Abstract

Squeezing and its inverse magnification form a one-parameter group of linear canonical transformations of continuous signals in paraxial optics. We search for corresponding unitary matrices to apply on signal vectors in N-point finite Hamiltonian systems. The analysis is extended to the phase space representation by means of Wigner quasi-probability distribution functions on the discrete torus and on the sphere. Together with two previous studies of the fractional Fourier and Fresnel transforms, we complete the finite counterparts of the group of linear canonical transformations.

© 2007 Optical Society of America

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  1. K. B. Wolf and G. Krötzsch, "Geometry and dynamics in the fractional discrete Fourier transform," J. Opt. Soc. Am. A 24, 651-658 (2007).
    [CrossRef]
  2. K. B. Wolf and G. Krötzsch, "Geometry and dynamics in the Fresnel transform in finite systems," J. Opt. Soc. Am. A 24, 2568-2577 (2007).
    [CrossRef]
  3. M. Moshinsky and C. Quesne, "Linear canonical transformations and their unitary representation," J. Math. Phys. 12, 1772-1780 (1971).
    [CrossRef]
  4. K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, 1979).
  5. I. D. Ado, "The representation of Lie algebras by matrices," Trends Am. Math. Soc. 9, 308-327 (1962).
  6. E. P. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
    [CrossRef]
  7. M. Hillery, R. F. O'Connel, M. O. Scully, and E. P. Wigner, "Distribution functions in physics: fundamentals," Phys. Rep. 259, 121-167 (1984).
    [CrossRef]
  8. H.-W. Lee, "Theory and applications of the quantum phase-space distribution functions," Phys. Rep. 259, 147-211 (1995).
    [CrossRef]
  9. U. Leonhardt, "Discrete Wigner function and quantum-state tomography," Phys. Rev. A 53, 2998-3013 (1996).
    [CrossRef] [PubMed]
  10. W. K. Wooters, "A Wigner-function formulation of finite-state quantum mechanics," Ann. Phys. (N.Y.) 176, 1-21 (1987).
    [CrossRef]
  11. N. M. Atakishiyev and K. B. Wolf, "Fractional Fourier-Kravchuk transform," J. Opt. Soc. Am. A 14, 1467-1477 (1997).
    [CrossRef]
  12. N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, "Finite models of the oscillator," Phys. Part. Nucl. 36, 521-555 (2005).
  13. N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, "Wigner distribution function for finite systems," J. Math. Phys. 39, 6247-6261 (1998).
    [CrossRef]
  14. L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics, Encyclopedia of Mathematics and Its Applications, G.-C.Rota, ed. (Addison-Wesley, 1981).
  15. A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable (Springer-Verlag, 1991).
  16. S. T. Ali, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, "The Wigner function for general Lie groups and the wavelet transform," Ann. Inst. Henri Poincare 1, 685-714 (2000).
    [CrossRef]
  17. R. Gilmore, Lie Groups, Lie Algebras, and Some of their Applications (Wiley Interscience, New York, 1978).
  18. S. M. Chumakov, A. Frank, and K. B. Wolf, "Finite Kerr medium: macroscopic quantum superposition states and Wigner functions on the sphere," Phys. Rev. A 60, 1817-1823 (1999).
    [CrossRef]
  19. R. L. Stratonovich, "On distributions in representation space," JETP 31, 1012-1020 (1956) R. L. Stratonovich,[Sov. Phys. JETP 4, 891-898 (1957)].
  20. G. S. Agarwal, "Relation between atomic coherent-state representation, state multipoles, and generalized phase-space distributions," Phys. Rev. A 24, 2889-2896 (1981).
    [CrossRef]
  21. S. M. Chumakov, A. B. Klimov, and K. B. Wolf, "On the connection of two Wigner functions for spin systems," Phys. Rev. A 61, 034101(3), (2000).
    [CrossRef]
  22. K. B. Wolf, Geometric Optics on Phase Space (Springer-Verlag, 2004).
  23. A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, "Evolution under polynomial Hamiltonians in quantum and optical phase spaces," Phys. Rev. A 55, 876-889 (1997).
    [CrossRef]
  24. N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, "Contraction of the finite one-dimensional oscillator," Int. J. Mod. Phys. A 18, 317-327 (2003).
    [CrossRef]

2007

2005

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, "Finite models of the oscillator," Phys. Part. Nucl. 36, 521-555 (2005).

