superposition Evolution: Schrödinger Equ. &?? Measure in matter wave interferometry K Hornberger, S Gerlich, P Haslinger, S Nimmrichter, M Arndt RMP 84, 157 (2012)

9 Quantum Mechanical Tests of the Superposition Principle Ramsey Approach Create superposition Evolution: Schrödinger Equ. &?? Measure in matter wave interferometry H Müntinga W Schleich E Rasel et al. PRL (2013)

10 Quantum Mechanical Tests of the Superposition Principle Ramsey Approach Create superposition Evolution: Schrödinger Equ. &?? Measure in optomechanics with nanospheres Romero-Isart et al. PRL 107, (2011)

11 Quantum Mechanical Tests of the Superposition Principle Tests based on Continuous Measurements idea: use continuous (precision) measurement to simultaneously prepare and measure superposition states compare predictions according to (stochastic) Schrödinger equation with prediction from (Schrödinger equation & nonstandard model) Macroscopic Quantum Mechanics: Theory and Experimental Concepts of Optomechanics Y. Chen, arxiv: goal: create and monitor pure quantum states (superposition states) of mechanical oscillators via continuous measurements

12 This talk Quantum Nondemolition Measurements of Mechanical Oscillators Continuous Measurement: measurement sensitivity & purity of conditional quantum state Quantum control via time continuous teleportation tomorrow s talk: Nonclassical superposition states via nonlinear dynamics Quantum control of optomechanical systems via coupling to atomic systems

13 Quantum Description Mechanical oscillator creation/annihilation operator Hamiltonian quadrature operators physical position and momentum Cavity creation/annihilation operator Hamiltonian cavity frequency Length depends on mirror position!

14 cavity frequency for moving mirror Quantum Description Optomechanical Hamiltonian Coupling rate in a Fabry-Perot microcavity for 1ng and 1MHz oscillator

15 Equations of motion Hamiltonian Equations of motion cavity decay drive vacuum noise slowly varying variable cavity resonance laser detuning from cavity resonance

16 Equations of motion Hamiltonian Equations of motion cavity mechanical oscillator decay drive vacuum noise cavity resonance decay thermal force

17 Quantum equations of motion Equations of motion Classical equations of motion no vacuum fluctuations classical random force

18 Equations of motion Classical equations of motion at zero temperature: (deterministic) no classical random force Equations describe nonlinear, highly nontrivial physics: multistability, self induced oscillations, chaos F. Marquardt, M. Ludwig Assume stable steady state solution exist with mean values We want to analyze the quantum dynamics around these mean values due to classical random forces and quantum noise on mirror and cavity: Introduce fluctuation operators:

19 Fluctuations evolve according to Equations of motion coupling strength of order: for large intracavity amplitude drop small, nonlinear terms

20 Equations of motion Quantum fluctuations evolve according to linearized equations of motion Classical description: replace operators by C-numbers, drop vacuum noise

21 Equations of motion Quantum fluctuations evolve according to linearized equations of motion Equations of motion in terms of quadratures:

22 Equations of motion Quantum fluctuations evolve according to linearized equations of motion Equations of motion in terms of amplitude (creation/annihilation) operators: RWA for noise: Valid for high Q Large T Corresponds to effective optomechanical interaction Hamiltonian compare to:

23 Radiation Pressure Interactions in Optomechanics (linearized) radiation pressure interaction in optomechanical systems

24 Squeezing via Quantum Non Demolition Measurement in

25 Squeezing via Quantum Non Demolition Measurement in out

26 Squeezing via Quantum Non Demolition Measurement in out final conditioned on measurement squeezed state!

27 Squeezing via Quantum Non Demolition Measurement QND Hamiltonian Heisenberg picture conditional atomic variance projection noise level KH, A.S. Sorensen, E.S. Polzik, RMP. 82, 1041 (2010);

28 QND Measurement with Short Pulses linearized radiation pressure interaction for short pulses relevant decoherence rate of oscillator requires short pulses compatible with cavity optimization of pulse profile matching of LO profile to compensate for pulse distortion good cavity design M. Vanner et al. PNAS 108, (2011)

29 Mechanical Squeezing via QND Measurement Application: Cooling via successive QND measurements 1 st free evolution for measurement 2 nd measurement quarter period M. Vanner, KH, et al. PNAS 108, (2011) Experiment with musec pulses: M. Vanner et al. quant-ph/ Application: Improved position/force sensing in pulsed or stroboscopic measurements Braginsky, Khalili

