ua math 2010 cavity optomechanics … from the top down: review of optomechanical mirror cooling...
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UA Math 2010
Cavity optomechanics …
• From the top down: Review of optomechanical mirror cooling
• Coupling to ultracold atoms and molecules
• From the bottom up: Bosonic & fermionic atomic mirrors
• Outlook: All-optical optomechanics
ARO NSF ONR
Mishkat Bhattacharya ( U. Rochester)Wenzhou ChenDan GoldbaumRina Kanamoto ( Ochanomizu University)Greg PhelpsSwati SinghEwan WrightKey Zhang ( Shanghai)
UA Math 2010
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?
This talk:
• From the top down: Cooled nanoscale cantilevers for sensing and quantum control of AMO systems
• From the bottom up: optomechanics in quantum-degenerate atomic systems
UA Math 2010
From the top down…
UA Math 2010
micromechanical cantilevers –single-particle detectors
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
UA Math 2010
micro-mirrors
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)
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• Optical spring effect: - laser radiation increases - Requires cavity tuned below laser frequency
• Optical damping: - Requires cavity tuned above laser frequency
• Use two lasers!
eff
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
Number of quanta of vibrational motion:
[ H. Metzger and K. Karrai, Nature 432, 1002 (2004) ]
[ T. Corbitt et al, PRL 98, 150802 (2007) ]
UA Math 2010
optical spring effect
Goal: increasing
Trapping due to `optical spring’: rp
opt
( )dFk
x
dx
B. S. Sheard et al. PRA 69, 051801(R) (2004)
eff
Need red-detuned cavity
UA Math 2010
optical damping
0
0
0
0
F
Goal: decreasing
Cooling due to asymmetric pumping of sidebands (resolved sidebands limit)
M. Lucamarini et al., PRA 74, 063816 (2006)A. Schliesser et al. PRL 97, 243905 (2006)
env effeff ( / )MTT
Need blue-detuned cavity
00 0
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• Adiabatic and non-adiabatic (optical delay) contributions:
• Adiabatic effect – “optical spring” (also bistability and multistability)
• Non-adiabatic effect
a simple theory
2 22
eff RP2eff eff
1 | |( )
2 / 2
( ) / 2
d z dz az F t
dt Q dt m m c c L
dai x a a i s
d
m
t
2 2
2 /
4 / 1P
eff
2 2 2
2
16 /
(4 1/ )
Q
0( )z zR
• Linear coupling: , integrate equation for a formally, keeping terms up to
dz
dt
laser cavity( ) ( )z z
eff( )
UA Math 2010
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)
• C. M. Caves, Phys. Rev. D23, 1693 (1981). • A. Dorsel, J. D. McCullen, PM, E. Vignes and H. Walther,” Optical bistability and
mirror confinement induced by radiation pressure,” PRL 51, 1550 (1983).
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early history
18K
Cavity cooling of a microlever, C. Höhberger-Metzger & K. Karrai, Nature 432, 1002 (2004)
UA Math 2010
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)
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)
recent history
< 10K
10K
135 mK
11K
0.17 mK
6.9 mK
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hot off the press…
(17 March 2010)
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a broad range of systems
From T. Kippenberg and K. VahalaScience 321, 1172 (2008)
(CPW: coplanar plane wave)
UA Math 2010
cantilevers for quantum detection and control
• Pushing quantum mechanics to truly macroscopic systems• Quantum superpositions and entanglement in macroscopic systems• Novel detectors• Coherent control
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)]
Recent experimental results: Treutlein et al., coupling via Casimir-Polder forcehttp://www.munichatomchip.de/microresonator.html
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cantilever coupling to dipolar molecules
UA Math 2010
single dipolar molecule
0
4ddG
0
1
4ddG Electric dipoles
Magnetic dipoles
Used in Magnetic Force Microscopy
Frequency shift squeezing
1/22 2( )c mr R x x
1/2
2' 5
0
3m
m cm
d d
mR
ò
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motional squeezing
2 †2IV C b b
Classical cantilever motion, '2 mc