2003

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, "Contraction of the finite one-dimensional oscillator," Int. J. Mod. Phys. A 18, 317-327 (2003).
[CrossRef]

2000

S. T. Ali, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, "The Wigner function for general Lie groups and the wavelet transform," Ann. Inst. Henri Poincare 1, 685-714 (2000).
[CrossRef]

S. M. Chumakov, A. B. Klimov, and K. B. Wolf, "On the connection of two Wigner functions for spin systems," Phys. Rev. A 61, 034101(3), (2000).
[CrossRef]

1999

S. M. Chumakov, A. Frank, and K. B. Wolf, "Finite Kerr medium: macroscopic quantum superposition states and Wigner functions on the sphere," Phys. Rev. A 60, 1817-1823 (1999).
[CrossRef]

1998

N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, "Wigner distribution function for finite systems," J. Math. Phys. 39, 6247-6261 (1998).
[CrossRef]

1997

A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, "Evolution under polynomial Hamiltonians in quantum and optical phase spaces," Phys. Rev. A 55, 876-889 (1997).
[CrossRef]

N. M. Atakishiyev and K. B. Wolf, "Fractional Fourier-Kravchuk transform," J. Opt. Soc. Am. A 14, 1467-1477 (1997).
[CrossRef]

1996

U. Leonhardt, "Discrete Wigner function and quantum-state tomography," Phys. Rev. A 53, 2998-3013 (1996).
[CrossRef] [PubMed]

1995

H.-W. Lee, "Theory and applications of the quantum phase-space distribution functions," Phys. Rep. 259, 147-211 (1995).
[CrossRef]

1987

W. K. Wooters, "A Wigner-function formulation of finite-state quantum mechanics," Ann. Phys. (N.Y.) 176, 1-21 (1987).
[CrossRef]

1984

M. Hillery, R. F. O'Connel, M. O. Scully, and E. P. Wigner, "Distribution functions in physics: fundamentals," Phys. Rep. 259, 121-167 (1984).
[CrossRef]

1981

G. S. Agarwal, "Relation between atomic coherent-state representation, state multipoles, and generalized phase-space distributions," Phys. Rev. A 24, 2889-2896 (1981).
[CrossRef]

1971

M. Moshinsky and C. Quesne, "Linear canonical transformations and their unitary representation," J. Math. Phys. 12, 1772-1780 (1971).
[CrossRef]

1962

I. D. Ado, "The representation of Lie algebras by matrices," Trends Am. Math. Soc. 9, 308-327 (1962).

1956

R. L. Stratonovich, "On distributions in representation space," JETP 31, 1012-1020 (1956) R. L. Stratonovich,[Sov. Phys. JETP 4, 891-898 (1957)].

R. L. Stratonovich, "On distributions in representation space," JETP 31, 1012-1020 (1956) R. L. Stratonovich,[Sov. Phys. JETP 4, 891-898 (1957)].

1932

E. P. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Ado, I. D.

I. D. Ado, "The representation of Lie algebras by matrices," Trends Am. Math. Soc. 9, 308-327 (1962).

Agarwal, G. S.

G. S. Agarwal, "Relation between atomic coherent-state representation, state multipoles, and generalized phase-space distributions," Phys. Rev. A 24, 2889-2896 (1981).
[CrossRef]

Ali, S. T.

S. T. Ali, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, "The Wigner function for general Lie groups and the wavelet transform," Ann. Inst. Henri Poincare 1, 685-714 (2000).
[CrossRef]

Atakishiyev, N. M.

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, "Finite models of the oscillator," Phys. Part. Nucl. 36, 521-555 (2005).

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, "Contraction of the finite one-dimensional oscillator," Int. J. Mod. Phys. A 18, 317-327 (2003).
[CrossRef]

S. T. Ali, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, "The Wigner function for general Lie groups and the wavelet transform," Ann. Inst. Henri Poincare 1, 685-714 (2000).
[CrossRef]

N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, "Wigner distribution function for finite systems," J. Math. Phys. 39, 6247-6261 (1998).
[CrossRef]

N. M. Atakishiyev and K. B. Wolf, "Fractional Fourier-Kravchuk transform," J. Opt. Soc. Am. A 14, 1467-1477 (1997).
[CrossRef]

A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, "Evolution under polynomial Hamiltonians in quantum and optical phase spaces," Phys. Rev. A 55, 876-889 (1997).
[CrossRef]

Biedenharn, L. C.

