Introduction to the Rotating Wave Approximation (RWA): Two Coherent Oscillations ()
1. Introduction
When undergraduates study Quantum Mechanics, they encounter several approximation methods like the WKB, the Born-Oppenheimer, the Hartree-Fock, etc. In fact, exactly solvable models are very few in Quantum Mechanics, so (many) approximation methods play an important role.1
When we study Quantum Optics we again encounter the same situation. We often use a method called the rotating wave approximation (RWA), which means fast oscillating terms (in effective Hamiltonians) removed. Because
holds if n is large enough. We believe that there is no problem on this approximation.
However, in some models slow oscillating terms are removed. Let us show an example. The Euler formula gives
From this, we approximate
to be
because
goes away from
by two times speed, so we neglect this term. In our case n is 2! Read the text for more details.
Why is such a “rude” method used? The main reason is to obtain analytic approximate solutions for some important models in Quantum Optics. To the best of our knowledge, we cannot obtain such analytic solutions without RWA.
In this review note, I introduce the rotating wave approximation in details with two models for undergraduates or non-experts. I expect that they will overcome this “high wall” in the near future.
2. Principles of Quantum Mechanics
One of targets of the paper is to study and solve the time evolution of a quantum state (which is a superposition of two physical states).
In order to set the stage and to introduce proper notation, let us start with a system of principles of Quantum Mechanics (QM in the following for simplicity). See for example [1] [2] [3] [4] . That is:
System of Principles of QM
1) Superposition Principle
If
and
are physical states, then their superposition
is also a physical state where
and
are complex numbers.
2) Schrödinger Equation and Evolution
Time evolution of a physical state proceeds like
where
is the unitary evolution operator (
and
) determined by a Schrödinger Equation.
3) Copenhagen Interpretation2
Let a and b be the eigenvalues of an observable Q, and
and
be the normalized eigenstates corresponding to a and b. When a state is a superposition
and we observe the observable Q the state collapses like
where their collapsing probabilities are
and
respectively (
).
This is called the collapse of the wave function and the probabilistic interpretation.
4) Many Particle State and Tensor Product
A multiparticle state can be constructed by the superposition of the Knonecker products of one particle states, which are called the tensor products. For example,
are two particle states.
These will play an essential role in the later sections.
3. Two-Level System of an Atom
In order to treat the two-level system of an atom we make a short review of the two-dimensional complex vector space
and complex matrix space
within our necessity. See for example [5] .
First we introduce the (famous) Pauli matrices
defined by
(1)
and set the unit matrx
by
Moreover, we set
Note that
. Then it is easy to see
(2)
Comment The Pauli matrices
are generators of the (real) Lie algebra
of the special unitary group
because of
and
are generators of the (complex) Lie algebra
of the
special linear group
because of
For the sake of readers, we write a Lie diagram of these algebras and groups, see the following Figure 1.
Next, we define
a basis of
by use of the Dirac’s notation
(3)
Then, since
satisfies the relation
it is called the flip operation.
Note If we define
as above then
should be chosen
as
instead of (3). Because,
However, I use the conventional notations in this note.
For the later convenience we calculate the exponential map. For a square matrix A the exponential map is defined by
where E is the unit matrix and
is a constant.
Here, let us calculate
as an example. Noting
we obtain
(4)
Exercise Calculate
We discuss an atom trapped in a cavity and consider only two energy states, namely (in our case) the ground state and first excited state. That is, all the remaining states are neglected. This is usually called the two-level approximation. See for example [6] as a general introduction.
We set that energies of the ground state
and first excited state
are
and
(
) respectively. Under this approximation the space of all states is two-dimensional, so there is no problem to identify
with (3).
Then we can write the Hamiltonian in a diagonal form like
For the later convenience let us transform it. For
the energy difference we have
(5)
To this atom we subject LASER (Light Amplification by Stimulated Emission of Radiation) in order to controll it. As an image see the following Figure 2.
In this note we treat Laser as a classical wave for simplicity, which is not so bad as shown in the following. That is, we may set the laser field as
By the way, from several experiments we know that an atom subjected by Laser raises an energy level and vice versa. This is expressed by the property of the Pauli matrix
so we can use
as the interaction term of the Hamiltonian.
As a result our Hamiltonian (effective Hamiltonian) can be written as
(6)
where g is a coupling constant regarding an interaction of between an atom and
laser, and A is absorbed in g (
). We ignore the scalar term
for simplicity. Note that (6) is semi-classical and time-dependent.
Therefore, our task is to solve the Schrödinger equation
(7)
exactly (if possible).
4. Rotating Wave Approximation
Unfortunately we cannot solve (7) exactly at the present time. It must be non-integrable although we don’t know the proof (see the appendix). Therefore we must apply some approximate method in order to obtain an analytic approximate solution. Now we explain a method called the Rotating Wave Approximation (RWA). Let us recall the Euler formula
From this we approximate
to be
(8)
because
goes away from
by two times speed, so we neglect this term ! See the following Figure 3. We call this the rotating wave approximation.
Problem In general, fast oscillating terms may be neglected because
if n is large. Our question is: Is
large enough?
By noting that the Hamiltonian should be hermitian, we approximate
by use of (8), so (6) is reduced to
(9)
As a result our modified task is to solve the Schrödinger equation
(10)
exactly. Mysteriously enough, this equation can be solved easily.
