Quantum Physics of Light-Matter Interactions: Lecture on Jaynes-Cummings Hamiltonian and Optical Cavities

1. Claudiu Genes Max Planck Institute for the Science of Light (Erlangen, Germany) Lecture 7: Jaynes-Cummings Hamiltonian Laser theory Quantum Physics of Light-Matter Interactions, SS19, FAU

2. Optical cavities Combining this with complexity – collective dynamics plays a big role

3. Optical cavity

4. Optical cavity Electric field operator Zero point electric field per photon

5. Optical cavity Operator creating excitations in a given delocalized mode Electric field operator Zero point electric field per photon

6. Optical cavity • Multiple resonances • Lorentzian profile – enhanced density of optical modes around resonances Quasi-mode frequency Cavity mode linewidth

7. Optical cavity Coherent light source

8. Optical cavity Coherent light source Leakage of cavity photons modelled as an exchange interaction 2 equivalent approaches

9. Optical cavity Coherent light source Leakage of cavity photons modelled as an exchange interaction Eliminate vacuum modes – master equation for the cavity mode

10. Optical cavity Coherent light source Leakage of cavity photons modelled as an exchange interaction Eliminate vacuum modes – master equation for the cavity mode Cavity intensity decay rate

11. Optical cavity Coherent light source Leakage of cavity photons modelled as an exchange interaction Eliminate vacuum modes – master equation for the cavity mode Collapse operator

12. Optical cavity Coherent light source Leakage of cavity photons modelled as an exchange interaction Eliminate vacuum modes – master equation for the cavity mode

13. Optical cavity Coherent light source Leakage of cavity photons modelled as an exchange interaction Eliminate vacuum modes – master equation for the cavity mode Driving field amplitude

14. Optical cavity Coherent light source Leakage of cavity photons modelled as an exchange interaction Keep vacuum modes – quantum Heisenberg Langevin equations (quantum stochastic approach)

15. Optical cavity Coherent light source Leakage of cavity photons modelled as an exchange interaction quantum Heisenberg Langevin equations Heisenberg equations of motion and

16. Optical cavity Coherent light source Leakage of cavity photons modelled as an exchange interaction quantum Heisenberg Langevin equations Heisenberg equations of motion and Formal solution

17. Optical cavity Coherent light source Leakage of cavity photons modelled as an exchange interaction

18. Optical cavity Coherent light source Leakage of cavity photons modelled as an exchange interaction Effective decay

19. Optical cavity Coherent light source Leakage of cavity photons modelled as an exchange interaction Driving plus noise term Effective decay

20. Optical cavity Coherent light source Leakage of cavity photons modelled as an exchange interaction Result

21. Optical cavity Coherent light source Leakage of cavity photons modelled as an exchange interaction Result Input noise correlations

22. Optical cavity Coherent light source Leakage of cavity photons modelled as an exchange interaction Result Input noise correlations Input-output relations

23. Cavity quantum electrodynamics Cavity quantum electrodynamics with two-level systems

24. Atom in cavity- Jaynes Cummings dynamics The Jaynes-Cummings Hamiltonian

25. Atom in cavity- Jaynes Cummings dynamics The Jaynes-Cummings Hamiltonian

26. Master equation of cavity-emitter system The full master equation with the total Hamiltonian

27. Atom in cavity- Jaynes Cummings dynamics The Jaynes-Cummings Hamiltonian Matrix elements

28. Atom in cavity- Jaynes Cummings dynamics The Jaynes-Cummings Hamiltonian Matrix elements

29. Atom in cavity- Jaynes Cummings dynamics The Jaynes-Cummings Hamiltonian Matrix elements Up to two excitations

30. Atom in cavity- Jaynes Cummings dynamics The Jaynes-Cummings Hamiltonian Matrix elements Leads to photon blockade (as we‘ll see later) Up to two excitations

31. Atom in cavity- Jaynes Cummings dynamics Polaritons (single excitation subspace)

32. Atom in cavity- Jaynes Cummings dynamics Polaritons (single excitation subspace) Eigenvalues

33. Atom in cavity- Jaynes Cummings dynamics Polaritons (single excitation subspace) Eigenvalues On resonance

34. Atom in cavity- Jaynes Cummings dynamics Polaritons (single excitation subspace) Eigenvalues On resonance

35. Atom in cavity- Jaynes Cummings dynamics The polariton transformation Diagonalization of the Hamiltonian Diagonalization of the Linblad term

36. Master equation of cavity-emitter system The full master equation with the total Hamiltonian

37. Master equation of cavity-emitter system The full master equation with the total Hamiltonian Let’s derive equations of motion for averages

38. Master equation of cavity-emitter system The full master equation with the total Hamiltonian Let’s derive equations of motion for averages

39. Linear regime Low excitation regime: two driven coupled oscillators

40. Strong coupling Low excitation regime: two driven coupled oscillators • Emergence of strong coupling Decay rates Frequencies

41. Strong coupling Low excitation regime: two driven coupled oscillators • Emergence of strong coupling

42. The Purcell effect Bad cavity regime Decay rates Frequencies

43. The Purcell effect Bad cavity regime Cooperativity (or Purcell factor) Decay rates Frequencies

44. Photon blockade Photon blockade

45. Photon blockade

46. Photon blockade

47. The laser (quantum theory)

48. Source: youtube: How a laser works

49. The laser Model 1 Fast relaxation pump Lasing transition Incoherent pump

50. The laser Model 1 Fast relaxation pump Lasing transition Incoherent pump Effective inverse decay – incoherent pumping model ...like decay from the ground to the excited state