Chapter 2: Semi-Classical Light-Matter Interaction · PDF fileChapter 2: Semi-Classical...

1

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Chapter 2: Semi-Classical Light-Matter Interaction

In this lecture you will learn:

• Semi-classical light matter interaction • Rabi oscillations• Optical Bloch equations• Decoherence and relaxation• Photon echo experiments• Ramsey fringes• Atomic clocks


Two-Level System Interacting with Classical Radiation

1

1e

2e2

Eo cos( t + )

In the absence of E&M field the Hamiltonian is:

212211ˆ eeeeHo

In the presence of E&M field the potential energy of a charge q is:

12

tEnrqtErq o cosˆ.ˆ.ˆ

In the presence of E&M field the Hamiltonian is:

tEnrqHtH oo cosˆ.ˆˆˆ Explicitly time-dependent

Assuming: 0ˆˆ2211 ereere

The above Hamiltonian is:

1221222111 coscosˆ eeteeteeeetH R

2112 ˆ.ˆ. enreqEenreqE ooR

2



1

1e

2e2

Eo cos( t + )

2112 ereered

= dipole matrix element

Dipole Matrix Element:

Light-Matter Interaction Coupling Constant:

ndqEoR ˆ.




1

1e

2e2

Eo cos( t + )


In the rotating wave approximation we write:

êxpêxp2

ˆˆ

expexp2

ˆ

2211

1221222111

itiitiNN

eeitieeitieeeetH

R

R

Solution Using the Schrodinger Picture:

2211

21

eetceetcttiti

Assume:

ttHt

ti

ˆ

and plug into the Schrodinger equation:

3



1

1e

2e2

Eo cos( t + )

itiR

itiR

etcidt

tcd

etcidt

tcd

12

21

2

2

One gets two equations:

Detuning is:

12

Solution, subject to the boundary condition , is: 10 et

titetcti

2sin

2cos2

1

tietc Riti

2sin2

2

22 R



1

1e

2e2

Eo cos( t + )

2

2

1

2

2sin

2sin

2cos)(

21

etieetitet Ri

ti

ti

Suppose =0 (Zero Detuning):

2

cos12

sin

2cos1

2cos

222

221

ttte

ttte

RR

RR

Rabi oscillations

R = Rabi frequency

12

22 R

1Max 21

22 tctc Occurs when: .....5,3,1

mmt

R

Suppose ≠0 (Non-Zero Detuning):

1Max22

222

12

2

R

Rtctc Occurs when:

.....5,3,1

mmt

4


Transformation to a Time-Independent Hamiltonian

1

1e

2e2

Eo cos( t + )

12

êxpêxp2

ˆˆ

expexp2

ˆ

2211

1221222111

itiitiNN

eeitieeitieeeetH

R

R

Define a unitary operator as: tB̂

1ˆ1êxpˆ11 tieNtNitB 1

1ˆêxpˆ BtNitB

2211

21

eetceetcttiti

Action of on a state: tB̂

2211

21

ˆ eetceetcttBtiti



1

1e

2e2

Eo cos( t + )

12

ˆˆ2

ˆˆˆˆˆˆˆ22111

iiRR eeNNHtBtHNtB

Important Property:

Now start from: ttHt

ti

ˆ

Let: ttBtR ˆ

tH

ttBtHNtB

ttHNtBttHtBttBN

t

ttBit

ttB

it

ti

RR

R

R

ˆ

ˆˆˆˆ

ˆˆˆˆˆˆˆ

ˆˆ

1

11

Then:

tHt

ti RR

R ˆ

5



1

1e

2e2

Eo cos( t + )

12

ˆˆ

2ˆˆˆ

2211iiR

R eeNNH

tHt

ti RR

R ˆ

Need to solve a time-independent two-level problem:

Hamiltonian in matrix form is (and assume zero detuning for simplicity):

2

20

2

2

2

2

2

2ˆ

iR

iR

iR

iR

Re

e

e

eH

Eigenenergies and eigenvalues are:

22

122

1

2122

12

2

2122

12

1

Rii

Rii

eeeev

eeeev


Boundary condition: 100 ettR


1

1e

2e2

Eo cos( t + )

12

21

2

1 200 vv

eett

i

R

Solution for t >0 is:

Finally:

21 2sin

2cosˆ 21

etieetettBt Rit

iR

ti

R

Same as before!