Note For the latter convenience let us rewrite the method with formal notations:
In order to solve (10) we set
for simplicity. From (9) it is easy to see
so we transform the wave function
in (10) into
(11)
Then the Schrödinger Equation (10) becomes
(12)
by a straightforward calculation.
Here we set the resonance condition
(13)
Namely, we subject the laser field with
equal to the energy difference
. See the following Figure 4.
Then (12) becomes
and we have only to solve the equation
By (4) (
) the solution is
and coming back to
(from
) we obtain
(14)
by (11) (
) because the total phase
can be neglected in Quantum Mechanics.
As an initial condition, if we choose
we have
(15)
by (3). That is,
oscillates between the two states
and
. This is called the coherent oscillation or the Rabi oscillation, which plays an essential role in Quantum Optics.
Concerning an application of this oscillation to Quantum Computation see for example [4] .
Problem Our real target is to solve the Schrödinger equation
with
Present a new idea and solve the equation.
5. Quantum Rabi Model
In this section, we discuss the quantum Rabi model whose Hamiltonian is given by
(16)
where
is the identity operator on the Fock space
generated by the Heisenberg algebra
, and
and
are constant, and g is a coupling constant. As a general introduction to this model see for example see [6] .
Let us recall the fundamental relations of the Heisenberg algebra
(17)
Here, the Fock space
is a Hilbert space over
given by
where
is the vacuum (
) and
is given by
On this space, the operators (=infinite dimensional matrices)
,
and
are represented as
(18)
by use of (17).
Note We can add a phase to
like
where
is constant. Then we have another Heisenberg algebra
Again, we would like to solve Schrödinger equation (
for simplicity)
(19)
exactly. To the best of our knowledge the exact solution has not been known, so we must use some approximation in order to obatin an analytic solution.
Since
we neglect the middle terms
and set
(20)
This is called the rotating wave approximation and the resultant Hamiltonian is called the Jaynes-Cummings one3, [7] .
Therefore, our modified task is to solve the Schrödinger equation
(21)
exactly. Mysteriously enough, to solve the equation is very easy.
For a unitary operator
we set
Then it is easy to see
by (21). If we choose U as
(we use
in place of
for simplicity), a straightforward calculation gives
(22)
and we have a simple equation
(23)
Note that in the process of calculation we have used the relations
The proof is easy by use of the formula
(24)
for square matrices X, A and (17).
Here we set the resonance condition
(25)
then (23) becomes
Let us solve this equation. By setting
we calculate the term
. Noting
we have
(26)
Therefore, the solution is given by
and coming back to
(from
) we finally obtain
(27)
where
.
As an initial condition, if we choose a simple state
(see Appendix B or more generally [5] about the tensor product) we have
(28)
because
and
, or
where the total phase
has been removed.
Problem Our real target is to solve the Schrödinger equation
with
Present a new idea and solve the equation.
As a developed version of the Jaynes-Cummings model see for example [8] and [9] .
6. Concluding Remarks
In this note, I introduced the rotating wave approximation which plays an important role in Quantum Optics with two examples. The problem is that the method is used even in a subtle case. As far as I know, it is very hard to obtain an analytic approximate solution without RWA.
I don’t know the reason why it is so. However, such a “temporary” method must be overcome in the near future. I expect that young researchers will attack and overcome this problem.
Concerning a recent criticism to RWA see [10] and its references, and concerning recent applications to the dynamical Casimir effect see [11] [12] [13] [14] ( [13] and [14] are highly recommended).
Appendix
[A] Another Approach
Let us give another approach to the derivation (4), which may be smart enough. It is easy to see the diagonal form
where W is the Walsh-Hadamard matrix (operation) given by
Note that
Then we obtain
Readers should remark that the Walsh-Hadamard matrix W plays an essential role in Quantum Computation. See for example [15] (note:
in this paper).
[B] Tensor Product
Let us give a brief introduction to the tensor product of matrices. For
and
the tensor product is defined by
Precisely, in case of
and
When I was a young student in Japan this product was called the Kronecker one. Nowadays, it is called the tensor product in a unified manner, which may be better.
Note that
while
The blanks in the matrices above are of course zero.
Readers should recognize the difference. See for example [5] for more details.
[C] Beyond the RWA
Let us try to solve the Equation (7). For the purpose it is convenient to assume a form for some solution
where we set
for simplicity. Note that this form called the disentangling form (a kind of Gauss decomposition of some matrices) is very popular in Quantum Physics.
By setting
for simplicity in (7) we must calculate
where
.
Then we have
From (2)
and the formula (24) it is easy to see
Therefore
By comparing two equations above we obtain a system of differential equations
By deforming them we have
This is a simple exercise for young students.
If we can solve the first equation then we obtain solutions like
The first equation
is a (famous) Riccati equation of general type. Unfortunately, we don't know how to solve it explicitly at the present time.
[D] Full Calculation
Let us give the full calculation to the Equation (23). We set
and calculate
without assuming
in (25). Again, noting
(
) we obtain
where we have set
for simplicity. See (26). This is a good exercise for young students.
NOTES
1As a text book of Quantum Mechanics I recommend [1] although it is not necessarily standard.
2There are some researchers who are against this terminology, see for example [4] . However, I don’t agree with them because the terminology is nowadays very popular in the world.
3In [6] , it is called the Jaynes-Cummings-Paul one.