L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics, Encyclopedia of Mathematics and Its Applications, G.-C.Rota, ed. (Addison-Wesley, 1981).

Chumakov, S. M.

S. T. Ali, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, "The Wigner function for general Lie groups and the wavelet transform," Ann. Inst. Henri Poincare 1, 685-714 (2000).
[CrossRef]

S. M. Chumakov, A. B. Klimov, and K. B. Wolf, "On the connection of two Wigner functions for spin systems," Phys. Rev. A 61, 034101(3), (2000).
[CrossRef]

S. M. Chumakov, A. Frank, and K. B. Wolf, "Finite Kerr medium: macroscopic quantum superposition states and Wigner functions on the sphere," Phys. Rev. A 60, 1817-1823 (1999).
[CrossRef]

N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, "Wigner distribution function for finite systems," J. Math. Phys. 39, 6247-6261 (1998).
[CrossRef]

A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, "Evolution under polynomial Hamiltonians in quantum and optical phase spaces," Phys. Rev. A 55, 876-889 (1997).
[CrossRef]

Frank, A.

S. M. Chumakov, A. Frank, and K. B. Wolf, "Finite Kerr medium: macroscopic quantum superposition states and Wigner functions on the sphere," Phys. Rev. A 60, 1817-1823 (1999).
[CrossRef]

Gilmore, R.

R. Gilmore, Lie Groups, Lie Algebras, and Some of their Applications (Wiley Interscience, New York, 1978).

Hillery, M.

M. Hillery, R. F. O'Connel, M. O. Scully, and E. P. Wigner, "Distribution functions in physics: fundamentals," Phys. Rep. 259, 121-167 (1984).
[CrossRef]

Klimov, A. B.

S. M. Chumakov, A. B. Klimov, and K. B. Wolf, "On the connection of two Wigner functions for spin systems," Phys. Rev. A 61, 034101(3), (2000).
[CrossRef]

Krötzsch, G.

Lee, H.-W.

H.-W. Lee, "Theory and applications of the quantum phase-space distribution functions," Phys. Rep. 259, 147-211 (1995).
[CrossRef]

Leonhardt, U.

U. Leonhardt, "Discrete Wigner function and quantum-state tomography," Phys. Rev. A 53, 2998-3013 (1996).
[CrossRef] [PubMed]

Louck, J. D.

L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics, Encyclopedia of Mathematics and Its Applications, G.-C.Rota, ed. (Addison-Wesley, 1981).

Moshinsky, M.

M. Moshinsky and C. Quesne, "Linear canonical transformations and their unitary representation," J. Math. Phys. 12, 1772-1780 (1971).
[CrossRef]

Nikiforov, A. F.

A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable (Springer-Verlag, 1991).

O'Connel, R. F.

M. Hillery, R. F. O'Connel, M. O. Scully, and E. P. Wigner, "Distribution functions in physics: fundamentals," Phys. Rep. 259, 121-167 (1984).
[CrossRef]

Pogosyan, G. S.

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, "Finite models of the oscillator," Phys. Part. Nucl. 36, 521-555 (2005).

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, "Contraction of the finite one-dimensional oscillator," Int. J. Mod. Phys. A 18, 317-327 (2003).
[CrossRef]

Quesne, C.

M. Moshinsky and C. Quesne, "Linear canonical transformations and their unitary representation," J. Math. Phys. 12, 1772-1780 (1971).
[CrossRef]

Rivera, A. L.

A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, "Evolution under polynomial Hamiltonians in quantum and optical phase spaces," Phys. Rev. A 55, 876-889 (1997).
[CrossRef]

Scully, M. O.

M. Hillery, R. F. O'Connel, M. O. Scully, and E. P. Wigner, "Distribution functions in physics: fundamentals," Phys. Rep. 259, 121-167 (1984).
[CrossRef]

Stratonovich, R. L.

R. L. Stratonovich, "On distributions in representation space," JETP 31, 1012-1020 (1956) R. L. Stratonovich,[Sov. Phys. JETP 4, 891-898 (1957)].

R. L. Stratonovich, "On distributions in representation space," JETP 31, 1012-1020 (1956) R. L. Stratonovich,[Sov. Phys. JETP 4, 891-898 (1957)].

