What differentiates quantum interference of light from classical? What are the advantages of using quantum states of light?

1. The interferometric signatures of
quantum and classical states of light
Saroosh Shabbir
Quantum Electronics & Quantum Op�cs
KTH Royal Ins�tute of Technology, Stockholm

2. Interferometric signatures of quantum and classical states
Do interferometric signals of quantum states differ fundam-
entally from classical states, in terms of shape and visibility?
How do the interferometric signals vary as states are trans-
formed from quantum to classical?
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 1 / 38

3. Outline
Interference - From Huygens & Young to Hanbury Brown & Twiss
Higher order interference
Two-mode projec�on measurements
Quantum interference from semi-classical states
Engineered interference
Projec�on measurements of increasingly dis�nguishable states
Summary and Conclusions
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 2 / 38

4. Classical Interference - First order
Unfiltered 605 nm, FWHM 5 nm HeNe laser
Chris�an Huygens
(1629-1659)
Thomas Young
(1773-1829)
All single mode states display first order interference.
First order interference does not discriminate between states.
It thus does not separate classical from quantum.
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 3 / 38

5. Second order interference
Robert Hanbury Brown
(1916-2002)
Richard Twiss
(1920-2005)
Intensi�es can also be correlated and have a coherence length
associated to the emi�ng light source.
R. Hanbury Brown & R. Twiss, Nature 178, 1046-1048 (1956)
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 4 / 38

6. Quantum Interference - Hong-Ou-Mandel effect
2,0 1,1 0,2
25% 50% 25%
Classically we would get:
2,0 1,1 0,2
50% 0% 50%
When wave packets overlap:
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 5 / 38

7. Quantum Interference - Hong-Ou-Mandel effect
2,0 1,1 0,2
25% 50% 25%
Classically we would get:
2,0 1,1 0,2
50% 0% 50%
When wave packets overlap:
Hong et al., PRL (1987)
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 6 / 38

8. Higher order (multiphoton) quantum interference
Y.-S Kim et al., Opt. Express 19, 24956 (2011)
Both the genera�on and the det-
ec�on of mul�-photon states is
complicated.
Non-linear op�cs is required to
generate states.
Polarisa�on op�cs and coincide-
nce detec�on is required to
detect states.
The measurement is probabilis�c.
Only when N photodetectors click
in coincidence the result is rec-
orded.
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 7 / 38

9. N00N states, de Broglie waves and quantum phase
super-resolution
1 2 3 4 5 6
0.2
0.4
0.6
0.8
1.0
phase shi�
No. of counts
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 8 / 38

10. N00N states, de Broglie waves and quantum phase
super-resolution
phase shi�
No. of counts
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
0.8
1.0
Phase difference
Countrate
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 9 / 38

11. N00N states, de Broglie waves and quantum phase
super-resolution
2 oscillations where we
would classically expect 1!
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 10 / 38

12. N00N states, de Broglie waves and quantum phase
super-resolution
N oscillations where we
would classically expect 1!
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 11 / 38

13. N00N states, de Broglie waves and quantum phase
super-resolution
N oscillations where we
would classically expect 1!
Phase super-resolu�on : Resolve features
�mes smaller than with ordinary light
Beyond Rayleigh diffraction limit
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 12 / 38

14. N00N states, de Broglie waves and quantum phase
super-resolution
N oscillations where we
would classically expect 1!
Phase super-sensi�vity : Uncertainty
in phase measurement
Phase super-resolu�on : Resolve features
�mes smaller than with ordinary light
Beyond Rayleigh diffraction limit
Heisenberg limit
J. Jacobson, G. Björk, I. Chuang, and Y. Yamamoto, PRL 74, 4835-4838 (1995)Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 13 / 38

15. Measurement post-selection of N00N states
given that we have only available.
Suppose we want to project out the state
Write the wanted state as
Form the polynomial and factorise over complex numbers
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 14 / 38

16. Measurement post-selection of N00N states
given that we have only available.
Suppose we want to project out the state
Write the wanted state as
Form the polynomial and factorise over complex numbers
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 15 / 38

