Quantum Optics

1. Quantum Optics Introduction Lasers Interaction of Light With Matter Field Quantization Applications of Quantum Optics M. S. Scully and M. O. Zubairy, Quantum Optics, Cambridge University Press, Cambridge (1997). L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms, Dover, NY (1975). V. Vedral,Modern Foundations of Quantum Optics, Oxford University Press (2006). W. Demtroder, Atoms, Molecules and Photons, Springer-Verlag, Berlin, Heidelberg (2006).

2. Non-linear optics • Multiphoton processes • Interaction of light with matter II • Field quantization • Interaction of light with matter III • Saturation phenomena • Optical cooling and trapping of atoms • Quantum computation • New trends in quantum optics • The optical Bloch equation

3. Optics • Geometrical Optics (GO) • Physical Optics (PO) - light is an electro-magnetic wave. Contains GO as an approximation • Quantum Optics - light is quantized in chunks of energy (photons). Contains PO (and hence GO) as an approximation

4. Geometrical Optics • in a homogeneous and uniform medium light travels in a straight line • the angle of incidence is the same as that of reflection • the law of refraction is governed by the sines rule • they can be derived from a more fundamental principle - Fermat's principle

5. Fermat's principle • Light travels such that the timeis extremized • in a homogeneous medium light travels in a straight line - the speed of light is the same everywhere and therefore a straight line - the shortest path between two points  the shortest travel time • same reasoning applies for the incidence and reflection angles • law of sines refraction

6. A y1 n = 1 qi x n qr y2 d B Suppose that light is going from a medium of n =1 to a medium of n n = c/v the time for travel from A to B The laws of GO can be derived from Fermat's principle d

7. Newton believed that light is made up of particles • Hyugens believed that light is a wave - If light is of particles collision of two beams will lead to some interesting effect – does not happen • interference - the key property that won the argument for Huygens against Newton - demonstrated by Young’s "double slit" experiment http://www.whatthebleep.com/trailer/doubleslit.wm.low.html

8. Photoelectric effect the energy of the ejected electrons proportional to the light frequency ejection energy was independent of the total energy of illumination - the interaction must be like that of a particle which gave all of its energy to the electron

9. Maxwell's equations in vacuum 0 - permeability of free space 0-permitivity of free space

10. E and B fields propagate at the speed of light Maxwell - light is an electro-magnetic wave! – interference

11. Interference • a light beam of wavelength l, passes through a single slit of width a. A distance Dafter the slit we will obtain a bright spot of diameter s lD = sa The Fraunhofer limit - the distance after which the light starts to spread, a = s D = a2/llight starts to behave like a wave For a 1 mm wide slit, l = 500 nmD= ? At this D light behaves like a wave

12. How is GO reconciled with the fact that light is a wave? • a beam of light encounters two atoms • the initial wavevector of light, k, also determines the propagation direction • suppose that the light changes its wave vector to k’after scattering • What is the final amplitude? the intensity is maximum,i.e., a straight line

13. What changes in QO? • light is again composed of particles (photons), behaving like waves - they interfere both Newton and Huygens were right

14. basic properties of quantum behavior of light • Mach-Zehnder interferometer • beam-splitter transformation The imaginary phase in front of c – when the light is reflected from a mirror at 90o it picks up a phase of the light comes out only in one arm

15. Interaction free measurement Detector 2 Detector 1 Absorber if the photon is absorbed – neither of the detectors clicks if the photon takes the other path – at the last bs it has an equal chance to be transmitted or reflected – the two detectors click with equal frequencies - the interference has been destroyed by the presence of the absorber the presence of an absorber in path 5 can be detected, without the photon even been absorbed by it - interaction free measurement! Thus, if detector 2 clicks, then the photon has gone to path 6 - an obstacle in path 5 else only detector 1 would be clicking

16. Elitzur-Vaidman bomb-testing problem 23, 987 (1993)..Found. Phys

17. Introduction to Lasers Laser - Light Amplification by Stimulated Emission of Radiation The invention and development of lasers -paramount to the understanding of interaction between light and matter

18. Normal modes in a cavity Laser is a cavity with a certain lasing medium inside bunch of atoms oscillating inside a box with highly reflecting mirrors What do Maxwell's equations tell us about the radiation?

