qnd measurement of photons quantum zeno effect & schrödingers cat julien bernu yep 2007

QND measurement of photonsQND measurement of photons

Quantum Zeno EffectQuantum Zeno Effect&&

Schrödinger’s CatSchrödinger’s Cat

Julien BERNU

YEP 2007

Historical Zeno ParadoxHistorical Zeno Paradox

« At a given time, an arrow has a well defined position. Thus, it is

motionless, since a motion is a change of position. A short time after,

you will then find it at the same position. Motion is so impossible! »

Historical Zeno ParadoxHistorical Zeno Paradox

« At a given time, an arrow

since a motion is a change of position. A short time after,

you will t

has

hen

a well defined position. Thus, it is

moti

find it at the same position. Motion is

onless,

so impossible! »

Quantum Zeno EffectQuantum Zeno Effect

2,1st meas

measmeas

P right meas T

TN

T

2,0 meas meas

meas

P right T N T

T T

left right

Time

Posi

tion

T

measT

<Posi

tion>

P(right)

Quantum Zeno EffectQuantum Zeno Effect

2,1st meas

measmeas

P right meas T

TN

T

A continuous measurement "freezes"

the motion of the nitrogen atom !

2,0

0meas meas

meas

P right T N T

T T

left right

Time

P(right) 2

measT

4measT

16measT 0measT

2,1 measst

measmeas

P right m Teas

TN

T

R1 R2

Classicalsource

Our experimental setupOur experimental setup

QND measurement of the photon number

0 100 200 300 400 500 600 700

0

1

2

3

4

5

nu

mb

er

of

ph

oto

ns

time (ms)

Coupling the cavity to a Coupling the cavity to a classical sourceclassical source

Classicalsource

The field gets acomplex amplitude

in phase space

The value depends on the power of the source, the coupling

proportionnal to the i

efficie

njection

ncy

du

,

and rati .s ion

Complex phase space

0

Coupling the cavity to a Coupling the cavity to a classical sourceclassical source

2

2

0

2

n!

n

n

en

n

Time

Mean p

hoto

n n

um

ber

Quadratic start

Zeno Effect !

Coherent field:

2 2n t t t

Experimental difficultiesExperimental difficulties

To be able to build up a field in the cavity

before showing we can freeze its evolution

Hudge to reach a few photons

field

Control the cavity and th

coherently

e source frequen

130

cies at

1

t

cavT ms s

111 Hz @ 50Ghe level of (precision )Hz 10

Why 1 Hz precision?Why 1 Hz precision?

Effect of a frequency noise or sideband picks on the source or the cavity:

random phase for injection pulses.

Complex phase space

How?How?

Source:Source: Anritsu generator locked on a Anritsu generator locked on a (very) good quartz locked on a (very) good quartz locked on a commercial atomic clockcommercial atomic clock

Cavity:Cavity: position of the mirrors must be position of the mirrors must be stable at the range of stable at the range of 1010-13-13m m (10(10-3-3 atomic atomic radius)! radius)!

Sensitivity to accoustic vibrations, Sensitivity to accoustic vibrations, pressure, temperature, voltage, hudge pressure, temperature, voltage, hudge field…field…

V

4He

Recycling

0.1 mbar @ 1 bar 0.2 Hz

0.1 mV @ ~100V = 0.2 Hz

P

PumpThermal contractions:

(1kg) 100 µK @ 0.8 K 0.2 Hz

ResultsResults

Injection watched with QND measurements:Injection watched with QND measurements:

0 20 40 60 80 1000,0

0,5

1,0

1,5

2,0

2,5

N1 0.0018 ± 0.0001

0.3 ± 0.5Yo 0.17 ± 0.02

Mea

n ph

oton

num

ber

Number of injection pulses

time

Injection pulses

(Zeno Effect)Measurement

ResultsResults

Injection watched with QND measurements:Injection watched with QND measurements:

0 20 40 60 80 1000,0

0,5

1,0

1,5

2,0

2,5

N1 0.0018 ± 0.0001

0.3 ± 0.5Yo 0.17 ± 0.02

Mea

n ph

oton

num

ber

Number of injection pulses

time

ResultsResults

Then with continuous measurement: Then with continuous measurement: Injection watched with QND measurements:Injection watched with QND measurements:

1

2

1 1 1

1

1, inj inj

inj

P N P N

P n

n t n N

0 20 40 60 80 100

0,0

0,5

1,0

1,5

2,0

2,5

N1 0.0018 ± 0.0001

0.3 ± 0.5Yo 0.17 ± 0.02

Mea

n ph

oton

num

ber

Number of injection pulses

0 20 40 60 80 1000,0

0,5

1,0

1,5

2,0

2,5

N1 0.0018 ± 0.0001

0.3 ± 0.5Yo 0.17 ± 0.02

Mea

n ph

oton

num

ber

Number of injection pulses

1

2

1 1

N

n

0,0 0,1 0,2 0,3 0,4 0,50,0

0,5

1,0

1,5

2,0

2,5

N1 0.0018 ± 0.0001

0.3 ± 0.5Yo 0.17 ± 0.02

Mea

n ph

oton

num

ber

Time (s)

