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Quantum Optics Leture at the Optics and Photonics Winter School Poul Jessen Brian Anderson Rolf Binder Jason Jones Galina Khitrova Miroslav Kolesik Masud Mansuripur Jerome Moloney Ewan Wright What is Quantum Optics? – Not classical optics, but something more and different – Science of non-classical light – Any science combining light and quantum mechanics What is Light? – Electromagnetic waves? – Photons? tour sign in / Author / ISBN What is Quantum Optics? Home My Books Friends Recommendations Explore Genres Listopia Giveaways Popular Goodreads Voice Ebooks Fun Trivia Quizzes Quotes Community Groups Creative Writing People Events dreads helps you keep track of books you want to read. by marking “Dancing with Photons” as Want to Read: nt to Read saving… Want to Read Currently Reading Read Not so fast! © Springer-Verlag 1995 Not so fast! Anti-photon W.E. Lamb, Jr. Optical Sciences Center, Universityof Arizona, Tucson, AZ 85721, USA Received: 23 July 1994/Accepted: 18 September 1994 Abstract. It should be apparent from the title of this article that the author does not like the use of the word "photon", which dates from 1926. In his view, there is no such thing as a photon. Only a comedy of errors and historical accidents led to its popularity among physicists and optical scientists. I admit that the word is short and convenient. Its use is also habit forming. Similarly, one might find it convenient to speak of the "aether" or "vacuum" to stand for empty space, even if no such thing existed. There are very good substitute words for "photon", (e.g., "radiation" or "light"), and for "photonics" (e.g., "optics" or "quantum optics"). Similar objections are possible to use of the word "phonon", which dates from 1932. Objects like electrons, neutrinos of finite rest mass, or helium atoms can, under suitable conditions, be considered to be particles, since their theories then have viable non-relativistic and non-quantum limits. This paper outlines the main features of the quantum theory of radiation and indicates how they can be used to treat problems in quantum optics. PACS: 12.20.-m; 42.50.-p afterward, there was a population explosion of people engaged in fundamental research and in very useful technical and commercial developments of lasers. QTR was available, but not in a form convenient for the problems at hand. The photon concepts as used by a high percentage of the laser community have no scientific justification. It is now about thirty-five years after the making of the first laser. The sooner an appropriate reformulation of our educational processes can be made, the better. 1 A short history of pre-photonic radiation Modern optical theory [2] began with the works of Ch. Huyghens and I. Newton near the end of the seventeenth century. Huyghen's treatise on wave optics was published in 1690. Newton's "Optiks', which appeared in 1704, dealt with his corpuscular theory of light. A decisive work in 1801 by T. Young, on the two-slit diffraction pattern, showed that the wave version of optics was much to be preferred over the corpuscular form. However, so high was the prestige of I. Newton, that the With all due resepect to W. Lamb, let us try again What is light? – a wave? – a stream of particles (photons)? Take the question seriously – test each hypothesis through experimentation! Key signature of wave behavior? – Interference! Double-slit experiment Single-slit diffraction Double-slit diffraction Key signature of particle behavior? Einstein: photo-electric effect Electrons are released only for light with a frequency ! such that h! is greater than the work function of w the metal in question But the quantum theory of electron excitation can explain this based on classical electromagnetic fields, so the photo-electric effect only confirms that electrons are particles. BS Indivisibility single photon input A particle incident on