Light dressed state
Encyclopedia
In the fields of atomic
, molecular
, and optical
science, the term light dressed state refers to a quantum state of an atomic or molecular system interacting with a laser
light
in terms of the Floquet picture, i.e. roughly like an atom
or a molecule
plus a photon
. The Floquet picture is based on the Floquet theorem in differential equations with periodic coefficients.
of a system of charged particles interacting with a laser light can be expressed as
where
is the vector potential
of the electromagnetic field of the laser;
is periodic in time as 
.
The position and momentum of the i-th
particle are denoted as
and 
, respectively,
while its mass and charge are symbolized as
and 
, respectively.
is the speed of light.
By virtue of this time-periodicity of the laser field, the total Hamiltonian is also
periodic in time as
The Floquet theorem guarantees that any solution
of the
Schrödinger equation
with this type of Hamiltonian,
can be expressed in the form
where
has the same time-periodicity as the Hamiltonian,
Therefore, this part can be expanded in a Fourier series
, obtaining
where
is the frequency of the laser field.
This expression (2) reveals that a quantum state of the system governed by the Hamiltonian (1)
can be specified by a real number
and an integer 
.
The integer n in eq. (2) can be regarded as the number of photons
absorbed from (or emitted to) the laser field.
In order to prove this statement, we clarify the correspondence between the solution (2),
which is derived from the classical expression of the electromagnetic field where there
is no concept of photons, and one which is derived from a quantized electromagnetic field (see quantum field theory
).
(It will be verified that n is equal to the expectation value of the absorbed photon number
at the limit of n<, where N is the initial number of total photons:
This part is under construction.)
Atomic physics
Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. It is primarily concerned with the arrangement of electrons around the nucleus and...
, molecular
Molecular physics
Molecular physics is the study of the physical properties of molecules, the chemical bonds between atoms as well as the molecular dynamics. Its most important experimental techniques are the various types of spectroscopy...
, and optical
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...
science, the term light dressed state refers to a quantum state of an atomic or molecular system interacting with a laser
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...
light
Light
Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...
in terms of the Floquet picture, i.e. roughly like an atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...
or a molecule
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
plus a photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
. The Floquet picture is based on the Floquet theorem in differential equations with periodic coefficients.
Mathematical formulation
The HamiltonianHamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
of a system of charged particles interacting with a laser light can be expressed as
where
is the vector potentialVector potential
In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a scalar potential, which is a scalar field whose negative gradient is a given vector field....
of the electromagnetic field of the laser;
is periodic in time as .The position and momentum of the i-th
particle are denoted as
and , respectively,while its mass and charge are symbolized as
and , respectively. is the speed of light.By virtue of this time-periodicity of the laser field, the total Hamiltonian is also
periodic in time as
The Floquet theorem guarantees that any solution
of theSchrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
with this type of Hamiltonian,
can be expressed in the form
where
has the same time-periodicity as the Hamiltonian,Therefore, this part can be expanded in a Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
, obtaining
where
is the frequency of the laser field.This expression (2) reveals that a quantum state of the system governed by the Hamiltonian (1)
can be specified by a real number
and an integer .The integer n in eq. (2) can be regarded as the number of photons
absorbed from (or emitted to) the laser field.
In order to prove this statement, we clarify the correspondence between the solution (2),
which is derived from the classical expression of the electromagnetic field where there
is no concept of photons, and one which is derived from a quantized electromagnetic field (see quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
).
(It will be verified that n is equal to the expectation value of the absorbed photon number
at the limit of n<
This part is under construction.)