Quantum Trajectory Method in Quantum Optics

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Quantum Trajectory Method in Quantum Optics

• Tarek Ahmed Mokhiemer
• Graduate StudentKing Fahd University of
  Petroleum and Minerals

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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

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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)

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

2000 Trajectories
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The quantum trajectory method is equivalent to
solving the master equation
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Advantages of QTM

• Computationally efficient
• Physically Insightful !

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A single quantum trajectory
Initial state
Non-Unitary Evolution
Quantum Jump
Non-Unitary Evolution
Quantum Jump
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The Master Equation
(Lindblad Form)
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Two level atom interacting with a classical field
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.
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Initial state
The probability of spontaneous emission of a
photon at ?t is
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Applying Weisskopf-Wigner approximations
( Valid for small ?t)
? spontaneous decay rate
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Deriving the conditional evolution Hamiltonian
Hcond
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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.
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Non-Hermetian Hamiltonian


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

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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

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Questions
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