Opacity of electromagnetically induced transparency for quantum fluctuations

Pablo Barberis Blostein y Marc Bienert

Instituto Nacional de Astrofisica Optica y electronica. Tonantzintla,

Mexico.

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.

Two level atom illuminated with a laser

Laser

When =0, the electron realizes Rabi oscillations between levels |0 y |1 with frequency:

Laser frequency = Atomic transition frequency.

|| g

|0

|1

Laser

Probability of finding the atom in the excited state:

|0

|1

{

Laser

Light Absorption by the atoms

Medium composed of Three level

atoms.

Laser

The linear response of the absorption is proportional to the imaginary part of electric dipole

operator.

Electromagnetically induced transparency (EIT)

|2

|0

|1 1

2 1

2

Laser 2Laser 1

{

Dark States

Perpendicular states to the dark state.

Dark state

|1 |2

|0

12

iii g

If the system is initially in state |0

|1 |2

|0

12

Dark states and EIT

Dark state:

0- 1

Laser 1 (pump)2{

Laser 2 (probe)

probe

|0

|1 |2

|2

|0

|1 1

2 1

2

Laser 2Laser 1

2 {1 {

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

Capturing the light

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.

If both fields are treated quantum mechanically:

First quantum EIT experiment:

What I want to answer:

Pump: Coherent state (Ideal Laser)

Three level atomspump

probe

Probe: Quantum state (Squeezed state)

Both fields are treated quantum mechanically, and the Rabi frequencies associated with each field are comparable.

pump

probe

|0

|1 |2

What are the squeezed and coherent states?In the quantum harmonic oscillator:

In a coherent state:

Squeezed state in x

Field quadratures:Annihilation and creation operators of one field

mode.

Analog to position operator

Analog to momentum operator

In the harmonic oscillator:

The quadrature is defined as

Uncertainty relation:

Squeezed state in quadrature =0:

Coherent state:

A mode vacuum is a coherent state with =0

A mode squeezed vacuum is a mode where

Resuming: we want:

Three level atomspump

probe

pump

probe

Initial condition of pump field

Mode in resonance with transition |0-|1 in coherent state |1.

The other modes in state |0.

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.

The mean values after interaction are the same

as before interaction.

What happens with the initial quantum fluctuations?

|0

|1 |2

Equations:

If 2=0 we have:

If 2=1= we have:

Noise spectrum of the probe field quadrature:

Noise spectrum of the pump field quadrature :

|2

|0

|1

2 1

12

P. Barberis-Blostein, M. Bienert, Phys. Rev. Lett. 98, 033602 (2007)

Cavity version: P. Barberis-Blostein, Phys. Rev. A 74, 013803 (2006)

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.

Resuming: we want:

Three level atomspump

probe

pump

probe

Initial condition of pump field

Mode with detuning with transition |0-|1 in coherent state |1.

The other modes in state |0.

Initial condition of probe field.

Mode with detuning with transition |0-|2 in a squeezed state such that the

field mean value is 2.

The other modes in a squeezed vacuum.

The mean values after interaction are the same

as before interaction.

What happens with the initial quantum fluctuations?

|0

|1 |2

{ {

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

Small two mode Resonance, equal Rabi frecuencies

implies

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

Influence of Doppler effect.Vacuum Squeezed state as probe field

Influence of Doppler effect.Squeezed state as probe field

Conclusions

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.

In EIT media: