Chapter 1
Coherent interaction of atoms and
light
Interaction of radiation with multilevel atomic systems is of importance to
many physical phenomena. A widely used approach is a semi-classical description where the atomic system with its discrete levels is treated quantum mechanically and the radiation field is treated classically. Within the
semiclassical approximation there are several possible approaches. The most
straight forward is to solve the Schr¨odinger equation for the levels coupled by
the optical field. This approach is useful for many situations, but does not
allow spontaneous emission to be treated consistently. The effects of spontaneous emission can be included using a non-Hermitian Hamiltonian and
Monte Carlo wave function methods.
Alternatively a density matrix formalism leading to a master equation
allows for a full description of spontaneous emission. This approach will
allow us to identify and characterize decay times for the population and the
coherence, commonly referred to as T1 and T2 times.
1.1
Rabi Oscillations
We will start with the simpler Schr¨odinger equation approach and afterwards show that the results obtained are in agreement with those found
from a density matrix treatment. Interaction of a two level atom with a
monochromatic field results in deterministic rotation of the atomic state
within the basis of the optically coupled levels. Here we assume a single
atom interacting with the field and solve the Schr¨odinger equation directly.
Writing |ψi = cg (t)e−ıωg t |gi + ce(t)e−ıωe t |ei the Hamiltonian can be written as
April 22, 2015 M. Saffman
2
1 Coherent interaction of atoms and light
ˆ 0 = ~ωg |gihg| + ~ωe |eihe|.
ˆ =H
ˆ 0 + Vˆd, with the unperturbed Hamiltonian H
H
ˆ with
The atom-field electric dipole interaction is described by Vˆd = −dE
dˆ = −eˆ
r ( the charge on an electron is −e) and the electric field is
E = (E/2)e−ıωt + c.c. . Using
rˆ =
=
=
=
IˆrˆIˆ
(|gihg| + |eihe|)ˆ
r(|gihg| + |eihe|)
he|ˆ
r|gi|eihg| + hg|ˆ
r|ei|gihe|
reg |eihg| + rge |gihe|
the interaction Hamiltonian is written Vˆd = −Edeg |eihg| + H.c. and the
Schr¨odinger equation then gives
i~c˙g + ~ωg cg = ~ωg cg − Edge cee−ı(ωe −ωg )t ,
i~c˙e + ~ωe ce = ~ωe ce − Edeg cg eı(ωe −ωg )t .
(1.1a)
(1.1b)
We are interested in the near resonant situation where the detuning ∆ =
ω − ωeg ωeg = ωe − ωg . Dropping therefore the counterrotating terms on
the r.h.s. of Eqs. (1.1) we get
dcg
Ω∗ ı∆t
= i cee ,
dt
2
dce
Ω
= i cg e−ı∆t
dt
2
(1.2a)
(1.2b)
with the complex Rabi frequency Ω = deg E/~.
To solve Eqs. (1.2) it is convenient to use the variables c˜g = e−ı∆t/2 cg ,
c˜e = eı∆t/2 ce , which satisfy the set
d˜
cg
∆
Ω∗
= −i c˜g + i c˜e
dt
2
2
d˜
ce
∆
Ω
= i c˜e + i c˜g
dt
2
2
(1.3a)
(1.3b)
The solutions with initial amplitudes c˜g (0) = c˜g0 , c˜e(0) = c˜e0 are
0 0 0 Ωt
∆
Ωt
Ω∗ ı∆t/2
Ωt
c˜g (t) = cos
− i 0 sin
c˜g0 + i 0 e
sin
c˜e0
2
Ω
2
Ω
2
0 0 0 Ω −ı∆t/2
Ωt
Ωt
∆
Ωt
c˜e (t) = i 0 e
sin
c˜g0 + cos
+ i 0 sin
c˜e0
Ω
2
2
Ω
2
(1.4a)
(1.4b)
1.1 Rabi Oscillations
3
p
with Ω0 = |Ω|2 + ∆2 the effective off-resonance Rabi frequency. We can
˜ t0 )˜c(t0) with
write the solution as a rotation matrix ˜c(t) = U(t,
 0
0
0

Ω (t−t0 )
Ω (t−t0 )
∆
Ω∗
0)
−
i
sin
i
sin
cos Ω (t−t
0
0
2
2
˜ t0) = 
Ω0
2
0 Ω 0
 .
U(t,
Ω (t−t0 )
Ω (t−t0 )
Ω (t−t0 )
Ω
∆
i Ω0 sin
cos
+
i
sin
2
2
Ω0
2
(1.5)
Transforming back to the variables without tildes we have c(t) = U(t, t0 )c(t0)
where
ı∆t/2
−ı∆t /2
0
e
0
e
0
˜
U(t, t0 ) =
U(t, t0 )
0
e−ı∆t/2
0
eı∆t0 /2
0
i
0

 ∆(t−t0 ) h
0
∆(t+t0 )
Ω (t−t0 )
Ω (t−t0 )
Ω (t−t0)
∆
ı
Ω∗
2
−
i
sin
ie
sin
eı 2
cos
0
0
2
2
Ω0
2
0
i .
h
0 Ω = 
∆(t+t0 )
∆(t−t0 )
Ω (t−t0 )
Ω (t−t0 )
Ω (t−t0 )
−ı
Ω
−ı
∆
2
2
ie
sin
e
+
i
sin
cos
0
0
Ω
2
2
Ω
2
(1.6)
Although the phase factors in (1.6) look odd as they depend on both t−t0 and
t + t0 they are needed to track the full phase dependence of the wavefunction.
We can easily verify that for t2 > t1 > t0 , U (t2, t0 ) = U (t2 , t1 )U (t1 , t0 ) which
must be true for a unitary transformation. If we are only interested in final
probabilities, and not the amplitudes, it is sufficient to propagate the state
with the simpler U˜ (t, t0 ).
When the driving field is resonant so ∆ = 0 and the field amplitude is
constant the interaction Hamiltonian has no explicit time dependence and
U(t, t0 ) = U(t − t0 ), so the matrix takes on the simple form


|Ω|(t−t0 )
|Ω|(t−t0 )
−ıφ
cos
ie sin
2
2

U (t − t0 ) = 
(1.7)
|Ω|(t−t
)
|Ω|(t−t
0
0)
ıφ
ie sin
cos
2
2
where Ω = |Ω|eıφ .
If the atom is initially in the ground state the time dependent probabilities
to be in the ground and excited states are
0 0 ∆2
2
2 Ωt
2 Ωt
|cg (t)| = cos
+
sin
,
2
|Ω|2 + ∆2
2
0 |Ω|2
2
2 Ωt
|ce(t)| =
sin
.
(1.8)
|Ω|2 + ∆2
2
The population oscillations are shown in Fig. 1.1. If the driving field is
resonant, ∆ = 0, Ω0 = Ω and the amplitude to be in the excited state will
April 22, 2015 M. Saffman
4
1 Coherent interaction of atoms and light
1.0
∆=ω− ωeg
|e>
ω
Ω1
Pe
0.8
0.6
0.4
0.2
0.0
0.0
|g>
1.0
2.0
3.0
t
Figure 1.1: Rabi oscillations of a two-level system with (Ω = 2π, ∆ = 0) and (Ω = 2π, ∆ =
4π).
reach unity at time Ωt/2 = π/2 or t = π/Ω. This defines a so-called π pulse
of duration tπ = π/Ω which inverts the population between the two states.
It is also interesting to examine the phase evolution of a quantum state
under Rabi oscillations. Suppose we start with the state |ψi = cg |gi + ce |ei.
A resonant 2π pulse, Ωt = 2π, has the evolution matrix
−1 0
U (t − t0 ) =
(1.9)
0 −1
which results in the transformation
|ψi → −|ψi = eıπ |ψi.
