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Three-Level Atom Micromaser
Analysis Using Quantum
Trajectory Method
Thesis Propsal
by
Tareq Ahmed Mokhiemer
Electronics Research Institute
• 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
An overview of Micromaser
•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.
Typical parameters:
10
Quality of cavity ~ 10
ِ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 35s
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)
ِQuantization of the electromagnetic field:
Beginning with Maxwell’s field equations in free
space we find that the vector potential A(r,t) satisfies
the homogenous wave equation :
2
 
1 
( 2 . 2  ) A(r , t )  0
c t
Expanding A(r,t) into a complete set of
 
wave functions uk, (r )
 
A( r , t ) 

 Al ql (t )u (r )

l
l
such that
1
VlVl '


3
 ( r )u  ( r )dr
u
  l ,l '
l'
 l
From Maxwell’s equations time evoultion of the basis functions is
given by :
 2
q l (t )  c | kl | ql (t )  0
..
2
(A harmonic oscillator equation)
The expansion of the electric and magnetic fields is given by:
.
  


E   A(r , t )   Al q l (t )ul (r )
t
  

B    A   Al ql (t )  u l (r )
The total energy inside the resonator is given by
1
1 2 3
2
H  ( 0E 
B )dr
2
20
1
Taking Al 
it follows directly from the columb gauge
 0Vl

1. 2 1 2 2
H   H l   q l  l ql ; where l  c | kl |
2
2
From the Harmonic Oscillator theory:
.
(ql , q l )  (qˆl , pˆ l )
[qˆl , pˆ l ' ]  i l ,l '
ql  qˆl
.
ˆl
ql  p
Field operators are given by :
ˆ 
 
E (r , t )   Al pˆ l (t ) ul (r )
ˆ 
  
B(r , t )   Al qˆl (t )  u l (r )
2
1
1 2
ˆ
ˆ
H   H l   pˆ l   l qˆl
2
2
2
Creation and Annihilati on operators



a l  l / 2 (ql  i pl / l )




a l  l / 2 (ql  i pl / l )

ˆ
ˆ
Let nl  a l aˆ l
Hˆ   Hˆ l    l (nˆl  1 / 2)
l
ˆ 
l 
  
E (r , t )  i  Al
(aˆl  aˆl )ul (r )
2
ˆ 
  
2

B(r , t )   Al
(aˆl  aˆl )  ul (r )
l
Notes :
 The electromagnetic energy is quantized in terms of photons
whose energy equals 
 Field operators don' t commute, so E and B can' t be measured
simultaneously upto arbitrary accuracies .
 The electromagnetic field in vacuum is not identicall y null
[ When n  0  H  0 (zero point energy ) ]

The physical meaning of the aˆ, aˆ operators
ˆa, aˆ  are called annihilati on and creation operators
respectively
where nks is the eigenstate of the number operator
nˆ correspond ing to eigenvalue n
So what is created or annihilated is actually photons
Interaction of an atom with a quantized
field:
• Consider 2-level atom
e
g
Define the atomic transition operators :
ˆ   e g ( excitation operator)
ˆ   g e ( decay operator)
̂ 3  e e  g g (atomic transition operator)
e
g
H total  H atom  H field  H atom field interaction
If we define the zero energy level to be halfway between e , g
1
1
ˆ
H Atom  ( Ee  E g )ˆ 3  0ˆ 3
2
2
Hˆ Field  aˆ  aˆ ( neglecting the zero - point energy term)
Using the electric dipole approximation : Hˆ
 dˆ.Eˆ
Interaction
1/ 2
  
ˆ
 (aˆ  aˆ  ) sin( kz)
E  e 
  0V 

where e is an arbitrary oriented polarizati on vector
dˆ  d (ˆ   ˆ  )
 Hˆ Interaction   (ˆ   ˆ  )( aˆ  aˆ  )
 
  d 
  0V
1/ 2



sin( kz) / 
1


ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
H total  0 3  a a   (     )( a  a )
2
The approximate time dependence of
the operator products is as follows :
i (0  ) t
ˆ
ˆ
 a ~ e
ˆ  aˆ ~ e

