Quantum Trajectory Method in Quantum Optics Tarek Ahmed

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 • 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

A single quantum trajectory represents the evolution of the system wavefunction conditioned to a series of quantum jumps at random times 1 0. 8 0. 6 0. 4 0. 2 Time 0. 05 0. 15 0. 2

The evolution of the system density matrix is obtained by taking the average over many quantum trajectories. 0. 8 0. 7 0. 6 0. 5 0. 4 0. 3 0. 2 0. 1 0 Time 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

The Master Equation (Lindblad Form)

Two level atom interacting with a classical field

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

Deriving the conditional evolution Hamiltonian Hcond

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

A single Quantum Trajectory time

Average of 2000 Trajectories: 0. 8 0. 7 0. 6 0. 5 0. 4 0. 3 0. 2 0. 1 0 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

The more general case…

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 ”

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 thermal reservoir The field loses a photon to 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…