INFRARED AND MILLIMETER WAVES: COHERENT SOURCES AND APPLICATIONS
CONTENTS 1 INTRODUCTION 2 DIRAC’S PHYSICS 3 THE EINSTEIN–PODOLSKY–ROSEN (EPR) PAPER 4 THE SCHRÖDINGER PAPERS 5 WHEELER’S PAPER 6 THE PROBABILITY AMPLITUDE FOR QUANTUM ENTANGLEMENT 7 THE QUANTUM ENTANGLEMENT EXPERIMENT 8 THE ANNIHILATION QUANTUM ENTANGLEMENT EXPERIMENTS 9 THE BOHM AND AHARONOV PAPER 10 BELL’S THEOREM 11 FEYNMAN’S HAMILTONIANS 12 THE SECOND WU QUANTUM ENTANGLEMENT EXPERIMENT 13 THE HIDDEN VARIABLE THEORY EXPERIMENTS 14 THE OPTICAL QUANTUM ENTANGLEMENT EXPERIMENTS 15 THE QUANTUM ENTANGLEMENT PROBABILITY AMPLITUDE 1947–1992 16 THE GHZ PROBABILITY AMPLITUDES 17 THE INTERFEROMETRIC DERIVATION OF THE QUANTUM ENTANGLEMENT PROBABILITY AMPLITUDE FOR N = N = 2 18 THE INTERFEROMETRIC DERIVATION OF THE QUANTUM ENTANGLEMENT PROBABILITY AMPLITUDE FOR N = N = 21, 22, 23, 24… 2R 19 THE INTERFEROMETRIC DERIVATION OF THE QUANTUM ENTANGLEMENT PROBABILITY AMPLITUDES FOR N = N = 3, 6 20 QUANTUM ENTANGLEMENT AT N = 1 AND N = 2 21 QUANTUM ENTANGLEMENT PROBABILITY AMPLITUDES APPLIED TO BELL’S THEOREM 22 QUANTUM ENTANGLEMENT VIA MATRIX NOTATION 23 CRYPTOGRAPHY VIA QUANTUM ENTANGLEMENT 24 QUANTUM ENTANGLEMENT AND TELEPORTATION 25 QUANTUM ENTANGLEMENT AND QUANTUM COMPUTING 26 SPACE-TO-SPACE AND SPACE-TO-EARTH COMMUNICATIONS VIA QUANTUM ENTANGLEMENT 27 SPACE-TO-SPACE QUANTUM INTERFEROMETRIC COMMUNICATIONS 28 QUANTA PAIR SOURCES FOR QUANTUM ENTANGLEMENT EXPERIMENTS 29 QUANTUM INTERFEROMETRIC PRINCIPLES 30 ON THE INTERPRETATION OF QUANTUM MECHANICS CHAPTER INTRODUCTION THE EXPERIMENTAL ORIGIN OF QUANTUM MECHANICS IS HIGHLIGHTED AND ESSENTIAL FEATURES OF THE PHOTON ARE DESCRIBED THE PHILOSOPHICAL PATH TO QUANTUM ENTANGLEMENT, AS DE?NED BY THE WORKS OF EPR ?BOHM AND AHARONOV ?BELL, AND THE PHYSICS PATH TO QUANTUM ENTANGLEMENT, AS DE?NED BY THE WORKS OF DIRAC ?WHEELER ?PRYCE AND WARD, ARE INTRODUCED AND DISCUSSED THE PROBABILITY AMPLITUDE FOR QUANTUM ENTANGLEMENT ? ??= ? ??? ? X Y Y X ( , , ) 2 , DISCOVERED IN 1947, IS THEN INTRODUCED PRIOR TO A HIGH-LEVEL OVERVIEW OF THE ?ELD THE INTRODUCTION THEN CONCLUDES WITH A CHAPTER- BY-CHAPTER DESCRIPTION OF THIS BOOK AND ITS AIMS 1 1 INTRODUCTION FEW SUBJECTS IN PHYSICS EVOKE THE INTEREST AND WONDER CREATED BY QUANTUM ENTANGLEMENT THE INTEREST IS INTRINSIC, FUELED BY INHERENT CURIOSITY IN EVERYTHING ‘QUANTUM’ AUTHORS AND WRITERS, WITHIN AND OUTSIDE PHYSICS, TRYING TO EXPLAIN QUANTUM ENTANGLEMENT EFFECTS USING WORDS SUCH AS ‘STRANGE’, ‘MYSTERIOUS’, AND EVEN ‘WEIRD’, ADD TO THE MYSTIQUE THE AIM OF THIS MONOGRAPH IS TO ILLUMINATE, AND ELUCIDATE, THE