30 Continuous Measurement and Back Action Full dynamics is not of QND type cavity phase quadrate reads out position position evolves freely + cavity decay + vacuum noise momentum couples to amplitude quadrature (radiation pressure) + decay + thermal noise Amplitude noise couples via mechanics into phase quadrature: back action of light

31 Continuous Measurement Sensitivity phase quadrature in frequency domain shot noise (phase noise) signal noise spectral density referred to signal strength back action noise (amplitude noise) mechanical susceptibility on resonance : sensitivity

32 Continuous Measurement Sensitivity sensitivity power standard quantum limit: shot noise = back action noise achieved for power such that sensitivity thermal noise back action larger than thermal noise achieved for power such that ~ Power strong optomechanical cooperativity Introduction to Quantum Noise, Measurement and Amplification Clerk, et al, RMP 82, 1155 (2010)

33 Strong Optomechanical Cooperativity Back action noise recently observed with micromechanical oscillators for the first time Purdy, Peterson, Regal, Science 339, 801 (2013) [observed before with atomic oscillators Murch, Stamper-Kurn, Nat. Phys. 4, 561 (2008)] optomechanical ground state cooling also requires Chan, et al., Nature 478, 89 (2011). Teufel, et al., Nature 475, 359 (2011). Marquardt, Wilson-Rae

34 Strong Optomechanical Cooperativity Back action noise recently observed with micromechanical oscillators for the first time Purdy, Peterson, Regal, Science 339, 801 (2013) [observed before with atomic oscillators Murch, Stamper-Kurn, Nat. Phys. 4, 561 (2008)] optomechanical cooperativity per single photon (linearized vs single photon coupling ) M. Aspelmeyer, F. Marquardt, T. Kippenberg, RMP, arxiv:

35 Conditional Quantum State Conditional quantum state according to stochastic master equation unconditional dynamics conditional dynamics solution: Monte Carlo Quantum Measurement and Control (2009) Wiseman, Milburn For linearized interaction and homodyne detection of light: Gaussian states and dynamics Stochastic Master Equation is equivalent to stochastic evolution of first moments deterministic evolution of second moments

36 Conditional Quantum State covariance matrix obeys Ricatti equation: drift matrix uncondtional dynamics diffusion matrix measurement term (nonlinear!) conditional dynamics Belavkin et al, arxiv: Genoni et al., PRA (2013) Vasilyev, Muschik, KH, PRA, arxiv: stochastic (but known) mean values and deterministic covariance matrix fully determine quantum state, e.g. Wigner function etc Quantum Optics in Phase Space W. Schleich purity of state uncertainty product pure state! even for

37 Continuous Preparation and Monitoring of Pure Quantum States the conditional state is pure, Gaussian position and momentum are effectively probed with equal sensitivity if in this case the conditional quantum state will be a coherent state a squeezed state is generated if per single photon arxiv:

38 Universal Quantum Control via Measurement and Feedback idea: (superposition) state engineering via continuous quantum teleportation drive optomechanical system on upper sideband entangling interaction between light & oscillator perform Bell measurement on light linear feedback transfers quantum statistics of light on oscillator: non-gaussian statistics of light generates non-gaussian state of oscillator pulsed teleportation protocols: (here: cw!) Romero-Isart, Pflanzer, Cirac, PRA 83, (2011) Hofer et al PRA 84, (2011)

39 Continuous Bell measurement with Gaussian Input Field linear interaction of arbitrary system with light field conditional master equation for system under continuous Bell measurement linear feedback with forces measurement terms depend on unconditional feedback master equation Hofer, Vasilyev, Aspelmeyer, KH, arxiv:

40 Continuous Quantum Teleportation select an entangling interaction by tuning to the upper sideband feedback master equation (ideally) i.e. with jump operator such that including thermal noise & counterrotating terms: mechanical squeezing input squeezing - 6dB Hofer, Vasilyev, Aspelmeyer, KH, arxiv:

41 Continuous Quantum Teleportation & Entanglement Swapping more general (non Gaussian) states: use field emitted from a second, source system and transfer to target system target field emitted by an cavity QED is a (continuos) Matrix Product State of the contiunuous 1D EM field: Barrett, KH et al. PRL 110, (2013) source if both systems have an entangling interaction with light: time continuous entanglement swapping generates EPR state of oscillators generates entangled state of micromechanical oscillators in steady state entangled states with EPR squeezing superposition size scaling as Hofer, Vasilyev, Aspelmeyer, KH, arxiv:

