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Optomechanics with Blue Detuning
Klemens Hammerer
Centre for Quantum Engineeringand Space-Time Research
Leibniz University HannoverInstitute for Theoretical Physics
Institute for Gravitational Physics (AEI)
Summer School on Optomechanics – Les Houches – Aug 2015
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Optomechanics in 1920s
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Gedankenexperiment #1: Heisenberg microscope
Heisenberg, The Physical Principles of the Quantum Theory, 1930
uncertainty in Compton recoil
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Standard Quantum Limit in Position Sensing
free mass
measure position to precision
momentum has uncertainty
uncertainty in position at later time
amplitude
phase
measurement back action
observed in cavity optomechanical system: Purdy et al, Science 339, 801 (2013)
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Gedankenexperiment #2: Optical Cooling of Mirror & its Quantum Limits
“On the development of our conception regarding the nature and constitution of radiation.”
Talk at “81st meeting of the Society of Natural Scientists and Medics” Salzburg 1909
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Doppler shift of reflected wave
force due to momentum transfer on mirror
Radiation provides friction
Gedankenexperiment #2: Optical Cooling of Mirror & its Quantum Limits
for power
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Doppler shift of reflected wave
force due to momentum transfer on mirror
Radiation provides friction
limitation is given by (quantum) fluctuations of EM field subject to Plancks law
Gedankenexperiment #2: Optical Cooling of Mirror & its Quantum Limits
for power
temperature
Doppler cooling
observed in cavity optomechanical system (with resolved sideband cooling):Chan Nature 478, 89 (2011), Teufel, Nature 475, 359 (2011).
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Quantum effects so far in optomechanics (incl. μw electromechanics)
» ground state cooling
» Quantum coherent coupling
» ponderomotive squeezing
» back action noise in position sensing
» quantum coherent state transfer
» optomechanical entanglement
» feedback control within decoherence time
» Quantum squeezing of motion
Chan Nature 478, 89 (2011).Teufel, Nature 475, 359 (2011).
Safavi-Naeini, Nature 500, 185 (2013).Brooks, Nature 488, 476 (2012).
Purdy, Science 339, 801 (2013).
O’Connell, Nature 464, 697 (2010)Palomaki, Nature 495, 210 (2013)
Palomaki, Science 342, 710 (2013)
Roukes, Schwab (2005)
KH, Science 342 702 (2013)
Verhagen, Nature 482, 63 (2012).
Wilson, arXiv:1410.6191 (2014)
Wollman, arXiv:1507,01662 (2015)
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Optomechanical systems
radiation pressure interaction
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Optomechanical systems
LIGO – Laser Interferometer Gravitational Wave Observatory
D. Bowmeester, Santa Barbara/Leiden
up to
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Mechanical SystemsCoupled to Light
Quantum OptomechanicsM. Aspelmeyer, F. Marquardt, T. Kippenberg,RMP, arXiv:1303.0733
Macroscopic Quantum Mechanics: Theory and Experimental Conceptsof OptomechanicsY. ChenJ Phys. B 46 104001 (2013)
Quantum Optomechanics - throwing a glanceM. Aspelmeyer, S. Groeblacher, KH, N. KieselJOSA B 27, A189 (2010)
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Optomechanical systems
radiation pressure interaction
for
M. Aspelmeyer, F. Marquardt, T. Kippenberg, RMP, arXiv:1303.0733
example:
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effective coupling strength
number of circulating photons
Optomechanical systems
for
linearized radiation pressure interaction
for 1mW and 1 MHz line width
strong coupling & normal mode splitting
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Optomechanical Cooperativity
condition for quantum coherent dynamics:
large optomechanical cooperativity
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Quantum effects so far in optomechanics (incl. μw electromechanics)
» ground state cooling
» Quantum coherent coupling
» ponderomotive squeezing
» back action noise in position sensing
» quantum coherent state transfer
» optomechanical entanglement
» feedback control within decoherence time
» Quantum squeezing of motion
Quantum Effects in Optomechanics
Chan Nature 478, 89 (2011).Teufel, Nature 475, 359 (2011).
Safavi-Naeini, Nature 500, 185 (2013).Brooks, Nature 488, 476 (2012).