a barrier is either transmitted or reflected. or single photon input Wave behavior BS M single photon input PD1 PD2 prob. of detection PD1 ! BS PD2 M phase shifter ! Particle behavior prob. of detection PD1 PD2 M single photon input BS PD1 PD2 ! M phase shifter ! – Evidently a single photon can behave like a wave or a particle, depending on the experiment we do. This is what we know as wave-particle duality. – Does the photon “know” when it hits the first BS if we are doing a wave or particle experiment and then behaves accordingly? – Wigner’s gedanken experiment: Delayed Choice! Decide at random whether to put in the second BS only after the photon has passed the first BS Wave behavior M single photon input BS Particle behavior BS PD1 PD2 M ! M phase shifter PD1 PD2 single photon input BS ! M phase shifter Wigners experiment was done in 2008 PRL 100, 220402 (2008) week ending 6 JUNE 2008 PHYSICAL REVIEW LETTERS Delayed-Choice Test of Quantum Complementarity with Interfering Single Photons Vincent Jacques,1 E Wu,1,2 Frédéric Grosshans,1 François Treussart,1 Philippe Grangier,3 Alain Aspect,3 and Jean-François Roch1,* 1 Laboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan, UMR CNRS 8537, Cachan, France 2 State Key Laboratory of Precision Spectroscopy, East China Normal University, Shanghai, China 3 Laboratoire Charles Fabry de l’Institut d’Optique, UMR CNRS 8501, Palaiseau, France (Received 12 February 2008; published 3 June 2008) We report an experimental test of quantum complementarity with single-photon pulses sent into a Mach-Zehnder interferometer with an output beam splitter of adjustable reflection coefficient R. In addition, the experiment is realized in Wheeler’s delayed-choice regime. Each randomly set value of R allows us to observe interference with visibility V and to obtain incomplete which-path information characterized by the distinguishability parameter D. Measured values of V and D are found to fulfill the complementarity relation V 2 ! D2 " 1. DOI: 10.1103/PhysRevLett.100.220402 As emphasized by Bohr [1], complementarity lies at the heart of quantum mechanics. A celebrated example is the illustration of wave-particle duality by considering single particles in a two-path interferometer [2], where one chooses either to observe interference fringes, associated to a wavelike behavior, or to know which path of the interferometer has been followed, according to a particlelike behavior [3]. Although interference has been observed at the individual particle level with electrons [4], neutrons [5], atoms [6,7], molecules [8], only a few experiments with massive particles have explicitly checked the mutual exclusiveness of which-path information (WPI) and interference [9–13]. PACS numbers: 03.65.Ta, 42.50.Ar, 42.50.Xa anced interferometer as used in some neutron interferometry experiments, one has partial WPI while keeping interference with limited visibility [24]. The complementary quantities —WPI and interference visibility —could then be partially determined simultaneously. Consistent theoretical analysis of both schemes, independently published by Jaeger et al. [25] and by Englert [26] leads to the inequality [27] V 2 ! D2 " 1 (1) which puts an upper bound to the maximum values of simultaneously determined interference visibility V and where V" is the half-wave voltage of the EOM. The parameters ! and V" have been independently measured for our experimental conditions and found equal to ! ! 24 " 1# and V" ! 217 " 1 V at the wavelength # ! 670 nm which is the emission peak of the negatively charged N-V color center [32]. This allows R to vary between 0 and 0.5 when VEOM is varied between 0 and 170 V. When R ! 