Cavity Optomechanics Florian Mar Marquardt quardt University of Erlangen-Nuremberg, and Max-Planck-Institute for the Science of Light
Radiation forces
baryon-photon fluid: sound speed
√ c/ 3
Johannes Kepler De Cometis, 1619
(Comet Hale-Bopp; by Robert Allevo)
Nichols and Hull, 1901 Lebedev, 1901
Nichols and Hull, Physical Review 13, 307 (1901)
Trapping and cooling • Optical tweezers • Optical lattices ...but usually no back-action from motion onto light!
Optomechanical Hamiltonian
Review “Cavity Optomechanics”: M. Aspelmeyer, T. Kippenberg, FM Rev. Mod. Phys. 2014 laser
optical cavity
mechanical mode
Review “Cavity Optomechanics”: M. Aspelmeyer, T. Kippenberg, FM Rev. Mod. Phys. 2014 laser
optical cavity
mechanical mode
laser detuning optomech. coupling
A bit of history
optomechanical change of mechanical damping rate
Braginsky, Manukin, Tikhonov JETP 1970
mechanical resonator microwave cavity
Static bistability in an optical cavity experiment Dorsel, McCullen, Meystre,Vignes, Walther PRL 1983 F rad(vs. x) mirror = 2I (xposition )/c force λ 2F
λ/2
Experimental proof of static bistability: A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes and H. Walther: Phys. Rev. Lett. 51, 1550 (1983)
V rad(x) V eff = V rad + V HO
x hysteresis
quasistatic F
finite sweep rate !
sweep x finite cavity ring-down rate γ ⇒ delayed response to cantilever motion
finite optical ringdown time – delayed response to cantilever motion F
F dx <0
>0
Höhberger-Metzger and Karrai 2004: x 300K to 17K [photothermal force]; cooling 0 heating Höhberger-Metzger and Karrai, 2006: radiation pressure cavity (amplification) Nature 432, Heidmann/ 1002 (2004): cooling [Aspelmeyer, C. Höhberger Metzger and K. Karrai, Nature 432, 1002 (2004) 300K to 17K [photothermal force]
A zoo of devices
...any dielectric moving inside a cavity generates an optomechanical interaction!
Karrai (Munich)
Bouwmeester (Santa Barbara)
LKB group (Paris)
Mavalvala (MIT)
Vahala (Caltech) Kippenberg (EPFL), Carmon, ...
Harris (Yale)
Teufel, Lehnert (Boulder (Boulder)) Painter (Caltech)
Schwab (Cornell)
Stamper-Kurn Stamper-K urn (Berkeley) ( Berkeley) cold atoms
Aspelmeyer (Vienna)
Why?
Fundamental tests of quantum mechanics in a new regime: entanglement with ‘macroscopic’ objects, unconventional decoherence? [e.g.: gravitationally induced?] Mechanics as a ‘bus’ for connecting hybrid components: superconducting qubits, spins, photons, cold atoms, .... Precision measurements small displacements, masses, forces, and accelerations
Painter lab
Tang lab (Yale)
Optomechanical circuits & arrays Exploit nonlinearities for classical and quantum information processing, storage, and amplification; study collective dynamics in arrays
Schwab and Roukes, Physics Today 2005
nano-electro-mechanical systems •Superconducting qubit coupled to nanoresonator: Cleland & Martinis 2010
• optomechanical systems
10
1x10 9 1x10 8 1x10 7 1x10 6 r x10 1 e b 100000 m 10000 u n 1000 n 100 o n 10 o h 1 p 0.1 0.01 0.001 0.0001 0.00001
T I M
e l a Y
B K L
l e f r u e 1 t 1 n 1 T e i 0 a n 2 P 1 h 0 J h r 2 o c a e r t k e l a s l d C O u
I Q Q A O P L Q I I M J
o B
ground state 0.01
0.1
analogy to (cavity-assisted)
m i n i p h o m u m p n o n o s s l n u m ib b e r e 1
10
FM et al., PRL 93, 093902 (2007)
Classical dynamics
optical cavity
input laser
mechanical frequency
mechanical damping
equilibrium position
cavity detuning from resonance decay rate
cantilever
radiation pressure
laser amplitude
(solve for arbitrary
)
Optomechanical frequency shift (“optical spring”) Effective optomechanical damping rate
Effective optomechanical damping rate
Optomechanical frequency shift (“optical spring”)
laser detuning
cooling
heating/ amplification
softer
stiffer
Quantum picture
Review “Cavity Optomechanics”: M. Aspelmeyer, T. Kippenberg, FM Rev. Mod. Phys. 2014 laser
optical cavity
mechanical mode
laser detuning optomech. coupling
† ˆ ˆ a ˆ a ˆ(b + b) †
photon
phonon
† ˆ ˆ ~ g0 a ˆ a ˆ(b + b ) †
ˆ = α + δ a ˆ
a
†
~ g0 (αδ a ˆ “laser-enhanced optomechanical coupling”:
g
=
+α
∗
† ˆ ˆ δ a ˆ)(b + b )
g0 α
α
bare optomechanical coupling (geometry, etc.: fixed!)
laser-driven cavity amplitude tuneable! phase!
