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Introduction to Quantum Optics for Cavity QED The semiclassical method Luis A. Orozco Joint Quantum Institute Department of Physics at UMD and NIST Cavity QED group: Present students (2011): Andres Cimmarusti David Norris Michael Scholten Undergraduate (2011) Joshua Crawford Postdocs: Daniela Manoel Daniel Freimund Jietai Jing Past members of my group: Juergen Gripp Stephen Mielke Gregory Foster Wade Smith Joseph Reiner Matthew Terraciano Rebecca Olson Knell Basudev Roy Collaborators: Prof. Howard Carmichael, University of Auckland, New Zealand. Prof. James Clemens, Miami University. Prof. Julio Gea Banacloche, University of Arkansas. Prof. Perry Rice, Miami University. Prof. Howard Wiseman, Griffith University, Brisbane, Australia. Prof. Pablo Barberis, UNAM, Mexico Visitors: Nicolas Leuillot, France, Jin Wang Australia, Stefan Kuhr Germany, Arturo Fernandez, Chile. Sociedad Mexicana de Fisica (summer experience for undergraduates). Gabriel Ramos, Manuel de la Cruz, Salvador Hernandez, Monserrat Bizarro, Edgar Vigil, Adonis Reyes, Marduk Bolaños, Ramon Ramos. Supported by NSF and NIST Coupled atoms and cavities: Drive Output Collection of N Two level atoms coupled to a single mode of the electromagnetic field. This is a far from equilibrium system. Driven with dissipation (atoms γ, cavity κ). Microwaves Visible light Micromaser Optical Bistability Cavity QED Implementations: Rydbergs on Superconducting cavities Alkali atoms on Optical Cavities Quantum dots on microcavities Superconducting qbits on microwave resonators .... Dipole coupling between the atom (fermion) and the cavity mode (boson). The dipole matrix element between two states is fixed by the properties of the states (radial part) and the Clebsh-Gordon coefficients from the angular part of the integral. It is of the order of a few times a0 (Bohr radius) times the electron charge e The field of one photon in a cavity with Volume Veff is: The electric field squared is an energy density. Single atom Cooperativity (measures the effect of one atom): g2 C1 = κγ Saturation photon number (measures the effect of one photon): 2 γ n0 = 2 3g Cooperativity (for N atoms): is the ratio of the atomic losses to the cavity losses or also can be read as the ratio between the good coupling (g) and the dissipation (κ,γ). α 0l C = C1N = T Are the two definitions equivalent? 2 2 2 g 1 d ω d ω 2 C1 = ; g = 2 = κγ 2ε 0V 2ε 0 Al 3 2 c 4ω d κ = T; γ = 2 2l 3 c 4πε 0c 2 α 0 l σρl 3 λ N C = C1N = = = T T 2π A T RATIO OF TWO AREAS times some ENHANCEMENT This is a very important figure of merit Cavity QED systems Optical free space Optical matter Microwave free space Microwave matter Typical system for optical experiments. Formulation of the problem: 1.- Free evolution of cavity mode and atoms, 2.- Coupling atom-cavity, 3.- Decay of atoms (reservoir), 4.- Decay of cavity field (reservoir), 5.- Drive of the cavity Use this Hamiltonian to find the equations of motion of the field <a>, the atomic polarization <σ+>, and atomic inversion <σz>. We assume N atoms distributed at the positions rj in the mode of the cavity. Maxwell Bloch Equations are then: State Equation. y is the normalized input field y=E/ n01/2, it is the field inside the cavity with no atoms. x is the normalized output field x=<a>/n01/2 , it is the field inside the cavity with atoms. Δ is the normalized atomic detuning Δ=(ωatom-ωlaser)/γ/2. θ is the normalized cavity detuning θ=(ωcavity-ωlaser)/κ. This equation makes explicit that the phase of the input and output fields need not be the same. Constructive interference happens when the phase is zero. What do we expect on resonance for the normalized fields (x,y) and the normalized intensities to the saturation photon number (X,Y)? For low intensity, the input and the output are linearly related, y = x (1+2C) ; Y=X(1+2C)2 X 1 = 2 Y (1 + 2C ) y = x (1+2C) For very high intensity, y = x ; Y=X +4C X 4C = 1+ Y Y 2C ⎤ ⎡ y = x ⎢1 + 2 ⎥ ⎣ 1 + x ⎦ At intermediate intensity, there can be saturation. It happens in this simple model for the case of C>4. C (Cooperativity) is the negative of the laser pump paramenter. The slope of the output x as a function of input y may be zero!