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

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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:

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[ 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

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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

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• 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)

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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

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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)

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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

= − ∆ − + +

= − ∆ − + +

&

&

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( )

( ) ( )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

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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

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/ 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

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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

+

= ∆ + +

− −−

′

− −

+ −+

+…

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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

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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

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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

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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

+= −

∑

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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

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• 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

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• Condensates in high-Q cavities• Esslinger et al, Science 322, 235 (2008)• Stamper-Kurn et al, Nature Physics 4, 561 (2008)