Abstract

We present a comparative study of the non-linear wave packet dynamics of two-mode coherent states of the Heisenberg-Weyl group, the SU(1,1) group and the SU(2) group under the action of a model anharmonic Hamiltonian. In each case, we find certain generic signatures of non-linear evolution such as quick onset of decoherence followed by Schrödinger cat formation and revival. We also report important differences in the evolution of coherent states belonging to different symmetry groups.

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References

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  1. I. Sh. Averbukh and N. F. Perelman, "Fractional revival: Universality in the long-term evolution of quantum wave packets beyond the correspondence principle dynamics," Phys. Lett. A 139, 449 (1989).
    [CrossRef]
  2. Z. D. Gaeta and C. R. Stroud Jr., "Classical and quantum-mechanical dynamics of a quasiclassical state of the hydrogen atom," Phys. Rev. A 42, 6308 (1990).
    [CrossRef] [PubMed]
  3. X. Chen and J. A. Yeazell, "Phase-conjugate picture of a wave-packet interference design for arbitrary target states," Phys. Rev. A 59, 3782 (1999).
    [CrossRef]
  4. J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, "Periodic spontaneous collapse and revival in a simple quantum model," Phys. Rev. Lett. 44, 1323 (1980).
    [CrossRef]
  5. B. Yurke and D. Stoler, "Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion," Phys. Rev. Lett. 57, 13 (1986).
    [CrossRef] [PubMed]
  6. K. Tara, G. S. Agarwal, and S. Chaturvedi, "Production of Schrodinger macroscopic quantum- superposition states in a Kerr medium," Phys. Rev. A 47, 5024 (1993).
    [CrossRef] [PubMed]
  7. C. Leichtle, I. Sh. Averbukh, and W. P. Schleich, "Multilevel quantum beats: an analytical approach," Phys. Rev. A 54, 5299 (1996).
    [CrossRef] [PubMed]
  8. R. Bluhm, V. A. Kostelecky and B. Tudose, "Wave-packet revivals for quantum systems with nondegenerate energies," Phys. Lett. A 222, 220 (1996).
    [CrossRef]
  9. G. S. Agarwal and J. Banerji, "Fractional revivals in systems with two time scales," Phys. Rev. A 57, 3880 (1998).
    [CrossRef]
  10. J. Banerji and G. S. Agarwal, "Revival and fractional revival in the quantum dynamics of SU(1,1) coherent states," Phys. Rev. A 59, 4777 (1999).
    [CrossRef]
  11. G. S. Agarwal, "Nonclassical statistics of fields in pair coherent states," J. Opt. Soc. Am. B 5, 1940 (1988).
    [CrossRef]
  12. A. M. Perelomov, Generalized Coherent States and Their Applications (Springer, Berlin 1986).
    [CrossRef]
  13. Coherent states, edited by J. R. Klauder and B. S. Skagerstam, (World Scientific, Singapore, 1985).
  14. C. M. Caves and B. L. Shumaker, "New formalism for two-photon quantum optics. I. Quadrature phases and squeezed states," Phys. Rev. A 31, 3068 (1985).
    [CrossRef] [PubMed]
  15. F. T. Arecchi, E. Courtens, R. Gilmore and H. Thomas, "Atomic coherent states in quantum optics," Phys. Rev. A 6, 2211 (1972).
    [CrossRef]
  16. K. Wódkiewicz and J. H. Eberly, "Coherent states, squeezed uctuations, and the SU(2) and SU(1,1,) groups in quantum-optics applications," J. Opt. Soc. Am B 2, 458 (1985).
    [CrossRef]
  17. V. Buzek and T. Quang, "Generalized coherent state for bosonic realization of SU(2) Lie algebra," J. Opt. Soc. Am B 6, 2447 (1989).
    [CrossRef]
  18. G. S. Agarwal, R. R. Puri and R. P. Singh, "Atomic Schrodinger cat states," Phys. Rev. A 56, 2249 (1997).
    [CrossRef]
  19. G. S. Agarwal, R. R. Puri and R. P. Singh, "Vortex states for the quantized radiation field," Phys. Rev. A 56, 4207 (1997).
    [CrossRef]
  20. L. Allen, M. J. Padgett and M. Babiker, "The orbital angular momentum of light" in Progress in Optics, edited by E. Wolf, (North Holland, Amsterdam), 39, 294 (1999).

