ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Chapter 11: Matter-Photon Interactions and Cavity QED
In this lecture you will learn:
• Electron-Photon Interactions• Cavity QED• Light-Matter Entanglement• Noise in Optical Transitions
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Interaction between a Charged Particle and Field
In quantum mechanics we have:
kjjk itvmtr ˆ,ˆ
i
vm This implies:
In the presence of an E&M field the canonical momentum is:
ttrAqtvmtp ,ˆˆˆ
kjjk itptr ˆ,ˆ
The canonical momentum satisfies the equal-time commutation relation:
i
p
This implies:
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
The classical expression for the momentum of a field is:
trHtrErdtP oo ,,3
),(),(),( trEtrEtrE TL
The E-field can be decomposed as:
0),(),(),( trEtrtrE LLo
0),( trETo
Where:
),(),(),( tkEtkEtkE TL
In Fourier-Space:
oL
oL
tkkkitkEtkitkEk
),(ˆ
),(),(),(
0),( tkEk T
Longitudinal component
Transverse component
Classical Field Momentum
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
),(),( trAtrHo
We also have:
),(),( tkAkitkHo
The vector potential can also be decomposed as:
),(),(),( trAtrAtrA TL
),(.ˆˆ),( tkAkktkAL
),(.ˆˆ),(),(ˆˆ1),( tkAkktkAtkAkktkAT
The field momentum becomes:
trHtrErdtrHtrErdtP TooLoo ,,,, 33
? ?
Since:
( , ) ( , ) ( , )L TE k t E k t E k t
( , ) 0TA r t
,( , )L
o
r tE r t
Classical Field Momentum
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
3
3 3
3 3
3 3
3 3
3
3
3
, ,
, , , ,2 2
( , ) ˆ ˆ ˆ, ( , ) 1 ,2 2
( , ) ,2
,
o o L
o o L o L
oo
T
T
d r E r t H r t
d k d kE k t H k t E k t ik A k t
d k k t d ki k ik A k t k t k k A k tk
d k k t A k t
d r r t A r
,t
Look at the first term in more detail:
trrqtr ),(
For a single particle with position and charge q the charge density is:
ttrAqtrHtrErd TLoo ,,,3
Therefore:
r t
Classical Field Momentum
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
trHtrErdttrAqtP TooT ,,, 3
Therefore the field momentum becomes:
The total momentum of the particle and the field is:
trHtrErdttrAqtvmtP TooTtotal ,,, 3
Kinetic momentum of the particle
Transverse field momentum
Longitudinal field momentum
Coordinate of the particle
Total Momentum of Particle and Field
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Total Momentum of Particle and FieldFor the particle interacting with the field, define a canonical momentum as:
,p t mv t qA r t t
The total momentum of the particle and the field is then:
trHtrErdttrAqtvmtP TooTtotal ,,, 3
Kinetic momentum of the particle
Transverse field momentum
Longitudinal field momentum
3
3
, , , ,
, , ,total L o o T
L o o T
P t mv t qA r t t qA r t t d r E r t H r t
p t qA r t t d r E r t H r t
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Total Angular Momentum of Particle and Field 3 , ,total o oJ t r t mv t d r r E r t H r t
3
3
3
3
3
3
, ,
, ,
, ,
, ,
, ,
,
o o
o T
o L T
o T T
T
o T
d r r E r t H r t
d r r E r t A r t
d r r E r t A r t
d r r E r t A r t
d r r r t A r t
d r r E r
,Tt A r t
, , ,
, 0L T
L
A r t A r t A r t
A r t
2
, , ,,
. ,
,,
L T
Lo
Lo
E r t E r t E r t
r tE r t
r tE k t i k
k
trrqtr
),(For a single charged particle:
3
,
, ,
T
o T T
r t mv t qA r t t
d r r E r t A r t
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Total Angular Momentum of Particle and FieldWe therefore get:
3, , ,total T o o TJ t r t mv t qA r t t d r r E r t H r t
Using the canonical momentum defined as:
,p t mv t qA r t t
We get:
3, , ,total L o o TJ t r t p t qA r t t d r r E r t H r t
3 , ,o T Td r r E r t A r t
3 3, ,, , , ,o T T o s T s T
sd r E r t A r t d r E r t r A r t
Photon spin Photon orbital
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Total Angular Momentum of Particle and Field
3
3
3, ,
, , ,
,
, ,
, ,
total L o o T
L
o T T
o s T s Ts
J t r t p t qA r t t d r r E r t H r t
r t p t qA r t t
d r E r t A r t
d r E r t r A r t
Particle angular momentum
Photon spin angular momentum
Photon orbital angular momentum
The above decomposition is gauge invariant
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Canonical Momentum: Interpretation in the Coulomb GaugeAssume Coulomb gauge:
0),( trAL
trHtrErdtptP Toototal ,,3
The expression for the total momentum (particle + field) is:
Momentum of photons in the Coulomb gauge (Chapter 5) – momentum associated with the transverse part of the field
v
q ttrAqtvmtp T ,
In the Coulomb gauge, the canonical momentum of a charged particle is seen to consist of two parts: i) The kinetic momentum of the charged particle ii) The momentum contributed by the longitudinal field associated with the charged particle, the source of which is the charged particle itselfiii) In other gauges, the above interpretation does not hold
ttrAqtvmtp T ,
Then canonical momentum of the particle becomes;