PHYSICS AND THE FUNDAMENTALS OF QUANTUM ENTANGLEMENT TO SCIENTISTS, ENGINEERS, AND EVEN PHYSICISTS ALL THAT IS REQUIRED IS A GOOD BACKGROUND IN ?RST YEAR PHYSICS AND ?RST YEAR MATHEMATICS, MATRIX ALGEBRA IN PARTICULAR THIS SECOND EDITION OF FUNDAMENTALS OF QUANTUM ENTANGLEMENT OFFERS A REVISED, IMPROVED, AND ENLARGED VERSION OF THE ORIGINAL (DUARTE 2019) WITH A TOTAL OF 30 CHAPTERS AND 10 APPENDICES A DISTINCTION WITH THE ?RST EDITION IS THAT NOW NOT ALL CHAPTERS ARE BRIEF SINCE SOME CHAPTERS HAVE BEEN EXTENDED AS NEEDED TWO NEW CHAPTERS ARE INCLUDED, CHAPTERS 22 AND 29 CHAPTER 22 IS TITLED ‘QUANTUM ENTANGLEMENT VIA MATRIX NOTATION’ AND PRESENTS A UTILITARIAN EXPOSITION ON THE MATRIX AND TENSOR REPRESENTATION OF THE QUANTUM ENTANGLEMENT PROBABILITY AMPLITUDES AS DEVELOPED SINCE 2019 (DUARTE ET AL 2020, DUARTE AND TAYLOR 2021) CHAPTER 29 IS TITLED ‘QUANTUM INTERFEROMETRIC PRINCIPLES’ AND PROVIDES A TRANSPARENT EXPOSITION OF QUANTUM MECHANICAL PRINCIPLES FROM AN INTERFEROMETRIC PERSPECTIVE IN THE COMPLETE ABSENCE OF MYSTERIES OR PARADOXES 10 34 JS IS PLANCK’S CONSTANT THIS EQUATION LINKS HIGH FREQUENCIES, OR WAVELENGTHS TOWARD THE BLUE END OF THE SPECTRUM, WITH HIGH ENERGIES ON THE OTHER HAND, LOW FREQUENCIES, OR WAVELENGTHS TOWARD THE RED END OF THE SPECTRUM, ARE ASSOCIATED WITH LOW ENERGIES PLANCK’S ENERGY EQUATION AND H ARE DISCUSSED FURTHER IN APPENDIX J THERE ARE SEVERAL EQUIVALENT WAYS TO APPROACH QUANTUM MECHANICS, THE MOST PREVALENT BEING: 1 HEISENBERG’S MATRIX MECHANICS (BORN AND JORDAN 1925, BORN ET AL 1926, HEISENBERG 1925) 2 SCHRÖDINGER’S EQUATION (SCHRÖDINGER 1926) 3 DIRAC’S BRA KET NOTATION (DIRAC 1939) 4 MOYAL’S STATISTICS (MOYAL 1949) 5 POSITION, MOMENTUM, AND ENERGY OPERATORS (DIRAC 1958, FEYNMAN ET AL 1965) 6 FEYNMAN’S PATH INTEGRALS (FEYNMAN AND HIBBS 1965) THE PRESENTATION IN THIS MONOGRAPH IS ENTIRELY BASED ON DIRAC’S BRA KET NOTATION (DIRAC 1939, 1958) THIS NOTATION IS EXPLAINED IN CHAPTER 2 1 2 1 THE MATHEMATICAL BASES OF QUANTUM MECHANICS FOLLOWING FEYNMAN’S LEAD AND USING A MINIMUM OF WORDS, HERE THE ESSENTIAL QUANTUM MECHANICAL PRINCIPLES NECESSARY TO UNDERSTAND QUANTUM INTERFERENCE AND QUANTUM ENTANGLEMENT ARE PRESENTED: 1 THEOREM: THE PROBABILITY AMPLITUDE FOR A SINGLE QUANTUM IS MATHEMATICALLY IDENTICAL TO THE PROBABILITY AMPLITUDE FOR AN ENSEMBLE OF INDISTINGUISHABLE QUANTA 2 THE PROBABILITY AMPLITUDE FOR A SINGLE QUANTUM, OR AN ENSEMBLE OF R INDISTINGUISHABLE QUANTA, IS GIVEN BY THE DIRAC–FEYNMAN INTERFEROMETRIC PRINCIPLE (FEYNMAN ET AL 1965) ? ???