Optomechanics: Hybrid Systems and Quantum Effects

Klemens Hammerer

Cavity Optomechanics – from the micro- to the macro scale – Innsbruck – Nov 06 2013

Centre for Quantum Engineering

and Space-Time Research

Leibniz University Hannover

Institute for Theoretical Physics

Institute for Gravitational Physics (Albert Einstein Institute)

Quantum effects so far in optomechanics (incl. μw electromechanics)

» ground state cooling

» ponderomotive squeezing

» back action noise in position sensing

» quantum coherent state transfer

» optomechanical entanglement

Quantum Optomechanics

Chan Nature 478, 89 (2011).

Teufel, Nature 475, 359 (2011).

Safavi-Naeini, arXiv:1302.6179 (2013).

Brooks, Nature 488, 476 (2012).

Purdy

Purdy, Science 339, 801 (2013).

O’Connell et al., Nature 464, 697 (2010)

Palomaki, Nature 495, 210 (2013)

Lehnert group (2013)

Roukes, Schwab (2005)

» first nonclassical state of (micro)mechanical oscillator

» resource for quantum state control of oscillator

» entanglement as resource in Q-networks

Optomechanical entanglement

Rabl, Lukin

Stannigel, Zoller

Steady state of continuously driven

optomechanical system can be entangled:

optomechanical cooperativity

Stationary Entanglement

Vitali, PRL 98, 030405 (2007)

Genes, Mari, Mancini, Tombesi

Paternostro

Meystre

Aspelmeyer, Zeilinger

Eisert

…

Genes, PRA 77, 033804 (2008)

entanglement between mechanical oscillator & intracavity field

Stationary Entanglement

- 2 - 1 0 1 2 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

coupling

strength

detuning

unstable regime

1

2

5

10

cooperativity

stationary entanglement: hard to produce/hard to verify

entanglement has to be verified by measurements on external field

entanglement with external modes required for applications in quantum information

Entanglement of mechanics and external field

Genes, Rev. A 78, 032316 (2008)

mechanical state conditioned on

homodyne detection of light

mean phonon number conditioned on photocurrent

necessary condition for correlations between mechanical oscillator & light (entanglement):

Entanglement of mechanics and external field

Wiseman, Milburn

Quantum Measurement and Control

Conditional Phonon Number

- 2 - 1 0 1 2 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4 unstable regime

measurement of phase quadrature

coupling

strength

cooperativity

detuning

1

2

5

10

Conditional Phonon Number

- 2 - 1 0 1 2 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4 unstable regime

measurement of amplitude quadrature

coupling

strength

detuning

1

2

5

10

cooperativity

measurement of amplitude quadrature

for drive on upper sideband conditional state is essentially pure!

- 2 - 1 0 1 2 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Conditional Phonon Number

conditional

phonon

number

detuning

/phase quadrature

Resonant interaction is entangling

Compare to parametric down-conversion in nonlinear optics:

Drive on first blue sideband

pump

Ou, Pereira, Kimble, Peng,

PRL 68, 3663 (1992)

optical

mode

optical

mode

optical

mode

Digression: EPR Correlations

for infinite squeezing this corresponds to the ideal EPR state

Center of mass position and relative momentum take sharp values

for states with finite entanglement (EPR squeezing) limit for

uncorrelated states

in ground state

Drive on upper sideband creates entanglemt

Problem: System is dynamically unstable for blue detuned drive

Parametric heating leads to self induced oscillations

use a pulsed drive: solve scattering problem

Pulsed entanglement

- 2 - 1 0 1 2

1 2

5

10

unstable regime Braginsky, Physics Letters A, 287:331, 2001

Marquardt, Ludwig, Khurgin, Armour, Nation

EXP:

Favero, Weig, Kippenberg, Vahala. Painter, Karrai, Khurgin…

Sebastian G. Hofer, Witlef Wieczorek, Markus Aspelmeyer, KH

Phys. Rev. A 84, 052327 (2011)