Recall that rotation of a spin 1/2 object about the z axis by 2π results in the
same transformation
ˆ
|ψi → e−ıJz θ/~ |ψi = e−ı(±1/2)(2π) |ψi = eı±π |ψi = eıπ |ψi.
A 2π pulse applied to a two-level system is analogous to rotation of a spin
1/2 through an angle of 2π. In both cases we are rotating between the two
quantum states of the object. Classically a 2π rotation has no observable
effect. In quantum mechanics the phase of the wave function can change.
The quantum phase can be observed using an interference experiment, and
this phase is an essential ingredient inmany protocols that are used to create
entanglement between different spins.
12
1 Coherent interaction of atoms and light
1.4
Density matrix description of two-level atom and
field
To calculate the interaction of a two-level atom with a radiation field we can
solve the Schr¨odinger equation directly as in the solution for Rabi flopping.
That solution neglected the finite lifetime of the excited state. For weak
decay we can add terms like −i(γ/2)|eihe| into the Hamiltonian. However,
the resulting Hamiltonian is non-Hermitian and does not strictly conserve
probability. A consistent approach uses the density matrix. A brief review
of density matrix theory is given in the QM primer notes.
The Hamiltonian can be written as
ˆ=H
ˆa + H
ˆ1.
H
(1.23)
ˆ a describes the unperturbed atom and H
ˆ 1 is the field-atom interaction
Here H
potential. Consider a two-level atom with lower level |gi and upper level
|ei. All atoms which spontaneously decay from the upper level populate the
lower level, and there is no spontaneous decay out of the lower level. Then
ˆ a )ij = δij Ui with Ui the unperturbed energy of level |ii (i = e, g). The
(H
interaction potential should account for shifts of the unperturbed levels due
to the ac Stark effect and dipole induced coupling between levels due to the
ˆ · E, with d
ˆ the dipole moment operator and E the
ˆ 1 = −d
interaction H
classical electric field. Since the energy eigenstates have definite parity the
ˆ vanish at first order in the field. Neglecting for the
diagonal elements of d
moment the 2nd order AC Stark shifts we find
ˆ1 =
H
0
−E · dge
−E · deg
0
(1.24)
ˆ
where deg = he|d|gi.
We can simplify the notation by writing E · deg = Ed so
ˆ 1 = −E
H
0 d∗
d 0
.
(1.25)
1.4 Density matrix description of two-level atom and field
13
Using Eqs. (2.41-1.24) we get the density matrix equations of motion
dρgg
i
= − E (dρge − d∗ ρeg )
(1.26a)
dt
~
dρee
i
= − E (d∗ ρeg − dρge )
(1.26b)
dt
~
dρeg
i
= −iωa ρeg − Ed (ρee − ρgg )
(1.26c)
dt
~
i
dρge
(1.26d)
= iωa ρge + Ed∗ (ρee − ρgg ) ,
dt
~
where ~ωa = Ue − Ug is the separation of the atomic levels.
dρ
dρ
We see that dtgg + dρdtee = 0 (which follows from ρgg + ρee = 1) and dteg =
dρ
( dtge )∗ (which follows from the fact that ρˆ is Hermitian). It is therefore
sufficient to keep only one of the off-diagonal elements, say ρeg = ρ∗ge .
We account for decay processes phenomenologically. The upper level population is assumed to have a radiative lifetime T1 . In a closed system all the
population that leaves the upper level enters the lower level which grows at a
rate ρee /T1 . The atomic polarization is assumed dephased with a characteristic time T2 . With these definitions the density matrix equations become
dρgg
ρee
i
=
− E dρ∗eg − d∗ ρeg
(1.27a)
dt
T1
~
dρee
ρee
i
− E d∗ ρeg − dρ∗eg
(1.27b)
= −
dt
T1
~
i
dρeg
1
(1.27c)
= −( + iωa )ρeg − Ed (ρee − ρgg ) .
dt
T2
~
We wish to solve for the atomic dynamics in the presence of a monochromatic driving field E = |E| cos(ωt + θ) = E2 e−ıωt + c.c., where E = |E|e−ıθ .
