Continuous and discrete variables:
the best of both worlds
Alexei Ourjoumtsev
CNRS
Laboratoire Charles Fabry,
Institut d’Optique, Palaiseau
Describing light
Discrete
Photons
Continuous
Wave
Describing light
Discrete
Look for :
Photons
Number & Coherence
Continuous
Wave
Amplitude & Phase (polar)
Quadratures X & P (cartesian)
Describing light
Discrete
Look for :
Describe with:
Photons
Continuous
Wave
Amplitude & Phase (polar)
Number & Coherence
Quadratures X & P (cartesian)
Wigner function
Density matrix
ρn,m
W(X,P)
ΔX. ΔP ≥ ½
Describing light
Discrete
Look for :
Describe with:
Measure by :
Photons
Number & Coherence
Density matrix
Counting:
Avalanche photodiodes,
Visible light photon counters,
Thermal edge sensors…
Continuous
Wave
Amplitude & Phase (polar)
Quadratures X & P (cartesian)
Wigner function
Describing light
Discrete
Look for :
Describe with:
Measure by :
Photons
Continuous
Amplitude & Phase (polar)
Number & Coherence
Quadratures X & P (cartesian)
Wigner function
Density matrix
Counting:
Avalanche photodiodes,
Visible light photon counters,
Thermal edge sensors…
Wave
Demodulating : Homodyne Detection
Local
oscillator
Quantum
state
V1-V2
 Xθ = X cosθ +P sinθ
Describing light
Discrete
Look for :
Describe with:
Measure by :
Photons
Continuous
Amplitude & Phase (polar)
Number & Coherence
Quadratures X & P (cartesian)
Wigner function
Density matrix
Counting:
Wave
Demodulating : Homodyne Detection
(€,  98%)
Avalanche photodiodes,
Visible light photon counters,
Thermal edge sensors…
Local
oscillator
Quantum
state
V1-V2
 Xθ = X cosθ +P sinθ
θ=0°
θ=45°
θ=90°
Describing light
Discrete
Look for :
Describe with:
Measure by :
« Simple »
States
Photons
Number & Coherence
Density matrix
Counting: APD, VLPC, TES...
Fock States
Sources :
- Single Atoms
- NV diamond centers
- Parametric fluorescence
....
Continuous
Wave
Amplitude & Phase (polar)
Quadratures X & P (cartesian)
Wigner function
Demodulating : Homodyne Detection
Describing light
Discrete
Look for :
Describe with:
Measure by :
« Simple »
States
Photons
Number & Coherence
Continuous
Amplitude & Phase (polar)
Quadratures X & P (cartesian)
Wigner function
Density matrix
Counting: APD, VLPC, TES...
Demodulating : Homodyne Detection
Fock States
Sources :
- Single Atoms
- NV diamond centers
- Parametric fluorescence
....
Wave
Gaussian States
Sources :
Lasers :
coherent states
P
Non-linear media :
squeezed states
P
α
α
θ
X
θ
X
Describing light
Discrete
Look for :
Describe with:
Measure by :
« Simple »
States
Photons
Number & Coherence
Density matrix
Counting: APD, VLPC, TES...
Fock States
Continuous
Wave
Amplitude & Phase (polar)
Quadratures X & P (cartesian)
Wigner function
Demodulating : Homodyne Detection
Gaussian States
The best of both worlds
Discrete :
• Easy to process
• Robust against noise & losses
• Allow postselection
Continuous
• Simple sources & detectors
• More bits per symbol
• Richer structure
The best of both worlds
Discrete :
Continuous
• Easy to process
• Robust against noise & losses
• Allow postselection
• Simple sources & detectors
• More bits per symbol
• Richer structure
Possible combination : QIP with coherent-state qubits
|+ =|α
|- = |-α
Features :
• Possible to distinguish all 4 Bell states
• Heisenberg-limited precision measurements
• Candidates for loophole-free Bell tests
Basic requirements :
• |α  |-α
• a|α + b|-α
• |α, α   |-α, - α 
• qubit rotations
•…
Discrete-variable operations
Continuous-variable states
Photon subtraction
from squeezed vacuum
Ideas
Dakna et al, PRA 55, 3184 (1997)
Optical
Parametric
Amplifier
â
R<<1
OPA
Non-resolving
counter (APD)
1 photon
subtracted :
|α - |-α
|α|²~1
Vacuum
Squeezed vacuum
Photon-Subtracted
Squeezed Vacuum
