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Cavity optomechanics
Pierre MeystrePierre Meystre
B2 Institute, Dept. of Physics and College of Optical Sciences
The University of Arizona
Mishkat BhattacharyaOmjyoti DuttaSwati Singh
ARO NSF ONR
quantum metrology
• How can quantum resources be exploited with maximal robustness and minimal technical overhead?
• What are the inherent sensitivity-reliability tradeoffs in quantum sensors?
• What is the role of particle statistics in quantum metrology?
• How do decoherence mechanisms set fundamental limits to measurement precision?
U. Virginia 2008
precision?
This talk:
• Cooled nanoscale cantilevers for quantum control of AMO systems
outline
• Brief review of optomechanical mirror cooling• Two- and three-mirror cavities• Cantilever coupling to dipolar molecules
– Single molecule– Phonon squeezing in molecular lattice
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– Phonon squeezing in molecular lattice– Coherent control
• Outlook
(J. Sankey, Yale University)
micromechanical cantilevers –single-particle detectors
U. Virginia 2008
Protein detectionThree cantilevers coated with antibodies to PSA, a prostate cancer marker found in the blood. The left cantilever bends as the protein PSA binds to the antibody. Photo courtesy of Kenneth Hsu/UC Berkeley & the Protein Data Bank
Micromechanical cantilever with a single E. Coli cellA micromechanical cantilever with a single E. Coli cell attached. The cantilever is used to detect changes in mass due to selectively attached biological agents present in small quantities.Craighead Research Group, Cornell University
micromirrors
U. Virginia 2008
D. Kleckner & D. Bouwmeester, Nature 444, 75 (2006)
SEM photograph of a vertical thermal actuator with integrated micromirror. Application of a current to the actuator arm produces vertical motion of the mirror, which can either reflect an optical beam or allow it to be transmitted.(Southwest Research Institute, http://www.swri.org/3pubs)
radiation pressure mirror cooling
Basic idea: change frequency and damping of a mirror mounted on a spring using radiation pressure
enveff
eff ef efff
MB Bk Tn
kT Γ Γ Ω Ω
= h h
Number of quanta of vibrational motion:
U. Virginia 2008
[ H. Metzger and K. Karrai, Nature 432, 1002 (2004) ]
• Cold damping: laser radiation increases mirror damping.• Requires laser tuned below cavity resonance• Little change in mirror frequency
• Increase effective mirror frequency• Requires laser tuned above cavity resonance
• Use two lasers!
[ T. Corbitt et al, PRL 98, 150802 (2007) ]
basic optomechanics − cold damping
Goal: increasing
Trapping due to `optical spring’: rp
opt
( )dFk
x
dx= −
effΩ
U. Virginia 2008
B. S. Sheard et al. PRA 69, 051801(R) (2004)
Need blue-detuned cavity
basic optomechanics − mirror cooling
Ω0ω
F
Goal: decreasing
Cooling due to asymmetric pumping of sidebands
env effeff ( / )MTT = Γ Γ
U. Virginia 2008
0ω + Ω
0ω
0ω − Ω
F
M. Lucamarini et al., PRA 74, 063816 (2006)A. Schliesser et al. PRL 97, 243905 (2006)
Need red-detuned cavity
competing requirements
• Blue detuned cavity
• Red detuned cavity
• Restoring force (optical spring)• Antidamping
• Anti-restoring force• Damping
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• Damping
two-wavelengths approach
T. Corbitt et al, PRL 98, 150802 (2007)
U. Virginia 2008
Spring constant
damping
k>0 Γ>0 Dynamically unstable
k>0 Γ<0 Stable
k<0 Γ>0 “anti-stable”
k<0 Γ<0 Statically unstable(carrier)
(sub
carr
ier)
Prehistory
• V. Braginsky, “Measurement of weak forces in physics experiments” (Univ. of Chicago Press, Chicago, 1977).
