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Quantum Trajectory Method in Quantum Optics. Tarek Ahmed Mokhiemer Graduate Student King Fahd University of Petroleum and Minerals. 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
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Quantum Trajectory Method in Quantum Optics Tarek Ahmed Mokhiemer Graduate StudentKing Fahd University of Petroleum and Minerals
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
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)
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
1 0.8 0.6 0.4 0.2 0.05 0.1 0.15 0.2 Time A single quantum trajectory represents the evolution of the system wavefunction conditioned to a series of quantum jumps at random times
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Time The evolution of the system density matrix is obtained by taking the average over many quantum trajectories. 2000 Trajectories
The quantum trajectory method is equivalent to solving the master equation
Advantages of QTM • Computationally efficient • Physically Insightful !
A single quantum trajectory Initial state Non-Unitary Evolution Quantum Jump Non-Unitary Evolution Quantum Jump
The Master Equation (Lindblad Form)
Initial state: The probability of spontaneous emission of a photon at Δt is:
Applying Weisskopf-Wigner approximations … ( Valid for small Δt) Г: spontaneous decay rate
Two methods Integrate the Schrödinger's equation till the probability of decay equals a random number. Compare the probability of decay each time step with a random number
Non-Hermetian Hamiltonian μ: Normalization Constant
Average of 2000 Trajectories: 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Time
Is a single trajectory physically realistic or is it just a “clever mathematical trick”?
Jump Superoperator: Applying the Dyson expansion
Initialstate Non-Unitary Evolution Quantum Jump Non-Unitary Evolution Quantum Jump
Different Unravellings A single number state A superposition of number states
The Micromaser “Single atoms interacting with a highly modified vacuum inside a superconducting resonator”
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
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.
References • 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