21

2

)2(

1

)2(2ˆ

2sin

2cos)(

2)()0()(

22

22

etieetet

vevee

ttet

Rit

iR

ti

R

ti

tii

RR

tHi

RRRR

6


Now suppose the initial state is:


1

1e

2e2

Eo cos( t + )

12

22

12

1 2

100 eeeevtt ii

R

22

12

2

1

)2(

1

)2(

21

2

2

2

ˆˆ

)(

eeeee

vetBttBt

vet

iti

ititi

ti

R

ti

R

R

R

R

Solution is:

And:

21

)0()(

21

)0()(

22

22

21

21

tete

tete

Occupation probabilities do not change with time for this initial state!!


Density Operator Equations

1

1e

2e2

Eo cos( t + )

12

Solution Using the Density Operator in the Schrodinger Picture:

êxpêxp2

ˆˆˆ2211 itiitiNNtH R

Density Operator Equation:

tHtttHttHdt

tdi ˆˆˆˆˆ,ˆˆ

Equations for the Density Matrix Elements:

ititititidt

td R

expexp2 1221

11

ititititi

dttd R

expexp

2 122122

ttitiitidt

td R112212

1212 exp2

ttitiitidt

td R112221

1221 exp2

Diagonal elements

Off-diagonal elements

7


Optical Bloch Equation

itiitix etettV 1221

itiitiy etetitV 1221

tttVz 1122

Define three quantities:

Related to diagonal elements

Related to off-diagonal elements

And then define a vector as: tV

ztVytVxtVtV zyx ˆˆˆ

tVdt

tVdy

x

One can then write the density matrix equations as:

tVtV

dt

tVdzRx

y

tVdt

tVdyR

z

tVdt

tVd

zxR ˆˆ




Magnetic Bloch Equation:

The classical equation of a spin magnetic moment in a magnetic field is: tM

B

tMXBdt

tMd

= Gyromagnetic ratio (ratio between the magnetic moment and the angular momentum of the spin)

Felix Bloch 1946B

tM

Optical Bloch Equation:

Cone

Plane of rotation (the front facet of the cone)

Circle of rotation (solid line)

tV tVX

dttVd

zxR ˆˆ

8



tV

tVdt

tVd

Some Properties of the Bloch Equation:

The following facts are not hard to prove and follow directly from the vector equation for :

i) The magnitude of the vector does not change with time

i) The vector executes a periodic motion and the angular frequency is equal to the magnitude of the vector :

i) The “plane of rotation” is the plane in which the tip of the vector lies during rotation. The vector is always normal to the plane of rotation

tV

tV

tV

tV

0.2.2.

tVtVdt

tVdtV

dttVtVd

constant.

.

22

2

22

2

tVtVdt

tVd

tVtVtVdt

tVd

tV

Angular frequency is:


Dynamics on the Bloch Sphere

x-axis

z-axisBloch Sphere

tV


itiitix etettV 1221

tttVz 1122

tVdt

tVd

zxR ˆˆ

1. tVtV

y-axis

9


x-axis

z-axis

Bloch Sphere tV

dttVd

Dynamics on the Bloch Sphere: Zero Detuning

And suppose:

10 et

xR ˆ

Assume no detuning:

ztV ˆ0

Then:

00

010ˆ t

00 tVx

00 tVy

1000 1122 tttVz

The state vector rotates in the z-y plane

The period of rotation is:

2

y-axis

ndqEoR ˆ.