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Classical Hamiltonian of Particles and Field
Assume several particles with:
Charges: Positions: Velocity:q tr tv
The classical Hamiltonian is:
2 31 1 1( , ). ( , ) ( , ). ( , )2 2 2o oH mv t d r E r t E r t H r t H r t
Or using the definition of the canonical momentum:
),().,(
21),().,(
21
2),( 3
2trHtrHtrEtrErd
mttrAqtpH oo
Note: the scalar electromagnetic field does not get included in the Hamiltonian
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Assume several particles with:
Charges: Positions: Velocity:q tr tv
The classical Hamiltonian is:
),().,(
21),().,(
21
2),( 3
2trHtrHtrEtrErd
mttrAqtpH oo
Note that:
),(),(),( trEtrEtrE TL
The longitudinal field energy is:
trtrqq
rrtrtrrd
ktktkkd
tkEtkEkdtrEtrErd
o
oo
LLoLLo
421
'4,',
21),(),(
21
2
),().,(21
2),().,(
21
323
3
3
33
Coulomb interaction energy of the charged particles
,( , )L
o
r tE r t
Classical Hamiltonian of Particles and Field
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
),().,(
21),().,(
21)(
2, 3
2trHtrHtrEtrErdtrqV
mttrAqtpH oTTo
Consider only a single particle moving in the static Coulomb potential of other particles and interacting with radiation:
Includes the static potential of all other fixed charges (not explicitly included)
Includes the longitudinal and transverse vector potential components
Quantizing Interacting Particle and Field
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Postulate the following equal-time commutation rules for the particle coordinates and canonical momenta:
kjjk itptr ˆ,ˆ
Choose Coulomb Gauge:
( , ) 0( , ) ( , )L
T
A r t
A r t A r t
,mv t p t qA r t t
i
p
Kinetic Momentum:
Quantizing Interacting Particle and Field
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
2
3ˆˆ ˆ , 1 1ˆ ˆ ˆ ˆˆˆ ( ) ( , ). ( , ) ( , ). ( , )2 2 2T
o T T o
p t qA r t tH qV r t d r E r t E r t H r t H r t
m
The Hamiltonian operator becomes (in the Heisenberg picture):
)(ˆ).(ˆ
21)(ˆ).(ˆ
21)ˆ(
2ˆˆˆˆ 3
2
rHrHrErErdrqVm
rAqpH oTToT
And in the Schrodinger picture:
Quantizing Interacting Particle and Field
The field can be quantized following the steps in Chapter 5
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Gauge Transformation
Suppose the electron is localized around or
or
rThe we can approximate the Hamiltonian as:
)(ˆ).(ˆ
21)(ˆ).(ˆ
21)ˆ(
2
ˆˆˆ 32
rHrHrErErdrqVm
rAqpH oTTooT
Now we will perform a gauge transformation to get rid of the vector potential
Suppose a quantum state of the particle-field system is:
Under a gauge transformation the new state is: Tnew ˆ
1ˆˆ TTThe operator has to be unitary:T
Any system operator will transform under as:T ˆ ˆˆ ˆnewO TOT O
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Gauge TransformationFor all physical observables, we must have:
TTHTTHH newnewnew ˆˆˆˆˆˆˆ
Hamiltonian is gauge invariant but will turn out to be not form invariant under the gauge transformation
ˆ ˆ ˆˆ ˆ ˆ ˆnew new newO O T TOT T
In particular, for the Hamiltonian we must have:
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
ˆˆ ˆˆˆˆ
new
T o
p TpT
p qA r
Suppose we choose:
ˆ ˆ( ).ˆ T o oi qA r r r
T e
Then:
3 .3
ˆ ˆˆ ˆ
1ˆ ˆˆ ˆ .2
1ˆ ˆ .
o
T new T
ik r rT j j o
jo
T T o oo
E r TE r T
d kE r k k q r r e
E r r r q r r
ˆ ˆˆ ˆ
ˆ
T new T
T
H r TH r T
H r
(Canonical momentum operator is not gauge invariant)
(E-field operator is not gauge invariant)
(Magnetic field operator is gauge invariant)
or
r
Gauge Transformation
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
ˆ ˆˆ ˆˆˆ
ˆˆ ˆˆ ˆˆ ˆ ˆ ˆ
new
T o
new T o
p TpT
p qA r
mv TmvT T p qA r T p
The relation:
In the new gauge, the kinetic momentum and the canonical momentum are the same!!!!
Therefore, in the new gauge the particle kinetic energy equals: 2
2 ˆ1 ˆ2 2new
pmvm
Implies:
In the old gauge:
2
2ˆˆ
1 ˆ2 2
T op qA rmv
m
Gauge Transformation
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
23
3
32
ˆ.ˆ2
12
)(ˆ).(ˆ21)(ˆ).(ˆ
21)(ˆ.ˆ)ˆ(
2ˆ
ˆˆˆˆ
ojoj
oTTooTo
new
rrkkd
rHrHrErErdrErrqrqVm
p
THTH
ˆ ˆ( ).ˆ T o oi qA r r r
T e
We have:
Dipole self-energy (infinite!)
We will use:
)(ˆ).(ˆ
21)(ˆ).(ˆ
21)(ˆ.ˆ)ˆ(
2ˆˆ 3
2rHrHrErErdrErrqrqV
mpH oTTooTo
PHIH FH
or
r
Gauge Transformation: The Dipole Hamiltonian
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Cavity Quantum Optics
21
Cavity
ˆ ˆ ˆ ˆP F IH H H H The Hamiltonian is:
2211
2221112
ˆˆ
ˆ2ˆˆ
NN
eeeerVqm
pHP
Consider a two-level system in an optical cavity
21ˆˆˆ aaH oF
)(ˆˆˆ ooI rErrqH
Recall that:
)(ˆˆ2
)(ˆ rUaairEo
o
)(ˆˆ2
)(ˆo
o
oo rUaairE
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Cavity Quantum Optics
21
Cavity 22112211
particleparticleˆ
1ˆ1ˆ
eeeeHeeee
HH
I
II
We can write the interaction Hamiltonian in a better form:
aakk
aaeekeekHI
ˆˆˆˆ
ˆˆˆ*
21*
12
2 1ˆ( ) .
