Lecture1 Qed Applic

8/6/2019 Lecture1 Qed Applic

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Optical cavity QED

Luis A. OrozcoJoint Quantum Institute

Department of PhysicsLecture 1: Atoms in cavities



Cavity QED group:Present students (2007):

David NorrisRebecca Olson KnellBasudev RoyMichael Scholten

Undergraduate (2007)Eric Cahoon

Postdocs:Dr. Daniela ManoelDr. Daniel FreimundDr. Jietai Jing

Past members of my group:Dr. Juergen GrippDr. Stephen MielkeDr. Gregory FosterDr. Wade SmithDr. Joseph ReinerDr. Matthew Terraciano

Collaborators:Prof. Howard Carmichael, University of Auckland, New Zealand.Prof. James Clemens, Miami University.Prof. Julio Gea Banacloche, University of Arkansas.Prof. Perry Rice, Miami University.Prof. Howard Wiseman, GriffithUniversity, Brisbane, Australia.

Visitors: Nicolas Leuillot, France, JinWang Australia, Stefan Kuhr Germany,Arturo Fernandez, Chile.

Sociedad Mexicana de Fisica (summerexperience for undergraduates).Gabriel Ramos, Manuel de la Cruz,Salvador Hernandez, Monserrat Bizarro,Edgar Vigil, Adonis Reyes.

Supported by NSF and NIST



Coupled atoms and cavities:

Drive Output

Collection of N Two level atoms coupled to a single mode of the electromagnetic field. This is a far from equilibriumsystem. Driven with dissipation (atoms γ, cavity κ).

Microwaves

Micromaser

Visible light

Optical Bistability

Cavity QED



Absorptive Element

• A saturable absorber (an atom) has an absorptioncoefficient which is a non-linear function of I:

• The cavity is setup for resonance.• At small intensities, the absorption due to the

element is high and the output is low.

• As the intensity is increase beyond Is, theabsorption rapidly decreases and the output goesto high.

intensitysaturationtheisIwhere

/ 1

s

s

o

I I +=α

α



The field inside the cavity comes from the addition

of the drive and what is already there

)0(Re)0(Let 1 niKLl

I n e E T ε ε α −+ +=

Where ε n+1 is the electric field after the n+1+ path aroundthe cavity, L is the round-trip length, α is the absorptioncoefficient and R=1-T the mirror reflectivity

non-linearEI, I I ET, IT

100 % 100 %

lε(0)

T T



:functioniontransmissamplitude

thegivesthesecombining

)( .distanceaatfieldelectricinternalthestime

nsmittancemirror traby thegivenisfieldoutputThe

)Re1( :givesthisrearanging

Re )0()0(thatsoconstantbemust

cavitytheinsidefieldelectricthestatesteadyAt

)(

)(

0

010

01

Re

Te

E

E

eT lT E l

E T

e E T

iKLl

LliK

I

T

liK oT

iKLl I

iKLl

nn

−=

•==

•−=

+=∴==

•

−

−

+−

+−

−+

α

α

α

α

ε ε

ε

ε ε ε ε ε



Absorptive BistabilityA saturable absorber, at resonance has an absorption

coefficient which is a non-linear function of I:

Ignoring the phase term in the numerator andassuming that αl<<1 so that e -αl~1- αl, gives:

intensitysaturationtheisIwhere

/ 1

s

s

o I I +=

α α

E

)(

I ReTe E

iKLl

LliK T

−= −

−

α T l E E

I

T

/ 11α +=

23 2

0 Al N λ

π σρ α ==



⎥⎦

⎤⎢⎣

⎡+

+=

=

T I I T l E E

T I

I

sT

oT I

T

/ 1 / 1

:toleadsusing

α

The ratio of losses: atomic losses per round trip ( αl)to cavity losses per round trip ( T ) is importantCooperativity

T l

C oα

=

22

1

2;

1

21

:and;with

x

Cx y x

x

C x y

TI

E x

I

E y

s

T

s

I

+−=⎥

⎦

⎤⎢

⎣

⎡

++=

==



Atomic Polarization

x

-2Cx1+x 2

Drive

Transmission

y



What do we expect on resonance for the normalized

fields (x,y) and the normalized intensities (X,Y)?

For low intensity, the input and the output arelinearly related,

y = x (1+2C) ; Y=X(1+2C)2

y = x (1+2C)

For very high intensity,

y = x ; Y=X +4C

( )221

1

C Y X

+=

Y

C

Y

X 41+=



⎥⎦

⎤

⎢⎣

⎡

++= 212

1 xC

x y

At intermediate intensity, there can be saturation, there is thepossibility of a phase transition. It happens in this simple model forthe case of C>4. C (Cooperativity) is the negative of the laser pumpparamenter. It is the ratio of the atomic losses to the cavity losses oralso can be read as the ratio between atom-cavity coupling (g) andthe dissipation ( κ,γ).

The slope of the output x as a function of input y may be zero!



Input-Output response of the atoms-cavity system for two differentcooperativities C=0 is with no atoms, C=60 has plenty of atoms, with a drivethat can saturate them and we recover the linear relationship with unit slopebetween Y and X. The hysteresis is clearly visible.



Typical system for optical experiments.

MHz12toupMHz7.42 =π g

MHz0.62 =π

γ MHz6.32 =π

κ



ηv E d

g

⋅

=Dipole coupling between the atom

and the cavity.