Other (20)

I. Sh. Averbukh and N. F. Perelman, "Fractional revival: Universality in the long-term evolution of quantum wave packets beyond the correspondence principle dynamics," Phys. Lett. A 139, 449 (1989).
[CrossRef]

Z. D. Gaeta and C. R. Stroud Jr., "Classical and quantum-mechanical dynamics of a quasiclassical state of the hydrogen atom," Phys. Rev. A 42, 6308 (1990).
[CrossRef] [PubMed]

X. Chen and J. A. Yeazell, "Phase-conjugate picture of a wave-packet interference design for arbitrary target states," Phys. Rev. A 59, 3782 (1999).
[CrossRef]

J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, "Periodic spontaneous collapse and revival in a simple quantum model," Phys. Rev. Lett. 44, 1323 (1980).
[CrossRef]

B. Yurke and D. Stoler, "Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion," Phys. Rev. Lett. 57, 13 (1986).
[CrossRef] [PubMed]

K. Tara, G. S. Agarwal, and S. Chaturvedi, "Production of Schrodinger macroscopic quantum- superposition states in a Kerr medium," Phys. Rev. A 47, 5024 (1993).
[CrossRef] [PubMed]

C. Leichtle, I. Sh. Averbukh, and W. P. Schleich, "Multilevel quantum beats: an analytical approach," Phys. Rev. A 54, 5299 (1996).
[CrossRef] [PubMed]

R. Bluhm, V. A. Kostelecky and B. Tudose, "Wave-packet revivals for quantum systems with nondegenerate energies," Phys. Lett. A 222, 220 (1996).
[CrossRef]

G. S. Agarwal and J. Banerji, "Fractional revivals in systems with two time scales," Phys. Rev. A 57, 3880 (1998).
[CrossRef]

J. Banerji and G. S. Agarwal, "Revival and fractional revival in the quantum dynamics of SU(1,1) coherent states," Phys. Rev. A 59, 4777 (1999).
[CrossRef]

G. S. Agarwal, "Nonclassical statistics of fields in pair coherent states," J. Opt. Soc. Am. B 5, 1940 (1988).
[CrossRef]

A. M. Perelomov, Generalized Coherent States and Their Applications (Springer, Berlin 1986).
[CrossRef]

Coherent states, edited by J. R. Klauder and B. S. Skagerstam, (World Scientific, Singapore, 1985).

C. M. Caves and B. L. Shumaker, "New formalism for two-photon quantum optics. I. Quadrature phases and squeezed states," Phys. Rev. A 31, 3068 (1985).
[CrossRef] [PubMed]

F. T. Arecchi, E. Courtens, R. Gilmore and H. Thomas, "Atomic coherent states in quantum optics," Phys. Rev. A 6, 2211 (1972).
[CrossRef]

K. Wódkiewicz and J. H. Eberly, "Coherent states, squeezed uctuations, and the SU(2) and SU(1,1,) groups in quantum-optics applications," J. Opt. Soc. Am B 2, 458 (1985).
[CrossRef]

V. Buzek and T. Quang, "Generalized coherent state for bosonic realization of SU(2) Lie algebra," J. Opt. Soc. Am B 6, 2447 (1989).
[CrossRef]

G. S. Agarwal, R. R. Puri and R. P. Singh, "Atomic Schrodinger cat states," Phys. Rev. A 56, 2249 (1997).
[CrossRef]

G. S. Agarwal, R. R. Puri and R. P. Singh, "Vortex states for the quantized radiation field," Phys. Rev. A 56, 4207 (1997).
[CrossRef]

L. Allen, M. J. Padgett and M. Babiker, "The orbital angular momentum of light" in Progress in Optics, edited by E. Wolf, (North Holland, Amsterdam), 39, 294 (1999).

Supplementary Material (8)

» Media 1: MOV (99 KB)     
» Media 2: MOV (394 KB)     
» Media 3: MOV (98 KB)     
» Media 4: MOV (668 KB)     
» Media 5: MOV (857 KB)     
» Media 6: MOV (262 KB)     
» Media 7: MOV (805 KB)     
» Media 8: MOV (842 KB)     

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Figures (9)

Fig. 1.
Fig. 1.

[102 KB] A frame of Movie (1) showing the erosion of the initial coherent structures in the quadrature distributions of |α, β〉 (Odd) and |α, β+ (Even). Shown here are the contour plots of these distributions as functions of x/α and y/β. The unit of time is T -. We have set α=2, β=3 and T +/T -=2=3. All the frames in this paper have been drawn in Mathematica using a customised ColorFunction in which the base is set at RGBColor[0.5,0.5,0.5] for better contrast.

Fig. 2.
Fig. 2.

[404 KB] A frame of Movie (2) showing the restoration of coherent structures in the quadrature distributions of |α, β- (Odd) and |α, β+ (Even) at later instants of time. All other details are as in Fig. 1.