1/2
6
2
15
4 2
/
m c
o t c c
B c c
d dC N
m R m
N k
t
T
u C
ò
S. Singh, M. Bhattacharya, O. Dutta, PM, PRL101, 263603 (2008)
UA Math 2010
… and from the bottom up…
UA Math 2010
cavity optomechanics with atoms
• Single atom J. Kimble, G. Rempe, …
• Utracold thermal gas D. Stamper-Kurn, Nature Phys. 4, 561 (2008)
• BEC T. Esslinger, Science 322, 235 (2008)
• Fermions R. Kanamoto & PM, PRL 104, 063601(2010) (theory)
(also: Interferometers, quantum simulators,, clocks,…)
Moving mirror mechanical center-of-mass motion of atom(s)
UA Math 2010
cantilever coupling to Bose condensates
2 2 2†
2† †† 2
2† 22 ( ) (
1c( )
2 2os )
2( )c m m
a a
ga
p da a a a a kxx xi a a m q dx
mH
mq
dx
light mirror rad. pressure condensate-light interaction
Condensate recoil: 0 2 2cos(2 ) cos(( ) 2 )x Nc kx c kx c
2 2† † 2
2† †2 †
2
1 2(
2( ))
2 4c m ma
p g Na a i a a ca m q
mH caq a a
rad. pressure rad. pressure on BEC “mirror”
T. Esslinger et al, Science 322, 235 (2008)D. Stamper-Kurn et al, Nature Physics 4, 56 (2008)
UA Math 2010
optomechanical coupling
† † †02 2 sm0
2ˆ ˆ ˆ ˆ ˆ ˆ
4ˆˆ
N Ua a c a Qc c aN
sm
sm
2
sm 2 2
0
†
4
1ˆ ˆ ˆ( )2
c
r
c
r
L m
mL NU
Q c c
Classical dynamics, adiabatic elimination of optical field:
22
22 2sm
22 sm
22 2sm
2
244
mm
rr
Q
Q
d
dt
d
dQ
Q
q
qt
Effective mirror:
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potential
UA Math 2010
quasi-periodic orbits
K. Zhang, W. Chen, M. Bhattacharya and PM, PRA 81, 013802 (2010)
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chaotic orbits
Next:
• quantum dynamics, entanglement• quantum control• quantum chaos (?)• …
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ring resonator system
† * †
1,2
† 2 † 22 2
†2 1 1 2
† †0 1 1 2 22
ˆ ˆ ˆ ˆ ˆ( )
ˆ ˆˆ ˆ ˆ ˆ( ) ( ),ˆ ˆ2
ˆ ˆikx i
i i i
kx
i i ii
H a a i a a
ddx x U a a a a x
m dxa a e a a e
† * †
†
† † † †
† † †
† †0
2 1 1
2 1 1 2 0 0
00
0
0
2
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( )( )
ˆ ˆ ˆ ˆ ˆ ˆ ˆ(
ˆ ˆ ˆ
) 4 ( )
2ˆ ˆ ˆ ˆ ˆ( )
2
)(
i i i i i r c
s
c
c s s
c
i
s
a a
H U N a a i a a c c c c
U
i U
a a c c c
a a c c c c
c
a a
0 cos(2 ) sin(2 )1 2 2
ˆ ˆ ˆ ˆ( ) c skx c c cL L
kL
x x Let
2 Counter-propagating running waves
1
2
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three-mode system†
†
0
ˆ ˆ ˆ( ) / 2
ˆ ˆ ˆ( ) / 2
ˆ
j j j
j j j
X c c
P i c c
c N
† †0 2 1
† * †0
2 2 2 2
1 2
† †0 2 1 1 2
ˆ ˆ ˆ ˆ( )
ˆ ˆ ˆ ˆ2 (
ˆ
ˆ ˆ ˆ ˆ( )
ˆ ˆ ˆ ˆ(
)
ˆ
ˆ )
i i i i ii
r c c
c
s s
sX
H
U N a a
U N a a i a a
X P X
a a
i U N
X
a a a a
P
Analog of 2 coupled optomechanical mirrors
(W. Chen, D. S. Goldbaum, M. Bhattacharya and PM, arXiv:1003.1812)
UA Math 2010
classical dynamics
* *2 1 1 2
*
2 20
2 20
1 0
*2 1 1
2 1 1
2
2
2 0 1 2
( )
( )
(
2 (16 ) 4
2 (16 ) 4
( )
( )
)
( )
c c
c
r c r
s
s
c
s r s
s
ri
X i
X X X U
X
N
X X X U N
i
X iX
U N i i
i U N i i
Dark states! Dark states!
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steady state intensity and phase
Spontaneous symmetry breaking
Symmetric pumping
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steady state “positions”
1 2
1 2
Spontaneous Symmetry breaking
1 2
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Outlook – toward all-optical optomechanics
optical tweezers
fixed FP mirror
Main advantage:
Optical spring is coupled to reservoir at T~ 0
(S. Singh, D. Goldbaum, E. M. Wright, PM, in preparation)
UA Math 2010
effects of laser linewidth
M
laser spectrum
2v
(G. Phelps)
UA Math 2010
UA Math 2010
nano-MRI
Cantilever force sensor at the heart of IBM's"nano-MRI" microscope. IBM scientists have used this nano-MRI to visualize structures at resolution 60,000 times better than current magnetic resonance imaging technology allows. This technique brings MRI capability to the nanoscale level for the first time and represents a major milestone in the quest to build a microscope that could "see" individual atoms in three dimensions. With further development, applications could include understanding how individual proteins interact with drugs for discovery and development, and analyzing computer circuits only a few atoms wide.