Suslov, S. K.

A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable (Springer-Verlag, 1991).

Uvarov, V. B.

A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable (Springer-Verlag, 1991).

Wigner, E. P.

M. Hillery, R. F. O'Connel, M. O. Scully, and E. P. Wigner, "Distribution functions in physics: fundamentals," Phys. Rep. 259, 121-167 (1984).
[CrossRef]

E. P. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Wolf, K. B.

K. B. Wolf and G. Krötzsch, "Geometry and dynamics in the Fresnel transform in finite systems," J. Opt. Soc. Am. A 24, 2568-2577 (2007).
[CrossRef]

K. B. Wolf and G. Krötzsch, "Geometry and dynamics in the fractional discrete Fourier transform," J. Opt. Soc. Am. A 24, 651-658 (2007).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, "Finite models of the oscillator," Phys. Part. Nucl. 36, 521-555 (2005).

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, "Contraction of the finite one-dimensional oscillator," Int. J. Mod. Phys. A 18, 317-327 (2003).
[CrossRef]

S. M. Chumakov, A. B. Klimov, and K. B. Wolf, "On the connection of two Wigner functions for spin systems," Phys. Rev. A 61, 034101(3), (2000).
[CrossRef]

S. T. Ali, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, "The Wigner function for general Lie groups and the wavelet transform," Ann. Inst. Henri Poincare 1, 685-714 (2000).
[CrossRef]

S. M. Chumakov, A. Frank, and K. B. Wolf, "Finite Kerr medium: macroscopic quantum superposition states and Wigner functions on the sphere," Phys. Rev. A 60, 1817-1823 (1999).
[CrossRef]

N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, "Wigner distribution function for finite systems," J. Math. Phys. 39, 6247-6261 (1998).
[CrossRef]

N. M. Atakishiyev and K. B. Wolf, "Fractional Fourier-Kravchuk transform," J. Opt. Soc. Am. A 14, 1467-1477 (1997).
[CrossRef]

A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, "Evolution under polynomial Hamiltonians in quantum and optical phase spaces," Phys. Rev. A 55, 876-889 (1997).
[CrossRef]

K. B. Wolf, Geometric Optics on Phase Space (Springer-Verlag, 2004).

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, 1979).

Wooters, W. K.

W. K. Wooters, "A Wigner-function formulation of finite-state quantum mechanics," Ann. Phys. (N.Y.) 176, 1-21 (1987).
[CrossRef]

Ann. Inst. Henri Poincare

S. T. Ali, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, "The Wigner function for general Lie groups and the wavelet transform," Ann. Inst. Henri Poincare 1, 685-714 (2000).
[CrossRef]

Ann. Phys. (N.Y.)

W. K. Wooters, "A Wigner-function formulation of finite-state quantum mechanics," Ann. Phys. (N.Y.) 176, 1-21 (1987).
[CrossRef]

Int. J. Mod. Phys. A

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, "Contraction of the finite one-dimensional oscillator," Int. J. Mod. Phys. A 18, 317-327 (2003).
[CrossRef]

J. Math. Phys.

N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, "Wigner distribution function for finite systems," J. Math. Phys. 39, 6247-6261 (1998).
[CrossRef]

M. Moshinsky and C. Quesne, "Linear canonical transformations and their unitary representation," J. Math. Phys. 12, 1772-1780 (1971).
[CrossRef]

J. Opt. Soc. Am. A

JETP

R. L. Stratonovich, "On distributions in representation space," JETP 31, 1012-1020 (1956) R. L. Stratonovich,[Sov. Phys. JETP 4, 891-898 (1957)].

Phys. Part. Nucl.

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, "Finite models of the oscillator," Phys. Part. Nucl. 36, 521-555 (2005).

Phys. Rep.

M. Hillery, R. F. O'Connel, M. O. Scully, and E. P. Wigner, "Distribution functions in physics: fundamentals," Phys. Rep. 259, 121-167 (1984).
[CrossRef]

H.-W. Lee, "Theory and applications of the quantum phase-space distribution functions," Phys. Rep. 259, 147-211 (1995).
[CrossRef]

Phys. Rev.

