Majorana Representation in Quantum Optics - SU(2) Interferometry and Uncertainty Relations
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The algebra of SU(2) is ubiquitous in physics, applicable both to the atomic spin states and the polarisation states of light. The method developed by Majorana to represent pure, symmetric spin-states of arbitrary value as a product of spin-1/2 states is a powerful tool allowing the representation of a qudit on the same geometry as a qubit, i.e., the Bloch sphere. An experimental implementation of the Majorana representation in the realm of quantum optics is presented. The technique allows the projection of arbitrary, two-mode, pure states from a coherent state input and the construction of arbitrary interference patterns. The representation also proves useful in delineating uncertainty limits of states with a particular spin value. Issues with traditional uncertainty relations involving the SU(2) operators, such as trivial bounds for certain states and non-invariance, are thereby resolved with the presented pictorial solution.
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Majorana Representation in Quantum Optics - SU(2) Interferometry and Uncertainty Relations
- 1. Majorana Representation in Quantum Optics SU(2) Interferometry and Uncertainty Relations SAROOSH SHABBIR QEO Supervisors: Prof. Gunnar Björk Dr. Marcin Swillo May 12, 2017 1/73
- 2. 2/73 The Birth of Experimental Quantum Optics Hanbury Brown and Twiss stellar interferometer
- 3. 3/73 The Birth of Experimental Quantum Optics Hanbury Brown and Twiss stellar interferometer Detector cable control building tower garage Intensity-intensity correlation
- 4. 4/73 The Birth of Experimental Quantum Optics Hanbury Brown and Twiss stellar interferometer Detector cable control building tower garage Intensity-intensity correlation
- 5. 5/73 The Birth of Experimental Quantum Optics Glauber's theory of optical coherence Higher order correlation measurements necessary to discriminate between classical and quantum states
- 6. 6/73 Outline Higher Order Measurements - Interferometry ARBITRARY INTERFERENCE PATTERNS - PAPERS I & II
- 7. 7/73 Outline Higher Order Measurements - Interferometry ARBITRARY INTERFERENCE PATTERNS - PAPERS I & II Quantum to Classical Transitions? NON-MONOTONIC PROJECTION PROBABILITIES - PAPER III & IV
- 8. 8/73 Outline Higher Order Measurements - Interferometry ARBITRARY INTERFERENCE PATTERNS - PAPERS I & II Quantum to Classical Transitions? NON-MONOTONIC PROJECTION PROBABILITIES - PAPER III & IV
- 9. 9/73 Outline Higher Order Measurements - Interferometry ARBITRARY INTERFERENCE PATTERNS - PAPERS I & II Geometry of Two-Mode States MAJORANA REPRESENTATION - PAPER V
- 10. 10/73 Outline Higher Order Measurements - Interferometry ARBITRARY INTERFERENCE PATTERNS - PAPERS I & II Geometry of Two-Mode States MAJORANA REPRESENTATION - PAPER V Uncertainty Relations of Angular Momentum Operators
- 11. 11/73 Outline Higher Order Measurements - Interferometry ARBITRARY INTERFERENCE PATTERNS - PAPERS I & II Geometry of Two-Mode States MAJORANA REPRESENTATION - PAPER V Uncertainty Relations of Angular Momentum Operators SU(2) UNCERTAINTY LIMITS - PAPER V
- 12. 12/73 Interferometry Beam splitter Phase-shifter Mirror Detector Measurement Observable i i i i i Mach-Zehnder Interferometer
- 15. 15/73 Interferometry 01-1 20 phase shi� (radians) 01-1 20 phase shi� (radians)
- 16. 16/73 Interferometry 01-1 20 phase shi� (radians) 01-1 20 phase shi� (radians) 2 oscillations where we would classically expect 1!
- 17. 17/73 Interferometry 01-1 20 phase shi� (radians) 01-1 20 phase shi� (radians) 2 oscillations where we would classically expect 1! N oscillations where we would classically expect 1! ?