1)0( et


x-axis

z-axis

Bloch Sphere

tV

ztV ˆ0



2

y-axis

21 2sin

2cos)(

21

etieeteti

ti

ti


itiitix etettV 1221

tttVz 1122

tVdt

tVd

xR ˆ


10


x-axis

z-axis

Bloch Sphere

tV

ztV ˆ0



2

y-axis

xR ˆ

tVdt

tVd

01221 itiitix etettV

11221 itiitiy etetitV

01122 tttVz

t = /2 :

2

21

221

2

1)2( eieeet

iii



x-axis

z-axis

Bloch Sphere

tV

ztV ˆ0



2

y-axis

xR ˆ

tVdt

tVd



11122 tttVz

t = / :

2

2

)( eietii


11


x-axis

z-axis

Bloch Sphere

tV

ztV ˆ0



2

y-axis

xR ˆ

tVdt

tVd



11122 tttVz

t = 3/2 :

2

2

3

12

3 21

2

1)23( eieeet

iii



x-axis

z-axis

Bloch Sphere

tV



2

y-axis

xR ˆ

tVdt

tVd



11122 tttVz

t = 2/ :

1

21

2 eeti


12


x-axis

z-axis

Dynamics on the Bloch Sphere: Non-Zero Detuning

Bloch Sphere

zxR ˆˆ

tVdt

tVd

ztV ˆ0

TOP VIEW

22

R

Plane of rotation when =0(the y-z plane)

Plane of rotation when is small

Plane of rotation when is large


x-axis

z-axis

Dynamics on the Bloch Sphere: Non-Zero Detuning

Bloch Sphere

zxR ˆˆ

tVdt

tVd

ztV ˆ0

TOP VIEW

22

R

Plane of rotation when =0(the y-z plane)

Plane of rotation when is small

Plane of rotation when is large

22

22max

R

RzV

13


Dynamics Including Decoherence and Population Decay

Equations for the Density Matrix Elements:

ititititi

Tt

dttd R

expexp

2 12211

2211

ititititi

Tt

dttd R

expexp

2 12211

2222

ttitiitiT

tdt

td R112212

12

2

1212 exp2

ttitiitiT

tdt

td R112221

12

2

2121 exp2

Diagonal elements

Off-diagonal elements

1

1e

2e2

Eo cos( t + ) 12

1T

Population relaxation

Decoherence


Dynamics Including Decoherence and Population Decay

tVtVTdt

tVdyx

x

2

1

tVtVtV

Tdt

tVdzRxy

y

2

1

tVTtV

dttVd

yRzz

1

1

One cannot write the simple compact equation anymore:

tVXdt

tVd

The equations for the components of the vector become: tV

14


x-axis

z-axis

Bloch Sphere

ztV ˆ0

y-axisDynamics with Decoherence (No Population Decay)

And suppose:

10 et

xR ˆ

Assume no detuning:

0

1

01

2

22

2

2

22

2

tVdt

tdV

Tdt

tVd

tVdt

tdVTdt

tVd

yRyy

zRzz

01

2 tV

TdttVd

xx

Equations are that of a damped harmonic oscillator

00 tVtV xx

00

010ˆ t


x-axis

z-axis

Bloch Sphere

ztV ˆ0

y-axisDynamics with Decoherence (No Population Decay)

And suppose:

10 et

xR ˆ

Assume no detuning:

0

1

01

2

22

2

2

22

2

tVdt

tdV

Tdt

tVd

tVdt

tdVTdt

tVd

yRyy

zRzz

01

2 tV

TdttVd

xx

0

0

0

tV

tV

tV

x

y

z

210

021ˆ t

Motion in the y-z plane

15


Dynamics with Decoherence and Population Decay

tVtVTdt

tVdyx

x

2

1 tVtVtV

Tdt

tVdzRxy

y

2

1

tVT

tV

dt

tVdyR

zz

1

1

In the most general case, the equations:

have a well defined steady state:

12

22

2

2

2

1

1

TTT

T

tV

R

z

12

22

2

2

1 TTT

TtV

R

Ry

12

22

2

22

1 TTT

TTtV

R

R

x


itiitiR eeeeeeeeeetH

1221222111 2ˆ

ndqEenreqEenreqE oooR ˆ.ˆ.ˆ. 2112

1e

2e

221 ee

221 ee

A superposition of the two eigenstates describe a state with a center of charge density not centered in the potential well

