Quantum micro-mechanics with

ultracold atoms

Dan Stamper-Kurn

UC Berkeley, Physics

Lawrence Berkeley National Laboratory, Materials Sciences

Quantum measurement of a mechanical oscillator

Motivations

Force detection (Caves, Braginski, Milburn, others)

• Apply force resonantly. Measure Z or E.

Basics of quantum metrology: e.g. what detector to use?

• Direct vs. post-amplification

• quadrature specific/non-specific

• destructive vs. non-destructive (QND)

Quantum mechanics of macroscopic bodies.

coszF t

Z

Detection

Some experimental realizations

Nanofabricated cantilevers

Schwab [(Nature 443, 193 (2006)]

also Roukes, Cleland, Wineland

Bouwmeester [(Nature 444, 75 (2006)]

also Aspelmayer/Zeilinger, Rugar, Pinard/Heidman, Harris

Some experimental realizations

Micro-toroidal resonators Ultracold atoms in a cavity

Vahala, Kippenberg

[(PRL 97, 243905 (2006)]this talk

Optical (bosonic) detection

Z

Detection

f Z

/ osc cav in outH H H H f Z nforce per photon:

2 1

2

hf

cL

L

Detection limits:

2

2 1

ph

ZN f

position measurement is photon-shot-noise limited: want more photons

2

2

2

ph

fP N

quantum back-action = radiation pressure fluctuations from shot noise: want fewer

photons

Optical (bosonic) detection

Z

Detection

f Z

/ osc cav in outH H H H f Z nforce per photon:

2 1

2

hf

cL

L

Detection limits:

const. Z P Uncertainty limit for quadrature specific measurement

2

SQL

z

ZM

Uncertainty limit for quadrature non-specific measurement

Micro-mechanical state of the art

Cannot passively cool to ground state

freq. = kHz – MHz < 10 mK = 200 MHz

Cooling by active feedback or by dynamical back-action demonstrated, though not to ground state

closest approach: n = 59 quanta (Schwab, 2006)

requires resolved sidebands:detector bandwidth < oscillator freq.[PRL 99, 093901-2 (2007)]

Measurement sensitivity approaching quantum limit

ΔZ = 6 x SQL (Schwab, 2004)

Back-action observed, but above quantum level

(ΔZ)(ΔP) > 15 x SQL (Schwab, 2006)

Trapped ions

sympathetic cooling

quantum states

prepared and

detected

resolved-sideband

laser cooling

Ultracold atoms in cavity

evaporative cooling

back-action heating

measured at quantum

level

cavity cooling of

single/many atoms

ΔZ = 5 x SQL

MOT

7.5 cm

Cavity mounted

through here

Trap

home of ring-trap

experiments

~200 mStabilized to sub picometer!

Max coupling strength g/2p 16 MHz

Cavity half-linewidth /2p 0.66 MHz

Atomic half-linewidth /2p 3 MHz

Finesse/Q 5.8x105/3x108

Beam waist 23 m

Critical atom number 0.02

Critical photon number 0.02

absorption image

through side of cavity

Cavity loading procedure

200m

Optical lattice

Magnetic trapping outside cavity (TOP trap)

Evaporative cooling

Translation of the magnetic trap to within the cavity mode

Transfer to 1D optical lattice inside cavity + turn off magnetic trap

Result: ~50,000 atoms trapped in ~200 sites of the in-cavity standing wave trap, at T ~ 1 μK (20 kHz)

Note: Following the lead of Vuletic, Chapman, Zimmermann, Hemmerich, Esslinger, Reichel; Walther, Blatt

initial TOP trap

Light-atom interactions in a cavity

0( ) sin( ) pg z g k z

p

c

cavity resonance

probe frequency

light-atom coupling

a

atomic resonance

atomic coherence decay rate

cavity coherence decay rate

Consider dispersive case of large cavity-atom detuning:

2

/

ˆˆ ˆ

i

c atoms in out

atoms ca

g zH n n H H

AC Stark shift per photon per atom Cavity resonance shift per atom

2

N

ca

gN

-100

-50

0

50

100

No

rma

l m

od

e (

GH

z)

-100 -50 0 50 100

Atom-cavity detuning (GHz)

-10

0

10

-10 0 10

Many-atom cavity QED

Normal mode spectrum:

100

99

10099

2

2

2 2

ca cac ca NN g

N = 60,000 atoms

2 2 10MHzg p

5 GHz

60 MHz

Dispersive regime:

Atom = index of refraction

Cavity frequency shift

expand to first order in atomic displacement from trap center

defines collective operators relevant to the cavity probe:

850 nmtrap

780 nmprobe

single photon counter

2

2

0

( )ig z

g 1 1/2 0 1/2 1

†

0 /ˆˆ ˆ ˆ ˆ c N eff z else in outH n N f Zn a a H H

2

/

ˆˆ ˆ

i

c atoms in out

atoms ca

g zH n n H H

1sin(2 ) i i

atomseff

Z k z zN

sin(2 )i i

atoms

P k z p 2sin (2 ) ieff

atoms

N k z

f f

Optomechanical bistability

“Coherent” effect: Optical forces in the cavity will displace the trapped atoms

Probing the cavity shifts the cavity resonance nonlinear cavity optics

0( ) sin( ) pg z g k z

“Cavity nonlinear optics at low photon numbers from collective atomic motion,”PRL 99, 213601 (2007)

probe probe

Transmission spectrum of nonlinear cavity

bare cavity

cavity shift for undriven cavity

probeprobe probe

Cavity t

ransm

issi

on

frequency

0 ca

probe light repels atoms

Nonlinear optics with n<1 photons

Requirements for nonlinear optics at n ≤ 1

medium whose optical properties are altered by single photons

many atoms held in loose trap; displaced by single photon optical force

medium for which this alteration persists longer than photon lifetime

cold atoms with long motional coherence times

11ms 100ns

obs ph

triggered sweep, Δa/2p = -30 GHz

n

- - - -

10

5

075 70 65 60

n

- - - -

0.4

0.2

0.0155 150 145 140

0.2

0.1

0.0-10 -5 0 5 10

probe freq (MHz) probe freq (MHz)

back+forth sweep, Δa/2p = -100 GHz

Optomechanical potential

†

0 /ˆˆ ˆ ˆ ˆ c N eff z else in outH n N f Zn a a H H

Consider force on collective variable: 2

0 Z eff zF N f n Z M Z

“Optical spring”Lorenzian vs. position

some effects:

