Optical cavity QED

Luis A. OrozcoJoint Quantum InstituteDepartment of Physics

Lecture 2

!

g

2"= 6 MHz

!

"

2#= 3.6 $ 106 s-1

!

"

2#= 6.0 $ 106 s-1

Typical system optical experiments.

!

C1=g2

"#$ 2 1.0

32

2

0!=

gn

"

Expansion parameters 1/n0 and C1

We want to study Quantum Phenomena in the TIMEDOMAIN: DynamicsWe looked at the frequency and time response to stepexcitation.

We want now to go to a situation where there is no explicit timedependence in the excitation. This is different from the stepexcitation.

What is strong coupling in cavity QED.

The quantum fluctuations are comparable greater than the mean.The size of the fluctuations is set by C1 and by 1/n0

Possibility to study an open quantum system.

The study of noisy signals is done with correlationfunctions. We have learned a lot about characteristictimes, sizes and some dynamics with such correlationfunctions in statistical mechanics. They have thefollowing form:

Example of a Noisy photocurrent:

<F(t) F(t+τ) ><F(t) G(t+τ)>

For Optical signals thevariables we want tocorrelate to themselves or toeach other can be:

Field and Intensity

G(1)(t+τ) = <E(t) E*(t+τ)> field-field

G(2)(t+τ) = <I(t) I(t+τ)> intensity-intensity

Η(t+τ) = <I(t) Ε(t+τ)> intensity-field

How do we measure these functions?

Wave-Wave CorrelationMichelson Interferomenter

)(

)()()(

*

)1(

tI

tEtEg

!!

+=

!!"!#

" dgixpeF )()(2

1)( )1(

$=

Spectrum of the source

Basis of Fourier Transform Spectroscopy

Classical Intensity Correlation Functions: HanburyBrown and Twiss. (HTB) two persons only!

At equal time:

With the variance:

The equal time correlation function is related to thevariance, and will be greater than 1.

How do we measure such correlations:

Construct the “Periodogram”

Take the photocurrent which is proportional to the intensity I(t)

ni

M

i

N

n

i

j

i

IItItI

ItI

ItI

+

= =

!!"+

"+

"

0 0

)()(

)(

)(

#

#

Discretize (digitize) the time series.Calculate the correlation by displacing the series by a fixednumber n, multiply and then sum and average.Careful to normalize each sum in a way to reveal that the size ofthe sample may be finite.

Ii

Ii+n

Multiply by the delayed time series

Discretize the time series:

Average the result and see the “correlation time”

Another way to calculate the correlation function is with thewaiting time distribution.

Measure the separation between two consecutive pulses (startand stop)

Histogram the distribution of separations.

This is the same as g(2)(τ) if the fluctuations are very rare.

Make sure the intensities are low, so that seldom you getcoincidences.

time

Intensity (photons)

Example of how to measure g(2)(τ) with the time series and thewaiting time distribution.

Digital storage oscilloscope(DO) captures thephotocurrent out of thePhotomultiplier tube (PMT)for a long time and thenprocess the time series.

Photon correlator withAvalanche Photo-Diodes (APD), watingtime distribution. TheTime to DigitalConverter (TDC) canregister up to 16 stops.

Average TTL pulseson oscilloscope as an

analog signal

APD A(trigger)

APD B(average)

BS

Auto- and cross-correlation with adigital storage oscilloscope

Comparison of g (2) (τ) from photon counting (a), andfrom a time series of intensities measured with a singlePMT (b).

Correlation functions in quantum optics are conditionalmeasurements.

The detection of the the first photon prepares a state that thenevolves in time.

The correlation functions can beField-Field: Mach-Zehnder, Interferogram.Intensity-Intensity: g(2)(t) Hanbury-Brown and Twiss.

When the dynamics of the conditioned state are slow enough itis possible to feedback to the system and modify its state.

Very brief and incomplete history of the intensity correlation functions.

1956 Hanbury-Brown and Twiss; astronomy to measure the size of a starlooking at the intensity with two different detectors, not interfering thefileds as Michelson had propossed and done.This is a classical effect of bunching.

