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Formation of Schrödinger-cat states in the Morse potential: Wigner function picture

Péter Földi, Attila Czirják, Balázs Molnár, and Mihály G. Benedict

Open Access Open Access

Abstract

We investigate the time evolution of Morse coherent states in the potential of the NO molecule. We present animated wave functions and Wigner functions of the system exhibiting spontaneous formation of Schrödinger-cat states at certain stages of the time evolution. These nonclassical states are coherent superpositions of two localized states corresponding to two different positions of the center of mass. We analyze the degree of nonclassicality as the function of the expectation value of the position in the initial state. Our numerical calculations are based on a novel, essentially algebraic treatment of the Morse potential.

©2002 Optical Society of America

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Supplementary Material (2)

Media 1: MOV (2071 KB)     
Media 2: MOV (2181 KB)     

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Figures (4)
Fig. 1.
Fig. 1. [2.1 MB] The absolute square of the wave functions corresponding to the Morse coherent states |x 0,0〉, with x 0 = 0.0 (ground state) and x 0 = 0.5. The time evolution of the latter initial wave function is shown in the attached movie file. These plots correspond to the case of the NO molecule, where s = 54.54.

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Fig. 2.
Fig. 2. The expectation value of the dimensionless position operator as a function of time. The initial states were |ϕ(t = 0)〉 = |x 0,0〉, with x 0 = 1.0, x 0 = 0.5 and x 0 = 0.06.

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Fig. 3.
Fig. 3. A) [1.3 MB] and B) [2.2 MB]. Frames of two movie files, showing Wigner functions of the Morse system at the initial stage of the time evolution and the formation of a Schrödinger-cat state. The plots correspond to t/T = 0 and t/T = 30, respectively. The initial state was |ϕ(t = 0)〉 = |x 0,0), with x 0 = 0.5.

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Fig. 4.
Fig. 4. Nonclassicality as a function of time. The initial state was |ϕ(t = 0)〉 = |x 0, 0〉, with x 0 = 1.0, x 0 = 0.5 and x 0 = 0.06.

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Equations (9)

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H = P 2 + ( s + 1 2 ) 2 [ exp ( − 2 X ) − 2 exp ( − X ) ] ,
d dt ∣ ϕ 〉 = − i 2 π 2 s + 1 H ∣ ϕ 〉 ,
〈 y ∣ β 〉 = ( 1 − ∣ β ∣ 2 ) s Γ ( 2 s ) ( 1 − β ) 2 s y s exp ( − y 2 1 + β 1 − β ) .
∣ β 〉 = ∑ n = 0 N c n ∣ ψ n 〉 = ∑ n = 0 [ s ] [ ( 2 s − 2 n ) Γ ( 2 s − n + 1 ) n ! Γ ( 2 s ) Γ ( 2 s − n ) Γ ( 2 s − 2 n + 1 ) ( 1 − ∣ β ∣ 2 ) s ( 1 − β ) n
× 2 F 1 ( − n , 2 s − n ; 2 s − 2 n + 1 ; 1 − β ) ∣ ψ n 〉 ] + ∑ n = [ s ] + 1 N c n ∣ ψ n 〉 ,
〈 X 〉 β = ln ( Re 1 + β 1 − β ) , 〈 P 〉 β = s Re [ ( 1 + β ) ( 1 − β ) ] Im [ ( 1 + β ) ( 1 − β ) ] ,
〈 X 〉 ( t ) = ∑ n , k = 0 N c n ( x ) c k * ( x ) 〈 ψ k ∣ X ∣ ψ n 〉 exp [ it 2 π 2 s + 1 ( E k ( s ) − E n ( s ) ) ]
W x p t = 1 2 π ∫ − ∞ ∞ ϕ * ( x + u 2 , t ) ϕ ( x − u 2 , t ) e iup du .
M nc ( ϕ ) = 1 − I + ( ϕ ) − I − ( ϕ ) I + ( ϕ ) + I − ( ϕ ) ,
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