Optomechanical frequency shift

Optomechanical bistability

Dynamical back-action (non-adiabatic) – heating, cooling, amplification

Bistability and atomic motion

bare cavity

Cavity t

ransm

issi

on

drive frequency

0 ca

2

1

3

45

Spectrum of driven cavity Potential seen by atoms

1

2

34

5

position Z

~ nm

Exciting collective motion from bistable switching

Direct observation of collective atomic motion

wait for cavity to come into resonance

excite collective motion diabatically and observe single-shot detection

(measured, rms) 1.1nm Z ( , rms) 0.22 nm

Z SQLNm

“Incoherent” effect: (quantum) fluctuations of the intracavity intensity cause momentum diffusion / diffusive heating

Atoms serve as “spectators,” i.e. non-destructive detectors, of quantum fluctuations of an intracavity field

Force fluctuations (-> momentum diffusion) represent backaction of position measurement

Quantum optomechanical fluctuations

0 †( )( ) 2 ( ) (0) ( ) (0) ( )

eff SQL

IN z z

N f ZLb b i a b a b

Atom-cavity dynamics and “granularity”

Equations of motion

0

0( ) (0)exp( ) ( )exp ( )

teff SQL

z z

N f Za t a i t i dt n t i t t

Look for small parameters (weak optomechanical coupling):

transient driven by “force history”

bare-cavity responsemodulations (mixing) due to

atomic motion

0

eff SQLN f Z

Generally: a hard problem! Some insights from Ritsch, Vuletic, Rempe, etc.

Granularity parameter

Scaled temperature†

z a a

collective atomic field

cavity optical field

Atom-cavity dynamics and “granularity”

Granularity parameter:

Scaled temperature:

Does a single photon’s kick significantly affect the cavity?

0

0

2

1

eff

eff

z

N fN f

Mline shift from atomic displacement

cavity linewidth

0

1

eff

SQL

N f

Z

(single-photon force) x (residence time of photon)

zero-point momentum spread

Does a single photon’s kick significantly perturb the trapped atom?

atom temperature

cavity-cooling equilibrium temperature

†

z a a

Is cavity cooling/anti-cooling significant compared to diffusive heating

Fluctuation and dissipation

… one obtains a general result

† 2 2 †ˆ ˆ ˆ ˆ nn nn nn

da a S S S a a

dt

with spectral density of photon number fluctuations

( ) ˆ ˆ ˆ ˆ( ) (0) ( ) (0)

zi t

nnS dt e n t n n t n

Diffusive heating + coherent amplification/cooling:

heating by red-detuned noise (promotes oscillator)

amplification/cooling = cavity anti-cooling/cooling (Ritsch, Vuletic, Rempe)

Resolved sidebands:

blue-detuned cavity: cooling to ground state (Zwerger/Kippenberg, Girvin/Harris)

red-detuned cavity: quantum-limited quadrature-non-specific amplifier

Quantum fluctuations of light within a cavity

For our system (non-granular, low temperature):

2 2

1

1 /

freespace

z

DdEC

dt m

Proportional to intensity

Note: At constant intensity, shot noise fluctuations of photon number in a resonant cavity are enhanced w.r.t. those in free space!

spectrum analyzer

monochromatic laser beam

vacuum in all other modes

2

0

2

gC

single-atomcooperativity

Cavity-induced heating: measured by atom loss

1

10Bk T U

( )B

dE dNN U k T

dt dt

wait and see

bare cavity

N

probe

Prepares atoms in well defined initial state

Cavity-induced heating: measured by atom loss

What this means:

Quantum metrology: back-action heating of macroscopic object at level prescribed by quantum measurement limits

Quantum optics: Shot-noise spectrum is enhanced/colored by a cavity

Cavity-light+atom interactions: “theory” looks good

Quantum measurement of atom number or spin: spells trouble for some non-destructive measurements

Micro-mechanics summary

Cavity mode structure selects a single collective mode in a large N gas of atoms.

Evaporative cooling cools this mode to the ground state

Detection: Atomic motion affects cavity properties

Actuation: Cavity light exerts concerted forces on trapped ultracold atoms

Back-action: Probe-light fluctuations cause momentum diffusion

Ahead:

Measurement: optimal method? fundamental limit?

Bistability: Quantum effects? Use as quadrature-specific amplifier?(cf Josephson bistability amplifier)

Amplification/cooling: Observe directly and test theories. Quantum amplifier?

Granular regime: quantum optics. Can we detect a single photon by radiation pressure?

E1: Spinor BECJennie GuzmanSabrina Leslie

Christopher SmallwoodMukund Vengalattore

(James Higbie)(Lorraine Sadler)

E2,E3: Cavity QEDThierry BotterDaniel Brooks

Joseph LowneyZhao-Yuan MaKater MurchTom Purdy

(Kevin Moore)(Subhadeep Gupta)

E4: Ring-trap interferometryJoanne Daniels

Ed MartiRyan Olf

Tony OettlEnrico VogtTiger Wu