Quantum Quantum Trajectory Method Trajectory Method in in Quantum Optics Quantum Optics Tarek Ahmed Mokhiemer Tarek Ahmed Mokhiemer Graduate Student Graduate Student King Fahd University of Petroleum King Fahd University of Petroleum and Minerals and Minerals
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Quantum Trajectory Method in Quantum Optics Tarek Ahmed Mokhiemer Graduate Student King Fahd University of Petroleum and Minerals Graduate Student King.
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King Fahd University of Petroleum King Fahd University of Petroleum and Mineralsand Minerals
OutlineOutline
• General overview
• QTM applied to a Two level atom interacting with a classical field
• A more formal approach to QTM
• QTM applied to micromaser
• References
The beginningThe beginning……• J. Dalibard, Y. Castin and K. Mølmer,
Phys. Rev. Lett. 68, 580 (1992)
• R. Dum, A. S. Parkins, P. Zoller and C. W. Gardiner, Phys. Rev. A 46, 4382 (1992)
• H. J. Carmichael, “An Open Systems Approach to Quantum Optics”, Lecture Notes in Physics (Springer, Berlin , 1993)
Quantum Trajectory Method is a Quantum Trajectory Method is a numerical Monte-Carlo analysis numerical Monte-Carlo analysis used to solve the master equation used to solve the master equation describing the interaction between describing the interaction between a quantum system and a Markovian a quantum system and a Markovian reservoir.reservoir.
system
Reservoir
A single quantum trajectory represents the evolution of the system wavefunction conditioned to a series of quantum jumps at random times
0.05 0.1 0.15 0.2
0.2
0.4
0.6
0.8
1
Time
( )ee t
The evolution of the system density matrix is obtained by taking the average over many quantum trajectories.
0.05 0.1 0.15 0.2
0.2
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1
0.05 0.1 0.15 0.2
0.2
0.4
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0.8
1
0.05 0.1 0.15 0.2
0.2
0.4
0.6
0.8
1
2000 Trajectories
00.10.20.30.40.50.60.70.8
Time
( ) ( ) ( )t Avg t t
( ) ( ) ( )t Avg t t
ˆ ˆ( ) ( ) ( ) ( )A t Tr A t Avg t A t
The quantum trajectory method is equivalent to solving the master equation
Advantages of QTMAdvantages of QTM• Computationally efficient
• Physically Insightful !
A single quantum trajectoryA single quantum trajectory
Initial state
Non-Unitary Evolution
Quantum Jump
Non-Unitary Evolution
Quantum Jump
The Master EquationThe Master Equation
((Lindblad FormLindblad Form))
Two level atom Two level atom interacting with a interacting with a
classical field classical field
s
, [ ]ss relax s
d iH L
dt
,2 2
d iH S S S S S S
dt
. 1, : Rabi Frequency
2H S S
e
g
: Spontaneous Decay Rate
, S e g S g e
0 0( ) ( )*int , , , ,
1,2
1
2k ki t i t
k k k kk
H s s g a s e g a s e
0 0vac vac g e
( ) ( ) 0I vact U t
int
0
( ) ( )t
I
iU t H t dt
The probability of spontaneous emission of a photon at Δt is: 2
1 ,1,2
( ) ,1 ( ) 0photon k I vack
P t g U t
Initial state:
2
1 , int1,2 0
( ) ,1 ( ) 0t
photon k vack
iP t g H t dt
02( ) *
, , ,1,2 0
,1 1k
ti t
k k kk
ig e g s dt
022 ( )( ')
,1,20 0
' k
t ti t t
kk
dt dt e g
2
1 ( ) . .photonP t t
Г: spontaneous decay rate
Applying Weisskopf-Wigner approximations …
( Valid for small Δt)
Deriving the conditional evolution Hamiltonian Hcond
( ) 0 ( ) 0cond vac I vacU t U t
int int int20 0 0
1( ) 0 ( ) ( ) ( ') ' 0
t t t
cond vac vac
iU t H t dt dt H t H t dt
0 0( ) ( )*int , , , ,
1,2
1
2k ki t i t
laser env k k k kk
H H H s s g a s e g a s e
2( ) . ( )cond cond
iU t I H t t
O
.2cond laser
iH H s s
Two methodsTwo methods
is fixedtCompare the probability of decay each time step with a random number
is varyingt
Integrate the Schrödinger's equation till the probability of decay equals a random number.
( ) (1 . ) (0)cond cond cond
it dt H
Non-Hermetian Hamiltonian
( ) (1 . ) (0)cond cond cond
idt dt H
μ: Normalization Constant
1
1
1 photonP
A single Quantum TrajectoryA single Quantum Trajectory
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0.2
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1
time
2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Average of 2000 Trajectories:
Time
11
Spontaneous decay in the absence of the driving field
time
11
0.05 0.1 0.15 0.2
0.2
0.4
0.6
0.8
1
Is a single trajectory physically Is a single trajectory physically realistic or is it just a “clever realistic or is it just a “clever
mathematical trickmathematical trick?”?”
0: QTM
A more formal A more formal approach…approach…
starting from the master starting from the master equationequation
L̂
1ˆ ˆ ˆ ˆ ˆ ˆˆ ,2
iH C C CC C C
J C C Jump Superoperator:
Applying the Dyson expansion
1
22 1 1
ˆ ˆ ˆ ˆ( )( ) ( )( )10 0
0
ˆ ˆ ˆ ˆ( )( ) ( )10
ˆ ˆ( ) .....
ˆ (0)
n n nt tL J t t L J t t
n nn
t L J t t L J t
t dt e J dt e J
dt e Je
L̂
1 1ˆ ˆ ˆ ˆ ˆ ˆ( )( ) ( )( ) ( )( )
1 2ˆ ˆ ˆ( ; , ,...... ) ....... (0)n n nL J t t L J t t L J tn
c nt t t t e Je J Je
ˆ ˆ ˆ ˆL L J J
Initial state
Non-Unitary Evolution
Quantum Jump
Non-Unitary Evolution
Quantum Jump
1
22 1 1
ˆ ˆ ˆ ˆ( )( ) ( )( )10 0
0
ˆ ˆ ˆ ˆ( )( ) ( )10
ˆ ˆ( ) .....
ˆ (0)
n n nt tL J t t L J t t
n nn
t L J t t L J t
t dt e J dt e J
dt e Je
1 1
ˆ ˆ ˆ ˆ ˆ ˆ( )( ) ( )( ) ( )( )1 2
ˆ ˆ ˆ( ; , ,...... ) ....... (0)n n nL J t t L J t t L J tnc nt t t t e Je J Je
ˆ ˆ / /( ) eff effiH t iH tL J te e e
2eff
iH H C C
2 1 1 0/ ( ) / ( ) / ( )0
ˆ ˆ( ) ...... ( )eff n eff effi H t t i H t t i H t tc ct e C e Ce t