Transient Theory of Cavity Spontaneous Emission

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Transient Theory of Cavity Spontaneous Emission

• Chris Takacs
• Physics 215C Quantum Presentation
• June 11th, 2009

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Quantizing the Fields in Cavities

• Identical to what we did in class except now L
  remains finite
• Excludes certain modes
• Atoms in cavity can put energy into the field and
  it can be re-absorbed later.
• In vacuum, this doesnt happen.
• Atom to Cavity coupling dependent on emission
  wavelength being close to a cavity mode.

L
L
Hydrogen Atom in a Cavity of dimension L
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Experimental Reality

• Lets say we are dealing with a Hydrogen atom
  decaying spontaneously from the first excited
  state into the ground state.
• This corresponds to a 10.4eV photon with
  wavelength 120nm
• This is a physically accessible length-scale
  phenomena for experimental physics (No Strings
  required)
• Strong effect in high-Q / small optical resonator
  cavities
• Diode lasers
• Micromasers
• Even changes spontaneous emission rate of atoms
  near a single mirror

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Jaynes-Cummings Model

• Can be used to describe an atom radiating inside
  a cavity
• Solvable model describing interaction of a
  2-level system with a single-mode harmonic
  oscillator
• Very general problem applicable in many areas
• Developed by Electrical Engineers

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Jaynes-Cummings Model
of Photons in Field
Occupation of the atomic states
Coupling between the field and atomic states
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Jaynes-Cummings Model

• This was a HW problem. It generalizes to a
  2-level system with off-diagonal terms
• Solution is Rabi Oscillations with frequency
• Detuning of field and cavity energies determines
  amplitude of oscillations
• The classical analog of this is a harmonic
  oscillator (Field) being driven (Atom) at some
  frequency

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Shortcomings of the JCM

• Developed by Electrical Engineers
• Not much physical content
• A phenomenological model
• Doesnt take into account finite speed of light
• Instantaneous interactions with the field
• Atom cannot immediately dump energy into the
  field
• Thought of in terms of normal modes which are
  standing wave solutions
• The atom does not know it is in a cavity!
• We should be able to think of this problem in
  terms of retarded waves

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Expected Behavior of Transient Cavity Spontaneous
Emission

• Fix the atom at the center of a spherical cavity
• Q infinity
• At t0, put atom into excited state
• Atomic dipole starts radiating at the free-space
  rate

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Expected Behavior of Transient Cavity Spontaneous
Emission
tau/2

• Cavity roundtrip time tau 2 R /c
• At tau/2, the radiation reaches the walls.
• The walls exactly radiate a wave inwards to
  exactly satisfy the boundary condition

tau/4
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Expected Behavior of Transient Cavity Spontaneous
Emission

• At t tau, the radiated wave finally reaches the
  atom
• This modifies the emission rate
• Depending on the phase of the reflected wave
  relative to the emitting dipole
• Process repeats and we see Rabi Oscillations
  after some number of roundtrips

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The Hamiltonian

• Ha is the atomic Hamiltonian
• We will turn this into a 2-level system like JCM
• p is the momentum of coulomb interaction of the
  electron
• Hf is the free-field Hamiltonian
• This is for all of free space. Not just cavity
  modes
• A is the usual Vector potential in the Coulomb
  gauge
• Note that the interaction only takes place at the
  origin where we consider the atom to be
• The () / (-) operator extracts the
  positive/negative frequency components
• Positive frequencies are terms like exp(-i w t)
  and Negative frequencies go as exp(i w t)
• Makes some simplifications transparent but clumsy
  when first looking at it
• b b() b(-) and b(-) adjoint( b() )
• Since we want explicit time-dependent solutions
• Work in Heisenberg picture
• Abuse the Heisenberg equations of motion
• Make a lot of approximations along the way!

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The Vector Potential

• This is the simplified form of the vector
  potential
• p has an exact solution given some reasonable
  approximations
• First part is responsible for free-space
  radiation.
• Second part is the radiation emitted by the atom
  at earlier times.
• These are proportional to the acceleration of
  the electron.
• With this expression and the Heisenberg equations
  of motion we can solve for the Electric field and
  Excited State Population

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Numerical Solutions to Equations of Motion for
E(r,t)2

• Atom is in resonance with the 10th mode of the
  cavity
• Left is t tau/4
• Emission is at free-space rate
• Right is t tau/2

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Numerical Solutions to Equations of Motion for
E(r,t)2

• Left is t (3 / 4) tau
• We see that boundary reflection causes
  interference with earlier wave
• Right is t tau
• This is the first point when the electric field
  at the origin changes

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Numerical Solutions to Equations of Motion for
E(r,t)2

• Left is t (5 / 4) tau
• Right is t (6/4) tau

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Numerical Solutions of Excited State Population
TR
2 TR
0
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Numerical Solutions of Excited State Population
TR ( 10 tau)
2 TR
0
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Numerical Solutions of Excited State Population
Detuned Cavity
TR
2 TR
3 TR
0
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Transient Theory vs. JCM
Tuned to 10th Mode
Slightly Detuned from 10th Mode
Numerical Solutions
JCM
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Limits of the Transient Theory

• Can be modified to include cavity losses
• As the number of roundtrips per Rabi period
  becomes large, the transient theory converges to
  the JCM
• Authors claim this is not unique
• Multimode corrections are interesting but can be
  mostly ignored in practice
• Setting up an experiment where the kinky nature
  of the oscillations is detectable would be
  difficult
• Light travels 0.3 m / ns
• Need very large, high-Q cavity

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Conclusions

• Transient Theory gives nice physical
  interpretation
• Transient Theory agrees well with JCM
• JCM is more useful tool in both experiment and
  theory
• Elegant and often exactly solvable
• Designed by Electrical Engineers
• Engineers win!

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The Vector Potential

• Similar to the vector potential of the free-field
  we solved in class EXCEPT we get an additional
  term related to the mechanical momentum
• The equations of motion for ak(t) can be
  formally integrated to give

• Where ak(0) is as the lowering operator
• Summing over the polarization and integrating
  over wavevectors up to some cutoff wavevector we
  get an expression for the vector potential

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The Vector Potential (continued)

• We modify the Vector Potential to include
  radiation reflected at the walls with the
  expression K(t t)
• K(t- t) acts as a series of delta-functions
• n0 corresponds to the mode closest to being in
  resonance with the Cavity
• M is the number of modes we are including around
  n0
• For this plot, the atom is in resonance with the
  10th cavity mode

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The Atomic Hamiltonian

• We specialize to a 2-level system and write it in
  terms of atomic transition operators e is
  excited and g is ground state.
• Not Pauli-spin matrices
• p represents the Coulomb interaction of the
  electron.
• Not perturbed by either the vector potential or
  the vacuum field to zeroth-order
• We can then write down the solution for the time
  dependence of p

where matrix at t0 is one at (i, j) and
zero everywhere else

• This is the key starting point and allows us to
  generate solutions for the other operators