Light dressed state
{{More footnotes|date=April 2021}}
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.
Mathematical formulation
The Hamiltonian of a system of charged particles interacting with a laser light can be expressed as
{{NumBlk|::|
H=\sum_i \frac{1}{2m_i}\left[\mathbf{p}_i-\frac{z_i}{c}\mathbf{A(\mathbf{r}_i, t)}\right]^2
+V(\{\mathbf{r}_i\}),
|{{EquationRef|1}}}}
where is the vector potential of the electromagnetic field of the laser;
is periodic in time as .
The position and momentum of the -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
:
H(t+T) = H(t) \, .
The Floquet theorem guarantees that any solution of the
Schrödinger equation with this type of Hamiltonian,
:
i\hbar \frac{\partial}{\partial t} \psi(\{\mathbf{r}_i\},t) = H(t)\psi(\{\mathbf{r}_i\},t)
can be expressed in the form
:
\psi(\{\mathbf{r}_i\},t) = \exp[-iEt/\hbar]\phi(\{\mathbf{r}_i\},t)
where has the same time-periodicity as the Hamiltonian,
\phi(\{\mathbf{r}_i\},t+T) = \phi(\{\mathbf{r}_i\},t).
Therefore, this part can be expanded in a Fourier series, obtaining
{{NumBlk|::|
\psi(\{\mathbf{r}_i\},t) = \exp[-iEt/\hbar]
\sum_{n=-\infty}^{\infty}\exp[in\omega t]\phi_n(\{\mathbf{r}_i\})
|{{EquationRef|2}}}}
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 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 can be verified that is equal to the expectation value of the absorbed photon number
at the limit of , where is the initial number of total photons.)
References
- {{cite journal|last1=Shirley|first1=Jon H.|title=Solution of the Schrödinger Equation with a Hamiltonian Periodic in Time|journal=Physical Review|volume=138|issue=4B|year=1965|pages=B979–B987|issn=0031-899X|doi=10.1103/PhysRev.138.B979|bibcode=1965PhRv..138..979S }}
- {{cite journal|last1=Sambe|first1=Hideo|title=Steady States and Quasienergies of a Quantum-Mechanical System in an Oscillating Field|journal=Physical Review A|volume=7|issue=6|year=1973|pages=2203–2213|issn=0556-2791|doi=10.1103/PhysRevA.7.2203|bibcode=1973PhRvA...7.2203S }}
- {{cite journal|last1=Guérin|first1=S|last2=Monti|first2=F|last3=Dupont|first3=J-M|last4=Jauslin|first4=H R|title=On the relation between cavity-dressed states, Floquet states, RWA and semiclassical models|journal=Journal of Physics A: Mathematical and General|volume=30|issue=20|year=1997|pages=7193–7215|issn=0305-4470|doi=10.1088/0305-4470/30/20/020|bibcode=1997JPhA...30.7193G}}
- {{cite journal|last1=Cardoso|first1=G.C.|last2=Tabosa|first2=J.W.R.|title=Four-wave mixing in dressed cold cesium atoms|journal=Optics Communications|volume=185|issue=4–6|year=2000|pages=353–358|issn=0030-4018|doi=10.1016/S0030-4018(00)01033-6|bibcode=2000OptCo.185..353C }}
- {{cite book|last1=Guérin|first1=S.|last2=Jauslin|first2=H. R.|title=Advances in Chemical Physics |chapter=Control of Quantum Dynamics by Laser Pulses: Adiabatic Floquet Theory|year=2003|pages=147–267|issn=1934-4791|doi=10.1002/0471428027.ch3|isbn=9780471214526 }}
- F.H.M. Faisal, Theory of Multiphoton Processes, Plenum (New York) 1987 {{ISBN|0-306-42317-0}}.