Quantum physics and the beam splitter mystery
Quantum physics and the beam splitter mystery François Hénault Institut de Planétologie et d’Astrophysique de Grenoble Université Joseph Fourier Centre National de la Recherche Scientifique BP 53, 38041 Grenoble – France
Conf. 9570 – The Nature of Light: What are Photons? VI
San Diego, 08-11-15
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Quantum physics and the beam splitter mystery
Plan of presentation • Part 1 Beamsplitter theoretical models – Quantum physics – Classical wave optics
• Part 2 Beamsplitter experimental setups – Hanbury Brown and Twiss experiment – Mach-Zehnder interferometer
Conf. 9570 – The Nature of Light: What are Photons? VI
San Diego, 08-11-15
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Quantum physics and the beam splitter mystery
Quantum view of the beamsplitter • Macroscopic, "black-box" matrix model – Energy conservation AT2 AR1
A′ A′ + A2′ A2′ = 1 * 1 1
*
Exit port 2’
– Unitary operator (TBC)
A1′A2′ + A2′ A′ = 0 *
* 1
– Beamsplitter matrix:
M BS
1 = 2
i 1 1 i
Conf. 9570 – The Nature of Light: What are Photons? VI
= A2’
Exit port 1’
A1
AT1
Entrance port 1
AR2
= A’1
BS
A2
Entrance port 2
San Diego, 08-11-15
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M BS =
1 2
i 1 1 i
Quantum physics and the beam splitter mystery
Real world beamsplitters • Here are essentially studied "symmetric" beamsplitters AR1 1 AR1
1
AT1
Cube beamsplitter 1 i 1 M BS = 2 1 i
AR1
AT1
M BS =
1 1 1 AR1 2 − 1 1
1
Pellicle beamsplitter
1 AT1
Asymmetric beamsplitters
Conf. 9570 – The Nature of Light: What are Photons? VI
AT1
1 i 1 M BS = 2 1 i
San Diego, 08-11-15
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Quantum physics and the beam splitter mystery
Wave optics model of the beamsplitter • Multi-interference effect as in Fabry-Perot interferometers t t exp(iϕ ) AT 1 = 12 212 1 − r21 exp(2iϕ ) AR1 =
Internal phase ϕ may depend on λ, θ...
r12
AR1
t21
r12 + r21 (t12 t 21 − r12 r21 )exp(2iϕ ) 1 − r212 exp(2iϕ )
• Lossless beamsplitter: 1 − exp(2iϕ ) AR1 = − r21 1 − r212 exp(2iϕ )
r21
e θ
1
O
– Energy conservation
AT 1 + AR1 = 1 2
2
φR1 − φT 1 = Arg [AR1 AT*1 ] = Arg [i sin ϕ ] = ±
AT1 O’
– Achromatic phase-shift
Conf. 9570 – The Nature of Light: What are Photons? VI
t12
π
ϕ = k ne cosθ =
2π
λ
ne cosθ
2 San Diego, 08-11-15
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Quantum physics and the beam splitter mystery
BS transmitted amplitudes and phase-shift Achromatic phase-shift of ±π/2
Lossless beamsplitter 1
6
0.6
Output phase (rad)
Output power (a. u.)
