Elitzur-Vaidman bomb-testing problem

1. Elitzur-Vaidman bomb-testing problem suppose you have a super-sensitive bomb that explodes upon the slightest interaction can you make sure that such a bomb is reliable without detonating it? the light source A emits a single photon

2. the photon (i) passes through the bs or (ii) is reflected • if the bomb is dud • the system reduces to the Mach-Zehnder interferometer otherwise: • logically, the bomb is real • if the photon took the lower route  the photon triggers the bomb  explosion otherwise: • logically, the photon took the upper route • the photon on the upper route passes through the bs and is reflected • the C detector monitors the photon, while D not otherwise: • logically, the photon passed through • the D detector monitors the photon, while C not • the bomb exploded  real • the bomb did not explode and C detected the photon  logically the bomb must be real • the bomb did not explode and D detected the photon  either dud or real if the third observation is made the experiment has to be run many times  sometimes good bombs explode but at other times good bombs can be found

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

4. 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 the 4psolid angle centered on the atom important consequences in laser coolingof atoms

5. 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

6. Steady State from the rate equation

7. 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

8. 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

9. 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!

10. 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

11. Basic optical processes Diffraction Interference Coherence Light pressure Optical absorption Amplification Atom-light interaction classical Spectral lines Mode locking

12. Diffraction • diffraction of plane parallel light of wavelength λ from a single slit of width d far-field (Fraunhofer) limit - L ≥d2/λ near-field (Fresnel) regime - L ≤d2/λ In the Fraunhofer limit, the pattern on the screen observed at angle θ is obtained by summing the field contributions over the slit:

13. where principal maximum at θ = 0, and there are minima whenever β = mπ, m being an integer. Subsidiary maxima - below β = (2m+1)π/2, for m ≥ 1. The intensity at the first subsidiary maximum is less than 5% of that of the principal maximum The angle at which the first minimum occurs is or diffraction from a slit causes an angular spread of ~λ/d fora circular hole of diameter D. The intensity pattern has circular symmetry about the axis, with a principal maximum at θ = 0 and the first minimum at θmin calculates the resolving power of optical telescopes and microscopes

14. Interference • Interference patterns - a light wave is divided and then recombined with a phase difference between the two paths Michelson interferometer – inputparallel rays from a linearly polarized monochromatic source of wavelength λ and amplitude E0 where ΔL = L2−L1 and k = 2π/λ, Δ- phase shifts between the two paths even when L1 = L2 The field maxima and minima are As L2 is scanned, bright and dark fringes appear at with a period equal to λ/2

15. Interference interference - the effect from a number of individual sources (electromagnetic waves), is larger or smaller than the sum of individual effects (constructive or destructive) suppose N atoms each emitting a light wave ak is the (time-independent) amplitude of the k-th wave Let's add up all the contributions to obtain the total amplitude and then take the square to obtain the total effect (intensity)

16. when atomic emission is not coordinated - phases vary randomly and in the end average to zero this is the same as if there was no interference and the total effect (intensity) is the same as the sum of individual effects (intensities) if all the phases are the same (laser) this is N times more intense than when the phases are random the sources are coherent

17. Coherence light is a 3D spatial wave evolving in time - two different coherences: • temporal coherence concerns the same "beam of light" at the same spatial point, but two different times • spatial coherence concerns two or more different points of the wave, but at the same time a monochromatic wave with infinite coherence time a wave whose phase drifts – short coherence time coherence - very important for obtaining interference

18. temporal coherence Michelson interferometer monochromatic source – the spread in frequency of light involved, dn < n if the time delay between the beams is dt than fringes are formed only if dtdn < 1 the same beam interferes with itself -at two different times for a lamp dn ~ 108 s-1 so that dt ~108 s-1 the spatial coherence length isdl = c x 10-8 s = 3 m for a laser, dn ~ 104 s-1 so thatdl =30 x 103 m

19. Spatial coherence – Young’s double slit experiment the same light illuminates two slits “close” to each other the light is of size ds and the angle between the source and the slits dq interference fringes will form only if dqds < l the source is composed of many point sources – different fringe patterns since all the sources are out of phase with each other we have to add all the point intensities at the end to obtain the total fringe pattern if the distance between the slits increases – the patterns become more and more out of step – fringe disappearance