++

+

Electron Charge Density in a Two-Level System

16


Electron Oscillations in Two-Level SystemsConsider the following state of a two level system:

2

20

21

21

21

eeeeeet

eeeet

tiiti

i

ii

1

1e

2e2

The mean position of the electron in such a state is:

tdtrt

12cosThe electron position is oscillating The phase of the oscillation is

0

sin

cos

tV

ttV

ttV

z

y

x

12

itiitix etettV 1221


tttVz 1122

Suppose one wants to represent the above state on the Bloch Sphere:

The argument represents the phase relative to t+


Photon Echo ExperimentsThe mean position of the electron is:

tdtrt

12cos

The oscillating electron can radiate E&M energy just like a dipole antenna…..and it does! But the radiation from a single electron is weak

Consider a collection of such two level systems:

1

1e

2e2

It is difficult to get the energy level separation same for all Detuning will be different for all

Suppose one prepares all two-level systems in the identical state:

20 21 eee

eti

i

The oscillations of electron in different two-level systems would soon go out of phase and their radiations would not add up constructively

17


Photon Echo Experiments: Step 1Suppose the collection of two level systems is excited by a strongradiation pulse that lasts for time /2 (a /2 pulse):

x-axis

z-axis 2tV

ztV ˆ0

y-axisR

12

After this pulse, all the two-level system have the state (up to an irrelevant phase factor) :

2

212

1)2( eieet

ii

RR 22

Assume no decoherence or relaxation

tVdt

tVd


Photon Echo Experiments: Step 2

x-axis

z-axis

y-axis

After the pulse, let the system evolve by itself for time Td :

12

tVdt

tVdy

x

tV

dt

tVdx

y

0

dt

tVd z

0

2

2

2

tV

dt

tVdxy

xy

dTt 2

State vectors of systems with different detunings will move apart in the x-y plane

Why is there action happening in the Bloch Sphere even though no pulse is present?

18


Photon Echo Experiments: Step 3After time Td , the collection of two-level systems is excited by a second strong radiation pulse that lasts for time / (a pulse):

12

z-axis

y-axis

tVdt

tVd

R

RR 22

All the state vector rotate in their circles of rotation

x-axis


Photon Echo Experiments: Step 3After time Td , the collection of two-level systems is excited by a second strong radiation pulse that lasts for time / (a pulse):

12

z-axis

y-axis

tVdt

tVd

R

RR 22

All the state vector rotate in their circles of rotation

x-axis

19



12

z-axis

y-axis

The -pulse flipped the sign of the Vy component but let Vx component unchanged

The result is that the state vectors now all start converging towards the y-axis!

After the second pulse, let the system evolve by itself for time Td :

x-axis

tVdt

tVdy

x

tV

dt

tVdx

y

0

dt

tVd z



12

z-axis

y-axis

At exactly time Td after the second pulse, all the state vectors come together, as shown:

x-axis

At this point all the two-level systems (irrespective of their detunings) have the state (up to an irrelevant phase factor):

2

223

12

1)2

23

( eieeTtiTi

dd

At this time:

tdtrt

12cos2

223

dT

20


Photon Echo Experiments

The in-phase oscillations of all the two level systems results in the constructive addition of their dipole radiation which can be detected:

time

0t

dTt

dTt 2

Echo pulse

Photon Echo Technique can be used to measure the decoherence time T2:

dT

Mag. Of Echo Pulse

The decoherence time can be extracted from this curve


The History and Science of Time KeepingThe Longitude Problem:

The problem of establishing the East-West position or longitude of a ship at sea, thus revolutionizing and extending the possibility of safe long distance sea travel

British Government: “20,000 Pounds for a method that could determine longitude for position accuracy within 30 nautical miles “ (1714 by the Board of Longitude)

John Harrison(1693-1776)

Inventor of the Marine Chronometer

Dava Sobel's 1996 bestseller: “Longitude”

Today we have GPS: Position accuracy: ~10 cm

21


The History and Science of Time Keeping

Mechanical Clocks:

T ~ 1 sec/day

Quartz Clocks:

