RÉSONATEURS NANOMÉCANIQUESDANS LE RÉGIME QUANTIQUE

NANOMECHANICAL RESONATORSIN QUANTUM REGIME

Chaire de Physique MésoscopiqueMichel Devoret

Année 2012, 15 mai - 19 juin

Troisième leçon / Third lecture

12-III-1

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Last update: 120531

PROGRAM OF THIS YEAR'S LECTURES

Lecture I: Introduction to nanomechanical systems

Lecture II: How do we model the coupling between electro-magnetic modes and mechanical motion?

Lecture III: Is zero-point motion of the detector variables a limita-tion in the measurement of the position of a mechanicaloscillator?

Lecture IV: In the active cooling of a nanomechanical resonator,is measurement-based feedback inferior to autono-mous feedback?

Lecture V: How close to the ground state can we bring a nano-resonator?

Lecture VI: What oscillator characteristics must we choose to con-vert quantum information from the microwave domain tothe optical domain? 12-III-2

May 15: Rob Schoelkopf (Yale University, USA)Quantum optics and quantum computation with superconducting circuits.

May 22: Konrad Lehnert (JILA, Boulder, USA)Micro-electromechanics: a new quantum technology.

May 29: Olivier Arcizet (Institut Néel, Grenoble)A single NV defect coupled to a nanomechanical oscillator: hybrid nanomechanics.

June 5: Ivan Favero (MPQ, Université Paris Diderot)From micro to nano-optomechanical systems: photons interactingwith mechanical resonators.

June 12: A. Douglas Stone (Yale University, USA)Lasers and anti-lasers: a mesoscopic physicist’s perspective on scatteringfrom active and passive media.

June 19: Tobias J. Kippenberg (EPFL, Lausanne, Suisse)Cavity optomechanics: exploring the coupling of light and micro- andnanomechanical oscillators.

CALENDAR OF 2012 SEMINARS

12-III-3

LECTURE III : POSITION MEASUREMENT OF AMECHANICAL RESONATOR IN QUANTUM LIMIT

OUTLINE

1. Quadrature representation of harmonic oscillator

2. Imprecision of single shot interference measurement of position

3. Continuous monitoring of position and associated backaction

12-III-4

QUADRATURE REPRESENTATION OF A QUANTUM HARMONIC OSCILLATOR

mω

eω0 X= +

ˆ eE z X⋅ →

22 † †

ˆ 1ˆ ˆ ˆ ˆˆ ˆ ; ; , 12 2 2m mP K KH X b b b bM M

ω ω⎛ ⎞ ⎡ ⎤= + = + = =⎜ ⎟ ⎣ ⎦⎝ ⎠

( )† / 2ˆ ˆˆ ; ; ;2ZPF ZPF m ZPF

m ZPF

X X b b X Z KM PZ X

= + = = =

Review:

Define dimensless hermitian operators*:

†ˆ ˆˆ

2b bI +

≡ a.k.a. ( ) 0ˆ ,b Xφ=R

†ˆ ˆˆ2

b bQi

−≡ a.k.a. ( ) / 2

ˆ ,b Xφ π=I

* see S. Haroche & J.M. Raimond, "Exploring the Quantum", Cambridge 2006

ˆˆ,2iI Q⎡ ⎤ =⎣ ⎦

14

I QΔ ⋅Δ ≥

These definitions yield properties useful later.

22ˆ ˆI I IΔ = −22ˆ ˆQ Q QΔ = −

standard deviations

ˆ ˆˆˆ ,2 2ZPF ZPF

X PI QX P

≡ ≡

12-III-5

COHERENT STATE IN QUADRATURE REPRESENTATION

( ) ( )2 / 2

0

e ; ; 0!

m

ni t

n

n b en

tβ ωβ β β ββ β β∞

− −

=

= = =∑

( )2 2†ˆ ˆ ˆ; ; ; in b b n n n n n n ne θβ β β ββ β= = = Δ = − = =

nθRe mIβ =

Im

mQβ

=

Im

Qm

Im

Qm

mtω

"Fresnel lollypop"

2 1mIΔ =

For coherentstate:

12m mI QΔ = Δ =

Gaussiandistributionof Im values

12-III-6

THE VACUUM STATE

Im

Qm

mIΔ

12m mI QΔ = Δ =

mQΔ0m mI Q= =

00 nβ = = =

All coherent states can be thought of as a translated vacuum state in quadrature space.

