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```The Nobel Prize in Physics 2012
Serge Haroche
David J. Wineland
Prize motivation: "for ground-breaking experimental methods that enable
measuring and manipulation of individual quantum systems"
The Nobel Prize in Physics 2012
BCIT
Magnetooptical atom trap used in atomic physics experiments
The Nobel Prize in Physics 2012
Cavity Quantum Electrodynamics, SCIENTIFIC AMERICAN’1993
Experimental
demonstration of
cavity induced
modification of
spontaneous
emissoin rate of
Rydberg atoms
BCIT
The Nobel Prize in Physics 2012
Quantum non-demolition measurement
Credit: Nobel Prize
The Nobel Prize in Physics 2012
Quantum Dots in Photonic Structures
Lecture 2: Basics of Quantum Cavity Electrodynamics
Jan Suffczyński
Wednesdays, 17.00, SDT
Projekt Fizyka Plus nr POKL.04.01.02-00-034/11 współfinansowany przez Unię Europejską ze środków Europejskiego
Funduszu Społecznego w ramach Programu Operacyjnego Kapitał Ludzki
Plan for today
1. Cavity
quality factor
2. Weak
coupling regime
3. Strong
coupling regime
Reminder
• Optical cavity: an arrangement of optical components
which allows a beam of light to circulate in a closed path.
• Optical cavity mode: EM field distribution which
reproduce itself (if no losses) after one cavity round trip.
A condtion: a phase shift after one round trip = an
integer multiple of 2π.
∙
Mode frequencies:  = N
d

• Quality factor: a measure of the rate at which optical
energy decays from the cavity (absorption,scattering,
leakage due to imperfect mirrors).
Lifetime of the photon within the cavity: τ = 1/Γ= Q/ωc
• Quantum fluctuations of the vacuum
and the Casimir effect
Q  c / 

c
Quality factor Q
R
R
Blackboard calculation
Quality factor Q
Decay of the photon from a cavity due to absorption,scattering, leakage due
to imperfect mirrors. Consider electric field at a given point inside a cavity:
E
1
E t   cos c t e
E = Electric field magnitude
u = Energy density
1
 t
2
1/e
0
t
2/
Optical period T = 1/fc = 2/c
u t 
  1 t
 e 2


2

  e  t


Γ – optical energy decay time
dut 
Energy density decay: 
 e t
dt
1. Definition of Q via energy storage:
c
StoredEnergy
u t  1
2
Q  2
 2


du t  T   T 
EnergyLostPerOptCycle

dt
2. Definition of Q via resonance bandwidth:
Fourier transform

Time domain
1
E
E t   cos c t e
Frequency domain
2
1 
 
2
2 

I    E   
I
1 
2
  c     
t
2 
1
 t
2
1/e
2/
Lorentzian
Q
c


c
• The two definitions for Q are equivalent
• This is how on can measure Q (not in the case of microcavities with QDs!)

Quality factor vs. Finesse
F - a measure of the rate at which optical energy decays from the cavity, but the
optical cycle time T (in the case of Q) is replaced by round trip time tRT:
Δ =
∙


c
F
c 1
StoredEnergy
 2
EnergyLostPerRoundTrip
Finesse: the ratio of free spectral range Δω (the frequency separation between
successive longitudinal cavity modes) to the linewidth Γ of a cavity mode:
F

=

„resolving power or spectral
resolution of the cavity”
Quality factor vs. Finesse
 Quality factor: number of optical
cycles (times 2) before stored energy
decays to 1/e of original value.
 Finesse: number of round trips (times
2) before stored energy decays to
1/e of original value.
Q

=

F

=

When mirror losses dominate cavity losses:
• F and Q similar in the case of micrometer size cavities
(as Δω~ωc in that case)
• Q can be increased by increasing cavity length
• F is independent of cavity length !
Quality factor and typical values
Number of bounces
Decay time ps
8
6
4
2
0
0
5000 10 000 15 000 20 000
Quality factor
75 000
50 000
25 000
0
0
50
Quality factor 103
100
For Q = 5000 and λ = 700 nm, cavity length = λ/2 = 350 nm:
• photon decay time τ = Q/ωc = 1.86 ps
• Total run = τ *(speed of the light) = 557 µm
• Number of bounces = 2*TotalRun/(λ/2) = 2Q/π = 3183
• Number of the field oscillations: 7854
Light-matter coupling:
Weak coupling regime
Spontaneous emission
in a free space
1887 (Wiena) – 1961 (Wiena)
Nobel Prize 1933
Helium emission spectrum
Spontaneous emission
in a free space
1887 (Wiena) – 1961 (Wiena)
Nobel Prize 1933
Helium emission spectrum
Spontaneous emission in a free space:
• Exponential decay with time:  −Γ
• Characterstic decay constant 1/Γ
• Irreversible process
Perturbation
necessary!
An emitter in the simplest case :
a two level system
Excited
state
E
Fundamental
+ Photon
state
E1
E1
+
Spontaneous
emission
E0
ℏ = 1 − 0
E0
Density of modes in a free space
Let’ consider a LxLxL box of vacuum:
it
E ( x, t )  Ae sin( k  r ) with k 

