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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