Purdy, Science 339, 801 (2013).
O’Connell, Nature 464, 697 (2010)Palomaki, Nature 495, 210 (2013)
Palomaki, Science 342, 710 (2013)
Verhagen, Nature 482, 63 (2012).
Wilson, arXiv:1410.6191 (2014)
Wollman, arXiv:1507,01662 (2015)
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Optomechanical “Phase Diagramm”
C. Genes, D. Vitali et al. Phys. Rev. A 77, 033804 (2008)
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unstable
unstable
stable
couplingstrength
detuning
cooperativity
1
2
5
10
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Resonant Drive: Position Sensing
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couplingstrength
detuning
cooperativity
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measurement back action: Purdy, Science 339, 801 (2013).
“QuantumNon-Demolitioninteraction”
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Red Detuned Drive: Optomechanical Cooling
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1.2
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couplingstrength
detuning
cooperativity
1
2
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ground state cooling
Chan Nature 478, 89 (2011)Teufel, Nature 475, 359 (2011)
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Blue Detuned Drive
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couplingstrength
detuning
cooperativity
1
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?
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Resonant interaction is entangling
Compare to parametric down-conversion in nonlinear optics:
Blue detuned drive: Optomechanical Heating
pump
Ou, Pereira, Kimble, Peng,PRL 68, 3663 (1992)
opticalmode
opticalmode
opticalmode
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Digression: Two Mode Squeezing
Squeezed states of two modes
each mode looks like it was in a thermal state
overall state is pure with corresponding wave function
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Digression: EPR Correlations
for infinite squeezing this corresponds to the ideal EPR state
Center of mass position and relative momentum take sharp values
for two mode squeezed states with finite squeezing limit for uncorrelated states in ground state
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Digression: Entanglement
Two mode squeezed states are entangled: cannot be written as product state
a mixed states is entangled if it can not be written as a mixture of product states
general EPR entanglement criterion for two modes
for Gaussian states this is necessary & sufficient (slightly generalized)
a state is entangled if
limit for uncorrelated states in ground state
Duan PRL (2000)Simon PRL (2000)
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Drive on upper sideband creates entanglement
Problem: System is dynamically unstable for blue detuned drive
use a pulsed drive: solve scattering problem
Blue detuned drive unstable regime
Sebastian G. Hofer, Witlef Wieczorek, Markus Aspelmeyer, KHPhys. Rev. A 84, 052327 (2011) O. Romero-Isart et al., Physical Review A 83, 013803 (2011)
noise
optically induced damping and frequency shifty
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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)
Sebastian G. Hofer, Witlef Wieczorek, Markus Aspelmeyer, KHPhys. Rev. A 84, 052327 (2011)
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will generate photons at cavity frequency in precise temporal mode
input-output relations for scattered pulse
squeezing parameter
Pulsed Generation of Entanglement
two mode squeezed state!
mode profile
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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
Sebastian G. Hofer, Witlef Wieczorek, Markus Aspelmeyer, KHPhys. Rev. A 84, 052327 (2011)
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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
Precoolingon red sideband
entangling pulseon blue sideband
readout pulseon red sideband
entanglement
red out
1st pulse
2nd pulse
mec
Sebastian G. Hofer, Witlef Wieczorek, Markus Aspelmeyer, KHPhys. Rev. A 84, 052327 (2011)
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mw optomechanical system:
Experiment by Lehnert group
Entangling mechanical motion with microwave fieldsT. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert
Science 342 6159 (2013)
news & views:KH, Science 342 6159 (2013)
Einstein-Podolsky-Rosen variance:
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Optomechanical Entanglement