42 Continuous Quantum Teleportation & Entanglement Swapping can also be applied to spins for correct feedback steady state is including tramsmision losses

43 Continuous Quantum Teleportation & Entanglement Swapping setup is similar to a GWD Michelson interferometer: entangled state of mirror test masses predicted in Müller-Ebhardt, Rehbein, Schnabel, Danzmann, Chen, PRL (2008) Miao, PRA (2010) continuous test of quantum mechanics with macroscopic objects: parametrize how the dynamics of the object would deviate from standard quantum mechanics (for a given collapse model) monitor true dynamics and exclude parameters Macroscopic Quantum Mechanics: Theory and Experimental Concepts of Optomechanics Y. Chen, arxiv:

44 Requirements & Summary Experiments need to have a strong optomechanical cooperativity the conditional state predicted by the stochastic master equation has to be reconstructed from measured photocurrent via Kalman/Wiener filtering applied to optomechanical system in free space Furusawa et al continuous measurement at the quantum limit is getting feasible in state of the art optomechanical systems toolbox for universal quantum state engineering detector of non-standard physics based on macroscopic superposition states (alternative to Ramsey-like approaches)

45 Gaussian States and Interactions The optomechanical toolbox linearized interaction: homodyne detection of light: measurement of injection of squeezed, coherent (Gaussian) light preserves Gaussian character of states. We can select the dominant interaction by choosing the detuning red blue resonant cooling, state exchange entanglement displacement detection QND measurement

46 Non Gaussian States and Interactions How can we leave the Gaussian world? The Non Gaussian toolbox inject Non-Gaussian light F.Khalili, S.Danilishin, H.Miao, H.Muller-Ebhardt, H.Yang, Y.Chen, arxiv: U. Akram, N. Kiesel, M. Aspelmeyer, G. J. Milburn, arxiv: photon counting (instead of homodyne detection) Marshall, C.Simon, R.Penrose, D.Bouwmeester, PRL 91, (2003) non-bilinear Hamiltonians can be engineered with micromembranes radiation pressure interaction is fundamentally nonlinear, but notoriously weak:

47 Quantum Optomechanics Radiation pressure Hamiltonian is nonlinear For sufficiently strong coupling per single photon g 0 the dynamics will be Non-Gaussian. Do Non-Gaussian features persist in steady state? Negative Wigner Functions? Master Equation thermal contact driving field cavity decay

48 Steady State Solve master equation for steady state for (rather extreme) parameters Wigner function A detuning Qian, Clerk, KH, Marquardt PRL 109, , 2012

49 Steady State Solve master equation for steady state for (rather extreme) parameters Wigner function B detuning Qian, Clerk, KH, Marquardt PRL 109, , 2012

50 Steady State Solve master equation for steady state for (rather extreme) parameters Wigner function C detuning Qian, Clerk, KH, Marquardt PRL 109, , 2012

51 Steady State Solve master equation for steady state for (rather extreme) parameters Wigner function D detuning Qian, Clerk, KH, Marquardt PRL 109, , 2012

52 Steady State Wigner functions Phonon number and Fano factor Negative Wigner functions heating cooling Qian, Clerk, KH, Marquardt PRL 109, , 2012

53 Limit Cycles Quantum equations of motion + noise + noise limit cycles are classical nonlinear dynamics Marquardt, Ludwig RMP, arxiv:

54 Limit Cycles Classical equations of motion slowly varying amplitude solution intensity off resonant terms Marquardt, Ludwig RMP, arxiv:

55 Limit Cycles mechanical amplitude nonlinear optical damping o no limit cycles for Δ > 0! o negative Wigner functions? Marquardt, Ludwig RMP, arxiv:

56 Conditional state Conditional state as calculated from one trajectory of Monte-Carlo simulation Loerch, KH, in prep

57 Collaborators: M. Aspelmeyer & company M.S. Kim, G.J. Milburn, C. Brukner, I. Pikovski T. Osborne, S. Harrison, S. Barrett, T. Northup C. Muschik Group: Sebastian Hofer arxiv: Denis Vasilyev Niels Loerch Sergey Tarabrin Centre for Quantum Engineering and Space- Time Research Institute for Theoretical Physics Thank you! Albert Einstein Institute Support through: DFG (QUEST), EC (MALICIA, iquoems) Winter School on Rydberg Physics and Quantum Information Obergurgl Feb