• V.B. Braginsky and F.Y. Khalili, "Quantum measurement'' Cambridge University Press, Cambridge, 1992
U. Virginia 2008
• C. M. Caves, “Quantum mechanical noise in an interferometer,” Phys. Rev. D23, 1693 (1981).
prehistory
U. Virginia 2008
early history
“Cavity cooling of a microlever,” C. Höhberger-Metzger & K. Karrai, Nature 432, 1002 (2004)
U. Virginia 2008
18K
S. Gigan et al. (Zeilinger group) “Self-cooling of a micromirror by radiation pressure”, Nature 444, 67 (2006)
O. Arcizet et al. (Pinard/Heidman group) “Radiation pressure cooling and optomechanical instability of a micromirror”, Nature 444, 71 (2006)
D. Kleckner et al. (Bouwmeester group) “Sub-Kelvin optical cooling of a micromechanical resonator”, Nature 444, 75 (2006)
recent history
< 10K
10K
135 mK
U. Virginia 2008
A. Schliesser et al. (Vahala/Kippenberg groups) “Radiation pressure cooling of a micromechanical oscillator using dynamical back-action” PRL 97, 243905 (2006)
T. Corbitt et al. (LIGO) “An all-optical trap for a gram-scale mirror”, PRL 98, 150802 (2007), “Optical dilution and feedback cooling of a gram-scale oscillator”, PRL 99, 160801 (2007)
A. Vinante et al (AURIGA gravitational wave detector mirror), Meff= 1,100 Kg, PRL 101, 033601 (2008)
135 mK
11K
0.17 mK
6.9 mK
systems
U. Virginia 2008
From T. Kippenberg and K. VahalaScience 321, 1172 (2008)
three-mirror geometry
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[ A. Dorsel et al., PRL 51, 1550 (1983); PM et al, JOSA B11, 1830 (1985); J. D. McCullen et al., Optics Lett. 9, 193 (1984) ]
three-mirror cavity
…
Nature 452, 72 (2008)
U. Virginia 2008
See also: M. Bhattacharya and PM, PRL 99, 073601 (2007)M. Bhattacharya et al, PRA 77, 033819 (2008)
Early discussion: PM et al., JOSA B2, 1830 (1985)
basic optomechanics − quantum approach
2† 2 21
( )2 2n M
pq a a m q
mH ω ω+= +h
U. Virginia 2008
1( ) 1
1 /n n n
n c qq
L q q L L
πω ω ω = = − + +
† †2
2 21
2 2n M a ap
a a mm
qH qω ω ξ−+ + h h
But:
So:
three-mirror cavity, R=1
,
,
(1 / )
(1 / )n l n
n r n
q L
q L
ω ωω ω
−+
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† † †2
† 2 21( ( ))
2 2n M
pa a b b m q a a b b qH
mω ω ξ−+ −+= +h h
L− L0 q
three-mirror cavity, R<1
, 0( )n e n e L q qω ω ξδ− − −
, 0( )n o n o L q qω ω ξδ+ + −
0 0q ≠
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/ 2q L
2† † 2 † †21
( ) ))2
((2n e n o M L
pa a b b m a a bH q b q
mω δ ω δ ω ξ−= + + + −− +h h h
W. J. Fader, IEEE J. Quant. Electron. 21, 1838 (1985)
three-mirror cavity, R< 1
20, ( )Qn e n q qξω ω − −
20, ( )Qn o n o q qω ξω + −∆+
0 0q
U. Virginia 2008
/ 2q L
2† † 2 2 † † 21
( ) (( )2 2
)Qn e n o MHp
a a b b m a aqm
b b qω δ ω δ ξω −−= + −+ + + hh h
allows preparation of energy eigenstates& observation of quantum jumps
linear vs. quadratic coupling
Linear coupling Quadratic coupling
• back-action: mirror motion → changed cavity frequency → changed intracavity power → changed radiation pressure
• Important when storage time of light
• Quadratic opto-mechanical coupling simply modifies the effective frequency of oscillating mirror
• Purely dispersive
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• Important when storage time of light comparable to inverse mirror oscillation frequency