Two-mode squeezed (Gaussian), entangled state of a macroscopic (micron-sized) mechanical oscillator and a travelling pulse of (mw) light
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quantum effects so far…
– demonstrate large cooperativity
– rely on linear dynamics
– use homodyne detections
– preserve Gaussian states
– therefore, have an equivalent classical interpretation (with some level of noise)
Quantum Optomechanics
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Galland, Sangouard, Piro, Gisin, Kippenberg, PRL 112, 143602 (2014)
Optomechanical Entanglement
Two-mode squeezed (Gaussian), entangled state of a macroscopic (micron-sized) mechanical oscillator and a travelling pulse of (mw) light
detection of single photon projects mechanical oscillator in single phonon Fock state
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Photon-Counting in Electro-Mechanics
qubit cavity
Lecocq, arXiv: 1409.0872
qubittransition
frequency
photonnumberin cavity
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Photon-Counting in Electro-Mechanics
qubit cavity
Lecocq, arXiv: 1409.0872
qubittransition
frequency
photonnumberin cavity
0 1 2 3 4
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Photon-Counting in Electro-Mechanics
qubit cavity
Lecocq, arXiv: 1409.0872
qubittransition
frequency
photonnumberin cavity
+ measurement of qubit in e0 1 2 3 4
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Galland, Sangouard, Piro, Gisin, Kippenberg, PRL 112, 143602 (2014)
Optomechanical Entanglement
Two-mode squeezed (Gaussian), entangled state of a macroscopic (micron-sized) mechanical oscillator and a travelling pulse of (mw) light
detection of single photon projects mechanical oscillator in single phonon Fock state
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Bell Inequality for Binary Observables
source of entangled systems A & B
CHSH Inequality (implied by realism & locality)
For two (effective) spin ½ systems in singlet state:
rules out (realism and locality)
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Add coherent amplitude before detection
define the binary observable
Binary Observable for Bosonic Modes
no click
click!no click
Banaszek, PRL 82, 2009 (1999)
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Bell Inequality in Optomechanics
0) Initialization of mechanics in ground state by red sideband cooling
1) Optomechanical entanglement by blue sideband pulse
2) Displacement by amplitude α & photon counting
3) Swap of mechanical state to photons by red sideband pulse
4) Displacement by amplitude β & photon counting
Repeat for various measurement setting α and β and infer
S.G. Hofer, K.W. Lehnert, KH, arXiv:1506.08097V. Caprara Vivoli, T. Barnea, C. Galland, N. Sangouard, arXiv:1506.06116
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Realization in Electro-Mechanics
qubit cavity
Lecocq, arXiv: 1409.0872
optomechanical master equation for cascaded cavity setup:
S.G. Hofer, K.W. Lehnert, KH, arXiv:1506.08097
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Bell Inequality in Optomechanics
cooperativity
Bellviolation
transmittivity
For parameters of: T. A. Palomaki, Science 342 6159 (2013) with solid (dashed)
Includes thermal decoherence, finite detection efficiency, transmission lossesS.G. Hofer, K.W. Lehnert, KH, arXiv:1506.08097
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Beyond pulsed entanglement: Continuous Drive on Blue Sideband
• Nonlinear quantum equations of motion
• consider corresponding nonlinear classical nonlinear dynamics
+ noise
+ noise
Experiment:Anetsberger, Nat. Phys. (2009)
Theory:Marquardt, Ludwig, see RMP, arXiv:1303.0733
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Limit Cycles
• Classical equations of motion
• slowly varying amplitude
solution
intensity
Marquardt, Ludwig…RMP, arXiv:1303.0733
off resonant terms
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Limit Cycles
• mechanical amplitude
• nonlinear optical damping
classical:Marquardt, Ludwig…RMP, arXiv:1303.0733
quantum:J. Qian, A. Clerk, KH, F. Marquardt, PRL 109, 253601 (2012)N. Lörch, J. Qian, A. Clerk, F. Marquardt, KH, arXiv:1310.1298, PRX (2014)
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Quantum Theory of Limit Cycles
• 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 decaydriving field
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Quantum Mechanical Steady State
• Solve master equation for steady state
for (rather extreme) parameters
• Wigner function
detuning
Qian, Clerk, KH, MarquardtPRL 109, 253601, 2012
A