2o
o oo
k q i U r e r r e
21
12
ˆˆ
eeee
Define:
We have:
Now make the rotating wave approximation:
ˆˆˆˆˆˆˆˆˆ *2211 akakaaNNH o
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Cavity Quantum Optics
21
Cavity
ˆˆˆˆˆˆˆˆˆ *2211 akakaaNNH o
The above time-independent Hamiltonian must have eigenstates and eigenenergies
Assume a solution for the eigenstate of the form:
2 1 1e n e n Then:
*2 2 1
1 1 2
*2 2 1 1
ˆ 1 1
( 1) 1 1
1 1 ( 1) 1
o
o
o o
H n e n k n e n
n e n k n e n
n k n e n k n n e n
Take the bra first with: and then with: ne 2 11 ne
EH ˆ
2*
1
11 ( 1)o
o
n k nE
k n n
EH ˆ
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Cavity Quantum Optics: Eigenenergies and Eigenstates
21
Cavity Detuning: oo )( 12
We will assume zero detuning from now onwards
2
2
( 1) 1
( 1) 1o
o
E n n k n
E n n k n
The two eigenvalues and eigenstates are:
*2 1
*2 1
1( 1) 12
1( 1) 12
kn e n e nk
kn e n e nk
...............3,2,1,0n
Each value of “n” will give two eigenstates and eigenenergies
2*
1
11 ( 1)o
o
n k nE
k n n
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Spectrum of Eigenenergies
21
Cavity
Suppose there was no electron-photon interactionSuppose zero-detuning
Then we would have the following eigenstates and eigenenergies:
1 1
2 1
2 2
2 1
2 2
2 1
2 2
Energies States
0
1 0
2 1
1 1 1
o
o
o
o
ε e
ε eε e
ε eε e
ε n e nε n e n
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Spectrum of Eigenenergies: Rabi Splitting
21
Cavity
1 1
2
2
2
2
2
2
Energies States
0
11
2 22 2
1 1
o
o
o
o
ε e
ε kε k
ε kε k
ε n k n nε n k n n
When the electron-photon interaction is turned on, we have the following eigenstates and eigenenergies:
The interaction lifts the degeneracy between the previously degenerate pair of states and splits their energies
This is called Rabi splitting and equals: 2R k n
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Quantum Rabi Oscillations
21
Cavity
Suppose: 20t e n
n
210 1 12
t e n n n
Need to find: t
Proceed as follows:
2
2
ˆ
1 1
2 1
0
12
cos 1 sin 1 1
o
o
Hi t
k ki n t i n t i n t
i n t
t e t
e e n e n
k kke n t e n i n t e nk
tnk
tne
tnk
tne
1sin1
1cos
221
222
This implies:
12 nk
R
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Vacuum Rabi Oscillations
21
Cavity n=0
Suppose:
210 0 1 12
t e
1sin0cos 12
2et
kkkiet
ket
tεi
Then:
These are called vacuum Rabi oscillations!
Vacuum Rabi frequency is:
2VRk
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Single Two-Level System in a Cavity
21
ˆˆˆˆˆˆˆˆˆ *2211 akakaaNNH o
* 1 2ˆ ˆ ˆ ˆ ˆˆ ˆ2 2z oH a a k a k a
2 1ˆ ˆz N N
Write this as:
Where:
*ˆ ˆ ˆ ˆ ˆˆ ˆ2 z oH a a k a k a
Just use:
The commutator algebra of is similar to that of Pauli matrices for spin-1/2 systems
ˆ ˆ, ,z
ˆ ˆˆ
2x yi
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ, 2 , 2 , 2
ˆ ˆ ˆ, 2x y z y z x z x y
z
i i i
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
21
21
*
2 2 2
1 1 1
ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ2
ˆ ˆ ˆˆ ˆ
z o
z zj j jj j j
H J k J a k a J a a
J J J
2 Two-Level Systems in a Cavity and Spin Systems
Any state of the atoms (or the two-level systems) can be written using the following basis set:
,
,
,
,
But it is more insightful to use a different basis set …………
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
2 Two-Level Systems in a Cavity and Spin Systems
21
21
*
2 2 2
1 1 1
ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ2
ˆ ˆ ˆˆ ˆ
z o
z zj j jj j j
H J k J a k a J a a
J J J
Think “addition of angular momenta” in undergraduate quantum mechanics textbooks
Consider the three triplet states and the one singlet state:
ˆ , 2 ,
1 1ˆ , , 0 , ,2 2
ˆ , 2 ,
z
z
z
J
J
J
1 1ˆ , , 0 , ,2 2zJ
2 2 2 2
2
ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ 2z x y
z
J J J J
J J J J J
Multiplet with eigenvalue 8 for the operator2J
Multiplet with eigenvalue 0 for the operator2J
Triplets:
Singlet:
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
2 Two-Level Systems in a Cavity: Radiation Cascades
,
1 , ,2
,
1 , ,2
n
21
21
Triplet Multiplet Singlet
J
J
Multiplet with eigenvalue 8 for the operator2J
Multiplet with eigenvalue 0 for the operator2J
*
2 2 2
1 1 1
ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ2
ˆ ˆ ˆˆ ˆ
z o
z zj j jj j j
H J k J a k a J a a
J J J
Dark entangled state !!
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
2 Two-Level Systems in a Cavity
21
21
1 , ,2
ˆ ˆ 0
S
J S J S
The singlet state is a “dark eigenstate” – it does not interact with photons inside the cavity
Within the multiplet of the three triplet states, we can write the Hamiltonian as:
*
*
2 1 0ˆ 2 1 0 1 2 2
0 2 2 2
o
o
o
n k n
H k n n k n
k n n
,
1 , , 12, 2
n
n
n
1 , ,2
n
System eigenstates
The three eigenenergies, assuming zero detuning, are:
2 2 * 3
2 2 * 3
o
o
o
n k nn
n k n
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
2 Two-Level Systems in a Cavity: Matter Entanglement Generation
21
21
Suppose at time t=0, the quantum state was prepared such that both two-level systems were in the upper state:
Photon
detector
0 , 0t
Suppose that the Rabi period and the decoherence times are both much longer than then photon lifetime in the cavity
Suppose at time t = T a photon is detected at the cavity output waveguide
Then what is post-detection?
1 , , 02
t T
t T
(with a very high likelihood)
, 1t T (less likely, under the stated assumptions)
*ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ2 z oH J k J a k a J a a
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
N Two-Level Systems in a Cavity
21
21
21
21
21
*
1 1 1
ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ2
ˆ ˆ ˆˆ ˆ
z o
N N Nz zj j j
j j j
H J k J a k a J a a
J J J
Several multiplets are possible corresponding to all possible eigenvalues of the operator:
2 2 2 2 2ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ2z x y zJ J J J J J J J J
One multiplet is particularly interesting, whose first state is:
2ˆ , 2 ,ˆ , ,
2 2, 2,
2
z
J M N N N M N
J M N M M N
N N M MJ M N M N
, , , ,.......N N
2,N N
4,N N
,N N
Within this multiplet:
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
N Two-Level Systems in a Cavity
21
21
21
21
21
Suppose at time t=0, the quantum state was prepared such that all two-level systems were in the lower state and there was one photon inside the cavity:
Find the system eigenstates, and then the state at time = t ?
