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Quantum localization in circularly polarized electromagnetic field in ultra-strong field limit

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Abstract

We predict analytically and confirm numerically the existence of sharply localized quantum states in an ultra-strong circularly polarized electromagnetic field with the probability density that represents non-classical wave packets moving around strongly unstable classical circular orbits.

©2001 Optical Society of America

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Supplementary Material (4)

Media 1: MOV (7608 KB)     
Media 2: MOV (7563 KB)     
Media 3: MOV (1348 KB)     
Media 4: MOV (1142 KB)     

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Figures (5)

Fig. 1.
Fig. 1. Frequencies of small oscillations of the resonant circular orbit as the function of the parameter q=1/r 3 ω 2. The orbit is stable for 8=9≤q≤1. It formally regains stability for q → 0 where the harmonic wavefunction is singular and we find the new states. The weakly unstable orbit supports non-dispersing wavepackets for q<1+3/4ω -1/3.
Fig. 2.
Fig. 2. Eigenvalues of the equation [-d 2 x/2dx 2+Veff (x)]ϕ= for as functions of the potential radius a. States with the energy EVeff (0) are strongly localized around x=0 (the bold green line). The density of levels exhibists abrupt change across the energy line E=Veff (0)
Fig. 3.
Fig. 3. First few eigenstates of the equation [-d 2 x=2dx 2+Veff (x)]ϕ= for a=50 a.u. The states for which the energy EVeff (0) are strongly localized around x=0 (the red plot).
Fig. 4.
Fig. 4. Snapshots of the time evolution of the packet for E=0.27 and ω=0.082. a) t=0, b) t=4, c) t=9 and d) t=19 for the hydrogen in CP field. The radius of packet motion is a=40 a.u. The movie (7.5MB) linked to this figure shows the full 20-cycle evolution.
Fig. 5.
Fig. 5. Snapshots of the time evolution of the packet for ε=0:27 and ω=0.082. a) t=0, b) t=4, c) t=6 and d) t=9 for the hydrogen in CP field and with removed Coulomb potential. The radius of packet motion is a=40 a.u. The movie (7.5MB) linked to this figure shows the full 20-cycle evolution. [Media 1] [Media 2] [Media 3] [Media 4]

Equations (14)

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H osc = ω + a + a ω b + b ,
H = 2 2 1 r + E ( t ) · r ,
H K H = e i α ( t ) · e id α ( t ) dt · r H e i α ( t ) · e id α ( t ) dt · r ,
H K H Ψ = i d Ψ dt , H K H = 2 2 1 r + α ( t ) , d 2 α ( t ) dt 2 = E ( t ) ,
1 r + α ( t ) = n V n ( r , ω ) e in ω t
V eff ( r ) = V 0 ( r , ω ) = 1 2 π 0 2 π 1 r + α ( τ ) d τ
= 2 π r a E [ 4 ar ( r a ) 2 ] , a = α ( t ) , τ = ω t
( 1 2 2 r 2 1 2 r r 1 2 m 2 r 2 + V eff ( r ) ) ϕ = E ϕ
V eff ( r ) Z δ ( r a ) , Z = 1 2 π a
ϕ k ( r ) = J 0 ( kr ) , J 0 ( k n a ) = 0 , E = k n 2 2 ,
ϕ k ( r ) = m ( 1 ) m e i δ m R m ( r ) e im ϕ ,
R m ( r ) = c m J m ( kr ) r < a ,
R m ( r ) = γ ( k ) ( cos δ m J m ( kr ) sin δ m N m ( kr ) ) r > 0 ,
Ψ k ( r t ) = C e id α ( t ) dt · r J 0 [ k 0 r α ( t ) ] ,
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