19. light in a cavity cannot be a free propagating wave the highly reflecting mirrors - electric field on the surface is very nearly zero where l, n, m are integers and L is the cavity dimension in all three directions the wavevector comes in each direction in discrete units of p/L

20. temporal dependence of the field The number of states with the wavevector klying in the interval (k, k+dk) is now proportional to the surface of the sphere so that This is a continuous number (cannot be true since the wavevector is discrete units of p/L) and has incorrect dimensions density of modes density of modes including polarization is twice higher

21. modes in a cavity are defined by their wavelengths wavelengths can only assume certain sizes such that an integral number of half wavelength is equal to the length of the cavity L= nl/2 - a consequence of the fact that the electric field has to disappear at the cavity walls

22. Basic properties of lasers • Directed - a beam of a diameter of a few centimeters directed at the Moon surface would generate a spot of a size of a few hundred meters • Intense - intensities can easily reach 1010 W • Monochromatic - The beam is nearly of one color - frequency spread of 106 Hz, compared to the 1015 Hzfrequency of light produced • Coherent - spatial and temporal coherence • Short

23. In 1954 Townes in the USA and Basov and Prokorov in Russia, suggested a method of achieving lasing - using ammonia to produce amplified Microwave radiation (MASER) - 1964 Nobel prize for Physics In 1958 Townes and Schawlow calculated the conditions to produce visible laser light In 1960 the first LASER was demonstrated by Maiman, using a Ruby crystal

24. Properties of light: blackbody radiation heat propagates in a given medium: conduction and radiation conduction is governed by a diffusion equation once the temperature is the same everywhere there is no conduction radiation is independent of temperature: a body that reflects all radiation that falls on it – white body a body that absorbs all the radiation that falls on it - blackbody rate of change of Ta to T gradient

25. from Maxwell's equation we know that light is a wave - a harmonic oscillator what is the energy per oscillator in thermal equilibrium? statistical physics tells us that every independent degree of freedom gets aenergyofkT/2 the energy of HO of frequency w energy density - Rayleigh and Jeans the intensity becomes larger and larger the higher the frequency, growing at a rate proportional the frequency square

26. Planck postulated that harmonic oscillators can have energies only in the "packets" of he knew from experiments the shape for blackbody radiation curve and extracted a formula it does not blow-up for large frequencies - approaches zero as a way to derive this formula was to assume that the HO energies are quantized

27. To complete the derivation he assumed that the probability to occupy the level with energy E is The average energy is given by N0is the total number of osc. The average energy per oscillator

28. Blackbody spectrum there is nothing strange at large frequencies the quantity I has to be multiplied by some frequency interval to obtain intensity light is quantized the total output intensity from a Black-body is obtained by integrating Planck's expression The value of the integral is p4/15 so that

29. Interaction of light with matter - Einstein's treatment • Einstein thought about the interaction of light and matter in 1917, before the advent of proper quantum mechanics in 1925 • what kind of information was available to him to attack the problem? he knew • about Planck's quantum assumption and the correct derivation of a blackbody's spectrum • that atoms were also quantized and that electrons occupied stationary states as in Bohr's atomic model no clue about how atoms interact with photons • an atom can absorb a photon to move from a lower energy level to a higher energy level – stimulated absorption • an atom can spontaneously emit a photon and jump from a higher energy level to a lower one – spontaneous emission • he reckoned that at thermal equilibrium the rate of emission and absorption have to be equalized

30. two level model • stimulated absorption from state 1 and spontaneous emission from state 2 • if an atom is left for long enough in an excited state it will naturally (spontaneously) emit energy • the emission rate - A21(Einstein's A coefficient), so that the total number of atoms spontaneously emitting per unit of time is A21N2 (N2 is the No. of atoms in level 2) • the absorption rate is also proportional to the density of radiation u(w12) so that the total number of atoms absorbed per unit of time is u(w12)B12N1 (B12is the Einstein B coefficient and N1is the No. of atoms in the level 1)