Putting ρeg = ρ˜eg e−ıωt gives
dρgg
ρee
i
=
−
Ed˜
ρ∗eg − E ∗ d∗ ρ˜eg + E ∗ deı2ωtρ˜∗eg − Ed∗ e−ı2ωtρ˜eg
dt
T1
2~
dρee
ρee
i
= −
+
Ed˜
ρ∗eg − E ∗ d∗ ρ˜eg + E ∗ deı2ωtρ˜∗eg − Ed∗ e−ı2ωtρ˜eg
dt
T1
2~
d˜
ρeg
1
d
= (i∆ − )˜
ρeg − i
E + E ∗ e2ıωt (ρee − ρgg ) .
dt
T2
2~
(1.28a)
(1.28b)
(1.28c)
We have introduced the detuning ∆ = ω − ωa .
Exact solutions to these equations are difficult to come by. When the
driving field is near resonant with the atomic transition |∆| ω, ωa and it is
April 22, 2015 M. Saffman
14
1 Coherent interaction of atoms and light
a good approximation to neglect all the rapidly oscillating terms on the right
hand side of Eqs. (1.28). This is known as the rotating wave approximation
(RWA) which takes the form
dρgg
ρee
i
=
−
Ω˜
ρ∗eg − Ω∗ ρ˜eg
dt
T1
2
dρee
ρee i
= −
+ Ω˜
ρ∗eg − Ω∗ ρ˜eg
dt
2
T1
Ω
d˜
ρeg
1
ρ˜eg − i (ρee − ρgg ) .
= i∆ −
dt
T2
2
(1.29a)
(1.29b)
(1.29c)
Here we have introduced the complex Rabi frequency Ω = dE/~. These
density matrix equations are often referred to as optical Bloch equations
since they have the same form as Bloch’s description of nuclear magnetic
resonance2.
Defining the inversion by w = ρee −ρgg we can reduce the two level problem
in the RWA to two equations
1+w
dw
= −
+ i Ω˜
ρ∗eg − Ω∗ ρ˜eg
dt
T1
1
d˜
ρeg
Ω
= i∆ −
ρ˜eg − i w.
dt
T2
2
(1.30a)
(1.30b)
When there is no driving field Ω = 0 and the steady state solutions are
w = −1 and ρ˜eg = 0. Although we are working with a model of a closed
transition it is often desirable to allow for an incoherent pumping mechanism
that creates a net inversion in the absence of a driving field. To do so we
generalize Eqs.(1.30) to
dw
w − w (0)
= −
+ i Ω˜
ρ∗eg − Ω∗ ρ˜eg
dt
T1 d˜
ρeg
1
Ω
= i∆ −
ρ˜eg − i w.
dt
T2
2
(1.31a)
(1.31b)
where w (0) is the zero field inversion. We see that the population inversion
w acts as a source term for the coherence and that the coherence acts as a
source term for changes in the population inversion. The coherent dynamics
evolves on the frequency scale set by Ω.
2
F. Bloch, Nuclear induction, Phys. Rev. 70, 460 (1946).
1.4 Density matrix description of two-level atom and field
15
The steady state solutions are a balance between the coherent dynamics
and decay processes governed by the time constants T1 , T2 . Setting the time
derivatives to zero we find the stationary solutions
1 + ∆2 T22
w = w
1 + ∆2 T22 + T1 T2 |Ω|2
w ΩT2
ρeg = −i
e−ıωt .
2 1 + i∆T2
(0)
(1.32)
The classical macroscopic polarization density is the sum of the individual atomic polarizations. Introducing the atomic density na we get for the
polarization density
ˆ = na Tr[ˆ
ˆ = na d(ρge + ρeg )
P (t) = na hdi
ρd]

= na w(0)

d2
1 + ∆2T22
 ∆ − i/T2

−ıωt
+ c.c. .