20. Light pressure a charge q interacts with an electric field E through the force and with the magnetic field B through the Lorentz force v is the charge velocity the combined effect of the two forces leads to radiation pressure (P) - the force on an area Aper that area force is also the rate of change of momentum (Dp)

21. the momentum density is given by the Poynting vector, S, divided by c2. The volume is AcDt considered just normal incidence, for isotropic radiation in a cavity

22. radiation pressure using Einstein's rate equations the photons that make up an EM wave of wavevector k each carry a momentum in a medium of refractive index the photon momentum is given Suppose now that a photon is absorbed by an atom of mass M. It then gains the velocity of

23. the atom then decays via stimulated emission - the emitted photon carries away the momentum in the same direction as the original photon spontaneous emission - direction of the momentum of the emitted photon anywhere within a 4p solid angle. The atom therefore recoils in some random direction. On average – there is no cancellation of the momentum previously gained as the net spontaneous emission transfer averages to zero

24. so in a cycle of absorption and emission we have the net transfer of from the photons to the atoms, in the direction of the incident beam • every absorption = one momentum kick in laser direction • • net effect: laser pushes atom • • the momentum transfer to atoms gives rise to radiation pressure

25. the number of atoms Nis large enough to produce small time dependencies in the atomic populations and that Pis the total atom momentum • The rate of change of Pin the presence of radiative energy density <W> at a frequency w(resonant with the ground and the excited state of the atoms) is proportional to the difference between the absorbed and the stimulated emission rates the rate of change of momentum is negative for N2 > N1 the number of atoms in steady state

26. for strong fields we have that WS << <W> this is the saturation value for transfer rate for very strong beams once saturation has been reached, any increase of beam strength produces very little change in the momentum transfer rate the steady state momentum transfer rate is equal to the force acting on each atom how can this force be measured?

27. suppose that an atom beam passes perpendicularly to a strong laser beam • the atoms in the beam interact with light and absorb and emit radiation • deflection occurs when atoms absorb light followed by spontaneous emission, in which case they gain momentum perpendicular to their direction of motion • the resulting deflection is about 10-5rad • the two isotopes have different transition frequencies • tuning the laser to a resonant transition of one isotope • only on-resonant atoms will be deflected, and therefore the two isotopes would separate into two different beams

28. Optical absorption light propagating through a medium is absorbed and reemitted by atoms - its intensity will decrease the intensity of radiation is I= uv u- energy density of the radiation field v- field velocity for small distances traveled by light, the change in intensity has to be proportional to the traveled medium length and to the intensity itself Kn - absorption coefficient (frequency and medium dependent)

29. Beer’s law the intensity changes exponentially as it propagates Using the fact that I= vu we can write the equation for dI from Einstein’s relationship ( instead of n)

30. Fuchbauer- Ladendurg formula for N1 N2we obtain dI = 0 no absorption - saturation for N2 >> N1amplification– thebasis of laser operation this condition, cannot be achieved by exciting a two level atom - when we have saturation then the No. of ground state atoms > No. of excited state atoms can be obtained in a three level atom

31. Amplification: three level system the atom population can be stored in the third level - decay to the second level and lase to the first level

32. S – transition probability due to both radiative and non-radiative processes total No. of atoms conserved use levels 1 and 2 for lasing, so the equations are solved for the steady state case (i.e. , dNi /dt = 0)

33. where N= N1+ N2+ N3is the total number of atoms in all three levels The number of atoms arriving at the level 2 per unit time is: Let’s express the numerator of N2 - N1 (the denominator is positive at all times). It is N( 􀀀 - A21) The condition for lasing Population inversion 3-level systems

34. Ruby-laser Maiman (1960): cavity L =n l Ruby: Al2O3 + Cr Xe t=0.003 s coherent monochromatic collimated

35. four level laser • atoms are pumped from ground state to level 4, rapid decay to level 3, creating population inversion with respect to level 2 • the pumping to level 4 can be optical (from a flashlamp or another laser) or electrical • the decay rate from level 2 to ground state (level 1) must be fast to prevent atoms accumulating in that level and destroying the population inversion

36. Nd:YAG (Nd:Y3AI5O12) laser

37. HeNe laser • pump He to metastable state (20.61 eV) • transfer excitation to Ne metastable state (20.66 eV) • laser transition • spontaneous emission (2 times) to deplete lower level ( low pumping) • not very efficient! (20.6 eV vs 2 eV)

