Quantum Trajectory Method in Quantum Optics

Quantum Quantum Trajectory Method Trajectory Method

in in Quantum OpticsQuantum Optics

Tarek Ahmed MokhiemerTarek Ahmed Mokhiemer

Graduate StudentGraduate Student

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

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Time

( )ee t

The evolution of the system density matrix is obtained by taking the average over many quantum trajectories.

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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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time

2

0

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0.2

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0.8

Average of 2000 Trajectories:

Time

11

Spontaneous decay in the absence of the driving field

time

11

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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

21 2 1 1

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( )( ) ( )( ) ( )( ) ( )1 1 0 00 0 0

0

ˆ ˆ ˆ( ) ..... ( ) ( )n n nt t tL J t t L J t t L J t t L J t

n nn

t dt e J dt e J dt e Je t t

2

1 1 1 2 1 20 0 00

( ) ..... ( ; , ,...... ) ( ; , ,...... )t t t

n n c n c nn

t dt dt dt t t t t t t t t

( )c t

ˆ( ) ( )c ct dt C t

( ) 1 . ( )c eff c

it dt dt H t

ˆ ˆ( ) ( )( )

( ) ( )c c

cc c

t C C tdp t dt

t t

( )cdp t

1 ( )cdp t

The more general caseThe more general case……

1

1

1

22 1 1

1

1

ˆ ˆ ˆ ˆ( )( ) ( )( )10 0

0

ˆ ˆ ˆ ˆ( )( ) ( )10

ˆ ˆ( )

ˆ ..... (0)

n n n

n n

n n

t tL J t t L J t tn i n i

n i i

t L J t t L J ti

i

t dt e J dt e J

dt e J e

2

1

1 10 0 00

( ) ( )1 1 2 2 1 1 2 2

( ) .... .....

( ; , , , ,...... , ) ( ; , , , ,...... , )

n

t t t

n nn i i

n nc n n n n

t dt dt dt

t i t i t i t P t i t i t i t

L̂ ˆ ˆ ˆ ˆL L J J

Different UnravellingsDifferent Unravellings

n

1n

1n

A single number state

n nn

n

A superposition of number states

The MicromaserThe Micromaser

“Single atoms interacting with a

highly modified vacuum inside

a superconducting resonator”

Quantum Semiclass. Opt. 8, 73–104 (1996)

L̂

int int

0

int int

sin sinˆ

cos 1 cos 1

g N g NL R R a a

N N

R g N g N

0

1ˆ ˆ , 1 221

22

c b

c b

iL H n a a a a a a

n aa aa a a

0 int

int

1

2

1

ˆ cos 1

sinˆ

ˆ

ˆ 1

c b

c b

C R g N

g NC R a

N

C n a

C n a

Atom passing without emitting a photon

Atom emits a photon while passing through the cavity

The field acquires a photon from the thermal reservoir

The field loses a photon to the thermal reservoir

ˆ ˆi i i

i i

J C C J Jump superoperator

ˆ ˆ ˆ ˆL L J J

1

1

1

22 1 1

1

1

ˆ ˆ ˆ ˆ( )( ) ( )( )10 0

0

ˆ ˆ ˆ ˆ( )( ) ( )10

ˆ ˆ( )

ˆ ..... (0)

n n n

n n

n n

t tL J t t L J t tn i n i

n i i

t L J t t L J ti

i

t dt e J dt e J

dt e J e

1 1

1 1

1 1

1 1

ˆ ˆ ˆ ˆ ˆ ˆ( )( ) ( )( ) ( )( )

1 1 2 2 ˆ ˆ ˆ ˆ ˆ ˆ( )( ) ( )( ) ( )

ˆ ˆ ˆ....... (0)( ; , , , ,...... , )

ˆ ˆ ˆ....... (0)

n n n

n n

n n n

n n

L J t t L J t t L J ti i in

c n n L J t t L J t t L J ti i i

e J e J J et i t i t i t

Tr e J e J J e

1 1

1 1

ˆ ˆ ˆ ˆ ˆ ˆ( )( ) ( )( ) ( )( )1 1 2 2

ˆ ˆ ˆ( ; , , , ,...... , ) ....... (0)n n n

n n

L J t t L J t t L J tnn n i i iP t i t i t i t Tr e J e J J e

2

1

1 10 0 00

( ) ( )1 1 2 2 1 1 2 2

( ) .... .....

( ; , , , ,...... , ) ( ; , , , ,...... , )

n

t t t

n nn i i

n nc n n n n

t dt dt dt

t i t i t i t P t i t i t i t

( )1 1 2 2

1 1 2 2 1 1 2 2

( ; , , , ,...... , )

( ; , , , ,...... , ) ( ; , , , ,...... , )

nc n n

c n n c n n

t i t i t i t

t i t i t i t t i t i t i t

1

0

1

2

ˆ

1

ˆ

0

ˆ

1

ˆ

2

1 ,

,

1 ,

1 ,

C

C

C

C

m m P

m m P

m m P

m m P

/ effi H te m m

5 10 15 20

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n

Comparison between QTM and the analytical solution

int=g / cR

The power of the Quantum The power of the Quantum Trajectory MethodTrajectory Method

time

n

Transient Evolution of the Transient Evolution of the Probability DistributionProbability Distribution

p(n)

n

Limitation of the methodLimitation of the method

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0

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1n

ConclusionConclusion• Quantum Trajectory Method can be

used efficiently to simulate transient and steady state behavior of quantum systems interacting with a markovian reservoir.

• They are most useful when no simple analytic solution exists or the dimensions of the density matrix are very large.

ReferencesReferences• A quantum trajectory analysis of the one-atom micromaser, J D

Cressery and S M Pickles, Quantum Semiclass. Opt. 8, 73–104 (1996)

• Wave-function approach to dissipative processes in quantum optics,Phys. Rev. Lett., 68, 580 (1992)

• Quantum Trajectory Method in Quantum Optics, Young-Tak Chough

• Measuring a single quantum trajectory, D Bouwmeester and G Nienhuis, Quantum Semiclass. Opt. 8 (1996) 277–282

QuestionsQuestions……

0 0 ( ) 0 0 ( ) 0 0 ( )vac vac cond vac vac I vac vac condt U t U t