Download Introductory Quantum Optics Section 2. A laser driven two

Introductory Quantum Optics
Section 2. A laser driven two-level atom
Almut Beige
(Dated: November 29, 2007)
Summary. The topic of this section is a single laser driven two-level atom. We consider different
models to predict its behaviour. The probably simplest one is Einstein’s rate equations, which
predicts many characteristic features of the system correctly. A more accurate prediction of the
possible trajectories of a single laser driven atom can be obtained using the quantum jump approach.
The master equations are especially suited for the calculation of expectation values when averaging
over an ensemble of single systems. This Section introduces and illustrates the standard tools for
the description of quantum optical systems undergoing spontaneous emissions.
2.0 Introduction
Once it was discovered that one can actually trap single ions (or atoms) in the laboratory, a whole new
class of experiments became feasible. Testing quantum mechanical laws no longer relied on experiments,
which are only indirectly based on quantum mechanical effects and in which the quantum mechanical
effects are partially washed out by the coupling of the system to the classical world. Instead one could,
for example, shine a laser field on a single atom and see what happens. Atomic traps like the one shown
in Figure 1 became a standard tool in hundreds of laboratories worldwide.
Apart from trapping single atoms in the potential minima created by a rapidly rotating electric field,
atoms emitted by a heated cathode (an oven) can also be slowed down using the potential created by their
interaction with red-detuned laser fields (laser cooling). To transfer, for example, a cold cloud of atoms
into a Bose-Einstein condensate, a large number of them is initially trapped inside a magneto-optical
trap. Afterwards, the hottest atoms are selectively removed from the trap, leaving the remaining ones at
a low enough temperature to form a condensate. Other applications of ion/atom trapping experiments
are very precise clocks (all radio controlled clocks are based on ion trap experiments in national physics
laboratories) and the first demonstration of the main building blocks of a quantum computer.
The aim of this section is to give a relatively accurate description of a single trapped two-level atom
driven by a resonant laser field, which spontaneously emits photons. We will see that the laser causes
oscillations between the ground and the excited state of the atom. Moreover, at random times, the
atom jumps into its ground state while emitting a photon. These photons can be registered as clicks
at a photon detector. A possible trajectory, as it may be recorded in such an experiment, is shown
FIG. 1: Experimental setup for the trapping of single ions. On the left side you see a schematic figure of the
electrodes and the applied fields, which then create a potential minima of the electric field in the middle of the
trap. Once an ion got stuck in this trap, it can stay there for a really long time. On the right side is a picture of
an experimental realisation of a Paul trap, which has been used in actual experiments.
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laser
field
.
atom
a spontaneously
possible trajectory:
emitted photon
random clicks at a
single photon detector
trapping
potential
0
time
FIG. 2: Schematic view of a simple experiment of a laser driven atom and a possible trajectory, as it might be
observed in a quantum optics experiment. On the right hand sight, we see a random sequence of possible photon
emission times causing clicks at a single photon detector.
in Figure 2. In this section, we review the main quantum optics tools for the description of such a system.
2.1 The Schrödinger picture
Up to now, we were interested in observables and probabilities for certain measurement outcomes given
a certain quantum state. However, the state of a quantum system is in general not constant in time.
Instead it evolves according to the Schrödinger equation
ih̄
d
|ψ(t)i = H |ψ(t)i ,
dt
(1)
which implies
d
i
|ψ(t)i = − H |ψ(t)i .
dt
h̄
(2)
Here H is the corresponding Hamiltonian and the observable for the energy in the setup. The formal
solution of Eq. (1) for a time independent Hamiltonian is
i
|ψ(t)i = U (t, 0) |ψ(0)i = exp − Ht |ψ(0)i .
h̄
(3)
The operator U (t, 0) in Eq. (3) is called the time evolution operator and is a function of the Hamiltonian
H. Once we know how to calculate a function of an operator, we have the solution to the Schrödinger
equation (1).
We now consider again an operator A with eigenvalues λi and eigenstates |λi i,
X
A=
λi |λi ihλi | .
(4)
i
P
For this operator, one can easily calculate A2 = A · A = i λ2i |λi ihλi | and show that (please do so)
X
Am = A · ... · A =
λm
(5)
i |λi ihλi | .
i
This equation implies the spectral theorem
f (A) =
X
f (λi ) |λi ihλi |
(6)
i
for any function f with a Taylor expansion. Note that the spectral theorem allows us to calculate the
function of any operator, as long as we know its eigenvalues and eigenstates.
