introduction to optomechanics - cqom-itn.net • cavity optomechanics: mechanical degree of freedom...

Introduction to Optomechanics

Klemens Hammerer

IMPRS Lecture Week – Mardorf -- 19

Centre for Quantum Engineering and Space-Time Research

Leibniz University Hannover Institute for Theoretical Physics

Institute for Gravitational Physics (AEI)

Optomechanics • Cavity Optomechanics: Mechanical degree of freedom coupled to a cavity mode

thermal force

radiation pressure force

Can be used to cool the mirror:

Mean energy in thermal equilibrium:

“Quantum Optomechanics”

Can it be used to observe quantum effects with macroscopic systems?

Optomechanical Systems

LIGO – Laser Interferometer Gravitational Wave Observatory

D. Bowmeester, Santa Barbara/Leiden


Aspelmeyer (Vienna) Heidmann (Paris) Bouwmeester (St Barbara, Leiden)

Kippenberg (MPQ) Weig (LMU) Vahala (Caltech) Bowen (UQ)

Harris (Yale) Kimble (Caltech) Treutlein (Basel)

Micromembranes Micromirrors Microtoroids


Optomechanical Crystals

Painter (Caltech) Tang (Yale)

Optomechanics with BEC

Esslinger (Caltech) Stamper-Kurn (Berkeley)

Theory: Chang (Caltech) Romero-Isart (MPQ, ICFO) Experiment: Raizen (Austin)

Levitated Nanoobjects

Mechanical Systems Coupled to Light

Quantum Optomechanics M. Aspelmeyer, F. Marquardt, T. Kippenberg, RMP, arXiv:1303.0733 Macroscopic Quantum Mechanics: Theory and Experimental Concepts of Optomechanics Y. Chen arXiv:1302.1924

Quantum MECHANICAL Tests of the Superposition Principle

Marshall Bouwmeester Penrose PRL 2003

Quantum Mechanical Tests of the Superposition Principle

• “Ramsey – Approach”

• in matter wave interferometry

Evolution: Schrödinger Equ. & ?? Create superposition Measure

K Hornberger, S Gerlich, P Haslinger, S Nimmrichter, M Arndt RMP 84, 157 (2012)



• in matter wave interferometry

Create superposition Measure

H Müntinga W Schleich E Rasel et al. PRL 110 093602 (2013)

Evolution: Schrödinger Equ. & ??



• in optomechanics with nanospheres

Create superposition Measure

Romero-Isart et al. PRL 107, 020405 (2011)

Evolution: Schrödinger Equ. & ??


• 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)

• goal: create and monitor pure quantum states (superposition states) of mechanical oscillators via continuous measurements

Macroscopic Quantum Mechanics: Theory and Experimental Concepts of Optomechanics

Y. Chen, arXiv:1302.1924

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

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!

Quantum Description

• cavity frequency for moving mirror

• Optomechanical Hamiltonian

• Coupling rate in a Fabry-Perot microcavity

for 1ng and 1MHz oscillator

Equations of motion

• Hamiltonian

• Equations of motion cavity slowly varying variable

decay drive vacuum noise

cavity resonance

laser detuning from cavity resonance

Equations of motion

• Hamiltonian

• Equations of motion cavity mechanical oscillator

decay drive vacuum noise

decay thermal force

cavity resonance

Equations of motion

• Quantum equations of motion

• Classical equations of motion

no vacuum fluctuations

classical random force

Equations of motion

• Classical equations of motion at zero temperature: (deterministic)

Equations describe nonlinear, highly nontrivial physics: multistability, self induced oscillations, chaos… 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:

no classical random force

F. Marquardt, M. Ludwig

Equations of motion

• Fluctuations evolve according to

coupling strength of order:

for large intracavity amplitude drop small, nonlinear terms

Equations of motion

• Quantum fluctuations evolve according to linearized equations of motion

• Classical description: replace operators by C-numbers, drop vacuum noise

Equations of motion


• Equations of motion in terms of quadratures:

Equations of motion


• Equations of motion in terms of amplitude (creation/annihilation) operators:

Corresponds to effective optomechanical interaction Hamiltonian

RWA for noise: Valid for high Q Large T

compare to:

Radiation Pressure Interactions in Optomechanics

• (linearized) radiation pressure interaction in optomechanical systems

Squeezing via Quantum Non Demolition Measurement

in


out in


squeezed state!

conditioned on measurement

out in final


• QND Hamiltonian

• Heisenberg picture

• conditional atomic variance

projection noise level

KH, A.S. Sorensen, E.S. Polzik, RMP. 82, 1041 (2010);

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, 16182 (2011)

Mechanical Squeezing via QND Measurement

• Application: Cooling via successive QND measurements

• Application: Improved position/force sensing in pulsed or stroboscopic measurements

1st measurement 2nd measurement free evolution for

quarter period

Braginsky, Khalili…

M. Vanner, KH, et al. PNAS 108, 16182 (2011)

Experiment with musec pulses: M. Vanner et al. quant-ph/1211.7036

Continuous Measurement and Back Action

• Full dynamics is not of QND type

• cavity phase quadrate reads out position position evolves freely momentum couples to amplitude quadrature (radiation pressure) Amplitude noise couples via mechanics into phase quadrature: back action of light

+ cavity decay + vacuum noise

+ decay + thermal noise

Continuous Measurement Sensitivity

• phase quadrature in frequency domain

• noise spectral density referred to signal strength on resonance :

shot noise (phase noise)

back action noise (amplitude noise)

signal

…mechanical susceptibility

sensitivity

Continuous Measurement Sensitivity

• sensitivity

• standard quantum limit: shot noise = back action noise achieved for power such that

• back action larger than thermal noise achieved for power such that “strong optomechanical cooperativity”

thermal noise

~ Power se

nsiti

vity

Introduction to Quantum Noise, Measurement and Amplification Clerk, et al, RMP 82, 1155 (2010)

power

Strong Optomechanical Cooperativity

• Back action noise recently observed with micromechanical oscillators for the first time

• optomechanical ground state cooling also requires

Purdy, Peterson, Regal, Science 339, 801 (2013) [observed before with atomic oscillators Murch, Stamper-Kurn, Nat. Phys. 4, 561 (2008)]

Chan, et al., Nature 478, 89 (2011). Teufel, et al., Nature 475, 359 (2011).

Marquardt, Wilson-Rae

Strong Optomechanical Cooperativity

• Back action noise recently observed with micromechanical oscillators for the first time

• optomechanical cooperativity per single photon

(linearized vs single photon coupling )

M. Aspelmeyer, F. Marquardt, T. Kippenberg, RMP, arXiv:1303.0733

Purdy, Peterson, Regal, Science 339, 801 (2013) [observed before with atomic oscillators Murch, Stamper-Kurn, Nat. Phys. 4, 561 (2008)]

Conditional Quantum State

• Conditional quantum state according to stochastic master equation solution: Monte Carlo

• 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

Quantum Measurement and Control (2009) Wiseman, Milburn

unconditional dynamics conditional dynamics

Conditional Quantum State

• covariance matrix obeys Ricatti equation:

• stochastic (but known) mean values and deterministic covariance matrix fully determine quantum state, e.g. Wigner function etc purity of state uncertainty product

drift matrix diffusion matrix measurement term (nonlinear!)

uncondtional dynamics conditional dynamics

Belavkin et al, arXiv:0506018 Genoni et al., PRA 87 042333 (2013)

Vasilyev, Muschik, KH, PRA, arXiv:1303.5888

Quantum Optics in Phase Space W. Schleich

pure state! even for

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:1303.0733

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, 013803 (2011) Hofer et al PRA 84, 052327 (2011)

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”

