Discrete photon statistics
from continuous measurements
Stéphane Virally, Jean Olivier Simoneau,
Christian Lupien and Bertrand Reulet
INTRIQ, April 28, 2015
Context
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Discrete photon statistics from continuous measurements
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Context
1.6
û2 =
Rescaled correlators
1.4
v̂2 =
1.2
2
(x̂1 −x̂2 )
Vac =0
2
2
(p̂1 +p̂2 )
2
x̂21
1.0
0.8
0.6
0.4
0.2
f0
Vac
phase
coherent
I
RF
LO
Q
RF
Q
I
A
B
(a)
300 K
3K
-20 dB
6
4
2
0
-2
-4
-6
(c)
Triplexer
<4 GHz
Tunnel
Junction
4-8 GHz
P1
>8 GHz
6
4
2
0
-2
-4
-6
-6 -4 -2 0 2 4 6
40
6
4
2
0
-2
-4
-6
60
80
0.10
0.08
0.06
0.04
0.02
-6 -4 -2 0 2 4 6
(d)
X2
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20
0.00
×10−6
-36 dB
(b)
-6 -4 -2 0 2 4 6
20 mK
1.9 MHz
0
Vdc (µV)
X1
f2
Vdc
LO
Digitizer
f1
0.0
-80 -60 -40 -20
-0.02
6
4
2
0
-2
-4
-6
-0.04
-0.06
-0.08
-0.10
-6 -4 -2 0 2 4 6
Discrete photon statistics from continuous measurements
P2
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Photon pair detection
f0
Vdc
f1
Multiplex.
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f2
Discrete photon statistics from continuous measurements
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Photon pair detection
f0
Vdc
f1
Multiplex.
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f2
Discrete photon statistics from continuous measurements
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Photon pair detection
f0
Vdc
f1
Multiplex.
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f2
Discrete photon statistics from continuous measurements
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Photodetector properties
Amplification
Discretization
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Discrete photon statistics from continuous measurements
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Photodetector properties
Amplification 3
Discretization
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Discrete photon statistics from continuous measurements
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Photodetector properties
Amplification 3
Discretization 7
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Discrete photon statistics from continuous measurements
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Josephson parametric amplifier
ω, 2ω
Non-linear LC resonator.
Theoretically quantum-limited:
p
√
â → g â + e iϕ g − 1 ↠.
In practice, ∼1 noise photon
added.
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Discrete photon statistics from continuous measurements
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Discrete vs. continuous representations
Resolution of the identity
Î =
X
n
|Ψi =
X
n
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1
|nihn| =
π
hn|Ψi |ni =
Z
1
π
Z
d 2 α |αihα|
d 2 α hα|Ψi |αi
Discrete photon statistics from continuous measurements
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Quadratures
P
Quantized EM field
Ê (t) ∝
X i âk e −iωk t − âk† e iωk t
ωt
X
k
Ê (t) ∝
i
Xh
X̂k cos(ωk t) + P̂k sin(ωk t)
k
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Discrete photon statistics from continuous measurements
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Photon pairs and quadratures
ω1
ω0
P
ω2
(ω2 + δω)t
X
(ω1 − δω)t
ω1
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ω2
ω0
Discrete photon statistics from continuous measurements
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Quadrature measurements
Quadratures can be measured with a homodyne/heterodyne setup
Sig.
LO
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Simon Jolly, Ph.D. Thesis,
Exeter College, Oxford (2003)
Discrete photon statistics from continuous measurements
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State reconstruction
G. Breitenbach et al., Nature 387, 471-475 (1997)
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Discrete photon statistics from continuous measurements
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Setup
7 mK
3K
300 K
Q
RF
LO
I
Acq.
∼2 noise photons added to signal
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Discrete photon statistics from continuous measurements
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Discrete statistics from cumulants
We measure X̂θ =
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1 iθ †
e â + e −iθ â
2
Discrete photon statistics from continuous measurements
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Discrete statistics from cumulants
1 iθ †
We measure X̂θ =
e â + e −iθ â
2
DD EE
1
2
hn̂i → X̂θ ≡ C2 : hni = C2 −
2
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Discrete photon statistics from continuous measurements
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Discrete statistics from cumulants
1 iθ †
We measure X̂θ =
e â + e −iθ â
2
DD EE
1
2
hn̂i → X̂θ ≡ C2 : hni = C2 −
2
DD EE
2
1
2
n̂ → X̂θ4 ≡ C4 :
δn2 = C4 + C22 −
3
4
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Discrete photon statistics from continuous measurements
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Discrete statistics from cumulants
1 iθ †
We measure X̂θ =
e â + e −iθ â
2
DD EE
1
2
hn̂i → X̂θ ≡ C2 : hni = C2 −
2
DD EE
2
1
2
n̂ → X̂θ4 ≡ C4 :
δn2 = C4 + C22 −
3
4
DD EE
3
3 2
1
6
δn = C6 + 4C2 C4 + 2C23 − C2 · · ·
n̂ → X̂θ ≡ C6 :
5
2
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Discrete photon statistics from continuous measurements
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Results for a coherent state
0.7
0.6
­
®
δn2
®
δn3
­ ®
n
­
0.5
­
δnk
®
0.4
0.3
0.2
0.1
0.0
0.0
0.1
0.2
0.3
­ ®
0.4
0.5
0.6
0.7
n
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Discrete photon statistics from continuous measurements
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Classical limits
Poisson distribution for constant intensity.
hni 2= ηI
δn = hni
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Discrete photon statistics from continuous measurements
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Classical limits
Poisson distribution for constant intensity.
hni 2= ηI
δn = hni
Added fluctuations for varying intensity
hni
2= η hI i
δn = hni + η 2 δI 2
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Discrete photon statistics from continuous measurements
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Classical limits
Poisson distribution for constant intensity.
hni
2= ηI
δn = hni
Added fluctuations for varying intensity
hni
2= η hI i
δn = hni + η 2 δI 2
If η → 0, δn2 → hni.
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Discrete photon statistics from continuous measurements
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Results for a modulated coherent state
0.6
0.5
Sine
Triangle
Square
­ ®
n
0.3
­
δn2
®
0.4
0.2
0.1
0.0
0.0
0.1
0.2
­ ®
0.3
0.4
0.5
n
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Classical vs. non-classical
10
8
Non class.
®
6
­
δn2
Class.
4
2
Quant.
0
0.0
0.5
1.0
1.5
2.0
­ ®
2.5
3.0
3.5
4.0
n
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Discrete photon statistics from continuous measurements
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Signature of photon pairs
Poster by J.O. Simoneau
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Discrete photon statistics from continuous measurements
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Conclusion
Discrete statistics can be obtained from continuous
measurements.
Good alternative to photodetectors.
Signatures of non-classicality can be observed (sub-poissonian
statistics, pairs, ...).
There is probably more to explore: correlations, “post-selection”,
...
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Discrete photon statistics from continuous measurements
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