0 , , ,...... 1 , 1t N N
The state will couple only with the state when interaction with photons is turned on
So in this subspace, the Hamiltonian is:
, 1N N 2, 0N N
*
2ˆ
2 o
N NkH
NNk
Eigenenergies, assuming zero detuning, are:
2
2
o
o
N Nk
N Nk
Rabi splitting = 2VR N k
Coupling is now times stronger!!N
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
N Two-Level Systems in a Cavity
1
2cos , 1 sin 2, 0
N
i t N k N kkt e t N N i t N Nk
The state at time t is then:
21
21
21
21
21
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
N Two-Level Systems and Dicke’s Superradiance
21
21
21
21
21
Consider N two-level systems all prepared in the upper state
The rate of spontaneous emission observed from this collection is expected to be proportional to N
But certain highly correlated and entangled matter states can radiate at a rate proportional to ~N2
Robert H. Dicke, "Coherence in Spontaneous Radiation Processes,” Physical Review, 93, 99–110 (1954). From Chapter 10:
Suppose 0 ,t N N ˆ ˆ0 0t J J t N
Therefore:
Photon Flux ˆ ˆ ˆ ˆ, ,r rE r t E r t J t J tc c
ˆ ˆ, ,r rE r t E r t Nc c
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
N Two-Level Systems and Dicke’s Superradiance
21
21
21
21
21
Consider N two-level systems now all prepared in the highly correlated state (assuming N is even):
Robert H. Dicke, "Coherence in Spontaneous Radiation Processes,” Physical Review, 93, 99–110 (1954).
0 0,t M N
21
What does this state looks like:
ˆ0 0, , , , ,.... , , , , ,....t M N S
N/2 N/2The operator makes a superposition such that the resulting state is completely symmetrical across all the two-level systems and is properly normalized
S
22ˆ ˆ0 04 4
N N Nt J J t
Now:
And:
Why is this the case?
2ˆ ˆ, ,r rE r t E r t Nc c
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
N Two-Level Systems and Dicke’s Superradiance
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Optical Transitions and Fermi’s Golden Rule
L
m
Stimulated Absorption:
mmmmmo
mmmLL akakaaNNH ˆˆˆˆˆˆˆˆˆ *
net L 0
mmoLm
mmoLLmmm
mmoLLIm
nk
neaken
neHenR
2
2
2
2
ˆˆ12
ˆ12n
Stimulated and Spontaneous Emission: Hmn
mmmmmo
mmmHH akakaaNNH ˆˆˆˆˆˆˆˆˆ *
net H 0
mmoHm
mmoHHmmm
mmoHHIm
nk
neaken
neHenR
12
ˆˆ12
ˆ12
2
2*
2
Agrees with the Einstein’s thermodynamic argument
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Spontaneous Emission in Free Space
1 2
z
y
x 2 1
kεV
etkatkaε
ikdVtrE jrki
jjj
o
k
ˆ,ˆ,ˆ22
,ˆ .
3
3
31 1 2 2 3
*
,
ˆ ˆ ˆ ˆ ˆ2
ˆ ˆˆ ˆ +
k j jj
j j j jj k
d kH N N V a k a k
k a k k a kV
The Hamiltonian is:
The coupling parameter is (assuming only the z-component of the field couples):
Initial and final states:
02initial e
jke ,1final 1 Many possible final states
2 1ˆ ˆ ˆ.2
kj j z
ok i ε k e q e z e
ε
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Spontaneous Emission in Free Space1 2
z
y
x 2 1
22 2 2 1,
,3 22
2 1 2 13
222 1
2
2 ˆ( ) 1 0
2 1ˆ ˆ .22
2 3 2
sp I kk jj k
kj k
j o
o
R FS e H e
d kV q ε k e z eε V
q e z eD
ε
Use Fermi’s Golden Rule:
Photon Density of States:
032
2
03
2
3
3
242
22
cdkcdkkkd
32
2
cD
Number of modes (of both polarizations) per unit volume per unit frequency interval
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Spontaneous Emission in a Cavity: The Purcell Effect
21
Cavity
E. M. Purcell "Spontaneous emission probabilities at radio frequencies" Phys. Rev. 69, 681 (1946)
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Spontaneous Emission in a Cavity: The Purcell Effect
21
CavityConsider a lossless optical cavity containing a two level system
Can we find the spontaneous emission rate inside the cavity?
The above question does not make sense because in a lossless closed cavity there are only Rabi oscillations (no Markovian optical transitions)
We need to incorporate loss into the cavity!
So now consider a lossy optical cavity containing a two level system
And now we ask the same question: what is the spontaneous emission rate for the two level system inside the cavity?