Pulsed Generation of Entanglement

integrate for pulse suration

central frequency at upper sideband

assuming

weak thermal decoherence

sideband resolved limit for suppression of Anti-Stokes scattering

weak coupling: adiabatic elimination of cavity mode (avoid memory effects)

will generate photons at cavity frequency in precise temporal mode

input-output relations for scattered pulse in RWA, neglecting thermal noise

squeezing parameter

Pulsed Generation of Entanglement

two mode squeezed state!

mode profile

Pulsed Generation of Entanglement

EPR variance, taking into account initial thermal occupation of mirror

if

large EPR squeezing requires large cooperativity:

for pulse length

squeezing parameter

for

drive system on first red sideband:

mechanical state is swapped to light

entanglement preparation and verification:

measure EPR quadratures of 1st and 2nd pulse and correlate

Verification of entanglement

Palomaki, Nature 495, 210 (2013)

time

Precooling

on red sideband

entangling pulse

on blue sideband

readout pulse

on red sideband

entanglement

red out

1st pulse

2nd pulse

mec

Sebastian G. Hofer, Witlef Wieczorek, Markus Aspelmeyer, KH

Phys. Rev. A 84, 052327 (2011)

mw optomechanical system:

Experiment by Lehnert group

Teufel, Nature 475, 359-363 (2011)

entangling

pulse

read-out

pulse

(= mechanics)

Experiment by Lehnert group

Variances

variances

covariances

e.g.

mw optomechanical system:

mw optomechanical system:

Experiment by Lehnert group

T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert

Entangling mechanical motion with microwave fields

Science (2013) (to be published)

Extension I: Quantum Teleportation

feedback

V B A entangled

Bell measurement feedback

Bennett PRL 1993

continuous variables:

Braunstein, Kimble PRL 2003

Vaidman PRL 2003

Sebastian G. Hofer, Witlef Wieczorek, Markus Aspelmeyer, KH

Phys. Rev. A 84, 052327 (2011)

teleportation in optomechanics:

talk by Paolo Tombesi

Romero-Isart, Pflanzer, Cirac

Extension II: Time Continuous Quantum Teleportation

cw drive on upper sideband, continuous Bell measurement & (stabilizing) feedback?

is special case of: system coupled to 1D field & continuous Bell measurement

with system operator s for optomechanical system (with adiabatically eliminated cavity)

Time Continuous Bell Measurements & Teleportation

master equation for general case

Stochastic Master Equation for Continuous Bell Measurement

Gaussian input

Hofer, Vasilyev, Aspelmeyer, KH, PRL 111, 170404 (2013)

unconditional master equation including feedback

Feedback Master Equation for Continuous Bell Measurement

Gaussian input

Wiseman, Milburn, Quantum Measurement

applied to optomechanics

time continuous quantum remote control: Non-Gaussian states…

more on quantum feedback control

talk by Mauro Paternostro

Time Continuous Teleportation in Optomechanics

cooperativity

6dB input

squeezing

Hofer, Vasilyev, Aspelmeyer, KH, PRL 111, 170404 (2013)

Gaussian input

squeezing:

• parametric drive

• reservoir engineering

• QND probe

• feedback

Paternostro, Vitali,

Clerk, Marquardt

Braginsky, Aspelmeyer,

Schwab, Nunnenlamp…

Quantum effects so far in optomechanics (incl. μw electromechanics)

» ground state cooling

» ponderomotive squeezing

» back action noise in position sensing

» quantum coherent state transfer

» optomechanical entanglement

Quantum Optomechanics

Chan Nature 478, 89 (2011).

Teufel, Nature 475, 359 (2011).

Safavi-Naeini, arXiv:1302.6179 (2013).

Brooks, Nature 488, 476 (2012).

Purdy

Purdy, Science 339, 801 (2013).

O’Connell et al., Nature 464, 697 (2010)

Palomaki, Nature 495, 210 (2013)

Lehnert group (2013)

Hybrid Quantum Systems

Atoms

talk by

Peter Rabl

Patrick Maletisnky talk by

Mika Sillanpää

coherent control & two level defects

talk by Eva Weig

Hybrid Quantum Systems

How can we coherently couple atomic ensembles (or single atoms) to solid state

quantum systems?

