1. 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

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. 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)

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. system Reservoir

5. A single quantum trajectory represents the evolution of the system wavefunction conditioned to a series of quantum jumps at random times 0.050.10.150.2 0.4 0.6 0.8 1 Time

6. The evolution of the system density matrix is obtained by taking the average over many quantum trajectories. 2000 Trajectories 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Time

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

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

15. Deriving the conditional evolution Hamiltonian H cond

16. 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.

17. Non-Hermetian Hamiltonian μ: Normalization Constant

19. A single Quantum Trajectory time

20. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Average of 2000 Trajectories: 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 approach … starting from the master equation

24. Jump Superoperator: Applying the Dyson expansion

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

28. The more general case …

30. Different Unravellings A single number stateA superposition of number states

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

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

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

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, 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

43. Questions …