Quantum Optics Ottica Quantistica Fabio De Matteis fabio

Quantum Optics Ottica Quantistica Fabio De Matteis fabio. dematteis@uniroma 2. it Sogene room D 007 - phone 06 7259 4521 Didatticaweb http: //didattica. uniroma 2. it/informazioni/index/insegnamento/182313 -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 Model Matter Light Classical Hertzian dipoles Waves Semi-classical Quantized levels Waves Full Quantum Quantized levels Photons F. De Matteis Quantum Optics 1/38

Quantum optics • Spontaneous emission • Photoelectric effect (Most of the experiments involving photoelectric effect can be interpreted by semi-classical theory) • Lamb shift (“radiative corrections due to the interaction between the atomic electron ant the vacuum”) • Casimir effect (attractive force between two parallel conducting mirrors placed in a vacuum. ) F. De Matteis Quantum Optics 2/35

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 3/35

Light as electromagnetic radiation Electric displacement field Electric field E Electric dipole moment per unit volume Electric susceptibility Magnetic (induction) field B F. De Matteis Quantum Optics 4/35

Light as electromagnetic radiation No free charges r=0 nor currents j=0 Maxwell Equation Wave Equation F. De Matteis Quantum Optics 5/35

Light as electromagnetic radiation Perfectly conductive walls z L Tangential component of E field is vanishing y x Independently from volume, shape and nature. Matter plays a role, however. Confinement F. De Matteis Quantum Optics 6/35

Field Modes z L y x Stationary Waves F. De Matteis Quantum Optics 7/35

Field Modes x=0 (x=L) z L y Only normal component different from 0 x Stationary Waves F. De Matteis Quantum Optics 8/35

Field Modes (ny=nz=0) z L y x Not more than one can be null at once (Otherwise ) Stationary Waves F. De Matteis Quantum Optics 9/35

Field Modes E 2 polarization for each k value kz k Divergence eq. /L _ How many field modes for each frequency interval : +d ky kx F. De Matteis Quantum Optics 10/35

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 _ 2 polarizzation for each k /L ky kx F. De Matteis Quantum Optics 11/35

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 _ 2 polarizzation for each k /L ky kx F. De Matteis Quantum Optics 12/35

Energy of harmonic oscillator field Time dependency of e. m. field Harmonic Oscillator F. De Matteis Quantum Optics 13/35

Harmonic oscillator field Harmonic Oscillator F. De Matteis Quantum Optics 14/35

Energy of harmonic oscillator field Time dependency of e. m. field Harmonic Oscillator Cycle-average theorem F. De Matteis Quantum Optics 15/35

Energy of harmonic oscillator field Time dependency of e. m. field Harmonic Oscillator “First quantization” F. De Matteis Quantum Optics 16/35

Planck’s Law At thermal equilibrium* Temperature T Excitation probability of nth-state Let’s set U = exp(- ħ /k. BT) 1/(1 -U) *Once more we need to resort to some matter. Thermalization F. De Matteis Quantum Optics 17/35

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 18/35

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 19/35

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 20/35

Wien’s displacement law Stefan-Boltzmann’s Law F. De Matteis Quantum Optics 21/35

Fluctuations in photon number We stated the probability Absorption and emission will distribution of the mode cause the fluctuation of the occupation for the cavity photon number in each mode of field 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 22/35

Fluctuations in photon number U = exp(- ħ /k. BT) F. De Matteis Quantum Optics 23/35

Fluctuations in photon number The r-th factorial moment is defined Therefore the second moment is The root mean square deviation n of the distribution is The fluctuation is always larger than the mean value F. De Matteis Quantum Optics 24/35

Emission and Absorption ħ =E 2 -E 1 An electron in an atom can make transition between two energy state absorbing or emitting a photon of frequency = E/ħ with E = E 2 – N 2 +1 EE 22 Energy E 1 energy difference between the two levels. The processes that can occur are: ħ N 1 -1 E 11 Absorption Spontaneous Emission Stimulated Emission F. De Matteis An electron occupying the lower energy state E 1 in presence of a photon of energy ħ = E 2 -E 1 can be excited to a level E 2 absorbing the energy of the photon. Quantum Optics 25/35