0, the VBS is equivalent to a perfectly transparent (or absent) beam splitter. Then, each ‘‘click’’ of one of the two photodetectors (P1 or P2 ) placed on the output EOM [see Eq. (2)]. In the laboratory framework, the random choice of VBS reflectivity is realized simultaneously with the entering of the corresponding emitted photon into the interferometer, ensuring the required spacelike separation [22]. As meaningful illustration of complementarity requires the use of single particles, the quantum behavior of the light field is first tested using the two output detectors feeding single and coincidence counters with no voltage applied to the EOM. The output beam splitter is then absent, and each detector is therefore univocally associated Wigners experiment was done in 2008 Clock 4.2 MHz QRNG VEOM S P FIG. 1 (color online). Delayed-choice complementarity-test experiment. A single-photon pulse is sent into a MachZehnder interferometer, composed of a 50=50 input beam splitter (BS) and a variable output beam splitter (VBS). The reflection coefficient is randomly set either to the null value or to an adjustable value R, after the photon has entered the interferometer. The single-photon photodetectors P1 and P2 allow to record both the interference and the WPI. EOM Path 1 Path 2 P2 WP PBS Φ P1 VBS FIG. 2 (color online). Variable output beam splitter (VBS) implementation. The optical axis of the polarization beam splitter (PBS) and the polarization eigenstates of the Wollaston prism (WP) are aligned, and make an angle ! with the optical axis of the EOM. The voltage VEOM applied to the EOM is randomly chosen accordingly to the output of a Quantum Random Number Generator (QRNG), located at the output of the interferometer and synchronized on the 4.2-MHz clock that triggers the singlephoton emission. 220402-2 1000 (c) R=0 Counts tion beam splitter PBS of VBS with a piezoelectric actuator (see Fig. 2). For each photon, we record the chosen configuration of the interferometer, the detection events, and the actuator position. All raw data are saved in real time and are processed only after a run is completed. The events corresponding to each configuration of the interferometer are finally sorted. For a given value R, the wavelike inforweek ending mation the L REVIE W L of E Tthe T Elight R S field is obtained by measuring visibility of the interference, predicted to be 6 JUNE 2008 1000 p!!!!!!!!!!!!!!!!!!! (a) R=0.43 elation V # 2 R!1 % R": (3) second n ideal The500 results, depicted in Fig. 3, show a reduction of V when perfect the randomly applied value of R decreases. clelike To test inequality (1), a value of the distinguishability D usly in is then0required, to qualitatively qualify the amount of WPI e [32], 4π which can be πextracted for2π each value of3πR. We introduce Phase shift Φ (rad) than 1, the quantity D1 (resp. D2 ), associated to the WPI on path 1 1000path 2): photon (resp. (b) R=0.05 D1 # jp!P1 ; path 1" % p!P2 ; path 1"j (4) single500 ndomly correD2 # jp!P1 ; path 2" % p!P2 ; path 2"j (5) and R he two folwhere 0p!Pi ; path j" πis the probability 2π that the particle 3π olarizalows path j and is detected onshift detector Phase Φ (rad)Pi . For a single ctuator particle arriving on the output beam splitter, one obtains 1000 n con" " (c) R=0 " " " " 1 " " ts, and % R" D1 # D2 # " : (6) " " " " " " al time 2 500 events The distinguishability parameter D is finally defined as ometer [26] e infor0 π 2π 3π ng the D # D1 & D2 # j1 Φ %(rad) 2Rj: (7) Phase shift 500 Counts Counts Counts Wigners experiment was done in 2008 (3) In order to test this relation, we visibility estimate Vthe values in of the D1 FIG. 3 (color online). Interference measured and D2 by blocking path of the interferometer and delayed-choice regime one for different values of VEOM . (a)– ' 150 V N(R # 0:43 and V # (c) correspond to VEOM measuring the number of detections and N on detectors 1 2 0 π 2π Phase shift Φ (rad) 3π FIG. 3 (color online). Interference visibility V measured in the delayed-choice regime for different values of VEOM . (a)– (c) correspond to VEOM ' 150 V (R # 0:43 and V # 93 $ 2%), VEOM ' 40 V (R # 0:05 and V # 42 $ 2%), and VEOM # 0 (R # 0 and V # 0). Each point is recorded with 1.9 s acquisition time. Detectors dark counts, corresponding to a rate of 60 s%1 for each, have been substracted to the data. # 1 jN1 % N2 j D1 # is both 2 N1 &aN2particle path 2 blocked Light and a(8) wave at the same time. # 1 jN1 % N2 j D2 # : (9) 2 N1 & N2 path 1 blocked What property we see depends on These measurements are also performed in the delayedwhat we described decideabove.toWe choice regime,property using the procedure finally obtain independent measurements of D and V for measure. different values of the reflection coefficient R, randomly to the interferometer. The final results, depicted on applied 2 Fig.This 4, lead to & D2 # 0:97 0:03, close to the maxiisV totally in $line with our mal value permitted by inequality (1) even though each general quantum quantity varies from 0 to 1. theory. The effects observed in this delayed-choice experiment are in perfect agreement with quantum mechanics predictions. No change is observed between a so-called ‘‘normalchoice’’ experiment and the ‘‘delayed-choice’’ version. It demonstrates that the complementarity principle cannot be interpreted in a naive way, assuming that the photon at the BTW, it works for ultracold atoms too! We need a quantum theory of light where behaviors such as wave-particle duality is built in – A formal procedure exists for obtaining a quantum theory from a known classical theory based on Lagrange – Hamilton formalism. – One cannot “prove” that this procedure is correct. It is justified only by repeated observation that quantum theories obtained in this fashion “work”, in the sense that their predictions agree with experiment. – This should not surprise us. Quantum Mechanics contains new physics that is absent from Classical Mechanics and cannot be derived from it. – An unfortunate consequence is that we often seem to pull ideas out of thin air when we teach quantum mechanics. The problem gets worse when we try to do things quickly, e. g., a 50 minute lecture on the quantum theory of light. Quantization of the Electromagnetic Field Starting point: Maxwell’s equations from classical Electromagnetism 1 ! " E(r,t) = $ (r,t) #0 ! " B(r,t) = 0 ' ! % E(r,t) = & B(r,t) 't 1 ' 1 ! % B(r,t) = 2 E(r,t) + j(r,t) 2 c 't # 0c In Quantum Electrodynamics (QED) the ME’s are still valid, but the fields E and B become operators that depend on space and time just like the classical electric and magnetic fields. For simplicity we consider empty space without charges or currents, and use the two last Maxwell equations to derive a wave equation $ 2 1 #2 ' &%! " c 2 #t 2 )( E(r,t) = 0 # !2 1 !2 & % !z 2 " c 2 !t 2 ( E(z,t) = 0 $ ' paraxial waves propagating along the z-axis Electromagnetic Field in a 1-Dimensional Cavity Cavity of length L, cross section A, and volume V = L A, field E = x E x x z y M M Any field inside the cavity can be expressed as a superposition of the cavity normal modes, which are standing wave solutions to the wave equation with nodes on the mirror surfaces. ! E x (z,t) = " A j q j (t)sin(k j z) , j=1 ! j j" kj = = , c L 2! 2j m j Aj = " 0V Here q j (t) are the time varying amplitudes for the different modes, and A j is chosen so E x has units of electric field (V/m) and q j (t) has units of length (m). Each normal mode is an independent degree of freedom that can be quantized by itself. For mode number j we have electric and magnetic fields (the latter can be found from E x using Maxwells equations) E x( j ) (z,t) = A j q j (t)sin(k j z), By( j ) (z,t) = Aj d ! ! q (t)cos(k z), q (t) = q j (t) j j 2 j k jc dt Using classical E & M we can show that the energy of the field in mode j is H = ! 