A detailed mathematical description of the effect can be found on the page for the Rabi problem. For example, for a two-state atom (an atom in which an electron can either be in the excited or ground state) in an electromagnetic field with frequency tuned to the excitation energy, the probability of finding the atom in the excited state is found from the Bloch equations to be:
|
c
b
(
t
)
|
2
∝
sin
2
(
ω
t
/
2
)
,
where
ω
is the Rabi frequency.
More generally, one can consider a system where the two levels under consideration are not energy eigenstates. Therefore, if the system is initialized in one of these levels, time evolution will make the population of each of the levels oscillate with some characteristic frequency, whose angular frequency is also known as the Rabi frequency. The state of a two-state quantum system can be represented as vectors of a two-dimensional complex Hilbert space, which means every state vector
|
ψ
⟩
is represented by two complex coordinates.
|
ψ
⟩
=
(
c
1
c
2
)
=
c
1
(
1
0
)
+
c
2
(
0
1
)
;
where
c
1
and
c
2
are the coordinates.
If the vectors are normalized,
c
1
and
c
2
are related by
|
c
1
|
2
+
|
c
2
|
2
=
1
. The basis vectors will be represented as
|
1
⟩
=
(
1
0
)
and
|
2
⟩
=
(
0
1
)
All observable physical quantities associated with this systems are 2
×
2 Hermitian matrices, which means the Hamiltonian of the system is also a similar matrix.
One can construct an oscillation experiment consisting of following steps:
(1) Prepare the system in a fixed state say
|
1
⟩
(2) Let the state evolve freely, under a Hamiltonian H for time t
(3) Find the probability P(t), that the state is in
|
1
⟩
If
|
1
⟩
was an eigenstate of H, P(t)=1 and there are no oscillations. Also if two states are degenerate, every state including
|
1
⟩
is an eigenstate of H. As a result, there are no oscillations.
On the other hand, if H has no degenerate eigenstates, neither of which is
|
1
⟩
, then there will be oscillations. The most general form of the Hamiltonian of a two-state system is given
H
=
(
a
0
+
a
3
a
1
−
i
a
2
a
1
+
i
a
2
a
0
−
a
3
)
here,
a
0
,
a
1
,
a
2
and
a
3
are real numbers. This matrix can be decomposed as,
H
=
a
0
⋅
σ
0
+
a
1
⋅
σ
1
+
a
2
⋅
σ
2
+
a
3
⋅
σ
3
;
The matrix
σ
0
is the 2
×
2 identity matrix and the matrices
σ
k
(
k
=
1
,
2
,
3
)
are the Pauli matrices. This decomposition simplifies the analysis of the system especially in the time-independent case where the values of
a
0
,
a
1
,
a
2
and
a
3
are constants. Consider the case of a spin-1/2 particle in a magnetic field
B
=
B
z
^
. The interaction Hamiltonian for this system is
H
=
−
μ
⋅
B
=
−
γ
S
⋅
B
=
−
γ
S
z
B
,
S
z
=
ℏ
2
σ
3
=
ℏ
2
(
1
0
0
−
1
)
,
where
μ
is the magnitude of the particle's magnetic moment,
γ
is the Gyromagnetic ratio and
σ
is the vector of Pauli matrices. Here the eigenstates of Hamiltonian are eigenstates of
σ
3
, that is
|
1
⟩
and
|
2
⟩
. The probability that a system in the state
|
ψ
⟩
can be found in the arbitrary state
|
ϕ
⟩
is given by
|
⟨
ϕ
|
ψ
⟩
|
2
.
Let the system initially (at
t
=
0
) be in
|
+
X
⟩
which is an eigenstate of
σ
1
:
|
ψ
(
0
)
⟩
=
1
2
(
1
1
)
=
1
2
(
1
0
)
+
1
2
(
0
1
)
.
Here the Hamiltonian is time independent. Thus by solving the stationary Schrödinger equation, the state after time t is given by
|
ψ
(
t
)
⟩
=
e
−
i
E
t
ℏ
|
ψ
(
0
)
⟩
, with total energy of the system
E
. So the state after time t is given by:
|
ψ
(
t
)
⟩
=
e
−
i
E
+
t
ℏ
1
2
|
1
⟩
+
e
−
i
E
−
t
ℏ
1
2
|
2
⟩
.
Now suppose the spin is measured in x-direction at time t. The probability of finding spin-up is given by:
|
⟨
+
X
|
ψ
(
t
)
⟩
|
2
=
|
1
2
(
⟨
1
|
+
⟨
2
|
)
(
e
−
i
E
+
t
ℏ
1
2
|
1
⟩
+
e
−
i
E
−
t
ℏ
1
2
|
2
⟩
)
|
2
=
cos
2
(
ω
t
2
)
,
where
ω
is a characteristic angular frequency given by
ω
=
E
−
−
E
+
ℏ
=
γ
B
, where it has been assumed that
E
−
≥
E
+
. So in this case the probability of finding spin-up in x-direction is oscillatory in time
t
when the system is initially in
|
+
X
⟩
direction. Similarly, if we measure the spin in z-direction, the probability of finding
ℏ
2
of the system is
1
2
. In case of
E
+
=
E
−
, that is when the Hamiltonian is degenerate, there is no oscillation.