2 Ee
2
2
2
2~ 1 + ∆ T2 + T1T2 |Ω|
∆2 + T12
(1.33)
It is apparent from Eq. (1.33) that P (t) includes terms at frequencies ω and
−ω proportional to all odd powers of E. Before proceeding let us specialize the
notation. At zero applied field the FWHM absorption linewidth (imaginary
part of the polarization) is given by γ = 2/T2 . Also we have quite generally
1
1 1
=
+ γc
T2
2 T1
where γc is the dipole dephasing rate due to collisions. This equation says
that in the absence of collisions the decay rate of the coherence is one half
the decay rate of the population. We can understand this qualitatively since
the coherence ∼ ce and the population ∼ |ce|2 . Neglecting collisions we have
T1 = 1/γ, T2 = 2/γ, so
2
1
2∆/γ − i
(0) d
−ıωt
P (t) = na w
Ee
+ c.c. .
(1.34)
~γ 1 + 2|Ω|22/γ22 1 + 4∆2 /γ 2
1+4∆ /γ
The ground and excited state populations for an initially unexcited atom
(w = −1) are
(0)
ρgg
ρee
1−w
1 + 4∆2 /γ 2 + |Ω|2/γ 2
=
=
2
1 + 4∆2 /γ 2 + 2|Ω|2 /γ 2
1+w
1
2|Ω|2 /γ 2
=
=
2
2 1 + 4∆2 /γ 2 + 2|Ω|2 /γ 2
April 22, 2015 M. Saffman
16
1 Coherent interaction of atoms and light
For the purpose of practical calculations the following definitions are useful. The inversion is given by
w (0)
w=
.
2
1 + γ22Ω
2
+4∆
(1.35)
On resonance this can be written in the form w = w (0)/(1 + I/Isat ) where
Isat = 0 nc~2 γ 2 /(4d2 ) and n is the refractive index. Using the result d2 =
3π0 c3 ~γ/ωa3 , which is true for electric dipole transitions, we can write the
saturation intensity as
Isat
n~γωa3
=
.
12πc2
(1.36)
Apart from the factor of n we have included here this is the same result found
from our earlier treatment based on the rate equations with A, B coefficients.
1.4.1
Comparison with Schr¨
odinger equation solution
We can compare the dynamics found from the density matrix equations with
the Schr¨odinger equation Rabi oscillation solutions. When T1 , T2 → ∞ the
two descriptions give the same results. With finite decay times significant
differences are seen as shown in Fig. 1.3. The Schr¨odinger equation solution
0.010
ρee
0.008
0.006
Ω2/2∆2
0.004
Ω2/4∆2
0.002
0.000
0.0
0.5
1.0
1.5
2.0
t
Figure 1.3: Excited state population from solution of density matrix equations for a two-level
atom with Ω = 2π, ∆ = 2π × 10, T1 = 2.5, T2 = 5.
1.5 Rate equations
17
for the average excited state population at large detuning is
hPei =
|Ω|2 1
|Ω|2
'
.
|Ω|2 + ∆2 2
2∆2
The factor of 1/2 comes from taking the time average of sin2 (Ω0 t/2). The
stationary solution of the density matrix equations at large detuning is
|Ω|2
,
ρee '
4∆2
which is twice smaller. We see that at short times the average of ρee is
indeed close to hPe i. However, at long times, there is a loss of coherence,
and the population oscillations decay towards a stationary solution of ρee =
hPei /2. Accurate prediction of the excited state population requires solving
the density matrix equations, even at time scales several times shorter than
T1 .
1.5
Rate equations
Using T1 = 1/γ, T2 = 2/γ the density matrix equations (1.30) take the form
dw
(1.37a)
= −γ(1 + w) + i Ω˜
ρ∗eg − Ω∗ ρ˜eg ,
dt
d˜
ρeg
γ
Ω
= i∆ −
ρ˜eg − i w.
