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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Two-Level System Interacting with Classical Radiation
1
1e
2e2
Eo cos( t + )
2112 ereered
= dipole matrix element
Dipole Matrix Element:
Light-Matter Interaction Coupling Constant:
ndqEoR ˆ.
1221222111 coscosˆ eeteeteeeetH R
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Two-Level System Interacting with Classical Radiation
1
1e
2e2
Eo cos( t + )
1221222111 coscosˆ eeteeteeeetH R
In the rotating wave approximation we write:
ˆexpˆexp2
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Two-Level System Interacting with Classical Radiation
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Two-Level System Interacting with Classical Radiation
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Transformation to a Time-Independent Hamiltonian
1
1e
2e2
Eo cos( t + )
12
ˆexpˆexp2
ˆˆ
expexp2
ˆ
2211
1221222111
itiitiNN
eeitieeitieeeetH
R
R
Define a unitary operator as: tB̂
1ˆ1ˆexpˆ11 tieNtNitB 1
1ˆˆexpˆ BtNitB
2211
21
eetceetcttiti
Action of on a state: tB̂
2211
21
ˆ eetceetcttBtiti
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Transformation to a Time-Independent Hamiltonian
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Transformation to a Time-Independent Hamiltonian
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Boundary condition: 100 ettR
Transformation to a Time-Independent Hamiltonian
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Now suppose the initial state is:
Transformation to a Time-Independent Hamiltonian
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!!
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Density Operator Equations
1
1e
2e2
Eo cos( t + )
12
Solution Using the Density Operator in the Schrodinger Picture:
ˆexpˆexp2
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
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ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
Optical Bloch Equation
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Optical Bloch Equation
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
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ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Optical Bloch Equation
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:
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Dynamics on the Bloch Sphere
x-axis
z-axisBloch Sphere
tV
itiitiy etetitV 1221
itiitix etettV 1221
tttVz 1122
tVdt
tVd
zxR ˆˆ
1. tVtV
y-axis
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ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
x-axis
z-axis
Bloch Sphere
tV
ztV ˆ0
The state vector rotates in the z-y plane
The period of rotation is:
2
y-axis
21 2sin
2cos)(
21
etieeteti
ti
ti
itiitiy etetitV 1221
itiitix etettV 1221
tttVz 1122
tVdt
tVd
xR ˆ
Dynamics on the Bloch Sphere: Zero Detuning
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ECE 407 – Spring 2009 – Farhan Rana – Cornell University
x-axis
z-axis
Bloch Sphere
tV
ztV ˆ0
The state vector rotates in the z-y plane
The period of rotation is:
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
Dynamics on the Bloch Sphere: Zero Detuning
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
x-axis
z-axis
Bloch Sphere
tV
ztV ˆ0
The state vector rotates in the z-y plane
The period of rotation is:
2
y-axis
xR ˆ
tVdt
tVd
01221 itiitix etettV
01221 itiitiy etetitV
11122 tttVz
t = / :
2
2
)( eietii
Dynamics on the Bloch Sphere: Zero Detuning
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ECE 407 – Spring 2009 – Farhan Rana – Cornell University
x-axis
z-axis
Bloch Sphere
tV
ztV ˆ0
The state vector rotates in the z-y plane
The period of rotation is:
2
y-axis
xR ˆ
tVdt
tVd
01221 itiitix etettV
01221 itiitiy etetitV
11122 tttVz
t = 3/2 :
2
2
3
12
3 21
2
1)23( eieeet
iii
Dynamics on the Bloch Sphere: Zero Detuning
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
x-axis
z-axis
Bloch Sphere
tV
The state vector rotates in the z-y plane
The period of rotation is:
2
y-axis
xR ˆ
tVdt
tVd
01221 itiitix etettV
01221 itiitiy etetitV
11122 tttVz
t = 2/ :
1
21
2 eeti
Dynamics on the Bloch Sphere: Zero Detuning
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ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
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ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
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ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
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ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
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ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
itiitiy etetitV 1221
tttVz 1122
Suppose one wants to represent the above state on the Bloch Sphere:
The argument represents the phase relative to t+
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
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ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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?
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ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Photon Echo Experiments: Step 4
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Photon Echo Experiments: Step 5
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
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ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
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ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
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ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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!
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Ramsey Fringes: Step 2
?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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Ramsey Fringes: Step 2
?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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Ramsey Fringes: Step 3
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:
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Ramsey Fringes: Step 4
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Cesium Atomic Clocks
Ramsey fringes for a Cesium clcok (NIST)
NIST