Quantum Trajectory Method in Quantum Optics Tarek Ahmed Mokhiemer Graduate Student King Fahd University of Petroleum and Minerals 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 References

The beginning … J. Dalibard, Y. Castin and K. M ø lmer, Phys. Rev. Lett. 68, 580 (1992)Phys. Rev. Lett. 68, 580 (1992) R. Dum, A. S. Parkins, P. Zoller and C. W. Gardiner, Phys. Rev. A 46, 4382 (1992)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. system Reservoir

A single quantum trajectory represents the evolution of the system wavefunction conditioned to a series of quantum jumps at random times Time

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

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)

Two level atom interacting with a classical field

.

The probability of spontaneous emission of a photon at Δt is: Initial state:

Г: spontaneous decay rate Applying Weisskopf-Wigner approximations … ( Valid for small Δt)

Deriving the conditional evolution Hamiltonian H cond

Two methods Compare the probability of decay each time step with a random number Integrate the Schr ö dinger's equation till the probability of decay equals a random number.

Non-Hermetian Hamiltonian μ: Normalization Constant

A single Quantum Trajectory time

Average of 2000 Trajectories: Time

Spontaneous decay in the absence of the driving field time

Is a single trajectory physically realistic or is it just a “ clever mathematical trick ” ?

A more formal approach … starting from the master equation

Jump Superoperator: Applying the Dyson expansion

Initial state Non-Unitary Evolution Quantum Jump Non-Unitary Evolution Quantum Jump

The more general case …

Different Unravellings A single number stateA superposition of number states

The Micromaser “ Single atoms interacting with a highly modified vacuum inside a superconducting resonator ”

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

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

Comparison between QTM and the analytical solution

The power of the Quantum Trajectory Method time

Transient Evolution of the Probability Distribution p(n) n

Limitation of the method

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

Questions …