• Requires asymmetry in frequency change for two directions of mirror motion
• Purely dispersive
a flavor of the theory (R=1)
( ) ( )† † †2 22
† 1
2 2 Mc qp
H a a b b m a a b b qm
ω ξ= + + + − −Ωh h
Quantum Langevin equations of motion, input-output formalism:
( )( )
in
in
/ 2
/ 2
a i Q a
b i Q b
a
b
ξ γ
ξ γ
γ
γ
= − ∆ − + +
= − ∆ − + +
&
&
U. Virginia 2008
( )
( ) ( )2
in
in† †2
/ 2
/
/ 2M M
b i Q b
p m
m a a b b m p
b
q
p q
ξ γ
ξ
γ
ε
= − ∆ − + + =
+ − − ΓΩ= − +
&
h&
Noise operators:
( ) ( ) ( ) ( )†in in, in in in, ' ( ')sa t a a t a t a t t tδ δ δ δ = + = −
( ) ( ) ( ) ( )'in in in0 ' 1 coth
2 2 2i t tM
B
dt t t e
m k Tωω ωε ε ε ω
π
∞− −
−∞
Γ= = +
∫h
h
steady-state − linear coupling
2 2 22 2 ineff
4
[( 2
1
/ ) ]M
P
ML
ξγγ δ
δ +
Ω Ω
−
in2 2 3
2
eff
4 1
[( / 2) ]M
P
ML
ξγγ δ
δ Γ Γ +
+
• Effective frequency:
• Cooling:
U. Virginia 2008
“stiffening” field:in
in
0
0
s s
c c
P P
P P
δ δ
δ δ
→ > →
→ < →“cooling” field:
M. Bhattacharya and PM, PRL 99, 073601 (2007)M. Bhattacharya, H. Uys and PM, PRA 77, 033819 (2008)
steady-state − quadratic coupling
in2ff 2 2
2e
2
( / 2)
1QM
n
P
Mω γξ γ
δ
Ω Ω + +
eff MΓ Γ
• Effective frequency:
• Effective stiffness:maximized on resonance
U. Virginia 2008
cooling of rotational motion
† †2
2 21
2 2z
c
La a I
IH a aφφω ω φ ξ φ−+ += hh
2
[ , ]
/ 2z i
I M
L
R
φ = −
=
h
U. Virginia 2008
/ 2I MR=
2in2 2
eff 2 2 2
2 2in
eff 2 2 3
2
[ ( ]/ 2)
]
2
[ ( / 2)
c
c
P
I
PD D
I
φφ
φφ
ξ γω ω
ω γξ γ
ω γ
∆−∆ +
∆+∆ +
2
2
| |sc
a
Iφ φφ
ω ω ξω
∆ = − − h
M. Bhattacharya & PM, PRL 99, 153603 (2007)
Spiral phase elements with opposite windings on both sides
rovibrational entanglement
: / :
: / :
z z z
z
z M p M
I L Iφ φ
ω ω
φ ω ω
h h
h h
scaling:
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† 2 2 2 2 † †( ) ( )2 2
zc z z z
cz
z
H a a p z L g a az g a a
cg g
L M L I
φφ
φφ
ωωω φ φ
ωω ω
+ += + + − +
= =
hhh h h
h l h
Hamiltonian (dimensionless form)
rovibrational entanglement
U. Virginia 2008
Entanglement measure:
M. Bhattacharya P.-L Giscard and PM, PRA 77, 030303(R) (2008)
2 2
2
( ) ( )( )
| [ ( ), ( )] |z
u v
z p
R R
R R
ω ωωω ω
⟨ ⟩⟨ ⟩≡⟨ ⟩
E
[ ( ) ( )] / 2z z
uR
u z
v p L
u u
δ δ δφδ δ δ
δ ω δ ω
= −= += + −
Rovibrational entanglement for ( ) 1ω <E
“Schrödinger drum”
Relative motion:
1 2
eff
eff
/ 2
3 m
q
m
q q
m
ω ω
−=
=
=
Center-of-mass motion:
1 2
eff
eff
( ) / 2
2
m
Q q q
mM
ω
+=
Ω =
=
U. Virginia 2008
23 ) sin sin 2 (3 ) 2sin cos( )cos 2
s
sin 2(
in
Qq qkL k L k k
R
θ θ θ θ
θ
+ + − = +
=
• Cavity spectrum (allowed values of k)
M. Bhattacharya and PM, PRA78, 041801(R) (2008)
cavity spectrum
20 0
0
20 ,
0
, , ,
, ,
,
(
( ) (
( , ) ) ( )
( ))
)
(
'
n i n i n i n i
n i n i
i
o
n
q q
Q
q q
Q Q Q
Q Q
B
q Q B M
M
q qP
ω
+
= ∆ + +
− −−
′
− −
+ −+
+…
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• Can cool normal modes separately
• can perform QND energy measurements on normal modes separately
• Can use mode coupling to control one mode with the other
• Can generate quantum entanglement of modes(Hartmann and Plenio, PRL to be published)
cantilevers for quantum detection and control
• Pushing quantum mechanics to truly macroscopic systems• Quantum superpositions and entanglement in macroscopic systems• Novel detectors• Coherent control
U. Virginia 2008