0 " dr 3 ( E 2 + c 2 B 2 ) V & !0A L # 2 A 2j 2 2 2 2 = dz A j q j (t) sin (k j z) + 2 q! j (t) cos (k j z) ( 2 "0 %$ kj ' = 1 1 m j) 2j q 2j + m j q! 2j 2 2 Writing the field energy in this form is highly suggestive. Harmonic Oscillator 1 1 2 2 2 H = m! x + mv 2 2 1 1 2 = m! 2 x 2 + p 2 2m EM field in mode j Vs. H= 1 1 m j! 2j q 2j + m j q! 2j 2 2 Quantum Theory Ê x ! q j " q̂ j x ! x̂ = x p ! p̂ = "i! [ x̂, p̂ ] = i! d dx B̂y ! m j q! j " p̂ j [ q̂ j , p̂ j ] = i!, !" Ê x , B̂y #$ % 0 Postulate (leap of faith): EM field in a normal mode is a harmonic oscillator e n u c le u a c la s s ic s Quick review of the Quantum Harmonic Oscillator x Standard paradigm: mass on a spring ! = K /m n e K ! ! Observables: x̂, p̂ Hamiltonian:! Ĥ = m 1 1 2 m! 2 x̂ 2 + p̂ 2 2m origin Energy eigenvalues and eigenfunctions: N En = !! (n + 1 / 2), no"t! 0e: Ĥ! n (z) = En! n (z) " V (x) ! W e shou ld regard E = !! (n + 1 / 2) a c la s s ic th is a s a # $ x /2 al m o d e ! n (z) " e H n ($ x) l ofE t=h!!e(3 /a2) model to m its e !E = !" H n : Hermite polynomial lf E = !! (1 / 2) W e can ju s tif y th is re can n 2 1 0 EXPLORE SEARCH SEARCH EXPLORE LATEST ABOUT EXPLORE SEARCH LATEST LATEST ABOUT ABOUT Harmonic Oscillator Eigenfunctions Harmonic Oscillator Eigenfunctions Harmonic Oscillator Eigenfunctions Energy eigenfunctions (stationary states): look at probability density ! n (x) 2 Files require Files require Files require Related Demonstrations Related Demonstrations More by Author More by Author Related Dem More by Au n=5 n=0 n = 30 Absolute value of the harmonic oscillator eigenfunctions. The harmonic oscillator is the most important exactly solvable model of quantum mechanics. The ground state eigenfunction minimizes the uncertainty product. With increasing quantum Absolute value of the harmonic oscillator eigenfunctions. The harmonic oscillator is the most important exactly solvable Absolute value of the harmonic oscillator eigenfunctions. The harmonic oscillator is the most important exactly solvable Topics Related Topics number, the square ofRelated the absolute value of the eigenfunctions approaches the probability distribution of a classical particle model of quantum The ground state eigenfunction minimizes the uncertainty product. With increasing quantum Chemistry model of quantum mechanics. The ground state eigenfunction minimizes the uncertainty product. With increasing quantum mechanics. Chemistry in a harmonic potential with inverse square root singularities at the turning points. theparticle square ofCollege the absolute number, the square of the absolute value of the eigenfunctions approaches the probability distribution of number, a classical Physics value of the eigenfunctions approaches the probability distribution of a classical particle College Physics in a harmonic potential with inverse square root singularities at the turning points. in a harmonic potential with inverse square root singularities at the turning points. General Chemistry General Chemistry Mechanics Mechanics Physical Chemistry Physical Chemistry Mean position in state ! n (x) Contributed by: Michael Trott Mechanics Quantum Quantum Mechanics Unique features: Contributed by: Michael Trott http://demonstrations.wolfram.com/HarmonicOscillatorEigenfunctions/ Quantum Physics Quantum Physics Contributed by: Michael Trott Quantum fluctuations Heisenberg uncertainty relation Page 1 of 6 !xn = Chemistry College Phy General Ch Mechanics Physical Ch Quantum M Quantum Ph x̂ = 0 http://demonstrations.wolfram.com/HarmonicOscillatorEigenfunctions/ http://demonstrations.wolfram.com/HarmonicOscillatorEigenfunctions/ Page 1 of 6 Related T ! n +1/ 2 m" !x!p = !