31. In equilibrium the two rates are equal (by definition)

32. the two level atom is sitting in the wall of the blackbody cavity - it has to produce the same kind of radiation that exists inside since the walls and the radiation are in equilibrium • the two expression for radiation density coincide only at small temperature or high frequency • from the correspondence at low T • right relationship between spontaneous emission and stimulated absorption even though we made a mistake, - we haven't been able to reproduce the right Planck radiation formula

33. The correct rate equation • we need to write a new detailed balance that includes another process to obtain the right blackbody formula • stimulated emission is the number of atoms that emit in a stimulated fashion per unit of time Einstein concluded thatstimulated emission must also exist

34. the equilibrium condition implies dN2/dt = 0 B21 = B12-the rate of stimulated emission and absorption are equal to each other what about the actual values of A and B? Einstein was unable to say anything about them - he lacked a more precise quantum mechanical formulation

35. Saturation density Suppose that the density of light is made larger and larger The atoms would follow this increase for some time absorbing more and more, but would ultimately reach their maximum capacity when all atoms become excited At this point light would just continue to propagate through material without being absorbed by the atoms as they are saturated The saturation radiative density, WS, is defined so that the rate of spontaneous and stimulated emissions are equal similar to

36. a two-level atom therefore either emits or absorbs light - total energy is in this case conserved • how about the momentum conservation? in stimulated emission the light is emitted in the same direction as the absorbed light - the total net momentum transfer is zero for an initially stationary atom - since the momentum has to be conserved - spontaneously emitted light cannot be in any particular direction for example a gas of atoms - suppose that spontaneous emission was directed in a particular way - then you would expect the gas to drift in a particular direction (i.e., the centre of mass would be moving) - this never happens in reality - spontaneous emission has to be random - uniformly distributed over a 4psolid anglecentered on the atom important consequences in laser coolingof atoms

37. Optical Excitation of Two-Level Atoms rate equation the radiation density <W> has two independent components the thermal black body density <WT> and the external density <WE> (the latter did not exist in Einstein's treatment) at room temperature, blackbody radiation will be much smaller than from an external source  blackbody radiation is absent suppose that all the population is initially in the ground state (i.e., state 1), then by solving the rate equation the number of excited atoms increases linearly with time for short times for times long enough, the number of atoms approaches its steady state value

38. Steady State from the rate equation

39. Steady state rates for emission and absorption when the laser is initially turned on, the population in the excited state starts to increase linearly, finally reaching its steady state value in the steady state the populations do not change any more and the total amount of energy stored in the atoms is given by once the laser action stops, these atoms release this energy through spontaneous emission

40. Lifetime and amplification • three different processes involved in the interaction between a two level atom and light • we may get the wrong impression that stimulated processes are continuous in time, while the spontaneous emission is an abrupt • this is not correct – spontaneous emission is also continuous the rate equation that we had previously without external field, <W> = 0 The lifetime is i.e., inversely proportional to the rate of spontaneous emission. The population decreases exponentially

41. Amplification criterion • for amplification the rate of stimulated emission should be much bigger than for spontaneous emission • Let's look at two different regimes: the microwave and the visible light at the room temperature (T = 300K) for = 0.1 m, the right hand side is close to 0 for = 500 nm it is huge, e100 Masers are possible Lasers are impossible!

42. Population inversion • the ultimate condition for amplification is population inversion between two levels, i.e., N2 > N1 • in thermal inversion equilibrium population inversion is impossible as the weights of states go as so that N2is always less populated than N1 • However, the steady state rate for N2is given by always less then N/2 (it approaches this limit for high intensities as the population inversion is impossible not only in thermal equilibrium, but also under the presence of an external coherent source - independent of the frequency of radiation - wrong conclusion