(1.37b)
dt
2
2
When |∆| γ, |Ω| the coherence ρ˜eg is slaved to the inversion and we can
dρ˜
adiabatically eliminate the coherence by setting dteg = 0 which gives a closed
equation for the inversion
dw
= −γ(1 + w) −
dt
1
2|Ω|2
γ
2 w.
+ 4∆
γ2
Even when the detuning is not large this effective rate equation for w
provides an efficient method for finding asymptotic solutions of the dynamics.
Rewriting in terms of populations Pe = ρee , Pg = ρgg we find
|Ω|2
dPe
γ
= −γPe −
2 (Pe − Pg ),
dt
1 + 4∆
2
γ
dPg
= γPe +
dt
1
April 22, 2015 M. Saffman
|Ω|2
γ
2 (Pe
+ 4∆
2
γ
− Pg ).
(1.38a)
(1.38b)
18
1 Coherent interaction of atoms and light
We see that when |∆| is large the coherent redistribution of population
proceeds at a rate proportional to γ|Ω|2 /∆2 = |Ω| × γ|Ω|/∆2 . For large
detuning this is slower than the resonant rate of Ω.
The rate equations (1.40) which were derived from the optical Bloch equations reproduce the Einstein description in terms of A,B coefficients. Putting
Ng = Pg , Ne = Pe, γ = A, and
|Ω|2
γ
2
1+ 4∆
γ2
= Bρω we get
dNe
(1.39a)
= −ANe − Bρω Ne + Bρω Ng ,
dt
dNg
= ANe + Bρω Ne − Bρω Ng ,
(1.39b)
dt
which are just the rate equations (1.1) from the atoms and radiation chapter.
1.5.1
Multilevel rate equations
Let’s generalize the rate equations to more than two levels. Assume there
are Ng lower states, all radiatively stable, and Ne excited states which can
decay to the lower states with decay rate γ. The total number of states is
N = Ng + Ne. The multilevel rate equations are
dPe,k
= −γPe,k −
dt
dPg,j
= γ
dt
Ne
X
|Ωjk |2
γ
(Pe,k
4∆2
1
+
γ2
j=1
Ng
X
bjk Pe,k +
k=1
− Pg,j ),
|Ωjk |2
γ
(Pe,k
4∆2
1
+
γ2
k=1
Ne
X
k = 1..Ne
− Pg,j ).
(1.40a)
j = 1...Ng (1.40b)
The fractional branching coefficients are
|hg, j|rmj −mk |e, ki|2
bjk = P
2
j |hg, j|rmj −mk |e, ki|
P
where q = −1, 0, 1. These are normalized so that j bjk = 1.
We can also go one step further and eliminate the excited state populations. If the excited state excitation and decay occur on fast time scales
relevant to the ground state dynamics the excited state populations are slaved
and we can put dPe,k dt ≈ 0 to get
Pe,k = −
|Ωjk |2
γ2
(Pe,k
4∆2
1
+
2
γ
j=1
Ng
X
− Pg,j ),
1.6 Multilevel density matrix equations
19
which can be written as
Pe,k =
|Ωjk |2
j=1 γ 2 Pg,j
PNg |Ωjk|2 .
4∆2
+
j=1 γ 2
γ2
PNg
1+
(1.41)
This is then inserted into (1.40b) to give a closed set of equations for the
ground state populations.
1.6
Multilevel density matrix equations
The Born-Markov approximation applied to a two-level atom interacting with
a radiation bath leads to a master equation in Lindblad form
dˆ
ρ
i ˆ
γ †
ˆ ρˆ + ρˆ
ˆσ†σˆ − 2ˆ
σ ρˆσ
ˆ† .
(1.42)
= − [H,
ρ]
ˆ −
σ
ˆσ
dt
~
2
ˆ = √γˆ
The Lindblad operators are L
σ with σ† = |eihg| a raising operator from
|gi → |ei, i.e. σ
ˆ † |gi = |ei and γ is the population decay rate from |ei → |gi.