(n + 1 / 2) Page 1 of 6 Creation and annihilation operators Introduce dimensionless variables Define Can show that X̂ = x̂ m! 2!, P̂ = p̂ â = X̂ + iP̂ annihilation operator â † = X̂ ! iP̂ creation operator ! n"1 # â ! n ! n+1 " â †! n annihilation/creation of an excitation â † â 2m!! ! n+1 !n ! n"1 â †â ! n = n ! n (# of excitations) What is a “phonon”? A quantum of exitation in a Harmonic Oscillator Back to… Harmonic Oscillator 1 1 2 2 2 H = m! x + mv 2 2 1 1 2 = m! 2 x 2 + p 2 2m EM field in mode j Vs. H= 1 1 m j! 2j q 2j + m j q! 2j 2 2 Quantum Theory Ê x ! q j " q̂ j x ! x̂ = x p ! p̂ = "i! [ x̂, p̂ ] = i! d dx B̂y ! m j q! j " p̂ j [ q̂ j , p̂ j ] = i!, !" Ê x , B̂y #$ % 0 Postulate (leap of faith): EM field in a normal mode is a harmonic oscillator x QED paradigm: normal mode j in cavity z y Observables: q̂ (! Ê x ), p̂ (! B̂y ) Hamiltonian: Ĥ = Energy eigenvalues and eigenfunctions: En = !! (n + 1 / 2), n " 0 M 1 1 2 m! 2 q̂ 2 + p̂ 2 2m Ĥ! n = En! n " V (x) En = !! (n + 1 / 2) Number states ! n No “wavefunction”, use Dirac notation ! n " ! n M !E = !" E1 = !! (3 / 2) E0 = !! (1 / 2) Creation and annihilation operators Q̂ = q̂ m! 2!, Introduce dimensionless variables Define Can show that P̂ = p̂ â = Q̂ + iP̂ annihilation operator â † = Q̂ ! iP̂ creation operator ! n"1 # â ! n ! n+1 " â ! n † annihilation/creation of an excitation â † â 2m!! ! n+1 !n ! n"1 â †â ! n = n ! n number states What is a “photon”? A quantum of exitation in a Normal Mode of the EM field Photons as particles Standing wave normal modes ! photons are delocalized in space We can make superpositions of standing waves that correspond to wavepackets & use these as our normal modes ! photons become localized in space x z y M M It is in this sense than we can talk about, e. g, a photon traveling along a specific path in an interferometer, as in the first part of the lecture. More about number states (Foch states): Mean field in state ! n Quantum fluctuations Ê x ! q̂ = 0 !q̂n = q0 n + 1 / 2 " (!Ê x )n = E0 n + 1 / 2 !q̂n=0 = q0 / 2 " Vacuum fluctuations (!Ê x )n=0 = E0 / 2 Does a laser emit light in a number state with a well defined number of photons? Back to… Harmonic Oscillator 1 1 2 2 2 H = m! x + mv 2 2 1 1 2 = m! 2 x 2 + p 2 2m EM field in mode j Vs. H= 1 1 m j! 2j q 2j + m j q! 2j 2 2 Quantum Theory Ê x ! q j " q̂ j x ! x̂ = x p ! p̂ = "i! [ x̂, p̂ ] = i! d dx B̂y ! m j q! j " p̂ j [ q̂ j , p̂ j ] = i!, !" Ê x , B̂y #$ % 0 Postulate (leap of faith): EM field in a normal mode is a harmonic oscillator EXPLORE SEARCH LATEST ABOUT EXPLORE SEARCH LATEST ABOUT Harmonic Oscillator Eigenfunctions Harmonic Oscillator Eigenfunctions Harmonic Oscillator Eigenfunctions Number states are highly non-classical – look at the probability density ! n (x) 2 Files require Files require Files require Related Demonstrations Related Demonstrations More by Author More by Author Related Dem More by Au n=5 n=0 n = 30 Absolute value of the harmonic oscillator eigenfunctions. The harmonic oscillator is the most important exactly solvable model of quantum mechanics. The ground state eigenfunction minimizes the uncertainty product. With increasing quantum Absolute value of the harmonic oscillator eigenfunctions. The harmonic oscillator is the most important exactly solvable Absolute value of the harmonic oscillator eigenfunctions. The harmonic oscillator is the most important exactly solvable Topics Related Topics number, the square ofRelated the absolute value of the eigenfunctions approaches the probability distribution of a classical particle model of quantum The ground state eigenfunction minimizes the uncertainty product. With increasing quantum Chemistry model of quantum mechanics. The ground state eigenfunction minimizes the uncertainty product. With increasing quantum mechanics. Chemistry in a harmonic potential with inverse square root singularities at the turning points. theparticle square ofCollege the absolute number, the square of the absolute value of the eigenfunctions approaches the probability distribution of number, a classical Physics value of the eigenfunctions approaches the probability distribution of a classical particle College Physics in a harmonic potential with inverse square root singularities at the turning points. in a harmonic potential with inverse square root singularities at the turning points. General Chemistry General Chemistry Physical Chemistry Physical Chemistry Contributed by: Michael Trott Mechanics Quantum