As an example, let us consider the relatively simple Hamiltonian
X
H=
h̄ωn |nihn| .
n
(7)
3
In this case, the time evolution operator U (t, 0) equals
i
X
U (t, 0) = exp − Ht =
exp − iωn t |nihn| .
h̄
n
(8)
For example, if the system is initially in the energy eigenstate and |ψ(0)i = |mi, then
|ψ(t)i = exp − iωm t |mi .
(9)
This shows, that the energy eigenstates of a system do not evolve in time. A system initially prepared
in a state |ni only accumulates an overall phase factor with no physical consequences. Eq. (8) is one of
the motivations, why people are interested in finding the energy eigenstates of a system.
2.2 The interaction picture
It is not always possible to calculate the time evolution of a system analytically by solving of the
Schrödinger equation (1). Often it is necessary to change first into a different reference frame. In
quantum mechanics, these reference frames are called pictures, with the usual one being the Schrödinger
picture. In the Schrödinger picture the Schrödinger equation is the one given in Eq. (1), the time
evolution operator is the one given in Eq. (3) and a system and its observables are described by the state
vector |ψ(t)i and Hermitian operators A.
An alternative to the Schrödinger picture is the interaction picture with respect to a Hamiltonian H0 .
Here H0 which can be chosen arbitrarily. Suppose
H = H0 + H1 .
(10)
We then define the state vector |ψI (t)i for the description of the system in the interaction picture as
|ψI (t)i = U0† (t, 0) |ψ(t)i
(11)
i
U0 (t, 0) = exp − H0 t
h̄
(12)
with
and
U0† (t, 0) = exp
i
h̄
H0 t .
(13)
Then one can show that
d
d
d
|ψI (t)i =
U0† (t, 0) |ψ(t)i + U0† (t, 0) ·
|ψ(t)i
dt
dt
dt
i
i
=
H0 U0† (t, 0) |ψ(t)i − U0† (t, 0) H |ψ(t)i
h̄
h̄
i †
= − U0 (t, 0) H − H0 |ψ(t)i
h̄
i
= − U0† (t, 0) H − H0 U0 (t, 0) U0† (t, 0) |ψ(t)i ,
h̄
(14)
since U0 (t, 0) U0† (t, 0) = 1. Using the above definitions, this can be simplified to
d
i
|ψI (t)i = − HI (t) |ψI (t)i
dt
h̄
(15)
HI (t) ≡ U0† (t, 0) H1 U0 (t, 0) .
(16)
with
Here HI (t) is the so-called interaction Hamiltonian. The motivation for the introduction of the interaction picture is that, in many cases, one can find a Hamiltonian H0 such that solving the Schrödinger
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∆
h ω0
2
h ωL
laser field
1
FIG. 3: Level scheme of a single two-level atom with energy level separation h̄ω0 driven by a laser field with
frequeny ωL and detuning ∆ = ω0 − ωL .
equation (15) is much easier than solving the original equation (1). This only works if H0 is chosen such
that one can easily calculate U0 (t, 0) and U0† (t, 0). Finally we remark that the initial condition in the
interaction picture are the same as in the Schrödinger picture. The initial states |ψ(0)i and |ψI (0)i are
the same and HI (0) = H1 , since U (0, 0) = 1.
2.3 A laser driven two-level atom in the absence of photon emisions
The laser driven two-level atom, which became experimentally accessible with the development of the
Paul trap by Wolfgang Paul in 1956, is the main topic of this section. Dehmelt and Paul won the Nobel
price for their contribution to the development of ion trap techniques and laser cooling in 1989 together
with Ramsey. Trapping of single ions (and atoms) has many applications, like precision spectroscopy and
quantum computing, but is especially exciting since it allows to directly observe quantum mechanical
effects on the single atom level instead of observing quantum mechanics indirectly as in experiments
measuring, for example, the black body radiation.
In the following, |1i denotes the ground state and |2i denotes the excited state of the atom, as shown
in Figure 3. The energy difference between the two states is h̄ω0 . Our aim is to calculate the effect of a
resonant laser field with frequency ωL = ω0 onto the state of the atom given the initial state |ψ(0)i = |1i
in the absence of spontaneous emission.
