Download Introduction to Quantum Optics for Cavity QED The semiclassical

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 2ε 0 Al
3
2
c
4ω
d
κ = T; γ = 2
2l
3 c 4πε 0c
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!