- 18. 18/73 Metrology 01-1 20 phase shi� (radians) 01-1 20 phase shi� (radians) 2 oscillations where we would classically expect 1! N oscillations where we would classically expect 1! ? Limits in Metrology Resolve features times smaller than with ordinary light Phase super-resolution
- 19. 19/73 Metrology 01-1 20 phase shi� (radians) 01-1 20 phase shi� (radians) 2 oscillations where we would classically expect 1! N oscillations where we would classically expect 1! ? Limits in Metrology Resolve features times smaller than with ordinary light Phase super-resolution Beyond Rayleigh diffraction limit resolved Rayleigh limit unresolved
- 20. 20/73 Metrology 01-1 20 phase shi� (radians) 01-1 20 phase shi� (radians) 2 oscillations where we would classically expect 1! N oscillations where we would classically expect 1! ? Limits in Metrology Resolve features times smaller than with ordinary light Phase super-resolution Beyond Rayleigh diffraction limit resolved Rayleigh limit unresolved Standard Quantum Limit (SQL) Phase sensitivity Uncertainty in phase measurement 20 phase shi� (radians) N(no.ofquanta)
- 21. 21/73 Metrology 01‐1 20 phase shi� (radians) 01‐1 20 phase shi� (radians) 2 oscillations where we would classically expect 1! N oscillations where we would classically expect 1! ? Limits in Metrology Resolve features times smaller than with ordinary light Phase super-resolution Beyond Rayleigh diffraction limit resolved Rayleigh limit unresolved Standard Quantum Limit (SQL) Phase sensitivity Uncertainty in phase measurement 20 phase shi� (radians) N (no. of quanta) super-resolution microscopy
- 22. 22/73 Metrology Standard Quantum Limit (SQL) Phase sensitivity Uncertainty in phase measurement 20 phase shi� (radians) N(no.ofquanta) phase super-resolution microscopy Heisenberg limitUncertainty in phase measurement Phase super-sensitivity Heisenberg Limit ?
- 23. 23/73 Metrology Standard Quantum Limit (SQL) Phase sensitivity Uncertainty in phase measurement 20 phase shi� (radians) N(no.ofquanta) phase super-resolution microscopy Heisenberg limitUncertainty in phase measurement Phase super-sensitivity Heisenberg Limit ? Quantum Metrology
- 24. 24/73 Issues in Metrology 01-1 20 phase shi� (radians) 01-1 20 phase shi� (radians) 2 oscillations where we would classically expect 1! N oscillations where we would classically expect 1! ? Limits in Metrology Resolve features times smaller than with ordinary light Phase super-resolution Beyond Rayleigh diffraction limit resolved Rayleigh limit unresolved Standard Quantum Limit (SQL) Phase sensitivity Uncertainty in phase measurement 2 0 phase shi� (radians) N(no.ofquanta) phase super-resolution microscopy Heisenberg limitUncertainty in phase measurement Phase super-sensitivity Heisenberg Limit ? Quantum Metrology Difficult! Requires : - Single Photon Sources - Photon no. resolving detectors etc
- 25. 25/73 Issues in Metrology - Approaches 01-1 20 phase shi� (radians) 01-1 20 phase shi� (radians) 2 oscillations where we would classically expect 1! N oscillations where we would classically expect 1! ? Limits in Metrology Resolve features times smaller than with ordinary light Phase super-resolution Beyond Rayleigh diffraction limit resolved Rayleigh limit unresolved Standard Quantum Limit (SQL) Phase sensitivity Uncertainty in phase measurement 2 0 phase shi� (radians) N(no.ofquanta) phase super-resolution microscopy Heisenberg limitUncertainty in phase measurement Phase super-sensitivity Heisenberg Limit ? Quantum Metrology Difficult! Requires : - Single Photon Sources - Photon no. resolving detectors etc Sol: Clever measurement scheme
- 28. 28/73 Projection Measurement D A L R
- 29. 29/73 Projection Measurement D A L R Coincident detection in all detectors Projects out the NOON4 state! (provided weak coherent state) D L R A
- 30. 30/73 Projection Measurement D A L R Coincident detection in all detectors Projects out the NOON4 state! (provided weak coherent state) D L R A
- 31. 31/73 Projection Measurement D A L R Coincident detection in all detectors Projects out the NOON4 state! (provided weak coherent state) D L R A Overlap with phase-shifted coherent state
- 32. 32/73 Projection Measurement D A L R Coincident detection in all detectors Projects out the NOON4 state! (provided weak coherent state) D L R A Overlap with phase-shifted coherent state 01 20 phase shi� (radians)
- 33. 33/73 Arbitrary Interference D A L R Coincident detection in all detectors Projects out the NOON4 state! (provided weak coherent state) D L R A 01 20 phase shi� (radians) Generalisation FUndamental theorem of Algebra complex number i
- 34. 34/73 Arbitrary Interference D A L R Coincident detection in all detectors Projects out the NOON4 state! (provided weak coherent state) D L R A Overlap with phase-shifted coherent state 01 20 phase shi� (radians) Generalisation FUndamental theorem of Algebra complex number i polariser polariser polariser polariser Quartz wedge Quartz wedge i i i
- 35. 35/73 Arbitrary Interference D A L R Coincident detection in all detectors Projects out the NOON4 state! (provided weak coherent state) D L R A Overlap with phase-shifted coherent state 01 20 phase shi� (radians) Generalisation FUndamental theorem of Algebra complex number i polariser polariser polariser polariser Quartz wedge Quartz wedge i i i Laser Laser Laser A B C Coherent state remains a cohrerent state whether split in space or time!