2 / 32 / 1 55 Pr Sed ρ=

The dipole matrix element between two states is fixed by theproperties of the states (radial part) and the Clebsh-Gordoncoefficients from the angular part of the integral. It is of theorder of a few times a 0 (Bohr radius) times the electron charge e

eff v V

E 02ε ω η=The field of one photon in a

cavity with Volume V eff is:

The electric field squared is an energy density.



Single atom Cooperativity (measures the effect of one atom):

κγ

2

1gC =

Saturation photon number (measures the effect of one photon):

2

2

0 3gnγ

=

T l N C C 0

1α ==Cooperativity (for N atoms): is the ratioof the atomic losses to the cavity losses

or also can be read as the ratio betweenthe good coupling (g) and the dissipation(κ,γ).



Are the two definitions equivalent?

cd

cT lc

Al

d

V

d ggC

η

η

η

η

0

2

2

30

2

0

2

2

22

1

434

;2

22

1;

πε ω

γ κ

ε

ω

ε

ω

κγ

==

===

23 20

1 AT N

T l

T l N C C λ

π σρ α ====

RATIO OF TWO AREAS



ηv E d

g

⋅

=Dipole coupling between the atom

and the cavity.

2 / 32 / 1 55 Pr Sed ρ=

The dipole matrix element between two states is fixed by theproperties of the states (radial part) and the Clebsh-Gordoncoefficients from the angular part of the integral. It is of theorder of a few times a 0 (Bohr radius) times the electron charge e

eff v V

E 02ε ω η=The field of one photon in a

cavity with Volume V eff is:

The electric field squared is an energy density.



To reach the strong coupling regime in the optical regime it is

necessary to make the coupling between the atoms and the cavity glarger than the decay avenues of the system (cavity κ) or (atoms γ).

The way to achieve this is making g larger, in the optical regime, bymaking the volume smaller. For microwaves, make R closer tounity.

However this increases linearly the decay of the cavity, as itdepends on the length L of the cavity but also on the reflectivity R,keeping other losses low, such that R+T=1

( ) T c

Rc

2l1

2l=−=κ



Formulation of the problem: 1.- Free evolution of cavity mode and atoms,2.- Coupling atom-cavity, 3.- Decay of atoms (reservoir), 4.- Decay of cavityfield (reservoir), 5.- Drive of the cavity

Use this Hamiltonian to find the equations of motion of the field <a>, the atomic

polarization < σ+>,and atomic inversion <σ

z>. We assume N atoms distributed atthe positions r j in the mode of the cavity.



Maxwell Bloch Equations are then:



State Equation of Optical Bistability , (Cavity QED).

y is the normalized input field y = E/ n 01/2 , it is the field

inside the cavity with no atoms.x is the normalized output field x=<a>/n

0

1/2 , it is the fieldinside the cavity with atoms.∆ is the normalized atomic detuning ∆=(ωatom −ωlaser )/γ/2.θ is the normalized cavity detuning θ=(ωcavity −ωlaser )/κ.

This equation makes explicit that the phase of the input andoutput fields need not be the same.Constructive interference happens when the phase is zero.



Transmission of light of different frequencies close to

resonance and phase delay foratoms alone

Transmission of light of different frequencies close toresonance and phase delay for

cavity alone

Transmission of light of different

frequencies close to resonanceand phase delay for atoms andcavity combined. Note that thepeaks happen where the phase

crosses zero.

Th t i i i th l i t it li it



The transmission in the low intensity limit canbe written in the following form to stress the

two normal modes present:



We are going to probe the eigenvalue structure of the system:

The first excited state is split from the coupling betweenatoms and cavity so the energy levels are:

In the spectroscopy we should see two peaks corresponding to thetwo possible transitions between the ground state and the excited

states. There is no difference using the Maxwell Bloch or the fullHamiltonian .



Some experimental considerations:

The atoms are optically pumped into the highest m sublevel of the F=3 ground state of 85Rb.

The atomic beam is highly collimated and is perpendicular to the

mode of the cavity.

The cavity can have a Finesse above 10 4.



Transmission spectroscopy of atoms-cavity system.

This is a way to probe the normal modes and see the eigenvaluestructure of the system.

For low intensity, we have two coupled harmonic oscillators: Thecavity mode and the polarization of the atoms (neglect any atomicinversion). We can observe the so-called Vacuum Rabi peaks.

For high intensity, the atoms saturate and so one only sees the FabryPerot fringe from the cavity.

For intermediate intensity, we have two anharmonic coupledoscillators that show frequency hysteresis.

E i t l t t t d th t i i t f th t it t



Experimental apparatus to study the transmission spectra of the atom cavity system .

Transmission Spectra at low intensity for different atomic



a s ss o Spect a at ow te s ty o d e e t ato cdetunings. Note the Vacuum Rabi peaks and the “avoided crossing”of the two coupled modes.

Calculation of the low intensity transmission spectra for



Calculation of the low intensity transmission spectra fordifferent atomic detunings

Transmission spectra for different intensities no detuning C=78



Transmission spectra for different intensities no detuning, C=78.

Input-Output hysteresis curve for the parameters of the transmission spectroscopy,



p p y p p py,indicating qhere the different behavior appears. (a) Vacuum Rabi peaks, (b)anharmonic oscillator, (c) broanened single peak, and (d) Fabry Perot resonance

with the atoms saturated.

Theoretical calculation of the transmission spectra as a



pfunction of the driving intensity.



Hysteresis of the light from the coupled atoms-cavity system. Two different scanswith equal input intensities are shown. Filled circles mark the scans with increasing

laser frequency; open circles mark scans with decreasing laser frequency. The linesare theoretical calculations from a semiclassical theory.

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