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| {{Expert-subject|date=February 2009}}
| | Oscar is what my spouse loves to contact me and I totally dig that name. To gather badges is what her family and her appreciate. Minnesota is exactly where he's been living for years. Supervising is my profession.<br><br>Here is my web page; [http://relatebook.com/index.php?do=/profile-32192/info/ home std test] |
| In the fields of [[atomic physics|atomic]], [[molecular physics|molecular]], and [[optics|optical]] science, the term '''light dressed state''' refers to a [[quantum state]] of an atomic or molecular system interacting with a [[laser]] [[light]]
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| 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.
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| ==Mathematical formulation==
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| The [[Hamiltonian (quantum mechanics)|Hamiltonian]] of a system of charged particles interacting with a laser light can be expressed as
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| :<math>
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| H=\sum_i \frac{1}{2m_i}\left[\mathbf{p}_i-\frac{z_i}{c}\mathbf{A(\mathbf{r}_i, t)}\right]^2
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| +V(\{\mathbf{r}_i\}),
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| \ \ \ \ \ \ \ \ \ \ \ (1)
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| </math>
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| where <math>\mathbf{A}</math> is the [[vector potential]] of the electromagnetic field of the laser;
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| <math>\mathbf{A}</math> is periodic in time as <math>\mathbf{A}(t+T)=\mathbf{A}(t)</math>.
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| The position and momentum of the <math>i\,</math>-th
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| particle are denoted as <math>\mathbf{r}_i \,</math> and <math>\mathbf{p}_i \,</math>, respectively,
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| while its mass and charge are symbolized as <math>m_i \,</math> and <math>z_i \,</math>, respectively.
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| <math>c \,</math> is the speed of light.
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| By virtue of this time-periodicity of the laser field, the total Hamiltonian is also
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| periodic in time as
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| :<math>
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| H(t+T) = H(t) \, .
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| </math>
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| The [[Floquet theorem]] guarantees that any solution <math>\psi(\mathbf{r},t)</math> of the
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| [[Schrödinger equation]] with this type of Hamiltonian, | |
| :<math> | |
| i\hbar \frac{\partial}{\partial t} \psi(\{\mathbf{r}_i\},t) = H(t)\psi(\{\mathbf{r}_i\},t)
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| </math>
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| can be expressed in the form
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| :<math>
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| \psi(\{\mathbf{r}_i\},t) = \exp[-iEt/\hbar]\phi(\{\mathbf{r}_i\},t)
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| </math>
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| where <math>\phi\,</math> has the same time-periodicity as the Hamiltonian,
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| <math>
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| \phi(\{\mathbf{r}_i\},t+T) = \phi(\{\mathbf{r}_i\},t).
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| </math>
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| Therefore, this part can be expanded in a [[Fourier series]], obtaining
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| :<math>
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| \psi(\{\mathbf{r}_i\},t) = \exp[-iEt/\hbar]
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| \sum_{n=-\infty}^{\infty}\exp[in\omega t]\phi_n(\{\mathbf{r}_i\})
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| \ \ \ \ \ \ \ \ \ \ \ (2)
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| </math>
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| where <math>\omega (=2\pi/T)\,</math> is the frequency of the laser field.
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| This expression (2) reveals that a quantum state of the system governed by the Hamiltonian (1)
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| can be specified by a real number <math>E\,</math> and an integer <math>n\,</math>.
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| The integer <math>n\,</math> in eq. (2) can be regarded as the number of photons
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| absorbed from (or emitted to) the laser field.
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| In order to prove this statement, we clarify the correspondence between the solution (2),
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| which is derived from the classical expression of the electromagnetic field where there
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| is no concept of photons, and one which is derived from a quantized electromagnetic field (see [[quantum field theory]]).
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| (It will be verified that <math>n\,</math> is equal to the expectation value of the absorbed photon number
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| at the limit of <math>n<<N\,</math>, where <math>N\,</math> is the initial number of total photons:
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| This part is under construction.)
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| == References ==
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| #J.H. Shirley, Phys. Rev. '''138''', B979 (1965).
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| #H. Sambe, Phys. Rev. A '''7''', 2203 (1973).
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| #S. Guerin, F. Monti, J-M. Dupont, and H.R. Jauslin, J. Phys. A '''30''', 7193 (1997).
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| #S. Guerin and H.R. Jauslin, Adv. Chem. Phys. '''125''' 147 (2003).
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| #F.H.M. Faisal, ''Theory of Multiphoton Processes,'' Plenum (New York) 1987 ISBN 0-306-42317-0.
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| == See also ==
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| *[[Quantum mechanics]]
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| *[[Hamiltonian (quantum mechanics)]]
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| [[Category:Quantum mechanics]]
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Oscar is what my spouse loves to contact me and I totally dig that name. To gather badges is what her family and her appreciate. Minnesota is exactly where he's been living for years. Supervising is my profession.
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