Quantum Optics Ottica Quantistica Fabio De Matteis fabio.dematteis@uniroma2.it Sogene room D007 - phone 06 7259 4521 Didatticaweb didattica.uniroma2.it/files/index/insegnamento/166336-Ottica-Quantistica

Quantum optics Quantum optics deals with phenomena that can only be explained by treating light as a stream of photons Light-matter interaction is the only way to «experience» light properties Theory of absorption of light needs (it is sufficient) the semi-classical model Model Matter Light Classical Hertzian dipoles Waves Semi-classical Quantized Full Quantum Photons F. De Matteis Quantum Optics

Quantum optics Spontaneous emission Photoelectric effect Lamb shift Casimir effect Photoelectric effect describes the ejection of electrons from a metal under influence of light. Usually it is reported as proof of the corpuscolar nature of light (Einstein 1905) Energy transferred to atoms in quantized packets But it could be seen as well as the probabilistic ejection of individual electron under the influence of a classical wave Most of the experiments involving photoelectric effect can be interpreted by semi-classical theory F. De Matteis Quantum Optics

On the Light side Tight binding between Light and Matter We are matter, light is a mean to investigate matter properties Effect of light on matter (photoelectric effect, state transition, diffraction grating, ecc.) Let’s start adopting light’s point of view F. De Matteis Quantum Optics

Light as electromagnetic radiation Electric field E Magnetic (induction) field B Electric displacement field Electric dipole moment per unit volume Electric susceptibility F. De Matteis Quantum Optics

Light as electromagnetic radiation No free charges r=0 nor currents j=0 Maxwell Equation Wave Equation F. De Matteis Quantum Optics

Light as electromagnetic radiation Perfectly conductive walls L y x z Tangential component of E field is vanishing Independently from volume, shape and nature. Matter plays a role, however. Confinement F. De Matteis Quantum Optics

Field Modes Stationary Waves L y x z F. De Matteis Quantum Optics X=0 - L  Ex#0 Ey=0 Ez=0 Stationary Waves F. De Matteis Quantum Optics

Only normal component different from 0 Field Modes x=0 (x=L) L y x z Only normal component different from 0 X=0 - L  Ex#0 Ey=0 Ez=0 Stationary Waves F. De Matteis Quantum Optics

Field Modes (ny=nz=0) Stationary Waves L y x z X=0 - L  Ex#0 Ey=0 Ez=0 Not more than one can be null at once (Otherwise ) Stationary Waves F. De Matteis Quantum Optics

Field Modes k E 2 polarization for each k value /L ky kx kz Divergence eq. _ How many field modes for each frequency interval  : +d F. De Matteis Quantum Optics

N modes? Spectral density of modes _ Number of lattice point in the first octant of a spherical shell defined by radius k : k+dk Each point occupies a volume (/L)3 kz /L 2 polarizzation for each k ky kx F. De Matteis Quantum Optics

N modes? Spectral density of modes _ Number of lattice point in the first octant of a spherical shell defined by radius k : k+dk k=/c Each point occupies a volume (/L)3 kz /L 2 polarizzation for each k Density of mode increases with the frequency ky kx F. De Matteis Quantum Optics

Energy of harmonic oscillator field Time dependency of e.m. field Harmonic Oscillator So much for the spatial dependence of EM. Amount of energy stored in each field mode at T F. De Matteis Quantum Optics

Energy of harmonic oscillator field Time dependency of e.m. field Harmonic Oscillator Planck’s quantization hypothesis Classical field but quantized energy of the field “First quantization” Cycle-average theorem F. De Matteis Quantum Optics

Planck’s Law At thermal equilibrium* Temperature T Excitation probability of nth-state Let’s set U = exp(- ħw/kBT) 1/(1-U) *Once more we need to resort to some matter. Thermalization F. De Matteis Quantum Optics

Mean Energy Density WT() Mean number of excited photons (for  mode) Photon energy (for  mode) Mode density in the interval  ÷d F. De Matteis Quantum Optics

Mean Energy Density WT() Classic Limit (Rayleigh 1900) Wien’s displacement law Stefan-Boltzmann’s Law At low temperature Wien’s Formula F. De Matteis Quantum Optics

Fluctuations in photon number We stated the probability distribution of the mode occupation for the cavity field Absorption and emission will cause the fluctuation of the photon number in each mode of the radiation field in the cavity with characteristic times Neglecting for now the nature of the characteristic times, we can infer some general properties making use of the ergodic theorem F. De Matteis Quantum Optics

Fluctuations in photon number U = exp(- ħw/kBT) F. De Matteis Quantum Optics

Fluctuations in photon number The root mean square deviation Dn of the distribution is The r-th factorial moment is defined Therefore the second moment is Taylor expansion of the root square The fluctuation is always larger than the mean value F. De Matteis Quantum Optics

Emission and Absorption An electron in an atom can make transition between two energy state absorbing or emitting a photon of frequency  = E/ħ with E = E2 – E1 energy difference between the two levels. The processes that can occur are: ħw=E2-E1 N2 +1 E2 N2 E2 Energy ħw N1 E1 N1-1 E1 Let’s now consider the basic interaction processes between em radiation and atoms Based on physically reasonable postulates. (Demonstrated by quantum-mechanical treatment) Absorption An electron occupying the lower energy state E1 in presence of a photon of energy ħ= E2-E1 can be excited to a level E2 absorbing the energy of the photon. Spontaneous Emission Stimulated Emission F. De Matteis Quantum Optics