Perfect control!to be removed…

Zeno Effect!

measn t t T

0 20 40 60 80 100

0,12

0,14

0,16

0,18

0,20

0,22

0,24

0,26

0,28

Me

an

ph

oto

n n

um

be

r

Number of injection pulses

ResultsResults

0.5

4 cav

t s

T

ResultsResults

0 20 40 60 80 100

0,12

0,14

0,16

0,18

0,20

0,22

0,24

0,26

0,28

y = Yo + N1*x

Yo 0.153 ± 0.006N1 0.0025 ± 0.0006

N1 0.0018 ± 0.0001

0.3 ± 0.5Yo 0.17 ± 0.02

Me

an

ph

oto

n n

um

be

r

Number of injection pulses

Perfect agreement!

QND detection of atomsQND detection of atoms

Re()

Im()

e

g

g

. ig eg

. ie ee

e

a single atom controls the phase of the field

R1 R2 ie e

ie g

QND detection of atomsQND detection of atoms

Re()

Im()

e

g

e

/2 pulse R1

0 01

2..

1

2i ie e ge eg

The field phase "points" on the atomic state

R1 R2 ie e

ie g

. ig eg

. ie ee a single atom controls the phase of the field

0 01

2..

1

2i ie e ge eg

This is a "Schrödinger cat state"

on

on

off

0-1 +1

on

on

off

0-1 +1

1 1

2 2, ,g ee g

Schrödinger’s CatSchrödinger’s Cat

Schrödinger’s CatSchrödinger’s Cat

0 01

2..

1

2i ie e ge eg

2

Main source of decoherence: atom detection.

A pulse in mixes the information carried by the a2

tom:R

0

0

0 0 0

0

2

1

2

. . . .

2

..

2

i i

i

i i

i

Pulsee e e e

g

e

ee e

g

2

2 2

2

0

2 2 12 2

0 0

Parity Parity 11

2 2 !

2 ! 2 112

!2

n

n

n n

n n

e nn

e enn n

n

Production of Schrödinger’s Cat by a simple photon number parity measurement ( phase shift per photon):

Schrödinger’s CatSchrödinger’s Cat

e

g

2n2 1n

1Pulse R2

2Pulse R2

2Pulse R2

Wigner FunctionWigner Function

(Phase space)

Wigner FunctionWigner Function

0

1

2(Phase space)

Wigner FunctionWigner Function

Wigner FunctionWigner Function

Statistical mixture

Wigner FunctionWigner Function

Schrödinger Cat

Wigner FunctionWigner Function

2

2

Wigner FunctionWigner Function

†

†

ˆ ˆ

ˆ1

1ˆ

2 ˆ ˆˆ 1

i a a

a a

W e Tr e d

Tr D D P

Simple parity measurement !

Size of the catSize of the cat

int 0 in

The decoherence time of a cat is: , where is

damping time of the cavity.

The needed interaction time to prepare or to "see" the cat with the probing

atoms

1

2 2

must be n

cavdecoh cav

TT T

n

T T T

t

2 3

0

.

We then must have , much larger than previous ones!2202

decoh

cav

T

Tn

T

Observing the decoherenceObserving the decoherence

0t 0.005 cavt T 0.010 cavt T 0.015 cavt T 0.020 cavt T 0.025 cavt T 0.050 cavt T 0.075 cavt T 0.100 cavt T 100n

2200

0t

Size of the catSize of the cat

int 0 in

The decoherence time of a cat is: , where is

damping time of the cavity.

The needed interaction time to prepare or to "see" the cat with the probing

atoms

1

2 2

must be n

cavdecoh cav

TT T

n

T T T

t

2 3

0

.

We the must have , much larger than previous ones!2202

decoh

cav

T

Tn

T

Size of the catSize of the cat

i nnt 0 i

The decoherence time of a cat is: , where is

damping time of the cavity.

The needed interaction time to prepare or to "see" the cat with the probing

atoms must

1

2

2

e n b

cavdecoh cav

T

n

T

TT T

T

t

2 3

0

.

We the must have , much larger than previous ones!2202

decoh

cav

T

Tn

T

Atom chip experiment

ConclusionConclusion

Using our QND Using our QND measurement procedure, we measurement procedure, we have been able to prevent have been able to prevent the building up of a coherent the building up of a coherent field by Quantum Zeno field by Quantum Zeno Effect.Effect.

