Title: Quantum Trajectory Method in Quantum Optics
1Quantum Trajectory Method in Quantum Optics
- Tarek Ahmed Mokhiemer
- Graduate StudentKing Fahd University of
Petroleum and Minerals
2Outline
- 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
3The 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)
4Quantum Trajectory Method is a numerical
Monte-Carlo analysis used to solve the master
equation describing the interaction between a
quantum system and a Markovian reservoir.
Reservoir
system
5A single quantum trajectory represents the
evolution of the system wavefunction conditioned
to a series of quantum jumps at random times
6- The evolution of the system density matrix is
obtained by taking the average over many quantum
trajectories.
2000 Trajectories
7The quantum trajectory method is equivalent to
solving the master equation
8Advantages of QTM
- Computationally efficient
- Physically Insightful !
9A single quantum trajectory
Initial state
Non-Unitary Evolution
Quantum Jump
Non-Unitary Evolution
Quantum Jump
10The Master Equation
(Lindblad Form)
11Two level atom interacting with a classical field
12.
13Initial state
The probability of spontaneous emission of a
photon at ?t is
14 Applying Weisskopf-Wigner approximations
( Valid for small ?t)
? spontaneous decay rate
15Deriving the conditional evolution Hamiltonian
Hcond
16Two methods
Compare the probability of decay each time step
with a random number
Integrate the Schrödinger's equation till the
probability of decay equals a random number.
17Non-Hermetian Hamiltonian
µ Normalization Constant
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19A single Quantum Trajectory
time
20Average of 2000 Trajectories
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Time
21Spontaneous decay in the absence of the driving
field
time
22Is a single trajectory physically realistic or is
it just a clever mathematical trick?
23A more formal approachstarting from the master
equation
24Jump Superoperator
Applying the Dyson expansion
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28The more general case
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30Different Unravellings
A single number state
A superposition of number states
31The Micromaser
Single atoms interacting with a highly modified
vacuum inside a superconducting resonator
32Quantum Semiclass. Opt. 8, 73104 (1996)
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34Atom 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
Jump superoperator
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37Comparison between QTM and the analytical solution
38The power of the Quantum Trajectory Method
time
39Transient Evolution of the Probability
Distribution
p(n)
n
40Limitation of the method
41Conclusion
- 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.
42References
- A quantum trajectory analysis of the one-atom
micromaser, J D Cressery and S M Pickles, Quantum
Semiclass. Opt. 8, 73104 (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) 277282
43Questions
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