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Light Matter Interaction - Semi-classical
The picture where atoms are quantized, but light is still kept classical –
semi-classical approximation
The time-dependent Schrodinger equation
The weak-field limit: Einstein’s B coefficient
The strong-field limit: Rabi oscillations
The Bloch sphere
Introduction
1913 - Bohr postulated that a quantum of light of
angular frequency ω is absorbed or emitted
whenever an atom jumps between two quantized
energy levels E1 and E2
E2 − E1 = ħω
1916–7 Einstein introduced the Einstein
coefficients to quantify the rate at which the
absorption and emission of quanta occur
Also, discovered the process of stimulated emission
- the basis for laser operation
What happens to the irradiated atom before the
absorption transition is complete?
The two-level atom approximation
When light with angular frequency
ω coincides with one of the optical transitions of the atom, a
resonant interaction between the transition and the light field exists
The other levels of the atom, which only weakly interact with the light
because they are off-resonance – are neglected
In the classical picture the light beam induces dipole oscillations in
the atom, which then re-radiate at the same frequency
If the light frequency corresponds to the natural frequency of the
atom, the magnitude of the dipole oscillations will be large and the
interaction between the atom and the light will be strong
Coherent superposition states
• resonant interaction between an atom and a light field involves the
concept of coherent superposition states
• Consider a quantum system with two levels: a two-level atom
or a spin 1/2 nucleus in a magnetic field
• The wave function of the system is of the form
c1 and c2 describe the wave function amplitude coefficients for the
two states of the atom or nucleus
• a measurement would give with probability |c1|2 for level 1 and |c2|2 for
level 2
• consider a gas of N0 identical two-level particles (e.g. two-level
atoms or spin 1/2 nuclei) with N1 particles in the lower level and N2
in the upper level - a statistical mixture
• by setting |c1|2 = N1/N0 and |c2|2 = N2/N0 we would obtain the same
results as for repeated measurements on a gas of N0 particles in the
superposition state
• what, then, is the difference?
• each of the particles in the superposition state is in some sense
simultaneously in the |1> and |2> states, and this leads to the
possibility of wave function interference
• in the statistical mixture, by contrast, a given particle is either in level
1 or in level 2, and no wave function interference can occur
• analogy can be made with interference effects in light beams
• consider two overlapping light beams of the same frequency with
phases φ1 and φ2, respectively. The resultant field is:
the beams interfere if they are coherent, when Δφ = φ2 − φ1 = cons. at a point in
space
at some positions where Δφ = (even integer x π) - a bright fringe and where Δφ = (odd
integer x π) - a dark fringe
if the beams are incoherent - phases vary randomly with time - no interference
the powers of the beams just add together giving |ε|2 = | ε1|2 + | ε2|2
analogy with the two-level superposition states can be made by setting c1 → ε1
exp(−iφ1) and c2 → ε2 exp(−iφ2)
wave function interference can occur for a definite phase relationship for c1 and c2
this occurs in the superposition state, but not in a statistical mixture, where the
different particle wavefunctions have random phases with respect to each other
in the treatment, the light pulse links the phases of the upper and lower levels of an
The density matrix
• the elements of the density matrix are ρij = <cic∗j>
ci is the wave function amplitude for the ith quantum level, and i and
j run over all the quantum states of the atom
for a two level system
the difference between statistical mixtures and coherent superpositions
are the off-diagonal terms (ρ12 and ρ21) in the density matrix - non-zero
off-diagonal elements
in a statistical mixture, each atom will either have |c1| = 1 and |c2| = 0 or
vice versa
The time-dependent Schr¨odinger equation
We want to solve the time-dependent Schr¨odinger equation for a two-level
(E1 and E2) atom in the presence of a light wave of angular frequency ω
assume that the light is very close to resonance with the transition
Let’s split the Hamiltonian into a time-indpendent Ĥ0 - the atom in dark and a
perturbation for light-atom interaction
For the two-level atom two solutions for the unperturbed system
The general solution for the time dependent eq. is
and for a two level system
Substituting this wavefunction into the time-dependent Schr¨odinger
equation and subsituting for Ĥ
implying that
On multiplying by ψ∗1, integrating over space, and making use of the
orthonormality of the eigenfunctions, requiring that
dij is the Kronecker delta function
Similarly, on multiplying by ψ∗2 and integrating
let’s consider explicitly the form of the perturbation
in the semi-classical approach, the light–atom interaction is given
by the energy shift of the atomic dipole in the electric field of the light
e - magnitude of the electron charge
e0 - amplitude of the light wave
arbitrarily choose the x-axis as the
polarization direction
the perturbation matrix elements are
the dipole matrix element mij is
and
x is an odd parity operator and atomic states have either even or odd parities,
therefore, μ11 = μ22 = 0
the dipole matrix element represents a measurable quantity
and must be real, implying μ21 = μ12, because μ21 = μ∗12
let’s introduce the Rabi frequency
These equations have to be solved to understand the behavior
of the atom in the light field
two distinct types of solution: the weakfield limit and the strong-field limit
The weak-field limit: Einstein’s B coefficient
• The weak-field limit applies to low-intensity light sources such as
blackbody lamps
• The atom is initially in the lower level and the lamp is turned on at t = 0,
implying that c1(0) = 1 and c2(0) = 0
• With a low-intensity source, the electric field amplitude will be small and
the perturbation weak
• The number of transitions expected is therefore small, and it will always
be the case that c1(t) >> c2(t)
• Therefore, c1(t) = 1 for all t
according to the rotating wave approximation, the second term is neglected
since δω<<(ω + ω0), the second term is much smaller than the first
For a beam tuned to exact resonance with the transition, δω is equal to zero
the probability that the atom is in the upper
level increases as t2
in Einstein’s approach the transition
probability is time independent
• Let’s re-examine the assumptions of our analysis
• It was assumed that the atomic transition line is perfectly sharp
• we know that all spectral lines have a finite width Δω
• we consider the interaction between the atom and a broad-band
• source such as a black-body lamp
• Such a broad-band source can be specified by the spectral energy
density u(ω), which must satisfy
Let’s integrate the eq. for the probability over the spectral line
• making the approximation that the spectral line is sharp compared to the
broad-band spectrum of the lamp
• u(ω) does not vary significantly within the integral, allowing to replace
u(ω) by a constant value u(ω0) and to evaluate the integral
• the limiting value for tΔω→∞ is u(ω0)2πt
• Hence we obtain
the probability that the atom is in the upper level increases linearly with
time
this eq. can be related to that defining the Einstein B coefficient
which implies that the transition probability per unit time per atom is
Bω12u(ω0)
• in the analysis it was assumed that the atomic dipole moment was
aligned parallel to the polarization vector of the light
• however, in a gas of atoms, the direction of the atomic dipoles
is random
• if the angle between the polarization and a particular dipole is θ,
then we need to take the average of (μ12 cos θ)2 for all the atoms in
the gas
• using <cos2 θ> = 1/3, we then replace μ212 by μ212/3 to obtain the
transition probability rate W12
the weak-field limit is equivalent
to the Einstein analysis
enables calculation of the B
coefficient from the atomic
wave functions
The strong-field limit: Rabi oscillations
• previously, assumed that the light field was weak so that the population
of the excited state was always small and the approximation c1(t) ≈ 1
was valid for all t
• the population of the upper level is significant
• this condition applies when the light–atom interaction is strong,
meaning strong electric fields - powerful laser beams
• to find a solution in the strong-field limit we make two simplifications:
• the rotating wave approximation to neglect the terms that oscillate at
±(ω +ω0)
• we only consider the case of exact resonance with δω = 0
• with the simplifications
differentiate the first line and substitute from the second to find
describes oscillatory motion
at angular frequency ΩR/2
• if the particle is in the lower level at t = 0 so that c1(0) = 1 and c2(0) = 0
the solution is
and the probability for finding the electron in the upper and lower levels is
The time dependence of the
probabilities is shown in the Fig.
At t = π/ΩR the electron is in the
upper level, whereas at t = 2π/ΩR it
is back in the lower level
The process then repeats itself
with a period equal to 2π/ΩR
The electron thus oscillates back and forth between the lower and
upper levels at a frequency equal to ΩR/2π
The oscillatory behavior in response to the strong-field is called Rabi
oscillation or Rabi flopping
• I. I. Rabi, Phys. Rev. 51, 652 (1937).
• Rabi’s original derivation applied to oscillating electromagnetic fields
tuned to resonance with the Zeeman-split levels of a spin-1/2 nucleus
• The RF field tips the spin vector from down to up and then back to
down again, a process equivalent to the Rabi flopping
• Rabi’s work was the precursor NMR techniques
• Nobel Prize for Physics in 1944.
For light that is not exactly resonant with the transition
Δω is the detuning
the frequency of the Rabi oscillations increases but their amplitude decreases
as the light is tuned away from resonance
for transitions in the visible-frequency range – observation of Rabi flopping
requires powerful laser beams - pulsed, so that the electric field amplitude varies
with time
therefore, the Rabi frequency ΩR/2π also varies with time, and so it is useful to
define the pulse area Θ
A pulse with an area equal to π is a π-pulse
An atom with c1 = 1 at t = 0 is promoted to the excited
state with c2 = 1 by a π-pulse
will end in the ground state if it interacts with a 2π-pulse
what are the time-scales involved in Rabi flopping processes?
• Example A powerful beam of light is incident on monatomic hydrogen and is
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tuned to resonance with the 1s → 2p transition at 137 nm.
(a) Calculate the Rabi oscillation period when the optical intensity is 10 kWm−2
and the light is polarized in the z-direction
(b) Calculate the optical intensity required to make the Rabi oscillation period
equal to the radiative lifetime of the 2p level, namely 1.6 ns.
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Solution
(a) The atomic dipole moment for this transition is −0.74ea0 = −6.32x10-30 Cm
The intensity of a light beam is related to its electric field amplitude
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In a gas we may take n ≈ 1, and so we find E0 = 2.7x 103 Vm−1
On substituting into the Rabi frequency:
the Rabi flopping period is 2π/ΩR = 38 ns
•
(b) A Rabi oscillation period of 1.6 ns corresponds to ΩR =3.9 x 109 rad s−1
E0 = 6.5 x104 Vm−1, and I = 5.7 MWm−2
Damping
• the example shows why it is difficult to observe Rabi oscillations
at low powers - the oscillation period is longer than the radiative
lifetime, and we would expect random spontaneous emission to
destroy the coherence of the superposition states
• thus higher powers are needed to shorten the Rabi flopping period
• Spontaneous emission is one of the damping mechanisms for the
Rabi oscillations
• consideration of damping is very important for determining the
experimental conditions under which Rabi oscillations can be
observed
The damping processes for Rabi flopping are characterized by two time
constants, T1 and T2 called - longitudinal relaxation and transverse
relaxation, respectively
• T1 damping is determined by population decay
• T2 damping is related to dephasing processes
• the T1 (longitudinal) damping processes is related to the
spontaneous tendency of the atom (in the excited state) to decay to
lower levels and randomly break the coherence of the electronic
wavefunction - interrupt the Rabi flopping
• The rate of these types of damping process is governed by the
lifetime t of the upper level, which itself is determined by both the
radiative and non-radiative decay rates
The upper limit on T1 is set by the radiative lifetime tR of the excited state includes
transitions both to the resonant lower level and to other non-resonant levels
The T2 (transverse) damping processes is related to elastic or near-elastic
collisions which breaks the phase of the wavefunction without altering the
population of the excited state mechanism
In a gas, collisions can occur between the atoms or with the walls
whereas in a solid there can be interactions with impurities or phonons
By randomizing the phase of the wavefunction, the collisions destroy any
Rabi flopping which rely on phase coherence
dephasing can occur by two distinct mechanisms:
population decay and population-conserving scattering processes
the total dephasing rate is:
in solids at room temperature, the pure dephasing rate is much faster than the
population decay rate (T2’ <<T1) decoherence is by scattering
When T2’ >> T1, the decoherence is then governed primarily by T1, which itself
is determined by the lifetime of the upper level
•
If the damping rate is g, the probability that the electron is in the upper level
|c2(t)|2 is given by
when g the formula reduces to the undamped case
For light damping - the electron performs a
few damped oscillations and approaches
the asymptotic limit |c1|2 = |c2|2 =1/2
expected from the Einstein analysis of a
pure two-level system in the strong-field limit
At high optical power levels the spontaneous
emission rate is negligible and the rates of
stimulated emission and absorption equal identical upper and lower level populations
for strong damping - equivalent to the weak-field limit, we can make g/ΩR large
by turning down the electric field of the light beam
Experimental observations of Rabi oscillations
• Rabi oscillations are strongly damped except when
In gases typical values of g for optical-frequency transitions 107–109 s−1
In solids the dephasing times are often shorter and g can be 1012 s−1
high damping rates make demonstration of Rabi oscillations difficult
The observation of the oscillations requires a time resolution shorter than 1/ΩR,
while the short Rabi oscillation periods require large electric field amplitudes
•
The first experimental evidence of Rabi oscillations came from the
observation of self-induced transparency - S. L. McCall and E. L. Hahn,
Phys. Rev. 183, 457 (1969).
•
if the pulse area is equal to 2π, then the atoms are left in the ground state
at the end of the pulse
•
This implies that there is no net absorption, and so a medium that absorbs
strongly at low powers would become transparent to a 2π-pulse: hence
‘self-induced transparency’
•
The condition to observe the phenomenon is that the pulse duration
should be shorter than the damping time, and that the pulse area should be
equal to an integer multiple of 2π
McCall and Hahn performed their experiments on the absorption of a ruby
crystal excited resonantly with nanosecond pulses from a ruby laser. The
ruby crystal was held to suppress damping by phonon scattering
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They confirmed that the crystal did indeed become more transparent as the
pulse area (determined by the energy of the pulse) approached 2π
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The first direct evidence of Rabi oscillations came from experiments of Gibbs H. M. Gibbs, Phys. Rev. Lett. 29,459
(1972) and Phys. Rev. A 8, 446 (1973)
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fluorescence emitted by Rb atoms excited resonantly by short pulses from a mercury laser
By placing the Rb atoms in a superconducting magnet, one of the hyperfine components of the MJ =
−1/2 → +1/2 line of the 5 2S1/2 → 5 2P1/2 transition with a dipole moment of 1.45 x 10-29 Cm could be
tuned to resonance with the laser by the Zeeman effect
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The upper level could decay either to the +1/2 or −1/2 levels of the 5s state, with radiative lifetimes of
42 and 84 ns, respectively, - a total radiative lifetime of (1/42 + 1/84)−1 = 28 ns
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With a low density of atoms to prevent dephasing by collisions, the right conditions to observe Rabi
flopping were present for pulses significantly shorter than 28 ns. The oscillations were then detected
by measuring the fluorescence from the upper level as a function of the pulse area Θ
•
When the pulse area was equal to odd integer multiples of π the atoms ended up in the excited state
at the completion of the pulse, and then decayed to the ground state by spontaneous emission,
producing a strong fluorescence signal
•
when the pulse area was equal to even integer multiples of π the atoms were in the ground state at
the completion of the pulse, and no fluorescence occurred
Mollow triplets - the frequency space equivalent of the
Rabi oscillations in the time domain
B. R. Mollow, Phys. Rev. 188 (1969).
• Mollow demonstrated that the coherent oscillatory motion of the
electrons in the strong-field limit beat with the fundamental transition
angular frequency ω0 and produce side bands in the emission
spectrum at ω0±ΩR
The AC Stark interaction between a two-level atom and an intense resonant
light field splits the bare atom states into doublets separated by the Rabi
frequency ΩR
R. E. Grove, F. Y. Wu, and S. Ezekiel, Phys. Rev. A 15, 227 (1977).