- 36. 36/73 Arbitrary Interference D A L R Coincident detection in all detectors Projects out the NOON4 state! (provided weak coherent state) D L R A Overlap with phase-shifted coherent state 01 20 phase shi� (radians) Generalisation FUndamental theorem of Algebra complex number i polariser polariser polariser polariser Quartz wedge Quartz wedge i i i Laser Laser Laser A B C Coherent state remains a cohrerent state whether split in space or time! polariser polariser polariser polariser Quartz wedge Quartz wedge i i i Recipe! - Impelement only one arm - Adjust polariser and phase-shifter setting - Record projection probabilities - Repeat
- 37. 37/73 Arbitrary Interference D A L R Coincident detection in all detectors Projects out the NOON4 state! (provided weak coherent state) D L R A Overlap with phase-shifted coherent state 01 20 phase shi� (radians) Generalisation FUndamental theorem of Algebra complex number i polariser polariser polariser polariser Quartz wedge Quartz wedge i i i Laser Laser Laser A B C Coherent state remains a cohrerent state whether split in space or time! polariser polariser polariser polariser Quartz wedge Quartz wedge i i i Recipe! - Impelement only one arm - Adjust polariser and phase-shifter setting - Record projection probabilities - Repeat Overlap with phase-shifted coherent state
- 38. 38/73 Arbitrary Interference D A L R Coincident detection in all detectors Projects out the NOON4 state! (provided weak coherent state) D L R A Overlap with phase-shifted coherent state 01 20 phase shi� (radians) Generalisation FUndamental theorem of Algebra complex number i polariser polariser polariser polariser Quartz wedge Quartz wedge i i i Laser Laser Laser A B C Coherent state remains a cohrerent state whether split in space or time! polariser polariser polariser polariser Quartz wedge Quartz wedge i i i Recipe! - Impelement only one arm - Adjust polariser and phase-shifter setting - Record projection probabilities - Repeat Overlap with phase-shifted 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 Phase difference (π radians) ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 0.6 0.7 0.8 0.9 1.0 1.6 ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 1.7 1.8 1.9 2.0 ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 0.6 0.7 0.8 0.9 1.0 Visibility Max 88 % Min 57.5 % NOON 60 PAPER I - S. Shabbir, M. Swillo, G. Björk, Phys. Rev. A 87, 053821 ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 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
- 39. 39/73 Arbitrary Interference D A L R Coincident detection in all detectors Projects out the NOON4 state! (provided weak coherent state) D L R A Overlap with phase-shifted coherent state 01 20 phase shi� (radians) Generalisation FUndamental theorem of Algebra complex number i polariser polariser polariser polariser Quartz wedge Quartz wedge i i i Laser Laser Laser A B C Coherent state remains a cohrerent state whether split in space or time! polariser polariser polariser polariser Quartz wedge Quartz wedge i i i Recipe! - Impelement only one arm - Adjust polariser and phase-shifter setting - Record projection probabilities - Repeat Overlap with phase-shifted 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 Phase difference (π radians) ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 0.6 0.7 0.8 0.9 1.0 1.6 ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 1.7 1.8 1.9 2.0 ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 0.6 0.7 0.8 0.9 1.0 Visibility Max 88 % Min 57.5 % NOON 60 PAPER I - S. Shabbir, M. Swillo, G. Björk, Phys. Rev. A 87, 053821
- 40. 40/73 Arbitrary Interference D A L R Coincident detection in all detectors Projects out the NOON4 state! (provided weak coherent state) D L R A Overlap with phase-shifted coherent state 01 20 phase shi� (radians) Generalisation FUndamental theorem of Algebra complex number i polariser polariser polariser polariser Quartz wedge Quartz wedge i i i Laser Laser Laser A B C Coherent state remains a cohrerent state whether split in space or time! polariser polariser polariser polariser Quartz wedge Quartz wedge i i i Recipe! - Impelement only one arm - Adjust polariser and phase-shifter setting - Record projection probabilities - Repeat Overlap with phase-shifted 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 Phase difference (π radians) ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 0.6 0.7 0.8 0.9 1.0 1.6 ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 1.7 1.8 1.9 2.0 ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 0.6 0.7 0.8 0.9 1.0 Visibility Max 88 % Min 57.5 % NOON 60 PAPER I - S. Shabbir, M. Swillo, G. Björk, Phys. Rev. A 87, 053821 ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 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 N.B. - Can be implemented for any two-mode pure state - Any arbitrary interference pattern can be constructed with unit visibility - Neither visibility nor the shape of the pattern a good indicator of quantumness - Completely extendible to the classical regime PAPER I - S. Shabbir, M. Swillo, G. Björk, Phys. Rev. A 87, 053821 PAPER II - S. Shabbir, M. Swillo, G. Björk, JOSA A 30, 1921