This form of the master equation is valid for a single atom. For a many atom
ensemble collisions also cause population redistribution and coherence decay.
To describe these effects additional terms need to be added to the right hand
side. A derivation of (1.42) can be found in many books on quantum optics3 .
We can generalize the two-level master equation to N atomic levels |ii,
i = 1, ...N interacting with a set of M fields Ej , j = 1, ...M. The density
matrix evolution satisfies
Ng
Ne X
dˆ
ρ
i ˆ
1X
lj
†
†
†
ˆki σ
= − [H, ρ]
ˆ −
γki σ
ˆlj ρˆ + ρˆσ
ˆki σ
ˆlj − 2ˆ
σlj ρˆσ
ˆki .
dt
~
2 i,j=1
(1.43)
k,l=1
In this equation we divide the atomic state space into Ng ground state levels and Ne excited levels and assume that spontaneous emission only occurs from levels labeled with i or j to levels labeled with k or l 4. Thus
†
σki
= σik = |iihk| is a raising operator and σlj = |lihj| is a lowering operator.
γ
·γ l←j
lj
The decay factor is γki
= √k←i
γk←i γl←j where the population decay rate from
i → k is γk←i and the vectorial decay constant is γ k←i = γk←ieki with eki a
3
See for example W. H. Louisell Quantum statistical properties of radiation (Wiley, New York, 1973).
This can be generalized in a straightforward way to several groups of levels decaying to more than one
group of lower levels.
4
April 22, 2015 M. Saffman
20
1 Coherent interaction of atoms and light
spherical unit vector connecting states |ii and |ki. For example in a hyperfine basis |ii = |ai, fi , mii, |ki = |ak , fk , mk i with ai, ak additional quantum
numbers and fi, mi , fk , mk the total coupled angular momenta and angular
momentum projections. The decay unit vectors in a spherical basis are then
eki = eq=mi −mk , and eq ·eq0 = δqq0 . This formulation of the multilevel Lindblad
equation can be derived using standard methods5.
Equation (1.43) accounts for the decay of population and coherence from
excited states as well as transfer of coherence from a pair of excited states
to a pair of ground states. This process turns out to be important in subDoppler laser cooling. Coherence transfer by spontaneous emission was first
described by Barrat and Cohen-Tannoudji in 19616 . Several recent books
treat multilevel atomic dynamics and optical pumping in great detail7
It is instructive to evaluate (1.43) for a specific density matrix element
ρˆmn . Let us first rewrite (1.43) in a more convenient form
Ng
Ne Ng
Ne X
X
dˆ
ρ
i ˆ
1 X X kj
lj
= − [H, ρˆ] −
γki (|iihj|ˆ
ρ + ρˆ|iihj|) +
γki
ρˆji |lihk|.
dt
~
2 i,j=1 k=1
i,j=1 k,l=1
(1.44)
Consider now the three cases: (m, n) excited states, (m, n) one excited and
one ground state, and (m, n) ground states. When (m, n) are excited states
we find
N
Ng
N
Ng
e X
e X
dˆ
ρmn
i
1X
1X
kj
kn
ˆ
= − hm|[H, ρˆ]|ni −
γkm ρˆjn −
γki
ρˆmi .
dt
~
2 j=1
2 i=1
k=1
k=1
If we specialize to a single excited hyperfine level fm = fj = fi = fn then
γ k←m · γ k←j = γk←mγk←j δjm , γ k←i · γ k←n = γk←iγk←n δin , and
Ng
dˆ
ρmn
i
1X
ˆ ρ]|ni
= − hm|[H,
ˆ
−
(γk←m + γk←n)ˆ
ρmn
dt
~
2
(m, n) excited states.
k=1
When (m, n) are an excited and a ground state we find
N
Ng
e X
X
dˆ
ρmn
i
1
kj
ˆ ρ]|ni
= − hm|[H,
ˆ
−
γkm
ρˆjn .
dt
~
2
j=1 k=1
5
see for example Z. Ficek and S. Swain, Quantum interference and coherence, Springer (2005).
