1. Three-Level Atom Micromaser Analysis Using Quantum Trajectory Method Thesis Propsal by Tareq Ahmed Mokhiemer Electronics Research Institute

2. An overview of Micromaser ( Single atom maser) Quantization of electromagnetic field Interaction of an atom with a quantized field Quantum Trajectory analysis Three level atom Research proposal

3. An overview of Micromaser

4. a collimated beam of Rubidium atoms is injected inside a high quality microwave cavity. The cavity dimensions are set to select only one radiation mode that is resonant with the maser transition. The system can be used to study the interaction between a single two level atom and a photon of the electromagnetic field inside the cavity.  The atom field interaction is probed by observing the population in the upper and lower maser levels after the atoms have left the cavity.

5. Typical parameters: Quality of cavity ~ ِAverage photon lifetime ~ 0.2 s Cavity Temperature:~0.3 K ِAverage number of thermal photons: 0.1 Rydeberg atoms : 63 p 1/2 to 61d 3/2. Corresponds to a transition frequency of 21.5GHz Interaction time

6. The importance of the micromaser A basic prototype to study the interaction between light and matter (in the level of single atoms and photons). A tool to generate non-classical states of electromagnetic fields, e.g. Fock states and single photon on demand. Has applications in quantum information processing, e.g. a source to generate entangled photons. ( The photon gun ) A platform for testing the postulates of quantum mechanics (e.g. testing the nonlocal features of quantum mechanics)

7. Beginning with Maxwell’s field equations in free space we find that the vector potential A(r,t) satisfies the homogenous wave equation : َQuantization of the electromagnetic field:

8. From Maxwell’s equations time evoultion of the basis functions is given by : The expansion of the electric and magnetic fields is given by:

9. From the Harmonic Oscillator theory:

13. So what is created or annihilated is actually photons

14. Interaction of an atom with a quantized field: Consider 2-level atom

17. The micromaser is a system in which single atoms and single modes of a cavity field interact via the Jaynes-Cummings Hamiltonian So what is a micromaser?

18. Consequences of atom-field interactions: Spontaneous emission in free space produces monotonic and irreversible decay of upper-level amplitude, whereas here we find the so-called vacuum Rabi oscillation,i.e, reversible spontaneous emission

19. Vacuum Rabi Oscillations:

22. Rabi Oscillations:

23. Collapse and revival of rabi oscillatio In the case of a coherent field, where the photon distribution is characterized by a poissonian distribution, Rabi Oscillations show a collapse and revival in the amplitude of the oscillations:

24. The semiconductor analog of the micromaser [ A quantum dot embedded in a microcavity ] One or more InAs quantum dots, surrounded by a GaAs matrix, are embedded in a micropillar optical cavity.

25. The optical microcavity serves to modify the spontaneous emission properties of the quantum dot through the Purcell effect The DBR mirrors consist of alternating layers of GaAs and AlAs

26. General notes Much of the interest in the micromaser lies in determining the properties of the field inside the cavity, but experimentally, this field is not amenable to observation, at the very least because there do not exist good photon detectors at microwave frequencies. By performing measurements on the atoms as they leave the cavity, information about the cavity field is obtained, since the states of the atoms and cavity field are entangled

27. For a given sequence of atomic measurements, the evolution of the state of the cavity field conditioned on the outcome of these measurements has been termed a ‘quantum trajectory’ Dissipative processes are coupled with the micromaser, such as the loss of photons through the low reflecting mirror and the coupling to free space modes ( considered as a large reservoir )

28. Density operator approach

29. Each operator A in quantum mechanics is represented by a matrix whose elements are given by: So the density operator is represented by the density matrix

30. Interpretation of matrix elements Diagonal elements = probabilities Off-diagonal elements = "coherences" (provide info. about relative phase)

31. All the information about the system is contained in the density matrix : The time evolution of the system is obtained from the evolution of the elements of the density matrix ( The Master Equation ) The expectaion value of any observable (a) is obtained from the density matrix

32. In many cases the system consists of two parts A, B (i.e, the atom and the field) whose basis functions : The combined system has eigenstates denoted by: And a general wavefunction is given by:

33. If the total density operator is p AB then the density operator of A only is called the reduced density operator p A and is given by Example: If p is the operator of the system+reservoir then p s,,the operator of the system, is given by: And the master equation of its development is given by:

34. Notes: It is assumed here to be local in time, which means that p s (t) depends only on p s at the same time (Markov approximation). All the system dynamics can be deduced from the master equation. The number of variables involved in solving the Master equation for the density matrices is ~ N 2

35. P(n) is the probability of finding n photons (stored) in the micromaser cavity. Nex is the average number of atoms that enter the cavity during,  is the vacuum Rabi flopping frequency, and t int is the atom ( cavity interaction time. One of the main results obtained from the master equation is the photon statistics of the one atom micromaser:

37. Non-Poissonian photon statistics

38. Quantum Trajectory Method Rather than solving the master equation for the density operator itself, an ensemble of random state functions is constructed. Each member of the ensemble evolves stochastically, typically by intervals of smooth non-unitary evolution interrupted at random by discontinuous changes of state—‘quantum jumps Examples of quantum jumps are atoms undergoing photon emissions, photons being lost from the cavities,…etc. Parameters of the system are obtained by averaging over the whole ensemble.

39. Example on a 2-level atom The Hamiltonian and the Master equation are given by The effective non-Hermitian Hamiltonian is then given by Assume a wavefunction in the form :

40. First we calculate the probability of spontaneous emission between t and t + dt which is given by A random number  is chosen to determine whether or not the spontaneous emission occurs

41. Three-level atoms: New Phenemoena and applications Fano interference:

42. Coherent population trapping

43. . CPT state serves as the basis for many other applications such as stimulated rapid adiabatic passage (STIRAP), lasing without inversion (LWI), electro- magnetically induced transparancy (EIT), etc…

44. Electromagnetically Induced Transparancy (EIT) It says propagate one laser beam through a medium and it will get absorbed; propagate two laser beams through the same medium and neither will be bsorbed. Qualitatively,the interactions of the two beams (conventionally called the pump and the probe) with the atoms pumps the latter into the dark state or CPT state. Once the atoms are in the CPT state, no light absorption can take place. (i.e the the medium becomes transparent)

45. Refractive index changes in EIT Modification of the absorptive properties of a medium will result in a change in the refractive index properties as well, hence the group velocity of the light waves through it will change as well.

46. Lasing Without Inversion In conventional lasers the medium suffers from stimulated emission as well as stimulated absorption then the medium can never experience laser action without a population inversion. However if the stimulated absorption is turned off, or significantly decreased then it should be possible to have inversionless laser.

47. stimulated rapid adiabatic passage STIRAP The technique of STIRAP is used to coherently transfer populations from one state to another (lower) state. Spontaneous transitions are not allowed to occur, hence coherence of emitted photons is ensured. Serves as a single photon source for quantum information systems

48. Vacuum-Stimulated Raman Scattering Based on Adiabatic Passage in a High-Finesse Optical Cavity [Similar to Raman Laser ]

49. Studying the micromaser theory for a three-level atom mainly : 1- Photon statistics 2- Coherent effects 3- Collective effects using the Quantum Trajectory Analysis as a numerical tool. Research Propsal

50. Thank you