L
(l , m, n)
(l,m,n are positive integers)
Blackboard calculation
2
2

dk

N ( )d  L3 2 2
d  L3 2 3 d
 c d
 c
Density of modes in a free space
2
 2 dk
3 
N ( )d  L 2 2
d  L 2 3 d
 c d
 c
3
N(ω)
Frequency ω
Density of states in a free space - example
-1
Density of photon states per unit volume (s )
Consider 1m3 of vacuum and l =500 nm
~50000 photon states per 1 Hz
:
150000
100000
~ 50000 states
50000
l nm
0
0
2
4
15
6
-1
Frequency (10 s )
Density of modes inside cavity
•
•
Cavity modifies density of states of the field
Energy of emitter emission counts much more then in free space!
Emitter in the cavity
Completely different situation than in a free space!
Spontaneous emission in the cavity:
• Exponential, irreversible decay with a modified decay rate or
• Reversible process
mirror
mirror
Spontaneous emission inhibited
Spontaneous emission enhanced
Spatial position of the emitter counts!
Fermi’s Golden Rule
• Spontaneous emission rate is not an inherent property of the emitter
• It depends on:
Dipol moment
Density of
of the emitter Electric field intensity
photon states
at emitter position
at emitter wavelength
Γ ∝ ()·|· emitter
Emission rate
2
|
Enrico Fermi
1901 (Rzym) – Chicago (1954)
Nobel Prize 1938
Fermi’s Golden Rule
Rate of decay from the initial, excited state of the emitter , 0|
to its ground s , 1| :
Γ
2
= 2 , 0 − ∙ (emitter ) , 1
ℏ
Spatial matching:
2
What is a mode intensity at the
emitter spatial position?
( −  ) ∙
Spectral matching:
How many final states are there
for the photon?
(+ a constraint: photon energy =
excited-ground energy level
difference)
Energy
Light-matter interaction: Weak coupling
S1
Emitter
S2
Cavity Mode
Optical Modes
outside the cavity
When S1 < S2 and Emitter in resonance with the Cavity Mode:
• photon „quickly” decays to the outside of the cavity
• Increased rate of the spontaneous emission into the cavity mode
Density of modes inside cavity
Ecav – energy position of the mode
Emitter
Cavity
+
density
of states
Outside
Purcell effect
Purcell effect
Probability that emiter
in excited state
 Purcell effect: acceleration of spontaneous emission for a
factor of FP
Edward M. Purcell
(1912–1997)
Nobel Prize 1952
Time
Spontaneous emission
into leaky modes
FP =
Spontaneous emission
to resonant cavity mode
 1
0
=
3 Q l03
1
42
0
V
n3
+
Spontaneous emission to
nonresonant modes
Purcell effect – the first observation
Europium ions
Spacer thickness d
Silver mirror
Emission in front of a mirror –
„almost” cavity case
Europium ions
Spacer thickness d
Silver mirror
Drexhage (1966):
Europium ions depends on
source position relative to a
silver mirror (l=612 nm)
• The better cavity, the larger emision rate enhancement
What if further improve cavity
parameters?
Light-matter interaction:
Strong coupling regime
Energy
Light-matter interaction : Strong coupling
S1
Emitter
S2
Cavity Mode
Optical Modes
outside the cavity
When S1 > S2 and Emitter in resonance with the Cavity Mode:
Photon preserved in the cavity „for long”
Reabsorption and reemission of the photon by the mitter
Strong coupling –Rabi splitting
Out of the resonence:
|1,0> :
Excited
emitter
Empty cavity
|0,1> :
Photon
Emitter
in ground state inside cavity
Strong coupling –Rabi splitting
In resonance:
|1,0> :
Excited
emitter
Empty cavity
Energy
Out of the resonence:
(|0,1>  |1,0>)/2
Rabbi
Splitting DR
|0,1> :
Photon
Emitter
in ground state inside cavity
(|0,1> + |1,0>)/2
Eigenstates :
Entengled states
emitter-photon
Oscillations |0,1> ↔ |1,0>
with Rabi frequency  = DR / h
Strong coupling regime
When emitter in the resonance with the cavity mode:
Probability that emitter
in excited state
1
0
Time
Emitter and cavity mode levels anticrossed for DE
Oscillations with Rabi frequency  = DE / h:
|emitter in a ground state, photon in the cavity>
|excited emitter, empty cavity>
Isidor Isaac Rabi
1898 (Rymanów) –
1988 (New York)
Nobel Prize 1944
Strong coupling regime
emitter – cavity mode detuning
Energy levels versus detuning:
• Anticrossing of levels at emitter – cavity mode resonance
Strong coupling regimethe first experiments
• Evidence for the
strong lightmatter coupling
[R. J. Thompson et al.,
Phys. Rev. Lett. (1992).
• Increasing light
matter cupling
wiht increasing
number of
atoms inside
cavity
Summary
• Spontaneous emission rate depends on the photonic
environment:
• it can be enhanced or supressed (weak coupling), or
• reversed!(strong coupling)
• Fermi’s Golden Rule: spontaneous emission rate depends on:
• availability of final states (spectral overlap emittermode) and
• spatial position of the emitter with respect to the
mode distribution and
• emitter dipol moment
Jaynes, F.W. Cummings model
It describes the system of a two-level atom
interacting with a quantized mode of an
optical cavity, with or without the presence of
light (in the form of a bath of electromagnetic
radiation that can cause spontaneous emission
and absorption).
E.T. Jaynes, F.W. Cummings (1963).
"Comparison of quantum and semiclassical
radiation theories with application to the
beam maser". Proc. IEEE 51 (1): 89–109.
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```