- 42. 42/73 Majorana Representation Two-mode state Geometric representation N points on the surface of the Bloch sphere
- 43. 43/73 Majorana Representation Two-mode state Geometric representation N points on the surface of the Bloch sphere N-level system on the same geometry as a 2-level system
- 44. 44/73 Majorana Representation Two-mode state Geometric representation N points on the surface of the Bloch sphere N-level system on the same geometry as a 2-level system Example - NOON 4 state Geometric representation 4 points distributed symmetrically along the equator
- 45. 45/73 Majorana Representation Two-mode state Geometric representation N points on the surface of the Bloch sphere N-level system on the same geometry as a 2-level system Example - NOON 4 state Geometric representation 4 points distributed symmetrically along the equator Linear optical elements perform rotations along the axes Phase-shifter Beam-splitter
- 46. 46/73 Majorana Representation Two-mode state Geometric representation N points on the surface of the Bloch sphere N-level system on the same geometry as a 2-level system Example - NOON 4 state Geometric representation 4 points distributed symmetrically along the equator Linear optical elements perform rotations along the axes Phase-shifter Beam-splitter Points rotate rigidly!
- 47. 47/73 Majorana Representation Two-mode state Geometric representation N points on the surface of the Bloch sphere N-level system on the same geometry as a 2-level system Example - NOON 4 state Geometric representation 4 points distributed symmetrically along the equator Linear optical elements perform rotations along the axes Phase-shifter Beam-splitter Points rotate rigidly! SU(2) coherent state NOON state
- 48. 48/73 Majorana Representation Two-mode state Geometric representation N points on the surface of the Bloch sphere N-level system on the same geometry as a 2-level system Example - NOON 4 state Geometric representation 4 points distributed symmetrically along the equator Linear optical elements perform rotations along the axes Phase-shifter Beam-splitter Points rotate rigidly! SU(2) coherent state NOON state Heuristic no. 1: The more symmetric a Majorana Representation, the more metrologically useful a state
- 49. 49/73 Majorana Representation Example - NOON 4 state Geometric representation 4 points distributed symmetrically along the equator Linear optical elements perform rotations along the axes Phase-shifter Beam-splitter Points rotate rigidly! SU(2) coherent state NOON state Heuristic no. 1: The more symmetric a Majorana Representation, the more metrologically useful a state AbraCaDabra! (post-selection)
- 50. 50/73 Majorana Representation Two-mode state Geometric representation N points on the surface of the Bloch sphere N-level system on the same geometry as a 2-level system Example - NOON 4 state Geometric representation 4 points distributed symmetrically along the equator Linear optical elements perform rotations along the axes Phase-shifter Beam-splitter Points rotate rigidly! Recall! Angular momentum operators generate rotations
- 51. 51/73 Majorana Representation Two-mode state Geometric representation N points on the surface of the Bloch sphere N-level system on the same geometry as a 2-level system Example - NOON 4 state Geometric representation 4 points distributed symmetrically along the equator Linear optical elements perform rotations along the axes Phase-shifter Beam-splitter Points rotate rigidly! Recall! Angular momentum operators generate rotations Schwinger boson representation
- 52. 52/73 Uncertainty Relations Two-mode state Geometric representation N points on the surface of the Bloch sphere N-level system on the same geometry as a 2-level system Example - NOON 4 state Geometric representation 4 points distributed symmetrically along the equator Linear optical elements perform rotations along the axes Phase-shifter Beam-splitter Points rotate rigidly! Recall! Angular momentum operators generate rotations Schwinger boson representation
- 53. 53/73 Uncertainty Relations Two-mode state Geometric representation N points on the surface of the Bloch sphere N-level system on the same geometry as a 2-level system Example - NOON 4 state Geometric representation 4 points distributed symmetrically along the equator Linear optical elements perform rotations along the axes Phase-shifter Beam-splitter Points rotate rigidly! Recall! Angular momentum operators generate rotations Schwinger boson representation Recall Heisenberg Uncertainty Rel.