Fig. 3.
Fig. 3.

[100 KB] A frame of Movie (3) showing how the ratio T +/T - affects the evolution of the quadrature distributions for |α, β〉. Shown here are the contour plots of these distributions as functions of x/α and y/β for T +=T -=2=3 (left) and T +=T -=3/5 (right). The unit of time is T -. We have set α=2 and β=3.

Fig. 4.
Fig. 4.

[684 KB] A frame of Movie (4) showing the evolution of the quadrature distribution for a Perelomov coherent state (ξ=0, η=-i tanh π=4). for q=0 (Even) and q=1 (Odd). Shown here are the contourplots of these distributions. The unit of time is T -. In order to erase any ambiguity from our nomenclature, we stress that the distribution for different values of q will be different even if the parity of q is the same.

Fig. 5.
Fig. 5.

[878 KB] A frame of Movie (5) showing the evolution of the quadrature distribution for a pair coherent state (ξ=3, η=0). All other details are as in Fig. 4.

Fig. 6.
Fig. 6.

Contour plots of the quadrature distributions for the SU(2) coherent state |τ, 11〉 for τ=1 (left picture) and τ=i (right picture).

Fig. 7.
Fig. 7.

[269 KB] A frame of Movie (6) showing how the pattern for the quadrature distribution of the SU(2) coherent state |τ, 11〉 changes when one varies the phase of τ while keeping |τ|=1.

Fig. 8.
Fig. 8.

[825 KB] A frame of movie (7) showing the evolution of the quadrature distributions of the SU(2) coherent state |1, N〉 for N=10 (Even) and N=11 (Odd). Shown here are the contourplots of these distributions. The unit of time is T +. Lest there be any ambiguity in our nomenclature, we stress that the distribution for different values of N will be different even if the parity of N is the same.

Fig. 9.
Fig. 9.

[863 KB] A frame of movie (8) showing the evolution of the quadrature distributions of the SU(2) coherent state |i, N〉. All other details are as in Fig. 8.

Equations (19)

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α , β ± = ( α , β ± α , β ) 2 P ± , P ± = 1 ± exp [ 2 ( α 2 + β 2 ) ] .
η , ξ , q = N ( ξ , q ) exp ( ξ η * ) n ' = 0 η n ' Γ ( n ' + 1 )
× n = 0 ξ n ( 1 η 2 ) n + q + 1 2 ( n + n ' ) ! ( n + n ' + q ) ! n ! ( n + q ) ! n + n ' + q , n + n ' ,
N ( ξ , q ) = [ n = 0 ξ 2 n n ! ( n + q ) ! ] 1 2 .
θ , ϕ , N τ , N , τ = tan θ 2 e i ϕ ;
= ( 1 + τ 2 ) N 2 K = 0 N ( N K ) 1 2 τ K K , N K .
H = c 1 [ ( a a ) 2 + ( b b ) 2 ] c 2 a a b b ,
H = π 4 [ ( a a + b b ) 2 T + ( a a b b ) 2 / T + ] , where , T ± = π ( 2 c 1 ± c 2 ) .
ϕ ± ( x , y ) = π 1 2 exp [ 1 2 { α 2 α 2 + β 2 β 2 + ( x ± α 2 ) 2 + ( y ± β 2 ) 2 } ] .
ψ ( x , y , 0 ) = 1 π 2 N N ! [ 1 + τ 2 1 + τ 2 ] N 2 e ( x 2 + y 2 ) 2 H N ( τ x + y 1 + τ 2 ) , τ = tan θ 2 e i ϕ .
ψ ( x , y , 0 ) τ = ± i = ( x 2 + y 2 ) N 2 π N ! exp [ ( x 2 + y 2 ) 2 ± i N η ] , η = tan 1 ( x y ) .
u ( r s T + ) τ , N = exp [ i π 4 r s N 2 ( 1 + T + T ) ]
× j = 0 l 1 α j ( r , s ) τ exp ( i π [ 2 j l r N s ] ) , N ,
l = { s , if r s ( mod 2 ) ; 2 s , if r = s = 1 ( mod 2 ) .
α j ( r , s ) = 1 l p = 0 l 1 exp ( i π r p 2 s + 2 i π p j l ) ,
u ( T + 2 ) τ , N τ e i π N 2 , N + i τ e i π N 2 , N .
u ( T + 2 ) 1 , 10 1 , 10 + i 1 , 10 ,
u ( T + 2 ) 1 , 11 i , 11 + i i , 11 .
sin 2 η N = 1 , i. e . η = π 4 N + π m N ;

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