1. Quantum Optics Ottica Quantistica
Fabio De Matteis
Sogene room D007 - phone
Didatticaweb
didattica.uniroma2.it/files/index/insegnamento/ Ottica-Quantistica

2. 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
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Quantum Optics

3. 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
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Quantum Optics

4. 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
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Quantum Optics

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

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

7. 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
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Quantum Optics

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

9. 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
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Quantum Optics

10. 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
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Quantum Optics

11. 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
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Quantum Optics

12. 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
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Quantum Optics

13. 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
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Quantum Optics

14. 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
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Quantum Optics

15. 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
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Quantum Optics

16. 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
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Quantum Optics

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

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

19. 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
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Quantum Optics

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

21. 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
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Quantum Optics

22. 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
N 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
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Quantum Optics

23. 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
N E2
N 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
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Quantum Optics

24. 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
N E2
N 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
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Quantum Optics

25. 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.
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Quantum Optics

26. 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.
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Quantum Optics

27. 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
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Quantum Optics

28. 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
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Quantum Optics

29. 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
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Quantum Optics

30. 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.
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Quantum Optics

31. 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
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Quantum Optics

32. 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.
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Quantum Optics

33. 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
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Quantum Optics

34. 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
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Quantum Optics

35. 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
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Quantum Optics

36. 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
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Quantum Optics