- 54. 54/73 Uncertainty Relations Schwinger boson representation Recall Heisenberg Uncertainty Rel. Pairwise UncertaintyTrivial bounds Not SU(2) invariant
- 55. 55/73 Uncertainty Relations Two-mode state Geometric representation N points on the surface of the Bloch sphere N-level system on the same geometry as a 2-level system Example - NOON 4 state Geometric representation 4 points distributed symmetrically along the equator Linear optical elements perform rotations along the axes Phase-shifter Beam-splitter Points rotate rigidly! Recall! Angular momentum operators generate rotations Schwinger boson representation Recall Heisenberg Uncertainty Rel. Involves only 2 of 3 operators Trivial bounds Not SU(2) invariant Covariance matrix approach
- 56. 56/73 Uncertainty Relations Schwinger boson representation Recall Heisenberg Uncertainty Rel. Pairwise UncertaintyTrivial bounds Not SU(2) invariant Covariance matrix approach Three invariants: Determinant Sum of principal minors Trace
- 57. 57/73 Uncertainty Relations Schwinger boson representation Recall Heisenberg Uncertainty Rel. Pairwise UncertaintyTrivial bounds Not SU(2) invariant Covariance matrix approach Three invariants: Determinant Sum of principal minors Trace - Non-trivial bounds - SU(2) invariant - Involves all three operators
- 58. 58/73 Uncertainty Relations Schwinger boson representation Recall Heisenberg Uncertainty Rel. Pairwise UncertaintyTrivial bounds Not SU(2) invariant Covariance matrix approach Three invariants: Determinant Sum of principal minors Trace - Non-trivial bounds - SU(2) invariant - Involves all three operators
- 59. 59/73 Uncertainty Relations Schwinger boson representation Recall Heisenberg Uncertainty Rel. Pairwise UncertaintyTrivial bounds Not SU(2) invariant Covariance matrix approach Three invariants: Determinant Sum of principal minors Trace - Non-trivial bounds - SU(2) invariant - Involves all three operators NOON stateSU(2) coherent state
- 60. 60/73 Uncertainty Relations Schwinger boson representation Recall Heisenberg Uncertainty Rel. Pairwise UncertaintyTrivial bounds Not SU(2) invariant Covariance matrix approach Three invariants: Determinant Sum of principal minors Trace - Non-trivial bounds - SU(2) invariant - Involves all three operators NOON stateSU(2) coherent state Heuristic no. 2: The higher the sum variance, the more metrologically useful a state
- 61. 61/73 Uncertainty Relations Schwinger boson representation Recall Heisenberg Uncertainty Rel. Pairwise UncertaintyTrivial bounds Not SU(2) invariant Covariance matrix approach Three invariants: Determinant Sum of principal minors Trace - Non-trivial bounds - SU(2) invariant - Involves all three operators NOON stateSU(2) coherent state Heuristic no. 2: The higher the sum variance, the more metrologically useful a state N=2
- 62. 62/73 Uncertainty Relations Schwinger boson representation Recall Heisenberg Uncertainty Rel. Pairwise UncertaintyTrivial bounds Not SU(2) invariant Covariance matrix approach Three invariants: Determinant Sum of principal minors Trace - Non-trivial bounds - SU(2) invariant - Involves all three operators NOON stateSU(2) coherent state Heuristic no. 2: The higher the sum variance, the more metrologically useful a state N=2
- 63. 63/73 Uncertainty Relations Schwinger boson representation Recall Heisenberg Uncertainty Rel. Pairwise UncertaintyTrivial bounds Not SU(2) invariant Covariance matrix approach Three invariants: Determinant Sum of principal minors Trace - Non-trivial bounds - SU(2) invariant - Involves all three operators NOON stateSU(2) coherent state Heuristic no. 2: The higher the sum variance, the more metrologically useful a state N=2
- 64. 64/73 Uncertainty Relations Schwinger boson representation Recall Heisenberg Uncertainty Rel. Pairwise UncertaintyTrivial bounds Not SU(2) invariant Covariance matrix approach Three invariants: Determinant Sum of principal minors Trace - Non-trivial bounds - SU(2) invariant - Involves all three operators NOON stateSU(2) coherent state Heuristic no. 2: The higher the sum variance, the more metrologically useful a state N=2 All states on the boundary and inside correspond to physical states
- 65. 65/73 Uncertainty Relations Schwinger boson representation Recall Heisenberg Uncertainty Rel. Pairwise UncertaintyTrivial bounds Not SU(2) invariant Covariance matrix approach Three invariants: Determinant Sum of principal minors Trace - Non-trivial bounds - SU(2) invariant - Involves all three operators NOON stateSU(2) coherent state Heuristic no. 2: The higher the sum variance, the more metrologically useful a state N=2 All states on the boundary and inside correspond to physical states N=2 SU(2) coherent state NOON state ?