• unconditional feedback master equation

measurement terms depend on

Hofer, Vasilyev, Aspelmeyer, KH, arXiv:1303.4976

mechanical squeezing

Continuous Quantum Teleportation

• select an entangling interaction by tuning to the upper sideband

• feedback master equation (ideally) with jump operator such that

• including thermal noise & counterrotating terms:


i.e.

input squeezing - 6dB

Continuous Quantum Teleportation & Entanglement Swapping

• more general (non Gaussian) states: use field emitted from a second, source system and transfer to target system field emitted by an cavity QED is a (continuos) Matrix Product State of the contiunuous 1D EM field:

• 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

source

target

Barrett, KH et al. PRL 110, 090501 (2013)



• can also be applied to spins for correct feedback steady state is

• including tramsmision losses


• setup is similar to a GWD Michelson interferometer: entangled state of mirror test masses predicted in

• 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

Müller-Ebhardt, Rehbein, Schnabel, Danzmann, Chen, PRL 100 013601 (2008) Miao, PRA 81 012114 (2010)

Macroscopic Quantum Mechanics: Theory and Experimental Concepts of Optomechanics Y. Chen, arXiv:1302.1924

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

• 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)

Furusawa et al 1305.0066

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

cooling, state exchange

blue

entanglement

resonant

displacement detection QND measurement

Non Gaussian States and Interactions

• How can we leave the Gaussian world?

• The Non Gaussian toolbox

– inject Non-Gaussian light

– photon counting (instead of homodyne detection)

– non-bilinear Hamiltonians

radiation pressure interaction is fundamentally nonlinear, but notoriously weak:

F.Khalili, S.Danilishin, H.Miao, H.Muller-Ebhardt, H.Yang, Y.Chen, arxiv:1001.3738 U. Akram, N. Kiesel, M. Aspelmeyer, G. J. Milburn, arxiv:1002.1517

can be engineered with micromembranes

Marshall, C.Simon, R.Penrose, D.Bouwmeester, PRL 91, 130401 (2003)

Quantum Optomechanics

• Radiation pressure Hamiltonian is nonlinear For sufficiently strong coupling per single photon g0 the dynamics will be Non-Gaussian.

• Do Non-Gaussian features persist in steady state? Negative Wigner Functions? Master Equation

thermal contact

cavity decay driving field

Steady State

• Solve master equation for steady state for (rather extreme) parameters

• Wigner function

detuning

Qian, Clerk, KH, Marquardt PRL 109, 253601, 2012

A

Steady State


• Wigner function

detuning


B

Steady State


• Wigner function

detuning

C


Steady State


• Wigner function

detuning


D

Steady State

• Wigner functions

• Phonon number and Fano factor Negative Wigner

functions

cooling heating


Limit Cycles

• Quantum equations of motion

• limit cycles are classical nonlinear dynamics

+ noise

+ noise

Marquardt, Ludwig… RMP, arXiv:1303.0733

Limit Cycles

• Classical equations of motion

• slowly varying amplitude solution intensity


off resonant terms

Limit Cycles

• mechanical amplitude

• nonlinear optical damping

o no limit cycles for Δ > 0 ! o negative Wigner functions?


Conditional state

• Conditional state as calculated from one trajectory of Monte-Carlo simulation

Loerch, KH, in prep

Albert Einstein Institute

Institute for Theoretical Physics

Centre for Quantum Engineering and Space-

Time Research Group: Sebastian Hofer Denis Vasilyev Niels Loerch Sergey Tarabrin

Collaborators: M. Aspelmeyer & company M.S. Kim, G.J. Milburn, C. Brukner, I. Pikovski T. Osborne, S. Harrison, S. Barrett, T. Northup C. Muschik

Thank you!

Support through: DFG (QUEST), EC (MALICIA, iQUOEMS)

Winter School on Rydberg Physics and Quantum Information Obergurgl Feb 10-14 1013 www.rqi.uni-hannover.de

arXiv:1303.4976

introduction to optomechanics - cqom-itn.net • cavity optomechanics: mechanical degree of freedom...

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