Atomic ensembles/

single atoms

Solid state systems

e.g. mechanical oscillators

?

Hybrid Mechanical Systems (review)

Philipp Treutlein, Claudiu Genes, KH, Martino Poggio, Peter Rabl

arXiv:1210.4151

Related work I: Hybrid systems of Atoms and Micromechanical Oscillator

Atoms in optical lattice Micromembrane

Experiment:

S. Camerer, M. Korppi, A. Jöckel, D. Hunger, T.W. Hänsch, P. Treutlein

Phys. Rev. Lett. 107, 223001 (2011)

Theory:

KH, K. Stannigel, C. Genes, P. Zoller, P. Treutlein, S. Camerer, D. Hunger, T. W. Hänsch PRA 82, 021803 (2010)

Berit Vogell et al. PRA 87, 023816 (2013) see POSTER

Quantum Treatment

• Hamiltonian: including membrane, atoms and electro-magnetic field as dofs

• laser drive will give rise

in quadratic order to lattice potential for atoms and mean force on membrane

in linear order to coupling of position fluctuations to EM vacuum fluctuations

kinetic energy

of atoms

optical potential

→ emission and reabsorption of sideband photons will give rise to

effective coupling & quantum noise

Markovian Master Equation

• Resulting Markovian Master Equation

Hamiltonian term for

coherent atom-membrane

interaction at strength

Lindblad terms describing radiation pressure

induced momentum diffusion of membrane

and momentum diffusion of atoms at rates

K. Karrai PRL 100, 240801 (2008)

optical “spring” between

membrane and atomic COM

motion (requires 3D

treatment)

Gordon, Ashkin, Cohen-Tannoudji

Markovian Master Equation

• Resulting Markovian Master Equation

Non-Lindblad term

due to non-zero membrane transmittivity t

effect is to reduce action of atoms on membrane

Asymmetric coupling characteristic for

cascaded quantum systems

atom → membrane:

membrane → atom:

C.W. Gardiner, PRL 70, 2269 (1993)

H. Carmichael, PRL 70, 2273 (1993)

Extension: Cavity enhancement

𝑔 𝑔 × 𝐹

Berit Vogell et al. PRA 87, 023816 (2013) see POSTER

enhances effective coupling by Finesse:

Berit Vogell et al in prep.

saves a Lamb-Dicke factor in coupling

𝑔 𝑔/(𝑘𝑥𝑍𝑃𝐹)

& coupling to internal state

Related Work II

hybrid coupling inside one cavity

KH, M. Wallquist, C. Genes, P. Zoller,

M. Ludwig, F. Marquardt, P. Treutlein, J. Ye, H.J. Kimble,

PRL 103, 063005 (2009), PRA 81 023816 (2010)

cf

talk by Aurelien Dantan

talk by Darrick Chang

other work:

Meiser, Meystre

Genes, Vital, Tombesi

Ritsch

Paternostro

Sun, Nori

…

Albert Einstein Institute

Institute for Theoretical Physics

Centre for Quantum

Engineering and Space-

Time Research

Group:

Sebastian Hofer

Denis Vasilyev

Niels Loerch

Sergey Tarabrin

Klemens Hammerer

Collaborators:

M. Aspelmeyer, W. Wieczorek

P. Treutlein, S. Camerer, M. Korppi, A. Jöckel, D. Hunger, T.W. Hänsch

B. Vogell, C. Genes, K. Stannigel, P. Zoller

Thank you!

Support through:

DFG (QUEST), EC (MALICIA, iQUOEMS)

WWTF

Continuous Bell Measurements

Hofer, Vasilyev, Aspelmeyer, KH,

arXiv:1303.4976

Quantum entanglement and teleportation in pulsed cavity-optomechanics

Hofer, Wieczorek, Aspelmeyer, KH

Phys. Rev. A 84, 052327 (2011)