- 66. 66/73 Uncertainty Relations Schwinger boson representation Recall Heisenberg Uncertainty Rel. Pairwise UncertaintyTrivial bounds Not SU(2) invariant Covariance matrix approach Three invariants: Determinant Sum of principal minors Trace - Non-trivial bounds - SU(2) invariant - Involves all three operators NOON stateSU(2) coherent state Heuristic no. 2: The higher the sum variance, the more metrologically useful a state N=2 All states on the boundary and inside correspond to physical states N=2 SU(2) coherent state NOON state ?
- 67. 67/73 Uncertainty Relations Schwinger boson representation Recall Heisenberg Uncertainty Rel. Pairwise UncertaintyTrivial bounds Not SU(2) invariant Covariance matrix approach Three invariants: Determinant Sum of principal minors Trace - Non-trivial bounds - SU(2) invariant - Involves all three operators NOON stateSU(2) coherent state Heuristic no. 2: The higher the sum variance, the more metrologically useful a state N=2 All states on the boundary and inside correspond to physical states N=2 SU(2) coherent state NOON state ?
- 68. 68/73 Uncertainty Relations Schwinger boson representation Recall Heisenberg Uncertainty Rel. Pairwise UncertaintyTrivial bounds Not SU(2) invariant Covariance matrix approach Three invariants: Determinant Sum of principal minors Trace - Non-trivial bounds - SU(2) invariant - Involves all three operators NOON stateSU(2) coherent state Heuristic no. 2: The higher the sum variance, the more metrologically useful a state N=2 All states on the boundary and inside correspond to physical states N=2 SU(2) coherent state NOON state ?
- 69. 69/73 Uncertainty Relations Schwinger boson representation Recall Heisenberg Uncertainty Rel. Pairwise UncertaintyTrivial bounds Not SU(2) invariant Covariance matrix approach Three invariants: Determinant Sum of principal minors Trace - Non-trivial bounds - SU(2) invariant - Involves all three operators NOON stateSU(2) coherent state Heuristic no. 2: The higher the sum variance, the more metrologically useful a state N=2 All states on the boundary and inside correspond to physical states N=2 SU(2) coherent state NOON state ? N=2 PAPER V - S. Shabbir, G. Björk, Phys. Rev. A 93, 052101 NOON SU(2) coherent
- 70. 70/73 Uncertainty Relations NOON stateSU(2) coherent state N=2 All states on the boundary and inside correspond to physical states N=2 SU(2) coherent state NOON state ? N=2 PAPER V - S. Shabbir, G. Björk, Phys. Rev. A 93, 052101 NOON SU(2) coherent N=3 NOON SU(2) coherent
- 71. 71/73 Uncertainty Relations NOON stateSU(2) coherent state N=2 All states on the boundary and inside correspond to physical states N=2 SU(2) coherent state NOON state ? N=2 PAPER V - S. Shabbir, G. Björk, Phys. Rev. A 93, 052101 NOON SU(2) coherent N=3 NOON SU(2) coherent Extendible to higher N, but computationally intensive
- 72. 72/73 Summary N=2 All states on the boundary and inside correspond to physical states N=2 SU(2) coherent state NOON state ? N=2 PAPER V - S. Shabbir, G. Björk, Phys. Rev. A 93, 052101 NOON SU(2) coherent N=3 NOON SU(2) coherent Majorana Representation ARBITRARY INTERFERENCE Interferometry Angular momentum operators SU(2) Uncertainty Limits
- 73. 73/73 Acknowlegements Thank you! Gunnar Björk Marcin Swillo