http://demonstrations.wolfram.com/HarmonicOscillatorEigenfunctions/ http://demonstrations.wolfram.com/HarmonicOscillatorEigenfunctions/ Page 1 of 6 EXPLORE LATEST ABOUT SEARCH 1/4/16 7:45 PM Files require EXPLORE LATEST ABOUT quasi-classical state Related Demonstrations More by Author Harmonic Oscillator Eigenfunctions Files require Related Demonstrations More by Author Absolute value of the harmonic oscillator eigenfunctions. The harmonic oscillator is the most important exactly solvable model of quantum mechanics. The ground state eigenfunction minimizes the uncertainty product. With increasing quantum number, the square of the absolute value of the eigenfunctions approaches the probability distribution of a classical particle in a harmonic potential with inverse square root singularities at the turning points. Related Topics Chemistry College Physics General Chemistry Mechanics Physical Chemistry Quantum Mechanics Contributed by: Michael Trott http://demonstrations.wolfram.com/HarmonicOscillatorEigenfunctions/ Absolute value of the harmonic oscillator eigenfunctions. The harmonic oscillator is the most important exactly solvable model of quantum mechanics. The ground state eigenfunction minimizes the uncertainty product. With increasing quantum number, the square of the absolute value of the eigenfunctions approaches the probability distribution of a classical particle in a harmonic potential with inverse square root singularities at the turning points. Related Topics Chemistry College Physics General Chemistry Mechanics Physical Chemistry Quantum Mechanics Contributed by: Michael Trott http://demonstrations.wolfram.com/HarmonicOscillatorEigenfunctions/ Quantum Physics Page 1 of 6 Physical Ch Page 1 of 6 Page 1 of 6 P Harmonic Oscillator Eigenfunctions SEARCH Mechanics Quantum Physics Quantum Physics Contributed by: Michael Trott Wolfram Demonstrations Project General Ch Quantum Ph A quasi-classical state is a minimum-uncertainty oscillating wavepacket http://demonstrations.wolfram.com/HarmonicOscillatorEigenfunctions/ College Phy Quantum M 1/4/16 7:45 PM Quantum Mechanics Contributed by: Michael Trott Chemistry Mechanics Mechanics Wolfram Demonstrations Project Related T Quantum Physics Page 1 of 6 ground state X We can make a quasi-classical state ! " (t) as a superposition of number states ! " (t) = e# " 2 i$ t n ( " e ) 2 % n n! ! n ! X̂ " cos(# t), P̂ " sin(# t) A coherent state is the equivalent superposition of photon number states ! " (t) = e# " 2 i$ t n ( " e ) 2 % n n! ! n Ê x ! Q̂ ! cos(" t), & P, By coherent state B̂y ! P̂ ! sin(" t) ground state Probability of detecting n photons P(n) = e !" 2 " n! 2n (shot noise) Q, E x An ideal laser comes very close to emitting a coherent state. This is the closest we can come to a classical, monochromatic light field. By coherent state ground state Ex Other Interesting Topics in Quantum Optics – The quantum beam splitter (photons are bosons) – Quantum theory of interferometers – How to make number states, squeezed states, coherent states, etc. – Two-level atoms in single-mode cavities (Jaynes-Cummings model) – Generalizations of the Jaynes-Cummings model – Excited atoms interacting with the vacuum, decoherence & decay – Quantum theory of photodetection – Quasi-probability distributions and non-classical light Atoms and Photons: Confessions of a Self-Admitted Control Freak Poul Jessen Daniel Hemmer Nathan Lysne David Melchior Enrique Montano Hector Sosa-Martinez Kyle Taylor Light, Atoms and Quantum Control Quantum Mechanics is our reigning theory of everything - behavior of light and matter on scales from micro- to macroscopic ✓ Quantum Computing, Quantum Information Science - quantum mechanics is a resource that allows us to do different things ✓ Central Challenge: Quantum Control - how to make quantum systems do what we want, not what comes naturally ? Experimental Setup – “NMR” w/Cold Atoms Quantum Control for Fun and Science