After linearization: two linearly coupled harmonic oscillators!
mechanical oscillator
driven optical cavity
red-detuned p m u p
blue-detuned
y t i v a c
beam-splitter (cooling)
QND
squeezer (entanglement)
“The slopes of Optomechanics”
Linear Optomechanics Displacement detection Optical Spring Cooling & Amplification Two-tone drive: “Optomechanically induced transparency” State transfer, pulsed operation Wavelength conversion Radiation Pressure Shot Noise Squeezing of Light Squeezing of Mechanics Entanglement Precision measurements
Optomechanical Circuits Bandstructure in arrays Synchronization/patterns in arrays Transport & pulses in arrays
Nonlinear Optomechanics Self-induced mechanical oscillations Synchronization of oscillations Chaos
Nonlinear Quantum Optomechanics Phonon number detection Phonon shot noise Photon blockade Optomechanical “which-way” experiment Nonclassical mechanical q. states Nonlinear OMIT Noncl. via Conditional Detection Single-photon sources Coupling to other two-level systems
Note: Yesterday’s red is today’s green!
Optomechanical wavelength conversion
ω2
signal out
ω1
signal in
r e t r e v n o c h t g n e l e a v w
ω2
signal out
a2 ˆ
g2
g1 ω1
signal in Ω
a1 ˆ
b
ˆ
Ω
optics to optics:
Painter 2012
Wang 2012
microwave/RF to optics:
Cleland 2013
Lehnert, Regal 2014
Schliesser, Polzik 2014
Detecting the phonon number
y c n e u q e r f l a c i t p o
Thompson, Zwickl, Jayich, Marquardt, Girvin, Harris, Nature 72, 452 (2008)
y c n e u q e r f
membrane transmission
membrane displacement
50 nm SiN membrane
Mechanical frequency: !M=2"!134 kHz Mechanical quality factor: Q=106÷107 Optomechanical cooling from 300K to 7mK
Thompson, Zwickl, Jayich, Marquardt,
Detection of displacement x: y c n e u q e r f
what we need!
not
membrane transmission
membrane displacement phase shift of measurement beam: (Time-average over cavity ring-down time)
QND measurement of phonon number!
t f i ) h r s e e b s m a u h n p d n o e t n a o l u h p m ( i s
! g n i g n l e l a h c y r v e
Signal-to-noise ratio: Optical freq. shift per phonon: Noise power of freq. measurement:
Ground state lifetime: Time
Ideal single-sided cavity: Can observe only phase of reflected light, i.e. x 2: good Two-sided cavity: Can also observe transmitted vs. reflected intensity: linear in x!
- need to go back to two-mode Hamiltonian! - transitions between Fock states!
Single-sided cavity, but with losses: same story Detailed analysis (Yanbei Chen’s group, PRL 2009) shows: need
absorptive part of photon decay (or 2nd mirror)
Sensing mechanical motion at the ultimate precision limit
input laser optical cavity
reflection phase shift
cantilever
Classical equipartition theorem:
Possibilities: •Direct time-resolved detection •Analyze fluctuation spectrum of x
extract temperature!
“Wiener-Khinchin theorem”
area yields variance of x:
General relation between noise spectrum and linear response susceptibility susceptibility
(classical limit) for the damped oscillator:
T=300 K
Experimental curve: Gigan et al., Nature 2006
meas
Two contributions to
phase noise of 1. measurement imprecision laser beam (shot noise limit!) 2. measurement back-action: fluctuating force on system noisy radiation pressure force
(measured)
(measured)
+ backaction noise + imprecision noise true spectrum
i m full noise p r e n o c i s i s e i o n
n i o t a c e k i s a c o b n
intrinsic fluctuations coupling to detector (intensity of measurement beam)
Best case allowed by quantum mechanics: “Standard quantum limit (SQL) of displacement detection” ...as if adding the zero-point fluctuations a second time: “adding half a photon”
“weak measurement”: integrating the signal over time to suppress the noise trying to detect slowly varying “quadratures of motion”: Heisenberg is the reason for SQL! no limit for instantaneous measurement of x(t)! SQL means: detect down to on a time scale Impressive: !
