mishkat bhattacharya omjyoti dutta swati singh · 2008-11-21 · mishkat bhattacharya omjyoti dutta...

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

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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

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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)

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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

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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

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) ]

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

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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

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/ 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:

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“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 =

=

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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

+

= ∆ + +

− −−

′

− −

+ −+

+…

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)

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

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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

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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

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

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

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(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)