Emission and Absorption ħ =E 2 -E 1 An electron in an atom can make transition between two energy state absorbing or emitting a photon of frequency = E/ħ with E = E 2 – Energy N 2 -1 E 2 E 1 energy difference between the two levels. ħ The processes that can occur are: Absorption Spontaneous Emission Stimulated Emission F. De Matteis N 1+1 E 1 An electron occupying the higher energy state E 2 can decay to the lower energy state (E 1) releasing the energy difference as a photon of energy ħ = E 2 -E 1 and a random direction (k) Quantum Optics 26/35

Emission and Absorption ħ =E 2 -E 1 An electron in an atom can make transition between two energy state absorbing or emitting a photon of frequency = E/ħ with E = E 2 – E 1 energy difference between the two levels. N 2 -1 E 2 Energy ħ ħ ħ The processes that can occur are: N 11+1 E E 11 An electron occupying the higher energy state E 2 can decay to the lower energy state (E 1) releasing the energy difference as a photon of energy ħ = E 2 -E 1 Differently from the Spontaneous Emission previous case the process is stimulated by the presence of a photon. The process is coherent, the emitted photon is Stimulated Emission coherent in phase and direction (k) with the stimulating one 27/38 F. De Matteis Quantum Optics Absorption

Einstein’s coefficients At thermal equilibrium the transition rate from state E 1 to E 2 has to be equal to that from state E 2 to E 1. N 1 number of atoms per unit of volume with energy E 1, Absorption rate proportional to N 1 and to the energy density at frequency W to promote the transition. N 1 WT B 12 N 2 W B 21 with B 21 constant called coefficient of stimulated emission. N 2 A 21 with A 21 constant called spontaneous emission. The coefficients B 12, B 21 and A 21 are called Einstein’s coefficients. F. De Matteis Quantum Optics 28/35

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 jlevel of energy Ej N 0 total population density gj j-level degeneracy. F. De Matteis Quantum Optics 29/35

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 F. De Matteis R~1 if k. T~ħ Quantum Optics T=12000 K 30/35

Other no-thermal radiation Which energy density does it need in order to get a ratio R~1 ? Visible l~600 nm ~3 x 1015 s-1 h~6, 626 x 10 -34 Js For a typical linewidth d ~2 p 1010 s-1 Mercury lamp CW laser Pulsed laser F. De Matteis I (W/m 2) 104 105 1013 E (V/m) 103 104 108 Quantum Optics n/V (m-3) 1014 1015 1023 Photons/mode 10 -2 1010 1018 31/35

Optical excitation of atoms Atomic level population achieved by light irradiation (N 2(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 For BW>>A system reaches saturation N 2=N/2 Powerful lasers F. De Matteis Quantum Optics 32/35

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 33/35

Absorption Let’s consider a collimated monochromatic beam of unitary area flowing through an absorbing medium with a single transition between level E 1 and E 2. 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: I(x) I(x+ x) x F. De Matteis and by integration, we obtain: where I 0 is the input radiative intensity. Quantum Optics 34/35

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 35/35

Macroscopic Theory of Absorption Considering a travelling plane wave solution 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 I 0 is the intensity at z=0 and is the absorption coefficient F. De Matteis Quantum Optics 36/35

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/h 2 net loss of photons for k -mode per unit of volume I(x) F. De Matteis Quantum Optics Wk(t) I(x+dx) dx 37/35

Absorption Coefficient at thermal equilibrium (g 2/g 1)N 1>N 2 hence the coefficient is positive. Population Inversion (g 2/g 1)N 1<N 2 → negative absorption coefficient Increment of the intensity passing through the medium F. De Matteis Quantum Optics 38/35

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 39/35