1963 Glauber and others (Mandel, Sudarshan, Wolf) formalize thequantum correlation funcions “Quantum theory of Coherence”.

1976 Experiment and Theory of single atom resonance fluorescenceKimble, Dagenais, Mandel; Carmichael and Walls.

Shows the non-classical effect of antibunching.

Earlier experiments for Bell Inequalities by Clauser with a cascadedatomic source can be interpreted as well as measurements of non-classical properties of light.

The variance of the number of photons is related to the probabilityof coincident photons given by g(2)(0). Light with Poissonianstatistics has g(2)(0) = 1. Light with a super-Poissonian statisticaldistribution has g(2)(0)> 1. A sub-Poissonian distribution has g(2)(0)< 1, a clear signature of a nonclassical field.

Quantum Mechanically (time and normal order):

Intensity correlation function measurements:

2

)2(

)(ˆ

)(ˆ)(ˆ

)(tI

tItIg

!!

+=

Gives the probability of detecting a photon attime t + τ given that one was detected at time t.This is a conditional measurement:

I

Ig c

ˆ

)(ˆ

)()2(!

! =

Experimental considerations

Collimated atomic beam: thermal, high velocityonly a few atoms are maximally coupled.

Atomic number fluctuations are small.

7 663 536 starts 1 838 544 stops

The values of the coefficients are:

The state of the cavity QED system for N atoms is:

The probability density of two simultaneous transmissionof photons is then:

!

00 ˆ a 2"

2

= # 2pq

2

This can be zero if p is zero

!

"c =ˆ a "

00 ˆ a "= 00 + # q 01 +$ pq10 ,

g(2)(%) = 1+

&$

$

'

( )

*

+ , exp -

(. + / /2)

2%

0

1 2 3

4 5

6 cos7% +(. + / /2)

27sin7%

'

( )

*

+ ,

2

&$

$

'

( )

*

+ , = -2C1

' 2C

1+ 2C - 2C1

'( )

0

1 2 2

3

4 5 5

Regression of the field to steady state after thedetection of a photon.

Classically g(2)(0)> g(2)(τ) and also |g(2)(0)-1|> |g(2)(τ)-1|

antibunched

Non-classical

Steady State:

Exchange of Excitation:

The conditional field prepared by the click is:

A(t)|0> + B(t)|1> with A(t) ≈ 1 and B(t) << 1

Mostly one prepares the vacuum!

Conditioned measurements in the language of correlationfunctions allow the study of the dynamics of the system.

Quantum conditioning, with photodetections, provides themost ideal times for controlling the evolution of the system.

Feedback.

Trigger the intensity-step with a fluctuation (photon) and measurethe time evolution of the intensity as in g(2) (τ). Can we feedback?

eqg

gpq

eg

ggss

,12

,22

,02

,1,0

22

!

"""

!" #+#+=$

),,( and ),,( ,ˆ !"!"# gqqgppa ===

A photodetection collapses the steady state into the following non-steady state from which the system evolves.

eqg

gpqga collapsess ,02

,1,0ˆ!

"" #+=$%$

( ) ( ) ( )[ ] ( )221

,0,1,0 !""!" Oefgfg +++=#

Conditional dynamics from the system wavefunction

Field Atomic Polarization

Use passive feedback to stabilize the wavefunction

( ) ( )Tfg

Tf12

2

!"=

For times when the following relation holds, we see that thestate resembles a steady state.

Problem: The driving field at times, Τ, will not stabilize thestate.

Solution: Change the driving field so that it will.

( )( )Tf

T

T1

) (intensity driving

intensity driving=

<

>

!

!

Plot to find times where quantum control will work.(g, κ, γ)/2π = (30.0, 7.9, 6.0) MHz

Theoretical prediction.

How long can we hold the system and then release it?As long as we can!

How sensitive is it to detunings?With our protocol we only operate well on resonance.

Where is the information stored?New steady state.

What is quantum about this?The detection of the first photon.

Deterministic source?No, we mostly create the vacuum: |0,g> + λ|1,g> + …