 Schrödinger’s kitten
Ideas
Dakna et al, PRA 55, 3184 (1997)
Optical
Parametric
Amplifier
ân
R<<1
OPA
Resolving
counter
Vacuum
Squeezed vacuum
n photons
subtracted :
|α|-α
Photon-Subtracted
Squeezed Vacuum
 Schrödinger’s cat
Experiments
Femtosecond pulses
OPA
Broadband
Multimode
Continuous-wave
OPA
n
Easy
Timing
Narrowband
Singlemode
n
Complex
Timing
n=1 : 2006, IOGS (Palaiseau)
Science 312, 83
Fidelity = 70% |α|² = 0.8
n=1 : 2006, NBI (Copenhagen)
PRL 97, 083604
Fidelity = 53% |α|² = 1.7
n3 : 2010, NIST (Boulder)
Arxiv: 1004.3656
Fidelity = 59% |α|² = 3
n=2 : 2008, NICT (Tokyo)
PRL 101, 233605
Fidelity = 60% |α|² = 2
Compromizes (fs)
Pulses : Not too short / Not too long
Repetition rate : Not too fast / Not too slow
Crystal : Not too thin / Not too thick
APD filters : Not too narrow / Not too broad…
a,b,c,d
simple functions of the
experimental parameters
Analytic model
Radon
transform
Correct for
Homodyne
Losses
Coherent-state qubits
â
R<<1
|α -|-α
|α+|-α
APD
Coherent-state qubits
A*â + B*1
R<<1
|α+|-α
D()
R<<1
A(|α -|-α)
+ B(|α+|-α)
= a|α + b|-α
|/R
APD
Experiment : |α+|-α |sqz
2010, NICT (Tokyo)
arXiv:1002.3211
Coherent Bell states
Local preparation
R=50%
|α2 -|-α 2 
|α,α-|-α,-α
Coherent Bell states
Local preparation
Losses
R=50%
|α2 -|-α 2 
|α,α-|-α,-α
Coherent Bell states
Local preparation
Non-local preparation
Losses
R=50%
|α2 -|-α 2 
| α  + |-α 
|α,α-|-α,-α
â1-â2
R<<1
50/50
| α  + |-α 
APD
R<<1
|ψ0 = |α,α-|-α,-α
Coherent Bell states
Local preparation
Non-local preparation
Losses
R=50%
| α  + |-α 
|α,α-|-α,-α
|α2 -|-α 2 
â1-eifâ2
R<<1
50/50
| α  + |-α 
f
APD
R<<1
|ψf = cos f |α,α -isin f |α,-α
2
2
+isin f |-α,α -cos f |-α,-α
2
2
Experiment : |α+|-α |sqz
|α|²=0.65
f=/2
2009, IOGS (Palaiseau)
Nature Phys. 5, 189
Coherent Bell states
Local preparation
Non-local preparation
| α  + |-α 
Losses
R=50%
|α,α-|-α,-α
â1-eifâ2
R<<1
50/50
| α  + |-α 
|α2 -|-α 2 
f
APD
R<<1
|ψf = cos f |α,α -isin f |α,-α
2
2
+isin f |-α,α -cos f |-α,-α
2
2
Teleportation:
n1
n2
|ψf
a|α + b|-α
[cos f a-isin f b] |α
2
2
+[cos f b-isin f a] |-α
2
2
Qubit rotation
Beyond coherent qubits
Beyond coherent qubits
[â,â†]0
â
â†
R<<1
G-1<<1
OPA
APD
APD
2007, LENS (Florence)
Science 317, 1890
Beyond coherent qubits
Gaussian entanglement distillation
[â,â†]0
â
â†
R<<1
G-1<<1
OPA
APD
APD
2007, LENS (Florence)
Science 317, 1890
R<<1
OPA
R<<1
2010, NICT (Tokyo)
Nature Phot. 4, 178
APD
APD
Beyond coherent qubits
Gaussian entanglement distillation
[â,â†]0
â
â†
R<<1
G-1<<1
OPA
APD
R<<1
OPA
APD
2007, LENS (Florence)
Science 317, 1890
R<<1
2010, NICT (Tokyo)
Nature Phot. 4, 178
Noiseless amplification
2010, QOIL (Brisbane)
Nature Phot. 4, 316
APD
|α
R’=50%
|1
R
APD
2010, IOGS (Palaiseau)
|gα PRL 104, 123603
APD
APD
Continuous-variable operations
Discrete-variable states
Homodyne measurements
on photon-number states
Fock state qubits
a|0+ b|1
|1
2004, Univ. Konstanz
PRL 92, 047903
a|0+ b|1
Homodyne
X
|0

|1
X
Squeezed Schrödinger cats
R=50%
Homodyne
|X/2 |<
|2n
|2n+1
/2
|n
X /2
Squeezed Schrödinger cats
|α|-α
(x)=xne-x²/2
R=50%
Homodyne
|X/2 |<
|2n
|2n+1
/2
|n
X /2
• Ideally, F>99%
• Squeezed by 3dB
• ||²n
• parityn
Squeezed Schrödinger cats
Experiment:
|2
R=50%
2007, IOGS (Palaiseau)
Nature 448, 784
Homodyne
|X/2 |<0.1
Homodyne preparation or photon subtration?
Theory : Homodyne exponentially more efficient
Experiments :
Homodyne
||²2.6
6Hz
Photon subtraction
||² 2
<1Hz
||² 3
5mHz
Conclusion
Discrete :
Continuous
• Easy to process
• Robust against noise & losses
• Allow postselection
• Simple sources & detectors
• More bits per symbol
• Richer structure
Combining the best of both worlds becomes possible in practice
Growing number of groups involved
Collaborations welcome!