Emission and Absorption An electron in an atom can make transition between two energy state absorbing or emitting a photon of frequency  = E/ħ with E = E2 – E1 energy difference between the two levels. The processes that can occur are: ħw=E2-E1 N2 E2 N2-1 E2 Energy ħw N1 E1 N1+1 E1 Absorption An electron occupying the higher energy state E2 can decay to the lower energy state (E1) releasing the energy difference as a photon of energy ħ= E2-E1 and a random direction (k) Spontaneous Emission Stimulated Emission F. De Matteis Quantum Optics

Emission and Absorption An electron in an atom can make transition between two energy state absorbing or emitting a photon of frequency  = E/ħ with E = E2 – E1 energy difference between the two levels. The processes that can occur are: ħw=E2-E1 N2-1 E2 N2 E2 Energy ħw ħw ħw N1 E1 N1+1 E1 An electron occupying the higher energy state E2 can decay to the lower energy state (E1) releasing the energy difference as a photon of energy ħ= E2-E1 Differently from the previous case the process is stimulated by the presence of a photon. The process is coherent, the emitted photon is coherent in phase and direction (k) with the stimulating one Absorption Spontaneous Emission Stimulated Emission F. De Matteis Quantum Optics

Einstein’s coefficients At thermal equilibrium the transition rate from state E1 to E2 has to be equal to that from state E2 to E1. N1 number of atoms per unit of volume with energy E1, Absorption rate proportional to N1 and to the energy density at frequency  W to promote the transition. N1 WT B12 N2 W B21 with B21 constant called coefficient of stimulated emission. N2 A21 with A21 constant called spontaneous emission. The coefficients B12, B21 and A21 are called Einstein’s coefficients. F. De Matteis Quantum Optics

Thermal Equilibrium At equilibrium the processes must equilibrate: The population of a generic energy level j of a system at thermal equilibrium is expressed by Boltzmann statistic: Nj population density of j-level of energy Ej N0 total population density gj j-level degeneracy. F. De Matteis Quantum Optics

Is Thermal Radiation Coherent? Thermal equilibrium WT () equal to black body with h refractive index of the medium. Ratio between rate of spontaneous emission and stimulated emission at thermal equilibrium If R>>1 Spontaneous emission dominates Incoherent R<<1 Stimulated emission dominates Coherent R~1 if kT~ħw T=12000 K F. De Matteis Quantum Optics

Other no-thermal radiation Which energy density does it need in order to get a ratio R~1 ? Visible l~600nm w~3x1015 s-1 h~6,626x10-34Js For a typical linewidth ~10-2 nm or dw~2p1010 s-1   I (W/m2) E (V/m) n/V (m-3) Photons/mode Mercury lamp 104 103 1014 10-2 CW laser 105 1015 1010 Pulsed laser 1013 108 1023 1018 F. De Matteis Quantum Optics

Optical excitation of atoms Atomic level population achieved by light irradiation (N2(t=0)=0) Thin cavity crossed by a light beam (negligible light intensity losses) Atoms are lifted into the excited state  energy is stored in the atomic system W not function of time For BW>>A system reaches saturation N2=N/2 Powerful lasers F. De Matteis Quantum Optics

Optical excitation of atoms When the light is switched off (t=0), the atomic system relaxes to its ground state (thermal equilibrium) The energy stored in the matter is re-emitted as photons. Reciprocal of A is the radiative lifetime of the transition. F. De Matteis Quantum Optics

Absorption Let’s consider a collimated monochromatic beam of unitary area flowing through an absorbing medium with a single transition between level E1 and E2. The intensity variation of the beam as a function of the distance will be: For a homogeneous medium I(x) is proportional to the intensity I(x) and to the travelled distance (x). Hence I(x) = -I(x)x with  absorption coefficient. Writing the differential equation: and by integration, we obtain: where I0 is the input radiative intensity. I(x) I(x+x) x F. De Matteis Quantum Optics

Macroscopic theory of absorption When the e.m. wave propagate in a dielectric medium it generates a polarizzation field P. For a not too intense field (linear response regime) where c is the linear elettric susceptibility The electric displacement vector D is connected to the electric field E by With the generalization of the dispersion relation The susceptibility is a complex quantity: We define the square root of the dielectric coefficient as complex refractive index where h is the refractive index and k is the extinction coefficient. F. De Matteis Quantum Optics

Macroscopic Theory of Absorption Let’s skip to a travelling plane wave solution rather than a stazionary one The intensity I of the electromagnetic wave, defined as the energy crossing the unit area in unit of time, is represented by the value, averaged over a cycle, of the flux of the Poynting vector The dependence on the space-time variables of all fields is that of a plane wave propagating along the z-axis, i.e. the intensity is Where I0 is the intensity at z=0 and a is the absorption coefficient F. De Matteis Quantum Optics

Microscopic Theory of Absorption Relation between absorption coefficient (macro) and Einstein’s coefficients (atomicmicro) Einstein’s Coefficients deal with em radiation incident on a 2 level system in vacuum W represents em energy density in the dielectric. Therefore we must substitute WW/h2 net loss of photons for k-mode per unit of volume Wk(t) I(x) I(x+dx) dx F. De Matteis Quantum Optics

Absorption Coefficient at thermal equilibrium (g2/g1)N1>N2 hence the coefficient is positive. Population Inversion (g2/g1)N1<N2 → negative absorption coefficient Increment of the intensity passing through the medium F. De Matteis Quantum Optics

Absorption Coefficient In stationary condition, the two level populations does not vary. For all ordinary light beams the second term in brackets is negligible with respect to the first one F. De Matteis Quantum Optics