i (0  ) t
i (  0  ) t

ˆ
ˆ
 a ~ e
ˆ  aˆ ~ e
 i (  0  ) t
When the time - dependent Schroding er' s equation is integrated
we can neglect the terms containing (0   ) in the denominator
w.r.t terms containing (0   ) , assuming near resonance condition
This is called Rotating Wave Approximation
1
So H total  0ˆ 3  aˆ  aˆ   (ˆ  aˆ  ˆ  aˆ  )
2
This is called Jayness - Cumming model
So what is a micromaser?
• The micromaser is a system in which single atoms
and single modes of a cavity field interact via the
Jaynes-Cummings Hamiltonian
Consequences of atom-field interactions:
The sponatneous emission rate of the atom
inside the cavity can be modified ( Purcell effect )
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
Vacuum Rabi Oscillations:
if we call the ground state b , the excited state a
then the combined eigen - states of the atom field unperturbe
system will be either b n  1 or a n
These are called bare - atom states.
The total hamiltonia n of the system (J - C Hamiltonia n)
is given by :
1
H  0ˆ 3  aˆ  aˆ  g (ˆ  aˆ  ˆ  aˆ  )
2
writing the total hamiltonia n in matrix notation based
on the bare atom states for a given n we obtain
1 1 0   
2 g n  1
H n   ( n  ) 
 


2 0 1  2  2 g n  1
 
where  is the detuning  - 
The eigen states of this Hamiltonia n are given by :
2n  cos  n a n  sin  n b n  1
1n  sin  n a n  cos  n b n  1
where cos  n 
Rn  
( Rn   )  4 g (n  1)
2
2
;
Rn (the quantized general Rabi flopping frequency)
  2  4 g 2 (n  1)
These are called dressed states.
The relation between the bare state coeffecientcan , cbn1
and dresssed state coeffecients c1n , c2 n is given by
the transformation matrix
c2n (t ) cos  n  sin  n   can (t ) 
c (t )   sin 
 c (t )
cos

n
n   bn 1
 2n  

for the resonant case (   0) the time evolution of the
bare state coeffecirnts is given by :
can (t )  cos 2 ( g n  1t )
2
cbn1 (t )  sin ( g n  1t )
2
2
which shows that the atom flops between the
lower and upper states with the Rabi frequency : g n  1
Rabi Oscillations:
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:
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.
• 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
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
• 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 )
Density operator approach
Usually a state of a quantum system
is expressed as a weighted sum of
the basis functions :
But some times a state vector is not enough
when what is known about the system is only
the probabilit ies of each state,then we use
the density operator to describe the state of
the system :
where Pm is the probabilit y of being
in state  m
Each operator A in quantum mechanics is represented by a
matrix whose elements are given by:
aij   i A  j
So the density operator is represented by the density matrix
Interpretation of matrix elements
Diagonal elements = probabilities
Off-diagonal elements = "coherences"
(provide info. about relative phase)
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
A  Tr ( . A)
• In many cases the system consists of two parts
A, B (i.e, the atom and the field)
whose basis functions :
  , 
A
jB

• The combined system has eigenstates denoted
by:

A
 jB

• And a general wavefunction is given by:
 AB   a , j  A  jB
, j
• If the total density operator is pAB then the
density operator of A only is called the reduced
density operator pA and is given by
A 
a


*
'j
a j  A  ' A
, ', j
• Example: If p is the operator of the
system+reservoir then ps , ,the operator of the
system, is given by:
And the master equation of its development is
given by:
Notes:
• It is assumed here to be local in time,
which means that ps(t) depends only on ps
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 ~ N2
One of the main results obtained from the master
equation is the photon statistics of the one atom
micromaser:
•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 tint is the atom ( cavity interaction
time.
Non-Poissonian photon statistics
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.
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 :
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
Three-level atoms:
New Phenemoena and applications
• Fano interference:
Fano Interference : Two excitation paths to the same ionising
state within a continuum leads to a cancellati on in absorption
as the two paths destructively interfere.
Coherent population trapping
 Levels 1 , 2 are hyperfine states within the same ground state
and both are populated
 The excitation to level 3 can occur through 2 routes
1 3, 2 3
 If both interfere destructively , population would
be trapped in the ground state and no excitation will
occur ( hence it' s called " Dark state" )
CPT state serves as the basis for many
other applications such as stimulated rapid
adiabatic passage (STIRAP), lasing
without inversion (LWI), electromagnetically induced transparancy (EIT),
etc…
.
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)
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.
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.
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
Vacuum-Stimulated Raman Scattering Based on Adiabatic
Passage in a High-Finesse Optical Cavity
[Similar to Raman Laser ]
Research Propsal
• 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.
Thank you