UA Math 2010
two-wavelengths approach
T. Corbitt et al, PRL 98, 150802 (2007)
Spring constant
damping
k>0 >0 Dynamically unstable
k>0 <0 Stable
k<0 >0 “anti-stable”
k<0 <0 Statically unstable(carrier)
(su
bca
rrie
r)
UA Math 2010
basic optomechanics − quantum approach
2† 2 21
( )2 2n M
pq a a m q
mH
1( ) 1
1 /n n n
n c qq
L q q L L
† †2
2 21
2 2n M a ap
a a mm
qH q
But:
So:
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three-mirror geometry
[ 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) ]
UA Math 2010
three-mirror cavity
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)
…
Nature 452, 72 (2008)
UA Math 2010
three-mirror cavity, R=1
,
,
(1 / )
(1 / )n l n
n r n
q L
q L
† † †2
† 2 21( ( ))
2 2n M
pa a b b m q a a b b qH
m
L L0 q
UA Math 2010
three-mirror cavity, R<1
, 0( )n e n e L q q
, 0( )n o n o L q q
2† † 2 † †21
( ) ))2
((2n e n o M L
pa a b b m a a bH q b q
m
0 0q
W. J. Fader, IEEE J. Quant. Electron. 21, 1838 (1985)
/ 2q L0
freq
uenc
y
0q
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three-mirror cavity, R<1
20, ( )Qn e n q q
20, ( )Qn o n o q q
2† † 2 2 † † 21
( ) (( )2 2
)Qn e n o MHp
a a b b m a aqm
b b q
0 0q
allows preparation of energy eigenstates& observation of quantum jumps
/ 2q L0
freq
uenc
y
UA Math 2010
a flavor of the full theory (R=1)
† † †2 22
† 1
2 2 Mc qp
H a a b b m a a b b qm
Quantum Langevin equations of motion, input-output formalism:
laser cavity
2 2
in
in
in† †
/ 2
/ 2
/
/ 2M M
a i Q a
b i Q b
p m
m a a b b m p
q
a
p
b
q
Noise operators:
†in in, in in in, ' ( ')sa t a a t a t a t t t
'in in in0 ' 1 coth
2 2 2i t tM
B
dt t t e
m k T
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experimental signatures
2
2eff
eff
e f
f
f
e f
( )
1 1
2 2
B
q
B
m
k
d q
k T
S
q
Tn
k
UA Math 2010
future ring cavity work
• Small quantum fluctuations
• Feeble quantized fields
• Revisit the Fabry-Pérot case
• Damping of the matter-wave side modes Heating and cooling mechanisms
• Optical monitoring of matter-wave fluctuations via dark states
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asymmetric pumping
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spin-polarized fermions in Fabry-Pérot
Rina KanamotoRina Kanamoto
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elementary excitations – bosons vs. fermions
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bosonization of fermionic field
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cantilever state measurement
How can we measure the quantum state of the cantilever? … No phonon counter (yet)!
Idea: monitor the state of a detector atom coupled to both the field to be characterized and to an additional classical field
Needed:
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field state measurement in the microwave regime
Wigner state tomography
[Banaszek and Wodkiewicz, PRL 76, 4344 (1996)]
Atomic homodyne detection
[Wilkens and PM PRA 43, 3832 (1991)]
MW pulse
( )D
qubit
Measurement
LO
qubit
Measurement
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nonlinear atomic homodyning detection of cantilever state
40
2
3 / 4
ac B F Fx Bc c
B B
g g m Gm
G r
†2 0Raman ( )
2e
k k
gH a a
†( )ac B F Fx B c acg m G x g cH c
Need:2 0
2e
acL
gg
Zeeman interaction:
(magnetic field gradient)
(see P. Treutlein et al)
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Wigner characteristic function measurement
cantilev
†
er
2ex
*
1 1. .
2 4
( ) ( )
( )
ai
ig I
c cW
W
ig e
P e h c
C Tr e
C
• classical electromagnetic field• weak cantilever excitation• interaction time
† † †k k k kc c a a a a I
S. Singh + PM., arXiv 0908.1415v1
† †ex
1 1 1cos(2 2 1) cos(2 2 )
2 2 2
( ) / 2k
P g A A g A A
A a c
atom | | | | Initial atomic state:
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measurement back-action
, 1,
| ( ) | ;|
( )c i
c i
g
g U g
P
State of field after detection of atom in ground state:
,
, 1
1| cos( ) ( / 2) ( / 2)
2
1sin( ) ( / 2) ( / 2) |
2
c i
g
c i
e
g I D DP
i g I D DP
Cantilever initially in ground state
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Dissipation – some numbers
6Li atoms coupled to Si resonator: 65 s c1/ eg
Typical lifetime of trapped atoms: 100 ms to a few seconds
Dipole-allowed transitions rates: MHz to GHz
Raman transitions are to virtual state: population weighting factor
2/L k L
Raman decay time: a few msec
MECHANICAL COUPLING
Thermal dissipation: thermal thermal thermalc B
c
k Tn n n
Q
OPTICAL COUPLING
6bath1MHz Q 10 10mKc T 3
dissipation 4 10 sec