1. Opacity of electromagnetically induced transparency for quantum fluctuations
Pablo Barberis Blostein y Marc Bienert
Instituto Nacional de Astrofisica Optica y electronica. Tonantzintla, Mexico.
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2. Plan
Introduction
Electromagnetically induced transparency (EIT)
Storing a light pulse in an atomic medium
Quantum memories
Propagation of quantum states (squeezed states) in EIT.
Resonance case
Two photon detuning case.

3. Two level atom illuminated with a laser
|0
Laser
Laser 
|1
Laser frequency = Atomic transition frequency.
Átomo de dois níveis interagindo com um continuo de modos.
Semiclássico equivalente à condição inicial do campo, sendo um estado coerente.
Campo elétrico proporcional ao operador de levantamento.
Estado estacionario.
Quando delta é zero, máxima probabilidade de achar o sistema no estado excitado. Também máxima emissão de fluorescência.
When =0, the electron realizes Rabi oscillations between levels |0 y |1 with frequency:

4. Probability of finding the atom in the excited state:
|0
|1
 {
Laser 
Átomo de dois níveis interagindo com um continuo de modos.
Semiclássico equivalente à condição inicial do campo, sendo um estado coerente.
Campo elétrico proporcional ao operador de levantamento.
Estado estacionario.
Quando delta é zero, máxima probabilidade de achar o sistema no estado excitado. Também máxima emissão de fluorescência.

5. Light Absorption by the atoms
Laser
Medium composed of Three level atoms.
The linear response of the absorption is proportional to the imaginary part of electric dipole operator.

6. Electromagnetically induced transparency (EIT)
|0
Laser 1
 {
Laser 2 2 1 1 2
|1
|2
Quando chega ao estado estacionário temos que a absorção é zero quando os dois lasers estão na ressonância.

7. EIT Experimental

8. Dark States |0 1 2 |1 |2 Dark state
Perpendicular states to the dark state.

9. |1
|2
|0 1 2
If the system is initially in state |0

10. Dark states and EIT Dark state: |0 2{ Laser 1 (pump) Laser 2 (probe)
Laser 1 (pump)
Laser 2 (probe) 
probe
0- 1
|1
|2
O estado estacionário é o estado escuro.
Explicar método intuitivo de porque o estado escuro é o estado estácionario.
Dark state:

11. 1 { |0 2 { Laser 1 2 1 Laser 2 1 2 |1 |2
Quando chega ao estado estacionário temos que a absorção é zero quando os dois lasers estão na ressonância.

13. Group velocity of a light pulse inside a medium showing EIT
If the pump Rabi frequency is much bigger than the probe Rabi frequency, the light pulse velocity is given by

14. Capturing the light

15. What happens if the field is treated quantum mechanically?
Probe field treated quantum mechanically
Classical pump field with Rabi frequency much bigger than probe field.
Adiabatic approximation.

16. If both fields are treated quantum mechanically:
First quantum EIT experiment:

17. What I want to answer: |0 |1 |2  probe probe Three level atoms
pump
pump
Pump: Coherent state (Ideal Laser)
Probe: Quantum state (Squeezed state)
Both fields are treated quantum mechanically, and the Rabi frequencies associated with each field are comparable.

18. What are the squeezed and coherent states?
In the quantum harmonic oscillator:
In a coherent state:
Squeezed state in x

19. Field quadratures:
Annihilation and creation operators of one field mode.
The  quadrature is defined as:
In the harmonic oscillator:
Analog to position operator
Analog to momentum operator
Uncertainty relation:
Coherent state:
Squeezed state in quadrature =0:

20. A mode vacuum is a coherent state with a=0
A mode squeezed vacuum is a mode where

21. Initial condition of probe field. Initial condition of pump field
Resuming: we want:
Initial condition of probe field.
Mode in resonance with transition |0-|2 in a squeezed state such that the field mean value is 2.
The other modes in a squeezed vacuum. 
|0
|1
|2
Three level atoms
pump
probe
Initial condition of pump field
Mode in resonance with transition |0-|1 in coherent state |1.
The other modes in state |0.
The mean values after interaction are the same as before interaction.
What happens with the initial quantum fluctuations?

22. Equations:

23. If 2=0 we have:

24. If 2=1= we have: Noise spectrum of the probe field  quadrature:
Noise spectrum of the pump field  quadrature :

25. General results: Noise spectrum of the probe field  quadrature :
Noise spectrum of the pump field  quadrature :
Where:

26. |2
|0
|1 g2 g1
g12
P. Barberis-Blostein, M. Bienert, Phys. Rev. Lett. 98, (2007)
Cavity version: P. Barberis-Blostein, Phys. Rev. A 74, (2006)

28. Partial Conclusions
When the Rabi frequencies are comparable, the media is not transparent for the initial quantum fluctuations.
There are two scales:
One, that depends on the atomic decayment rate, and is responsible of the lost of information (absorption) and behaves similar to the usual EIT transparency curve.
Other, that depends on the Rabi frequencies, and is responsible of the oscillation of quantum properties between the pump and probe field.

29. Initial condition of probe field. Initial condition of pump field
Resuming: we want: g
|0
|1
|2
 {
Initial condition of probe field.
Mode with detuning d with transition |0-|2 in a squeezed state such that the field mean value is 2.
The other modes in a squeezed vacuum.
Three level atoms
pump
probe
The mean values after interaction are the same as before interaction.
Initial condition of pump field
Mode with detuning d with transition |0-|1 in coherent state |1.
The other modes in state |0.
What happens with the initial quantum fluctuations?

32. The probe field is a vacuum squeezed state and the pump field is a coherent detuned state

33. Resonance, equal Rabi frecuencies
Small two mode
Resonance, equal Rabi frecuencies

34. implies

36. The carrier frequencies of the Fields are in a large two mode resonance

37. Influence of Doppler effect. Vacuum Squeezed state as probe field

38. Influence of Doppler effect. Squeezed state as probe field

39. Conclusions In EIT media:
The propagation of a squeezed probe state is very sensitive to two photon detuning. When the detuning is small there are three scales.
A vacuum squeezed state as a probe rotates its squeezed quadrature as it propagates, when the pump field is detuned.
The Doppler effect has a lot of impact in the propagation of squeezed states, preventing the possibility of making EIT experiments with quantum states in thermal clouds.