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Quantum Trajectory Method in Quantum Optics

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Title: Quantum Trajectory Method in Quantum Optics


1
Quantum Trajectory Method in Quantum Optics
  • Tarek Ahmed Mokhiemer
  • Graduate StudentKing Fahd University of
    Petroleum and Minerals

2
Outline
  • 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

3
The 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)

4
Quantum 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
5
A 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
7
The quantum trajectory method is equivalent to
solving the master equation
8
Advantages of QTM
  • Computationally efficient
  • Physically Insightful !

9
A single quantum trajectory
Initial state
Non-Unitary Evolution
Quantum Jump
Non-Unitary Evolution
Quantum Jump
10
The Master Equation
(Lindblad Form)
11
Two level atom interacting with a classical field
12
.
13
Initial state
The probability of spontaneous emission of a
photon at ?t is
14

Applying Weisskopf-Wigner approximations
( Valid for small ?t)
? spontaneous decay rate
15
Deriving the conditional evolution Hamiltonian
Hcond
16
Two 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.
17
Non-Hermetian Hamiltonian


µ Normalization Constant
18
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19
A single Quantum Trajectory
time
20
Average of 2000 Trajectories
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Time
21
Spontaneous decay in the absence of the driving
field
time
22
Is a single trajectory physically realistic or is
it just a clever mathematical trick?
23
A more formal approachstarting from the master
equation
24
Jump Superoperator
Applying the Dyson expansion
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28
The more general case
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30
Different Unravellings
A single number state
A superposition of number states
31
The Micromaser
Single atoms interacting with a highly modified
vacuum inside a superconducting resonator
32
Quantum Semiclass. Opt. 8, 73104 (1996)
33
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34
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
Jump superoperator
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37
Comparison between QTM and the analytical solution
38
The power of the Quantum Trajectory Method
time
39
Transient Evolution of the Probability
Distribution
p(n)
n
40
Limitation of the method
41
Conclusion
  • 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.

42
References
  • 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

43
Questions
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