17. Measurement post-selection of N00N states
given that we have only available.
Suppose we want to project out the state
Write the wanted state as
Form the polynomial and factorise over complex numbers
D A R L
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 16 / 38

18. Measurement post-selection of N00N states
given that we have only available.
Suppose we want to project out the state
Write the wanted state as
Form the polynomial and factorise over complex numbers
D A R L
A beam spli�er has the transforma�on law
and addi�onal phase-shi� gives the transforma�on
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 17 / 38

19. Measurement post-selection of N00N states
D A R L
A beam spli�er has the transforma�on law
and addi�onal phase-shi� gives the transforma�on
Coincident detection in all 4 SPDs
projects outs the NOON4 state
from the input!
R
L
A
D
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 18 / 38

20. Post-selection using coherent state input
A linearly polarised coherent state
also has a non-zero overlap with
NOON4 state!
R
L
A
D
With very weak excita�on, probability of having 5 or more photons << probability
of having exactly 4 photons
If 4 detectors click in coincidence, we are pre�y sure we've detected NOON4 state!
K. J. Resch et al., Phys. Rev. Le�. 98, 223601 (2007)
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 19 / 38

21. Quantum optics from semi-classical states
A linearly polarised coherent state
also has a non-zero overlap with
NOON4 state!
R
L
A
D
With very weak excita�on, probability of having 5 or more photons << probability
of having exactly 4 photons
If 4 detectors click in coincidence, we are pre�y sure we've detected NOON4 state!
K. J. Resch et al., Phys. Rev. Le�. 98, 223601 (2007)
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 20 / 38

22. Generalising the projection measurement method
Any complex polynomial can be factored over the field of complex numbers.
Mathematical Theorem:
Implication:
The corresponding projector to any N-photon, two-mode state can be implemented
through a series of beam-spli�ers, polarising op�cs, and signle photon coincidence
measurements!
Example:
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 21 / 38

23. Coherent state - temporal instead of spatial splitting
Uncorrelated (product state)!
R. J. Glauber, Phys. Rev. 131, 2766 (1963)
where
Laser
Laser
Laser
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24. Coherent state - temporal instead of spatial splitting
Uncorrelated (product state)!
R. J. Glauber, Phys. Rev. 131, 2766 (1963)
where
Laser
Laser
Laser
Switch spa�al spli�ng for temporal spli�ng!
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25. Coherent state - temporal instead of spatial splitting
R
L
A
D
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26. N00N states projected from a coherent state
0.0 0.5 1.0 1.5 2.0
0.0
0.5
1.0
0.0 0.5 1.0 1.5 2.0
0.0
0.5
1.0
N=30
N=60
Countrate(arb.units) 0.0 0.5 1.0 1.5 2.0
0.0
0.5
1.0 N=15
Phase difference (π radians)
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0.6 0.7 0.8 0.9 1.0
Visibility
Max 88 %
Min 57.5 %
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1.0
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N=60
Countrate(arb.units)
0.0 0.5 1.0 1.5 2.0
0.0
0.5
1.0 N=15
Phase difference (π radians)
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0.6 0.7 0.8 0.9 1.0
Visibility
Max 88 %
Min 57.5 %
phase shi� radians) (
S. Shabbir, M. Swillo, G. Björk, Phys. Rev. A 87, 053821
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 25 / 38

27. Arbitrary interference a using coherent state
S. Shabbir, M. Swillo, G. Björk, Phys. Rev. A 87, 053821
Birefringence
N-photon
coincident
detec�on
General two-mode state:
Overlap with phase-shifted coherent state:
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 26 / 38

28. Arbitrary interference a using coherent state
S. Shabbir, M. Swillo, G. Björk, Phys. Rev. A 87, 053821
Fourier series
Birefringence
N-photon
coincident
detec�on
General two-mode state:
Overlap with phase-shifted coherent state:
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 27 / 38

29. Arbitrary interference a using coherent state
S. Shabbir, M. Swillo, G. Björk, Phys. Rev. A 87, 053821
Fourier series
Birefringence
N-photon
coincident
detec�on
General two-mode state:
Overlap with phase-shifted coherent state:
Engineer any interference pa�ern!
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 28 / 38