Single-domain ferromagnet with oscillatory component B(t) couples to atomic spin F
Described by Tavis-Cummings Hamiltonian[P. Treutlein et al., PRL 99, 140403 (2007)]
Tavis-Cummings Hamiltonian
Magnetic field at center of microtrap: ( )( ) xr mt G a t=B e$
Cantilever-condensate interaction: ( ) ( )· r b xFH t a tg Fµ µ= − =B
Detuning: 0| |B FL c c
g Bµδ ω ω ω= − −=h
U. Virginia 2008
Hamiltonian: † †BEC ( )r L z cH H H V S a a g S a S aω ω +
−= + + = + + +h h h
spin N/22 2
mB
r
gG
m
µω
= h
h
L c cδ ω ω ω= − −=h
[P. Treutlein et al., PRL 99, 140403 (2007)]
cantilever coupling to dipolar molecules
U. Virginia 2008
single molecule
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0
4ddGµπ
=
0
1
4ddGπε
= Electric dipoles
Magnetic dipoles
Used in Magnetic Force Microscopy
Frequency shift squeezing
1/22 2( )c mr R x x = + +
1/2
25
0
3 m ct t
d d
mRω ω
π′
= + ε
motional squeezing
( )2 †2IV C b b= − +h
Classical cantilever motion, 2c tω ω ′=
1/2
6
15
4 2m c
o t c c
d dC N
m R mπ ω ω′
=
h
ε
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2
/
o t c c
B c cN k
t
T
u C
ω
=
=
h
S. Singh, M. Bhattacharya, O. Dutta, PM, arXiv:0805.3735
lattice of molecular dipoles
One-dimensional molecular chain:E-field
l
2 2
3
1
2 4
N Ni m
p ti i jo i j
p dH V
m x xπ <
= + +−
∑ ∑ε
U. Virginia 2008
Acoustic phonons † 1
2p k k kk
b bH ω = +
∑h
0
5 1/20 0
2 | sin( / 2) |
(3 / 2 )k
m m
k
d
ω ωω π
=
=
l
lε• Small displacements:
P. Rabl and P. Zoller PRA 76 04230 (2007)
l
coupling to micromechanical oscillator
† 1
2cck
a aH ω = +
∑h
• Lengths hierarchy: iq
N R
l
l
2
3 2
3( )1m c c
Ii i i
d d R xV
r r
+= −
∑
U. Virginia 2008
N Rl
Frequency shift:
Phonons-oscillator interaction
50
61
4m c
k kk
d d
m Rω ω
π ω→ +
ε
Squeezing!
( )( )† † † † †I k k k k k k k k k
k
V C a a b b b b b b b b− − − −= − + + + +∑h
1/2
6
3
2 2m c
ko k c c
d dC
m R mπ ω ω′
= −
h
ε
phonon squeezing
Classical cantilever motion, RWA,… ( )† †I k k k k kV C b bN b b− −= − +h
Two-mode squeezing:† †
1
† †2
1( )
21
( )2
k k k k
k k k k
s b b b b
s b b b bi
− −
− −
= + + +
= − − +
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SrO molecule 16
21
2MHz
Kg
C 1.
Cm
10
100
10
m
200 nm
u= 2
0
2
8
C
c
c
c
k
k
N
d
R
Nt
m
ω
µ
−
−
=
=
=
=
===
l
coherent control
2 2 2 2com rel
2 2 3 2com com rel rel com
2r
70
el rel com
24 ( )
48 48 140 6
12
0
060
8
]
[ cm
c
c
c
c
c
R x x x
R x x R x x x x x
d d
RV
xx xx x
π+ +
− − +
+
=
−
− l
l l
l
ε
U. Virginia 2008
• Center-of-mass and relative modes of motion,classical cantilever
†,a a †,b b
†rel comexp[ ( )] . .( )ccL ab i h ctV ω ω ω∝ − − +
1/2
2cc c
Nm
Lω
=
h
Varied for coherentcontrol
simple case
com rel
com rel com rel
| (0) |1 ,0
| ( ) |1 ,0 | 0 ,1t
ψψ α β
⟩ = ⟩ →⟩ = ⟩ + ⟩
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Cantilever frequency dependence –last generation
More interesting – start from a thermal stateA
vera
ge o
ccup
atio
n
Can
tilev
er fr
eque
ncy
U. Virginia 2008
(Preliminary results)
generation
generation
time
Mea
n en
ergy
• Limits of cooling• Preparation of exotic states• Quantized cantilever motion• Quantum phase transitions
• Wenzhou Chen & PM (under construction)
outlook
U. Virginia 2008
• Condensates in high-Q cavities• Esslinger et al, Science 322, 235 (2008)• Stamper-Kurn et al, Nature Physics 4, 561 (2008)