Notice that if a system is in an eigenstate of a given Hamiltonian, the system remains in that state.
This is true even for time dependent Hamiltonians. Taking for example
H
=
−
γ
S
z
B
sin
(
ω
t
)
, the probability that a measurement of the system in y-direction at time
t
results in
+
ℏ
2
is
|
⟨
+
Y
|
ψ
(
t
)
⟩
|
2
=
cos
2
(
γ
B
2
ω
cos
ω
t
)
, where the initial state is in
|
+
Y
⟩
.
Let us consider a Hamiltonian in the form
H
=
E
0
⋅
σ
0
+
W
1
⋅
σ
1
+
W
2
⋅
σ
2
+
Δ
⋅
σ
3
=
(
E
0
+
Δ
W
1
−
i
W
2
W
1
+
i
W
2
E
0
−
Δ
)
.
The eigen values of this matrix are given by
λ
+
=
E
+
=
E
0
+
Δ
2
+
W
1
2
+
W
2
2
=
E
0
+
Δ
2
+
|
W
|
2
and
λ
−
=
E
−
=
E
0
−
Δ
2
+
W
1
2
+
W
2
2
=
E
0
−
Δ
2
+
|
W
|
2
.Where
W
=
W
1
+
ı
W
2
and
|
W
|
2
=
W
1
2
+
W
2
2
=
W
W
∗
. so we can take
W
=
|
W
|
e
−
ı
ϕ
.
Now eigen vector for
E
+
can be found from equation :
(
E
0
+
Δ
W
1
−
i
W
2
W
1
+
i
W
2
E
0
−
Δ
)
(
a
b
)
=
E
+
(
a
b
)
.
So
b
=
−
a
(
E
0
+
Δ
−
E
+
)
W
1
−
ı
W
2
.
Using normalisation condition of eigen vector,
|
a
|
2
+
|
b
|
2
=
1
.So
|
a
|
2
+
|
a
|
2
(
Δ
|
W
|
−
Δ
2
+
|
W
|
2
|
W
|
)
2
=
1
.
Let
sin
θ
=
|
W
|
Δ
2
+
|
W
|
2
and
cos
θ
=
Δ
Δ
2
+
|
W
|
2
. so
tan
θ
=
|
W
|
Δ
.
So we get
|
a
|
2
+
|
a
|
2
(
1
−
cos
θ
)
2
sin
2
θ
=
1
. That is
|
a
|
2
=
cos
2
θ
/
2
. Taking arbitrary phase angle
ϕ
,we can write
a
=
exp
(
ı
ϕ
/
2
)
cos
θ
/
2
. Similarly
b
=
exp
(
−
ı
ϕ
/
2
)
sin
θ
/
2
.
So eigen vector for
E
+
eigen value is given by
|
E
+
⟩
=
e
ı
ϕ
/
2
(
cos
θ
/
2
e
−
ı
ϕ
sin
θ
/
2
)
.
As overall phase is immaterial so we can write
|
E
+
⟩
=
(
cos
θ
/
2
e
−
ı
ϕ
sin
θ
/
2
)
=
cos
(
θ
/
2
)
|
1
⟩
+
e
−
ı
ϕ
sin
(
θ
/
2
)
|
2
⟩
.
Similarly we can find eigen vector for
E
−
value and we get
|
E
−
⟩
=
−
e
ı
ϕ
sin
(
θ
/
2
)
|
1
⟩
+
cos
(
θ
/
2
)
|
2
⟩
.
From these two equations, we can write
|
1
⟩
=
cos
(
θ
/
2
)
|
E
+
⟩
−
sin
(
θ
/
2
)
e
−
ı
ϕ
|
E
−
⟩
and
|
2
⟩
=
e
ı
ϕ
sin
(
θ
/
2
)
|
E
+
⟩
+
cos
(
θ
/
2
)
|
E
−
⟩
.
Let at time t=0, system be in
|
1
⟩
That Is
|
ψ
(
0
)
⟩
=
|
1
⟩
=
cos
(
θ
/
2
)
|
E
+
⟩
−
sin
(
θ
/
2
)
e
−
ı
ϕ
|
E
−
⟩
.
State of system after time t is given by
|
ψ
(
t
)
⟩
=
e
−
i
E
t
ℏ
|
ψ
(
0
)
⟩
=
cos
(
θ
/
2
)
e
−
ı
E
+
t
ℏ
|
E
+
⟩
−
sin
(
θ
/
2
)
e
−
ı
ϕ
e
−
ı
E
−
t
ℏ
|
E
−
⟩
.
Now a system is in one of the eigen states
|
1
⟩
or
|
2
⟩
, it will remain the same state, however in a general state as shown above the time evolution is non trivial.