= ?????? = D S D J J S (1 2) I N INFRARED AND MILLIMETER WAVES THE N IN THIS SUPERPOSITION SERIES MEANS THAT EVERY PROPAGATION PATH MUST BE INCORPORATED IN THE CALCULATION 3 THE PRINCIPLE OF SUPERPOSITION: ‘ANY STATE MAY BE CONSIDERED AS THE RESULT OF A SUPERPOSITION OF TWO OR MORE OTHER STATES… CONVERSELY ANY TWO OR MORE STATES MAY BE SUPERPOSED TO GIVE A NEW STATE’ (DIRAC 1958) FEYNMAN REFERS TO THE J STATES AS BASE STATES… ‘ANY QUANTUM MECHANICAL STATE CAN BE DESCRIBED COMPLETELY BY THE AMPLITUDE ??? J S TO GO INTO THE BASE STATES… ANY STATE IN THE WORLD CAN BE EXPRESSED AS A SUPERPOSITION… OF BASE STATES’ (FEYNMAN ET AL 1965) THE DIRAC–FEYNMAN INTERFEROMETRIC PRINCIPLE, EXPRESSED IN EQUATION (1 2), IS INHERENTLY LINKED TO THE PRINCIPLE OF SUPERPOSITION FURTHER EXAMPLES OF SUPERPOSITION ARE THE INTERFEROMETRIC PROBABILITY AMPLITUDES ? ? ? ??= ??+ ?? + ? ? ? ? ??= ????? ? ? 4 THE PROBABILITY AMPLITUDES ARE MATHEMATICALLY REPRESENTED BY COMPLEX WAVE FUNCTIONS OF THE FORM (DIRAC 1958) ? = ? ? ? ? X T E ( , ) (1 5) I T KX ( ) OR MORE SPECI?CALLY ???= ? ? ? ? J S R E ( ) (1 6) J S I ???= ? ? ? ? D J R E ( ) (1 7) D J I IMAGINARY NUMBERS AND COMPLEX NUMBERS PLAY A CRUCIAL ROLE IN QUANTUM MECHANICS 5 ADDITIONAL DIRAC PRINCIPLES AS OUTLINED BY FEYNMAN ET AL (1965): ???= ??? ? D S S D (1 8) ? ???= J I (1 9) JI 6 REVERSIBILITY: THE PHYSICS OF QUANTUM MECHANICS IS REVERSIBLE FOR INSTANCE, THE PROBABILITY AMPLITUDE FOR QUANTUM ENTANGLEMENT CAN BE DERIVED FROM THE DIRAC–FEYNMAN PRINCIPLE (DUARTE 2013A, 2013B, 2014, 2019): ? ? ???= ?????????= ????± ???? = ? D S D J J S X Y Y X ( ) (1 10) J N OR VICE VERSA (DUARTE 2019) ? ? ??= ????± ???? ????= ?????? = ? X Y Y X D S D J J S ( ) (1 11) J N 7 DIRAC’S IDENTITY FOR INDISTINGUISHABLE QUANTA IN DIFFERENT STATES (DIRAC 1958): ? ?= ?????? ?? X A B C G (1 12) N 8 NONLOCALITY: ‘PHOTONS CANNOT BE LOCALIZED… AND THEY DO NOT BEHAVE AT ALL LIKE PARTICLES’ (LAMB 1995) THIS IS TOTALLY COMPATIBLE WITH HEISENBERG’S UNCERTAINTY PRINCIPLE 9 HEISENBERG’S UNCERTAINTY PRINCIPLE (HEISENBERG 1927): ? ? ? P X H (1 13) 10 BORN’S RULE: THE PROBABILITY IS CALCULATED BY MULTIPLYING THE PROBABILITY AMPLITUDE WITH ITS COMPLEX CONJUGATE (BORN ET AL 1926): ?????= ?????? ? D S D S D S (1 14) PROBABILITY AMPLITUDES SUCH AS ??? D S ARE COMPLEX NUMBERS THE STRAIGHT- FORWARD MULTIPLICATION OF THE PROBABILITY AMPLITUDE WITH ITS COMPLEX CONJUGATE VIA BORN’S RULE ?????? ? D S D S PROVIDES A PROBABILITY IN THE RANGE ???????? ? D S D S 1 11 MEASURABLES: THE MEASURED INTENSITY, AT A GIVEN FREQUENCY, IS PROPORTIONAL TO THE QUANTUM PROBABILITY ??????? ? I D S D S (1 15) PROBABILITIES AND POLARIZATION ARE ALSO