To achieve this, we will solve the Schrödinger equation (1) but first we have to find the corresponding
Hamiltonian H. This Hamiltonian consists of two parts,
H = Hatom + Hlaser .
(17)
The Hamiltonian describing the energy of the atomic states given by
Hatom = h̄ω0 |2ih2| .
(18)
Here we assumed, without loss of generality, that the state |1i has zero energy. The energy of the excited
state |2i is then h̄ω0 . Moreover, we treat the laser field in a classical way and assume that it generates
an electromagnetic field with
E0 (t) = E0 e−iω0 t + E∗0 eiω0 t ,
(19)
at the position of the atom. Here E0 is the amplitude of the applied laser field. The term describing the
atom-laser interaction then equals in analogy to the classical expression for the energy of a single electron
moving in a classical electromagnetic field
Hlaser = e E0 (t) · x .
(20)
Here x is the three dimensional quantum mechanical position operator. As one can see from this equation,
the interaction between the atom and the laser is a dipole interaction. To bring the Hamiltonian (20) in
a form, which is useful for us, we do the following trick and write
x = 1 · x · 1 = |1ih1| + |2ih2| x |1ih1| + |2ih2|
= h1| x |2i |1ih2| + h2| x |1i |2ih1|
= D |1ih2| + D∗ |2ih1| .
(21)
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Here we used the fact that the energy eigenstates |1i and |2i of an atom are (due to the symmetry of their
potential) always either symmetric or antisymmetric such that the matrix elements h1| x |1i = h2| x |2i =
0. This is the case, since the application of the position operator x changes a symmetric energy eigenstate
into an antisymmetric one and vice versa, which has no overlap with the original state. Moreover, we
introduced the dipole moment D = h1| x |2i characterising the coupling strength of the laser to the 1-2
transition of the atom. The calculation in Eq. (21) finally yields the Hamiltonian
Hlaser = e E0 e−iω0 t + E∗0 eiω0 t · D |1ih2| + D∗ |2ih1|
= e E0 · D e−iω0 t |1ih2| + e E0 · D∗ e−iω0 t |2ih1| + e E∗0 · D eiω0 t |1ih2| + e E∗0 · D∗ eiω0 t |2ih1| .
(22)
Due to the time dependence of Eq. (22), a straightforward solution of the corresponding Schrödinger
equation is not possible. We therefore now move into an interaction picture.
For reasons that become obvious later we choose
H0 = h̄ω0 |2ih2| .
(23)
The corresponding time evolution operator is then given by
U0 (t, 0) = exp − iω0 t |2ih2|
(24)
U0† (t, 0) = exp iω0 t |2ih2| .
(25)
and, hence,
Using Eq. (16), we find that the interaction Hamiltonian of the laser driven atom equals
HI (t) ≡ U0† (t, 0) H − H0 U0 (t, 0)
= e E0 · D e−2iω0 t |1ih2| + e E0 · D∗ |2ih1| + e E∗0 · D |1ih2| + e E∗0 · D∗ e2iω0 t |2ih1| .
(26)
Experience shows that the contribution of fast oscillating terms in a Hamiltonian to the time evolution
of a system is negligible, when compared to time independent terms. This approximation is known as
the rotating wave approximation. Applying this approximation here, we obtain
(27)
HI (t) = HI = 12 h̄ Ω |2ih1| + Ω∗ |1ih2| ,
with the Rabi frequency Ω defined as
Ω≡
2e E0 · D∗
.
h̄
(28)
The Rabi frequency Ω is a measure for the amplitude, i.e. the strength, of the applied laser field although
it is called a frequency.
The motivation for the choice of the interaction picture above was to obtain the time independent
Hamiltonian (27). We now have all the tools to calculate the time evolution of the atom. Just one final
remark. Without any restrictions, one can always consider the Rabi frequency Ω as real. the reason is
that any possible phase factor eiϕ can be compensated by adding an overall phase factor to the definition
of the state |2i. It is possible to replace |2i by eiϕ |2i with no other consequences than simplifying our
calculations, since global phase factors have no physical effect.
The next step is to calculate the time evolution operator UI (t, 0) in the interaction picture. Using
Eq. (27) and assuming that Ω is real we obtain
i
UI (t, 0) = exp − HI t = exp − 2i Ωt A
(29)
h̄
with
A = |2ih1| + |1ih2| .