38. Classical treatment of atom-light interaction the atom - a mass (electron) on a spring (attached to the nucleus) this spring is then contracted and extended as it interacts with light - an EM wave as the spring extends the energy from the EM field gets stored -absorption of radiation and is then released when the spring contracts - radiation emission F is the force on the electron due to the field F = qE0 cos(wt) the atom oscillates at the frequency of the driving field the highest amplitude of oscillation is when the field is on resonance - the driving frequency is the same as the natural oscillator

39. Radiation damping the solution decays exponentially to zero - all oscillations must eventually die away as energy dissipated into the environment the solution is not a monochromatic wave - more than one frequency component is present in its expansion We look at the Fourier spectrum by taking a Fourier Transform

40. there are many frequencies in the spectrum and not just that of the driving field the intensities of various frequencies is given by Lorentzian broadening

41. Spectral lines • Atomic states have in principle well defined energies - these energy levels, when analyzed spectroscopically, appear to be broadened • The shape of the emission line is is described by the spectral lineshape function gw(w), which peaks at the line center defined by hw0 = E2 - E1 • Where do these come from? atomic collisions Doppler broadening lifetime (natural) broadening

42. lifetime broadening light is emitted when an electron in an excited state drops to a lower level by spontaneous emission The rate of decay is determined by the Einstein A coefficient  determines the lifetime t the finite lifetime of the excited state leads to broadening of the spectral line according to uncertainty principle DEDt > /2p Dw =AE/ > 1/t this broadening is intrinsic to the transition – natural broadening and the spectrum corresponds to Lorentzian lineshape

43. collisional (pressure) broadening The atoms in a gas frequently collide with each other and with walls of the containing vessel, interrupting the light emission and shortening the effective lifetime of the excited state If the mean time between collisions, tcol , is shorter than the radiative lifetimethan we need to replacet by tcol in Dwlifetime = 1/t – resulting in additional broadening Based on the kinetic theory of gases tcol is given by ss is the collision cross section and P the pressure 1/tcol and Dw are proportional to P collisional broadening  pressure broadening

44. Emission of all the atoms atoms moving toward the observer Doppler broadening originates from the random motion of the atoms in the gas Doppler shifts in the observed frequencies the Maxwell-Boltzmann velocity dis. where N(v) is the number of atoms moving with v The line shape is a Gaussian the Doppler broadening gives a Gaussian profile rather than a Lorentzian its half width at half maximum is The dominant broadening in low pressure gases at room temperature is usually Doppler broadening and the lineshape is closer to Gaussian

45. Line broadening in solids • the spectra will be subject to lifetime broadening as in gases – a fundamental property of radiative emission • the atoms are locked in their positions – neither pressure nor Doppler broadening are relevant • The emission and absorption lines can be broadened by other mechanisms non-radiative transitions (phonons) The non-radiative transitions shorten the lifetime of the excited state according to the phonon emission times in solids are often very fast – substantial broadening inhomogeneity of the host medium

46. More detailed principles of laser laser elements: • a cavity with two or more highly reflecting mirrors • a gain medium - support inverted atomic population • an energy source which can excite the atoms in the gain medium to achieve the population inversion • a loss mechanism by which the stored energy is dissipated

47. absorption and amplification in a certain medium, but now • an electro-magnetic field propagates through a medium, the medium responds to it – polarization • the medium is composed of a bunch of classical oscillating springs • the microscopic dipole is then p= -ex P= Np= -Nex the driven harmonic oscillator equation for polarization is the proportionality constant between P and E - a complex number the measured quantity is the intensity which is the mod square of E the real part represents the intensity decay due to the absorption in the medium, while the imaginary part the oscillatory behavior of E solutions

48. the dielectric susceptibilty E a plane wave the response of the medium is linear, therefore the wavevector is also is a complex number, so is the refractive index the real part n’ is the "normal" refractive index, the imaginary component absorption or gain in vacuum in medium

49. the electric field is modified as it propagates. It has the form the last term is equal to which proves Beer’s law in terms of intensity the absorption (gain) coefficient

50. the laser cavity has two laser mirrors if intensity Iimpinges on a mirror of reflectivity r will return with the intensity rI the gain due to the medium for combination of one round trip reflections and the gain coefficient where For oscillations to build up The maximum value of the gain coefficient is threshold population inversion