5
0.8
Transmitted
0.5
Reflected
0.4 0.2
4 3 2
+π/2
Transmitted
1
Reflected
0
[2π π] Difference [2Pi]
-1
-π/2
-2
0 0
0.25
0.5
0.75
1
1.25
1.5
1.75
0
2
0.25
0.5
0.75
1
1.25
1.5
1.75
2
Internal phase (Fractions of π Pi)
Internal phase (Fractions of π Pi)
6
Absorbing beamsplitter
Output phase (rad)
5 4 3 2
+π/2
Transmitted
1
Reflected
0
[2π π] Difference [2Pi]
-1
-π/2
-2 0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
Internal phase (Fractions of Pi) π
Conf. 9570 – The Nature of Light: What are Photons? VI
San Diego, 08-11-15
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Quantum physics and the beam splitter mystery
BS correlation experiments • Inspired from Hanbury Brown and Twiss experiment on intensity interferometry (1956) • Used in coincidence counting mode by Grangier, Roger and Aspect (GRA) Anti-correlation at low light levels (1986) • Demonstrates the particle nature of light (indivisible photon) IR1
Correlator / Coincidence counter
CRT D2
A1
IT1 BS
D1
Measurement apparatus Conf. 9570 – The Nature of Light: What are Photons? VI
Experimental results San Diego, 08-11-15
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Quantum physics and the beam splitter mystery
GRA experiment – Classical model • Uses classical notions of coherence length, generated currents… Transmitted amplitude Reflected amplitude
π/2 π i 1 AT 1 (t , k ) = exp(− ikct ) phase AR1 (t , k ) = exp(− ikct ) shifted 2 2
• First integration on spectral domain [k − δ k 2 , k + δ k 2] δk
C RT (t ) =
k +δ k 2
1 2δ k
∫ AT 1 (t , k ′) AR1 (t , k ′) dk ′ =
k −δ k 2
i exp(− 2ikct ) sin c(δ k ct ) 2
• Second integration on time domain [− τ ,+τ ] C RT
1 = 2τ
+τ
∫[
−τ
(
)]
1 Re al C RT (t ) dt = 2τ δk
Conf. 9570 – The Nature of Light: What are Photons? VI
2
+τ
∫τ −
sin 2 (2kct )sin c 2 (δk ct ) dt
San Diego, 08-11-15
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Quantum physics and the beam splitter mystery
GRA experiment – Classical model Final expression C RT = (1 − sin c (2 k cτ )) 4
• • Not in excellent agreement due to drastic approximations • But accounts for experimental photon anti-correlation Correlation factor C
0.4 0.3
Classical limit
0.25
0.2 Model
0.1
GRA data
0 0
10
20
30
40
50
Integration time (ns)
Conf. 9570 – The Nature of Light: What are Photons? VI
San Diego, 08-11-15
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Quantum physics and the beam splitter mystery
The Mach-Zehnder interferometer • Originally used as metrology tool in optics, gas dynamics etc. A’ 2
A2’ Exit port 2’
Exit port 2’
BS2
BS2
M2
M2 Semireflective coatings
A1’
A1’
Exit port 1’
1
Exit port 1’
1 M1
M1 BS1
Non symmetric, one reflexion only configuration Conf. 9570 – The Nature of Light: What are Photons? VI
BS1
Symmetric configuration, double Fabry-Perot effect San Diego, 08-11-15
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Quantum physics and the beam splitter mystery
MZ interferometer – OPD modulation δ • Achromatic phase-shift ∆φ = ±π/2 when δ = 0 • Equal to 0 [π] otherwise I 2′ = 1 − 4 AT 1 AR1 + 4 AT 1 AR1 sin 2 (kδ 2) 2
2
D2
δ0
2 II1’′ = 4 A A (kδ 2) cos 1 T 1 R1
M2
2
D1
BS2
δ0+δ OPD modulation
A1 BS1
M1
Conf. 9570 – The Nature of Light: What are Photons? VI
• Energy conservation OK • In agreement with quantum optics
San Diego, 08-11-15
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Quantum physics and the beam splitter mystery
MZ interferometer – Wave optics model At zero optical path difference Output power (a. u.)
1
With OPD modulation 1.2
Arm 1
Arm 2
1
0.8
0.8 0.6 0.6 0.5 0.4
0.5
0.4 Arm 1
0.2
0.2
Arm 2
0
0 0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
As measured in GRA experiment
-1
-0.75
-0.5
BS internal phase (Fractions of Pi) π
Output phase (rad)
14
Arm 1
12
Arm 2
10
π] Difference [2π [2Pi]
-0.25
0
0.25
0.5
0.75
1
0.75
1
OPD (Fractions of Lambda) 10 Arm 1
8
Arm 2
[2π π] Difference [2Pi]
6
8 4
+π
6
+3π/2
4 2
2 0
+π/2
0
-2 0
0.25
0.5
0.75
1
1.25
1.5
1.75
π BS internal phase (Fractions of Pi)
Conf. 9570 – The Nature of Light: What are Photons? VI
2
-1
-0.75
-0.5
-0.25
0
0.25
0.5
OPD (Fractions of Lambda)
San Diego, 08-11-15
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Quantum physics and the beam splitter mystery
Conclusion • Quantum and wave optics BS theories are in global agreement. They both describe a ±π/2 ±π phase shift between transmitted/reflected electric fields – Quantum physics is a macroscopic "black-box" model – Classical optics evidences a multi-interference effect
• 4th-order interference (HBT) experiments show anticorrelation of BS outputs (GRA) – Quantum physics Interpretation confirms photon existence – Can also be explained with classical wave optics model including the ±π/2 phase shift
• Future work on other interference experiments – Mach-Zehnder, Hong-Ou-Mandel… Conf. 9570 – The Nature of Light: What are Photons? VI
San Diego, 08-11-15
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