The probability amplitude of finding the system at time t in the state
|
2
⟩
is given by
|
⟨
2
|
ψ
(
t
)
⟩
|
=
e
−
ı
ϕ
sin
(
θ
/
2
)
cos
(
θ
/
2
)
(
e
−
ı
E
+
t
ℏ
−
e
−
ı
E
−
t
ℏ
)
.
Now the probability that a system in the state
|
ψ
(
t
)
⟩
will be found to be in the arbitrary state
|
2
⟩
is given by
P
1
→
2
(
t
)
=
|
⟨
2
|
ψ
(
t
)
⟩
|
2
=
e
+
ı
ϕ
sin
(
θ
/
2
)
cos
(
θ
/
2
)
(
e
+
ı
E
+
t
ℏ
−
e
+
ı
E
t
ℏ
)
e
−
ı
ϕ
sin
(
θ
/
2
)
cos
(
θ
/
2
)
(
e
−
ı
E
+
t
ℏ
−
e
i
ı
E
t
ℏ
)
=
sin
2
θ
4
(
2
−
2
cos
(
(
E
+
−
E
−
)
t
ℏ
)
)
By simplifying
P
1
→
2
(
t
)
=
sin
2
(
θ
)
sin
2
(
(
E
+
−
E
−
)
t
2
ℏ
)
=
|
W
|
2
Δ
2
+
|
W
|
2
sin
2
(
(
E
+
−
E
−
)
t
2
ℏ
)
.........(1).
This shows that there is a finite probability of finding the system in state
|
2
⟩
when the system is originally in the state
|
1
⟩
. The probability is oscillatory with angular frequency
ω
=
E
+
−
E
−
2
ℏ
=
Δ
2
+
|
W
|
2
ℏ
, which is simply unique Bohr frequency of the system and also called Rabi frequency. The formula(1) is known as Rabi formula. Now after time t the probability that the system in state
|
1
⟩
is given by
|
⟨
+
X
|
ψ
(
t
)
⟩
|
2
=
1
−
sin
2
(
θ
)
sin
2
(
(
E
+
−
E
−
)
t
2
ℏ
)
, which is also oscillatory. This type of oscillations between two levels are called Rabi oscillation and seen in many problems such as Neutrino oscillation, ionized Hydrogen molecule, Quantum computing, Ammonia maser etc.
Any two state quantum system can be used to model a qubit. Consider a spin
1
2
system with magnetic moment
μ
placed in a classical magnetic field
B
=
B
0
z
^
+
B
1
(
cos
ω
t
x
^
−
sin
ω
t
y
^
)
. Let
γ
be the gyromagnetic ratio for the system. The magnetic moment is thus
μ
=
ℏ
2
γ
σ
. The Hamiltonian of this system is then given by
H
=
−
μ
⋅
B
=
−
ℏ
2
ω
0
σ
z
−
ℏ
2
ω
1
(
σ
x
cos
ω
t
−
σ
y
sin
ω
t
)
where
ω
0
=
γ
B
0
and
ω
1
=
γ
B
1
. One can find the eigenvalues and eigenvectors of this Hamiltonian by the above-mentioned procedure. Now, let the qubit be in state
|
1
⟩
at time
t
=
0
. Then, at time
t
, the probability of it being found in state
|
2
⟩
is given by
P
1
→
2
(
t
)
=
(
ω
1
Ω
)
2
sin
2
(
Ω
t
2
)
where
Ω
=
(
ω
−
ω
0
)
2
+
ω
1
2
. This phenomenon is called Rabi oscillation. Thus, the qubit oscillates between the
|
1
⟩
and
|
2
⟩
states. The maximum amplitude for oscillation is achieved at
ω
=
ω
0
, which is the condition for resonance. At resonance, the transition probability is given by
P
1
→
2
(
t
)
=
sin
2
(
ω
1
t
2
)
. To go from state
|
1
⟩
to state
|
2
⟩
it is sufficient to adjust the time
t
during which the rotating field acts such that
ω
1
t
2
=
π
2
or
t
=
π
ω
1
. This is called a
π
pulse. If a time intermediate between 0 and
π
ω
1
is chosen, we obtain a superposition of
|
1
⟩
and
|
2
⟩
. In particular for
t
=
π
2
ω
1
, we have a
π
2
pulse, which acts as:
|
1
⟩
→
|
1
⟩
+
i
|
2
⟩
2
. This operation has crucial importance in quantum computing. The equations are essentially identical in the case of a two level atom in the field of a laser when the generally well satisfied rotating wave approximation is made. Then
ℏ
ω
0
is the energy difference between the two atomic levels,
ω
is the frequency of laser wave and Rabi frequency
ω
1
is proportional to the product of the transition electric dipole moment of atom
d
→
and electric field
E
→
of the laser wave that is
ω
1
∝
d
→
⋅
E
→
ℏ
. In summary, Rabi oscillations are the basic process used to manipulate qubits. These oscillations are obtained by exposing qubits to periodic electric or magnetic fields during suitably adjusted time intervals.