T ~ 1 msec/day

First built by Warren Marrison and J.W. Horton, 1927, Bell Labs

Atomic Clocks

T < 1 nsec/day

The idea of using atomic transitions to measure time was first suggested by Lord Kelvin in 1879


Atomic Clocks

i) The idea of using atomic transitions to measure time was first suggested by Lord Kelvin in 1879

ii) Magnetic resonance, developed in the 1930s by IsidorRabi, became the practical method for doing this. In 1945, Rabi first publicly suggested that atomic beam magnetic resonance might be used as the basis of a clock

iii) The first atomic clock was an ammonia maser device built in 1949 at the U.S. National Bureau of Standards (Now NIST)

iv) The first accurate atomic clock, using Cesium atoms, was built by Louis Essen in 1955 at the National Physical Laboratory in the UK

The History and Science of Time Keeping

1 second = Duration of 9 192 631 770 periods of the radiation corresponding to the transition between thetwo hyperfine levels of the ground state of the 133-Cs atom

22


Determination of Frequencies with High Precision

Question: How does one determine the frequency of a radiation source with high precision (one part in 1015 or better)?

Option: Try interacting the radiation with two-level atoms of different energy level separations and see which one absorbs

x-axis

z-axis

tV

ztV ˆ0

y-axis

?12

1

1e

2e2

Eo cos( t + )

A two-level system is excited by a radiation pulse that lasts for time / (a pulse):

Problem: don’t know a-priori?

22Time

R

Try a strong pulse: R

RR 22

1

22

2

22

R

Rt

The final upper state occupation then is always unity!


Ramsey Fringes: Step 1

Consider a two-level system prepared in the ground state:

10 et

?12

1

1e

2e2

Eo cos( t + )

The two-level system is excited by a strong radiation pulse that lasts for time /2 (a /2 pulse):

x-axis

z-axis tV

ztV ˆ0

y-axisR

RR 22

21

2

0

2

2

22

22

2

R

RzV

Assume no decoherence and population relaxation

After the pulse:

23



?12

x-axis

z-axis

tV

ztV ˆ0

y-axis

After the pulse, let the system evolve by itself for time T :

tVdt

tVdy

x

tV

dt

tVdx

y

0

dttVd z

The vector rotates in the x-y plane with a frequency equal to for a duration T

tV

0

2

2

2

tV

dt

tVdxy

xy

1

1e

2e2



?12

1

1e

2e2

x-axis

z-axis

tV

ztV ˆ0

y-axis

The vector rotates in the x-y plane with a frequency equal to for a duration T

tV

.....5,3,1odd

mmT

x-axis

z-axis

tV

ztV ˆ0

y-axis

.....6,4,2,0even

mmT

Consider two possible values of the duration T:

24



x-axis

z-axis

tV

ztV ˆ0

y-axis

.....5,3,1odd

mmT

x-axis

z-axis

tV

ztV ˆ0

y-axis

.....6,4,2,0even

mmT

After time T, the two-level system is excited by a second strong radiation pulse that lasts for time /2 (a /2 pulse):

Consider two possible values of the duration T:

1

1

22

zV

0

1

22

zV

After the /2 pulse: After the /2 pulse:



In general the value of Vz and 22 at the end of the second pulse are given approximately by:

T

TVz

cos121

cos

22

-20 -10 0 10 200

0.2

0.4

0.6

0.8

1

(/h) T

22

If the occupation of the upper level is measured after the second pulse then this can be used to determine the value of the detuning to a precision given by:

T1

~min

T can be chosen to very large in atomic systems (~ 1 second) and so the frequency can be determined to a high precision

If instead makes a measurement on Na two-level systems then the frequency precision is improved by aN

1

1e

2e2

Think of it as a very high-Q system

25


Cesium Atomic Clocks

~Controlloop

Microwavesynthesizer

Cesium fountain

RF Oscillator

Erro

r sign

al

Clock output

Cesium Fountain

1

1e

2e2

GHz 192.92

a

c

N

T

T11


Cesium Atomic Clocks

Ramsey fringes for a Cesium clcok (NIST)

NIST

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