(30)
6
E= h ω0
level 2
energy of
the atom
E=0
time
t=0
level 1
FIG. 4: Sketch of the time evolution of a two-level atom driven by a resonant laser field (∆ = 0).
To calculate the right hand side of Eq. (25), we have to determine the eigenvalues and eigenvectors of A.
They equal
λ1/2 = ±1 and |λ1/2 i = √12 |1i ± |2i .
(31)
Proceeding as proposed in the beginning of this section, we find
X
UI (t, 0) =
exp − 2i Ω λn t |λn ihλn |
n=1,2
exp − 2i Ωt |1i + |2i h1| + h2| + 21 exp 2i Ωt |1i − |2i h1| − h2|
h
i
= 12 exp − 2i Ωt + exp 2i Ωt
|1ih1| + |2ih2|
h
i
+ 21 exp − 2i Ωt − exp 2i Ωt
|1ih2| + |2ih1|
= cos 12 Ωt |1ih1| + |2ih2| − i sin 21 Ωt |1ih2| + |2ih1| .
=
1
2
(32)
Suppose, the atom is initially prepared in its ground state and |ψ(0)i = |1i. Then the state of the system
equals at time t
|ψI (t)i = UI (t, 0) |1i = cos 12 Ωt |1i − i sin 12 Ωt |2i .
(33)
The amount of population in level 2 at t is therefore given by
P2 (t) = | h2|ψI (t)i |2 = sin2
1
2 Ωt
.
(34)
Note that this applies independent of the picture we are in. The excited state population is the same in
the interaction and in the Schrödinger picture.
In the presence of a laser field, the state of the atom starts to do so-called Rabi oscillations. As can
be seen from Eq. (34), the laser field creates a potential in which the electron starts oscillating between
the ground state and the excited state. The initial population in the ground state can be transferred
completely into the excited state (see Figure 4).
2.4 Einstein’s rate equations for a single atom
The simplest model for the description of a laser driven two-level atom undergoing spontaneous emission
are Einstein’s rate equations. Although being simple, they predict the behaviour of such a system, as it
is observed in quantum optical experiments, relatively well. The basic assumption in this model is that
the atom is always either in level 1 or in level 2. It is assumed that the laser induces sudden jumps from
one level to the other. The probability density for such a jump to take place is the constant rate B. We
have seen in the previous Section that this not too fare away from the truth (see Figure 4), since the
laser causes indeed transitions from the ground into the excited state and vice versa.
Moreover, Einstein’s rate equations take the possibility of the spontaneous emission of a photon
into account, when the atom is in the excited state. An emission can then occur with the probability
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level 2
E= hω0
energy of
the atom
E=0
time
t=0
level 1
FIG. 5: A possible trajectory of a two-level atom as it could be predicted using Einstein’s rate equation model.
At time t = 0, the atom is in its ground state but can go over in the excited state due to the presence of a laser
field. Once in the excited state, the atom returns in the ground state either via a spontaneous or the induced
emission of a photon.
density A and results in a transition from level 2 into the ground state |1i. The rate A is also called the
spontaneous decay rate of level 2. Using the A and B rates, one could use a random number generator
to calculate a possible single trajectory for the time evolution of the atom, like the one shown in Figure
5. Most of the time, the atom is in its ground state, since the rate for a jump from |2i into |1i, namely
A + B, is larger than the rate B for a jump from level 1 into level 2.
2.5 Einstein’s rate equations for an ensemble of atoms
Instead of predicting the possible time evolutions of a single atom, we could be interested in expectation
values, like populations averaged over an ensemble of single systems. Such an ensemble could be realised
by repeating a single-atom experiments many times or by storing a large number of atoms simultaneously
stored in the same trap (ergodicity). Suppose, we observe a whole ensemble of single atoms, each of them
driven by a laser field and each of them randomly emitting photons. We can then ask the question, what
is the probability to find any of these atoms in |ii at a time t. In the following, we note this probability
again by Pi . According to Einstein, the populations P1 and P2 change according to
d
dt
d
dt
P1 = −B P1 + (A + B) P2 ,
P2 = B P1 − (A + B) P2 .
