In ion trapping and atomic physics experiments, the Lamb Dicke regime (or Lamb Dicke limit) is a quantum regime within an atom-light field system in which the spatial extent of the atom's wavefunction is much smaller than the wavelength of the addressing light field. In this regime, the coupling between an ion or atom's internal qubit states and its motional states is sufficiently small so that transitions that change the motional quantum number by more than one are strongly suppressed. This condition is quantitively expressed by the inequality
η 2 ( 2 n + 1 ) ≪ 1 , {\displaystyle \eta ^{2}(2n+1)\ll 1,}
where η {\displaystyle \eta } is the Lamb–Dicke parameter and n {\displaystyle n} is the motional quantum number of the ion or atom's harmonic oscillator state.
Lamb Dicke parameter Considering the ion's motion along the direction of the static trapping potential of an ion trap (the axial motion in z {\displaystyle z} -direction), the trap potential can be validly approximated as quadratic around the equilibrium position and the ion's motion locally be considered as that of a quantum harmonic oscillator with quantum harmonic oscillator eigenstates | n ⟩ {\displaystyle |n\rangle } . In this case the position operator z ^ {\displaystyle {\hat {z}}} is given by
z ^ = z 0 ( a ^ + a ^ † ) . {\displaystyle {\hat {z}}=z_{0}({\hat {a}}+{\hat {a}}^{\dagger }).}
where
z 0 = ⟨ 0 | z ^ 2 | 0 ⟩ = ℏ 2 m ω z {\displaystyle z_{0}={\sqrt {\langle 0\vert {\hat {z}}^{2}\vert 0\rangle }}={\sqrt {\frac {\hbar }{2m\omega _{z}}}}}
is the spread of the zero-point wavefunction, ω z {\displaystyle \omega _{z}} is the frequency of the static harmonic trapping potential in z {\displaystyle z} -direction and a ^ , a ^ † {\displaystyle {\hat {a}},{\hat {a}}^{\dagger }} are the ladder operators of the harmonic oscillator. The Lamb Dicke regime corresponds to the condition
⟨ Ψ m o t i o n | k z 2 z ^ 2 | Ψ m o t i o n ⟩ ≪ 1 {\displaystyle {\sqrt {\langle \Psi _{\rm {motion}}\vert k_{z}^{2}{\hat {z}}^{2}\vert \Psi _{\rm {motion}}}}\rangle \ll 1}
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