J. P. Barrat and C. Cohen-Tannoudji, J. Phys. Rad. 22, 329 (1961).
7
W. Happer, Y.-Y. Jau, and T. G. Walker, Optically pumped atoms Wiley-VCH (2010); M. Auzinsh, S.
Budker, and S. M. Rochester, Optically polarized atoms, Oxford (2010).
6
1.6 Multilevel density matrix equations
21
Specializing again to a single excited hyperfine level this reduces to
Ng
dˆ
ρmn
i
1X
ˆ ρ]|ni
= − hm|[H,
ˆ
−
γk←mρˆmn
dt
~
2
(m, n) (excited, ground) states.
k=1
Finally when (m, n) are both ground states we find
Ne
X
dˆ
ρmn
i
mj
ˆ ρˆ]|ni +
γni
ρˆji
= − hm|[H,
dt
~
i,j=1
(m, n) ground states.
(1.45)
mj
When m = n and there is a single excited hyperfine level γni
→ γm←j δij and
we get for the ground state population
Ne
X
dˆ
ρmm
i
ˆ ρ]|mi
= − hm|[H,
ˆ
+
γm←j ρˆjj .
dt
~
j=1
mj
→
When m 6= n and there is a single excited hyperfine level γni
√
γn←iγm←j δmi −mn ,mj −mm and the second term in (1.45) reduces to three possible terms corresponding to emission of a π, σ+ , or σ− photon. These three
cases are the transfer of coherence to the ground state by spontaneous emission introduced by Barrat and Cohen-Tannoudji.
Two-level example
Let’s verify that (1.42) reproduces known results for a two-level atom (levels
|gi, |ei) interacting with a single field E. We have
ˆ = Ug |gihg| + Ue|eihe| − E dˆ
H
= Ug |gihg| + Ue|eihe| − E (|gihg| + |eihe|) dˆ(|gihg| + |eihe|)
= Ug |gihg| + Ue|eihe| − Edeg |eihg| − Edge |gihe|
ˆ and we have assumed |gi, |ei are states of definite parity
where deg = he|d|gi
ˆ =ihe|d|e|
ˆ =i0. Writing formally ρˆ = P ρij |iihj| = ρgg |gihg| +
so hg|d|g|
i,j
April 22, 2015 M. Saffman
22
1 Coherent interaction of atoms and light
ρee |eihe| + ρge |gihe| + ρeg |eihg| the commutator in (1.42) evaluates to givc
dρgg
dt
dρee
i~
dt
dρeg
i~
dt
dρge
i~
dt
i~
= E(deg ρge − dge ρeg )
= −E(deg ρge − dge ρeg )
= ~ωa ρeg + Edeg (ρee − ρgg )
= −~ωa ρge − Edge (ρee − ρgg )
where we have inserted ~ωa = Ue − Ug .
Using γgg = γee = γge = 0 and γeg = γ the second term on the right hand
side of (1.43) evaluates to
~γ
(ˆ
σeg σˆge ρˆ + ρˆσ
ˆeg σ
ˆge − 2ˆ
σge ρˆσ
ˆeg )
2
~γ
ρ + ρˆ|eihe| − 2|gihe|ˆ
ρ|eihg|)
= −i (|eihe|ˆ
2
~γ
= −i [2ρee (|eihe| − |gihg|) + ρeg |eihg| + ρge |gihe|] .
2
−i
Combining with the previous equations we get
dρgg
dt
dρee
i~
dt
dρeg
i~
dt
dρge
i~
dt
i~
= E(deg ρge − dge ρeg ) + i~γρee
= −E(deg ρge − dge ρeg ) − i~γρee
~γ
ρeg + ~ωa ρeg + Edeg (ρee − ρgg )
2
~γ
= −i ρge − ~ωa ρge − Edge (ρee − ρgg ).
2
= −i
We can easily verify that these equations are the same as (1.27) with the
identifications T1 = 1/γ and T2 = 2/γ.