21
CavityWaveguide
z = 0
ˆˆˆˆˆˆˆˆˆ *2211 akakaaNNH o
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Spontaneous Emission in a Cavity: The Purcell Effect
21
CavityWaveguide
z = 0
Consider a cavity coupled to a waveguide
Step 1: Assume there is no two-level system
tiL
p
g
po oetzb
vtai
dttad
,0ˆˆ
21ˆ
L
eadvL
LeadLetzb
tiL
g
titi
Lo
o
o
o
o
ˆ
2ˆ
2,0ˆ
2
2
2
2
L
eadLv
taidt
tad tiL
pgpo
o
o
ˆ
21ˆ
21ˆ 2
2
L
eea
iidL
veeata
poo
o
po
ttiti
Lpopg
tti
22
2
2 ˆ22
1ˆˆ
Solution is:
L
eai
idLv
tati
Lpopg
o
o
ˆ22
1ˆ2
2
As t→∞:
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
tiL
pg
tiR oo etzbta
vetzb
),0(ˆˆ1),0(ˆ 2
1
CavityWaveguide
z = 0
Lpo
poR
tiL
po
po
g
tiR
g
aii
a
Lea
iid
vL
Lead
vL o
o
o
o
ˆ22
ˆ
ˆ22
2ˆ
2
2
2
2
2
As t→∞:
L
eai
idLv
tati
Rpopg
o
o
ˆ22
1ˆ2
2
We had:
L
eai
idLv
tati
Lpopg
o
o
ˆ22
1ˆ2
2
L
ai
idLv
a Rpopg
o
o
1ˆ22
1ˆ2
2
SP
HP
Spontaneous Emission in a Cavity: The Purcell Effect
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
21
CavityWaveguide
z = 0
Now see how the interaction Hamiltonian gets modified:
L
ai
idLv
ik
La
iidL
vkakak
Rpopg
Rpopg
o
o
o
o
1ˆˆ22
1
1ˆˆ22
1ˆˆˆˆ
2
2
*
2
2
*
Step 2: Include the two-level system
Initial and final states:
02initial e
final 1 1 Re
Assume that:1
2p VRpk
Spontaneous Emission in a Cavity: The Purcell Effect
A continuum of possible final states
ˆˆˆˆˆˆˆˆˆ *2211 akakaaNNH o
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
2 22 1222
2
2 22
2Cavity2 1 2
1 2 2
1 2
o
o
psp
o p
p
o p
k dR
k
Spontaneous Emission in a Cavity
Using Fermi’s Golden Rule:
One may use Fermi’s Golden Rule provided:1
2p VRpk
212
2212 ˆ).(
2ˆ.)(
2ezerUqkezerUiqk o
o
oo
o
o
21
CavityWaveguide
z = 0
The coupling parameter was:
Define mode volume Vp as:
1)().(1)().( *3* rUrUrdV
rUrUp
oo
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Purcell Enhancement of Spontaneous Emission in a Cavity
21
CavityWaveguide
z = 0
2222 1
2 22
1 262Cavity3 2 1 2
pposp
o p o p
q e z eR
ε V
We have:
At resonance we have:
22
2 1 22
62Cavity3 2
posp VR p
o p
q e z eR
ε V
In free space we had:
22
2 12
23 2 o
q e z eR FS D
ε
The ratio of the spontaneous emission rate in the cavity w.r.t. free space is:
6 1p
pV D
Cavity quality factor:
pQ
1VR p
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Purcell Enhancement of Spontaneous Emission in a Cavity
21
CavityWaveguide
z = 0
2 23
43
p
QV
n
Purcell Enhancement Factors:
1) Spontaneous emission enhancement factor in a dielectric cavity of average index ncompared to in vacuum:
2) Spontaneous emission enhancement factor in a dielectric cavity of average index ncompared to in an infinite dielectric media of the same refractive index n :
2 33
43
p
QV
n
6 1p
pV D
pQ
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Spontaneous Emission in a Cavity with Decoherence Included
2 2
2 1222
2Cavity2 1 2
o
o
psp
o p
k dR
Using Fermi’s Golden Rule (our previous result):
The effect of decoherence of the material polarization can be included by broadening the energy conserving delta function:
2 22
2 2 2222 2
22
2 222
12 1Cavity2 11 2
1 1 22 1 1 2
o
o
psp
o p
p
o p
k d TRT
Tk
T
2
2
2 2VR p
p
T
T
on resonance
21
CavityWaveguide
z = 0
Assume: 21,
2pVR
Tk
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Spontaneous Emission Rate in a Cavity: Different Regimes
21
CavityWaveguide
z = 0
The more general case: 21,
2pVR
Tk
p1
2
1T
o
p1
2
1T
2
2
2 2VR p
p
T
T
on resonance
o
o
2 22
2 2 2222 2
22
2 222
12 1Cavity2 11 2
1 1 22 1 1 2
o
o
psp
o p
p
o p
k d TRT
Tk
T
Purcell RegimeIncreasing the cavity Q increases the spontaneous emission rate
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Spontaneous Emission Rates in a Cavity: Different RegimesIn vacuum we had:
22
2 12
23 2sp
o
q e z eR FS D
ε
2222 1
2 22
1 262Cavity3 2 1 2
pposp
o p o p
q e z eR
ε V
In a cavity when:
we had:
21
CavityWaveguide
z = 0
2 21, ,
2p pVR
T Tk
In a cavity when:
we had:2 2
1, ,2 p p
VRT T
k
22
2 2 22
2 222 1 22
2 2 22
1Cavity 21
12 33 2 1
spo
o
o p o
k TRT
q e z e TTε V T
22
2VRT
on resonance
Purcell Regime
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Enhancement of Spontaneous Emission in a Cavity
21
CavityWaveguide
z = 0
2 23
2
24 13
p p
QV
nT
Cavity Spontaneous Enhancement Factors:
1) Spontaneous emission enhancement factor in a dielectric cavity of average index ncompared to in vacuum:
2) Spontaneous emission enhancement factor in a dielectric cavity of average index ncompared to in an infinite dielectric media of the same refractive index n :
2 33
2
24 13
p p
QV
nT
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Weak, Intermediate, and Strong Coupling in Cavity Quantum Optics
21
CavityWaveguide
z = 0
Weak Coupling:
2
2 1 1,2VR
p
kT
Strong Coupling:
2
21 1,2 VR
p
kT
Vacuum Rabi frequency is much smaller than the decoherence and the photon decay rates
Vacuum Rabi frequency is much larger than the decoherence and the photon decay rates
Intermediate Coupling:
2
21 12VR
p
kT
Vacuum Rabi frequency is much smaller than the decoherence and the photon decay rates
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Spontaneous Emission Spectrum in the Strong Coupling Limit
21
CavityWaveguide
z = 0 ˆ , oi tLb z t e
ˆ , oi tRb z t e
2
ˆ 1 ˆˆ ˆˆ dd t ii t k N t a t G t
dt T
ˆ 1 ˆˆ ˆ 0,
2og i t
o Lp p
vda t ii a t k t b t edt
Material polarization:
Cavity field equation:
ˆ
ˆ ˆˆ ˆ2dd N t i k t a t k a t tdt
Population equation:
2
21 1,2 VR
p
kT
Assume a strong coupling regime:
1ˆ ˆˆ0, 0,o oi t i tR L
g pb z t e a t b z t e
v
Equation for the field outside the cavity:
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
21
CavityWaveguide
z = 0 ˆ , oi tLb z t e
ˆ , oi tRb z t e
1ˆ ˆˆ0, 0,o oi t i tR L
g pb z t e a t b z t e
v
Equation for the field outside the cavity:
The outside the cavity field correlation function becomes:
ˆ ˆ ˆ ˆ0, 0, oo i ti tR Rb z t e b z t e a t a t
Initial state at time t:
10 1t e
Possible states of the cavity system as the time evolves:
2 1 10 1 0e e e
Within this reduced Hilbert space:
ˆ ˆ ˆ 0dN t a t a t
Spontaneous Emission Spectrum in the Strong Coupling Limit
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
21
CavityWaveguide
z = 0 ˆ , oi tLb z t e
ˆ , oi tRb z t e
therefore we can use the simpler linear equations:
2
2
ˆ 1 ˆˆ ˆˆ
1 ˆˆˆ
dd t ii t k N t a t G t
dt T
ii t k a t G tT
ˆ 1 ˆˆ ˆ 0,
2g i t
o Lp p
vda t ii a t k t b t edt
Spontaneous Emission Spectrum in the Strong Coupling Limit
Within this reduced Hilbert space:
ˆ ˆ ˆ 0dN t a t a t
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
21
CavityWaveguide
z = 0 ˆ , oi tLb z t e
ˆ , oi tRb z t e
Spontaneous Emission Spectrum in the Strong Coupling Limit
2
ˆ 1 ˆˆd t ii t k a t
dt T
ˆ 1 ˆ ˆ
2op
d a t ii a t k tdt
Consider the averaged equations:
Solution subject to the conditions:
is: ˆˆ 0 1 0 0t a t
*
ˆ sinoi t t kka t t i e t tk
2
2
21 1,2
1 122
0
VRp
p
o
kT
T
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
21
CavityWaveguide
z = 0 ˆ , oi tLb z t e
ˆ , oi tRb z t e
The correlation functions become:
2
ˆ ˆ 1 ˆ ˆ ˆˆ
ˆˆ
d a t t ii a t t k a t a td T
a t G t
ˆ ˆ 1 ˆ ˆ ˆ ˆ2
ˆˆ 0,
op
g i tL
p
d a t a t ii a t a t k a t td
va t b t e
0
0
Spontaneous Emission Spectrum in the Strong Coupling Limit
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
21
CavityWaveguide
z = 0 ˆ , oi tLb z t e
ˆ , oi tRb z t e
Spontaneous Emission Spectrum in the Strong Coupling Limit
2
ˆ ˆ 1 ˆ ˆ ˆˆd a t t ii a t t k a t a t
d T
ˆ ˆ 1 ˆ ˆ ˆ ˆ2o
p
d a t a t ii a t a t k a t td
Given the linearity of the equations, it is not difficult, but tedious, to show that:
*ˆ ˆa t a t t t
*
ˆ sinoi t t kka t t i e t tk
2
2
21 1,2
1 122
0
VRp
p
o
kT
T
Where:
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Spontaneous Emission Spectrum in the Strong Coupling Limit
21
CavityWaveguide
z = 0 ˆ , oi tLb z t e
ˆ , oi tRb z t e
*ˆ ˆa t a t t t
*
ˆ sinoi t t kka t t i e t tk
2
2
21 1,2
1 122
0
VRp
p
o
kT
T
The correlation function is not stationary!!!
How to find the spectrum??
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
21
CavityWaveguide
z = 0 ˆ , oi tLb z t e
ˆ , oi tRb z t e
Spontaneous Emission Spectrum in the Strong Coupling Limit
*
2
*
2
2
2
ˆ ˆ .
ˆ ˆ2
.
2
.
2
.
2
i
i
i ti t
i t
dt d a t a t e c cS
dt a t a t
dt d t t e c c
dt t
dt t e d t e c c
dt t
dt t e c c
dt t
2 2
2 22 2
4
o o
k
k k
2
2
21 1,2
1 122
0
VRp
p
o
kT
T
For non-stationary correlation functions, the spectral density is:
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
21
CavityWaveguide
z = 0 ˆ , oi tLb z t e
ˆ , oi tRb z t e
Spontaneous Emission Spectrum in the Strong Coupling Limit
2VRk
Spontaneous emission spectrum
Linewidth (FWHM) = 2
1 12 pT
2 2
2 22 2
4
o o
kS
k k
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Spontaneous Emission Spectrum in the Strong Coupling Limit
21
CavityWaveguide
z = 0 ˆ , oi tLb z t e
ˆ , oi tRb z t e
1n
1 0e
1n VR
o
2 0e
1 1e
1 1
2
2
2
2
2
2
Energies States
0
11
2 22 2
1 1
o
o
o
o
ε e
ε kε k
ε kε k
ε n k n nε n k n n
Dressed states
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Transmission through a Strongly Coupled Cavity
21
o
Consider an empty-cavity transmission experiment:
A resonance in transmission will be observed at the frequency o
Consider a cavity transmission experiment when the contains a two-level system:
What will be the frequencies of transmission resonances now??