MEASURABLE QUANTITIES THESE LAWS, AND/OR PRINCIPLES, OF QUANTUM MECHANICS PROVIDE THE FOUNDATIONS OF THE KNOWLEDGE NEEDED TO QUANTIFY GENERALIZED QUANTUM INTERFERENCE AND QUANTUM ENTANGLEMENT 1 2 2 THE PHOTON FROM A QUANTUM PERSPECTIVE THE CONCEPT OF A SINGLE PHOTON, OR QUANTUM, IS EXTRAORDINARILY IMPORTANT TO THE ?ELD OF QUANTUM ENTANGLEMENT EQUALLY IMPORTANT ARE CORRELATED PAIRS OF PHOTONS OR CORRELATED PAIRS OF QUANTA IN THIS SUBSECTION RELEVANT THOUGHTS AND RE?ECTIONS ON THE PHOTON BY THREE OF THE BRIGHTEST LUMINARIES IN THE ?ELD OF QUANTUM PHYSICS ARE INCLUDED IN AN EFFORT TO PROVIDE A MANIFOLD OF USEFUL CONCEPTS ON THE PHOTON: ‘THE WAVE FUNCTION GIVES INFORMATION ABOUT THE PROBABILITY OF ONE PHOTON BEING IN A PARTICULAR PLACE AND NOT THE PROBABLE NUMBER OF PHOTONS IN THAT PLACE… EACH PHOTON THEN INTERFERES ONLY WITH ITSELF INTERFERENCE BETWEEN TWO DIFFERENT PHOTONS NEVER OCCURS ’ (DIRAC 1958) ‘NEWTON THOUGHT THAT LIGHT WAS MADE UP OF PARTICLES, BUT THEN IT WAS DISCOVERED THAT IT BEHAVES LIKE A WAVE… WE SAY: IT IS LIKE NEITHER ’ (FEYNMAN ET AL 1965) ‘PHOTONS CANNOT BE LOCALIZED IN ANY MEANINGFUL MANNER, AND THEY DO NOT BEHAVE AT ALL LIKE PARTICLES, WHETHER DESCRIBED BY A WAVE FUNCTION OR NOT ’ (LAMB 1995) INFRARED AND MILLIMETER WAVES AND ? ?? = A SINGLE PHOTON EXHIBITS A QUANTUM ‘MOMENTUM’ OF (DE BROGLIE 1924) = ? P K, (1 18) WHERE ? ? = K 2 / IS KNOWN AS THE WAVE NUMBER AS MENTIONED BY DIRAC (1958) A SINGLE PHOTON IS ASSOCIATED WITH COMPLEX WAVE FUNCTIONS OF THE FORM ? ? = ? ? ? X T E ( , ) I T KX ( ) AND THESE COMPLEX WAVE FUNCTIONS ARE USED TO PROVIDE A MATHEMATICAL REPRESENTA- TION OF THE PROBABILITY AMPLITUDES FURTHERMORE, IT CAN BE ESTABLISHED FROM HEISENBERG’S UNCERTAINTY PRINCIPLE ? ? ? P X H (DIRAC 1958) THAT SINGLE PHOTONS CAN BE EXTREMELY NON-LOCAL AND CAN EXHIBIT ENORMOUS COHERENCE LENGTHS AS DESCRIBED BY ? ? ? ? X C (1 19) THE EXTREME NONLOCALITY COMES FROM THE FACT THAT A SINGLE QUANTUM CAN EXHIBIT AN EXTRAORDINARILY NARROW-LINEWIDTH ? ? AS IT WILL BE SEEN FURTHER IN THIS MONOGRAPH, THIS CONDITION OF NONLOCALITY IS CENTRAL TO QUANTUM ENTANGLEMENT ALSO, ‘ENSEMBLES OF INDISTINGUISHABLE PHOTONS EXHIBITING VERY NARROW-LINEWIDTH ? ? ORIGINATING FROM NEARLY MONOCHROMATIC SOURCES, SUCH AS NARROW-LINEWIDTH LASERS, APPROXIMATE THE BEHAVIOR OF A SINGLE-PHOTON’ (DUARTE 2014) NONLOCALITY AND QUANTUM ENTANGLEMENT CO-EXIST AS DESCRIBED IN THE FOLLOWING STATEMENT: ‘THE TWO PHOTONS ARE ENTANGLED AND ACCORDING TO LOCAL REALISM, THEIR POLARIZATIONS PLANES SHOULD BECOME INDEPENDENT… A TYPICAL EPR SITUATION ALREADY INFRARED AND MILLIMETER WAVES