Im

Qm

β

Driving an harmonicoscillator with an arbitrarytime-dependent force canonly result in displacingthe vacuum state.

iβ β β′ ′′= +

β ′

β ′′

12-III-7

FLYING OSCILLATORA wave-packet propagating in medium with constant phase velocity can be seen asas oscillator

Envelope ( ) varies slowlycompared with center frequency

( ) ( )ˆ ˆˆ2Env cos sind ZPF S c S cS t t t S I t Q tω ω⎡ ⎤= − +⎣ ⎦

y

k spatial mode

t

ω temporal mode

cω

dtdy

ck

Signal :(electric field,

voltage,etc...)

t

cω

dt

aT A

12-III-8

2

aTπ

HOMODYNE MEASUREMENT

LOα

Sα

–

Measurement of a coherent state:

signal @

@

λ/4

cω

cω

reference a.k.a."local oscillator"

Ideal photodetector

12-III-9

HOMODYNE MEASUREMENT

LOαˆLOa

ˆSa

Sα

1ˆ ˆˆ

2LO Sia iaa +

=

–

( )2

ˆ ˆˆ

2S LOa i ia

a+

=

Measurement of a coherent state:

signal @

@

[ ]1 1ˆ ˆ=Tr in nρ

ˆ , ,i LO S S LOρ α α α α=†

1 1 1ˆ ˆ ˆn a a=

[ ]2 2ˆ ˆ=Tr in nρ†

2 2 2ˆ ˆ ˆn a a=

ˆLOia

λ/4

2 2 + +=

2LO S LO S LO Sα α α α α α∗ ∗+

2 2

=2

LO S LO S LO Sα α α α α α∗ ∗+ − −

LO S LO S

LO S LO S

D α α α α

α α α α

∗ ∗= +

′ ′ ′′ ′′= +

Measurement of an arbitrary state:

Output is ( )0 0ˆˆ2 cos sinLO L S L SD I Qα θ θ= +

Ideal homodyne setup measures a generalized quadrature

1 2ˆ ˆ ˆD n n= −

cω

cω

reference a.k.a."local oscillator"

Ideal photodetector

density matrix at input of beams:

12-III-9a

FLUCTUATIONS OF A HOMODYNE MEASUREMENTThe photoelectron difference number is 1 2

ˆ ˆ ˆD n n= −We suppose that the efficiency of the detector is unity.

( )( ) ( )( )( )( )

† † † †

† †

† † †

1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ2

ˆ ˆ ˆ ˆ2

ˆ ˆ ˆ ˆ ˆ ˆ2

LO S LO S LO S LO S

LO S LO S

LO S LO S LO S LO S

D a a a a a a a a

a a a a

a a a a a aα α δ δ∗

⎡ ⎤= + + − − −⎣ ⎦

= +

= + + +

ˆ ˆLO LO LOa aα δ= +

Fluctuations are of order unity

Fluctuations are of order N1/2

12-III-10

We have thus, in the limit of large LO number of photons:

( )0 0ˆˆ ˆ2 cos sinLO S L S LD N I Qθ θ≅ +

FLUCTUATIONS OF A HOMODYNE MEASUREMENTThe photoelectron difference number is 1 2

ˆ ˆ ˆD n n= −We suppose that the efficiency of the detector is unity.

( )( ) ( )( )( )( )

† † † †

† †

† † †

1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ2

ˆ ˆ ˆ ˆ2

ˆ ˆ ˆ ˆ ˆ ˆ2

LO S LO S LO S LO S

LO S LO S

LO S LO S LO S LO S

D a a a a a a a a

a a a a

a a a a a aα α δ δ∗

⎡ ⎤= + + − − −⎣ ⎦

= +

= + + +

ˆ ˆLO LO LOa aα δ= +

Fluctuations are of order unity

Fluctuations are of order N1/2

We have thus, in the limit of large LO number of photons:

( )0 0ˆˆ ˆ2 cos sinLO S L S LD N I Qθ θ≅ +

2S

LO

DxN

=

( )SP x

Probability

0 0ˆˆcos sinL S L SI Qθ θ+

detector inefficiency finite LO strength resultin a further broadeningof this distribution.

2 SxΔ

For a coherent state, =1 12-III-10a

0 0ˆˆcos sinL S L SI Qθ θ+

0Lθ SI

SQ

HOMODYNE MEASUREMENTPERFORMS A PROJECTION IN

QUADRATURE PLANE

2S

LO

DxN

=

signal state

( ) ( )0 0ˆˆTr cos sinS L S L SP x I Qρ θ θ⎡ ⎤= +⎣ ⎦

( )ˆS LOx θ

12-III-11

In the ideal case,homodyne measurementperforms a noise-less measurement of an oscillatorgeneralized quadrature.

INTERFEROMETRIC MEASUREMENT OF POSITION

–

eiN θα =

φ

SI

eiδθα

( ) ( )

( )

ˆˆ ˆ ˆ ˆ2 cos 2 sin 2

ˆˆ ˆ2 4

LO S e S e

LO S e ZPF S m

D I k X t Q k X t

I k X Q I t

α

α

⎡ ⎤= +⎣ ⎦⎡ ⎤≅ +⎣ ⎦

/ 8m LO e ZPFx D N Nk X=

Single shot

1a mT ω−

mω

( )0 X t= +

0

14 e ZPFNk X

1ee

ZPF

kc n Xω

=

24

e

e ZPF m

k Xk X I

δθ δδ

==SQ

0mIδ

simplifiedrepresentationof separationof input andoutput beams

SI Nδ δθ=

12-III-12

Convolution of two fluctuations:mechanical and probe photon shot noise.