o
21
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Transmission through a Strongly Coupled Cavity
2
2 1 1,VRp
kT
Assume strong coupling:
Input state:
ˆ , i t i tR inb z t e e CW coherent state of frequency
Output state:
ˆ , i t i tR outd z t e e
Statement of the problem:
Find as a function of the frequency out
in
21
o ˆ , i tLb z t e
ˆ , i tRb z t e
ˆ , i tLd z t e
ˆ , i tRd z t e
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
2
ˆ 1 ˆˆ ˆˆ dd t ii t k N t a t G t
dt T
ˆ 1 1 ˆ ˆˆ ˆ 0, 0,
2 2g gi t i t
o R Lp p p p
v vda t ii a t k t b t e d t edt
Material polarization:
Cavity field equation:
1
ˆ ˆ 1 ˆˆ ˆˆ ˆ2 2d dN
d N t N t i k t a t k a t t F tdt T
Population equation:
1ˆ ˆˆ0, 0,
1ˆ ˆˆ0, 0,
i t i tR L
g p
i t i tL R
g p
d z t e a t d z t ev
b z t e a t b z t ev
Output field equations:
21
o ˆ , i tLb z t e
ˆ , i tRb z t e
ˆ , i tLd z t e
ˆ , i tRd z t e
Transmission through a Strongly Coupled Cavity
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Transmission through a Strongly Coupled Cavity: Small Input Power
Possible states of the cavity system when the input power is very small:
2 1 10 1 0e e e
Within this reduced Hilbert space:
ˆ ˆ ˆ 0dN t a t a t
2
2
ˆ 1 ˆˆ ˆˆ
1 ˆˆˆ
dd t ii t k N t a t G t
dt T
ii t k a t G tT
ˆ 1 ˆˆ ˆ 0,
2g i t
o Lp p
vda t ii a t k t b t edt
So we can write:
21
o ˆ , i tLb z t e
ˆ , i tRb z t e
ˆ , i tLd z t e
ˆ , i tRd z t e
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
ˆ i tt e
ˆ i ta t e
2
ˆ 1 ˆˆd t ii t k a t
dt T
ˆ 1 1 ˆ ˆ2 2
ˆ ˆ 0, 0,
op p
g gi t i tR L
p p
d a t ii a t k tdt
v vb t e d t e
Assume in steady state:
ˆ , i t i tR inb z t e e
21
o ˆ , i tLb z t e
ˆ , i tRb z t e
ˆ , i tLd z t e
ˆ , i tRd z t e
Transmission through a Strongly Coupled Cavity: Small Input Power
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
22
22
gin
po
p
ii vT
ki iT
2
22
gin
po
p
ki v
ki iT
21
o ˆ , i tLb z t e
ˆ , i tRb z t e
ˆ , i tLd z t e
ˆ , i tRd z t e
Transmission through a Strongly Coupled Cavity: Small Input Power
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
2
ˆ ˆ 1 ˆ ˆ ˆˆd a t t ii a t t k a t a t
d T
ˆ ˆ 1 1 ˆ ˆ ˆ ˆ2 2
ˆ ˆˆ ˆ 0, 0,
op p
g gi t i tR L
p p
d a t a t ii a t a t k a t td
v va t b t e a t d t e
* iine
0
Need:
1ˆ ˆ ˆ ˆ0, 0, i ti tR R
g pd z t e d z t e a t a t
v
21
o ˆ , i tLb z t e
ˆ , i tRb z t e
ˆ , i tLd z t e
ˆ , i tRd z t e
Transmission through a Strongly Coupled Cavity: Small Input Power
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
21
o ˆ , i tLb z t e
ˆ , i tRb z t e
ˆ , i tLd z t e
ˆ , i tRd z t e
Since a steady state does exist, and correlation function is going to be stationary, one can solve via Fourier transformation and obtain:
2
222
22
ˆ ˆ 2 giin
po
p
ivT
d a t a t eki i
T
Spectral density of the input:
2ˆ ˆ0, 0, 2i ti t iR R ind b z t e b z t e e
Spectral density of the output:
2ˆ ˆ0, 0, 2
1 ˆ ˆ
i ti t iR R out
i
g p
d d z t e d z t e e
d a t a t ev
Transmission through a Strongly Coupled Cavity: Small Input Power
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
21
o ˆ , i tLb z t e
ˆ , i tRb z t e
ˆ , i tLd z t e
ˆ , i tRd z t e
Spectral density of the inputSpectral density of the output
=
2
222
22
1
po
p
iT
ki iT
2
2
21 1,2
1 122
0
VRp
p
o
kT
T
222
22 22 2
1 1o
po o
T
k k
The transmission spectrum is given by:
Transmission through a Strongly Coupled Cavity: Small Input Power
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
21
o ˆ , i tLb z t e
ˆ , i tRb z t e
ˆ , i tLd z t e
ˆ , i tRd z t e
2VRk
The transmission at goes to zero as T2 goes to ∞
The zero transmission is a result of interference in transmission via the two Rabi-split cavity resonances
Transmission through a Strongly Coupled Cavity: Small Input Power
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Transmission through a Strongly Coupled Cavity: Photon Blockade
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Spectrum of Eigenenergies: Rabi Splitting
21
Cavity
When the electron-photon interaction is turned on, we have the following eigenstates and eigenenergies:
The interaction lifts the degeneracy between the previously degenerate pair of states and splits their energies
This is called Rabi splitting and equals: 2R k n
1 1
2
2
2
2
2
2
Energies States
0
11
2 22 2
1 1
o
o
o
o
ε e
ε kε k
ε kε k
ε n k n nε n k n n
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Transmission through a Strongly Coupled Cavity: First Photon Entry
21
21
Initial state Final state
0E oE k
o
k
Maximum transmission of the first incident photon if its energy is: oE k
So you choose your laser photon energy to be:
oE k
Abs
orpt
ion
spec
tra
o k Energy reqd:
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
21
Initial state
oE k
21
Final state
2 2oE k
o
Maximum transmission of the second incident photon if its energy is:
Transmission through a Strongly Coupled Cavity: Second Photon Entry
2 1o k
Energy reqd:
But you have already chosen your laser photon energy to be: oE k 2 1o k
k
2 1k
• Second photon is less likely to enter the cavity when the first photon is already present because the second photon is now detuned!
• Photons are more likely to enter the cavity one at a time
Abs
orpt
ion
spec
tra
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Transmission through a Strongly Coupled Cavity: Anti-Bunching and Bunching
o k
o Bunched light
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Decoherence in a Two-Level System: Heisenberg PictureConsider a two-level system with the Hamiltonian:
2211 ˆˆˆ NNH
1 2
2 1
2 1
ˆ ˆ0 0
ˆ ˆ
ˆ ˆ
dN t dN tdt dt
d t i tdt
d t i tdt
Suppose the system is not interacting with anything:
Now lets try to introduce pure decoherence that destroys the superpositon between the upper and lower levels:
2
2ˆˆˆ
ˆˆˆ
Ttti
dttd
Ttti
dttd
12
The decay terms destroy the quantum mechanical consistency of the equations!
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
12
t
Tittt
2
11ˆˆ
t
Tittt
2
11ˆˆ
tNtNttNttN
tT
tNtNttNttN
tT
tttttt
1212
21212
2
ˆˆˆˆ
21ˆˆˆˆ
21ˆ,ˆˆ,ˆ
Solve the equations to order t in time:
And find the equal-time commutator to order t in time:
Wrong!