E. P. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Phys. Rev. A

U. Leonhardt, "Discrete Wigner function and quantum-state tomography," Phys. Rev. A 53, 2998-3013 (1996).
[CrossRef] [PubMed]

G. S. Agarwal, "Relation between atomic coherent-state representation, state multipoles, and generalized phase-space distributions," Phys. Rev. A 24, 2889-2896 (1981).
[CrossRef]

S. M. Chumakov, A. B. Klimov, and K. B. Wolf, "On the connection of two Wigner functions for spin systems," Phys. Rev. A 61, 034101(3), (2000).
[CrossRef]

S. M. Chumakov, A. Frank, and K. B. Wolf, "Finite Kerr medium: macroscopic quantum superposition states and Wigner functions on the sphere," Phys. Rev. A 60, 1817-1823 (1999).
[CrossRef]

A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, "Evolution under polynomial Hamiltonians in quantum and optical phase spaces," Phys. Rev. A 55, 876-889 (1997).
[CrossRef]

Trends Am. Math. Soc.

I. D. Ado, "The representation of Lie algebras by matrices," Trends Am. Math. Soc. 9, 308-327 (1962).

Other

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, 1979).

K. B. Wolf, Geometric Optics on Phase Space (Springer-Verlag, 2004).

L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics, Encyclopedia of Mathematics and Its Applications, G.-C.Rota, ed. (Addison-Wesley, 1981).

A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable (Springer-Verlag, 1991).

R. Gilmore, Lie Groups, Lie Algebras, and Some of their Applications (Wiley Interscience, New York, 1978).

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Figures (7)

Fig. 1
Fig. 1

Squeezing S ( α ) classically produces a flow of phase space ( q , p ) R 2 along the hyperbolas q p = const . For α > 0 , positions are squeezed, momenta (wavenumbers) are magnified, and areas are conserved.

Fig. 2
Fig. 2

Left, the discrete rectangle signal Rect 4 , 4 ( m ) on N = 31 points ( j = 15 ) . Right, the corresponding Wigner function on the discrete torus ( m , k ) [ j , j ] modulo N; the upper and lower boundaries represent the same meridian, as do the left and right boundaries; the basic pattern at the center (0,0) is (approximately) repeated at its three antipodes in the torus.

Fig. 3
Fig. 3

Evolution of the discrete rectangle signal of the previous figure and the corresponding Wigner functions on the discrete torus, under the “geometric” version of squeezing and magnification (where position Q has a symmetric, equally spaced spectrum), with α g = ( 4 N ) 2 α for the α’s indicated between each pair.

Fig. 4
Fig. 4

Evolution of the rectangle signal and its Wigner function on the torus under the “dynamical” version of squeezing (based on momentum P 2 = Δ ), for the indicated values of α d α .

Fig. 5
Fig. 5

Left, the s u ( 2 ) Wigner function lives in the meta-phase space ( m , k , μ ) R 3 of position, momentum, and (displaced) energy. Spherical coordinates ( β , γ ) are referred to the position axis ( m ) . The thick arrowed lines represent the global flux due to squeezing (explained in the text). Right, the ( β , γ ) plane presents the s u ( 2 ) Wigner function for easier comparison with the previous figures (on the discrete torus). The origin of coordinates coincides with the “bottom pole” of ground energy at ( β = 1 2 π , γ = 0 ) ; the “top pole” is at ( β = 1 2 π , γ = ± π ) ; the points on the momentum axis are ( 1 2 π , ± 1 2 π ) . The flux lines under squeezing are indicated by the thick arrowed lines (cf. Fig. 1).

Fig. 6
Fig. 6

Left, the rectangle function Rect 5 , 5 ( m ) as an N = 31 -point signal. Right, the corresponding s u ( 2 ) Wigner function on the sphere, displayed by its level curves on a square. The vertical and horizontal axes are labeled by ( m , k ) for direct comparison with Fig. 2; this square is actually the ( β , γ ) plane of the sphere, as explained in Fig. 5. The Wigner function has a strong peak centered at the origin and elongated over the width of the signal, with shallow dimples of negative values. We display 40 level curves to emphasize the behavior of the s u ( 2 ) Wigner function where its values are small or negative.

Fig. 7
Fig. 7

Evolution of the rectange signal and its Wigner function on the sphere (as explained in Fig. 6), under s u ( 2 ) squeezing of positions by the factors e a , with a = 1 5 ( 4 N ) 2 α , for the indicated values of α (cf. Figs. 3, 4).