30. Engineered interference
S. Shabbir, M. Swillo, G. Björk, Phys. Rev. A 87, 053821
Fourier series
Birefringence
N-photon
coincident
detec�on
General two-mode state:
Overlap with phase-shifted coherent state:
Engineer any interference pa�ern!
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■
■
■
■
■
■
■
0.0 0.5 1.0 1.5 2.0
0.0
0.5
1.0
Phase difference (π radians)
Countrate(arb.units)
31 term Fourier expansion
of Saw function
Raw data
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 29 / 38

31. Engineered interference
S. Shabbir, M. Swillo, G. Björk, Phys. Rev. A 87, 053821
Fourier series
Birefringence
N-photon
coincident
detec�on
General two-mode state:
Overlap with phase-shifted coherent state:
Engineer any interference pa�ern!
■
■
■
■
■
■
■
■
■
■
■
0.0 0.5 1.0 1.5 2.0
0.0
0.5
1.0
Phase difference (π radians)
Countrate(arb.units)
■
■ ■
■
■
■
■
■
■
■
■
■
■
■
0.0 0.5 1.0 1.5 2.0
0.0
0.5
1.0
Phase difference (π radians)
Countrate(arb.units)
31 term Fourier expansion
of Saw function
Raw data 31 term Fourier expansion
of Rectangular function
Raw data
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 30 / 38

32. Distinguishability transitions
Normalized counts
Path delay
Completely
indistinguishable
Completely
distinguishable
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 31 / 38

33. Distinguishability transitions
Normalized counts
Completely
indistinguishable
Completely
distinguishable
Normalized counts
Path delay ( )
Path delay ( )
Y-S. Ra et al., PNAS 110, 1227 (2013)
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 32 / 38

34. Distinguishability transitions
Normalized counts
Completely
indistinguishable
Completely
distinguishable
Normalized counts
Path delay ( )
Path delay ( )
Y-S. Ra et al., PNAS 110, 1227 (2013)
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 33 / 38

35. Distinguishability transitions
Normalized counts
Completely
indistinguishable
Completely
distinguishable
Normalized counts
Path delay ( )
Path delay ( )
Y-S. Ra et al., PNAS 110, 1227 (2013)
Non-monotonic quantum to classical transi�on?
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 34 / 38

36. Distinguishability transitions
Normalized counts
Completely
indistinguishable
Completely
distinguishable
Path delay ( )
G. Björk, S. Shabbir, New J. Phys. 16, 013006 (2014)
Coincidence detec�on window projects the
output onto
In fact, one could write projectors for single photon
and classical states that also show non-monotonic
behaviour.
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 35 / 38

37. Distinguishability transitions
Normalized counts
Completely
indistinguishable
Completely
distinguishable
Path delay ( )
G. Björk, S. Shabbir, New J. Phys. 16, 013006 (2014)
Coincidence detec�on window projects the
output onto
In fact, one could write projectors for single photon
and classical states that also show non-monotonic
behaviour.
Non-monotonic projec�on probabili�es as a func�on of dis�nguishability
do not signal quantum to classical transi�on.
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 36 / 38

38. Summary & Conclusions
It is possible to demonstrate highly "non-classical" interference effects using coherent state input.
The special character of the coherent state allows the measurement to be done "in series" rather
than "in parallel", saving �me and material resources.
The measurement non-linearity creates the desired "non-classical" interference.
Mul�-photon interference can give highly unusual interference effects/pa�erns.
Using linear op�cs and single photon counters one can synthesize any two-mode projec�on
measurement.
It is also possible to implement engineered interference. Any "Fourier spectrum" can be obtained.
However, the measurement is probabilis�c, which means that it's not an efficient method in terms
of input photons. Photon number resolving detectors would improve the detec�on efficiency.
In general, neither the shape of the interference pa�ern nor the visibility are signatures of
quantum states.
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 37 / 38

39. Acknowledgements
Gunnar Björk Marcin Swillo
Thank you!
Saroosh Shabbir, Gunnar Bj¨ork Interferometric signatures June 3, 2014 38 / 38