Decoherence in a Two-Level System: Heisenberg Picture
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Modeling Decoherence: Langevin Sources12
ˆ ˆG t G t
2
2
ˆ 1 ˆˆ ˆ
ˆ 1 ˆˆ ˆ
d t i t t G tdt T
d t i t t G tdt T
Add Langevin noise sources:
We make the assumptions:
a) Noise operators act in their own Hilbert space
b) System operators at time t commute with the noise operators at time t’ if t’ > t
c)
d)
e)
f)
ˆ ˆ 0G t G t
1 2 1 1 2ˆ ˆG t G t A t t t 1 2 1 1 2
ˆ ˆG t G t B t t t
1 2 1 1 1 2ˆ ˆ,G t G t A t B t t t
1 2 1 2ˆ ˆ ˆ ˆ 0G t G t G t G t
Assume now that: 1 2ˆ ˆ0dN t dN t
dt dt
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Modeling Decoherence: Langevin Sources 12 The solution to order t in time is:
2
2
1 ˆˆ ˆ 1
1 ˆˆ ˆ 1
t t
t
t t
t
t t t i t G t dtT
t t t i t G t dtT
STEP 1: Multiply the second with the first and take the average: tttN ˆˆˆ 2
2 22
2 ˆ ˆˆ ˆ ˆ ˆ1
ˆ ˆ " "
t t t t
t tt t t t
t t
N t t N t t t G t dt G t dt tT
dt dt G t G t
22
2
22
2
22 2
2 2
ˆ 2 ˆ ˆˆ " "
ˆ 2 ˆ
ˆ2 2ˆ ˆ
t t t t
t t
dN tt N t t dt dt G t G t
dt T
dN tN t A t
dt T
dN tA t N t N t
T dt T
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Modeling Decoherence: Langevin Sources 12
STEP 2: Multiply the second with the first and take the average:
tttN ˆˆˆ1
tNTdt
tNdtNT
tB 12
11
2ˆ2ˆˆ2
STEP 3: Take the commutator of the first and the second:
2
2 1 2 12
2ˆ ˆ ˆ ˆ, , 1
ˆ ˆ,
2ˆ ˆ ˆ ˆ 1
ˆ ˆ,
t t t tt t
t t t tt t
t t t t t t tT
dt dt G t G t
N t t N t t N t N t tT
dt dt G t G t
2 12 1
2 2
ˆ ˆ' ' 2 2ˆ ˆ ˆ ˆ' , " ' ' ' "dN t dN tG t G t N t N t t t
dt dt T T
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Modeling Non-Radiative Transitions: Heisenberg Picture
Consider the Hamiltonian:
2211 ˆˆˆ NNH
Suppose we include non-radiative transitions:
1
21
122
ˆˆ
ˆˆ
TtN
dttNd
TtN
dttNd
12
1T
Check to see if the equations remain consistent:
122
122
1ˆˆ
ˆˆ
TttNttN
TtN
dttNd
122
22
21
22
22
21ˆˆ
ˆˆ21ˆˆ
TttNttN
tNtNT
ttNttNNow square both sides:
Not consistent!
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
12
1TTry fixing by adding Langevin noise sources:
tFT
tNdt
tNdNˆ
ˆˆ
122 tFtF NN ˆˆ
Since:
0ˆˆ 21 tNtNdtd
We must have:
tFT
tNdt
tNdNˆ
ˆˆ
121
The equations give:
111
22 ˆ1ˆˆ tFdtT
ttNttN Ntt
t
Now square both sides:
ttt NN
ttt
ttt N
tFtFdtdt
dttFT
ttNT
ttNttN
2121
111
21
22
ˆˆ
ˆ1ˆ221ˆˆ
1 2 1 1 2ˆ ˆN NF t F t C t t t
Modeling Non-Radiative Transitions: Heisenberg Picture
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
12
1T
ttt NN
ttt
ttt N
tFtFdtdt
dttFT
ttNT
ttNttN
2121
111
21
22
ˆˆ
ˆ1ˆ221ˆˆ
Take the average:
2 21
2ˆ ˆ 1 tN t t N t C t tT
2
1
ˆaverage relaxation rate
N tC t
T
The noise source therefore represents the shot noise associated with the non-radiative transitions
tFT
tNdt
tNdNˆ
ˆˆ
122
tFT
tNdt
tNdNˆ
ˆˆ
121
Modeling Non-Radiative Transitions: Heisenberg Picture
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Cavity Quantum Dynamics: Complete Set of Equations
ˆˆˆˆˆˆˆˆˆ *2211 akakaaNNH o
The Hamiltonian is:
The Heisenberg equations are:
2 2
1
1 2
1
ˆ ˆ ˆˆ ˆˆ ˆ
ˆ ˆ ˆˆ ˆˆ ˆ
N
N
d N t N t i k t a t k a t t F tdt T
dN t N t i k t a t k a t t F tdt T
2 12
2 12
ˆ 1 ˆˆ ˆˆˆ
ˆ 1 ˆˆ ˆ ˆˆ
d t ii t k a t N t N t G tdt T
d t ii t k N t N t a t G tdt T
21
Cavity
Waveguide
z = 0
ˆ , oi tLb z t e
ˆ , oi tRb z t e
1T
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Field Equations:
ˆ 1 1 ˆˆ ˆ2
ˆ 1 1 ˆˆ ˆ2
o inp p
o inp p
da t ii a t k t S tdt
da t ii a t k t S tdt
1ˆ ˆ ˆˆ0, 0,
1ˆ ˆ ˆˆ0, 0,
o o
o o
i t i tout g R g L
p
i t i tout g R g L
p
S t v b z t e a t v b z t e
S t v b z t e a t v b z t e
Cavity Quantum Dynamics: Complete Set of Equations
21
Cavity
Waveguide
z = 0
ˆ , oi tLb z t e
ˆ , oi tRb z t e
1T
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Cavity Quantum Dynamics: Complete Set of Equations
2
1
Cavity
Waveguide
z = 0
ˆ , oi tLb z t e
ˆ , oi tRb z t e
1TOptical pumping can be included in a similar fashion with a corresponding Langevin noise source: PT
The Heisenberg equations become:
2 1 2
1
1 1 2
1
ˆ ˆ ˆ ˆ ˆˆ ˆˆ ˆ
ˆ ˆ ˆ ˆ ˆˆ ˆˆ ˆ
N PP
N PP
d N t N t N t i k t a t k a t t F t F tdt T T
dN t N t N t i k t a t k a t t F t F tdt T T
ˆ ˆP PF t F t
1 2 1 1 2
1 11 average pumping rate
ˆ ˆ
ˆP P
P
F t F t D t t t
N tD t
T
ECE 407 – Spring 2009 – Farhan Rana – Cornell University