Equations (36)

Equations on this page are rendered with MathJax. Learn more.

[ a b c d ] = [ cos ω sin ω sin ω cos ω ] [ e α 0 0 e α ] [ 1 0 z 1 ] .
S ( α ) : f ( q , p ) exp ( α { q p , } Pb ) : f ( q , p ) = f ( e α q , e α p ) .
g Fr 1 2 p 2 , g Fou 1 2 ( p 2 + q 2 ) , g FL 1 2 q 2 ,
g S ( q , p ) = q p = { g FL , g Fr } Pb .
S ( α ) : f ( q ) exp ( i α { Q P } W ) : f ( q ) = e α 2 f ( e α q ) ,
{ Q P } W 1 2 ( Q P + P Q ) = i q d d q i 1 2 1 = i [ 1 2 Q 2 , 1 2 P 2 ] .
Q g diag ( m ) , m j j P g = Φ Q g Φ 1 ,
1 2 ( Q d ) 2 = diag ( 2 sin 2 ( π m N ) ) 1 2 ( P d ) 2 1 2 Δ .
G g 1 2 ( Q g P g + P g Q g ) ,
G d i [ 1 2 ( Q d ) 2 , 1 2 ( P d ) 2 ] ,
S g ( α g ) exp ( i α g G g ) , S d ( α d ) exp ( i α d G d ) ,
W ( f m , k ) 1 N n = j j f m n * exp ( 4 π i N k n ) f m + n .
position : Q = L 1 ,
momentum : P = L 2 .
( displaced ) energy : H = L 3 + ( j + 1 2 ) 1 .
[ H , Q ] = i P , [ P , H ] = i Q ,
[ Q , P ] = i [ H ( j + 1 2 ) 1 ] .
number : N H 1 2 1 = L 3 + j 1 .
ψ ν ( j ) ( m ) j , m j , ν j 3 1 = d m , ν j j ( 1 2 π ) = d ν j , m j ( 1 2 π ) ,
Q m , m m δ m , m , m , m [ j , j ] ,
P m , m i ( 1 2 ( j m ) ( j + m + 1 ) δ m + 1 , m 1 2 ( j + m ) ( j m + 1 ) δ m 1 , m ) ,
N m , m + j δ m , m 1 2 ( j m ) ( j + m + 1 ) δ m + 1 , m + 1 2 ( j + m ) ( j m + 1 ) δ m 1 , m .
Fourier Kravchuk : K ( κ ) exp ( i 1 2 π κ N ) ,
Fresnel Kravchuk : F ( z ) exp ( i 1 2 z P 2 ) .
G { Q P } W = ( L 1 L 2 + L 2 L 1 ) ,
G = 1 2 ( Q P + P Q ) .
G m , m = i ( ( m + 1 2 ) ( j m ) ( j + m + 1 ) δ m + 1 , m ( m 1 2 ) ( j + m ) ( j m + 1 ) δ m 1 , m ) .
S ( a ) exp ( i a G ) , S ( a ) exp ( i a G ) .
W ( v ) su ( 2 ) d g ( w ) exp i [ w 1 ( q Q ) + w 2 ( p P ) + w 3 ( μ L 3 ) ] ,
w ψ u ̂ ( θ , ϕ ) = ( w 1 w 2 w 3 ) = ( ψ sin θ sin ϕ ψ sin θ cos ϕ ψ cos θ ) , 0 ψ < 2 π , 0 θ π , 0 ϕ < 2 π .
d g ( ψ , θ , ϕ ) = 1 2 sin 2 1 2 ψ sin θ d ψ d θ d ϕ .
M Q ( q ) = R d p R d μ W ( q , p , μ ) = ( 2 π ) 2 π π d w exp [ i w ( q Q ) ] .
W ( f q , p , μ ) f W ( q , p , μ ) f = m , m f m * W m , m ( j ) ( q , p , μ ) f m .
M ( f q ) R d p R d μ W ( f q , p , μ ) = f M Q ( q ) f = ( 2 π ) 3 m f m 2 sinc π ( q m ) .
Q 2 + P 2 + L 3 2 = j ( j + 1 ) 1 ,
G [ 1 2 L 1 2 , 1 2 L 2 2 ] = i 1 2 ( L 1 L 2 L 3 + L 3 L 2 L 1 ) ,

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