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Quantum Mechanical Steady State
• Solve master equation for steady state
for (rather extreme) parameters
• Wigner function
detuning
Qian, Clerk, KH, MarquardtPRL 109, 253601, 2012
B
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Quantum Mechanical Steady State
• Solve master equation for steady state
for (rather extreme) parameters
• Wigner function
detuning
C
Qian, Clerk, KH, MarquardtPRL 109, 253601, 2012
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Quantum Mechanical Steady State
• Solve master equation for steady state
for (rather extreme) parameters
• Wigner function
detuning
Qian, Clerk, KH, MarquardtPRL 109, 253601, 2012
D
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Quantum Mechanical Steady State
• Wigner functions
• Phonon number and Fano factorNegativeWigner
functions
co
olin
gh
eatin
g
Qian, Clerk, KH, MarquardtPRL 109, 253601, 2012
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Laser theory approach provides Fokker-Planck equation for mechanical oscillator
Can exhibit sub-Poissonian phonon statistics & negative Wigner functions
Limit Cycles in the Quantum Regime
Oscillation amplitude r
dam
pin
gdiff
usio
n
J. Qian, A. Clerk, KH, F. Marquardt, PRL 109, 253601 (2012)N. Lörch, J. Qian, A. Clerk, F. Marquardt, KH, PRX 4, 011015 (2014)
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Optomechanical Entanglement
Two-mode squeezed (Gaussian), entangled state of a macroscopic (micron-sized) mechanical oscillator and a travelling pulse of (mw) light
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Use for Quantum Teleportation
feedback
VB Aentangled
Bell measurementfeedback
Bennett PRL 1993
continuous variables:Braunstein, Kimble PRL 2003
Vaidman PRL 2003
Sebastian G. Hofer, Witlef Wieczorek, Markus Aspelmeyer, KHPhys. Rev. A 84, 052327 (2011)
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Time Continuous Bell Measurements & Teleportation
continuous wave drive on upper sideband, continuous Bell measurement & (stabilizing) feedback?
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“time continuous quantum remote control”
Time Continuous Teleportation in Optomechanics
cooperativity
6dB inputsqueezing
Gaussian input
mechanical squeezing:• parametric drive• reservoir engineering• QND probeVitali, Clerk, MarquardtBraginsky, Aspelmeyer,Schwab, Nunnenlamp…
Hofer, Vasilyev, Aspelmeyer, KH, PRL 111, 170404 (2013)Hofer, KH PRA 91, 033822 (2015)
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Time Continuous Entanglement Swapping in Optomechanics
EPR squeezinglog negativity
generates deteriministic, stationary entanglement
Hofer, Vasilyev, Aspelmeyer, KH, PRL 111, 170404 (2013)Hofer, KH PRA 91, 033822 (2015)
cooperativity
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Time Continuous Entanglement Swapping in Optomechanics
EPR squeezinglog negativity
generates deteriministic, stationary entanglement tolerant to losses and finite detection efficiency
transmissivity
Hofer, Vasilyev, Aspelmeyer, KH, PRL 111, 170404 (2013)Hofer, KH arXiv:1411.1337
cooperativity
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Time Continuous Entanglement Swapping in Optomechanics
10 m prototype @ AEI
Michelson interferometer!
in context of GWD setups:Y. Chen, PRL 100 013601 (2008)
Hofer, Vasilyev, Aspelmeyer, KH, PRL 111, 170404 (2013)Hofer, KH arXiv:1411.1337
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Summary
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couplingstrength
detuning
cooperativity
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coolingstate swap
position sensingforce sensing
measurement back action
heatingentanglementFock state preparationBell testsquantum control via measurement & feedbacknonclassical limit cycle states
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former members:
Niels Loerch (Basel)
Sergey Tarabrin
Andre Xuereb (Malta)
Albert EinsteinInstitute
Institute forTheoretical Physics
Centre for Quantum Engineering and Space-
Time Research
Group:
Sebastian Hofer
Ondrej Cernotik
Alexander Roth
Jonas Lammers
Marius Schulte
Hashem Zoubi
Klemens Hammerer
Thank you!
Support through:DFG (QUEST, GRK 1991)EC (MALICIA, iQUOEMS)Vienna (WWTF)
optomechanical transducerCollaborators on optomechanics
Konrad Lehnert (JILA)Philipp Treutlein (Basel)Peter Zoller (Innsbruck)Eugene Polzik (Copenhagen)Florian Marquardt (Erlangen)Ash Clerk (McGill)Roman Schnabel (Hamburg)Farit Khalili (Moscow)Markus Aspelmeyer (Vienna)Klaus Hornberger (Duisburg)