(35)
These equations are called Einstein’s rate equations. They conserve the probability to find the valence
electron anywhere in the atoms, since Ṗ1 = −Ṗ2 . I know, we did not do this in the lecture but let me
give you also the analytical solution for these equations,
1
P1 (t) = P1 (0) −
1 − e−(2A+B)t A P1 (0) − (A + B) P2 (0) ,
2A + B
1
P2 (t) = P2 (0) +
1 − e−(2A+B)t A P1 (0) − (A + B) P2 (0) .
(36)
2A + B
Suppose, an atom is initially prepared in its ground state |1i. How likely is it to find the atom a time t
later in the excited state?
Even if the initial state of the atom is known, the system reaches after a relatively short time a
stationary state in which the populations Pi change no longer in time and
d
dt
P1 =
d
dt
P2 = 0 .
(37)
Calculating P1 and P2 using Einstein’s rate equations and the additional condition
P1 + P2 = 1
(38)
gives us the probability to find an atom of the ensemble in level 1 or 2, respectively, once the system
reached its stationary state. From Eqs. (35) and (38) we obtain the steady state populations
P1 =
A+B
B
and P2 =
.
A + 2B
A + 2B
(39)
8
level 2
E= h ω 0
energy of
the atom
E=0
"jump"
t=0
level 1
time
FIG. 6: Possible evolution of the energy of the trapped atom in the presence of spontaneous photon emission and
resonant laser driving.
For large enough times, Eq. (36) gives exactly the same result. In the limit of very weak laser driving,
the atom is most likely in its ground state, since P1 = 1 and P2 = 0 for B/A → 0. For strong driving,
about half of the atomic population accumulates in the excited state, since P1 = P2 = 21 for B/A → ∞.
2.6 The quantum jump approach
More accurately, the time evolution of a single two-level atom between photon emissions is described
by a Schrödinger equation. Whenever there is some population in the excited state, then there is a
probability density for the emission of a photon. In case of an emission, the atom returns into a ground
state. This sudden change of the state of the atom is known as a quantum jump and was the subject of
much debate between Schrödinger and Bohr.
If we know the quantitative details of the above described qualitative model, it is possible to use a
so-called Monte-Carlo simulation, to predict the possible trajectories of the trapped laser-driven atom
with more accuracy, than when using Einstein’s ate equation model. An approach which gives us
these details is the quantum jump approach, which was developed by several groups in parallel in the
beginning of the 90’s. The main assumption in the derivation of the quantum jump approach is that
the environment performs continuous measurements (i.e. measurements on a coarse grained time scale),
whether the atom emits a photon or not.
The result of this assumption is the time independent, conditional interaction Hamiltonian
Hcond = 12 h̄Ω |2ih1| + |1ih2| − 21 ih̄Γ |2ih2| .
(40)
This Hamiltonian differs from the laser Hamiltonian in Eq. (27) by the last term proportional to the
spontaneous photon emission rate Γ. This term continually reduces the population in the excited state
|2i. This takes into account that the observation of no photon results in a continual gain of information
about the state of the two-level atom. It tells us that there is probably not as much population in the
excited state as we assume initially. Under the condition of no photon emission in (0, t), the state of the
system becomes
|ψ 0 (t)i = Ucond (t, 0) |ψ(0)i/kUcond (t, 0) |ψ(0)ik ,
(41)
which emphasizes the population in the ground state |1i. Moreover,
P0 (t) = kUcond (t, 0) |ψ(0)ik2
(42)
is the probability for no photon in (0, t). In case of an emission, the state of the atom becomes |1i. This
occurs with probability density Γ · P2 (t).
2.5 The density matrix and the master equations
In this final subsection, we introduce the master equation. They allows us to take the effect of the applied
laser field (the Rabi oscillations) and the effect of spontaneous emission without any simplifications into
9
account. Master equations are the main quantum optical tool to predict the time evolution of laser
driven systems undergoing spontaneous emission. They are especially suited to calculate expectation
values, whose calculation would otherwise require an averaging over all the possible trajectories of the
system. However, before we can write down the master equations for the laser driven two-level atom, we
have to introduce the density matrix operator ρ.