IS IT POSSIBLE TO MAKE PHOTONSHOT NOISE NEGLIGIBLE COMPARED

TO POSITION FLUCTUATIONS?

IN OTHER WORDS, CAN WE HAVE1

4m

e ZPF

INk X

δ

OR EVEN 1ZPF

e

X Nλ

DIFFICULT SINCE 1510 mZPFX −∼

LAMB-DICKE PARAMETER ALWAYS SMALL!2 ZPF

e ZPFe

Xk X π

λ=

12-III-13

TWO HELPFUL FACTORS:

- CAVITY (ELECTROMAGNETIC RESONATOR)

- MANY PHOTONS (LONG ACQUISITION TIME)

12-III-14

ENHANCEMENT OF INTERFEROMETRICPHASE-SHIFT BY CAVITY

–

e riθα1 1a mTκ ω− −

mωeω

κ

0 ωeω

( )2rθ ωπ κ

1ω

D

( )0 X t= +

1ωκ

=F

( )†ˆ ˆ

ˆˆ 8

8

Xe

ZPF

e

b b

X

X

δδθλ

λ+

=

=

F

F

Enhancement of Lamb-Dickeparameter by finesse of cavity

2 cav

cL

1

2

3

κ

see Cohadon,. Heidmann, and M. Pinard, PRL83, 3177(1999)

α

12-III-15

μWAVE vs OPTICS

L.O.

φL.O. output

–

mωeω

κ

μwavegene-rator(s)

LASER

mω

eω

output

κ

12-III-16

MIXERS AND PHASE-SENSITIVE AMPLIFIERS

signal

L.O.

Mixers have conversion gains less than unity.Add at least 3dB of noise. Following amplifier adds also noise.

DC

signal

L.O.

φ

Josephson parametric amplifiersin phase sensitive mode amplify without

adding noise 1 quadrature of signal.Has enough gain to beat noise of following amplifier.

DC

L.O. DC

12-III-17

SPECTRAL DENSITY OF OSCILLATOR MOTIONReview:

( ) [ ]12

i tX t X e dωω ωπ

+∞ −

−∞= ∫ [ ] ( )1

2i tX X t e dtωω

π+∞ +

−∞= ∫

[ ] ( ) ( )ˆ ˆ 0 i tXXS X t X e dωω ω

+∞ +

−∞= ∫ [ ] [ ] [ ] ( )ˆ ˆ

XXX X Sω ω ω δ ω ω′ ′= +

[ ] [ ]2XX XX XXf S Sω ω ωπ

⎛ ⎞= = + −⎜ ⎟⎝ ⎠

S"Engineer" spectral density

mω+

γ[ ]XXS ω

ω

Total area under curve ~ number of phonons in mechanical resonator

mω−

t

t

X ~ n1~ γ −

~ n ( )~ 1n +

( ) 0X t =

12-III-18

temporal and ensemble averaging

measured by usual spectrum analyzer

SPECTRAL DENSITY OF HOMODYNE SIGNAL

SI

( ) ( )8 / et X tδθ λ= FSQ

0

We now consider a continuous homodyne measurement.

LOLON N→

N N→

( ) ( )1 1

4 4m ZPF m ZPF

LO LO

D tDx X x t XX XN N N N

θ θ− −∂ ∂⎛ ⎞ ⎛ ⎞= → =⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

Apparent oscillator motion in quadrature representation

ω

( )0xx ωS

mω

Spectral density of oscillatorapparent motion if photon shotnoise is negligible, as well as backaction.

γ 1 22

nγ

⎛ ⎞+⎜ ⎟⎝ ⎠

12-III-19

MEASURED NOISE: IMPRECISION AND BACKACTION

ω

( )totxx ωS

imprecision

backaction

mω

( )totxx mωS

N

(log scale)

number of photons incavity (log scale)

zero-pointfluctuationsof resonator

backactionimprecision

SQL

total measurednoise

Standard Quantum Limit (SQL): optimal compromise between imprecision and backaction noises

At SQL, the total noise energy in the resonator is equivalent to a full phonon.

intrinsic noise

12-III-21

IMPRECISION vs RESOLUTION

1) Measure fairness of coin by tossing it many times

2500 trials will determine fairness with 1% imprecision(@ 1 standard deviation)

2) Measure wavelength of incoming beam

width of slitdeterminesresolution

12-III-22

END OF LECTURE