mishkat bhattacharya omjyoti dutta swati singh · 2008-11-21 · mishkat bhattacharya omjyoti dutta...
TRANSCRIPT
![Page 1: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/1.jpg)
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
![Page 2: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/2.jpg)
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
![Page 3: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/3.jpg)
outline
• Brief review of optomechanical mirror cooling• Two- and three-mirror cavities• Cantilever coupling to dipolar molecules
– Single molecule– Phonon squeezing in molecular lattice
U. Virginia 2008
– Phonon squeezing in molecular lattice– Coherent control
• Outlook
(J. Sankey, Yale University)
![Page 4: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/4.jpg)
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
![Page 5: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/5.jpg)
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)
![Page 6: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/6.jpg)
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) ]
![Page 7: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/7.jpg)
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
![Page 8: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/8.jpg)
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
![Page 9: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/9.jpg)
competing requirements
• Blue detuned cavity
• Red detuned cavity
• Restoring force (optical spring)• Antidamping
• Anti-restoring force• Damping
U. Virginia 2008
• Damping
![Page 10: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/10.jpg)
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)
![Page 11: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/11.jpg)
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).
![Page 12: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/12.jpg)
prehistory
U. Virginia 2008
![Page 13: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/13.jpg)
early history
“Cavity cooling of a microlever,” C. Höhberger-Metzger & K. Karrai, Nature 432, 1002 (2004)
U. Virginia 2008
18K
![Page 14: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/14.jpg)
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
![Page 15: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/15.jpg)
systems
U. Virginia 2008
From T. Kippenberg and K. VahalaScience 321, 1172 (2008)
![Page 16: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/16.jpg)
three-mirror geometry
U. Virginia 2008
[ 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) ]
![Page 17: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/17.jpg)
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)
![Page 18: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/18.jpg)
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:
![Page 19: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/19.jpg)
three-mirror cavity, R=1
,
,
(1 / )
(1 / )n l n
n r n
q L
q L
ω ωω ω
−+
U. Virginia 2008
† † †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
![Page 20: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/20.jpg)
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 ≠
U. Virginia 2008
/ 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)
![Page 21: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/21.jpg)
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
![Page 22: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/22.jpg)
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
U. Virginia 2008
• 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
![Page 23: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/23.jpg)
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
![Page 24: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/24.jpg)
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)
![Page 25: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/25.jpg)
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
![Page 26: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/26.jpg)
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
![Page 27: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/27.jpg)
rovibrational entanglement
: / :
: / :
z z z
z
z M p M
I L Iφ φ
ω ω
φ ω ω
h h
h h
scaling:
U. Virginia 2008
† 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)
![Page 28: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/28.jpg)
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
![Page 29: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/29.jpg)
“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)
![Page 30: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/30.jpg)
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
+
= ∆ + +
− −−
′
− −
+ −+
+…
U. Virginia 2008
• 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)
![Page 31: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/31.jpg)
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)]
![Page 32: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/32.jpg)
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)]
![Page 33: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/33.jpg)
cantilever coupling to dipolar molecules
U. Virginia 2008
![Page 34: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/34.jpg)
single molecule
U. Virginia 2008
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
![Page 35: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/35.jpg)
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
U. Virginia 2008
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
![Page 36: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/36.jpg)
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
![Page 37: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/37.jpg)
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
![Page 38: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/38.jpg)
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
− −
− −
= + + +
= − − +
U. Virginia 2008
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
![Page 39: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/39.jpg)
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
![Page 40: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/40.jpg)
simple case
com rel
com rel com rel
| (0) |1 ,0
| ( ) |1 ,0 | 0 ,1t
ψψ α β
⟩ = ⟩ →⟩ = ⟩ + ⟩
U. Virginia 2008
Cantilever frequency dependence –last generation
![Page 41: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/41.jpg)
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
![Page 42: Mishkat Bhattacharya Omjyoti Dutta Swati Singh · 2008-11-21 · Mishkat Bhattacharya Omjyoti Dutta Swati Singh ARO NSF ONR. quantum metrology •How can quantum resources be exploited](https://reader034.vdocuments.net/reader034/viewer/2022042622/5fa0c4f29fa9006128634898/html5/thumbnails/42.jpg)
• 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)