In quantum optics we often come across situations, where it is only known that a system is with a
certain probability prepared in a certain state but it is not known in which one. For example, the single
laser driven two-level atom emits photons at random times. If someone keeps track of all emission times,
then the system can be described by a pure state. In general this is not possible. If the concrete history
of the system (its trajectory) is not known, the atom can only be described by the density matrix ρ. The
main motivation for the introduction of this operator is, that it allows us to calculate the probability to
find the atom at a time t in the excited state or the probability density for a photon emission at a time
t without having to know anything else.
Suppose, a system is prepared with probability Pi in a state |ψi i. Then its density matrix ρ is defined
as
ρ=
X
Pi |ψi ihψi | .
(43)
i
Like the state vector |ψi, the density matrix ρ contains all the information needed to calculate the
expectation value of any physical observable. This is shown below. If A is an observable with eigenvalues
λn and eigenvectors |λn i,
X
A=
λn |λn ihλn | ,
(44)
n
then the expectation value of A for a system prepared in ρ equals
X
XX
X
hAiρ =
Pi hAiψi =
Pi λn hλn |ψi ihψi |λn i =
λn hλn |ρ|λn i .
i
n
i
(45)
n
Calculating this can be done without knowing the probabilities Pi and states |ψi i, since the right hand
side of this equation depends only on ρ. In many quantum optics and quantum mechanics textbook you
find the equivalent expression
XX
hAiρ =
an hm|nihn|ρ|mi = Tr Aρ .
(46)
m
n
Here “Tr” stands for the trace of an operator. Eq. (46) applies, since the states |ni form a complete
basis and hm|ni = 1 for m = n and hm|ni = 1 for m 6= n.
We know that the time evolution of a pure state |ψi is governed by a Schrödinger equation (1). The
equivalent of this equation for a density matrix ρ is called a master equation. Let us first see what the
master equation looks like in the absence of processes like the spontaneous emission of photons (i.e. Γ = 0).
Using Eqs. (1) and (43) we find that
X
X
d
d
d
ρ =
Pi
|ψi i hψi | +
Pi |ψi i
hψi |
dt
dt
dt
i
i
X
i
= −
Pi H |ψi ihψi | − |ψi ihψi | H
h̄ i
i
H ρ − ρH
h̄
i = − H, ρ .
h̄
= −
However, in general the atoms cannot be described by such a simple equation.
(47)
10
Below we see the master equations taking spontaneous emission into account. It reads
i †
Hcond ρ − ρHcond
+ Γ |1ih2| ρ |2ih1|
h̄
i = − HI , ρ − 21 Γ |2ih2| ρ + ρ |2ih2| + Γ |1ih2| ρ |2ih1| .
h̄
ρ̇ = −
(48)
Again Γ is, as in the previous subsection, the spontaneous decay rate of the excited state |2i and Hcond
is the conditional Hamiltonian in Eq. (42). The first terms describe the time evolution of the atom
between photon emissions. The last term in Eq. (48) describes atoms that suddenly jump from the
excited into the ground state, thereby sending out a photon. The master equations model spontaneous
emission in a similar way as Einstein’s rate equations. If an atom is in |2i, the probability density to
create a photon equals the spontaneous decay rate Γ. However, an emission can also occur, when the
atom is only partially in the excited state.
Finally, we use the master equations (48) to calculate the stationary state of a single or an ensemble
of laser-driven two-level atoms. It can be calculated by simply applying the condition
ρ̇ = 0 .
(49)
Using the laser Hamiltonian (27) and solving a system of four linear equations (we don’t do this here but
we could), we find that the steady state population in ground and excited state is given by
P1 =
Γ2 + Ω2
Ω2
and P2 = 2
.
2
2
Γ + 2Ω
Γ + 2Ω2
(50)
A comparison with Eq. (39) in the previous subsection shows that this result is in very good agreement
with the predictions of Einstein’s rate equations, if we identify the laser Rabi frequency Ω with the rate
B and Γ with A.
2.6 Final remarks
There are many examples, where a more detailed analysis of the time evolution of a system, using
the master equations and not Einstein’s rate equations, is required to explain or predict the findings
of quantum optical experiments. We have already seen that Einstein’s rate equations cannot predict
Rabi oscillations between the ground and the excited state. They also do not allow the system to
be in a superposition of the ground state |1i and the excited state |2i. In general, the description of
any quantum optical system with more than two levels or involving a more complex, time dependent
Hamiltonian requires a solution of the corresponding master equations.