We can also use it to We can also use it to produce big produce big Schrödinger cats and Schrödinger cats and study their study their decoherence by decoherence by measuring their measuring their Wigner function.Wigner function.

0 20 40 60 80 1000,0

0,5

1,0

1,5

2,0

2,5

N1 0.0018 ± 0.0001

0.3 ± 0.5Yo 0.17 ± 0.02

Mea

n ph

oton

num

ber

Number of injection pulses

PerspectivesPerspectives

2 cavities for non-local 2 cavities for non-local experiments:experiments:

teleportation of atomsteleportation of atoms non-local Scrödinger’s catnon-local Scrödinger’s cat quantum corrector codesquantum corrector codes

Thank you!Thank you!

The team:

J. B.Samuel DelégliseChristine GuerlinClément Sayrin

Igor Dotsenko

Michel BruneJean-Michel RaimondSerge Haroche

Sebastien GleyzesStefan Kuhr

Atom chip team

The origin of decoherence:The origin of decoherence:entanglement with the entanglement with the

environmentenvironmentDecay of a coherent field:Decay of a coherent field:

the cavity field remains the cavity field remains coherentcoherent

the leaking field has the same the leaking field has the same phase as phase as

0 .

0

cav

envenv

t

tvacuum t

t e

Environment

Decay of a "cat" state:Decay of a "cat" state: cavity-environment entanglement:cavity-environment entanglement:

the leaking field "broadcasts" phase informationthe leaking field "broadcasts" phase information trace over the environmenttrace over the environment

decoherence (=diagonal field reduced density decoherence (=diagonal field reduced density matrix) as soon as:matrix) as soon as:

1 2env

ca

v

t

en

env

t

vacuum

ttt

Environment

0env

t t

The origin of decoherence:The origin of decoherence:entanglement with the entanglement with the

environmentenvironment

Wigner functions of Wigner functions of Schrödinger’s catsSchrödinger’s cats

2

2

7n

Residual problemResidual problem

20

2,0

020

,

20

2 20

, ,

, ,

11 14

Dispersive regime:

Expressions valid only if .

Real dressed states formula

=2 2 2

4

1

2

s:

e neff

g n

e n n

g n n

nE t

n nn

E

n

nE

E

2 20

2

n

Dephasi

ng

per

ph

oto

n /

Number of photons

200 400 600 800 1000

0.2

0.4

0.6

0.8

1

0 200

400

600

800

1000

1.0

0.8

0.6

0.4

0.2

0

No quantum Zeno effect for No quantum Zeno effect for thermal photons and decaysthermal photons and decays

1, 1, 1, 0 1, 2 1, 1,

1, 1, 0 1

Probability to lose the photon

1

during

If ,

But the probability to stay in after a finite time :1

: .

meas meas meas

cav

measmeas cav meas

cav

P T P T t P T T P N

dtdt dP

T

TT

T

T P T tT

0

1, 1

where

(classical expected va

1 e u l )

meas meas

meas

meas cav

meas

T T T

T Tmeas

cav

T N T

N T T

Te

T

Zeno Effect for quadratic Zeno Effect for quadratic growthgrowth

0We now measure the photon number between every two injections .

If we are able to reduce , we can keep the same global injection power with

weaker but closer injection pulses.

Then we can w

meas

meas

T T

T

2

2

rite: .

The probability to find after the first injection is:

The probabilit

1

1,1

1 y to jump to during a finite time is:

0 1,0 0 1,1 0 1,2 0 1,

inj meas

stinj

st nd th

injm

T

P inj

T

P T P inj P inj P N inj

TN

T

2

The quick repeated measurements "freeze" the building up of the fiel

0 with and

d

!

meas meas meas inj measeas

T T N T T T

2

2

2

0

2 2

Coherent field:

n!

1

n

n

en

P e

2

Time

2

2

2

measT measT

2measT 2measT 2measT

0 20 40 60 80 1000,0

0,5

1,0

1,5

2,0

Mea

n ph

oton

num

ber

Number of injection pulses

Results: injectionResults: injection

Effect of a small frequency Effect of a small frequency detuning between the source detuning between the source

and the cavity:and the cavity:

Complex phase space

Quantum Zeno EffectQuantum Zeno Effect

2

More precisely, the probability to stay on the left after the first measurement i :

1 ,1

s

,1 1st stmeasP left meas P right meas T

The probability to stay on the left till is:T

, 0 , , 2 , , , 1meas meas meas meas meas

P left T P left T P left T left T P left N T left N T

2112 2

ln 11 1 1

measmeasmeasmeas TT

meas measmeas

TN TT T e e

Graphes de wignerGraphes de wigner